<b>Transcriber’s notes</b>:
The text of this book has been preserved as closely as practicable
to its original form. However, the author used some unusual symbols,
and I have taken the liberty of using Unicode characters with similar
appearance (ꖌ ᔕ) as substitutes, disregarding their official meaning
and aware that they might not display on all devices. An archaic symbol
used by the author to indicate the mathematical ‘factorial’ function
has been replaced by the modern equivalent, viz. ! Unusual placements
of various sub- and superscripted symbols remain as in the original text.
In this plain-text version, italic text is denoted by *asterisks*;
superscripted and subscripted characters are enclosed within {curly}
brackets preceded by a caret (^) or underline (_) respectively.
Inconsistencies of punctuation have been corrected silently, but
inconsistent spellings such as *Roemer, Römer, Rœmer* have not been
altered. A list of corrected spellings is appended at the end of the
book.
Footnotes have been renumbered consecutively and positioned below the
relevant paragraphs. A missing footnote marker has been inserted on
p.751 after tracking down the original document. A missing negative
symbol has been added to an exponent in a formula on p.327.
There is a misleading calculation on p.194 and the table that follows,
regarding progressive powers of two: ((2^{2})^{2})^{2} is equivalent to
(16)^{2} which equals 256 not 65,356 as stated, but 2^{16} does equal
65,356.
[sic] has been inserted on p.179 alongside a statement that the
alphabet contains 24 letters; however, the statement may well be
correct given that it was written in 1704 by a Flemish author and the
language is not specified.
A large arithmetic triangle on p.184 cannot be displayed optimally in
this plain-text version for lack of space
THE PRINCIPLES OF SCIENCE.
[Illustration]
[Illustration: THE LOGICAL MACHINE.]
THE PRINCIPLES OF SCIENCE:
*A TREATISE ON LOGIC*
*AND*
*SCIENTIFIC METHOD.*
BY
W. STANLEY JEVONS,
LL.D. (EDINB.), M.A. (LOND.), F.R.S.
London:
MACMILLAN AND CO.
1883.
*The Right of Translation and Reproduction is Reserved.*
LONDON:
R. Clay, Sons, & Taylor, Printers,
BREAD STREET HILL.
Stereotyped Edition.
PREFACE
*TO THE FIRST EDITION*.
It may be truly asserted that the rapid progress of the physical
sciences during the last three centuries has not been accompanied by
a corresponding advance in the theory of reasoning. Physicists speak
familiarly of Scientific Method, but they could not readily describe
what they mean by that expression. Profoundly engaged in the study
of particular classes of natural phenomena, they are usually too
much engrossed in the immense and ever-accumulating details of their
special sciences to generalise upon the methods of reasoning which they
unconsciously employ. Yet few will deny that these methods of reasoning
ought to be studied, especially by those who endeavour to introduce
scientific order into less successful and methodical branches of
knowledge.
The application of Scientific Method cannot be restricted to the sphere
of lifeless objects. We must sooner or later have strict sciences of
those mental and social phenomena, which, if comparison be possible,
are of more interest to us than purely material phenomena. But it
is the proper course of reasoning to proceed from the known to the
unknown--from the evident to the obscure--from the material and
palpable to the subtle and refined. The physical sciences may therefore
be properly made the practice-ground of the reasoning powers,
because they furnish us with a great body-of precise and successful
investigations. In these sciences we meet with happy instances of
unquestionable deductive reasoning, of extensive generalisation, of
happy prediction, of satisfactory verification, of nice calculation of
probabilities. We can note how the slightest analogical clue has been
followed up to a glorious discovery, how a rash generalisation has at
length been exposed, or a conclusive *experimentum crucis* has decided
the long-continued strife between two rival theories.
In following out my design of detecting the general methods of
inductive investigation, I have found that the more elaborate and
interesting processes of quantitative induction have their necessary
foundation in the simpler science of Formal Logic. The earlier, and
probably by far the least attractive part of this work, consists,
therefore, in a statement of the so-called Fundamental Laws of
Thought, and of the all-important Principle of Substitution, of which,
as I think, all reasoning is a development. The whole procedure of
inductive inquiry, in its most complex cases, is foreshadowed in
the combinational view of Logic, which arises directly from these
fundamental principles. Incidentally I have described the mechanical
arrangements by which the use of the important form called the Logical
Alphabet, and the whole working of the combinational system of Formal
Logic, may be rendered evident to the eye, and easy to the mind and
hand.
The study both of Formal Logic and of the Theory of Probabilities has
led me to adopt the opinion that there is no such thing as a distinct
method of induction as contrasted with deduction, but that induction
is simply an inverse employment of deduction. Within the last century
a reaction has been setting in against the purely empirical procedure
of Francis Bacon, and physicists have learnt to advocate the use of
hypotheses. I take the extreme view of holding that Francis Bacon,
although he correctly insisted upon constant reference to experience,
had no correct notions as to the logical method by which from
particular facts we educe laws of nature. I endeavour to show that
hypothetical anticipation of nature is an essential part of inductive
inquiry, and that it is the Newtonian method of deductive reasoning
combined with elaborate experimental verification, which has led to all
the great triumphs of scientific research.
In attempting to give an explanation of this view of Scientific Method,
I have first to show that the sciences of number and quantity repose
upon and spring from the simpler and more general science of Logic.
The Theory of Probability, which enables us to estimate and calculate
quantities of knowledge, is then described, and especial attention
is drawn to the Inverse Method of Probabilities, which involves, as
I conceive, the true principle of inductive procedure. No inductive
conclusions are more than probable, and I adopt the opinion that the
theory of probability is an essential part of logical method, so
that the logical value of every inductive result must be determined
consciously or unconsciously, according to the principles of the
inverse method of probability.
The phenomena of nature are commonly manifested in quantities of
time, space, force, energy, &c., and the observation, measurement,
and analysis of the various quantitative conditions or results
involved, even in a simple experiment, demand much employment of
systematic procedure. I devote a book, therefore, to a simple and
general description of the devices by which exact measurement is
effected, errors eliminated, a probable mean result attained, and
the probable error of that mean ascertained. I then proceed to the
principal, and probably the most interesting, subject of the book,
illustrating successively the conditions and precautions requisite
for accurate observation, for successful experiment, and for the sure
detection of the quantitative laws of nature. As it is impossible to
comprehend aright the value of quantitative laws without constantly
bearing in mind the degree of quantitative approximation to the truth
probably attained, I have devoted a special chapter to the Theory of
Approximation, and however imperfectly I may have treated this subject,
I must look upon it as a very essential part of a work on Scientific
Method.
It then remains to illustrate the sound use of hypothesis, to
distinguish between the portions of knowledge which we owe to empirical
observation, to accidental discovery, or to scientific prediction.
Interesting questions arise concerning the accordance of quantitative
theories and experiments, and I point out how the successive
verification of an hypothesis by distinct methods of experiment yields
conclusions approximating to but never attaining certainty. Additional
illustrations of the general procedure of inductive investigations are
given in a chapter on the Character of the Experimentalist, in which
I endeavour to show, moreover, that the inverse use of deduction was
really the logical method of such great masters of experimental inquiry
as Newton, Huyghens, and Faraday.
In treating Generalisation and Analogy, I consider the precautions
requisite in inferring from one case to another, or from one part of
the universe to another part; the validity of all such inferences
resting ultimately upon the inverse method of probabilities. The
treatment of Exceptional Phenomena appeared to afford an interesting
subject for a further chapter illustrating the various modes in which
an outstanding fact may eventually be explained. The formal part of
the book closes with the subject of Classification, which is, however,
very inadequately treated. I have, in fact, almost restricted myself to
showing that all classification is fundamentally carried out upon the
principles of Formal Logic and the Logical Alphabet described at the
outset.
In certain concluding remarks I have expressed the conviction which
the study of Logic has by degrees forced upon my mind, that serious
misconceptions are entertained by some scientific men as to the
logical value of our knowledge of nature. We have heard much of
what has been aptly called the Reign of Law, and the necessity and
uniformity of natural forces has been not uncommonly interpreted as
involving the non-existence of an intelligent and benevolent Power,
capable of interfering with the course of natural events. Fears have
been expressed that the progress of Scientific Method must therefore
result in dissipating the fondest beliefs of the human heart. Even
the ‘Utility of Religion’ is seriously proposed as a subject of
discussion. It seemed to be not out of place in a work on Scientific
Method to allude to the ultimate results and limits of that method. I
fear that I have very imperfectly succeeded in expressing my strong
conviction that before a rigorous logical scrutiny the Reign of Law
will prove to be an unverified hypothesis, the Uniformity of Nature an
ambiguous expression, the certainty of our scientific inferences to a
great extent a delusion. The value of science is of course very high,
while the conclusions are kept well within the limits of the data on
which they are founded, but it is pointed out that our experience is
of the most limited character compared with what there is to learn,
while our mental powers seem to fall infinitely short of the task of
comprehending and explaining fully the nature of any one object. I
draw the conclusion that we must interpret the results of Scientific
Method in an affirmative sense only. Ours must be a truly positive
philosophy, not that false negative philosophy which, building on a few
material facts, presumes to assert that it has compassed the bounds
of existence, while it nevertheless ignores the most unquestionable
phenomena of the human mind and feelings.
It is approximately certain that in freely employing illustrations
drawn from many different sciences, I have frequently fallen into
errors of detail. In this respect I must throw myself upon the
indulgence of the reader, who will bear in mind, as I hope, that the
scientific facts are generally mentioned purely for the purpose of
illustration, so that inaccuracies of detail will not in the majority
of cases affect the truth of the general principles illustrated.
*December 15, 1873.*
PREFACE
*TO THE SECOND EDITION*.
Few alterations of importance have been made in preparing this second
edition. Nevertheless, advantage has been taken of the opportunity to
revise very carefully both the language and the matter of the book.
Correspondents and critics having pointed out inaccuracies of more
or less importance in the first edition, suitable corrections and
emendations have been made. I am under obligations to Mr. C. J. Monro,
M.A., of Barnet, and to Mr. W. H. Brewer, M.A., one of Her Majesty’s
Inspectors of Schools, for numerous corrections.
Among several additions which have been made to the text, I may mention
the abstract (p. 143) of Professor Clifford’s remarkable investigation
into the number of types of compound statement involving four classes
of objects. This inquiry carries forward the inverse logical problem
described in the preceding sections. Again, the need of some better
logical method than the old Barbara Celarent, &c., is strikingly shown
by Mr. Venn’s logical problem, described at p. 90. A great number
of candidates in logic and philosophy were tested by Mr. Venn with
this problem, which, though simple in reality, was solved by very few
of those who were ignorant of Boole’s Logic. Other evidence could
be adduced by Mr. Venn of the need for some better means of logical
training. To enable the logical student to test his skill in the
solution of inductive logical problems, I have given (p. 127) a series
of ten problems graduated in difficulty.
To prevent misapprehension, it should be mentioned that, throughout
this edition, I have substituted the name *Logical Alphabet* for
*Logical Abecedarium*, the name applied in the first edition to the
exhaustive series of logical combinations represented in terms of
*A*, *B*, *C*, *D* (p. 94). It was objected by some readers that
*Abecedarium* is a long and unfamiliar name.
To the chapter on Units and Standards of Measurement, I have added two
sections, one (p. 325) containing a brief statement of the Theory of
Dimensions, and the other (p. 319) discussing Professor Clerk Maxwell’s
very original suggestion of a Natural System of Standards for the
measurement of space and time, depending upon the length and rapidity
of waves of light.
In my description of the Logical Machine in the *Philosophical
Transactions* (vol. 160, p. 498), I said--“It is rarely indeed that
any invention is made without some anticipation being sooner or later
discovered; but up to the present time I am totally unaware of even
a single previous attempt to devise or construct a machine which
should perform the operations of logical inference; and it is only,
I believe, in the satirical writings of Swift that an allusion to an
actual reasoning machine is to be found.” Before the paper was printed,
however, I was able to refer (p. 518) to the ingenious designs of the
late Mr. Alfred Smee as attempts to represent thought mechanically. Mr.
Smee’s machines indeed were never constructed, and, if constructed,
would not have performed actual logical inference. It has now just
come to light, however, that the celebrated Lord Stanhope actually did
construct a mechanical device, capable of representing syllogistic
inferences in a concrete form. It appears that logic was one of the
favourite studies of this truly original and ingenious nobleman. There
remain fragments of a logical work, printed by the Earl at his own
press, which show that he had arrived, before the year 1800, at the
principle of the quantified predicate. He puts forward this principle
in the most explicit manner, and proposes to employ it throughout his
syllogistic system. Moreover, he converts negative propositions into
affirmative ones, and represents these by means of the copula “is
identic with.” Thus he anticipated, probably by the force of his own
unaided insight, the main points of the logical method originated in
the works of George Bentham and George Boole, and developed in this
work. Stanhope, indeed, has no claim to priority of discovery, because
he seems never to have published his logical writings, although they
were put into print. There is no trace of them in the British Museum
Library, nor in any other library or logical work, so far as I am
aware. Both the papers and the logical contrivance have been placed
by the present Earl Stanhope in the hands of the Rev. Robert Harley,
F.R.S., who will, I hope, soon publish a description of them.[1]
[1] Since the above was written Mr. Harley has read an account of
Stanhope’s logical remains at the Dublin Meeting (1878) of the
British Association. The paper will be printed in *Mind*. (Note added
November, 1878.)
By the kindness of Mr. Harley, I have been able to examine Stanhope’s
logical contrivance, called by him the Demonstrator. It consists of a
square piece of bay-wood with a square depression in the centre, across
which two slides can be pushed, one being a piece of red glass, and
the other consisting of wood coloured gray. The extent to which each
of these slides is pushed in is indicated by scales and figures along
the edges of the aperture, and the simple rule of inference adopted
by Stanhope is: “To the gray add the red and subtract the *holon*,”
meaning by holon (ὅλον) the whole width of the aperture. This rule
of inference is a curious anticipation of De Morgan’s numerically
definite syllogism (see below, p. 168), and of inferences founded on
what Hamilton called “Ultra-total distribution.” Another curious point
about Stanhope’s device is, that one slide can be drawn out and pushed
in again at right angles to the other, and the overlapping part of the
slides then represents the probability of a conclusion, derived from
two premises of which the probabilities are respectively represented
by the projecting parts of the slides. Thus it appears that Stanhope
had studied the logic of probability as well as that of certainty, here
again anticipating, however obscurely, the recent progress of logical
science. It will be seen, however, that between Stanhope’s Demonstrator
and my Logical Machine there is no resemblance beyond the fact that
they both perform logical inference.
In the first edition I inserted a section (vol. i. p. 25), on
“Anticipations of the Principle of Substitution,” and I have reprinted
that section unchanged in this edition (p. 21). I remark therein that,
“In such a subject as logic it is hardly possible to put forth any
opinions which have not been in some degree previously entertained. The
germ at least of every doctrine will be found in earlier writings, and
novelty must arise chiefly in the mode of harmonising and developing
ideas.” I point out, as Professor T. M. Lindsay had previously done,
that Beneke had employed the name and principle of substitution, and
that doctrines closely approximating to substitution were stated by the
Port Royal Logicians more than 200 years ago.
I have not been at all surprised to learn, however, that other
logicians have more or less distinctly stated this principle of
substitution during the last two centuries. As my friend and successor
at Owens College, Professor Adamson, has discovered, this principle can
be traced back to no less a philosopher than Leibnitz.
The remarkable tract of Leibnitz,[2] entitled “Non inelegans Specimen
Demonstrandi in Abstractis,” commences at once with a definition
corresponding to the principle:--
“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint
*A* et *B*, et *A* ingrediatur aliquam propositionem veram, et ibi in
aliquo loco ipsius *A* pro ipso substituendo *B* fiat nova propositio
æque itidem vera, idque semper succedat in quacunque tali propositione,
*A* et *B* dicuntur esse eadem; et contra, si eadem sint *A* et *B*,
procedet substitutio quam dixi.”
[2] Leibnitii *Opera Philosophica quæ extant*. Erdmann, Pars I.
Berolini, 1840, p. 94.
Leibnitz, then, explicitly adopts the principle of substitution, but
he puts it in the form of a definition, saying that those things are
the same which can be substituted one for the other, without affecting
the truth of the proposition. It is only after having thus tested
the sameness of things that we can turn round and say that *A* and
*B*, being the same, may be substituted one for the other. It would
seem as if we were here in a vicious circle; for we are not allowed
to substitute *A* for *B*, unless we have ascertained by trial that
the result is a true proposition. The difficulty does not seem to be
removed by Leibnitz’ proviso, “idque semper succedat in quacunque
tali propositione.” How can we learn that because *A* and *B* may
be mutually substituted in some propositions, they may therefore
be substituted in others; and what is the criterion of likeness of
propositions expressed in the word “tali”? Whether the principle
of substitution is to be regarded as a postulate, an axiom, or a
definition, is just one of those fundamental questions which it seems
impossible to settle in the present position of philosophy, but this
uncertainty will not prevent our making a considerable step in logical
science.
Leibnitz proceeds to establish in the form of a theorem what is
usually taken as an axiom, thus (*Opera*, p. 95): “Theorema I. Quæ
sunt eadem uni tertio, eadem sunt inter se. Si *A* ∝ *B* et *B* ∝ *C*,
erit *A* ∝ *C*. Nam si in propositione *A* ∝ *B* (vera ea hypothesi)
substituitur *C* in locum *B* (quod facere licet per Def. I. quia *B* ∝
*C* ex hypothesi) fiet *A* ∝ *C*. Q. E. Dem.” Thus Leibnitz precisely
anticipates the mode of treating inference with two simple identities
described at p. 51 of this work.
Even the mathematical axiom that ‘equals added to equals make equals,’
is deduced from the principle of substitution. At p. 95 of Erdmann’s
edition, we find: “Si eidem addantur coincidentia fiunt coincidentia.
Si *A* ∝ *B*, erit *A* + *C* ∝ *B* + *C*. Nam si in propositione *A*
+ *C* ∝ *A* + *C* (quæ est vera per se) pro *A* semel substituas *B*
(quod facere licet per Def. I. quia *A* ∝ *B*) fiet *A* + *C* ∝ *B* +
*C* Q. E. Dem.” This is unquestionably the mode of deducing the several
axioms of mathematical reasoning from the higher axiom of substitution,
which is explained in the section on mathematical inference (p. 162) in
this work, and which had been previously stated in my *Substitution of
Similars*, p. 16.
There are one or two other brief tracts in which Leibnitz anticipates
the modern views of logic. Thus in the eighteenth tract in Erdmann’s
edition (p. 92), called “Fundamenta Calculi Ratiocinatoris”, he says:
“Inter ea quorum unum alteri substitui potest, salvis calculi legibus,
dicetur esse æquipollentiam.” There is evidence, also, that he had
arrived at the quantification of the predicate, and that he fully
understood the reduction of the universal affirmative proposition to
the form of an equation, which is the key to an improved view of logic.
Thus, in the tract entitled “Difficultates Quædam Logicæ,”[3] he says:
“Omne *A* est *B*; id est æquivalent *AB* et *A*, seu *A* non *B* est
non-ens.”
[3] Erdmann, p. 102.
It is curious to find, too, that Leibnitz was fully acquainted with the
Laws of Commutativeness and “Simplicity” (as I have called the second
law) attaching to logical symbols. In the “Addenda ad Specimen Calculi
Universalis” we read as follows.[4] “Transpositio literarum in eodem
termino nihil mutat, ut *ab* coincidet cum *ba*, seu animal rationale
et rationale animal.”
“Repetitio ejusdem literæ in eodem termino est inutilis, ut *b* est
*aa*; vel *bb* est *a*; homo est animal animal, vel homo homo est
animal. Sufficit enim dici *a* est *b*, seu homo est animal.”
[4] Ibid. p. 98.
Comparing this with what is stated in Boole’s *Mathematical Analysis of
Logic*, pp. 17–18, in his *Laws of Thought*, p. 29, or in this work,
pp. 32–35, we find that Leibnitz had arrived two centuries ago at a
clear perception of the bases of logical notation. When Boole pointed
out that, in logic, *xx* = *x*, this seemed to mathematicians to be a
paradox, or in any case a wholly new discovery; but here we have it
plainly stated by Leibnitz.
The reader must not assume, however, that because Leibnitz correctly
apprehended the fundamental principles of logic, he left nothing for
modern logicians to do. On the contrary, Leibnitz obtained no useful
results from his definition of substitution. When he proceeds to
explain the syllogism, as in the paper on “Definitiones Logicæ,”[5]
he gives up substitution altogether, and falls back upon the notion
of inclusion of class in class, saying, “Includens includentis est
includens inclusi, seu si *A* includit *B* et *B* includit *C*, etiam
*A* includet *C*.” He proceeds to make out certain rules of the
syllogism involving the distinction of subject and predicate, and
in no important respect better than the old rules of the syllogism.
Leibnitz’ logical tracts are, in fact, little more than brief memoranda
of investigations which seem never to have been followed out. They
remain as evidence of his wonderful sagacity, but it would be difficult
to show that they have had any influence on the progress of logical
science in recent times.
[5] Erdmann, p. 100.
I should like to explain how it happened that these logical writings
of Leibnitz were unknown to me, until within the last twelve months. I
am so slow a reader of Latin books, indeed, that my overlooking a few
pages of Leibnitz’ works would not have been in any case surprising.
But the fact is that the copy of Leibnitz’ works of which I made
occasional use, was one of the edition of Dutens, contained in Owens
College Library. The logical tracts in question were not printed in
that edition, and with one exception, they remained in manuscript in
the Royal Library at Hanover, until edited by Erdmann, in 1839–40.
The tract “Difficultates Quædam Logicæ,” though not known to Dutens,
was published by Raspe in 1765, in his collection called *Œuvres
Philosophiques de feu M^{r.} Leibnitz*; but this work had not come to
my notice, nor does the tract in question seem to contain any explicit
statement of the principle of substitution.
It is, I presume, the comparatively recent publication of Leibnitz’
most remarkable logical tracts which explains the apparent ignorance of
logicians as regards their contents and importance. The most learned
logicians, such as Hamilton and Ueberweg, ignore Leibnitz’ principle
of substitution. In the Appendix to the fourth volume of Hamilton’s
*Lectures on Metaphysics and Logic*, is given an elaborate compendium
of the views of logical writers concerning the ultimate basis of
deductive reasoning. Leibnitz is briefly noticed on p. 319, but without
any hint of substitution. He is here quoted as saying, “What are the
same with the same third, are the same with each other; that is, if *A*
be the same with *B*, and *C* be the same with *B*, it is necessary
that *A* and *C* should also be the same with one another. For this
principle flows immediately from the principle of contradiction, and is
the ground and basis of all logic; if that fail, there is no longer any
way of reasoning with certainty.” This view of the matter seems to be
inconsistent with that which he adopted in his posthumous tract.
Dr. Thomson, indeed, was acquainted with Leibnitz’ tracts, and refers
to them in his *Outline of the Necessary Laws of Thought*. He calls
them valuable; nevertheless, he seems to have missed the really
valuable point; for in making two brief quotations,[6] he omits all
mention of the principle of substitution.
[6] Fifth Edition, 1860, p. 158.
Ueberweg is probably considered the best authority concerning the
history of logic, and in his well-known *System of Logic and History
of Logical Doctrines*,[7] he gives some account of the principle of
substitution, especially as it is implicitly stated in the *Port Royal
Logic*. But he omits all reference to Leibnitz in this connection,
nor does he elsewhere, so far as I can find, supply the omission.
His English editor, Professor T. M. Lindsay, in referring to my
*Substitution of Similars*, points out how I was anticipated by Beneke;
but he also ignores Leibnitz. It is thus apparent that the most learned
logicians, even when writing especially on the history of logic,
displayed ignorance of Leibnitz’ most valuable logical writings.
[7] Section 120.
It has been recently pointed out to me, however, that the Rev. Robert
Harley did draw attention, at the Nottingham Meeting of the British
Association, in 1866, to Leibnitz’ anticipations of Boole’s laws of
logical notation,[8] and I am informed that Boole, about a year after
the publication of his *Laws of Thought*, was made acquainted with
these anticipations by R. Leslie Ellis.
[8] See his “Remarks on Boole’s Mathematical Analysis of Logic.”
*Report of the 36th Meeting of the British Association, Transactions
of the Sections*, pp. 3–6.
There seems to have been at least one other German logician who
discovered, or adopted, the principle of substitution. Reusch, in his
*Systema Logicum*, published in 1734, laboured to give a broader basis
to the *Dictum de Omni et Nullo*. He argues, that “the whole business
of ordinary reasoning is accomplished by the substitution of ideas in
place of the subject or predicate of the fundamental proposition. This
some call the *equation of thoughts*.” But, in the hands of Reusch,
substitution does not seem to lead to simplicity, since it has to be
carried on according to the rules of Equipollence, Reciprocation,
Subordination, and Co-ordination.[9] Reusch is elsewhere spoken of[10]
as the “celebrated Reusch”; nevertheless, I have not been able to find
a copy of his book in London, even in the British Museum Library; it is
not mentioned in the printed catalogue of the Bodleian Library; Messrs.
Asher have failed to obtain it for me by advertisement in Germany; and
Professor Adamson has been equally unsuccessful. From the way in which
the principle of substitution is mentioned by Reusch, it would seem
likely that other logicians of the early part of the eighteenth century
were acquainted with it; but, if so, it is still more curious that
recent historians of logical science have overlooked the doctrine.
[9] Hamilton’s Lectures, vol. iv. p. 319.
[10] Ibid. p. 326.
It is a strange and discouraging fact, that true views of logic should
have been discovered and discussed from one to two centuries ago, and
yet should have remained, like George Bentham’s work in this century,
without influence on the subsequent progress of the science. It may be
regarded as certain that none of the discoverers of the quantification
of the predicate, Bentham, Hamilton, Thomson, De Morgan, and Boole,
were in any way assisted by the hints of the principle contained in
previous writers. As to my own views of logic, they were originally
moulded by a careful study of Boole’s works, as fully stated in my
first logical essay.[11] As to the process of substitution, it was
not learnt from any work on logic, but is simply the process of
substitution perfectly familiar to mathematicians, and with which I
necessarily became familiar in the course of my long-continued study of
mathematics under the late Professor De Morgan.
[11] *Pure Logic, or the Logic of Quality apart from Quantity;
with Remarks on Boole’s System, and on the Relation of Logic and
Mathematics.* London, 1864, p. 3.
I find that the Theory of Number, which I explained in the eighth
chapter of this work, is also partially anticipated in a single
scholium of Leibnitz. He first gives as an axiom the now well-known law
of Boole, as follows:--
“Axioma I. Si idem secum ipso sumatur, nihil constituitur novum, seu
*A* + *A* ∝ *A*.” Then follows this remarkable scholium: “Equidem in
numeris 4 + 4 facit 8, seu bini nummi binis additi faciunt quatuor
nummos, sed tunc bini additi sunt alii a prioribus; si iidem essent
nihil novi prodiret et perinde esset ac si joco ex tribus ovis facere
vellemus sex numerando, primum 3 ova, deinde uno sublato residua 2, ac
denique uno rursus sublato residuum.”
Translated this would read as follows:--
“Axiom I. If the same thing is taken together with itself, nothing new
arises, or *A* + *A* = *A*.
“Scholium. In numbers, indeed, 4 + 4 makes 8, or two coins added to two
coins make four coins, but then the two added are different from the
former ones; if they were the same nothing new would be produced, and
it would be just as if we tried in joke to make six eggs out of three,
by counting firstly the three eggs, then, one being removed, counting
the remaining two, and lastly, one being again removed, counting the
remaining egg.”
Compare the above with pp. 156 to 162 of the present work.
M. Littré has quite recently pointed out[12] what he thinks is an
analogy between the system of formal logic, stated in the following
pages, and the logical devices of the celebrated Raymond Lully. Lully’s
method of invention was described in a great number of mediæval books,
but is best stated in his *Ars Compendiosa Inveniendi Veritatem, seu
Ars Magna et Major*. This method consisted in placing various names
of things in the sectors of concentric circles, so that when the
circles were turned, every possible combination of the things was
easily produced by mechanical means. It might, perhaps, be possible to
discover in this method a vague and rude anticipation of combinational
logic; but it is well known that the results of Lully’s method were
usually of a fanciful, if not absurd character.
[12] *La Philosophie Positive*, Mai-Juin, 1877, tom. xviii. p. 456.
A much closer analogue of the Logical Alphabet is probably to be found
in the Logical Square, invented by John Christian Lange, and described
in a rare and unnoticed work by him which I have recently found in the
British Museum.[13] This square involved the principle of bifurcate
classification, and was an improved form of the Ramean and Porphyrian
tree (see below, p. 702). Lange seems, indeed, to have worked out his
Logical Square into a mechanical form, and he suggests that it might
be employed somewhat in the manner of Napier’s Bones (p. 65). There
is much analogy between his Square and my Abacus, but Lange had not
arrived at a logical system enabling him to use his invention for
logical inference in the manner of the Logical Abacus. Another work
of Lange is said to contain the first publication of the well known
Eulerian diagrams of proposition and syllogism.[14]
[13] *Inventum Novum Quadrati Logici*, &c., Gissæ Hassorum, 1714, 8vo.
[14] See *Ueberweg’s System of Logic*, &c., translated by Lindsay,
p. 302.
Since the first edition was published, an important work by Mr. George
Lewes has appeared, namely, his *Problems of Life and Mind*, which
to a great extent treats of scientific method, and formulates the
rules of philosophising. I should have liked to discuss the bearing
of Mr. Lewes’s views upon those here propounded, but I have felt it
to be impossible in a book already filling nearly 800 pages, to enter
upon the discussion of a yet more extensive book. For the same reason
I have not been able to compare my own treatment of the subject of
probability with the views expressed by Mr. Venn in his *Logic of
Chance*. With Mr. J. J. Murphy’s profound and remarkable works on
*Habit and Intelligence*, and on *The Scientific Basis of Faith*, I
was unfortunately unacquainted when I wrote the following pages. They
cannot safely be overlooked by any one who wishes to comprehend the
tendency of philosophy and scientific method in the present day.
It seems desirable that I should endeavour to answer some of the
critics who have pointed out what they consider defects in the
doctrines of this book, especially in the first part, which treats
of deduction. Some of the notices of the work were indeed rather
statements of its contents than critiques. Thus, I am much indebted
to M. Louis Liard, Professor of Philosophy at Bordeaux, for the very
careful exposition[15] of the substitutional view of logic which he
gave in the excellent *Revue Philosophique*, edited by M. Ribot.
(Mars, 1877, tom. iii. p. 277.) An equally careful account of the
system was given by M. Riehl, Professor of Philosophy at Graz, in
his article on “Die Englische Logik der Gegenwart,” published in the
*Vierteljahrsschrift für wissenschaftliche Philosophie*. (1 Heft,
Leipzig, 1876.) I should like to acknowledge also the careful and able
manner in which my book was reviewed by the *New York Daily Tribune*
and the *New York Times*.
[15] Since the above was written M. Liard has republished this
exposition as one chapter of an interesting and admirably lucid
account of the progress of logical science in England. After a brief
but clear introduction, treating of the views of Herschel, Mill, and
others concerning Inductive Logic, M. Liard describes in succession
the logical systems of George Bentham, Hamilton, De Morgan, Boole,
and that contained in the present work. The title of the book is as
follows:--*Les Logiciens Anglais Contemporains*. Par Louis Liard,
Professeur de Philosophie à la Faculté des Lettres de Bordeaux.
Paris: Librairie Germer Baillière. 1878. (Note added November, 1878.)
The most serious objections which have been brought against my
treatment of logic have regard to my failure to enter into an analysis
of the ultimate nature and origin of the Laws of Thought. The
*Spectator*,[16] for instance, in the course of a careful review, says
of the principle of substitution, “Surely it is a great omission not
to discuss whence we get this great principle itself; whether it is a
pure law of the mind, or only an approximate lesson of experience; and
if a pure product of the mind, whether there are any other products
of the same kind, furnished by our knowing faculty itself.” Professor
Robertson, in his very acute review,[17] likewise objects to the want
of psychological and philosophical analysis. “If the book really
corresponded to its title, Mr. Jevons could hardly have passed so
lightly over the question, which he does not omit to raise, concerning
those undoubted principles of knowledge commonly called the Laws of
Thought.... Everywhere, indeed, he appears least at ease when he
touches on questions properly philosophical; nor is he satisfactory
in his psychological references, as on pp. 4, 5, where he cannot
commit himself to a statement without an accompaniment of ‘probably,’
‘almost,’ or ‘hardly.’ Reservations are often very much in place, but
there are fundamental questions on which it is proper to make up one’s
mind.”
[16] *Spectator*, September 19, 1874, p. 1178. A second portion of
the review appeared in the same journal for September 26, 1874,
p. 1204.
[17] *Mind*: a Quarterly Review of Psychology and Philosophy. No. II.
April 1876. Vol. I. p. 206.
These remarks appear to me to be well founded, and I must state why
it is that I have ventured to publish an extensive work on logic,
without properly making up my mind as to the fundamental nature of the
reasoning process. The fault after all is one of omission rather than
of commission. It is open to me on a future occasion to supply the
deficiency if I should ever feel able to undertake the task. But I do
not conceive it to be an essential part of any treatise to enter into
an ultimate analysis of its subject matter. Analyses must always end
somewhere. There were good treatises on light which described the laws
of the phenomenon correctly before it was known whether light consisted
of undulations or of projected particles. Now we have treatises on the
Undulatory Theory which are very valuable and satisfactory, although
they leave us in almost complete doubt as to what the vibrating medium
really is. So I think that, in the present day, we need a correct and
scientific exhibition of the formal laws of thought, and of the forms
of reasoning based on them, although we may not be able to enter into
any complete analysis of the nature of those laws. What would the
science of geometry be like now if the Greek geometers had decided that
it was improper to publish any propositions before they had decided
on the nature of an axiom? Where would the science of arithmetic be
now if an analysis of the nature of number itself were a necessary
preliminary to a development of the results of its laws? In recent
times there have been enormous additions to the mathematical sciences,
but very few attempts at psychological analysis. In the Alexandrian and
early mediæval schools of philosophy, much attention was given to the
nature of unity and plurality chiefly called forth by the question of
the Trinity. In the last two centuries whole sciences have been created
out of the notion of plurality, and yet speculation on the nature of
plurality has dwindled away. This present treatise contains, in the
eighth chapter, one of the few recent attempts to analyse the notion of
number itself.
If further illustration is needed, I may refer to the differential
calculus. Nobody calls in question the formal truth of the results of
that calculus. All the more exact and successful parts of physical
science depend upon its use, and yet the mathematicians who have
created so great a body of exact truths have never decided upon the
basis of the calculus. What is the nature of a limit or the nature of
an infinitesimal? Start the question among a knot of mathematicians,
and it will be found that hardly two agree, unless it is in regarding
the question itself as a trifling one. Some hold that there are no such
things as infinitesimals, and that it is all a question of limits.
Others would argue that the infinitesimal is the necessary outcome of
the limit, but various shades of intermediate opinion spring up.
Now it is just the same with logic. If the forms of deductive and
inductive reasoning given in the earlier part of this treatise are
correct, they constitute a definite addition to logical science, and
it would have been absurd to decline to publish such results because I
could not at the same time decide in my own mind about the psychology
and philosophy of the subject. It comes in short to this, that my book
is a book on Formal Logic and Scientific Method, and not a book on
psychology and philosophy.
It may be objected, indeed, as the *Spectator* objects, that Mill’s
System of Logic is particularly strong in the discussion of the
psychological foundations of reasoning, so that Mill would appear to
have successfully treated that which I feel myself to be incapable of
attempting at present. If Mill’s analysis of knowledge is correct, then
I have nothing to say in excuse for my own deficiencies. But it is
well to do one thing at a time, and therefore I have not occupied any
considerable part of this book with controversy and refutation. What I
have to say of Mill’s logic will be said in a separate work, in which
his analysis of knowledge will be somewhat minutely analysed. It will
then be shown, I believe, that Mill’s psychological and philosophical
treatment of logic has not yielded such satisfactory results as some
writers seem to believe.[18]
[18] Portions of this work have already been published in my
articles, entitled “John Stuart Mill’s Philosophy Tested,” printed
in the *Contemporary Review* for December, 1877, vol. xxxi. p. 167,
and for January and April, 1878, vol. xxxi. p. 256, and vol. xxxii.
p. 88. (Note added in November, 1878.)
Various minor but still important criticisms were made by Professor
Robertson, a few of which have been noticed in the text (pp. 27, 101).
In other cases his objections hardly admit of any other answer than
such as consists in asking the reader to judge between the work and the
criticism. Thus Mr. Robertson asserts[19] that the most complex logical
problems solved in this book (up to p. 102 of this edition) might be
more easily and shortly dealt with upon the principles and with the
recognised methods of the traditional logic. The burden of proof here
lies upon Mr. Robertson, and his only proof consists in a single case,
where he is able, as it seems to me accidentally, to get a special
conclusion by the old form of dilemma. It would be a long labour to
test the old logic upon every result obtained by my notation, and I
must leave such readers as are well acquainted with the syllogistic
logic to pronounce upon the comparative simplicity and power of the
new and old systems. For other acute objections brought forward by Mr.
Robertson, I must refer the reader to the article in question.
[19] *Mind*, vol. i. p. 222.
One point in my last chapter, that on the Results and Limits of
Scientific Method, has been criticised by Professor W. K. Clifford in
his lecture[20] on “The First and the Last Catastrophe.” In vol. ii.
p. 438 of the first edition (p. 744 of this edition) I referred to
certain inferences drawn by eminent physicists as to a limit to
the antiquity of the present order of things. “According to Sir W.
Thomson’s deductions from Fourier’s *theory of heat*, we can trace down
the dissipation of heat by conduction and radiation to an infinitely
distant time when all things will be uniformly cold. But we cannot
similarly trace the Heat-history of the Universe to an infinite
distance in the past. For a certain negative value of the time, the
formulæ give impossible values, indicating that there was some initial
distribution of heat which could not have resulted, according to known
laws of nature, from any previous distribution.”
[20] *Fortnightly Review*, New Series, April 1875, p. 480. Lecture
reprinted by the Sunday Lecture Society, p. 24.
Now according to Professor Clifford I have here misstated Thomson’s
results. “It is not according to the known laws of nature, it is
according to the known laws of conduction of heat, that Sir William
Thomson is speaking.... All these physical writers, knowing what they
were writing about, simply drew such conclusions from the facts which
were before them as could be reasonably drawn. They say, here is a
state of things which could not have been produced by the circumstances
we are at present investigating. Then your speculator comes, he reads
a sentence and says, ‘Here is an opportunity for me to have my fling.’
And he has his fling, and makes a purely baseless theory about the
necessary origin of the present order of nature at some definite point
of time, which might be calculated.”
Professor Clifford proceeds to explain that Thomson’s formulæ only give
a limit to the heat history of, say, the earth’s crust in the solid
state. We are led back to the time when it became solidified from the
fluid condition. There is discontinuity in the history of the solid
matter, but still discontinuity which is within our comprehension.
Still further back we should come to discontinuity again, when the
liquid was formed by the condensation of heated gaseous matter. Beyond
that event, however, there is no need to suppose further discontinuity
of law, for the gaseous matter might consist of molecules which had
been falling together from different parts of space through infinite
past time. As Professor Clifford says (p. 481) of the bodies of the
universe, “What they have actually done is to fall together and get
solid. If we should reverse the process we should see them separating
and getting cool, and as a limit to that, we should find that all these
bodies would be resolved into molecules, and all these would be flying
away from each other. There would be no limit to that process, and we
could trace it as far back as ever we liked to trace it.”
Assuming that I have erred, I should like to point out that I have
erred in the best company, or more strictly, being a speculator, I
have been led into error by the best physical writers. Professor Tait,
in his *Sketch of Thermodynamics*, speaking of the laws discovered by
Fourier for the motion of heat in a solid, says, “Their mathematical
expressions point also to the fact that a uniform distribution of heat,
or a distribution tending to become uniform, must have arisen from some
primitive distribution of heat of a kind not capable of being produced
by known laws from any previous distribution.” In the latter words it
will be seen that there is no limitation to the laws of conduction,
and, although I had carefully referred to Sir W. Thomson’s original
paper, it is not unnatural that I should take Professor Tait’s
interpretation of its meaning.[21]
[21] Sir W. Thomson’s words are as follows (*Cambridge Mathematical
Journal*, Nov. 1842, vol. iii. p. 174). “When *x* is negative, the
state represented cannot be the result of any *possible* distribution
of temperature which has previously existed.” There is no limitation
in the sentence to the laws of conduction, but, as the whole paper
treats of the results of conduction in a solid, it may no doubt be
understood that there is a *tacit* limitation. See also a second
paper on the subject in the same journal for February, 1844, vol. iv.
p. 67, where again there is no expressed limitation.
In his new work *On some Recent Advances in Physical Science*,
Professor Tait has recurred to the subject as follows:[22] “A profound
lesson may be learned from one of the earliest little papers of Sir W.
Thomson, published while he was an undergraduate at Cambridge, where
he shows that Fourier’s magnificent treatment of the conduction of
heat [in a solid body] leads to formulæ for its distribution which
are intelligible (and of course capable of being fully verified by
experiment) for all time future, but which, except in particular
cases, when extended to time past, remain intelligible for a finite
period only, and *then* indicate a state of things which could
not have resulted under known laws from any conceivable previous
distribution [of heat in the body]. So far as heat is concerned, modern
investigations have shown that a previous distribution of the *matter*
involved may, by its potential energy, be capable of producing such
a state of things at the moment of its aggregation; but the example
is now adduced not for its bearing on heat alone, but as a simple
illustration of the fact that all portions of our Science, especially
that beautiful one, the Dissipation of Energy, point unanimously to a
beginning, to a state of things incapable of being derived by present
laws [of tangible matter and its energy] from any conceivable previous
arrangement.” As this was published nearly a year after Professor
Clifford’s lecture, it may be inferred that Professor Tait adheres to
his original opinion that the theory of heat does give evidence of “a
beginning.”
[22] Pp. 25–26. The parentheses are in the original, and show
Professor Tait’s corrections in the verbatim reports of his lectures.
The subject is treated again on pp. 168–9.
I may add that Professor Clerk Maxwell’s words seem to countenance the
same view, for he says,[23] “This is only one of the cases in which a
consideration of the dissipation of energy leads to the determination
of a superior limit to the antiquity of the observed order of things.”
The expression “observed order of things” is open to much ambiguity,
but in the absence of qualification I should take it to include the
aggregate of the laws of nature known to us. I should interpret
Professor Maxwell as meaning that the theory of heat indicates the
occurrence of some event of which our science cannot give any further
explanation. The physical writers thus seem not to be so clear about
the matter as Professor Clifford assumes.
[23] *Theory of Heat* 1871, p. 245.
So far as I may venture to form an independent opinion on the subject,
it is to the effect that Professor Clifford is right, and that the
known laws of nature do not enable us to assign a “beginning.” Science
leads us backwards into infinite past duration. But that Professor
Clifford is right on this point, is no reason why we should suppose
him to be right in his other opinions, some of which I am sure are
wrong. Nor is it a reason why other parts of my last chapter should be
wrong. The question only affects the single paragraph on pp. 744–5 of
this book, which might, I believe, be struck out without necessitating
any alteration in the rest of the text. It is always to be remembered
that the failure of an argument in favour of a proposition does not,
generally speaking, add much, if any, probability to the contradictory
proposition. I cannot conclude without expressing my acknowledgments
to Professor Clifford for his kind expressions regarding my work as a
whole.
2, The Chestnuts,
West Heath,
Hampstead, N. W.
*August 15, 1877.*
CONTENTS.
BOOK I.
FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE.
CHAPTER I.
INTRODUCTION.
SECTION PAGE
1. Introduction 1
2. The Powers of Mind concerned in the Creation of Science 4
3. Laws of Identity and Difference 5
4. The Nature of the Laws of Identity and Difference 6
5. The Process of Inference 9
6. Deduction and Induction 11
7. Symbolic Expression of Logical Inference 13
8. Expression of Identity and Difference 14
9. General Formula of Logical Inference 17
10. The Propagating Power of Similarity 20
11. Anticipations of the Principle of Substitution 21
12. The Logic of Relatives 22
CHAPTER II.
TERMS.
1. Terms 24
2. Twofold meaning of General Names 25
3. Abstract Terms 27
4. Substantial Terms 28
5. Collective Terms 29
6. Synthesis of Terms 30
7. Symbolic Expression of the Law of Contradiction 31
8. Certain Special Conditions of Logical Symbols 32
CHAPTER III.
PROPOSITIONS.
1. Propositions 36
2. Simple Identities 37
3. Partial Identities 40
4. Limited Identities 42
5. Negative Propositions 43
6. Conversion of Propositions 46
7. Twofold Interpretation of Propositions 47
CHAPTER IV.
DEDUCTIVE REASONING.
1. Deductive Reasoning 49
2. Immediate Inference 50
3. Inference with Two Simple Identities 51
4. Inference with a Simple and a Partial Identity 53
5. Inference of a Partial from Two Partial Identities 55
6. On the Ellipsis of Terms in Partial Identities 57
7. Inference of a Simple from Two Partial Identities 58
8. Inference of a Limited from Two Partial Identities 59
9. Miscellaneous Forms of Deductive Inference 60
10. Fallacies 62
CHAPTER V.
DISJUNCTIVE PROPOSITIONS.
1. Disjunctive Propositions 66
2. Expression of the Alternative Relation 67
3. Nature of the Alternative Relation 68
4. Laws of the Disjunctive Relation 71
5. Symbolic Expression of the Law of Duality 73
6. Various Forms of the Disjunctive Proposition 74
7. Inference by Disjunctive Propositions 76
CHAPTER VI.
THE INDIRECT METHOD OF INFERENCE.
1. The Indirect Method of Inference 81
2. Simple Illustrations 83
3. Employment of the Contrapositive Proposition 84
4. Contrapositive of a Simple Identity 86
5. Miscellaneous Examples of the Method 88
6. Mr. Venn’s Problem 90
7. Abbreviation of the Process 91
8. The Logical Alphabet 94
9. The Logical Slate 95
10. Abstraction of Indifferent Circumstances 97
11. Illustrations of the Indirect Method 98
12. Second Example 99
13. Third Example 100
14. Fourth Example 101
15. Fifth Example 101
16. Fallacies Analysed by the Indirect Method 102
17. The Logical Abacus 104
18. The Logical Machine 107
19. The Order of Premises 114
20. The Equivalence of Propositions 115
21. The Nature of Inference 118
CHAPTER VII.
INDUCTION.
1. Induction 121
2. Induction an Inverse Operation 122
3. Inductive Problems for Solution by the Reader 126
4. Induction of Simple Identities 127
5. Induction of Partial Identities 130
6. Solution of the Inverse or Inductive Problem, involving
Two Classes 134
7. The Inverse Logical Problem, involving Three Classes 137
8. Professor Clifford on the Types of Compound Statement
involving Four Classes 143
9. Distinction between Perfect and Imperfect Induction 146
10. Transition from Perfect to Imperfect Induction 149
BOOK II.
NUMBER, VARIETY, AND PROBABILITY.
CHAPTER VIII.
PRINCIPLES OF NUMBER.
1. Principles of Number 153
2. The Nature of Numbe 156
3. Of Numerical Abstraction 158
4. Concrete and Abstract Number 159
5. Analogy of Logical and Numerical Terms 160
6. Principle of Mathematical Inference 162
7. Reasoning by Inequalities 165
8. Arithmetical Reasoning 167
9. Numerically Definite Reasoning 168
10. Numerical meaning of Logical Conditions 171
CHAPTER IX.
THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS
AND PERMUTATIONS.
1. The Variety of Nature 173
2. Distinction of Combinations and Permutations 177
3. Calculation of Number of Combinations 180
4. The Arithmetical Triangle 182
5. Connexion between the Arithmetical Triangle and the
Logical Alphabet 189
6. Possible Variety of Nature and Art 190
7. Higher Orders of Variety 192
CHAPTER X.
THEORY OF PROBABILITY.
1. Theory of Probability 197
2. Fundamental Principles of the Theory 200
3. Rules for the Calculation of Probabilities 203
4. The Logical Alphabet in questions of Probability 205
5. Comparison of the Theory with Experience 206
6. Probable Deductive Arguments 209
7. Difficulties of the Theory 213
CHAPTER XI.
PHILOSOPHY OF INDUCTIVE INFERENCE.
1. Philosophy of Inductive Inference 218
2. Various Classes of Inductive Truths 219
3. The Relation of Cause and Effect 220
4. Fallacious Use of the Term Cause 221
5. Confusion of Two Questions 222
6. Definition of the Term Cause 224
7. Distinction of Inductive and Deductive Results 226
8. The Grounds of Inductive Inference 228
9. Illustrations of the Inductive Process 229
10. Geometrical Reasoning 233
11. Discrimination of Certainty and Probability 235
CHAPTER XII.
THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY
OF PROBABILITY.
1. The Inductive or Inverse Application of the Theory 240
2. Principle of the Inverse Method 242
3. Simple Applications of the Inverse Method 244
4. The Theory of Probability in Astronomy 247
5. The General Inverse Problem 250
6. Simple Illustration of the Inverse Problem 253
7. General Solution of the Inverse Problem 255
8. Rules of the Inverse Method 257
9. Fortuitous Coincidences 261
10. Summary of the Theory of Inductive Inference 265
BOOK III.
METHODS OF MEASUREMENT.
CHAPTER XIII.
THE EXACT MEASUREMENT OF PHENOMENA.
1. The Exact Measurement of Phenomena 270
2. Division of the Subject 274
3. Continuous quantity 274
4. The Fallacious Indications of the Senses 276
5. Complexity of Quantitative Questions 278
6. The Methods of Accurate Measurement 282
7. Conditions of Accurate Measurement 282
8. Measuring Instruments 284
9. The Method of Repetition 288
10. Measurements by Natural Coincidence 292
11. Modes of Indirect Measurement 296
12. Comparative Use of Measuring Instruments 299
13. Systematic Performance of Measurements 300
14. The Pendulum 302
15. Attainable Accuracy of Measurement 303
CHAPTER XIV.
UNITS AND STANDARDS OF MEASUREMENT.
1. Units and Standards of Measurement 305
2. Standard Unit of Time 307
3. The Unit of Space and the Bar Standard 312
4. The Terrestrial Standard 314
5. The Pendulum Standard 315
6. Unit of Density 316
7. Unit of Mass 317
8. Natural System of Standards 319
9. Subsidiary Units 320
10. Derived Units 321
11. Provisional Units 323
12. Theory of Dimensions 325
13. Natural Constants 328
14. Mathematical Constants 330
15. Physical Constants 331
16. Astronomical Constants 332
17. Terrestrial Numbers 333
18. Organic Numbers 333
19. Social Numbers 334
CHAPTER XV.
ANALYSIS OF QUANTITATIVE PHENOMENA.
1. Analysis of Quantitative Phenomena 335
2. Illustrations of the Complication of Effects 336
3. Methods of Eliminating Error 339
4. Method of Avoidance of Error 340
5. Differential Method 344
6. Method of Correction 346
7. Method of Compensation 350
8. Method of Reversal 354
CHAPTER XVI.
THE METHOD OF MEANS.
1. The Method of Means 357
2. Several Uses of the Mean Result 359
3. The Mean and the Average 360
4. On the Average or Fictitious Mean 363
5. The Precise Mean Result 365
6. Determination of the Zero Point 368
7. Determination of Maximum Points 371
CHAPTER XVII.
THE LAW OF ERROR.
1. The Law of Error 374
2. Establishment of the Law of Error 375
3. Herschel’s Geometrical Proof 377
4. Laplace’s and Quetelet’s Proof of the Law 378
5. Logical Origin of the Law of Error 383
6. Verification of the Law of Error 383
7. The Probable Mean Result 385
8. The Probable Error of Results 386
9. Rejection of the Mean Result 389
10. Method of Least Squares 393
11. Works upon the Theory of Probability 394
12. Detection of Constant Errors 396
BOOK IV.
INDUCTIVE INVESTIGATION.
CHAPTER XVIII.
OBSERVATION.
1. Observation 399
2. Distinction of Observation and Experiment 400
3. Mental Conditions of Correct Observation 402
4. Instrumental and Sensual Conditions of Correct Observation 404
5. External Conditions of Correct Observation 407
6. Apparent Sequence of Events 409
7. Negative Arguments from Non-Observation 411
CHAPTER XIX.
EXPERIMENT.
1. Experiment 416
2. Exclusion of Indifferent Circumstances 419
3. Simplification of Experiments 422
4. Failure in the Simplification of Experiments 424
5. Removal of Usual Conditions 426
6. Interference of Unsuspected Conditions 428
7. Blind or Test Experiments 433
8. Negative Results of Experiment 434
9. Limits of Experiment 437
CHAPTER XX.
METHOD OF VARIATIONS.
1. Method of Variations 439
2. The Variable and the Variant 440
3. Measurement of the Variable 441
4. Maintenance of Similar Conditions 443
5. Collective Experiments 445
6. Periodic Variations 447
7. Combined Periodic Changes 450
8. Principle of Forced Vibrations 451
9. Integrated Variations 452
CHAPTER XXI.
THEORY OF APPROXIMATION.
1. Theory of Approximation 456
2. Substitution of Simple Hypotheses 458
3. Approximation to Exact Laws 462
4. Successive Approximations to Natural Conditions 465
5. Discovery of Hypothetically Simple Laws 470
6. Mathematical Principles of Approximation 471
7. Approximate Independence of Small Effects 475
8. Four Meanings of Equality 479
9. Arithmetic of Approximate Quantities 481
CHAPTER XXII.
QUANTITATIVE INDUCTION.
1. Quantitative Induction 483
2. Probable Connexion of Varying Quantities 484
3. Empirical Mathematical Laws 487
4. Discovery of Rational Formulæ 489
5. The Graphical Method 492
6. Interpolation and Extrapolation 495
7. Illustrations of Empirical Quantitative Laws 499
8. Simple Proportional Variation 501
CHAPTER XXIII.
THE USE OF HYPOTHESIS.
1. The Use of Hypothesis 504
2. Requisites of a good Hypothesis 510
3. Possibility of Deductive Reasoning 511
4. Consistency with the Laws of Nature 514
5. Conformity with Facts 516
6. Experimentum Crucis 518
7. Descriptive Hypotheses 522
CHAPTER XXIV.
EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION.
1. Empirical Knowledge, Explanation and Prediction 525
2. Empirical Knowledge 526
3. Accidental Discovery 529
4. Empirical Observations subsequently Explained 532
5. Overlooked Results of Theory 534
6. Predicted Discoveries 536
7. Predictions in the Science of Light 538
8. Predictions from the Theory of Undulations 540
9. Prediction in other Sciences 542
10. Prediction by Inversion of Cause and Effect 545
11. Facts known only by Theory 547
CHAPTER XXV.
ACCORDANCE OF QUANTITATIVE THEORIES.
1. Accordance of Quantitative Theories 551
2. Empirical Measurements 552
3. Quantities indicated by Theory, but Empirically Measured 553
4. Explained Results of Measurement 554
5. Quantities determined by Theory and verified by
Measurement 555
6. Quantities determined by Theory and not verified 556
7. Discordance of Theory and Experiment 558
8. Accordance of Measurements of Astronomical Distances 560
9. Selection of the best Mode of Measurement 563
10. Agreement of Distinct Modes of Measurement 564
11. Residual Phenomena 569
CHAPTER XXVI.
CHARACTER OF THE EXPERIMENTALIST.
1. Character of the Experimentalist 574
2. Error of the Baconian Method 576
3. Freedom of Theorising 577
4. The Newtonian Method, the True Organum 581
5. Candour and Courage of the Philosophic Mind 586
6. The Philosophic Character of Faraday 587
7. Reservation of Judgment 592
BOOK V.
GENERALISATION, ANALOGY, AND CLASSIFICATION.
CHAPTER XXVII.
GENERALISATION.
1. Generalisation 594
2. Distinction of Generalisation and Analogy 596
3. Two Meanings of Generalisation 597
4. Value of Generalisation 599
5. Comparative Generality of Properties 600
6. Uniform Properties of all Matter 603
7. Variable Properties of Matter 606
8. Extreme Instances of Properties 607
9. The Detection of Continuity 610
10. The Law of Continuity 615
11. Failure of the Law of Continuity 619
12. Negative Arguments on the Principle of Continuity 621
13. Tendency to Hasty Generalisation 623
CHAPTER XXVIII.
ANALOGY.
1. Analogy 627
2. Analogy as a Guide in Discovery 629
3. Analogy in the Mathematical Sciences 631
4. Analogy in the Theory of Undulations 635
5. Analogy in Astronomy 638
6. Failures of Analogy 641
CHAPTER XXIX.
EXCEPTIONAL PHENOMENA.
1. Exceptional Phenomena 644
2. Imaginary or False Exceptions 647
3. Apparent but Congruent Exceptions 649
4. Singular Exceptions 652
5. Divergent Exceptions 655
6. Accidental Exceptions 658
7. Novel and Unexplained Exceptions 661
8. Limiting Exceptions 663
9. Real Exceptions to Supposed Laws 666
10. Unclassed Exceptions 668
CHAPTER XXX.
CLASSIFICATION.
1. Classification 673
2. Classification involving Induction 675
3. Multiplicity of Modes of Classification 677
4. Natural and Artificial Systems of Classification 679
5. Correlation of Properties 681
6. Classification in Crystallography 685
7. Classification an Inverse and Tentative Operation 689
8. Symbolic Statement of the Theory of Classification 692
9. Bifurcate Classification 694
10. The Five Predicates 698
11. Summum Genus and Infima Species 701
12. The Tree of Porphyry 702
13. Does Abstraction imply Generalisation? 704
14. Discovery of Marks or Characteristics 708
15. Diagnostic Systems of Classification 710
16. Index Classifications 714
17. Classification in the Biological Sciences 718
18. Classification by Types 722
19. Natural Genera and Species 724
20. Unique or Exceptional Objects 728
21. Limits of Classification 730
BOOK VI.
CHAPTER XXXI.
REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.
1. Reflections on the Results and Limits of Scientific Method 735
2. The Meaning of Natural Law 737
3. Infiniteness of the Universe 738
4. The Indeterminate Problem of Creation 740
5. Hierarchy of Natural Laws 742
6. The Ambiguous Expression--“Uniformity of Nature” 745
7. Possible States of the Universe 749
8. Speculations on the Reconcentration of Energy 751
9. The Divergent Scope for New Discovery 752
10. Infinite Incompleteness of the Mathematical Sciences 754
11. The Reign of Law in Mental and Social Phenomena 759
12. The Theory of Evolution 761
13. Possibility of Divine Interference 765
14. Conclusion 766
INDEX 773
THE PRINCIPLES OF SCIENCE.
CHAPTER I.
INTRODUCTION.
Science arises from the discovery of Identity amidst Diversity. The
process may be described in different words, but our language must
always imply the presence of one common and necessary element. In
every act of inference or scientific method we are engaged about a
certain identity, sameness, similarity, likeness, resemblance, analogy,
equivalence or equality apparent between two objects. It is doubtful
whether an entirely isolated phenomenon could present itself to our
notice, since there must always be some points of similarity between
object and object. But in any case an isolated phenomenon could be
studied to no useful purpose. The whole value of science consists
in the power which it confers upon us of applying to one object the
knowledge acquired from like objects; and it is only so far, therefore,
as we can discover and register resemblances that we can turn our
observations to account.
Nature is a spectacle continually exhibited to our senses, in which
phenomena are mingled in combinations of endless variety and novelty.
Wonder fixes the mind’s attention; memory stores up a record of each
distinct impression; the powers of association bring forth the record
when the like is felt again. By the higher faculties of judgment and
reasoning the mind compares the new with the old, recognises essential
identity, even when disguised by diverse circumstances, and expects to
find again what was before experienced. It must be the ground of all
reasoning and inference that *what is true of one thing will be true
of its equivalent*, and that under carefully ascertained conditions
*Nature repeats herself*.
Were this indeed a Chaotic Universe, the powers of mind employed in
science would be useless to us. Did Chance wholly take the place of
order, and did all phenomena come out of an *Infinite Lottery*, to use
Condorcet’s expression, there could be no reason to expect the like
result in like circumstances. It is possible to conceive a world in
which no two things should be associated more often, in the long run,
than any other two things. The frequent conjunction of any two events
would then be purely fortuitous, and if we expected conjunctions to
recur continually, we should be disappointed. In such a world we might
recognise the same kind of phenomenon as it appeared from time to time,
just as we might recognise a marked ball as it was occasionally drawn
and re-drawn from a ballot-box; but the approach of any phenomenon
would be in no way indicated by what had gone before, nor would it be a
sign of what was to come after. In such a world knowledge would be no
more than the memory of past coincidences, and the reasoning powers, if
they existed at all, would give no clue to the nature of the present,
and no presage of the future.
Happily the Universe in which we dwell is not the result of chance,
and where chance seems to work it is our own deficient faculties which
prevent us from recognising the operation of Law and of Design. In
the material framework of this world, substances and forces present
themselves in definite and stable combinations. Things are not in
perpetual flux, as ancient philosophers held. Element remains element;
iron changes not into gold. With suitable precautions we can calculate
upon finding the same thing again endowed with the same properties.
The constituents of the globe, indeed, appear in almost endless
combinations; but each combination bears its fixed character, and
when resolved is found to be the compound of definite substances.
Misapprehensions must continually occur, owing to the limited extent
of our experience. We can never have examined and registered possible
existences so thoroughly as to be sure that no new ones will occur and
frustrate our calculations. The same outward appearances may cover
any amount of hidden differences which we have not yet suspected. To
the variety of substances and powers diffused through nature at its
creation, we should not suppose that our brief experience can assign
a limit, and the necessary imperfection of our knowledge must be ever
borne in mind.
Yet there is much to give us confidence in Science. The wider our
experience, the more minute our examination of the globe, the
greater the accumulation of well-reasoned knowledge,--the fewer in
all probability will be the failures of inference compared with the
successes. Exceptions to the prevalence of Law are gradually reduced
to Law themselves. Certain deep similarities have been detected among
the objects around us, and have never yet been found wanting. As the
means of examining distant parts of the universe have been acquired,
those similarities have been traced there as here. Other worlds and
stellar systems may be almost incomprehensively different from ours in
magnitude, condition and disposition of parts, and yet we detect there
the same elements of which our own limbs are composed. The same natural
laws can be detected in operation in every part of the universe within
the scope of our instruments; and doubtless these laws are obeyed
irrespective of distance, time, and circumstance.
It is the prerogative of Intellect to discover what is uniform and
unchanging in the phenomena around us. So far as object is different
from object, knowledge is useless and inference impossible. But so
far as object resembles object, we can pass from one to the other. In
proportion as resemblance is deeper and more general, the commanding
powers of knowledge become more wonderful. Identity in one or other
of its phases is thus always the bridge by which we pass in inference
from case to case; and it is my purpose in this treatise to trace out
the various forms in which the one same process of reasoning presents
itself in the ever-growing achievements of Scientific Method.
*The Powers of Mind concerned in the Creation of Science.*
It is no part of the purpose of this work to investigate the nature
of mind. People not uncommonly suppose that logic is a branch of
psychology, because reasoning is a mental operation. On the same
ground, however, we might argue that all the sciences are branches
of psychology. As will be further explained, I adopt the opinion of
Mr. Herbert Spencer, that logic is really an objective science, like
mathematics or mechanics. Only in an incidental manner, then, need
I point out that the mental powers employed in the acquisition of
knowledge are probably three in number. They are substantially as
Professor Bain has stated them[24]:--
[24] *The Senses and the Intellect*, Second Ed., pp. 5, 325, &c.
1. The Power of Discrimination.
2. The Power of Detecting Identity.
3. The Power of Retention.
We exert the first power in every act of perception. Hardly can we have
a sensation or feeling unless we discriminate it from something else
which preceded. Consciousness would almost seem to consist in the break
between one state of mind and the next, just as an induced current of
electricity arises from the beginning or the ending of the primary
current. We are always engaged in discrimination; and the rudiment of
thought which exists in the lower animals probably consists in their
power of feeling difference and being agitated by it.
Yet had we the power of discrimination only, Science could not be
created. To know that one feeling differs from another gives purely
negative information. It cannot teach us what will happen. In such a
state of intellect each sensation would stand out distinct from every
other; there would be no tie, no bridge of affinity between them. We
want a unifying power by which the present and the future may be linked
to the past; and this seems to be accomplished by a different power of
mind. Lord Bacon has pointed out that different men possess in very
different degrees the powers of discrimination and identification. It
may be said indeed that discrimination necessarily implies the action
of the opposite process of identification; and so it doubtless does in
negative points. But there is a rare property of mind which consists
in penetrating the disguise of variety and seizing the common elements
of sameness; and it is this property which furnishes the true measure
of intellect. The name of “intellect” expresses the interlacing of the
general and the single, which is the peculiar province of mind.[25] To
*cogitate* is the Latin *coagitare*, resting on a like metaphor. Logic,
also, is but another name for the same process, the peculiar work of
reason; for λογος is derived from λεγειν, which like the Latin *legere*
meant originally to gather. Plato said of this unifying power, that if
he met the man who could detect *the one in the many*, he would follow
him as a god.
[25] Max Müller, *Lectures on the Science of Language*, Second
Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view
of the etymological meaning of “intellect” is given above on the
authority of Professor Max Müller. It seems to be opposed to the
ordinary opinion, according to which the Latin *intelligere* means to
choose between, to see a difference between, to discriminate, instead
of to unite.
*Laws of Identity and Difference.*
At the base of all thought and science must lie the laws which express
the very nature and conditions of the discriminating and identifying
powers of mind. These are the so-called Fundamental Laws of Thought,
usually stated as follows:--
1. The Law of Identity. *Whatever is, is.*
2. The Law of Contradiction. *A thing cannot both be and not be.*
3. The Law of Duality. *A thing must either be or not be.*
The first of these statements may perhaps be regarded as a description
of identity itself, if so fundamental a notion can admit of
description. A thing at any moment is perfectly identical with itself,
and, if any person were unaware of the meaning of the word “identity,”
we could not better describe it than by such an example.
The second law points out that contradictory attributes can never be
joined together. The same object may vary in its different parts;
here it may be black, and there white; at one time it may be hard and
at another time soft; but at the same time and place an attribute
cannot be both present and absent. Aristotle truly described this law
as the first of all axioms--one of which we need not seek for any
demonstration. All truths cannot be proved, otherwise there would be an
endless chain of demonstration; and it is in self-evident truths like
this that we find the simplest foundations.
The third of these laws completes the other two. It asserts that at
every step there are two possible alternatives--presence or absence,
affirmation or negation. Hence I propose to name this law the Law of
Duality, for it gives to all the formulæ of reasoning a dual character.
It asserts also that between presence and absence, existence and
non-existence, affirmation and negation, there is no third alternative.
As Aristotle said, there can be no mean between opposite assertions: we
must either affirm or deny. Hence the inconvenient name by which it has
been known--The Law of Excluded Middle.
It may be allowed that these laws are not three independent and
distinct laws; they rather express three different aspects of the
same truth, and each law doubtless presupposes and implies the other
two. But it has not hitherto been found possible to state these
characters of identity and difference in less than the threefold
formula. The reader may perhaps desire some information as to the
mode in which these laws have been stated, or the way in which they
have been regarded, by philosophers in different ages of the world.
Abundant information on this and many other points of logical history
will be found in Ueberweg’s *System of Logic*, of which an excellent
translation has been published by Professor T. M. Lindsay (see
pp. 228–281).
*The Nature of the Laws of Identity and Difference.*
I must at least allude to the profoundly difficult question concerning
the nature and authority of these Laws of Identity and Difference.
Are they Laws of Thought or Laws of Things? Do they belong to mind or
to material nature? On the one hand it may be said that science is a
purely mental existence, and must therefore conform to the laws of
that which formed it. Science is in the mind and not in the things,
and the properties of mind are therefore all important. It is true
that these laws are verified in the observation of the exterior world;
and it would seem that they might have been gathered and proved by
generalisation, had they not already been in our possession. But
on the other hand, it may well be urged that we cannot prove these
laws by any process of reasoning or observation, because the laws
themselves are presupposed, as Leibnitz acutely remarked, in the very
notion of a proof. They are the prior conditions of all thought and
all knowledge, and even to question their truth is to allow them true.
Hartley ingeniously refined upon this argument, remarking that if the
fundamental laws of logic be not certain, there must exist a logic of
a second order whereby we may determine the degree of uncertainty: if
the second logic be not certain, there must be a third; and so on *ad
infinitum*. Thus we must suppose either that absolutely certain laws of
thought exist, or that there is no such thing as certainty whatever.[26]
[26] Hartley on Man, vol. i. p. 359.
Logicians, indeed, appear to me to have paid insufficient attention to
the fact that mistakes in reasoning are always possible, and of not
unfrequent occurrence. The Laws of Thought are often called necessary
laws, that is, laws which cannot but be obeyed. Yet as a matter of
fact, who is there that does not often fail to obey them? They are
the laws which the mind ought to obey rather than what it always does
obey. Our thoughts cannot be the criterion of truth, for we often
have to acknowledge mistakes in arguments of moderate complexity,
and we sometimes only discover our mistakes by collision between our
expectations and the events of objective nature.
Mr. Herbert Spencer holds that the laws of logic are objective
laws,[27] and he regards the mind as being in a state of constant
education, each act of false reasoning or miscalculation leading to
results which are likely to prevent similar mistakes from being again
committed. I am quite inclined to accept such ingenious views; but at
the same time it is necessary to distinguish between the accumulation
of knowledge, and the constitution of the mind which allows of the
acquisition of knowledge. Before the mind can perceive or reason at
all it must have the conditions of thought impressed upon it. Before
a mistake can be committed, the mind must clearly distinguish the
mistaken conclusion from all other assertions. Are not the Laws of
Identity and Difference the prior conditions of all consciousness and
all existence? Must they not hold true, alike of things material and
immaterial? and if so, can we say that they are only subjectively true
or objectively true? I am inclined, in short, to regard them as true
both “in the nature of thought and things,” as I expressed it in my
first logical essay;[28] and I hold that they belong to the common
basis of all existence. But this is one of the most difficult questions
of psychology and metaphysics which can be raised, and it is hardly
one for the logician to decide. As the mathematician does not inquire
into the nature of unity and plurality, but develops the formal laws of
plurality, so the logician, as I conceive, must assume the truth of the
Laws of Identity and Difference, and occupy himself in developing the
variety of forms of reasoning in which their truth may be manifested.
[27] *Principles of Psychology*, Second Ed., vol. ii. p. 86.
[28] *Pure Logic, or the Logic of Quality apart from Quantity*, 1864,
pp. 10, 16, 22, 29, 36, &c.
Again, I need hardly dwell upon the question whether logic treats of
language, notions, or things. As reasonably might we debate whether a
mathematician treats of symbols, quantities, or things. A mathematician
certainly does treat of symbols, but only as the instruments whereby
to facilitate his reasoning concerning quantities; and as the axioms
and rules of mathematical science must be verified in concrete objects
in order that the calculations founded upon them may have any validity
or utility, it follows that the ultimate objects of mathematical
science are the things themselves. In like manner I conceive that
the logician treats of language so far as it is essential for the
embodiment and exhibition of thought. Even if reasoning can take place
in the inner consciousness of man without the use of any signs, which
is doubtful, at any rate it cannot become the subject of discussion
until by some system of material signs it is manifested to other
persons. The logician then uses words and symbols as instruments of
reasoning, and leaves the nature and peculiarities of language to the
grammarian. But signs again must correspond to the thoughts and things
expressed, in order that they shall serve their intended purpose. We
may therefore say that logic treats ultimately of thoughts and things,
and immediately of the signs which stand for them. Signs, thoughts, and
exterior objects may be regarded as parallel and analogous series of
phenomena, and to treat any one of the three series is equivalent to
treating either of the other series.
*The Process of Inference.*
The fundamental action of our reasoning faculties consists in inferring
or carrying to a new instance of a phenomenon whatever we have
previously known of its like, analogue, equivalent or equal. Sameness
or identity presents itself in all degrees, and is known under various
names; but the great rule of inference embraces all degrees, and
affirms that *so far as there exists sameness, identity or likeness,
what is true of one thing will be true of the other*. The great
difficulty doubtless consists in ascertaining that there does exist
a sufficient degree of likeness or sameness to warrant an intended
inference; and it will be our main task to investigate the conditions
under which reasoning is valid. In this place I wish to point out that
there is something common to all acts of inference, however different
their apparent forms. The one same rule lends itself to the most
diverse applications.
The simplest possible case of inference, perhaps, occurs in the use of
a *pattern*, *example*, or, as it is commonly called, a *sample*. To
prove the exact similarity of two portions of commodity, we need not
bring one portion beside the other. It is sufficient that we take a
sample which exactly represents the texture, appearance, and general
nature of one portion, and according as this sample agrees or not with
the other, so will the two portions of commodity agree or differ.
Whatever is true as regards the colour, texture, density, material of
the sample will be true of the goods themselves. In such cases likeness
of quality is the condition of inference.
Exactly the same mode of reasoning holds true of magnitude and figure.
To compare the sizes of two objects, we need not lay them beside each
other. A staff, string, or other kind of measure may be employed to
represent the length of one object, and according as it agrees or not
with the other, so must the two objects agree or differ. In this case
the proxy or sample represents length; but the fact that lengths can
be added and multiplied renders it unnecessary that the proxy should
always be as large as the object. Any standard of convenient size, such
as a common foot-rule, may be made the medium of comparison. The height
of a church in one town may be carried to that in another, and objects
existing immovably at opposite sides of the earth may be vicariously
measured against each other. We obviously employ the axiom that
whatever is true of a thing as regards its length, is true of its equal.
To every other simple phenomenon in nature the same principle of
substitution is applicable. We may compare weights, densities, degrees
of hardness, and degrees of all other qualities, in like manner. To
ascertain whether two sounds are in unison we need not compare them
directly, but a third sound may be the go-between. If a tuning-fork is
in unison with the middle C of York Minster organ, and we afterwards
find it to be in unison with the same note of the organ in Westminster
Abbey, then it follows that the two organs are tuned in unison. The
rule of inference now is, that what is true of the tuning-fork as
regards the tone or pitch of its sound, is true of any sound in unison
with it.
The skilful employment of this substitutive process enables us to
make measurements beyond the powers of our senses. No one can count
the vibrations, for instance, of an organ-pipe. But we can construct
an instrument called the *siren*, so that, while producing a sound of
any pitch, it shall register the number of vibrations constituting the
sound. Adjusting the sound of the siren in unison with an organ-pipe,
we measure indirectly the number of vibrations belonging to a sound
of that pitch. To measure a sound of the same pitch is as good as to
measure the sound itself.
Sir David Brewster, in a somewhat similar manner, succeeded in
measuring the refractive indices of irregular fragments of transparent
minerals. It was a troublesome, and sometimes impracticable work to
grind the minerals into prisms, so that the power of refracting light
could be directly observed; but he fell upon the ingenious device
of compounding a liquid possessing the same refractive power as the
transparent fragment under examination. The moment when this equality
was attained could be known by the fragments ceasing to reflect or
refract light when immersed in the liquid, so that they became almost
invisible in it. The refractive power of the liquid being then measured
gave that of the solid. A more beautiful instance of representative
measurement, depending immediately upon the principle of inference,
could not be found.[29]
[29] Brewster, *Treatise on New Philosophical Instruments*, p. 273.
Concerning this method see also Whewell, *Philosophy of the Inductive
Sciences*, vol. ii. p. 355; Tomlinson, *Philosophical Magazine*,
Fourth Series, vol. xl. p. 328; Tyndall, in Youmans’ *Modern
Culture*, p. 16.
Throughout the various logical processes which we are about
to consider--Deduction, Induction, Generalisation, Analogy,
Classification, Quantitative Reasoning--we shall find the one same
principle operating in a more or less disguised form.
*Deduction and Induction.*
The processes of inference always depend on the one same principle of
substitution; but they may nevertheless be distinguished according as
the results are inductive or deductive. As generally stated, deduction
consists in passing from more general to less general truths; induction
is the contrary process from less to more general truths. We may
however describe the difference in another manner. In deduction we are
engaged in developing the consequences of a law. We learn the meaning,
contents, results or inferences, which attach to any given proposition.
Induction is the exactly inverse process. Given certain results or
consequences, we are required to discover the general law from which
they flow.
In a certain sense all knowledge is inductive. We can only learn the
laws and relations of things in nature by observing those things. But
the knowledge gained from the senses is knowledge only of particular
facts, and we require some process of reasoning by which we may
collect out of the facts the laws obeyed by them. Experience gives
us the materials of knowledge: induction digests those materials, and
yields us general knowledge. When we possess such knowledge, in the
form of general propositions and natural laws, we can usefully apply
the reverse process of deduction to ascertain the exact information
required at any moment. In its ultimate foundation, then, all knowledge
is inductive--in the sense that it is derived by a certain inductive
reasoning from the facts of experience.
It is nevertheless true,--and this is a point to which insufficient
attention has been paid, that all reasoning is founded on the
principles of deduction. I call in question the existence of any method
of reasoning which can be carried on without a knowledge of deductive
processes. I shall endeavour to show that *induction is really the
inverse process of deduction*. There is no mode of ascertaining the
laws which are obeyed in certain phenomena, unless we have the power
of determining what results would follow from a given law. Just as the
process of division necessitates a prior knowledge of multiplication,
or the integral calculus rests upon the observation and remembrance
of the results of the differential calculus, so induction requires a
prior knowledge of deduction. An inverse process is the undoing of
the direct process. A person who enters a maze must either trust to
chance to lead him out again, or he must carefully notice the road by
which he entered. The facts furnished to us by experience are a maze of
particular results; we might by chance observe in them the fulfilment
of a law, but this is scarcely possible, unless we thoroughly learn the
effects which would attach to any particular law.
Accordingly, the importance of deductive reasoning is doubly supreme.
Even when we gain the results of induction they would be of no use
unless we could deductively apply them. But before we can gain them
at all we must understand deduction, since it is the inversion of
deduction which constitutes induction. Our first task in this work,
then, must be to trace out fully the nature of identity in all its
forms of occurrence. Having given any series of propositions we must be
prepared to develop deductively the whole meaning embodied in them, and
the whole of the consequences which flow from them.
*Symbolic Expression of Logical Inference.*
In developing the results of the Principle of Inference we require to
use an appropriate language of signs. It would indeed be quite possible
to explain the processes of reasoning by the use of words found in the
dictionary. Special examples of reasoning, too, may seem to be more
readily apprehended than general symbolic forms. But it has been shown
in the mathematical sciences that the attainment of truth depends
greatly upon the invention of a clear, brief, and appropriate system
of symbols. Not only is such a language convenient, but it is almost
essential to the expression of those general truths which are the very
soul of science. To apprehend the truth of special cases of inference
does not constitute logic; we must apprehend them as cases of more
general truths. The object of all science is the separation of what is
common and general from what is accidental and different. In a system
of logic, if anywhere, we should esteem this generality, and strive to
exhibit clearly what is similar in very diverse cases. Hence the great
value of *general symbols* by which we can represent the form of a
reasoning process, disentangled from any consideration of the special
subject to which it is applied.
The signs required in logic are of a very simple kind. As sameness or
difference must exist between two things or notions, we need signs to
indicate the things or notions compared, and other signs to denote the
relations between them. We need, then, (1) symbols for terms, (2) a
symbol for sameness, (3) a symbol for difference, and (4) one or two
symbols to take the place of conjunctions.
Ordinary nouns substantive, such as *Iron*, *Metal*, *Electricity*,
*Undulation*, might serve as terms, but, for the reasons explained
above, it is better to adopt blank letters, devoid of special
signification, such as A, B, C, &c. Each letter must be understood to
represent a noun, and, so far as the conditions of the argument allow,
*any noun*. Just as in Algebra, *x*, *y*, *z*, *p*, *q*, &c. are used
for *any quantities*, undetermined or unknown, except when the special
conditions of the problem are taken into account, so will our letters
stand for undetermined or unknown things.
These letter-terms will be used indifferently for nouns substantive
and adjective. Between these two kinds of nouns there may perhaps
be differences in a metaphysical or grammatical point of view.
But grammatical usage sanctions the conversion of adjectives into
substantives, and *vice versâ*; we may avail ourselves of this latitude
without in any way prejudging the metaphysical difficulties which may
be involved. Here, as throughout this work, I shall devote my attention
to truths which I can exhibit in a clear and formal manner, believing
that in the present condition of logical science, this course will lead
to greater advantage than discussion upon the metaphysical questions
which may underlie any part of the subject.
Every noun or term denotes an object, and usually implies the
possession by that object of certain qualities or circumstances common
to all the objects denoted. There are certain terms, however, which
imply the absence of qualities or circumstances attaching to other
objects. It will be convenient to employ a special mode of indicating
these *negative terms*, as they are called. If the general name A
denotes an object or class of objects possessing certain defined
qualities, then the term Not A will denote any object which does not
possess the whole of those qualities; in short, Not A is the sign for
anything which differs from A in regard to any one or more of the
assigned qualities. If A denote “transparent object,” Not A will denote
“not transparent object.” Brevity and facility of expression are of
no slight importance in a system of notation, and it will therefore
be desirable to substitute for the negative term Not A a briefer
symbol. De Morgan represented negative terms by small Roman letters,
or sometimes by small italic letters;[30] as the latter seem to be
highly convenient, I shall use *a*, *b*, *c*, ... *p*, *q*, &c., as the
negative terms corresponding to A, B, C, ... P, Q, &c. Thus if A means
“fluid,” *a* will mean “not fluid.”
[30] *Formal Logic*, p. 38.
*Expression of Identity and Difference.*
To denote the relation of sameness or identity I unhesitatingly adopt
the sign =, so long used by mathematicians to denote equality. This
symbol was originally appropriated by Robert Recorde in his *Whetstone
of Wit*, to avoid the tedious repetition of the words “is equal to;”
and he chose a pair of parallel lines, because no two things can be
more equal.[31] The meaning of the sign has however been gradually
extended beyond that of equality of quantities; mathematicians have
themselves used it to indicate equivalence of operations. The force
of analogy has been so great that writers in most other branches
of science have employed the same sign. The philologist uses it to
indicate the equivalence of meaning of words: chemists adopt it to
signify identity in kind and equality in weight of the elements which
form two different compounds. Not a few logicians, for instance
Lambert, Drobitsch, George Bentham,[32] Boole,[33] have employed it
as the copula of propositions. De Morgan declined to use it for this
purpose, but still further extended its meaning so as to include the
equivalence of a proposition with the premises from which it can be
inferred;[34] and Herbert Spencer has applied it in a like manner.[35]
[31] Hallam’s *Literature of Europe*, First Ed., vol. ii. p. 444.
[32] *Outline of a New System of Logic*, London, 1827, pp. 133, &c.
[33] *An Investigation of the Laws of Thought*, pp. 27, &c.
[34] *Formal Logic*, pp. 82, 106. In his later work, *The Syllabus of
a New System of Logic*, he discontinued the use of the sign.
[35] *Principles of Psychology*, Second Ed., vol. ii. pp. 54, 55.
Many persons may think that the choice of a symbol is a matter of
slight importance or of mere convenience; but I hold that the common
use of this sign = in so many different meanings is really founded
upon a generalisation of the widest character and of the greatest
importance--one indeed which it is a principal purpose of this work to
explain. The employment of the same sign in different cases would be
unphilosophical unless there were some real analogy between its diverse
meanings. If such analogy exists, it is not only allowable, but highly
desirable and even imperative, to use the symbol of equivalence with a
generality of meaning corresponding to the generality of the principles
involved. Accordingly De Morgan’s refusal to use the symbol in logical
propositions indicated his opinion that there was a want of analogy
between logical propositions and mathematical equations. I use the sign
because I hold the contrary opinion.
I conceive that the sign = as commonly employed, always denotes
some form or degree of sameness, and the particular form is usually
indicated by the nature of the terms joined by it. Thus “6,720 pounds =
3 tons” is evidently an equation of quantities. The formula - × - = +
expresses the equivalence of operations. “Exogens = Dicotyledons” is a
logical identity expressing a profound truth concerning the character
and origin of a most important group of plants.
We have great need in logic of a distinct sign for the copula, because
the little verb *is* (or *are*), hitherto used both in logic and
ordinary discourse, is thoroughly ambiguous. It sometimes denotes
identity, as in “St. Paul’s is the *chef-d’œuvre* of Sir Christopher
Wren;” but it more commonly indicates inclusion of class within class,
or partial identity, as in “Bishops are members of the House of
Lords.” This latter relation involves identity, but requires careful
discrimination from simple identity, as will be shown further on.
When with this sign of equality we join two nouns or logical terms, as
in
Hydrogen = The least dense element,
we signify that the object or group of objects denoted by one term is
identical with that denoted by the other, in everything except the
names. The general formula
A = B
must be taken to mean that A and B are symbols for the same object
or group of objects. This identity may sometimes arise from the mere
imposition of names, but it may also arise from the deepest laws of the
constitution of nature; as when we say
Gravitating matter = Matter possessing inertia,
Exogenous plants = Dicotyledonous plants,
Plagihedral quartz crystals = Quartz crystals causing
the plane of polarisation of light to rotate.
We shall need carefully to distinguish between relations of terms which
can be modified at our own will and those which are fixed as expressing
the laws of nature; but at present we are considering only the mode of
expression which may be the same in either case.
Sometimes, but much less frequently, we require a symbol to indicate
difference or the absence of complete sameness. For this purpose we
may generalise in like manner the symbol ~, which was introduced by
Wallis to signify difference between quantities. The general formula
B ~ C
denotes that B and C are the names of two objects or groups which are
not identical with each other. Thus we may say
Acrogens ~ Flowering plants.
Snowdon ~ The highest mountain in Great Britain.
I shall also occasionally use the sign ᔕ to signify in the most general
manner the existence of any relation between the two terms connected by
it. Thus ᔕ might mean not only the relations of equality or inequality,
sameness or difference, but any special relation of time, place, size,
causation, &c. in which one thing may stand to another. By A ᔕ B I
mean, then, any two objects of thought related to each other in any
conceivable manner.
*General Formula of Logical Inference.*
The one supreme rule of inference consists, as I have said, in the
direction to affirm of anything whatever is known of its like, equal
or equivalent. The *Substitution of Similars* is a phrase which seems
aptly to express the capacity of mutual replacement existing in any
two objects which are like or equivalent to a sufficient degree.
It is matter for further investigation to ascertain when and for
what purposes a degree of similarity less than complete identity is
sufficient to warrant substitution. For the present we think only of
the exact sameness expressed in the form
A = B.
Now if we take the letter C to denote any third conceivable object, and
use the sign ᔕ in its stated meaning of *indefinite relation*, then the
general formula of all inference may be thus exhibited:--
From A = B ᔕ C
we may infer A ᔕ C
or, in words--*In whatever relation a thing stands to a second thing,
in the same relation it stands to the like or equivalent of that second
thing.* The identity between A and B allows us indifferently to place
A where B was, or B where A was; and there is no limit to the variety
of special meanings which we can bestow upon the signs used in this
formula consistently with its truth. Thus if we first specify only the
meaning of the sign ᔕ, we may say that if *C is the weight of B*, then
*C is also the weight of A*. Similarly
If C is the father of B, C is the father of A;
If C is a fragment of B, C is a fragment of A;
If C is a quality of B, C is a quality of A;
If C is a species of B, C is a species of A;
If C is the equal of B, C is the equal of A;
and so on *ad infinitum*.
We may also endow with special meanings the letter-terms A, B, and C,
and the process of inference will never be false. Thus let the sign
ᔕ mean “is height of,” and let
A = Snowdon,
B = Highest mountain in England or Wales,
C = 3,590 feet;
then it obviously follows since “3,590 feet is the height of Snowdon,”
and “Snowdon = the highest mountain in England or Wales,” that, “3,590
feet is the height of the highest mountain in England or Wales.”
One result of this general process of inference is that we may in any
aggregate or complex whole replace any part by its equivalent without
altering the whole. To alter is to make a difference; but if in
replacing a part I make no difference, there is no alteration of the
whole. Many inferences which have been very imperfectly included in
logical formulas at once follow. I remember the late Prof. De Morgan
remarking that all Aristotle’s logic could not prove that “Because a
horse is an animal, the head of a horse is the head of an animal.” I
conceive that this amounts merely to replacing in the complete notion
*head of a horse*, the term “horse,” by its equivalent *some animal* or
*an animal*. Similarly, since
The Lord Chancellor = The Speaker of the House of Lords,
it follows that
The death of the Lord Chancellor = The death of the Speaker of the
House of Lords;
and any event, circumstance or thing, which stands in a certain
relation to the one will stand in like relation to the other. Milton
reasons in this way when he says, in his Areopagitica, “Who kills a
man, kills a reasonable creature, God’s image.” If we may suppose him
to mean
God’s image = man = some reasonable creature,
it follows that “The killer of a man is the killer of some reasonable
creature,” and also “The killer of God’s image.”
This replacement of equivalents may be repeated over and over again to
any extent. Thus if *person* is identical in meaning with *individual*,
it follows that
Meeting of persons = meeting of individuals;
and if *assemblage* = *meeting*, we may make a new replacement and show
that
Meeting of persons = assemblage of individuals.
We may in fact found upon this principle of substitution a most general
axiom in the following terms[36]:--
[36] *Pure Logic, or the Logic of Quality*, p. 14.
*Same parts samely related make same wholes.*
If, for instance, exactly similar bricks and other materials be used
to build two houses, and they be similarly placed in each house, the
two houses must be similar. There are millions of cells in a human
body, but if each cell of one person were represented by an exactly
similar cell similarly placed in another body, the two persons would
be undistinguishable, and would be only *numerically* different. It
is upon this principle, as we shall see, that all accurate processes
of measurement depend. If for a weight in a scale of a balance we
substitute another weight, and the equilibrium remains entirely
unchanged, then the weights must be exactly equal. The general test of
equality is substitution. Objects are equally bright when on replacing
one by the other the eye perceives no difference. Objects are equal in
dimensions when tested by the same gauge they fit in the same manner.
Generally speaking, two objects are alike so far as when substituted
one for another no alteration is produced, and *vice versâ* when alike
no alteration is produced by the substitution.
*The Propagating Power of Similarity.*
The relation of similarity in all its degrees is reciprocal. So far
as things are alike, either may be substituted for the other; and
this may perhaps be considered the very meaning of the relation. But
it is well worth notice that there is in similarity a peculiar power
of extending itself among all the things which are similar. To render
a number of things similar to each other we need only render them
similar to one standard object. Each coin struck from a pair of dies
not only resembles the matrix or original pattern from which the dies
were struck, but resembles every other coin manufactured from the same
original pattern. Among a million such coins there are not less than
499,999,500,000 *pairs of coins* resembling each other. Similars to
the same are similars to all. It is one great advantage of printing
that all copies of a document struck from the same type are necessarily
identical each with each, and whatever is true of one copy will be true
of every copy. Similarly, if fifty rows of pipes in an organ be tuned
in perfect unison with one row, usually the Principal, they must be
in unison with each other. Similarity can also reproduce or propagate
itself *ad infinitum*: for if a number of tuning-forks be adjusted in
perfect unison with one standard fork, all instruments tuned to any one
fork will agree with any instrument tuned to any other fork. Standard
measures of length, capacity, weight, or any other measurable quality,
are propagated in the same manner. So far as copies of the original
standard, or copies of copies, or copies again of those copies, are
accurately executed, they must all agree each with every other.
It is the capability of mutual substitution which gives such great
value to the modern methods of mechanical construction, according
to which all the parts of a machine are exact facsimiles of a fixed
pattern. The rifles used in the British army are constructed on the
American interchangeable system, so that any part of any rifle can be
substituted for the same part of another. A bullet fitting one rifle
will fit all others of the same bore. Sir J. Whitworth has extended
the same system to the screws and screw-bolts used in connecting
together the parts of machines, by establishing a series of standard
screws.
*Anticipations of the Principle of Substitution.*
In such a subject as logic it is hardly possible to put forth any
opinions which have not been in some degree previously entertained. The
germ at least of every doctrine will be found in earlier writers, and
novelty must arise chiefly in the mode of harmonising and developing
ideas. When I first employed the process and name of *substitution*
in logic,[37] I was led to do so from analogy with the familiar
mathematical process of substituting for a symbol its value as given in
an equation. In writing my first logical essay I had a most imperfect
conception of the importance and generality of the process, and I
described, as if they were of equal importance, a number of other laws
which now seem to be but particular cases of the one general rule of
substitution.
[37] *Pure Logic*, pp. 18, 19.
My second essay, “The Substitution of Similars,” was written shortly
after I had become aware of the great simplification which may be
effected by a proper application of the principle of substitution. I
was not then acquainted with the fact that the German logician Beneke
had employed the principle of substitution, and had used the word
itself in forming a theory of the syllogism. My imperfect acquaintance
with the German language had prevented me from acquiring a complete
knowledge of Beneke’s views; but there is no doubt that Professor
Lindsay is right in saying that he, and probably other logicians,
were in some degree familiar with the principle.[38] Even Aristotle’s
dictum may be regarded as an imperfect statement of the principle of
substitution; and, as I have pointed out, we have only to modify that
dictum in accordance with the quantification of the predicate in order
to arrive at the complete process of substitution.[39] The Port-Royal
logicians appear to have entertained nearly equivalent views, for
they considered that all moods of the syllogism might be reduced
under one general principle.[40] Of two premises they regard one as
the *containing proposition* (propositio continens), and the other as
the *applicative proposition*. The latter proposition must always be
affirmative, and represents that by which a substitution is made; the
former may or may not be negative, and is that in which a substitution
is effected. They also show that this method will embrace certain cases
of complex reasoning which had no place in the Aristotelian syllogism.
Their views probably constitute the greatest improvement in logical
doctrine made up to that time since the days of Aristotle. But a true
reform in logic must consist, not in explaining the syllogism in one
way or another, but in doing away with all the narrow restrictions of
the Aristotelian system, and in showing that there exists an infinite
variety of logical arguments immediately deducible from the principle
of substitution of which the ancient syllogism forms but a small and
not even the most important part.
[38] Ueberweg’s *System of Logic*, transl. by Lindsay, pp. 442–446,
571, 572. The anticipations of the principle of substitution to be
found in the works of Leibnitz, Reusch, and perhaps other German
logicians, will be noticed in the preface to this second edition.
[39] *Substitution of Similars* (1869), p. 9.
[40] *Port-Royal Logic*, transl. by Spencer Baynes, pp. 212–219. Part
III. chap. x. and xi.
*The Logic of Relatives.*
There is a difficult and important branch of logic which may be
called the Logic of Relatives. If I argue, for instance, that because
Daniel Bernoulli was the son of John, and John the brother of James,
therefore Daniel was the nephew of James, it is not possible to prove
this conclusion by any simple logical process. We require at any rate
to assume that the son of a brother is a nephew. A simple logical
relation is that which exists between properties and circumstances of
the same object or class. But objects and classes of objects may also
be related according to all the properties of time and space. I believe
it may be shown, indeed, that where an inference concerning such
relations is drawn, a process of substitution is really employed and an
identity must exist; but I will not undertake to prove the assertion
in this work. The relations of time and space are logical relations
of a complicated character demanding much abstract and difficult
investigation. The subject has been treated with such great ability by
Peirce,[41] De Morgan,[42] Ellis,[43] and Harley, that I will not in
the present work attempt any review of their writings, but merely refer
the reader to the publications in which they are to be found.
[41] *Description of a Notation for the Logic of Relatives, resulting
from an Amplification of the Conceptions of Boole’s Calculus of
Logic.* By C. S. Peirce. *Memoirs of the American Academy*, vol. ix.
Cambridge, U.S., 1870.
[42] *On the Syllogism No IV., and on the Logic of Relations.* By
Augustus De Morgan. *Transactions of the Cambridge Philosophical
Society*, vol. x. part ii., 1860.
[43] *Observations on Boole’s Laws of Thought.* By the late R. Leslie
Ellis; communicated by the Rev. Robert Harley, F.R.S. *Report of the
British Association*, 1870. *Report of Sections*, p. 12. Also, *On
Boole’s Laws of Thought*. By the Rev. Robert Harley, F.R.S., *ibid.*
p. 14.
CHAPTER II.
TERMS.
Every proposition expresses the resemblance or difference of the things
denoted by its terms. As inference treats of the relation between two
or more propositions, so a proposition expresses a relation between two
or more terms. In the portion of this work which treats of deduction
it will be convenient to follow the usual order of exposition. We will
consider in succession the various kinds of terms, propositions, and
arguments, and we commence in this chapter with terms.
The simplest and most palpable meaning which can belong to a term
consists of some single material object, such as Westminster Abbey,
Stonehenge, the Sun, Sirius, &c. It is probable that in early stages of
intellect only concrete and palpable things are the objects of thought.
The youngest child knows the difference between a hot and a cold body.
The dog can recognise his master among a hundred other persons, and
animals of much lower intelligence know and discriminate their haunts.
In all such acts there is judgment concerning the likeness of physical
objects, but there is little or no power of analysing each object and
regarding it as a group of qualities.
The dignity of intellect begins with the power of separating points of
agreement from those of difference. Comparison of two objects may lead
us to perceive that they are at once like and unlike. Two fragments of
rock may differ entirely in outward form, yet they may have the same
colour, hardness, and texture. Flowers which agree in colour may differ
in odour. The mind learns to regard each object as an aggregate of
qualities, and acquires the power of dwelling at will upon one or other
of those qualities to the exclusion of the rest. Logical abstraction,
in short, comes into play, and the mind becomes capable of reasoning,
not merely about objects which are physically complete and concrete,
but about things which may be thought of separately in the mind though
they exist not separately in nature. We can think of the hardness of
a rock, or the colour of a flower, and thus produce abstract notions,
denoted by abstract terms, which will form a subject for further
consideration.
At the same time arise general notions and classes of objects. We
cannot fail to observe that the quality *hardness* exists in many
objects, for instance in many fragments of rock; mentally joining these
together, we create the class *hard object*, which will include, not
only the actual objects examined, but all others which may happen to
agree with them, as they agree with each other. As our senses cannot
possibly report to us all the contents of space, we cannot usually
set any limits to the number of objects which may fall into any such
class. At this point we begin to perceive the power and generality of
thought, which enables us in a single act to treat of indefinitely or
even infinitely numerous objects. We can safely assert that whatever is
true of any one object coming under a class is true of any of the other
objects so far as they possess the common qualities implied in their
belonging to the class. We must not place a thing in a class unless
we are prepared to believe of it all that is believed of the class in
general; but it remains a matter of important consideration to decide
how far and in what manner we can safely undertake thus to assign
the place of objects in that general system of classification which
constitutes the body of science.
*Twofold Meaning of General Names.*
Etymologically the *meaning* of a name is that which we are caused
to think of when the name is used. Now every general name causes us
to think of some one or more of the objects belonging to a class; it
may also cause us to think of the common qualities possessed by those
objects. A name is said to *denote* the object of thought to which it
may be applied; it *implies* at the same time the possession of certain
qualities or circumstances. The objects denoted form the *extent*
of meaning of the term; the qualities implied form the *intent* of
meaning. Crystal is the name of any substance of which the molecules
are arranged in a regular geometrical manner. The substances or objects
in question form the extent of meaning; the circumstance of having the
molecules so arranged forms the intent of meaning.
When we compare general terms together, it may often be found that
the meaning of one is included in the meaning of another. Thus
all *crystals* are included among *material substances*, and all
*opaque crystals* are included among *crystals*; here the inclusion
is in extension. We may also have inclusion of meaning in regard to
intension. For, as all crystals are material substances, the qualities
implied by the term material substance must be among those implied by
crystal. Again, it is obvious that while in extension of meaning opaque
crystals are but a part of crystals, in intension of meaning crystal
is but part of opaque crystal. We increase the intent of meaning of a
term by joining to it adjectives, or phrases equivalent to adjectives,
and the removal of such adjectives of course decreases the intensive
meaning. Now, concerning such changes of meaning, the following
all-important law holds universally true:--*When the intent of meaning
of a term is increased the extent is decreased; and* vice versâ, *when
the extent is increased the intent is decreased*. In short, as one is
increased the other is decreased.
This law refers only to logical changes. The number of steam-engines
in the world may be undergoing a rapid increase without the intensive
meaning of the name being altered. The law will only be verified,
again, when there is a real change in the intensive meaning, and an
adjective may often be joined to a noun without making a change.
*Elementary metal* is identical with *metal*; *mortal man* with *man*;
it being a *property* of all metals to be elements, and of all men to
be mortals.
There is no limit to the amount of meaning which a term may have. A
term may denote one object, or many, or an infinite number; it may
imply a single quality, if such there be, or a group of any number
of qualities, and yet the law connecting the extension and intension
will infallibly apply. Taking the general name *planet*, we increase
its intension and decrease its extension by prefixing the adjective
*exterior*; and if we further add *nearest to the earth*, there remains
but one planet, *Mars*, to which the name can then be applied. Singular
terms, which denote a single individual only, come under the same law
of meaning as general names. They may be regarded as general names of
which the meaning in extension is reduced to a minimum. Logicians have
erroneously asserted, as it seems to me, that singular terms are devoid
of meaning in intension, the fact being that they exceed all other
terms in that kind of meaning, as I have elsewhere tried to show.[44]
[44] Jevons’ *Elementary Lessons in Logic*, pp. 41–43; *Pure Logic*,
p. 6. See also J. S. Mill, *System of Logic*, Book I. chap. ii.
section 5, and Shedden’s *Elements of Logic*, London, 1864, pp. 14,
&c. Professor Robertson objects (*Mind*, vol. i. p. 210) that I
confuse *singular* and *proper* names; if so, it is because I hold
that the same remarks apply to proper names, which do not seem to me
to differ logically from singular names.
*Abstract Terms.*
Comparison of objects, and analysis of the complex resemblances and
differences which they present, lead us to the conception of *abstract
qualities*. We learn to think of one object as not only different from
another, but as differing in some particular point, such as colour, or
weight, or size. We may then convert points of agreement or difference
into separate objects of thought which we call qualities and denote by
*abstract terms*. Thus the term *redness* means something in which a
number of objects agree as to colour, and in virtue of which they are
called red. Redness forms, in fact, the intensive meaning of the term
red.
Abstract terms are strongly distinguished from general terms by
possessing only one kind of meaning; for as they denote qualities
there is nothing which they cannot in addition imply. The adjective
“red” is the name of red objects, but it implies the possession by
them of the quality *redness*; but this latter term has one single
meaning--the quality alone. Thus it arises that abstract terms are
incapable of plurality. Red objects are numerically distinct each
from each, and there are multitudes of such objects; but redness is a
single quality which runs through all those objects, and is the same in
one as it is in another. It is true that we may speak of *rednesses*,
meaning different kinds or tints of redness, just as we may speak of
*colours*, meaning different kinds of colours. But in distinguishing
kinds, degrees, or other differences, we render the terms so far
concrete. In that they are merely red there is but a single nature in
red objects, and so far as things are merely coloured, colour is a
single indivisible quality. Redness, so far as it is redness merely,
is one and the same everywhere, and possesses absolute oneness. In
virtue of this unity we acquire the power of treating all instances of
such quality as we may treat any one. We possess, in short, general
knowledge.
*Substantial Terms.*
Logicians appear to have taken little notice of a class of terms which
partake in certain respects of the character of abstract terms and yet
are undoubtedly the names of concrete existing things. These terms are
the names of substances, such as gold, carbonate of lime, nitrogen, &c.
We cannot speak of two golds, twenty carbonates of lime, or a hundred
nitrogens. There is no such distinction between the parts of a uniform
substance as will allow of a discrimination of numerous individuals.
The qualities of colour, lustre, malleability, density, &c., by which
we recognise gold, extend through its substance irrespective of
particular size or shape. So far as a substance is gold, it is one and
the same everywhere; so that terms of this kind, which I propose to
call *substantial terms*, possess the peculiar unity of abstract terms.
Yet they are not abstract; for gold is of course a tangible visible
body, entirely concrete, and existing independently of other bodies.
It is only when, by actual mechanical division, we break up the uniform
whole which forms the meaning of a substantial term, that we introduce
number. *Piece of gold* is a term capable of plurality; for there may
be a great many pieces discriminated either by their various shapes and
sizes, or, in the absence of such marks, by simultaneously occupying
different parts of space. In substance they are one; as regards the
properties of space they are many.[45] We need not further pursue this
question, which involves the distinction between unity and plurality,
until we consider the principles of number in a subsequent chapter.
[45] Professor Robertson has criticised my introduction of
“Substantial Terms” (*Mind*, vol. i. p. 210), and objects, perhaps
correctly, that the distinction if valid is extra-logical. I am
inclined to think, however, that the doctrine of terms is, strictly
speaking, for the most part extra-logical.
*Collective Terms.*
We must clearly distinguish between the *collective* and the *general
meanings* of terms. The same name may be used to denote the whole body
of existing objects of a certain kind, or any one of those objects
taken separately. “Man” may mean the aggregate of existing men, which
we sometimes describe as *mankind*; it is also the general name
applying to any man. The vegetable kingdom is the name of the whole
aggregate of *plants*, but “plant” itself is a general name applying
to any one or other plant. Every material object may be conceived as
divisible into parts, and is therefore collective as regards those
parts. The animal body is made up of cells and fibres, a crystal
of molecules; wherever physical division, or as it has been called
*partition*, is possible, there we deal in reality with a collective
whole. Thus the greater number of general terms are at the same time
collective as regards each individual whole which they denote.
It need hardly be pointed out that we must not infer of a collective
whole what we know only of the parts, nor of the parts what we know
only of the whole. The relation of whole and part is not one of
identity, and does not allow of substitution. There may nevertheless be
qualities which are true alike of the whole and of its parts. A number
of organ-pipes tuned in unison produce an aggregate of sound which
is of exactly the same pitch as each separate sound. In the case of
substantial terms, certain qualities may be present equally in each
minutest part as in the whole. The chemical nature of the largest mass
of pure carbonate of lime is the same as the nature of the smallest
particle. In the case of abstract terms, again, we cannot draw a
distinction between whole and part; what is true of redness in any case
is always true of redness, so far as it is merely red.
*Synthesis of Terms.*
We continually combine simple terms together so as to form new terms
of more complex meaning. Thus, to increase the intension of meaning of
a term we write it with an adjective or a phrase of adjectival nature.
By joining “brittle” to “metal,” we obtain a combined term, “brittle
metal,” which denotes a certain portion of the metals, namely, such as
are selected on account of possessing the quality of *brittleness*.
As we have already seen, “brittle metal” possesses less extension and
greater intension than metal. Nouns, prepositional phrases, participial
phrases and subordinate propositions may also be added to terms so as
to increase their intension and decrease their extension.
In our symbolic language we need some mode of indicating this junction
of terms, and the most convenient device will be the juxtaposition of
the letter-terms. Thus if A mean brittle, and B mean metal, then AB
will mean brittle metal. Nor need there be any limit to the number of
letters thus joined together, or the complexity of the notions which
they may represent.
Thus if we take the letters
P = metal,
Q = white,
R = monovalent,
S = of specific gravity 10·5,
T = melting above 1000° C.,
V = good conductor of heat and electricity,
then we can form a combined term PQRSTV, which will denote “a white
monovalent metal, of specific gravity 10·5, melting above 1000° C., and
a good conductor of heat and electricity.”
There are many grammatical usages concerning the junction of words and
phrases to which we need pay no attention in logic. We can never say
in ordinary language “of wood table,” meaning “table of wood;” but we
may consider “of wood” as logically an exact equivalent of “wooden”; so
that if
X = of wood,
Y = table,
there is no reason why, in our symbols, XY should not be just as
correct an expression for “table of wood ” as YX. In this case indeed
we might substitute for “of wood ” the corresponding adjective
“wooden,” but we should often fail to find any adjective answering
exactly to a phrase. There is no single word by which we could express
the notion “of specific gravity 10·5:” but logically we may consider
these words as forming an adjective; and denoting this by S and metal
by P, we may say that SP means “metal of specific gravity 10·5.” It
is one of many advantages in these blank letter-symbols that they
enable us completely to neglect all grammatical peculiarities and to
fix our attention solely on the purely logical relations involved.
Investigation will probably show that the rules of grammar are mainly
founded upon traditional usage and have little logical signification.
This indeed is sufficiently proved by the wide grammatical differences
which exist between languages, though the logical foundation must be
the same.
*Symbolic Expression of the Law of Contradiction.*
The synthesis of terms is subject to the all-important Law of
Thought, described in a previous section (p. 5) and called the Law of
Contradiction, It is self-evident that no quality can be both present
and absent at the same time and place. This fundamental condition
of all thought and of all existence is expressed symbolically by a
rule that a term and its negative shall never be allowed to come
into combination. Such combined terms as A*a*, B*b*, C*c*, &c., are
self-contradictory and devoid of all intelligible meaning. If they
could represent anything, it would be what cannot exist, and cannot
even be imagined in the mind. They can therefore only enter into our
consideration to suffer immediate exclusion. The criterion of false
reasoning, as we shall find, is that it involves self-contradiction,
the affirming and denying of the same statement. We might represent
the object of all reasoning as the separation of the consistent and
possible from the inconsistent and impossible; and we cannot make any
statement except a truism without implying that certain combinations of
terms are contradictory and excluded from thought. To assert that “all
A’s are B’s” is equivalent to the assertion that “A’s which are not B’s
cannot exist.”
It will be convenient to have the means of indicating the exclusion of
the self-contradictory, and we may use the familiar sign for *nothing*,
the cipher 0. Thus the second law of thought may be symbolised in the
forms
A*a* = 0 AB*b* = 0 ABC*a* = 0
We may variously describe the meaning of 0 in logic as the
*non-existent*, the *impossible*, the *self-inconsistent*, the
*inconceivable*. Close analogy exists between this meaning and its
mathematical signification.
*Certain Special Conditions of Logical Symbols.*
In order that we may argue and infer truly we must treat our logical
symbols according to the fundamental laws of Identity and Difference.
But in thus using our symbols we shall frequently meet with
combinations of which the meaning will not at first sight be apparent.
If in one case we learn that an object is “yellow and round,” and in
another case that it is “round and yellow,” there arises the question
whether these two descriptions are identical in meaning or not. Again,
if we proved that an object was “round round,” the meaning of such an
expression would be open to doubt. Accordingly we must take notice,
before proceeding further, of certain special laws which govern the
combination of logical terms.
In the first place the combination of a logical term with itself is
without effect, just as the repetition of a statement does not alter
the meaning of the statement; “a round round object” is simply “a round
object.” What is yellow yellow is merely yellow; metallic metals cannot
differ from metals, nor circular circles from circles. In our symbolic
language we may similarly hold that AA is identical with A, or
A = AA = AAA = &c.
The late Professor Boole is the only logician in modern times who has
drawn attention to this remarkable property of logical terms;[46]
but in place of the name which he gave to the law, I have proposed
to call it The Law of Simplicity.[47] Its high importance will only
become apparent when we attempt to determine the relations of logical
and mathematical science. Two symbols of quantity, and only two, seem
to obey this law; we may say that 1 × 1 = 1, and 0 × 0 = 0 (taking 0
to mean absolute zero or 1 – 1); there is apparently no other number
which combined with itself gives an unchanged result. I shall point
out, however, in the chapter upon Number, that in reality all numerical
symbols obey this logical principle.
[46] *Mathematical Analysis of Logic*, Cambridge, 1847, p. 17. *An
Investigation of the Laws of Thought*, London, 1854, p. 31.
[47] *Pure Logic*, p. 15.
It is curious that this Law of Simplicity, though almost unnoticed
in modern times, was known to Boëthius, who makes a singular remark
in his treatise *De Trinitate et Unitate Dei* (p. 959). He says: “If
I should say sun, sun, sun, I should not have made three suns, but I
should have named one sun so many times.”[48] Ancient discussions about
the doctrine of the Trinity drew more attention to subtle questions
concerning the nature of unity and plurality than has ever since been
given to them.
[48] “Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed
uno toties prædicaverim.”
It is a second law of logical symbols that order of combination is a
matter of indifference. “Rich and rare gems” are the same as “rare and
rich gems,” or even as “gems, rich and rare.” Grammatical, rhetorical,
or poetic usage may give considerable significance to order of
expression. The limited power of our minds prevents our grasping many
ideas at once, and thus the order of statement may produce some effect,
but not in a simply logical manner. All life proceeds in the succession
of time, and we are obliged to write, speak, or even think of things
and their qualities one after the other; but between the things and
their qualities there need be no such relation of order in time or
space. The sweetness of sugar is neither before nor after its weight
and solubility. The hardness of a metal, its colour, weight, opacity,
malleability, electric and chemical properties, are all coexistent
and coextensive, pervading the metal and every part of it in perfect
community, none before nor after the others. In our words and symbols
we cannot observe this natural condition; we must name one quality
first and another second, just as some one must be the first to sign a
petition, or to walk foremost in a procession. In nature there is no
such precedence.
I find that the opinion here stated, to the effect that relations of
space and time do not apply to many of our ideas, is clearly adopted by
Hume in his celebrated *Treatise on Human Nature* (vol. i. p. 410). He
says:[49]--“An object may be said to be no where, when its parts are
not so situated with respect to each other, as to form any figure or
quantity; nor the whole with respect to other bodies so as to answer
to our notions of contiguity or distance. Now this is evidently the
case with all our perceptions and objects, except those of sight and
feeling. A moral reflection cannot be placed on the right hand or on
the left hand of a passion, nor can a smell or sound be either of a
circular or a square figure. These objects and perceptions, so far from
requiring any particular place, are absolutely incompatible with it,
and even the imagination cannot attribute it to them.”
[49] Book i., Part iv., Section 5.
A little reflection will show that knowledge in the highest perfection
would consist in the *simultaneous* possession of a multitude of
facts. To comprehend a science perfectly we should have every fact
present with every other fact. We must write a book and we must read
it successively word by word, but how infinitely higher would be
our powers of thought if we could grasp the whole in one collective
act of consciousness! Compared with the brutes we do possess some
slight approximation to such power, and it is conceivable that in the
indefinite future mind may acquire an increase of capacity, and be less
restricted to the piecemeal examination of a subject. But I wish here
to make plain that there is no logical foundation for the successive
character of thought and reasoning unavoidable under our present mental
conditions. *We are logically weak and imperfect in respect of the
fact that we are obliged to think of one thing after another.* We must
describe metal as “hard and opaque,” or “opaque and hard,” but in the
metal itself there is no such difference of order; the properties are
simultaneous and coextensive in existence.
Setting aside all grammatical peculiarities which render a substantive
less moveable than an adjective, and disregarding any meaning indicated
by emphasis or marked order of words, we may state, as a general law of
logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB =
BCA = &c.
Boole first drew attention in recent years to this property of logical
terms, and he called it the property of Commutativeness.[50] He not
only stated the law with the utmost clearness, but pointed out that
it is a Law of Thought rather than a Law of Things. I shall have in
various parts of this work to show how the necessary imperfection of
our symbols expressed in this law clings to our modes of expression,
and introduces complication into the whole body of mathematical
formulæ, which are really founded on a logical basis.
[50] *Laws of Thought*, p. 29. It is pointed out in the preface to
this Second Edition that Leibnitz was acquainted with the Laws of
Simplicity and of Commutativeness.
It is of course apparent that the power of commutation belongs only
to terms related in the simple logical mode of synthesis. No one can
confuse “a house of bricks” with “bricks of a house,” “twelve square
feet” with “twelve feet square,” “the water of crystallization” with
“the crystallization of water.” All relations which involve differences
of time and space are inconvertible; the higher must not be made to
change places with the lower, nor the first with the last. For the
parties concerned there is all the difference in the world between A
killing B and B killing A. The law of commutativeness simply asserts
that difference of order does not attach to the connection between the
properties and circumstances of a thing--to what I call *simple logical
relation*.
CHAPTER III.
PROPOSITIONS.
We now proceed to consider the variety of forms of propositions in
which the truths of science must be expressed. I shall endeavour to
show that, however diverse these forms may be, they all admit the
application of the one same principle of inference that what is true of
a thing is true of the like or same. This principle holds true whatever
be the kind or manner of the likeness, provided proper regard be had to
its nature. Propositions may assert an identity of time, space, manner,
quantity, degree, or any other circumstance in which things may agree
or differ.
We find an instance of a proposition concerning time in the
following:--“The year in which Newton was born, was the year in which
Galileo died.” This proposition expresses an approximate identity of
time between two events; hence whatever is true of the year in which
Galileo died is true of that in which Newton was born, and *vice
versâ*. “Tower Hill is the place where Raleigh was executed” expresses
an identity of place; and whatever is true of the one spot is true
of the spot otherwise defined, but in reality the same. In ordinary
language we have many propositions obscurely expressing identities
of number, quantity, or degree. “So many men, so many minds,” is a
proposition concerning number, that is to say, an equation; whatever
is true of the number of men is true of the number of minds, and
*vice versâ*. “The density of Mars is (nearly) the same as that of
the Earth,” “The force of gravity is directly as the product of the
masses, and inversely as the square of the distance,” are propositions
concerning magnitude or degree. Logicians have not paid adequate
attention to the great variety of propositions which can be stated by
the use of the little conjunction *as*, together with *so*. “As the
home so the people,” is a proposition expressing identity of manner;
and a great number of similar propositions all indicating some kind of
resemblance might be quoted. Whatever be the special kind of identity,
all such expressions are subject to the great principle of inference;
but as we shall in later parts of this work treat more particularly of
inference in cases of number and magnitude, we will here confine our
attention to logical propositions which involve only notions of quality.
*Simple Identities.*
The most important class of propositions consists of those which fall
under the formula
A = B,
and may be called *simple identities*. I may instance, in the first
place, those most elementary propositions which express the exact
similarity of a quality encountered in two or more objects. I may
compare the colour of the Pacific Ocean with that of the Atlantic, and
declare them identical. I may assert that “the smell of a rotten egg
is like that of hydrogen sulphide;” “the taste of silver hyposulphite
is like that of cane sugar;” “the sound of an earthquake resembles
that of distant artillery.” Such are propositions stating, accurately
or otherwise, the identity of simple physical sensations. Judgments of
this kind are necessarily pre-supposed in more complex judgments. If
I declare that “this coin is made of gold,” I must base the judgment
upon the exact likeness of the substance in several qualities to
other pieces of substance which are undoubtedly gold. I must make
judgments of the colour, the specific gravity, the hardness, and of
other mechanical and chemical properties; each of these judgments is
expressed in an elementary proposition, “the colour of this coin is the
colour of gold,” and so on. Even when we establish the identity of a
thing with itself under a different name or aspect, it is by distinct
judgments concerning single circumstances. To prove that the Homeric
χαλκός is copper we must show the identity of each quality recorded of
χαλκός with a quality of copper. To establish Deal as the landing-place
of Cæsar all material circumstances must be shown to agree. If the
modern Wroxeter is the ancient Uriconium, there must be the like
agreement of all features of the country not subject to alteration by
time.
Such identities must be expressed in the form A = B. We may say
Colour of Pacific Ocean = Colour of Atlantic Ocean.
Smell of rotten egg = Smell of hydrogen sulphide.
In these and similar propositions we assert identity of single
qualities or causes of sensation. In the same form we may also express
identity of any group of qualities, as in
χαλκός = Copper.
Deal = Landing-place of Cæsar.
A multitude of propositions involving singular terms fall into the same
form, as in
The Pole star = The slowest-moving star.
Jupiter = The greatest of the planets.
The ringed planet = The planet having seven satellites.
The Queen of England = The Empress of India.
The number two = The even prime number.
Honesty = The best policy.
In mathematical and scientific theories we often meet with simple
identities capable of expression in the same form. Thus in mechanical
science “The process for finding the resultant of forces = the process
for finding the resultant of simultaneous velocities.” Theorems in
geometry often give results in this form, as
Equilateral triangles = Equiangular triangles.
Circle = Finite plane curve of constant curvature.
Circle = Curve of least perimeter.
The more profound and important laws of nature are often expressible in
the form of simple identities; in addition to some instances which have
already been given, I may suggest,
Crystals of cubical system = Crystals not possessing the power of
double refraction.
All definitions are necessarily of this form, whether the objects
defined be many, few, or singular. Thus we may say,
Common salt = Sodium chloride.
Chlorophyl = Green colouring matter of leaves.
Square = Equal-sided rectangle.
It is an extraordinary fact that propositions of this elementary form,
all-important and very numerous as they are, had no recognised place
in Aristotle’s system of Logic. Accordingly their importance was
overlooked until very recent times, and logic was the most deformed
of sciences. But it is impossible that Aristotle or any other person
should avoid constantly using them; not a term could be defined
without their use. In one place at least Aristotle actually notices a
proposition of the kind. He observes: “We sometimes say that that white
thing is Socrates, or that the object approaching is Callias.”[51] Here
we certainly have simple identity of terms; but he considered such
propositions purely accidental, and came to the unfortunate conclusion,
that “Singulars cannot be predicated of other terms.”
[51] *Prior Analytics*, i. cap. xxvii. 3.
Propositions may also express the identity of extensive groups of
objects taken collectively or in one connected whole; as when we say,
The Queen, Lords, and Commons = The Legislature of the United Kingdom.
When Blackstone asserts that “The only true and natural foundation of
society are the wants and fears of individuals,” we must interpret him
as meaning that the whole of the wants and fears of individuals in the
aggregate form the foundation of society. But many propositions which
might seem to be collective are but groups of singular propositions or
identities. When we say “Potassium and sodium are the metallic bases of
potash and soda,” we obviously mean,
Potassium = Metallic base of potash;
Sodium = Metallic base of soda.
It is the work of grammatical analysis to separate the various
propositions often combined into a single sentence. Logic cannot be
properly required to interpret the forms and devices of language, but
only to treat the meaning when clearly exhibited.
*Partial Identities.*
A second highly important kind of proposition is that which I propose
to call *a partial identity*. When we say that “All mammalia are
vertebrata,” we do not mean that mammalian animals are identical with
vertebrate animals, but only that the mammalia form a *part of the
class vertebrata*. Such a proposition was regarded in the old logic as
asserting the inclusion of one class in another, or of an object in a
class. It was called a universal affirmative proposition, because the
attribute *vertebrate* was affirmed of the whole subject *mammalia*;
but the attribute was said to be *undistributed*, because not all
vertebrata were of necessity involved in the proposition. Aristotle,
overlooking the importance of simple identities, and indeed almost
denying their existence, unfortunately founded his system upon the
notion of inclusion in a class, instead of adopting the basis of
identity. He regarded inference as resting upon the rule that what is
true of the containing class is true of the contained, in place of the
vastly more general rule that what is true of a class or thing is true
of the like. Thus he not only reduced logic to a fragment of its proper
self, but destroyed the deep analogies which bind together logical and
mathematical reasoning. Hence a crowd of defects, difficulties and
errors which will long disfigure the first and simplest of the sciences.
It is surely evident that the relation of inclusion rests upon the
relation of identity. Mammalian animals cannot be included among
vertebrates unless they be identical with part of the vertebrates.
Cabinet Ministers are included almost always in the class Members of
Parliament, because they are identical with some who sit in Parliament.
We may indicate this identity with a part of the larger class in
various ways; as for instance,
Mammalia = part of the vertebrata.
Diatomaceæ = a class of plants.
Cabinet Ministers = some members of Parliament.
Iron = a metal.
In ordinary language the verbs *is* and *are* express mere inclusion
more often than not. *Men are mortals*, means that *men* form a part
of the class *mortal*; but great confusion exists between this sense
of the verb and that in which it expresses identity, as in “The sun is
the centre of the planetary system.” The introduction of the indefinite
article *a* often expresses partiality; when we say “Iron is a metal”
we clearly mean that iron is *one only* of several metals.
Certain recent logicians have proposed to avoid the indefiniteness
in question by what is called the Quantification of the Predicate,
and they have generally used the little word *some* to show that only
a part of the predicate is identical with the subject. *Some* is an
*indeterminate adjective*; it implies unknown qualities by which we
might select the part in question if the qualities were known, but
it gives no hint as to their nature. I might make use of such an
indeterminate sign to express partial identities in this work. Thus,
taking the special symbol V = Some, the general form of a partial
identity would be A = VB, and in Boole’s Logic expressions of the
kind were much used. But I believe that indeterminate symbols only
introduce complexity, and destroy the beauty and simple universality
of the system which may be created without their use. A vague word
like *some* is only used in ordinary language by *ellipsis*, and to
avoid the trouble of attaining accuracy. We can always employ more
definite expressions if we like; but when once the indefinite *some* is
introduced we cannot replace it by the special description. We do not
know whether *some* colour is red, yellow, blue, or what it is; but on
the other hand *red* colour is certainly *some* colour.
Throughout this system of logic I shall dispense with such indefinite
expressions; and this can readily be done by substituting one of the
other terms. To express the proposition “All A’s are some B’s” I shall
not use the form A = VB, but
A = AB.
This formula states that the class A is identical with the class AB;
and as the latter must be a part at least of the class B, it implies
the inclusion of the class A in that of B. We might represent our
former example thus,
Mammalia = Mammalian vertebrata.
This proposition asserts identity between a part (or it may be the
whole) of the vertebrata and the mammalia. If it is asked What part?
the proposition affords no answer, except that it is the part which is
mammalian; but the assertion “mammalia = some vertebrata” tells us no
more.
It is quite likely that some readers will think this mode of
representing the universal affirmative proposition artificial and
complicated. I will not undertake to convince them of the opposite
at this point of my exposition. Justification for it will be found,
not so much in the immediate treatment of this proposition, as in the
general harmony which it will enable us to disclose between all parts
of reasoning. I have no doubt that this is the critical difficulty in
the relation of logical to other forms of reasoning. Grant this mode of
denoting that “all A’s are B’s,” and I fear no further difficulties;
refuse it, and we find want of analogy and endless anomaly in every
direction. It is on general grounds that I hope to show overwhelming
reasons for seeking to reduce every kind of proposition to the form of
an identity.
I may add that not a few logicians have accepted this view of the
universal affirmative proposition. Leibnitz, in his *Difficultates
Quædam Logicæ*, adopts it, saying, “Omne A est B; id est æquivalent AB
et A, seu A non B est nonens.” Boole employed the logical equation *x*
= *xy* concurrently with *x* = *vy*; and Spalding[52] distinctly
says that the proposition “all metals are minerals” might be described
as an assertion of *partial identity* between the two classes. Hence
the name which I have adopted for the proposition.
[52] *Encyclopædia Britannica*, Eighth Ed. art. Logic, sect. 37,
note. 8vo. reprint, p. 79.
*Limited Identities.*
An important class of propositions have the form
AB = AC,
expressing the identity of the class AB with the class AC. In other
words, “Within the sphere of the class A, all the B’s are all the
C’s;” or again, “The B’s and C’s, which are A’s, are identical.” But
it will be observed that nothing is asserted concerning things which
are outside of the class A; and thus the identity is of limited extent.
It is the proposition B = C limited to the sphere of things called A.
Thus we may say, with some approximation to truth, that “Large plants
are plants devoid of locomotive power.”
A barrister may make numbers of most general statements concerning
the relations of persons and things in the course of an argument, but
it is of course to be understood that he speaks only of persons and
things under the English Law. Even mathematicians make statements which
are not true with absolute generality. They say that imaginary roots
enter into equations by pairs; but this is only true under the tacit
condition that the equations in question shall not have imaginary
coefficients.[53] The universe, in short, within which they habitually
discourse is that of equations with real coefficients. These implied
limitations form part of that great mass of tacit knowledge which
accompanies all special arguments.
[53] De Morgan, *On the Root of any Function*. Cambridge
Philosophical Transactions, 1867, vol. xi. p. 25.
To De Morgan is due the remark, that we do usually think and argue in
a limited universe or sphere of notions, even when it is not expressly
stated.[54]
[54] *Syllabus of a proposed System of Logic*, §§ 122, 123.
It is worthy of inquiry whether all identities are not really limited
to an implied sphere of meaning. When we make such a plain statement as
“Gold is malleable” we obviously speak of gold only in its solid state;
when we say that “Mercury is a liquid metal” we must be understood to
exclude the frozen condition to which it may be reduced in the Arctic
regions. Even when we take such a fundamental law of nature as “All
substances gravitate,” we must mean by substance, material substance,
not including that basis of heat, light, and electrical undulations
which occupies space and possesses many wonderful mechanical
properties, but not gravity. The proposition then is really of the form
Material substance = Material gravitating substance.
*Negative Propositions.*
In every act of intellect we are engaged with a certain identity or
difference between things or sensations compared together. Hitherto
I have treated only of identities; and yet it might seem that the
relation of difference must be infinitely more common than that of
likeness. One thing may resemble a great many other things, but then it
differs from all remaining things in the world. Diversity may almost be
said to constitute life, being to thought what motion is to a river.
The perception of an object involves its discrimination from all other
objects. But we may nevertheless be said to detect resemblance as often
as we detect difference. We cannot, in fact, assert the existence of
a difference, without at the same time implying the existence of an
agreement.
If I compare mercury, for instance, with other metals, and decide that
it is *not solid*, here is a difference between mercury and solid
things, expressed in a negative proposition; but there must be implied,
at the same time, an agreement between mercury and the other substances
which are not solid. As it is impossible to separate the vowels of the
alphabet from the consonants without at the same time separating the
consonants from the vowels, so I cannot select as the object of thought
*solid things*, without thereby throwing together into another class
all things which are *not solid*. The very fact of not possessing a
quality, constitutes a new quality which may be the ground of judgment
and classification. In this point of view, agreement and difference are
ever the two sides of the same act of intellect, and it becomes equally
possible to express the same judgment in the one or other aspect.
Between affirmation and negation there is accordingly a perfect
equilibrium. Every affirmative proposition implies a negative one, and
*vice versâ*. It is even a matter of indifference, in a logical point
of view, whether a positive or negative term be used to denote a given
quality and the class of things possessing it. If the ordinary state
of a man’s body be called *good health*, then in other circumstances
he is said *not to be in good health*; but we might equally describe
him in the latter state as *sickly*, and in his normal condition he
would be *not sickly*. Animal and vegetable substances are now called
*organic*, so that the other substances, forming an immensely greater
part of the globe, are described negatively as *inorganic*. But we
might, with at least equal logical correctness, have described the
preponderating class of substances as *mineral*, and then vegetable and
animal substances would have been *non-mineral*.
It is plain that any positive term and its corresponding negative
divide between them the whole universe of thought: whatever does not
fall into one must fall into the other, by the third fundamental Law
of Thought, the Law of Duality. It follows at once that there are
two modes of representing a difference. Supposing that the things
represented by A and B are found to differ, we may indicate (see p. 17)
the result of the judgment by the notation
A ~ B.
We may now represent the same judgment by the assertion that A agrees
with those things which differ from B, or that A agrees with the
not-B’s. Using our notation for negative terms (see p. 14), we obtain
A = A*b*
as the expression of the ordinary negative proposition. Thus if we
take A to mean quicksilver, and B solid, then we have the following
proposition:--
Quicksilver = Quicksilver not-solid.
There may also be several other classes of negative propositions, of
which no notice was taken in the old logic. We may have cases where
all A’s are not-B’s, and at the same time all not-B’s are A’s; there
may, in short, be a simple identity between A and not-B, which may be
expressed in the form
A = *b*.
An example of this form would be
Conductors of electricity = non-electrics.
We shall also frequently have to deal as results of deduction, with
simple, partial, or limited identities between negative terms, as in
the forms
*a* = *b*, *a* = *a**b*, *a*C = *b*C, etc.
It would be possible to represent affirmative propositions in the
negative form. Thus “Iron is solid,” might be expressed as “Iron is not
not-solid,” or “Iron is not fluid;” or, taking A and *b* for the terms
“iron,” and “not-solid,” the form would be A ~ *b*.
But there are very strong reasons why we should employ all propositions
in their affirmative form. All inference proceeds by the substitution
of equivalents, and a proposition expressed in the form of an identity
is ready to yield all its consequences in the most direct manner. As
will be more fully shown, we can infer *in* a negative proposition,
but not *by* it. Difference is incapable of becoming the ground of
inference; it is only the implied agreement with other differing
objects which admits of deductive reasoning; and it will always be
found advantageous to employ propositions in the form which exhibits
clearly the implied agreements.
*Conversion of Propositions.*
The old books of logic contain many rules concerning the conversion of
propositions, that is, the transposition of the subject and predicate
in such a way as to obtain a new proposition which will be true when
the original proposition is true. The reduction of every proposition to
the form of an identity renders all such rules and processes needless.
Identity is essentially reciprocal. If the colour of the Atlantic Ocean
is the same as that of the Pacific Ocean, that of the Pacific must
be the same as that of the Atlantic. Sodium chloride being identical
with common salt, common salt must be identical with sodium chloride.
If the number of windows in Salisbury Cathedral equals the number of
days in the year, the number of days in the year must equal the number
of the windows. Lord Chesterfield was not wrong when he said, “I will
give anybody their choice of these two truths, which amount to the
same thing; He who loves himself best is the honestest man; or, The
honestest man loves himself best.” Scotus Erigena exactly expresses
this reciprocal character of identity in saying, “There are not two
studies, one of philosophy and the other of religion; true philosophy
is true religion, and true religion is true philosophy.”
A mathematician would not think it worth while to mention that if
*x* = *y* then also *y* = *x*. He would not consider these to be
two equations at all, but one equation accidentally written in two
different manners. In written symbols one of two names must come first,
and the other second, and a like succession must perhaps be observed
in our thoughts: but in the relation of identity there is no need for
succession in order (see p. 33), each is simultaneously equal and
identical to the other. These remarks will hold true both of logical
and mathematical identity; so that I shall consider the two forms
A = B and B = A
to express exactly the same identity differently written. All need for
rules of conversion disappears, and there will be no single proposition
in the system which may not be written with either end foremost. Thus A
= AB is the same as AB = A, *a*C = *b*C is the same as *b*C = *a*C, and
so forth.
The same remarks are partially true of differences and inequalities,
which are also reciprocal to the extent that one thing cannot differ
from a second without the second differing from the first. Mars differs
in colour from Venus, and Venus must differ from Mars. The Earth
differs from Jupiter in density; therefore Jupiter must differ from the
Earth. Speaking generally, if A ~ B we shall also have B ~ A, and these
two forms may be considered expressions of the same difference. But
the relation of differing things is not wholly reciprocal. The density
of Jupiter does not differ from that of the Earth in the same way that
that of the Earth differs from that of Jupiter. The change of sensation
which we experience in passing from Venus to Mars is not the same as
what we experience in passing back to Venus, but just the opposite
in nature. The colour of the sky is lighter than that of the ocean;
therefore that of the ocean cannot be lighter than that of the sky, but
darker. In these and all similar cases we gain a notion of *direction*
or character of change, and results of immense importance may be shown
to rest on this notion. For the present we shall be concerned with the
mere fact of identity existing or not existing.
*Twofold Interpretation of Propositions.*
Terms, as we have seen (p. 25), may have a meaning either in extension
or intension; and according as one or the other meaning is attributed
to the terms of a proposition, so may a different interpretation be
assigned to the proposition itself. When the terms are abstract we
must read them in intension, and a proposition connecting such terms
must denote the identity or non-identity of the qualities respectively
denoted by the terms. Thus if we say
Equality = Identity of magnitude,
the assertion means that the circumstance of being equal exactly
corresponds with the circumstance of being identical in magnitude.
Similarly in
Opacity = Incapability of transmitting light,
the quality of being incapable of transmitting light is declared to be
the same as the intended meaning of the word opacity.
When general names form the terms of a proposition we may apply a
double interpretation. Thus
Exogens = Dicotyledons
means either that the qualities which belong to all exogens are the
same as those which belong to all dicotyledons, or else that every
individual falling under one name falls equally under the other.
Hence it may be said that there are two distinct fields of logical
thought. We may argue either by the qualitative meaning of names or
by the quantitative, that is, the extensive meaning. Every argument
involving concrete plural terms might be converted into one involving
only abstract singular terms, and *vice versâ*. But there are reasons
for believing that the intensive or qualitative form of reasoning is
the primary and fundamental one. It is sufficient to point out that the
extensive meaning of a name is a changeable and fleeting thing, while
the intensive meaning may nevertheless remain fixed. Very numerous
additions have been lately made to the extensive meanings both of
planet and element. Every iron steam-ship which is made or destroyed
adds to or subtracts from the extensive meaning of the name steam-ship,
without necessarily affecting the intensive meaning. Stage coach means
as much as ever in one way, but in extension the class is nearly
extinct. Chinese railway, on the other hand, is a term represented only
by a single instance; in twenty years it may be the name of a large
class.
CHAPTER IV.
DEDUCTIVE REASONING.
The general principle of inference having been explained in the
previous chapters, and a suitable system of symbols provided, we have
now before us the comparatively easy task of tracing out the most
common and important forms of deductive reasoning. The general problem
of deduction is as follows:--*From one or more propositions called
premises to draw such other propositions as will necessarily be true
when the premises are true.* By deduction we investigate and unfold the
information contained in the premises; and this we can do by one single
rule--*For any term occurring in any proposition substitute the term
which is asserted in any premise to be identical with it.* To obtain
certain deductions, especially those involving negative conclusions, we
shall require to bring into use the second and third Laws of Thought,
and the process of reasoning will then be called *Indirect Deduction*.
In the present chapter, however, I shall confine my attention to
those results which can be obtained by the process of *Direct
Deduction*, that is, by applying to the premises themselves the rule of
substitution. It will be found that we can combine into one harmonious
system, not only the various moods of the ancient syllogism but a great
number of equally important forms of reasoning, which had no recognised
place in the old logic. We can at the same time dispense entirely with
the elaborate apparatus of logical rules and mnemonic lines, which were
requisite so long as the vital principle of reasoning was not clearly
expressed.
*Immediate Inference.*
Probably the simplest of all forms of inference is that which has been
called *Immediate Inference*, because it can be performed upon a single
proposition. It consists in joining an adjective, or other qualifying
clause of the same nature, to both sides of an identity, and asserting
the equivalence of the terms thus produced. For instance, since
Conductors of electricity = Non-electrics,
it follows that
Liquid conductors of electricity = Liquid non-electrics.
If we suppose that
Plants = Bodies decomposing carbonic acid,
it follows that
Microscopic plants = Microscopic bodies decomposing
carbonic acid.
In general terms, from the identity
A = B
we can infer the identity
AC = BC.
This is but a case of plain substitution; for by the first Law of
Thought it must be admitted that
AC = AC,
and if, in the second side of this identity, we substitute for A its
equivalent B, we obtain
AC = BC.
In like manner from the partial identity
A = AB
we may obtain
AC = ABC
by an exactly similar act of substitution; and in every other case
the rule will be found capable of verification by the principle of
inference. The process when performed as here described will be quite
free from the liability to error which I have shown[55] to exist in
“Immediate Inference by added Determinants,” as described by Dr.
Thomson.[56]
[55] *Elementary Lessons in Logic*, p. 86.
[56] *Outline of the Laws of Thought*, § 87.
*Inference with Two Simple Identities.*
One of the most common forms of inference, and one to which I shall
especially direct attention, is practised with two simple identities.
From the two statements that “London is the capital of England” and
“London is the most populous city in the world,” we instantaneously
draw the conclusion that “The capital of England is the most populous
city in the world.” Similarly, from the identities
Hydrogen = Substance of least density,
Hydrogen = Substance of least atomic weight,
we infer
Substance of least density = Substance of least atomic weight.
The general form of the argument is exhibited in the symbols
B = A (1)
B = C (2)
hence A = C. (3)
We may describe the result by saying that terms identical with the
same term are identical with each other; and it is impossible to
overlook the analogy to the first axiom of Euclid that “things equal
to the same thing are equal to each other.” It has been very commonly
supposed that this is a fundamental principle of thought, incapable of
reduction to anything simpler. But I entertain no doubt that this form
of reasoning is only one case of the general rule of inference. We have
two propositions, A = B and B = C, and we may for a moment consider
the second one as affirming a truth concerning B, while the former one
informs us that B is identical with A; hence by substitution we may
affirm the same truth of A. It happens in this particular case that the
truth affirmed is identity to C, and we might, if we preferred it, have
considered the substitution as made by means of the second identity in
the first. Having two identities we have a choice of the mode in which
we will make the substitution, though the result is exactly the same in
either case.
Now compare the three following formulæ,
(1) A = B = C, hence A = C
(2) A = B ~ C, hence A ~ C
(3) A ~ B ~ C, no inference.
In the second formula we have an identity and a difference, and we are
able to infer a difference; in the third we have two differences and
are unable to make any inference at all. Because A and C both differ
from B, we cannot tell whether they will or will not differ from each
other. The flowers and leaves of a plant may both differ in colour from
the earth in which the plant grows, and yet they may differ from each
other; in other cases the leaves and stem may both differ from the
soil and yet agree with each other. Where we have difference only we
can make no inference; where we have identity we can infer. This fact
gives great countenance to my assertion that inference proceeds always
through identity, but may be equally well effected in propositions
asserting difference or identity.
Deferring a more complete discussion of this point, I will only mention
now that arguments from double identity occur very frequently, and are
usually taken for granted, owing to their extreme simplicity. In regard
to the equivalence of words this form of inference must be constantly
employed. If the ancient Greek χαλκός is our *copper*, then it must
be the French *cuivre*, the German *kupfer*, the Latin *cuprum*,
because these are words, in one sense at least, equivalent to copper.
Whenever we can give two definitions or expressions for the same term,
the formula applies; thus Senior defined wealth as “All those things,
and those things only, which are transferable, are limited in supply,
and are directly or indirectly productive of pleasure or preventive
of pain.” Wealth is also equivalent to “things which have value in
exchange;” hence obviously, “things which have value in exchange = all
those things, and those things only, which are transferable, &c.” Two
expressions for the same term are often given in the same sentence,
and their equivalence implied. Thus Thomson and Tait say,[57] “The
naturalist may be content to know matter as that which can be perceived
by the senses, or as that which can be acted upon by or can exert
force.” I take this to mean--
Matter = what can be perceived by the senses;
Matter = what can be acted upon by or can exert force.
[57] *Treatise on Natural Philosophy*, vol. i. p. 161.
For the term “matter” in either of these identities we may substitute
its equivalent given in the other definition. Elsewhere they often
employ sentences of the form exemplified in the following:[58] “The
integral curvature, or whole change of direction of an arc of a plane
curve, is the angle through which the tangent has turned as we pass
from one extremity to the other.” This sentence is certainly of the
form--
The integral curvature = the whole change of direction, &c. = the
angle through which the tangent has turned, &c.
[58] *Treatise on Natural Philosophy*, vol. i. p. 6.
Disguised cases of the same kind of inference occur throughout all
sciences, and a remarkable instance is found in algebraic geometry.
Mathematicians readily show that every equation of the form *y* = *mx*
+ *c* corresponds to or represents a straight line; it is also easily
proved that the same equation is equivalent to one of the general form
A*x* + B*y* + C = 0, and *vice versâ*. Hence it follows that every
equation of the form in question, that is to say, every equation of the
first degree, corresponds to or represents a straight line.[59]
[59] Todhunter’s *Plane Co-ordinate Geometry*, chap. ii. pp. 11–14.
*Inference with a Simple and a Partial Identity.*
A form of reasoning somewhat different from that last considered
consists in inference-between a simple and a partial identity. If we
have two propositions of the forms
A = B,
B = BC,
we may then substitute for B in either proposition its equivalent in
the other, getting in both cases A = BC; in this we may if we like make
a second substitution for B, getting
A = AC.
Thus, since “The Mont Blanc is the highest mountain in Europe, and
the Mont Blanc is deeply covered with snow,” we infer by an obvious
substitution that “The highest mountain in Europe is deeply covered
with snow.” These propositions when rigorously stated fall into the
forms above exhibited.
This mode of inference is constantly employed when for a term we
substitute its definition, or *vice versâ*. The very purpose of a
definition is to allow a single noun to be employed in place of a long
descriptive phrase. Thus, when we say “A circle is a curve of the
second degree,” we may substitute a definition of the circle, getting
“A curve, all points of which are at equal distances from one point, is
a curve of the second degree.” The real forms of the propositions here
given are exactly those shown in the symbolic statement, but in this
and many other cases it will be sufficient to state them in ordinary
elliptical language for sake of brevity. In scientific treatises a
term and its definition are often both given in the same sentence,
as in “The weight of a body in any given locality, or the force
with which the earth attracts it, is proportional to its mass.” The
conjunction *or* in this statement gives the force of equivalence to
the parenthetic phrase, so that the propositions really are
Weight of a body = force with which the earth attracts it.
Weight of a body = weight, &c. proportional to its mass.
A slightly different case of inference consists in substituting in a
proposition of the form A = AB, a definition of the term B. Thus from A
= AB and B = C we get A = AC. For instance, we may say that “Metals are
elements” and “Elements are incapable of decomposition.”
Metal = metal element.
Element = what is incapable of decomposition.
Hence
Metal = metal incapable of decomposition.
It is almost needless to point out that the form of these arguments
does not suffer any real modification if some of the terms happen to be
negative; indeed in the last example “incapable of decomposition” may
be treated as a negative term. Taking
A = metal
B = element
C = capable of decomposition
*c* = incapable of decomposition;
the propositions are of the forms
A = AB
B = *c*
whence, by substitution,
A = A*c*.
*Inference of a Partial from Two Partial Identities.*
However common be the cases of inference already noticed, there is
a form occurring almost more frequently, and which deserves much
attention, because it occupied a prominent place in the ancient
syllogistic system. That system strangely overlooked all the kinds of
argument we have as yet considered, and selected, as the type of all
reasoning, one which employs two partial identities as premises. Thus
from the propositions
Sodium is a metal (1)
Metals conduct electricity, (2)
we may conclude that
Sodium conducts electricity. (3)
Taking A, B, C to represent the three terms respectively, the premises
are of the forms
A = AB (1)
B = BC. (2)
Now for B in (1) we can substitute its expression as given in (2),
obtaining
A = ABC, (3)
or, in words, from
Sodium = sodium metal, (1)
Metal = metal conducting electricity, (2)
we infer
Sodium = sodium metal conducting electricity, (3)
which, in the elliptical language of common life, becomes
“Sodium conducts electricity.”
The above is a syllogism in the mood called Barbara[60] in the truly
barbarous language of ancient logicians; and the first figure of the
syllogism contained Barbara and three other moods which were esteemed
distinct forms of argument. But it is worthy of notice that, without
any real change in our form of inference, we readily include these
three other moods under Barbara. The negative mood Celarent will be
represented by the example
[60] An explanation of this and other technical terms of the old
logic will be found in my *Elementary Lessons in Logic*, Sixth
Edition, 1876; Macmillan.
Neptune is a planet, (1)
No planet has retrograde motion; (2)
Hence Neptune has not retrograde motion. (3)
If we put A for Neptune, B for planet, and C for “having retrograde
motion,” then by the corresponding negative term c, we denote “not
having retrograde motion.” The premises now fall into the forms
A = AB (1)
B = B*c*, (2)
and by substitution for B, exactly as before, we obtain
A = AB*c*. (3)
What is called in the old logic a particular conclusion may be deduced
without any real variation in the symbols. Particular quantity is
indicated, as before mentioned (p. 41), by joining to the term an
indefinite adjective of quantity, such as *some*, *a part of*,
*certain*, &c., meaning that an unknown part of the term enters into
the proposition as subject. Considerable doubt and ambiguity arise out
of the question whether the part may not in some cases be the whole,
and in the syllogism at least it must be understood in this sense.[61]
Now, if we take a letter to represent this indefinite part, we need
make no change in our formulæ to express the syllogisms Darii and
Ferio. Consider the example--
[61] *Elementary Lessons in Logic*, pp. 67, 79.
Some metals are of less density than water, (1)
All bodies of less density than water will float
upon the surface of water; hence (2)
Some metals will float upon the surface of
water. (3)
Let
A = some metals,
B = body of less density than water,
C = floating on the surface of water
then the propositions are evidently as before,
A = AB, (1)
B = BC; (2)
hence A = ABC, (3)
Thus the syllogism Darii does not really differ from Barbara. If the
reader prefer it, we can readily employ a distinct symbol for the
indefinite sign of quantity.
Let P = some,
Q = metal,
B and C having the same meanings as before. Then the premises become
PQ = PQB, (1)
B = BC; (2)
hence, by substitution, as before,
PQ = PQBC. (3)
Except that the formulæ look a little more complicated there is no
difference whatever.
The mood Ferio is of exactly the same character as Darii or Barbara,
except that it involves the use of a negative term. Take the example,
Bodies which are equally elastic in all directions do not doubly
refract light;
Some crystals are bodies equally elastic in all directions;
therefore, some crystals do not doubly refract light.
Assigning the letters as follows:--
A = some crystals,
B = bodies equally elastic in all directions,
C = doubly refracting light,
*c* = not doubly refracting light.
Our argument is of the same form as before, and may be concisely stated
in one line,
A = AB = AB*c*.
If it is preferred to put PQ for the indefinite *some crystals*, we have
PQ = PQB = PQB*c*.
The only difference is that the negative term c takes the place of C in
the mood Darii.
*Ellipsis of Terms in Partial Identities.*
The reader will probably have noticed that the conclusion which we
obtain from premises is often more full than that drawn by the old
Aristotelian processes. Thus from “Sodium is a metal,” and “Metals
conduct electricity,” we inferred (p. 55) that “Sodium = sodium, metal,
conducting electricity,” whereas the old logic simply concludes that
“Sodium conducts electricity.” Symbolically, from A = AB, and B = BC,
we get A = ABC, whereas the old logic gets at the most A = AC. It is
therefore well to show that without employing any other principles of
inference than those already described, we may infer A = AC from A =
ABC, though we cannot infer the latter more full and accurate result
from the former. We may show this most simply as follows:--
By the first Law of Thought it is evident that
AA = AA;
and if we have given the proposition A = ABC, we may substitute for
both the A’s in the second side of the above, obtaining
AA = ABC . ABC.
But from the property of logical symbols expressed in the Law of
Simplicity (p. 33) some of the repeated letters may be made to
coalesce, and we have
A = ABC . C.
Substituting again for ABC its equivalent A, we obtain
A = AC,
the desired result.
By a similar process of reasoning it may be shown that we can always
drop out any term appearing in one member of a proposition, provided
that we substitute for it the whole of the other member. This
process was described in my first logical Essay,[62] as *Intrinsic
Elimination*, but it might perhaps be better entitled the *Ellipsis
of Terms*. It enables us to get rid of needless terms by strict
substitutive reasoning.
[62] *Pure Logic*, p. 19.
*Inference of a Simple from Two Partial Identities.*
Two terms may be connected together by two partial identities in yet
another manner, and a case of inference then arises which is of the
highest importance. In the two premises
A = AB (1)
B = AB (2)
the second member of each is the same; so that we can by obvious
substitution obtain
A = B.
Thus, in plain geometry we readily prove that “Every equilateral
triangle is also an equiangular triangle,” and we can with equal ease
prove that “Every equiangular triangle is an equilateral triangle.”
Thence by substitution, as explained above, we pass to the simple
identity,
Equilateral triangle = equiangular triangle.
We thus prove that one class of triangles is entirely identical with
another class; that is to say, they differ only in our way of naming
and regarding them.
The great importance of this process of inference arises from the
fact that the conclusion is more simple and general than either of
the premises, and contains as much information as both of them put
together. It is on this account constantly employed in inductive
investigation, as will afterwards be more fully explained, and it is
the natural mode by which we arrive at a conviction of the truth of
simple identities as existing between classes of numerous objects.
*Inference of a Limited from Two Partial Identities.*
We have considered some arguments which are of the type treated by
Aristotle in the first figure of the syllogism. But there exist two
other types of argument which employ a pair of partial identities. If
our premises are as shown in these symbols,
B = AB (1)
B = CB, (2)
we may substitute for B either by (1) in (2) or by (2) in (1), and by
both modes we obtain the conclusion
AB = CB, (3)
a proposition of the kind which we have called a limited identity
(p. 42). Thus, for example,
Potassium = potassium metal (1)
Potassium = potassium capable of floating on water; (2)
hence
Potassium metal = potassium capable of floating on water. (3)
This is really a syllogism of the mood Darapti in the third figure,
except that we obtain a conclusion of a more exact character than
the old syllogism gives. From the premises “Potassium is a metal”
and “Potassium floats on water,” Aristotle would have inferred that
“Some metals float on water.” But if inquiry were made what the “some
metals” are, the answer would certainly be “Metal which is potassium.”
Hence Aristotle’s conclusion simply leaves out some of the information
afforded in the premises. It even leaves us open to interpret the
*some metals* in a wider sense than we are warranted in doing. From
these distinct defects of the old syllogism the process of substitution
is free, and the new process only incurs the possible objection of
being tediously minute and accurate.
*Miscellaneous Forms of Deductive Inference.*
The more common forms of deductive reasoning having been exhibited
and demonstrated on the principle of substitution, there still remain
many, in fact an indefinite number, which may be explained with nearly
equal ease. Such as involve the use of disjunctive propositions will
be described in a later chapter, and several of the syllogistic moods
which include negative terms will be more conveniently treated after
we have introduced the symbolic use of the second and third laws of
thought.
We sometimes meet with a chain of propositions which allow of repeated
substitution, and form an argument called in the old logic a Sorites.
Take, for instance, the premises
Iron is a metal, (1)
Metals are good conductors of electricity, (2)
Good conductors of electricity are useful for
telegraphic purposes. (3)
It obviously follows that
Iron is useful for telegraphic purposes. (4)
Now if we take our letters thus,
A = Iron, B = metal, C = good conductor of electricity, D = useful
for telegraphic purposes,
the premises will assume the forms
A = AB, (1)
B = BC, (2)
C = CD. (3)
For B in (1) we can substitute its equivalent in (2) obtaining, as
before,
A = ABC.
Substituting for C in this intermediate result its equivalent as given
in (3), we obtain the complete conclusion
A = ABCD. (4)
The full interpretation is that *Iron is iron, metal, good conductor of
electricity, useful for telegraphic purposes*, which is abridged in
common language by the ellipsis of the circumstances which are not of
immediate importance.
Instead of all the propositions being exactly of the same kind as
in the last example, we may have a series of premises of various
character; for instance,
Common salt is sodium chloride, (1)
Sodium chloride crystallizes in a cubical form, (2)
What crystallizes in a cubical form does not possess the power of
double refraction; (3)
it will follow that
Common salt does not possess the power of double refraction. (4)
Taking our letter-terms thus,
A = Common salt,
B = Sodium chloride,
C = Crystallizing in a cubical form,
D = Possessing the power of double refraction,
we may state the premises in the forms
A = B, (1)
B = BC, (2)
C = C*d*. (3)
Substituting by (3) in (2) and then by (2) as thus altered in (1) we
obtain
A = BC*d*, (4)
which is a more precise version of the common conclusion.
We often meet with a series of propositions describing the qualities or
circumstances of the one same thing, and we may combine them all into
one proposition by the process of substitution. This case is, in fact,
that which Dr. Thomson has called “Immediate Inference by the sum of
several predicates,” and his example will serve my purpose well.[63]
He describes copper as “A metal--of a red colour--and disagreeable
smell--and taste--all the preparations of which are poisonous--which is
highly malleable--ductile--and tenacious--with a specific gravity of
about 8.83.” If we assign the letter A to copper, and the succeeding
letters of the alphabet in succession to the series of predicates, we
have nine distinct statements, of the form A = AB (1) A = AC (2) A = AD
(3) ... A = AK (9). We can readily combine these propositions into one
by substituting for A in the second side of (1) its expression in (2).
We thus get
[63] *An Outline of the Necessary Laws of Thought*, Fifth Ed. p. 161.
A = ABC,
and by repeating the process over and over again we obviously get the
single proposition
A = ABCD ... JK.
But Dr. Thomson is mistaken in supposing that we can obtain in
this manner a *definition* of copper. Strictly speaking, the above
proposition is only a *description* of copper, and all the ordinary descriptions
of substances in scientific works may be summed up in this form. Thus
we may assert of the organic substances called Paraffins that they are
all saturated hydrocarbons, incapable of uniting with other substances,
produced by heating the alcoholic iodides with zinc, and so on. It may
be shown that no amount of ordinary description can be equivalent to a
definition of any substance.
*Fallacies.*
I have hitherto been engaged in showing that all the forms of
reasoning of the old syllogistic logic, and an indefinite number of
other forms in addition, may be readily and clearly explained on the
single principle of substitution. It is now desirable to show that the
same principle will prevent us falling into fallacies. So long as we
exactly observe the one rule of substitution of equivalents it will
be impossible to commit a *paralogism*, that is to break any one of
the elaborate rules of the ancient system. The one new rule is thus
proved to be as powerful as the six, eight, or more rules by which the
correctness of syllogistic reasoning was guarded.
It was a fundamental rule, for instance, that two negative premises
could give no conclusion. If we take the propositions
Granite is not a sedimentary rock, (1)
Basalt is not a sedimentary rock, (2)
we ought not to be able to draw any inference concerning the relation
between granite and basalt. Taking our letter-terms thus:
A = granite, B = sedimentary rock, C = basalt,
the premises may be expressed in the forms
A ~ B, (1)
C ~ B. (2)
We have in this form two statements of difference; but the principle
of inference can only work with a statement of agreement or identity
(p. 63). Thus our rule gives us no power whatever of drawing any
inference; this is exactly in accordance with the fifth rule of the
syllogism.
It is to be remembered, indeed, that we claim the power of always
turning a negative proposition into an affirmative one (p. 45); and it
might seem that the old rule against negative premises would thus be
circumvented. Let us try. The premises (1) and (2) when affirmatively
stated take the forms
A = A*b* (1)
C = C*b*. (2)
The reader will find it impossible by the rule of substitution to
discover a relation between A and C. Three terms occur in the above
premises, namely A, *b*, and C; but they are so combined that no term
occurring in one has its exact equivalent stated in the other. No
substitution can therefore be made, and the principle of the fifth rule
of the syllogism holds true. Fallacy is impossible.
It would be a mistake, however, to suppose that the mere occurrence of
negative terms in both premises of a syllogism renders them incapable
of yielding a conclusion. The old rule informed us that from two
negative premises no conclusion could be drawn, but it is a fact that
the rule in this bare form does not hold universally true; and I am not
aware that any precise explanation has been given of the conditions
under which it is or is not imperative. Consider the following example:
Whatever is not metallic is not capable of powerful
magnetic influence, (1)
Carbon is not metallic, (2)
Therefore, carbon is not capable of powerful magnetic
influence. (3)
Here we have two distinctly negative premises (1) and (2), and yet they
yield a perfectly valid negative conclusion (3). The syllogistic rule
is actually falsified in its bare and general statement. In this and
many other cases we can convert the propositions into affirmative ones
which will yield a conclusion by substitution without any difficulty.
To show this let
A = carbon, B = metallic,
C = capable of powerful magnetic influence.
The premises readily take the forms
*b* = *bc*, (1)
A = A*b*, (2)
and substitution for *b* in (2) by means of (1) gives the conclusion
A = A*bc*. (3)
Our principle of inference then includes the rule of negative premises
whenever it is true, and discriminates correctly between the cases
where it does and does not hold true.
The paralogism, anciently called *the Fallacy of Undistributed Middle*,
is also easily exhibited and infallibly avoided by our system. Let the
premises be
Hydrogen is an element, (1)
All metals are elements. (2)
According to the syllogistic rules the middle term “element” is here
undistributed, and no conclusion can be obtained; we cannot tell then
whether hydrogen is or is not a metal. Represent the terms as follows
A = hydrogen,
B = element,
C = metal.
The premises then become
A = AB, (1)
C = CB. (2)
The reader will here, as in a former page (p. 62), find it impossible
to make any substitution. The only term which occurs in both premises
is B, but it is differently combined in the two premises. For B we
must not substitute A, which is equivalent to AB, not to B. Nor must
we confuse together CB and AB, which, though they contain one common
letter, are different aggregate terms. The rule of substitution gives
us no right to decompose combinations; and if we adhere rigidly to the
rule, that if two terms are stated to be equivalent we may substitute
one for the other, we cannot commit the fallacy. It is apparent that
the form of premises stated above is the same as that which we obtained
by translating two negative premises into the affirmative form.
The old fallacy, technically called the *Illicit Process of the Major
Term*, is more easy to commit and more difficult to detect than any
other breach of the syllogistic rules. In our system it could hardly
occur. From the premises
All planets are subject to gravity, (1)
Fixed stars are not planets, (2)
we might inadvertently but fallaciously infer that, “Fixed stars are
not subject to gravity.” To reduce the premises to symbolic form, let
A = planet
B = fixed star
C = subject to gravity;
then we have the propositions
A = AC (1)
B = B*a*. (2)
The reader will try in vain to produce from these premises by
legitimate substitution any relation between B and C; he could not then
commit the fallacy of asserting that B is not C.
There remain two other kinds of paralogism, commonly known as the
fallacy of Four Terms and the Illicit Process of the Minor Term. They
are so evidently impossible while we obey the rule of the substitution
of equivalents, that it is not necessary to give any illustrations.
When there are four distinct terms in two propositions as in A = B
and C = D, there could evidently be no opening for substitution. As
to the Illicit Process of the Minor Term it consists in a flagrant
substitution for a term of another wider term which is not known to be
equivalent to it, and which is therefore not allowed by our rule to be
substituted for it.
CHAPTER V.
DISJUNCTIVE PROPOSITIONS.
In the previous chapter I have exhibited various cases of deductive
reasoning by the process of substitution, avoiding the introduction of
disjunctive propositions; but we cannot long defer the consideration of
this more complex class of identities. General terms arise, as we have
seen (p. 24), from classifying or mentally uniting together all objects
which agree in certain qualities, the value of this union consisting in
the fact that the power of knowledge is multiplied thereby. In forming
such classes or general notions, we overlook or abstract the points of
difference which exist between the objects joined together, and fix our
attention only on the points of agreement. But every process of thought
may be said to have its inverse process, which consists in undoing the
effects of the direct process. Just as division undoes multiplication,
and evolution undoes involution, so we must have a process which undoes
generalization, or the operation of forming general notions. This
inverse process will consist in distinguishing the separate objects or
minor classes which are the constituent parts of any wider class. If
we mentally unite together certain objects visible in the sky and call
them planets, we shall afterwards need to distinguish the contents of
this general notion, which we do in the disjunctive proposition--
A planet is either Mercury or Venus or the Earth or ... or Neptune.
Having formed the very wide class “vertebrate animal,” we may specify
its subordinate classes thus:--“A vertebrate animal is either a
mammal, bird, reptile, or fish.” Nor is there any limit to the
number of possible alternatives. “An exogenous plant is either a
ranunculus, a poppy, a crucifer, a rose, or it belongs to some one
of the other seventy natural orders of exogens at present recognized
by botanists.” A cathedral church in England must be either that of
London, Canterbury, Winchester, Salisbury, Manchester, or of one of
about twenty-four cities possessing such churches. And if we were to
attempt to specify the meaning of the term “star,” we should require
to enumerate as alternatives, not only the many thousands of stars
recorded in catalogues, but the many millions unnamed.
Whenever we thus distinguish the parts of a general notion we employ
a disjunctive proposition, in at least one side of which are several
alternatives joined by the so-called disjunctive conjunction or,
a contracted form of *other*. There must be some relation between
the parts thus connected in one proposition; we may call it the
*disjunctive* or *alternative* relation, and we must carefully inquire
into its nature. This relation is that of ignorance and doubt, giving
rise to choice. Whenever we classify and abstract we must open the way
to such uncertainty. By fixing our attention on certain attributes to
the exclusion of others, we necessarily leave it doubtful what those
other attributes are. The term “molar tooth” bears upon the face of
it that it is a part of the wider term “tooth.” But if we meet with
the simple term “tooth” there is nothing to indicate whether it is
an incisor, a canine, or a molar tooth. This doubt, however, may be
resolved by further information, and we have to consider what are the
appropriate logical processes for treating disjunctive propositions in
connection with other propositions disjunctive or otherwise.
*Expression of the Alternative Relation.*
In order to represent disjunctive propositions with convenience we
require a sign of the alternative relation, equivalent to one meaning
at least of the little conjunction *or* so frequently used in common
language. I propose to use for this purpose the symbol ꖌ. In my first
logical essay I followed the practice of Boole and adopted the sign
+; but this sign should not be employed unless there exists exact
analogy between mathematical addition and logical alternation. We shall
find that the analogy is imperfect, and that there is such profound
difference between logical and mathematical terms as should prevent
our uniting them by the same symbol. Accordingly I have chosen a sign
ꖌ, which seems aptly to suggest whatever degree of analogy may exist
without implying more. The exact meaning of the symbol we will now
proceed to investigate.
*Nature of the Alternative Relation.*
Before treating disjunctive propositions it is indispensable to decide
whether the alternatives must be considered exclusive or unexclusive.
By *exclusive alternatives* we mean those which cannot contain the same
things. If we say “Arches are circular or pointed,” it is certainly to
be understood that the same arch cannot be described as both circular
and pointed. Many examples, on the other hand, can readily be suggested
in which two or more alternatives may hold true of the same object. Thus
Luminous bodies are self-luminous or luminous by reflection.
It is undoubtedly possible, by the laws of optics, that the same
surface may at one and the same moment give off light of its own and
reflect light from other bodies. We speak familiarly of *deaf or dumb*
persons, knowing that the majority of those who are deaf from birth are
also dumb.
There can be no doubt that in a great many cases, perhaps the greater
number of cases, alternatives are exclusive as a matter of fact. Any
one number is incompatible with any other; one point of time or place
is exclusive of all others. Roger Bacon died either in 1284 or 1292; it
is certain that he could not die in both years. Henry Fielding was born
either in Dublin or Somersetshire; he could not be born in both places.
There is so much more precision and clearness in the use of exclusive
alternatives that we ought doubtless to select them when possible. Old
works on logic accordingly contained a rule directing that the *Membra
dividentia*, the parts of a division or the constituent species of a
genus, should be exclusive of each other.
It is no doubt owing to the great prevalence and convenience of
exclusive divisions that the majority of logicians have held it
necessary to make every alternative in a disjunctive proposition
exclusive of every other one. Aquinas considered that when this was not
the case the proposition was actually *false*, and Kant adopted the
same opinion.[64] A multitude of statements to the same effect might
readily be quoted, and if the question were to be determined by the
weight of historical evidence, it would certainly go against my view.
Among recent logicians Hamilton, as well as Boole, took the exclusive
side. But there are authorities to the opposite effect. Whately,
Mansel, and J. S. Mill have all pointed out that we may often treat
alternatives as *Compossible*, or true at the same time. Whately gives
us an example,[65] “Virtue tends to procure us either the esteem of
mankind, or the favour of God,” and he adds--“Here both members are
true, and consequently from one being affirmed we are not authorized to
deny the other. Of course we are left to conjecture in each case, from
the context, whether it is meant to be implied that the members are
or are not exclusive.” Mansel says,[66] “*We may happen to know* that
two alternatives cannot be true together, so that the affirmation of
the second necessitates the denial of the first; but this, as Boethius
observes, is a *material*, not a *formal* consequence.” Mill has also
pointed out the absurdities which would arise from always interpreting
alternatives as exclusive. “If we assert,” he says,[67] “that a man
who has acted in some particular way must be either a knave or a fool,
we by no means assert, or intend to assert, that he cannot be both.”
Again, “to make an entirely unselfish use of despotic power, a man must
be either a saint or a philosopher.... Does the disjunctive premise
necessarily imply, or must it be construed as supposing, that the same
person cannot be both a saint and a philosopher? Such a construction
would be ridiculous.”
[64] Mansel’s *Aldrich*, p. 103, and *Prolegomena Logica*, p. 221.
[65] *Elements of Logic*, Book II. chap. iv. sect. 4.
[66] Aldrich, *Artis Logicæ Rudimenta*, p. 104.
[67] *Examination of Sir W. Hamilton’s Philosophy*, pp. 452–454.
I discuss this subject fully because it is really the point which
separates my logical system from that of Boole. In his *Laws of
Thought* (p. 32) he expressly says, “In strictness, the words ‘and,’
‘or,’ interposed between the terms descriptive of two or more classes
of objects, imply that those classes are quite distinct, so that no
member of one is found in another.” This I altogether dispute. In the
ordinary use of these conjunctions we do not join distinct terms only;
and when terms so joined do prove to be logically distinct, it is by
virtue of a *tacit premise*, something in the meaning of the names and
our knowledge of them, which teaches us that they are distinct. If our
knowledge of the meanings of the words joined is defective it will
often be impossible to decide whether terms joined by conjunctions are
exclusive or not.
In the sentence “Repentance is not a single act, but a habit or
virtue,” it cannot be implied that a virtue is not a habit; by
Aristotle’s definition it is. Milton has the expression in one of
his sonnets, “Unstain’d by gold or fee,” where it is obvious that if
the fee is not always gold, the gold is meant to be a fee or bribe.
Tennyson has the expression “wreath or anadem.” Most readers would
be quite uncertain whether a wreath may be an anadem, or an anadem a
wreath, or whether they are quite distinct or quite the same. From
Darwin’s *Origin of Species*, I take the expression, “When we see any
*part or organ* developed in a remarkable *degree or manner*.” In this,
*or* is used twice, and neither time exclusively. For if *part* and
*organ* are not synonymous, at any rate an organ is a part. And it
is obvious that a part may be developed at the same time both in an
extraordinary degree and an extraordinary manner, although such cases
may be comparatively rare.
From a careful examination of ordinary writings, it will thus be found
that the meanings of terms joined by “and,” “or” vary from absolute
identity up to absolute contrariety. There is no logical condition of
distinctness at all, and when we do choose exclusive alternatives, it
is because our subject demands it. The matter, not the form of an
expression, points out whether terms are exclusive or not.[68] In
bills, policies, and other kinds of legal documents, it is sometimes
necessary to express very distinctly that alternatives are not
exclusive. The form and/or is then used, and, as Mr. J. J. Murphy has
remarked, this form coincides exactly in meaning with the symbol ꖌ.
[68] *Pure Logic*, pp 76, 77.
In the first edition of this work (vol. i., p. 81), I took the
disjunctive proposition “Matter is solid, or liquid, or gaseous,” and
treated it as an instance of exclusive alternatives, remarking that the
same portion of matter cannot be at once solid and liquid, properly
speaking, and that still less can we suppose it to be solid and
gaseous, or solid, liquid, and gaseous all at the same time. But the
experiments of Professor Andrews show that, under certain conditions
of temperature and pressure, there is no abrupt change from the liquid
to the gaseous state. The same substance may be in such a state as to
be indifferently described as liquid and gaseous. In many cases, too,
the transition from solid to liquid is gradual, so that the properties
of solidity are at least partially joined with those of liquidity.
The proposition then, instead of being an instance of exclusive
alternatives, seems to afford an excellent instance to the opposite
effect. When such doubts can arise, it is evidently impossible to treat
alternatives as absolutely exclusive by the logical nature of the
relation. It becomes purely a question of the matter of the proposition.
The question, as we shall afterwards see more fully, is one of
the greatest theoretical importance, because it concerns the true
distinction between the sciences of Logic and Mathematics. It is the
foundation of number that every unit shall be distinct from every other
unit; but Boole imported the conditions of number into the science of
Logic, and produced a system which, though wonderful in its results,
was not a system of logic at all.
*Laws of the Disjunctive Relation.*
In considering the combination or synthesis of terms (p. 30), we found
that certain laws, those of Simplicity and Commutativeness, must be
observed. In uniting terms by the disjunctive symbol we shall find that
the same or closely similar laws hold true. The alternatives of either
member of a disjunctive proposition are certainly commutative. Just as
we cannot properly distinguish between *rich and rare gems* and *rare
and rich gems*, so we must consider as identical the expression *rich
or rare gems*, and *rare or rich gems*. In our symbolic language we may
say
A ꖌ B = B ꖌ A.
The order of statement, in short, has no effect upon the meaning of an
aggregate of alternatives, so that the Law of Commutativeness holds
true of the disjunctive symbol.
As we have admitted the possibility of joining as alternatives terms
which are not really different, the question arises, How shall we treat
two or more alternatives when they are clearly shown to be the same?
If we have it asserted that P is Q or R, and it is afterwards proved
that Q is but another name for R, the result is that P is either R or
R. How shall we interpret such a statement? What would be the meaning,
for instance, of “wreath or anadem” if, on referring to a dictionary,
we found *anadem* described as a wreath? I take it to be self-evident
that the meaning would then become simply “wreath.” Accordingly we may
affirm the general law
A ꖌ A = A.
Any number of identical alternatives may always be reduced to, and are
logically equivalent to, any one of those alternatives. This is a law
which distinguishes mathematical terms from logical terms, because
it obviously does not apply to the former. I propose to call it the
*Law of Unity*, because it must really be involved in any definition
of a mathematical unit. This law is closely analogous to the Law of
Simplicity, AA = A; and the nature of the connection is worthy of
attention.
Few or no logicians except De Morgan have adequately noticed the close
relation between combined and disjunctive terms, namely, that every
disjunctive term is the negative of a corresponding combined term, and
*vice versâ*. Consider the term
Malleable dense metal.
How shall we describe the class of things which are not
malleable-dense-metals? Whatever is included under that term must have
all the qualities of malleability, denseness, and metallicity. Wherever
any one or more of the qualities is wanting, the combined term will not
apply. Hence the negative of the whole term is
Not-malleable or not-dense or not-metallic.
In the above the conjunction *or* must clearly be interpreted
as unexclusive; for there may readily be objects which are both
not-malleable, and not-dense, and perhaps not-metallic at the same
time. If in fact we were required to use *or* in a strictly exclusive
manner, it would be requisite to specify seven distinct alternatives
in order to describe the negative of a combination of three terms. The
negatives of four or five terms would consist of fifteen or thirty-one
alternatives. This consideration alone is sufficient to prove that the
meaning of *or* cannot be always exclusive in common language.
Expressed symbolically, we may say that the negative of
ABC
is not-A or not-B or not-C;
that is, *a* ꖌ *b* ꖌ *c*.
Reciprocally the negative of
P ꖌ Q ꖌ R
is *pqr*.
Every disjunctive term, then, is the negative of a combined term, and
*vice versâ*.
Apply this result to the combined term AAA, and its negative is
*a* ꖌ *a* ꖌ *a*.
Since AAA is by the Law of Simplicity equivalent to A, so *a* ꖌ *a* ꖌ
*a* must be equivalent to *a*, and the Law of Unity holds true. Each
law thus necessarily presupposes the other.
*Symbolic expression of the Law of Duality.*
We may now employ our symbol of alternation to express in a clear and
formal manner the third Fundamental Law of Thought, which I have called
the Law of Duality (p. 6). Taking A to represent any class or object
or quality, and B any other class, object or quality, we may always
assert that A either agrees with B, or does not agree. Thus we may say
A = AB ꖌ A*b*.
This is a formula which will henceforth be constantly employed, and it
lies at the basis of reasoning.
The reader may perhaps wish to know why A is inserted in both
alternatives of the second member of the identity, and why the law is
not stated in the form
A = B ꖌ *b*.
But if he will consider the contents of the last section (p. 73), he
will see that the latter expression cannot be correct, otherwise no
term could have a corresponding negative term. For the negative of B
ꖌ *b* is *b*B, or a self-contradictory term; thus if A were identical
with B ꖌ *b*, its negative *a* would be non-existent. To say the least,
this result would in most cases be an absurd one, and I see much reason
to think that in a strictly logical point of view it would always be
absurd. In all probability we ought to assume as a fundamental logical
axiom that *every term has its negative in thought*. We cannot think at
all without separating what we think about from other things, and these
things necessarily form the negative notion.[69] It follows that any
proposition of the form A = B ꖌ *b* is just as self-contradictory as
one of the form A = B*b*.
[69] *Pure Logic*, p. 65. See also the criticism of this point by De
Morgan in the *Athenæum*, No. 1892, 30th January, 1864; p. 155.
It is convenient to recapitulate in this place the three Laws of
Thought in their symbolic form, thus
Law of Identity A = A.
Law of Contradiction A*a* = 0.
Law of Duality A = AB ꖌ A*b*.
*Various Forms of the Disjunctive Proposition.*
Disjunctive propositions may occur in a great variety of forms, of
which the old logicians took insufficient notice. There may be any
number of alternatives, each of which may be a combination of any
number of simple terms. A proposition, again, may be disjunctive in one
or both members. The proposition
Solids or liquids or gases are electrics or conductors of electricity
is an example of the doubly disjunctive form. The meaning of such a
proposition is that whatever falls under any one or more alternatives
on one side must fall under one or more alternatives on the other side.
From what has been said before, it is apparent that the proposition
A ꖌ B = C ꖌ D
will correspond to
*ab* = *cd*,
each member of the latter being the negative of a member of the former
proposition.
As an instance of a complex disjunctive proposition I may give Senior’s
definition of wealth, which, briefly stated, amounts to the proposition
“Wealth is what is transferable, limited in supply, and either
productive of pleasure or preventive of pain.”[70]
[70] Boole’s *Laws of Thought*, p. 106. Jevons’ *Pure Logic*, p. 69.
Let A = wealth
B = transferable
C = limited in supply
D = productive of pleasure
E = preventive of pain.
The definition takes the form
A = BC(D ꖌ E);
but if we develop the alternatives by a method to be afterwards more
fully considered, it becomes
A = BCDE ꖌ BCD*e* ꖌ BC*d*E.
An example of a still more complex proposition is found in De Morgan’s
writings,[71] as follows:--“He must have been rich, and if not
absolutely mad was weakness itself, subjected either to bad advice or
to most unfavourable circumstances.”
[71] *On the Syllogism*, No. iii. p. 12. Camb. Phil. Trans. vol. x,
part i.
If we assign the letters of the alphabet in succession, thus,
A = he
B = rich
C = absolutely mad
D = weakness itself
E = subjected to bad advice
F = subjected to most unfavourable circumstances,
the proposition will take the form
A = AB{C ꖌ D (E ꖌ F)},
and if we develop the alternatives, expressing some of the different
cases which may happen, we obtain
A = ABC ꖌ AB*c*DEF ꖌ AB*c*DE*f* ꖌ AB*c*D*e*F.
The above gives the strict logical interpretation of the sentence, and
the first alternative ABC is capable of development into eight cases,
according as D, E and F are or are not present. Although from our
knowledge of the matter, we may infer that weakness of character cannot
be asserted of a person absolutely mad, there is no explicit statement
to this effect.
*Inference by Disjunctive Propositions.*
Before we can make a free use of disjunctive propositions in the
processes of inference we must consider how disjunctive terms can be
combined together or with simple terms. In the first place, to combine
a simple term with a disjunctive one, we must combine it with every
alternative of the disjunctive term. A vegetable, for instance, is
either a herb, a shrub, or a tree. Hence an exogenous vegetable is
either an exogenous herb, or an exogenous shrub, or an exogenous tree.
Symbolically stated, this process of combination is as follows,
A(B ꖌ C) = AB ꖌ AC.
Secondly, to combine two disjunctive terms with each other, combine
each alternative of one with each alternative of the other. Since
flowering plants are either exogens or endogens, and are at the
same time either herbs, shrubs or trees, it follows that there are
altogether six alternatives--namely, exogenous herbs, exogenous shrubs,
exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees.
This process of combination is shown in the general form
(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE.
It is hardly necessary to point out that, however numerous the
terms combined, or the alternatives in those terms, we may effect
the combination, provided each alternative is combined with each
alternative of the other terms, as in the algebraic process of
multiplication.
Some processes of deduction may be at once exhibited. We may
always, for instance, unite the same qualifying term to each side
of an identity even though one or both members of the identity be
disjunctive. Thus let
A = B ꖌ C.
Now it is self-evident that
AD = AD,
and in one side of this identity we may for A substitute its equivalent
B ꖌ C, obtaining
AD = BD ꖌ CD.
Since “a gaseous element is either hydrogen, or oxygen, or nitrogen,
or chlorine, or fluorine,” it follows that “a free gaseous element
is either free hydrogen, or free oxygen, or free nitrogen, or free
chlorine, or free fluorine.”
This process of combination will lead to most useful inferences when
the qualifying adjective combined with both sides of the proposition is
a negative of one or more alternatives. Since chlorine is a coloured
gas, we may infer that “a colourless gaseous element is either
(colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative
chlorine disappears because colourless chlorine does not exist. Again,
since “a tooth is either an incisor, canine, bicuspid, or molar,”
it follows that “a not-incisor tooth is either canine, bicuspid,
or molar.” The general rule is that from the denial of any of the
alternatives the affirmation of the remainder can be inferred. Now this
result clearly follows from our process of substitution; for if we have
the proposition
A = B ꖌ C ꖌ D,
and we insert this expression for A on one side of the self-evident
identity
A*b* = A*b*,
we obtain A*b* = AB*b* ꖌ A*b*C ꖌ A*b*D;
and, as the first of the three alternatives is self-contradictory, we
strike it out according to the law of contradiction: there remains
A*b* = A*b*C ꖌ A*b*D.
Thus our system fully includes and explains that mood of the
Disjunctive Syllogism technically called the *modus tollendo ponens*.
But the reader must carefully observe that the Disjunctive Syllogism of
the mood *ponendo tollens*, which affirms one alternative, and thence
infers the denial of the rest, cannot be held true in this system. If I
say, indeed, that
Water is either salt or fresh water,
it seems evident that “water which is salt is not fresh.” But this
inference really proceeds from our knowledge that water cannot be at
once salt and fresh. This inconsistency of the alternatives, as I have
fully shown, will not always hold. Thus, if I say
Gems are either rare stones or beautiful stones, (1)
it will obviously not follow that
A rare gem is not a beautiful stone, (2)
nor that
A beautiful gem is not a rare stone. (3)
Our symbolic method gives only true conclusions; for if we take
A = gem
B = rare stone
C = beautiful stone,
the proposition (1) is of the form
A = B ꖌ C
hence AB = B ꖌ BC
and AC = BC ꖌ C;
but these inferences are not equivalent to the false ones (2) and (3).
We can readily represent disjunctive reasoning by the *modus ponendo
tollens*, when it is valid, by expressing the inconsistency of the
alternatives explicitly. Thus if we resort to our instance of
Water is either salt or fresh,
and take
A = Water B = salt C = fresh,
then the premise is apparently of the form
A = AB ꖌ AC;
but in reality there is an unexpressed condition that “what is salt
is not fresh,” from which follows, by a process of inference to be
afterwards described, that “what is fresh is not salt.” We have then,
in letter-terms, the two propositions
B = B*c*
C = *b*C.
If we substitute these descriptions in the original proposition, we
obtain /* A = AB*c* ꖌ A*b*C; */
uniting B to each side we infer
AB = AB*c* ꖌ AB*b*C
or AB = AB*c*;
that is,
Water which is salt is water salt and not fresh.
I should weary the reader if I attempted to illustrate the multitude of
forms which disjunctive reasoning may take; and as in the next chapter
we shall be constantly treating the subject, I must here restrict
myself to a single instance. A very common process of reasoning
consists in the determination of the name of a thing by the successive
exclusion of alternatives, a process called by the old name *abscissio
infiniti*. Take the case:
Red-coloured metal is either copper or gold (1)
Copper is dissolved by nitric acid (2)
This specimen is red-coloured metal (3)
This specimen is not dissolved by nitric acid (4)
Therefore, this specimen consists of gold (5)
Let us assign the letter-symbols thus--
A = this specimen
B = red-coloured metal
C = copper
D = gold
E = dissolved by nitric acid.
Assuming that the alternatives copper or gold are intended to be
exclusive, as just explained in the case of fresh and salt water, the
premises may be stated in the forms
B = BC*d* ꖌ B*c*D (1)
C = CE (2)
A = AB (3)
A = A*e* (4)
Substituting for C in (1) by means of (2) we get
B = BC*d*E ꖌ B*c*D
From (3) and (4) we may infer likewise
A = AB*e*
and if in this we substitute for B its equivalent just stated, it
follows that
A = ABC*d*E*e* ꖌ AB*c*D*e*
The first of the alternatives being contradictory the result is
A = AB*c*D*e*
which contains a full description of “this specimen,” as furnished
in the premises, but by ellipsis asserts that it is gold. It will be
observed that in the symbolic expression (1) I have explicitly stated
what is certainly implied, that copper is not gold, and gold not
copper, without which condition the inference would not hold good.
CHAPTER VI.
THE INDIRECT METHOD OF INFERENCE.
The forms of deductive reasoning as yet considered, are mostly cases
of Direct Deduction as distinguished from those which we are now about
to treat. The method of Indirect Deduction may be described as that
which points out what a thing is, by showing that it cannot be anything
else. We can define a certain space upon a map, either by colouring
that space, or by colouring all except the space; the first mode is
positive, the second negative. The difference, it will be readily seen,
is exactly analogous to that between the direct and indirect modes
of proof in geometry. Euclid often shows that two lines are equal,
by showing that they cannot be unequal, and the proof rests upon the
known number of alternatives, greater, equal or less, which are alone
conceivable. In other cases, as for instance in the seventh proposition
of the first book, he shows that two lines must meet in a particular
point, by showing that they cannot meet elsewhere.
In logic we can always define with certainty the utmost number of
alternatives which are conceivable. The Law of Duality (pp. 6, 74)
enables us always to assert that any quality or circumstance whatsoever
is either present or absent. Whatever may be the meaning of the terms A
and B it is certainly true that
A = AB ꖌ A*b*
B = AB ꖌ *a*B.
These are universal tacit premises which may be employed in the
solution of every problem, and which are such invariable and necessary
conditions of all thought, that they need not be specially laid down.
The Law of Contradiction is a further condition of all thought and of
all logical symbols; it enables, and in fact obliges, us to reject from
further consideration all terms which imply the presence and absence
of the same quality. Now, whenever we bring both these Laws of Thought
into explicit action by the method of substitution, we employ the
Indirect Method of Inference. It will be found that we can treat not
only those arguments already exhibited according to the direct method,
but we can include an infinite multitude of other arguments which are
incapable of solution by any other means.
Some philosophers, especially those of France, have held that the
Indirect Method of Proof has a certain inferiority to the direct
method, which should prevent our using it except when obliged. But
there are many truths which we can prove only indirectly. We can
prove that a number is a prime only by the purely indirect method of
showing that it is not any of the numbers which have divisors, and the
remarkable process known as Eratosthenes’ Sieve is the only mode by
which we can select the prime numbers.[72] It bears a strong analogy to
the indirect method here to be described. We can prove that the side
and diameter of a square are incommensurable, but only in the negative
or indirect manner, by showing that the contrary supposition inevitably
leads to contradiction.[73] Many other demonstrations in various
branches of the mathematical sciences proceed upon a like method.
Now, if there is only one important truth which must be, and can only
be, proved indirectly, we may say that the process is a necessary and
sufficient one, and the question of its comparative excellence or
usefulness is not worth discussion. As a matter of fact I believe that
nearly half our logical conclusions rest upon its employment.
[72] See Horsley, *Philosophical Transactions*, 1772; vol. lxii.
p. 327. Montucla, *Histoire des Mathematiques*, vol. i. p. 239.
*Penny Cyclopædia*, article “Eratosthenes.”
[73] Euclid, Book x. Prop. 117.
*Simple Illustrations.*
In tracing out the powers and results of this method, we will begin
with the simplest possible instance. Let us take a proposition of the
common form, A = AB, say,
*A Metal is an Element,*
and let us investigate its full meaning. Any person who has had the
least logical training, is aware that we can draw from the above
proposition an apparently different one, namely,
*A Not-element is a Not-metal.*
While some logicians, as for instance De Morgan,[74] have considered
the relation of these two propositions to be purely self-evident, and
neither needing nor allowing analysis, a great many more persons, as
I have observed while teaching logic, are at first unable to perceive
the close connection between them. I believe that a true and complete
system of logic will furnish a clear analysis of this process, which
has been called *Contrapositive Conversion*; the full process is as
follows:--
[74] *Philosophical Magazine*, December 1852; Fourth Series, vol. iv.
p. 435, “On Indirect Demonstration.”
Firstly, by the Law of Duality we know that
*Not-element is either Metal or Not-metal.*
If it be metal, we know that it is by the premise *an element*; we
should thus be supposing that the same thing is an element and a
not-element, which is in opposition to the Law of Contradiction.
According to the only other alternative, then, the not-element must be
a not-metal.
To represent this process of inference symbolically we take the premise
in the form
A = AB. (1)
We observe that by the Law of Duality the term not-B is thus described
*b* = A*b* ꖌ *ab*. (2)
For A in this proposition we substitute its description as given in
(1), obtaining
*b* = AB*b* ꖌ *ab*.
But according to the Law of Contradiction the term AB*b* must be
excluded from thought, or
AB*b* = 0.
Hence it results that *b* is either nothing at all, or it is *ab*; and
the conclusion is
*b* = *ab*.
As it will often be necessary to refer to a conclusion of this kind I
shall call it, as is usual, the *Contrapositive Proposition* of the
original. The reader need hardly be cautioned to observe that from all
A’s are B’s it does not follow that all not-A’s are not-B’s. For by the
Law of Duality we have
*a* = *a*B ꖌ *ab*,
and it will not be found possible to make any substitution in this by
our original premise A = AB. It still remains doubtful, therefore,
whether not-metal is element or not-element.
The proof of the Contrapositive Proposition given above is exactly the
same as that which Euclid applies in the case of geometrical notions.
De Morgan describes Euclid’s process as follows[75]:--“From every not-B
is not-A he produces Every A is B, thus: If it be possible, let this A
be not-B, but every not-B is not-A, therefore this A is not-A, which
is absurd: whence every A is B.” Now De Morgan thinks that this proof
is entirely needless, because common logic gives the inference without
the use of any geometrical reasoning. I conceive however that logic
gives the inference only by an indirect process. De Morgan claims “to
see identity in Every A is B and every not-B is not-A, by a process
of thought prior to syllogism.” Whether prior to syllogism or not, I
claim that it is not prior to the laws of thought and the process of
substitutive inference, by which it may be undoubtedly demonstrated.
[75] *Philosophical Magazine*, Dec. 1852; p. 437.
*Employment of the Contrapositive Proposition.*
We can frequently employ the contrapositive form of a proposition by
the method of substitution; and certain moods of the ancient syllogism,
which we have hitherto passed over, may thus be satisfactorily
comprehended in our system. Take for instance the following syllogism
in the mood Camestres:--
“Whales are not true fish; for they do not respire water,
whereas true fish do respire water.”
Let us take
A = whale
B = true fish
C = respiring water
The premises are of the forms
A = A*c* (1)
B = BC (2)
Now, by the process of contraposition we obtain from the second premise
*c* = *bc*
and we can substitute this expression for *c* in (1), obtaining
A = A*bc*
or “Whales are not true fish, not respiring water.”
The mood Cesare does not really differ from Camestres except in the
order of the premises, and it could be exhibited in an exactly similar
manner.
The mood Baroko gave much trouble to the old logicians, who could
not *reduce* it to the first figure in the same manner as the other
moods, and were obliged to invent, specially for it and for Bokardo, a
method of Indirect Reduction closely analogous to the indirect proof
of Euclid. Now these moods require no exceptional treatment in this
system. Let us take as an instance of Baroko, the argument
All heated solids give continuous spectra (1)
Some nebulæ do not give continuous spectra (2)
Therefore, some nebulæ are not heated solids (3)
Treating the little word some as an indeterminate adjective of
selection, to which we assign a symbol like any other adjective, let
A = some
B = nebulæ
C = giving continuous spectra
D = heated solids
The premises then become
D = DC (1)
AB = AB*c* (2)
Now from (1) we obtain by the indirect method the contrapositive
proposition
*c* = *cd*
and if we substitute this expression for *c* in (2) we have
AB = AB*cd*
the full meaning of which is that “some nebulæ do not give continuous
spectra and are not heated solids.”
We might similarly apply the contrapositive in many other instances.
Take the argument, “All fixed stars are self-luminous; but some of the
heavenly bodies are not self-luminous, and are therefore not fixed
stars.” Taking our terms
A = fixed stars
B = self-luminous
C = some
D = heavenly bodies
we have the premises
A = AB, (1)
CD = *b*CD (2)
Now from (1) we can draw the contrapositive
*b* = *ab*
and substituting this expression for *b* in (2) we obtain
CD = *ab*CD
which expresses the conclusion of the argument that some heavenly
bodies are not fixed stars.
*Contrapositive of a Simple Identity.*
The reader should carefully note that when we apply the process of
Indirect Inference to a simple identity of the form
A = B
we may obtain further results. If we wish to know what is the term
not-B, we have as before, by the Law of Duality,
*b* = A*b* ꖌ *ab*
and substituting for A we obtain
*b* = B*b* ꖌ *ab* = *ab*.
But we may now also draw a second contrapositive; for we have
*a* = *a*B ꖌ *ab*,
and substituting for B its equivalent A we have
*a* = *a*A ꖌ *ab* = *ab*.
Hence from the single identity A = B we can draw the two propositions
*a* = *ab*
*b* = *ab*,
and observing that these propositions have a common term *ab* we can
make a new substitution, getting
*a* = *b*.
This result is in strict accordance with the fundamental principles of
inference, and it may be a question whether it is not a self-evident
result, independent of the steps of deduction by which we have reached
it. For where two classes are coincident like A and B, whatever is true
of the one is true of the other; what is excluded from the one must be
excluded from the other similarly. Now as *a* bears to A exactly the
same relation that *b* bears to B, the identity of either pair follows
from the identity of the other pair. In every identity, equality, or
similarity, we may argue from the negative of the one side to the
negative of the other. Thus at ordinary temperatures
Mercury = liquid-metal,
hence obviously
Not-mercury = not liquid-metal;
or since
Sirius = brightest fixed star,
it follows that whatever star is not the brightest is not Sirius, and
*vice versâ*. Every correct definition is of the form A = B, and may
often require to be applied in the equivalent negative form.
Let us take as an illustration of the mode of using this result the
argument following:
Vowels are letters which can be sounded alone, (1)
The letter *w* cannot be sounded alone; (2)
Therefore the letter *w* is not a vowel. (3)
Here we have a definition (1), and a comparison of a thing with that
definition (2), leading to exclusion of the thing from the class
defined.
Taking the terms
A = vowel,
B = letter which can be sounded alone,
C = letter *w*,
the premises are plainly of the forms
A = B, (1)
C = *b*C. (2)
Now by the Indirect method we obtain from (1) the Contrapositive
*b* = *a*,
and inserting in (2) the equivalent for *b* we have
C = *a*C, (3)
or “the letter *w* is not a vowel.”
*Miscellaneous Examples of the Method.*
We can apply the Indirect Method of Inference however many may be the
terms involved or the premises containing those terms. As the working
of the method is best learnt from examples, I will take a case of two
premises forming the syllogism Barbara: thus
Iron is metal (1)
Metal is element. (2)
If we want to ascertain what inference is possible concerning the
term *Iron*, we develop the term by the Law of Duality. Iron must be
either metal or not-metal; iron which is metal must be either element
or not-element; and similarly iron which is not-metal must be either
element or not-element. There are then altogether four alternatives
among which the description of iron must be contained; thus
Iron, metal, element, (α)
Iron, metal, not-element, (β)
Iron, not-metal, element, (γ)
Iron, not-metal, not-element. (δ)
Our first premise informs us that iron is a metal, and if we substitute
this description in (γ) and (δ) we shall have self-contradictory
combinations. Our second premise likewise informs us that metal
is element, and applying this description to (β) we again have
self-contradiction, so that there remains only (α) as a description of
iron--our inference is
Iron = iron, metal, element.
To represent this process of reasoning in general symbols, let
A = iron
B = metal
C = element,
The premises of the problem take the forms
A = AB (1)
B = BC. (2)
By the Law of Duality we have
A = AB ꖌ A*b* (3)
A = AC ꖌ A*c*. (4)
Now, if we insert for A in the second side of (3) its description in
(4), we obtain what I shall call the *development of A with respect to
B and C*, namely
A = ABC ꖌ AB*c* ꖌ A*b*C ꖌ A*bc*. (5)
Wherever the letters A or B appear in the second side of (5) substitute
their equivalents given in (1) and (2), and the results stated at full
length are
A = ABC ꖌ ABC*c* ꖌ AB*b*C ꖌ AB*b*C*c*.
The last three alternatives break the Law of Contradiction, so that
A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC.
This conclusion is, indeed, no more than we could obtain by the
direct process of substitution, that is by substituting for B in (1),
its description in (2) as in p. 55; it is the characteristic of the
Indirect process that it gives all possible logical conclusions, both
those which we have previously obtained, and an immense number of
others or which the ancient logic took little or no account. From the
same premises, for instance, we can obtain a description of the class
*not-element* or *c*. By the Law of Duality we can develop *c* into
four alternatives, thus
*c* = AB*c* ꖌ A*bc* ꖌ *a*B*c* ꖌ *abc*.
If we substitute for A and B as before, we get
*c* = ABC*c* ꖌ AB*bc* ꖌ *a*BC*c* ꖌ *abc*,
and, striking out the terms which break the Law of Contradiction, there
remains
*c* = *abc*,
or what is not element is also not iron and not metal. This Indirect
Method of Inference thus furnishes a complete solution of the following
problem--*Given any number of logical premises or conditions, required
the description of any class of objects, or of any term, as governed by
those conditions.*
The steps of the process of inference may thus be concisely stated--
1. By the Law of Duality develop the utmost number of alternatives
which may exist in the description of the required class or term as
regards the terms involved in the premises.
2. For each term in these alternatives substitute its description as
given in the premises.
3. Strike out every alternative which is then found to break the Law of
Contradiction.
4. The remaining terms may be equated to the term in question as the
desired description.
*Mr. Venn’s Problem.*
The need of some logical method more powerful and comprehensive than
the old logic of Aristotle is strikingly illustrated by Mr. Venn
in his most interesting and able article on Boole’s logic.[76] An
easy example, originally got, as he says, by the aid of my method as
simply described in the *Elementary Lessons in Logic*, was proposed in
examination and lecture-rooms to some hundred and fifty students as a
problem in ordinary logic. It was answered by, at most, five or six
of them. It was afterwards set, as an example on Boole’s method, to
a small class who had attended a few lectures on the nature of these
symbolic methods. It was readily answered by half or more of their
number.
[76] *Mind*; a Quarterly Review of Psychology and Philosophy;
October, 1876, vol. i. p. 487.
The problem was as follows:--“The members of a board were all of them
either bondholders, or shareholders, but not both; and the bondholders
as it happened, were all on the board. What conclusion can be drawn?”
The conclusion wanted is, “No shareholders are bondholders.” Now, as
Mr. Venn says, nothing can look simpler than the following reasoning,
*when stated*:--“There can be no bondholders who are shareholders; for
if there were they must be either on the board, or off it. But they
are not on it, by the first of the given statements; nor off it, by
the second.” Yet from the want of any systematic mode of treating such
a question only five or six of some hundred and fifty students could
succeed in so simple a problem.
By symbolic statement the problem is instantly solved. Taking
A = member of board
B = bondholder
C = shareholder
the premises are evidently
A = AB*c* ꖌ A*b*C
B = AB.
The class C or shareholders may in respect of A and B be developed into
four alternatives,
C = ABC ꖌ A*b*C ꖌ *a*BC ꖌ *ab*C.
But substituting for A in the first and for B in the third alternative
we get
C = ABC*c* ꖌ AB*b*C ꖌ A*b*C ꖌ *a*ABC ꖌ *ab*C.
The first, second, and fourth alternatives in the above are
self-contradictory combinations, and only these; striking them out
there remain
C = A*b*C ꖌ *ab*C = *b*C,
the required answer. This symbolic reasoning is, I believe, the exact
equivalent of Mr. Venn’s reasoning, and I do not believe that the
result can be attained in a simpler manner. Mr. Venn adds that he
could adduce other similar instances, that is, instances showing the
necessity of a better logical method.
*Abbreviation of the Process.*
Before proceeding to further illustrations of the use of this method,
I must point out how much its practical employment can be simplified,
and how much more easy it is than would appear from the description.
When we want to effect at all a thorough solution of a logical problem
it is best to form, in the first place, a complete series of all the
combinations of terms involved in it. If there be two terms A and B,
the utmost variety of combinations in which they can appear are
AB *a*B
A*b* *ab*.
The term A appears in the first and second; B in the first and third;
*a* in the third and fourth; and *b* in the second and fourth. Now if
we have any premise, say
A = B,
we must ascertain which of these combinations will be rendered
self-contradictory by substitution; the second and third will have to
be struck out, and there will remain only
AB
*ba*.
Hence we draw the following inferences
A = AB, B = AB, *a* = *ab*, *b* = *ab*.
Exactly the same method must be followed when a question involves a
greater number of terms. Thus by the Law of Duality the three terms A,
B, C, give rise to eight conceivable combinations, namely
ABC (α) *a*BC (ε)
AB*c* (β) *a*B*c* (ζ)
A*b*C (γ) *ab*C (η)
A*bc* (δ) *abc*. (θ)
The development of the term A is formed by the first four of these; for
B we must select (α), (β), (ε), (ζ); C consists of (α), (γ), (ε), (η);
*b* of (γ), (δ), (η), (θ), and so on.
Now if we want to investigate completely the meaning of the premises
A = AB (1)
B = BC (2)
we examine each of the eight combinations as regards each premise; (γ)
and (δ) are contradicted by (1), and (β) and (ζ) by (2), so that there
remain only
ABC (α)
*a*BC (ε)
*ab*C (η)
*abc*. (θ)
To describe any term under the conditions of the premises (1) and (2),
we have simply to draw out the proper combinations from this list;
thus, A is represented only by ABC, that is to say
A = ABC,
similarly *c* = *abc*.
For B we have two alternatives thus stated,
B = ABC ꖌ *a*BC;
and for *b* we have
*b* = *ab*C ꖌ *abc*.
When we have a problem involving four distinct terms we need to
double the number of combinations, and as we add each new term the
combinations become twice as numerous. Thus
A, B produce four combinations
A, B, C, " eight "
A, B, C, D " sixteen "
A, B, C, D, E " thirty-two "
A, B, C, D, E, F " sixty-four "
and so on.
I propose to call any such series of combinations the *Logical
Alphabet*. It holds in logical science a position the importance
of which cannot be exaggerated, and as we proceed from logical to
mathematical considerations, it will become apparent that there is
a close connection between these combinations and the fundamental
theorems of mathematical science. For the convenience of the reader
who may wish to employ the *Alphabet* in logical questions, I have
had printed on the next page a complete series of the combinations up
to those of six terms. At the very commencement, in the first column,
is placed a single letter X, which might seem to be superfluous. This
letter serves to denote that it is always some higher class which is
divided up. Thus the combination AB really means ABX, or that part of
some larger class, say X, which has the qualities of A and B present.
The letter X is omitted in the greater part of the table merely for
the sake of brevity and clearness. In a later chapter on Combinations
it will become apparent that the introduction of this unit class is
requisite in order to complete the analogy with the Arithmetical
Triangle there described.
The reader ought to bear in mind that though the Logical Alphabet seems
to give mere lists of combinations, these combinations are intended in
every case to constitute the development of a term of a proposition.
Thus the four combinations AB, A*b*, *a*B, *ab* really mean that any
class X is described by the following proposition,
X = XAB ꖌ XA*b* ꖌ X*a*B ꖌ X*ab*.
If we select the A’s, we obtain the following proposition
AX = XAB ꖌ XA*b*.
Thus whatever group of combinations we treat must be conceived as part
of a higher class, *summum genus* or universe symbolised in the term
X; but, bearing this in mind, it is needless to complicate our formulæ
by always introducing the letter. All inference consists in passing
from propositions to propositions, and combinations *per se* have no
meaning. They are consequently to be regarded in all cases as forming
parts of propositions.
THE LOGICAL ALPHABET.
I. II. III. IV. V. VI. VII.
X AX AB ABC ABCD ABCDE ABCDEF
*a*X A*b* AB*c* ABC*d* ABCD*e* ABCDE*f*
*a*B A*b*C AB*c*D ABC*d*E ABCD*e*F
*ab* A*bc* AB*cd* ABC*de* ABCD*ef*
*a*BC A*b*CD AB*c*DE ABC*d*EF
*a*B*c* A*b*C*d* AB*c*D*e* ABC*d*E*f*
*ab*C A*bc*D AB*cd*E ABC*de*F
*abc* Ab*cd* AB*cde* ABC*def*
*a*BCD A*b*CDE AB*c*DEF
*a*BC*d* A*b*CD*e* AB*c*DE*f*
*a*B*c*D A*b*C*d*E AB*c*D*e*F
*a*B*cd* A*b*C*de* AB*c*D*ef*
*ab*CD A*bc*DE AB*cd*EF
*ab*C*d* A*bc*D*e* AB*cd*E*f*
*abc*D A*bcd*E AB*cde*F
*abcd* A*bcde* AB*cdef*
*a*BCDE A*b*CDEF
*a*BCD*e* A*b*CDE*f*
*a*BC*d*E A*b*CD*e*F
*a*BC*de* A*b*CD*ef*
*a*B*c*DE A*b*C*d*EF
*a*B*c*D*e* A*b*C*d*E*f*
*a*B*cd*E A*b*C*de*F
*a*B*cde* A*b*C*def*
*ab*CDE A*bc*DEF
*ab*CD*e* A*bc*DE*f*
*ab*C*d*E A*bc*D*e*F
*ab*Cd*e* A*bc*D*ef*
*abc*DE A*bcd*EF
*abc*D*e* A*bcd*E*f*
*abcd*E A*bcde*F
*abcde* A*bcdef*
*a*BCDEF
*a*BCDE*f*
*a*BCD*e*F
*a*BCD*ef*
*a*BC*d*EF
*a*BC*d*E*f*
*a*BC*de*F
*a*BC*def*
*a*B*c*DEF
*a*B*c*DE*f*
*a*B*c*D*e*F
*a*B*c*D*ef*
*a*B*cd*EF
*a*B*cd*E*f*
*a*B*cde*F
*a*B*cdef*
*ab*CDEF
*ab*CDE*f*
*ab*CD*e*F
*ab*CD*ef*
*ab*C*d*EF
*ab*C*d*E*f*
*ab*C*de*F
*ab*C*def*
*abc*DEF
*abc*DE*f*
*abc*D*e*F
*abc*D*ef*
*abcd*EF
*abcd*E*f*
*abcde*F
*abcdef*
In a theoretical point of view we may conceive that the Logical
Alphabet is infinitely extended. Every new quality or circumstance
which can belong to an object, subdivides each combination or class,
so that the number of such combinations, when unrestricted by logical
conditions, is represented by an infinitely high power of two. The
extremely rapid increase in the number of subdivisions obliges us to
confine our attention to a few qualities at a time.
When contemplating the properties of this Alphabet I am often inclined
to think that Pythagoras perceived the deep logical importance of
duality; for while unity was the symbol of identity and harmony, he
described the number two as the origin of contrasts, or the symbol
of diversity, division and separation. The number four, or the
*Tetractys*, was also regarded by him as one of the chief elements of
existence, for it represented the generating virtue whence come all
combinations. In one of the golden verses ascribed to Pythagoras, he
conjures his pupil to be virtuous:[77]
“By him who stampt *The Four* upon the Mind,
*The Four*, the fount of Nature’s endless stream.”
[77] Whewell, *History of the Inductive Sciences*, vol. i. p. 222.
Now four and the higher powers of duality do represent in this logical
system the numbers of combinations which can be generated in the
absence of logical restrictions. The followers of Pythagoras may have
shrouded their master’s doctrines in mysterious and superstitious
notions, but in many points these doctrines seem to have some basis in
logical philosophy.
*The Logical Slate.*
To a person who has once comprehended the extreme significance and
utility of the Logical Alphabet the indirect process of inference
becomes reduced to the repetition of a few uniform operations of
classification, selection, and elimination of contradictories. Logical
deduction, even in the most complicated questions, becomes a matter
of mere routine, and the amount of labour required is the only
impediment, when once the meaning of the premises is rendered clear.
But the amount of labour is often found to be considerable. The mere
writing down of sixty-four combinations of six letters each is no small
task, and, if we had a problem of five premises, each of the sixty-four
combinations would have to be examined in connection with each premise.
The requisite comparison is often of a very tedious character, and
considerable chance of error intervenes.
I have given much attention, therefore, to lessening both the manual
and mental labour of the process, and I shall describe several devices
which may be adopted for saving trouble and risk of mistake.
In the first place, as the same sets of combinations occur over and
over again in different problems, we may avoid the labour of writing
them out by having the sets of letters ready printed upon small sheets
of writing-paper. It has also been suggested by a correspondent that,
if any one series of combinations were marked upon the margin of a
sheet of paper, and a slit cut between each pair of combinations, it
would be easy to fold down any particular combination, and thus strike
it out of view. The combinations consistent with the premises would
then remain in a broken series. This method answers sufficiently well
for occasional use.
A more convenient mode, however, is to have the series of letters shown
on p. 94, engraved upon a common school writing slate, of such a size,
that the letters may occupy only about a third of the space on the
left hand side of the slate. The conditions of the problem can then be
written down on the unoccupied part of the slate, and the proper series
of combinations being chosen, the contradictory combinations can be
struck out with the pencil. I have used a slate of this kind, which I
call a *Logical Slate*, for more than twelve years, and it has saved me
much trouble. It is hardly possible to apply this process to problems
of more than six terms, owing to the large number of combinations which
would require examination.
*Abstraction of Indifferent Circumstances.*
There is a simple but highly important process of inference which
enables us to abstract, eliminate or disregard all circumstances
indifferently present and absent. Thus if I were to state that “a
triangle is a three-sided rectilinear figure, either large or not
large,” these two alternatives would be superfluous, because, by the
Law of Duality, I know that everything must be either large or not
large. To add the qualification gives no new knowledge, since the
existence of the two alternatives will be understood in the absence of
any information to the contrary. Accordingly, when two alternatives
differ only as regards a single component term which is positive in one
and negative in the other, we may reduce them to one term by striking
out their indifferent part. It is really a process of substitution
which enables us to do this; for having any proposition of the form
A = ABC ꖌ AB*c*, (1)
we know by the Law of Duality that
AB = ABC ꖌ AB*c*. (2)
As the second member of this is identical with the second member of (1)
we may substitute, obtaining
A = AB.
This process of reducing useless alternatives may be applied again and
again; for it is plain that
A = AB (CD ꖌ C*d* ꖌ *c*D ꖌ *cd*)
communicates no more information than that A is B. Abstraction
of indifferent terms is in fact the converse process to that of
development described in p. 89; and it is one of the most important
operations in the whole sphere of reasoning.
The reader should observe that in the proposition
AC = BC
we cannot abstract C and infer
A = B;
but from
AC ꖌ A*c* = BC ꖌ B*c*
we may abstract all reference to the term C.
It ought to be carefully remarked, however, that alternatives which
seem to be without meaning often imply important knowledge. Thus if
I say that “a triangle is a three-sided rectilinear figure, with or
without three equal angles,” the last alternatives really express a
property of triangles, namely, that some triangles have three equal
angles, and some do not have them. If we put P = “Some,” meaning by the
indefinite adjective “Some,” one or more of the undefined properties of
triangles with three equal angles, and take
A = triangle
B = three-sided rectilinear figure
C = with three equal angles,
then the knowledge implied is expressed in the two propositions
PA = PBC
*p*A = *p*B*c*.
These may also be thrown into the form of one proposition, namely,
A = PBC ꖌ *p*B*c*;
but these alternatives cannot be reduced, and the proposition is quite
different from
A = BC ꖌ B*c*.
*Illustrations of the Indirect Method.*
A great variety of arguments and logical problems might be introduced
here to show the comprehensive character and powers of the Indirect
Method. We can treat either a single premise or a series of premises.
Take in the first place a simple definition, such as “a triangle is a
three-sided rectilinear figure.” Let
A = triangle
B = three-sided
C = rectilinear figure,
then the definition is of the form
A = BC.
If we take the series of eight combinations of three letters in the
Logical Alphabet (p. 94) and strike out those which are inconsistent
with the definition, we have the following result:--
ABC
*a*B*c*
*ab*C
*abc.*
For the description of the class C we have
C = ABC ꖌ *ab*C,
that is, “a rectilinear figure is either a triangle and three-sided, or
not a triangle and not three-sided.”
For the class *b* we have
*b* = *ab*C ꖌ *abc*.
To the second side of this we may apply the process of simplification
by abstraction described in the last section; for by the Law of Duality
*ab* = *ab*C ꖌ *abc*;
and as we have two propositions identical in the second side of each we
may substitute, getting
*b* = *ab*,
or what is not three-sided is not a triangle (whether it be rectilinear
or not).
*Second Example.*
Let us treat by this method the following argument:--
“Blende is not an elementary substance; elementary substances
are those which are undecomposable; blende, therefore, is
decomposable.”
Taking our letters thus--
A = blende,
B = elementary substance,
C = undecomposable,
the premises are of the forms
A = A*b*, (1)
B = C. (2)
No immediate substitution can be made; but if we take the
contrapositive of (2) (see p. 86), namely
*b* = *c*, (3)
we can substitute in (1) obtaining the conclusion
A = A*c*.
But the same result may be obtained by taking the eight combinations
of A, B, C, of the Logical Alphabet; it will be found that only three
combinations, namely,
A*bc*
*a*BC
*abc*,
are consistent with the premises, whence it results that
A = A*bc*,
or by the process of Ellipsis before described (p. 57)
A = A*c*.
*Third Example.*
As a somewhat more complex example I take the argument thus stated, one
which could not be thrown into the syllogistic form:--
“All metals except gold and silver are opaque; therefore what is not
opaque is either gold or silver or is not-metal.”
There is more implied in this statement than is distinctly asserted,
the full meaning being as follows:
All metals not gold or silver are opaque, (1)
Gold is not opaque but is a metal, (2)
Silver is not opaque but is a metal, (3)
Gold is not silver. (4)
Taking our letters thus--
A = metal C = silver
B = gold D = opaque,
we may state the premises in the forms
A*bc* = A*bc*D (1)
B = AB*d* (2)
C = AC*d* (3)
B = B*c*. (4)
To obtain a complete solution of the question we take the sixteen
combinations of A, B, C, D, and striking out those which are
inconsistent with the premises, there remain only
AB*cd*
A*b*C*d*
A*bc*D
*abc*D
*abcd*.
The expression for not-opaque things consists of the three combinations
containing *d*, thus
*d* = AB*cd* ꖌ A*b*C*d* ꖌ *abcd*,
or *d* = A*d* (B*c* ꖌ *b*C) ꖌ *abcd*.
In ordinary language, what is not-opaque is either metal which is
gold, and then not-silver, or silver and then not-gold, or else it is
not-metal and neither gold nor silver.
*Fourth Example.*
A good example for the illustration of the Indirect Method is to be
found in De Morgan’s *Formal Logic* (p. 123), the premises being
substantially as follows:--
From A follows B, and from C follows D; but B and D are inconsistent
with each other; therefore A and C are inconsistent.
The meaning no doubt is that where A is, B will be found, or that
every A is a B, and similarly every C is a D; but B and D cannot occur
together. The premises therefore appear to be of the forms
A = AB, (1)
C = CD, (2)
B = B*d*. (3)
On examining the series of sixteen combinations, only five are found to
be consistent with the above conditions, namely,
AB*cd*
*a*B*cd*
*ab*CD
*abc*D
*abcd*.
In these combinations the only A which appears is joined to *c*, and
similarly C is joined to *a*, or A is inconsistent with C.
*Fifth Example.*
A more complex argument, also given by De Morgan,[78] contains five
terms, and is as stated below, except that the letters are altered.
Every A is one only of the two B or C; D is both B and C, except
when B is E, and then it is neither; therefore no A is D.
[78] *Formal Logic*, p. 124. As Professor Croom Robertson has pointed
out to me, the second and third premises may be thrown into a single
proposition, D = D*e*BC ꖌ DE*bc*.
The meaning of the above premises is difficult to interpret, but seems
to be capable of expression in the following symbolic forms--
A = AB*c* ꖌ A*b*C, (1)
De = D*e*BC, (2)
DE = DE*bc*. (3)
As five terms enter into these premises it is requisite to treat their
thirty-two combinations, and it will be found that fourteen of them
remain consistent with the premises, namely
AB*cd*E *a*BCD*e* *ab*C*d*E
AB*cde* *a*BC*d*E *ab*C*de*
A*b*C*d*E *a*BC*de* *abc*DE
A*b*C*de* *a*B*cd*E *abcd*E
*a*B*cde* *abcde*.
If we examine the first four combinations, all of which contain A, we
find that they none of them contain D; or again, if we select those
which contain D, we have only two, thus--
D = *a*BCD*e* ꖌ *abc*DE.
Hence it is clear that no A is D, and *vice versâ* no D is A. We might
draw many other conclusions from the same premises; for instance--
DE = *abc*DE,
or D and E never meet but in the absence of A, B, and C.
*Fallacies analysed by the Indirect Method.*
It has been sufficiently shown, perhaps, that we can by the Indirect
Method of Inference extract the whole truth from a series of
propositions, and exhibit it anew in any required form of conclusion.
But it may also need to be shown by examples that so long as we follow
correctly the almost mechanical rules of the method, we cannot fall
into any of the fallacies or paralogisms which are often committed in
ordinary discussion. Let us take the example of a fallacious argument,
previously treated by the Method of Direct Inference (p. 62),
Granite is not a sedimentary rock, (1)
Basalt is not a sedimentary rock, (2)
and let us ascertain whether any precise conclusion can be drawn
concerning the relation of granite and basalt. Taking as before
A = granite,
B = sedimentary rock,
C = basalt,
the premises become
A = A*b*, (1)
C = C*b*. (2)
Of the eight conceivable combinations of A, B, C, five agree with these
conditions, namely
A*b*C *a*B*c*
A*bc* *ab*C
*abc*.
Selecting the combinations which contain A, we find the description of
granite to be
A = A*b*C ꖌ A*bc* = A*b*(C ꖌ *c*),
that is, granite is not a sedimentary rock, and is either basalt or
not-basalt. If we want a description of basalt the answer is of like
form
C = A*b*C ꖌ *ab*C = *b*C(A ꖌ *a*),
that is basalt is not a sedimentary rock, and is either granite or
not-granite. As it is already perfectly evident that basalt must be
either granite or not, and *vice versâ*, the premises fail to give us
any information on the point, that is to say the Method of Indirect
Inference saves us from falling into any fallacious conclusions. This
example sufficiently illustrates both the fallacy of Negative premises
and that of Undistributed Middle of the old logic.
The fallacy called the Illicit Process of the Major Term is also
incapable of commission in following the rules of the method. Our
example was (p. 65)
All planets are subject to gravity, (1)
Fixed stars are not planets. (2)
The false conclusion is that “fixed stars are not subject to gravity.”
The terms are
A = planet
B = fixed star
C = subject to gravity.
And the premises are A = AC, (1) B = *a*B. (2)
The combinations which remain uncontradicted on comparison with these
premises are
A*b*C *a*B*c*
*a*BC *ab*C
*abc*.
For fixed star we have the description
B = *a*BC ꖌ *a*B*c*,
that is, “a fixed star is not a planet, but is either subject or not,
as the case may be, to gravity.” Here we have no conclusion concerning
the connection of fixed stars and gravity.
*The Logical Abacus.*
The Indirect Method of Inference has now been sufficiently described,
and a careful examination of its powers will show that it is capable of
giving a full analysis and solution of every question involving only
logical relations. The chief difficulty of the method consists in the
great number of combinations which may have to be examined; not only
may the requisite labour become formidable, but a considerable chance
of mistake arises. I have therefore given much attention to modes
of facilitating the work, and have succeeded in reducing the method
to an almost mechanical form. It soon appeared obvious that if the
conceivable combinations of the Logical Alphabet, for any number of
letters, instead of being printed in fixed order on a piece of paper
or slate, were marked upon light movable pieces of wood, mechanical
arrangements could readily be devised for selecting any required class
of the combinations. The labour of comparison and rejection might thus
be immensely reduced. This idea was first carried out in the Logical
Abacus, which I have found useful in the lecture-room for exhibiting
the complete solution of logical problems. A minute description of
the construction and use of the Abacus, together with figures of the
parts, has already been given in my essay called *The Substitution of
Similars*,[79] and I will here give only a general description.
[79] Pp. 55–59, 81–86.
The Logical Abacus consists of a common school black-board placed in a
sloping position and furnished with four horizontal and equi-distant
ledges. The combinations of the letters shown in the first four columns
of the Logical Alphabet are printed in somewhat large type, so that
each letter is about an inch from the neighbouring one, but the letters
are placed one above the other instead of being in horizontal lines
as in p. 94. Each combination of letters is separately fixed to the
surface of a thin slip of wood one inch broad and about one-eighth
inch thick. Short steel pins are then driven in an inclined position
into the wood. When a letter is a large capital representing a positive
term, the pin is fixed in the upper part of its space; when the letter
is a small italic representing a negative term, the pin is fixed in
the lower part of the space. Now, if one of the series of combinations
be ranged upon a ledge of the black-board, the sharp edge of a flat
rule can be inserted beneath the pins belonging to any one letter--say
A, so that all the combinations marked A can be lifted out and placed
upon a separate ledge. Thus we have represented the act of thought
which separates the class A from what is not-A. The operation can be
repeated; out of the A’s we can in like manner select those which are
B’s, obtaining the AB’s; and in like manner we may select any other
classes such as the *a*B’s, the *ab*’s, or the *abc*’s.
If now we take the series of eight combinations of the letters A, B,
C, *a*, *b*, *c*, and wish to analyse the argument anciently called
Barbara, having the premises
A = AB (1)
B = BC, (2)
we proceed as follows--We raise the combinations marked *a*, leaving
the A’s behind; out of these A’s we move to a lower ledge such as
are *b*’s, and to the remaining AB’s we join the *a*’s which have
been raised. The result is that we have divided all the combinations
into two classes, namely, the A*b*’s which are incapable of existing
consistently with premise (1), and the combinations which are
consistent with the premise. Turning now to the second premise, we
raise out of those which agree with (1) the *b*’s, then we lower
the B*c*’s; lastly we join the *b*’s to the BC’s. We now find our
combinations arranged as below.
+---+-----+-----+-----+-----+-----+-----+-----+
| A | | | | *a* | | *a* | *a* |
| B | | | | B | | *b* | *b* |
| C | | | | C | | C | *c* |
+---+-----+-----+-----+-----+-----+-----+-----+
| | A | A | A | | *a* | | |
| | B | *b* | *b* | | B | | |
| | *c* | C | *c* | | *c* | | |
+---+-----+-----+-----+-----+-----+-----+-----+
The lower line contains all the combinations which are inconsistent
with either premise; we have carried out in a mechanical manner that
exclusion of self-contradictories which was formerly done upon the
slate or upon paper. Accordingly, from the combinations remaining in
the upper line we can draw any inference which the premises yield. If
we raise the A’s we find only one, and that is C, so that A must be C.
If we select the *c*’s we again find only one, which is *a* and also
*b*; thus we prove that not-C is not-A and not-B.
When a disjunctive proposition occurs among the premises the requisite
movements become rather more complicated. Take the disjunctive argument
A is either B or C or D,
A is not C and not D,
Therefore A is B.
The premises are represented accurately as follows:--
A = AB ꖌ AC ꖌ AD (1)
A = A*c* (2)
A = A*d*. (3)
As there are four terms, we choose the series of sixteen combinations
and place them on the highest ledge of the board but one. We raise
the *a*’s and out of the A’s, which remain, we lower the *b*’s. But
we are not to reject all the A*b*’s as contradictory, because by the
first premise A’s may be either B’s or C’s or D’s. Accordingly out
of the A*b*’s we must select the *c*’s, and out of these again the
*d*’s, so that only A*bcd* will remain to be rejected finally. Joining
all the other fifteen combinations together again, and proceeding to
premise (2), we raise the *a*’s and lower the AC’s, and thus reject
the combinations inconsistent with (2); similarly we reject the AD’s
which are inconsistent with (3). It will be found that there remain,
in addition to all the eight combinations containing *a*, only one
containing A, namely
AB*cd*,
whence it is apparent that A must be B, the ordinary conclusion of the
argument.
In my “Substitution of Similars” (pp. 56–59) I have described the
working upon the Abacus of two other logical problems, which it would
be tedious to repeat in this place.
*The Logical Machine.*
Although the Logical Abacus considerably reduced the labour of using
the Indirect Method, it was not free from the possibility of error.
I thought moreover that it would afford a conspicuous proof of the
generality and power of the method if I could reduce it to a purely
mechanical form. Logicians had long been accustomed to speak of Logic
as an Organon or Instrument, and even Lord Bacon, while he rejected
the old syllogistic logic, had insisted, in the second aphorism of his
“New Instrument,” that the mind required some kind of systematic aid.
In the kindred science of mathematics mechanical assistance of one kind
or another had long been employed. Orreries, globes, mechanical clocks,
and such like instruments, are really aids to calculation and are of
considerable antiquity. The Arithmetical Abacus is still in common use
in Russia and China. The calculating machine of Pascal is more than two
centuries old, having been constructed in 1642–45. M. Thomas of Colmar
manufactures an arithmetical machine on Pascal’s principles which
is employed by engineers and others who need frequently to multiply
or divide. To Babbage and Scheutz is due the merit of embodying the
Calculus of Differences in a machine, which thus became capable of
calculating the most complicated tables of figures. It seemed strange
that in the more intricate science of quantity mechanism should be
applicable, whereas in the simple science of qualitative reasoning, the
syllogism was only called an instrument by a figure of speech. It is
true that Swift satirically described the Professors of Laputa as in
possession of a thinking machine, and in 1851 Mr. Alfred Smee actually
proposed the construction of a Relational machine and a Differential
machine, the first of which would be a mechanical dictionary and the
second a mode of comparing ideas; but with these exceptions I have not
yet met with so much as a suggestion of a reasoning machine. It may be
added that Mr. Smee’s designs, though highly ingenious, appear to be
impracticable, and in any case they do not attempt the performance of
logical inference.[80]
[80] See his work called *The Process of Thought adapted to Words
and Language, together with a Description of the Relational and
Differential Machines*. Also *Philosophical Transactions*, [1870]
vol. 160, p. 518.
The Logical Abacus soon suggested the notion of a Logical Machine,
which, after two unsuccessful attempts, I succeeded in constructing
in a comparatively simple and effective form. The details of the
Logical Machine have been fully described by the aid of plates in the
Philosophical Transactions,[81] and it would be needless to repeat the
account of the somewhat intricate movements of the machine in this
place.
[81] *Philosophical Transactions* [1870], vol. 160, p. 497.
*Proceedings of the Royal Society*, vol. xviii. p. 166, Jan. 20,
1870. *Nature*, vol, i. p. 343.
The general appearance of the machine is shown in a plate facing
the title-page of this volume. It somewhat resembles a very small
upright piano or organ, and has a keyboard containing twenty-one keys.
These keys are of two kinds, sixteen of them representing the terms
or letters A, *a*, B, *b*, C, *c*, D, *d*, which have so often been
employed in our logical notation. When letters occur on the left-hand
side of a proposition, formerly called the subject, each is represented
by a key on the left-hand half of the keyboard; but when they occur on
the right-hand side, or as it used to be called the predicate of the
proposition, the letter-keys on the right-hand side of the keyboard are
the proper representatives. The five other keys may be called operation
keys, to distinguish them from the letter or term keys. They stand for
the stops, copula, and disjunctive conjunctions of a proposition. The
middle key of all is the copula, to be pressed when the verb *is* or
the sign = is met. The key to the extreme right-hand is called the Full
Stop, because it should be pressed when a proposition is completed,
in fact in the proper place of the full stop. The key to the extreme
left-hand is used to terminate an argument or to restore the machine to
its initial condition; it is called the Finis key. The last keys but
one on the right and left complete the whole series, and represent the
conjunction *or* in its unexclusive meaning, or the sign ꖌ which I have
employed, according as it occurs in the right or left hand side of the
proposition. The whole keyboard is arranged as shown on the next page--
+-+-----------------------------------+-+-----------------------------------+---+
| | |C| | |
|F| Left-hand side of Proposition. |o| Right-hand side of Proposition. |F S|
|i| |p| |u t|
|n+---+---+---+---+---+---+---+---+---+u+---+---+---+---+---+---+---+---+---+l o|
|i| | | | | | | | | |l| | | | | | | | | |l p|
|s|ꖌ|*d*| D |*c*| C |*b*| B |*a*| A |a| A |*a*| B |*b*| C |*c*| D |*d*|ꖌ| .|
|.|Or | | | | | | | | |.| | | | | | | | | Or| |
+-+---+---+---+---+---+---+---+---+---+-+---+---+---+---+---+---+---+---+---+---+
To work the machine it is only requisite to press the keys in
succession as indicated by the letters and signs of a symbolical
proposition. All the premises of an argument are supposed to be reduced
to the simple notation which has been employed in the previous pages.
Taking then such a simple proposition as
A = AB,
we press the keys A (left), copula, A (right), B (right), and full stop.
If there be a second premise, for instance
B = BC,
we press in like manner the keys--
B (left), copula, B (right), C (right), full stop.
The process is exactly the same however numerous the premises may be.
When they are completed the operator will see indicated on the face of
the machine the exact combinations of letters which are consistent with
the premises according to the principles of thought.
As shown in the figure opposite the title-page, the machine exhibits in
front a Logical Alphabet of sixteen combinations, exactly like that of
the Abacus, except that the letters of each combination are separated
by a certain interval. After the above problem has been worked upon the
machine the Logical Alphabet will have been modified so as to present
the following appearance--
+-------------------------------------------------------+
| |
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| A | A | | | | | | |*a*|*a*| | |*a*|*a*|*a*|*a*|
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| |
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| B | B | | | | | | | B | B | | |*b*|*b*|*b*|*b*|
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| |
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| C | C | | | | | | | C | C | | | C | C |*c*|*c*|
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| |
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| D |*d*| | | | | | | D |*d*| | | D |*d*| D |*d*|
+---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+
| |
+-------------------------------------------------------+
The operator will readily collect the various conclusions in the manner
described in previous pages, as, for instance that A is always C, that
not-C is not-B and not-A; and not-B is not-A but either C or not-C. The
results are thus to be read off exactly as in the case of the Logical
Slate, or the Logical Abacus.
Disjunctive propositions are to be treated in an exactly similar
manner. Thus, to work the premises
A = AB ꖌ AC
B ꖌ C = BD ꖌ CD,
it is only necessary to press in succession the keys
A (left), copula, A (right), B, ꖌ, A, C, full stop.
B (left), ꖌ, C, copula, B (right), D, ꖌ, C, D, full stop.
The combinations then remaining will be as follows
ABCD *a*BCD *abc*D
AB*c*D *a*B*c*D *abcd.*
A*c*CD *ab*CD
On pressing the left-hand key A, all the possible combinations which do
not contain A will disappear, and the description of A may be gathered
from what remain, namely that it is always D. The full-stop key
restores all combinations consistent with the premises and any other
selection may be made, as say not-D, which will be found to be always
not-A, not-B, and not-C.
At the end of every problem, when no further questions need be
addressed to the machine, we press the Finis key, which has the effect
of bringing into view the whole of the conceivable combinations of
the alphabet. This key in fact obliterates the conditions impressed
upon the machine by moving back into their ordinary places those
combinations which had been rejected as inconsistent with the premises.
Before beginning any new problem it is requisite to observe that the
whole sixteen combinations are visible. After the Finis key has been
used the machine represents a mind endowed with powers of thought,
but wholly devoid of knowledge. It would not in that condition give
any answer but such as would consist in the primary laws of thought
themselves. But when any proposition is worked upon the keys, the
machine analyses and digests the meaning of it and becomes charged with
the knowledge embodied in that proposition. Accordingly it is able
to return as an answer any description of a term or class so far as
furnished by that proposition in accordance with the Laws of Thought.
The machine is thus the embodiment of a true logical system. The
combinations are classified, selected or rejected, just as they should
be by a reasoning mind, so that at each step in a problem, the Logical
Alphabet represents the proper condition of a mind exempt from mistake.
It cannot be asserted indeed that the machine entirely supersedes the
agency of conscious thought; mental labour is required in interpreting
the meaning of grammatical expressions, and in correctly impressing
that meaning on the machine; it is further required in gathering
the conclusion from the remaining combinations. Nevertheless the
true process of logical inference is really accomplished in a purely
mechanical manner.
It is worthy of remark that the machine can detect any
self-contradiction existing between the premises presented to it;
should the premises be self-contradictory it will be found that one or
more of the letter-terms disappears entirely from the Logical Alphabet.
Thus if we work the two propositions, A is B, and A is not-B, and then
inquire for a description of A, the machine will refuse to give it
by exhibiting no combination at all containing A. This result is in
agreement with the law, which I have explained, that every term must
have its negative (p. 74). Accordingly, whenever any one of the letters
A, B, C, D, *a*, *b*, *c*, *d*, wholly disappears from the alphabet,
it may be safely inferred that some act of self-contradiction has been
committed.
It ought to be carefully observed that the logical machine cannot
receive a simple identity of the form A = B except in the double form
of A = B and B = A. To work the proposition A = B, it is therefore
necessary to press the keys--
A (left), copula, B (right), full stop;
B (left), copula, A (right), full stop.
The same double operation will be necessary whenever the proposition is
not of the kind called a partial identity (p. 40). Thus AB = CD, AB =
AC, A = B ꖌ C, A ꖌ B = C ꖌ D, all require to be read from both ends
separately.
The proper rule for using the machine may in fact be given in the
following way:--(1) *Read each proposition as it stands, and play
the corresponding keys*: (2) *Convert the proposition and read and
play the keys again in the transposed order of the terms.* So long
as this rule is observed the true result must always be obtained.
There can be no mistake. But it will be found that in the case of
partial identities, and some other similar forms of propositions, the
transposed reading has no effect upon the combinations of the Logical
Alphabet. One reading is in such cases all that is practically needful.
After some experience has been gained in the use of the machine, the
worker naturally saves himself the trouble of the second reading when
possible.
It is no doubt a remarkable fact that a simple identity cannot
be impressed upon the machine except in the form of two partial
identities, and this may be thought by some logicians to militate
against the equational mode of representing propositions.
Before leaving the subject I may remark that these mechanical devices
are not likely to possess much practical utility. We do not require in
common life to be constantly solving complex logical questions. Even in
mathematical calculation the ordinary rules of arithmetic are generally
sufficient, and a calculating machine can only be used with advantage
in peculiar cases. But the machine and abacus have nevertheless two
important uses.
In the first place I hope that the time is not very far distant when
the predominance of the ancient Aristotelian Logic will be a matter
of history only, and when the teaching of logic will be placed on a
footing more worthy of its supreme importance. It will then be found
that the solution of logical questions is an exercise of mind at least
as valuable and necessary as mathematical calculation. I believe that
these mechanical devices, or something of the same kind, will then
become useful for exhibiting to a class of students a clear and visible
analysis of logical problems of any degree of complexity, the nature
of each step being rendered plain to the eyes of the students. I often
used the machine or abacus for this purpose in my class lectures while
I was Professor of Logic at Owens College.
Secondly, the more immediate importance of the machine seems to consist
in the unquestionable proof which it affords that correct views of the
fundamental principles of reasoning have now been attained, although
they were unknown to Aristotle and his followers. The time must come
when the inevitable results of the admirable investigations of the late
Dr. Boole must be recognised at their true value, and the plain and
palpable form in which the machine presents those results will, I hope,
hasten the time. Undoubtedly Boole’s life marks an era in the science
of human reason. It may seem strange that it had remained for him first
to set forth in its full extent the problem of logic, but I am not
aware that anyone before him had treated logic as a symbolic method
for evolving from any premises the description of any class whatsoever
as defined by those premises. In spite of several serious errors into
which he fell, it will probably be allowed that Boole discovered the
true and general form of logic, and put the science substantially into
the form which it must hold for evermore. He thus effected a reform
with which there is hardly anything comparable in the history of logic
between his time and the remote age of Aristotle.
Nevertheless, Boole’s quasi-mathematical system could hardly be
regarded as a final and unexceptionable solution of the problem. Not
only did it require the manipulation of mathematical symbols in a very
intricate and perplexing manner, but the results when obtained were
devoid of demonstrative force, because they turned upon the employment
of unintelligible symbols, acquiring meaning only by analogy. I
have also pointed out that he imported into his system a condition
concerning the exclusive nature of alternatives (p. 70), which is
not necessarily true of logical terms. I shall have to show in the
next chapter that logic is really the basis of the whole science of
mathematical reasoning, so that Boole inverted the true order of proof
when he proposed to infer logical truths by algebraic processes. It is
wonderful evidence of his mental power that by methods fundamentally
false he should have succeeded in reaching true conclusions and
widening the sphere of reason.
The mechanical performance of logical inference affords a demonstration
both of the truth of Boole’s results and of the mistaken nature of his
mode of deducing them. Conclusions which he could obtain only by pages
of intricate calculation, are exhibited by the machine after one or
two minutes of manipulation. And not only are those conclusions easily
reached, but they are demonstratively true, because every step of the
process involves nothing more obscure than the three fundamental Laws
of Thought.
*The Order of Premises.*
Before quitting the subject of deductive reasoning, I may remark that
the order in which the premises of an argument are placed is a matter
of logical indifference. Much discussion has taken place at various
times concerning the arrangement of the premises of a syllogism; and it
has been generally held, in accordance with the opinion of Aristotle,
that the so-called major premise, containing the major term, or the
predicate of the conclusion, should stand first. This distinction
however falls to the ground in our system, since the proposition is
reduced to an identical form, in which there is no distinction of
subject and predicate. In a strictly logical point of view the order
of statement is wholly devoid of significance. The premises are
simultaneously coexistent, and are not related to each other according
to the properties of space and time. Just as the qualities of the same
object are neither before nor after each other in nature (p. 33), and
are only thought of in some one order owing to the limited capacity of
mind, so the premises of an argument are neither before nor after each
other, and are only thought of in succession because the mind cannot
grasp many ideas at once. The combinations of the logical alphabet
are exactly the same in whatever order the premises be treated on
the logical slate or machine. Some difference may doubtless exist as
regards convenience to human memory. The mind may take in the results
of an argument more easily in one mode of statement than another,
although there is no real difference in the logical results. But in
this point of view I think that Aristotle and the old logicians were
clearly wrong. It is more easy to gather the conclusion that “all A’s
are C’s” from “all A’s are B’s and all B’s are C’s,” than from the same
propositions in inverted order, “all B’s are C’s and all A’s are B’s.”
*The Equivalence of Propositions*.
One great advantage which arises from the study of this Indirect
Method of Inference consists in the clear notion which we gain of
the Equivalence of Propositions. The older logicians showed how from
certain simple premises we might draw an inference, but they failed
to point out whether that inference contained the whole, or only a
part, of the information embodied in the premises. Any one proposition
or group of propositions may be classed with respect to another
proposition or group of propositions, as
1. Equivalent,
2. Inferrible,
3. Consistent,
4. Contradictory.
Taking the proposition “All men are mortals” as the original, then
“All immortals are not men” is its equivalent; “Some mortals are men”
is inferrible, or capable of inference, but is not equivalent; “All
not-men are not mortals” cannot be inferred, but is consistent, that
is, may be true at the same time; “All men are immortals” is of course
contradictory.
One sufficient test of equivalence is capability of mutual inference.
Thus from
All electrics = all non-conductors,
I can infer
All non-electrics = all conductors,
and *vice versâ* from the latter I can pass back to the former. In
short, A = B is equivalent to *a* = *b*. Again, from the union of the
two propositions, A = AB and B = AB, I get A = B, and from this I
might as easily deduce the two with which I started. In this case one
proposition is equivalent to two other propositions. There are in fact
no less than four modes in which we may express the identity of two
classes A and B, namely,
FIRST MODE. SECOND MODE. THIRD MODE. FOURTH MODE.
A = B *a* = *b* A = AB } *a* = *ab* }
B = AB } *b* = *ab* }
The Indirect Method of Inference furnishes a universal and clear
criterion as to the relationship of propositions. The import of a
statement is always to be measured by the combinations of terms which
it destroys. Hence two propositions are equivalent when they remove
the same combinations from the Logical Alphabet, and neither more nor
less. A proposition is inferrible but not equivalent to another when
it removes some but not all the combinations which the other removes,
and none except what this other removes. Again, propositions are
consistent provided that they jointly allow each term and the negative
of each term to remain somewhere in the Logical Alphabet. If after all
the combinations inconsistent with two propositions are struck out,
there still appears each of the letters A, *a*, B, *b*, C, *c*, D, *d*,
which were there before, then no inconsistency between the propositions
exists, although they may not be equivalent or even inferrible.
Finally, contradictory propositions are those which taken together
remove any one or more letter-terms from the Logical Alphabet.
What is true of single propositions applies also to groups of
propositions, however large or complicated; that is to say, one group
may be equivalent, inferrible, consistent, or contradictory as regards
another, and we may similarly compare one proposition with a group of
propositions.
To give in this place illustrations of all the four kinds of relation
would require much space: as the examples given in previous sections or
chapters may serve more or less to explain the relations of inference,
consistency, and contradiction, I will only add a few instances of
equivalent propositions or groups.
In the following list each proposition or group of propositions is
exactly equivalent in meaning to the corresponding one in the other
column, and the truth of this statement may be tested by working out
the combinations of the alphabet, which ought to be found exactly the
same in the case of each pair of equivalents.
A = A*b* . . . . . . . B = *a*B
A = *b* . . . . . . . . *a* = B
A = BC . . . . . . . . *a* = *b* ꖌ *c*
A = AB ꖌ AC . . . . . . *b* = *ab* ꖌ A*b*C
A ꖌB = C ꖌ D . . . . . . . *ab* = *cd*
A ꖌ *c* = B ꖌ *d* . . . . . . *a*C = *b*D
A = AB*c* ꖌ A*b*C . . .{ A = AB ꖌ AC
{ AB = AB*c*
A = B } { A = B
B = C } . . . . . . . . . { A = C
A = AB } { A = AC
B = BC }. . . . . . . . . { B = A ꖌ *a*BC
Although in these and many other cases the equivalents of certain
propositions can readily be given, yet I believe that no uniform and
infallible process can be pointed out by which the exact equivalents
of premises can be ascertained. Ordinary deductive inference usually
gives us only a portion of the contained information. It is true that
the combinations consistent with a set of premises may always be thrown
into the form of a proposition which must be logically equivalent to
those premises; but the difficulty consists in detecting the other
forms of propositions which will be equivalent to the premises. The
task is here of a different character from any which we have yet
attempted. It is in reality an inverse process, and is just as much
more troublesome and uncertain than the direct process, as seeking is
compared with hiding. Not only may several different answers equally
apply, but there is no method of discovering any of those answers
except by repeated trial. The problem which we have here met is really
that of induction, the inverse of deduction; and, as I shall soon show,
induction is always tentative, and, unless conducted with peculiar
skill and insight, must be exceedingly laborious in cases of complexity.
De Morgan was unfortunately led by this equivalence of propositions
into the most serious error of his ingenious system of Logic. He held
that because the proposition “All A’s are all B’s,” is but another
expression for the two propositions “All A’s are B’s” and “All B’s
are A’s,” it must be a composite and not really an elementary form
of proposition.[82] But on taking a general view of the equivalence
of propositions such an objection seems to have no weight. Logicians
have, with few exceptions, persistently upheld the original error of
Aristotle in rejecting from their science the one simple relation of
identity on which all more complex logical relations must really rest.
[82] *Syllabus of a proposed system of Logic*, §§ 57, 121, &c.
*Formal Logic*, p. 66.
*The Nature of Inference.*
The question, What is Inference? is involved, even to the present day,
in as much uncertainty as that ancient question, What is Truth? I shall
in more than one part of this work endeavour to show that inference
never does more than explicate, unfold, or develop the information
contained in certain premises or facts. Neither in deductive nor
inductive reasoning can we add a tittle to our implicit knowledge,
which is like that contained in an unread book or a sealed letter. Sir
W. Hamilton has well said, “Reasoning is the showing out explicitly
that a proposition not granted or supposed, is implicitly contained in
something different, which is granted or supposed.”[83]
[83] Lectures on Metaphysics, vol. iv. p. 369.
Professor Bowen has explained[84] with much clearness that the
conclusion of an argument states explicitly what is virtually or
implicitly thought. “The process of reasoning is not so much a mode of
evolving a new truth, as it is of establishing or proving an old one,
by showing how much was admitted in the concession of the two premises
taken together.” It is true that the whole meaning of these statements
rests upon that of such words as “explicit,” “implicit,” “virtual.”
That is implicit which is wrapped up, and we render it explicit when
we unfold it. Just as the conception of a circle involves a hundred
important geometrical properties, all following from what we know, if
we have acuteness to unfold the results, so every fact and statement
involves more meaning than seems at first sight. Reasoning explicates
or brings to conscious possession what was before unconscious. It does
not create, nor does it destroy, but it transmutes and throws the same
matter into a new form.
[84] Bowen, *Treatise on Logic*, Cambridge, U.S., 1866; p. 362.
The difficult question still remains, Where does novelty of form begin?
Is it a case of inference when we pass from “Sincerity is the parent of
truth” to “The parent of truth is sincerity?” The old logicians would
have called this change *conversion*, one case of immediate inference.
But as all identity is necessarily reciprocal, and the very meaning
of such a proposition is that the two terms are identical in their
signification, I fail to see any difference between the statements
whatever. As well might we say that *x* = *y* and *y* = *x* are
different equations.
Another point of difficulty is to decide when a change is merely
grammatical and when it involves a real logical transformation. Between
a *table of wood* and a *wooden table* there is no logical difference
(p. 31), the adjective being merely a convenient substitute for the
prepositional phrase. But it is uncertain to my mind whether the
change from “All men are mortal” to “No men are not mortal” is purely
grammatical. Logical change may perhaps be best described as consisting
in the determination of a relation between certain classes of objects
from a relation between certain other classes. Thus I consider it a
truly logical inference when we pass from “All men are mortal” to “All
immortals are not-men,” because the classes *immortals* and *not-men*
are different from *mortals* and *men*, and yet the propositions
contain at the bottom the very same truth, as shown in the combinations
of the Logical Alphabet.
The passage from the qualitative to the quantitative mode of expressing
a proposition is another kind of change which we must discriminate
from true logical inference. We state the same truth when we say that
“mortality belongs to all men,” as when we assert that “all men are
mortals.” Here we do not pass from class to class, but from one kind
of term, the abstract, to another kind, the concrete. But inference
probably enters when we pass from either of the above propositions to
the assertion that the class of immortal men is zero, or contains no
objects.
It is of course a question of words to what processes we shall or shall
not apply the name “inference,” and I have no wish to continue the
trifling discussions which have already taken place upon the subject.
What we need to do is to define accurately the sense in which we use
the word “inference,” and to distinguish the relation of inferrible
propositions from other possible relations. It seems to be sufficient
to recognise four modes in which two apparently different propositions
may be related. Thus two propositions may be--
1. *Tautologous* or *identical*, involving the same relation between
the same terms and classes, and only differing in the order of
statement; thus “Victoria is the Queen of England” is tautologous with
“The Queen of England is Victoria.”
2. *Grammatically related*, when the classes or objects are the same
and similarly related, and the only difference is in the words; thus
“Victoria is the Queen of England” is grammatically equivalent to
“Victoria is England’s Queen.”
3. *Equivalents* in qualitative and quantitative form, the classes
being the same, but viewed in a different manner.
4. *Logically inferrible*, one from the other, or it may be
*equivalent*, when the classes and relations are different, but involve
the same knowledge of the possible combinations.
CHAPTER VII.
INDUCTION.
We enter in this chapter upon the second great department of logical
method, that of Induction or the Inference of general from particular
truths. It cannot be said that the Inductive process is of greater
importance than the Deductive process already considered, because the
latter process is absolutely essential to the existence of the former.
Each is the complement and counterpart of the other. The principles
of thought and existence which underlie them are at the bottom the
same, just as subtraction of numbers necessarily rests upon the same
principles as addition. Induction is, in fact, the inverse operation of
deduction, and cannot be conceived to exist without the corresponding
operation, so that the question of relative importance cannot arise.
Who thinks of asking whether addition or subtraction is the more
important process in arithmetic? But at the same time much difference
in difficulty may exist between a direct and inverse operation; the
integral calculus, for instance, is infinitely more difficult than the
differential calculus of which it is the inverse. Similarly, it must
be allowed that inductive investigations are of a far higher degree of
difficulty and complexity than any questions of deduction; and it is
this fact no doubt which led some logicians, such as Francis Bacon,
Locke, and J. S. Mill, to erroneous opinions concerning the exclusive
importance of induction.
Hitherto we have been engaged in considering how from certain
conditions, laws, or identities governing the combinations of
qualities, we may deduce the nature of the combinations agreeing
with those conditions. Our work has been to unfold the results of
what is contained in any statements, and the process has been one of
*Synthesis*. The terms or combinations of which the character has been
determined have usually, though by no means always, involved more
qualities, and therefore, by the relation of extension and intension,
fewer objects than the terms in which they were described. The truths
inferred were thus usually less general than the truths from which they
were inferred.
In induction all is inverted. The truths to be ascertained are more
general than the data from which they are drawn. The process by which
they are reached is *analytical*, and consists in separating the
complex combinations in which natural phenomena are presented to us,
and determining the relations of separate qualities. Given events
obeying certain unknown laws, we have to discover the laws obeyed.
Instead of the comparatively easy task of finding what effects will
follow from a given law, the effects are now given and the law is
required. We have to interpret the will by which the conditions of
creation were laid down.
*Induction an Inverse Operation*
I have already asserted that induction is the inverse operation of
deduction, but the difference is one of such great importance that I
must dwell upon it. There are many cases in which we can easily and
infallibly do a certain thing but may have much trouble in undoing
it. A person may walk into the most complicated labyrinth or the
most extensive catacombs, and turn hither and thither at his will;
it is when he wishes to return that doubt and difficulty commence.
In entering, any path served him; in leaving, he must select certain
definite paths, and in this selection he must either trust to memory
of the way he entered or else make an exhaustive trial of all possible
ways. The explorer entering a new country makes sure his line of return
by barking the trees.
The same difficulty arises in many scientific processes. Given any
two numbers, we may by a simple and infallible process obtain their
product; but when a large number is given it is quite another
matter to determine its factors. Can the reader say what two numbers
multiplied together will produce the number 8,616,460,799? I think it
unlikely that anyone but myself will ever know; for they are two large
prime numbers, and can only be rediscovered by trying in succession
a long series of prime divisors until the right one be fallen upon.
The work would probably occupy a good computer for many weeks, but it
did not occupy me many minutes to multiply the two factors together.
Similarly there is no direct process for discovering whether any number
is a prime or not; it is only by exhaustively trying all inferior
numbers which could be divisors, that we can show there is none, and
the labour of the process would be intolerable were it not performed
systematically once for all in the process known as the Sieve of
Eratosthenes, the results being registered in tables of prime numbers.
The immense difficulties which are encountered in the solution of
algebraic equations afford another illustration. Given any algebraic
factors, we can easily and infallibly arrive at the product; but given
a product it is a matter of infinite difficulty to resolve it into
factors. Given any series of quantities however numerous, there is very
little trouble in making an equation which shall have those quantities
as roots. Let *a*, *b*, *c*, *d*, &c., be the quantities; then
(*x* - *a*)(*x* - *b*)(*x* - *c*)(*x* - d) ... = 0
is the equation required, and we only need to multiply out the
expression on the left hand by ordinary rules. But having given a
complex algebraic expression equated to zero, it is a matter of
exceeding difficulty to discover all the roots. Mathematicians have
exhausted their highest powers in carrying the complete solution up to
the fourth degree. In every other mathematical operation the inverse
process is far more difficult than the direct process, subtraction than
addition, division than multiplication, evolution than involution;
but the difficulty increases vastly as the process becomes more
complex. Differentiation, the direct process, is always capable of
performance by fixed rules, but as these rules produce considerable
variety of results, the inverse process of integration presents
immense difficulties, and in an infinite majority of cases surpasses
the present resources of mathematicians. There are no infallible and
general rules for its accomplishment; it must be done by trial, by
guesswork, or by remembering the results of differentiation, and using
them as a guide.
Coming more nearly to our own immediate subject, exactly the same
difficulty exists in determining the law which certain things obey.
Given a general mathematical expression, we can infallibly ascertain
its value for any required value of the variable. But I am not aware
that mathematicians have ever attempted to lay down the rules of a
process by which, having given certain numbers, one might discover a
rational or precise formula from which they proceed. The reader may
test his power of detecting a law, by contemplation of its results, if
he, not being a mathematician, will attempt to point out the law obeyed
by the following numbers:
1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, 3617/510, 43867/798, etc.
These numbers are sometimes in low terms, but unexpectedly spring up
to high terms; in absolute magnitude they are very variable. They seem
to set all regularity and method at defiance, and it is hardly to be
supposed that anyone could, from contemplation of the numbers, have
detected the relations between them. Yet they are derived from the
most regular and symmetrical laws of relation, and are of the highest
importance in mathematical analysis, being known as the numbers of
Bernoulli.
Compare again the difficulty of decyphering with that of cyphering.
Anyone can invent a secret language, and with a little steady labour
can translate the longest letter into the character. But to decypher
the letter, having no key to the signs adopted, is a wholly different
matter. As the possible modes of secret writing are infinite in number
and exceedingly various in kind, there is no direct mode of discovery
whatever. Repeated trial, guided more or less by knowledge of the
customary form of cypher, and resting entirely on the principles of
probability and logical induction, is the only resource. A peculiar
tact or skill is requisite for the process, and a few men, such as
Wallis or Wheatstone, have attained great success.
Induction is the decyphering of the hidden meaning of natural
phenomena. Given events which happen in certain definite combinations,
we are required to point out the laws which govern those combinations.
Any laws being supposed, we can, with ease and certainty, decide
whether the phenomena obey those laws. But the laws which may exist
are infinite in variety, so that the chances are immensely against
mere random guessing. The difficulty is much increased by the fact
that several laws will usually be in operation at the same time, the
effects of which are complicated together. The only modes of discovery
consist either in exhaustively trying a great number of supposed laws,
a process which is exhaustive in more senses than one, or else in
carefully contemplating the effects, endeavouring to remember cases
in which like effects followed from known laws. In whatever manner we
accomplish the discovery, it must be done by the more or less conscious
application of the direct process of deduction.
The Logical Alphabet illustrates induction as well as deduction. In
considering the Indirect Process of Inference we found that from
certain propositions we could infallibly determine the combinations
of terms agreeing with those premises. The inductive problem is just
the inverse. Having given certain combinations of terms, we need to
ascertain the propositions with which the combinations are consistent,
and from which they may have proceeded. Now, if the reader contemplates
the following combinations,
ABC *ab*C
*a*BC *abc*,
he will probably remember at once that they belong to the premises
A = AB, B = BC (p. 92). If not, he will require a few trials before he
meets with the right answer, and every trial will consist in assuming
certain laws and observing whether the deduced results agree with the
data. To test the facility with which he can solve this inductive
problem, let him casually strike out any of the combinations of the
fourth column of the Logical Alphabet, (p. 94), and say what laws
the remaining combinations obey, observing that every one of the
letter-terms and their negatives ought to appear in order to avoid
self-contradiction in the premises (pp. 74, 111). Let him say, for
instance, what laws are embodied in the combinations
ABC *a*BC
A*bc* *ab*C.
The difficulty becomes much greater when more terms enter into the
combinations. It would require some little examination to ascertain the
complete conditions fulfilled in the combinations
AC*e* *ab*C*e*
*a*BC*e* *abc*E.
*a*B*cd*E
The reader may discover easily enough that the principal laws are
C = *e*, and A = A*e*; but he would hardly discover without some
trouble the remaining law, namely, that BD = BD*e*.
The difficulties encountered in the inductive investigations of
nature, are of an exactly similar kind. We seldom observe any law in
uninterrupted and undisguised operation. The acuteness of Aristotle and
the ancient Greeks did not enable them to detect that all terrestrial
bodies tend to fall towards the centre of the earth. A few nights of
observation might have convinced an astronomer viewing the solar system
from its centre, that the planets travelled round the sun; but the
fact that our place of observation is one of the travelling planets,
so complicates the apparent motions of the other bodies, that it
required all the sagacity of Copernicus to prove the real simplicity of
the planetary system. It is the same throughout nature; the laws may
be simple, but their combined effects are not simple, and we have no
clue to guide us through their intricacies. “It is the glory of God,”
said Solomon, “to conceal a thing, but the glory of a king to search
it out.” The laws of nature are the invaluable secrets which God has
hidden, and it is the kingly prerogative of the philosopher to search
them out by industry and sagacity.
*Inductive Problems for Solution by the Reader.*
In the first edition (vol. ii. p. 370) I gave a logical problem
involving six terms, and requested readers to discover the laws
governing the combinations given. I received satisfactory replies
from readers both in the United States and in England. I formed
the combinations deductively from four laws of correction, but my
correspondents found that three simpler laws, equivalent to the four
more complex ones, were the best answer; these laws are as follows:
*a* = *ac*, *b* = *cd*, *d* = E*f*.
In case other readers should like to test their skill in the inductive
or inverse problem, I give below several series of combinations forming
problems of graduated difficulty.
PROBLEM I.
A B *c*
A *b* C
*a* B C
PROBLEM II.
A B C
A *b* C
*a* B C
*a* B *c*
PROBLEM III.
A B C
A *b* C
*a* B C
*a* B *c*
*a* *b* *c*
PROBLEM IV.
A B C D
A *b* *c* D
*a* B *c* *d*
*a* *b* C *d*
PROBLEM V.
A B C D
A B C *d*
A B *c* *d*
A *b* C D
A *b* *c* D
*a* B C D
*a* B *c* D
*a* B *c* *d*
*a* *b* C *d*
PROBLEM VI.
A B C D E
A B C *d* *e*
A B *c* D E
A B *c* *d* *e*
A *b* C D E
*a* B C D E
*a* B C *d* *e*
*a* *b* C D E
*a* *b* *c* *d* *e*
PROBLEM VII.
A *b* *c* D *e*
*a* B C *d* E
*a* *b* C *d* E
PROBLEM VIII.
A B C D E
A B C D *e*
A B C *d* *e*
A B *c* *d* *e*
A *b* C D E
A *b* *c* *d* E
A *b* *c* *d* *e*
*a* B C D *e*
*a* B C *d* *e*
*a* B *c* D *e*
*a* *b* C D *e*
*a* *b* C *d* E
*a* *b* *c* D *e*
*a* *b* *c* *d* E
PROBLEM IX.
A B *c* D E F
A B *c* D *e* F
A *b* C D *e* *f*
A *b* *c* D E *f*
A *b* *c* D *e* *f*
A *b* *c* *d* E F
A *b* *c* *d* *e* F
*a* B *c* D E F
*a* B *c* D *e* F
*a* B *c* *d* E F
*a* *b* C D E F
*a* *b* C D *e* F
*a* *b* C D *e* *f*
*a* *b* *c* D *e* *f*
*a* *b* *c* D E *f*
*a* *b* *c* *d* *e* F
PROBLEM X.
A B C D *e* F
A B *c* D E *f*
A *b* C D E F
A *b* C D *e* F
A *b* *c* D *e* F
*a* B C D E *f*
*a* B *c* D E *f*
*a* *b* C D *e* F
*a* *b* C *d* *e* F
*a* *b* *c* D *e* *f*
*a* *b* *c* *d* *e* *f*
*Induction of Simple Identities*.
Many important laws of nature are expressible in the form of simple
identities, and I can at once adduce them as examples to illustrate
what I have said of the difficulty of the inverse process of induction.
Two phenomena are conjoined. Thus all gravitating matter is exactly
coincident with all matter possessing inertia; where one property
appears, the other likewise appears. All crystals of the cubical
system, are all the crystals which do not doubly refract light. All
exogenous plants are, with some exceptions, those which have two
cotyledons or seed-leaves.
A little reflection will show that there is no direct and infallible
process by which such complete coincidences may be discovered.
Natural objects are aggregates of many qualities, and any one of
those qualities may prove to be in close connection with some others.
If each of a numerous group of objects is endowed with a hundred
distinct physical or chemical qualities, there will be no less than
(1/2)(100 × 99) or 4950 pairs of qualities, which may be connected,
and it will evidently be a matter of great intricacy and labour to
ascertain exactly which qualities are connected by any simple law.
One principal source of difficulty is that the finite powers of the
human mind are not sufficient to compare by a single act any large
group of objects with another large group. We cannot hold in the
conscious possession of the mind at any one moment more than five or
six different ideas. Hence we must treat any more complex group by
successive acts of attention. The reader will perceive by an almost
individual act of comparison that the words *Roma* and *Mora* contain
the same letters. He may perhaps see at a glance whether the same is
true of *Causal* and *Casual*, and of *Logica* and *Caligo*. To assure
himself that the letters in *Astronomers* make *No more stars*, that
*Serpens in akuleo* is an anagram of *Joannes Keplerus*, or *Great gun
do us a sum* an anagram of *Augustus de Morgan*, it will certainly be
necessary to break up the act of comparison into several successive
acts. The process will acquire a double character, and will consist in
ascertaining that each letter of the first group is among the letters
of the second group, and *vice versâ*, that each letter of the second
is among those of the first group. In the same way we can only prove
that two long lists of names are identical, by showing that each name
in one list occurs in the other, and *vice versâ*.
This process of comparison really consists in establishing two partial
identities, which are, as already shown (p. 58), equivalent in
conjunction to one simple identity. We first ascertain the truth of the
two propositions A = AB, B = AB, and we then rise by substitution to
the single law A = B.
There is another process, it is true, by which we may get to exactly
the same result; for the two propositions A = AB, *a* = *ab* are also
equivalent to the simple identity A = B. If then we can show that
all objects included under A are included under B, and also that all
objects not included under A are not included under B, our purpose is
effected. By this process we should usually compare two lists if we are
allowed to mark them. For each name in the first list we should strike
off one in the second, and if, when the first list is exhausted, the
second list is also exhausted, it follows that all names absent from
the first must be absent from the second, and the coincidence must be
complete.
These two modes of proving an identity are so closely allied that it
is doubtful how far we can detect any difference in their powers and
instances of application. The first method is perhaps more convenient
when the phenomena to be compared are rare. Thus we prove that all the
musical concords coincide with all the more simple numerical ratios, by
showing that each concord arises from a simple ratio of undulations,
and then showing that each simple ratio gives rise to one of the
concords. To examine all the possible cases of discord or complex ratio
of undulation would be impossible. By a happy stroke of induction Sir
John Herschel discovered that all crystals of quartz which cause the
plane of polarization of light to rotate are precisely those crystals
which have plagihedral faces, that is, oblique faces on the corners of
the prism unsymmetrical with the ordinary faces. This singular relation
would be proved by observing that all plagihedral crystals possessed
the power of rotation, and *vice versâ* all crystals possessing this
power were plagihedral. But it might at the same time be noticed that
all ordinary crystals were devoid of the power. There is no reason
why we should not detect any of the four propositions A = AB, B = AB,
*a* = *ab*, *b* = *ab*, all of which follow from A = B (p. 115).
Sometimes the terms of the identity may be singular objects; thus we
observe that diamond is a combustible gem, and being unable to discover
any other that is, we affirm--
Diamond = combustible gem.
In a similar manner we ascertain that
Mercury = metal liquid at ordinary temperatures,
Substance of least density = substance of least atomic weight.
Two or three objects may occasionally enter into the induction, as when
we learn that
Sodium ꖌ potassium = metal of less density than water,
Venus ꖌ Mercury ꖌ Mars = major planet devoid of satellites.
*Induction of Partial Identities*.
We found in the last section that the complete identity of two classes
is almost always discovered not by direct observation of the fact,
but by first establishing two partial identities. There are also a
multitude of cases in which the partial identity of one class with
another is the only relation to be discovered. Thus the most common
of all inductive inferences consists in establishing the fact that
all objects having the properties of A have also those of B, or that
A = AB. To ascertain the truth of a proposition of this kind it is
merely necessary to assemble together, mentally or physically, all the
objects included under A, and then observe whether B is present in
each of them, or, which is the same, whether it would be impossible
to select from among them any not-B. Thus, if we mentally assemble
together all the heavenly bodies which move with apparent rapidity,
that is to say, the planets, we find that they all possess the property
of not scintillating. We cannot analyse any vegetable substance without
discovering that it contains carbon and hydrogen, but it is not true
that all substances containing carbon and hydrogen are vegetable
substances.
The great mass of scientific truths consists of propositions of
this form A = AB. Thus in astronomy we learn that all the planets
are spheroidal bodies; that they all revolve in one direction round
the sun; that they all shine by reflected light; that they all obey
the law of gravitation. But of course it is not to be asserted that
all bodies obeying the law of gravitation, or shining by reflected
light, or revolving in a particular direction, or being spheroidal
in form, are planets. In other sciences we have immense numbers of
propositions of the same form, as, for instance, all substances in
becoming gaseous absorb heat; all metals are elements; they are all
good conductors of heat and electricity; all the alkaline metals are
monad elements; all foraminifera are marine organisms; all parasitic
animals are non-mammalian; lightning never issues from stratous clouds;
pumice never occurs where only Labrador felspar is present; milkmaids
do not suffer from small-pox; and, in the works of Darwin, scientific
importance may attach even to such an apparently trifling observation
as that “white tom-cats having blue eyes are deaf.”
The process of inference by which all such truths are obtained may
readily be exhibited in a precise symbolic form. We must have one
premise specifying in a disjunctive form all the possible individuals
which belong to a class; we resolve the class, in short, into its
constituents. We then need a number of propositions, each of which
affirms that one of the individuals possesses a certain property. Thus
the premises must be of the forms
A = B ꖌ C ꖌ D ꖌ .... ꖌ P ꖌ Q
B = BX
C = CX
... ...
... ...
Q = QX.
Now, if we substitute for each alternative of the first premise its
description as found among the succeeding premises, we obtain
A = BX ꖌ CX ꖌ .... ꖌ PX ꖌ QX
or
A = (B ꖌ C ꖌ .... ꖌ Q)X
But for the aggregate of alternatives we may now substitute their
equivalent as given in the first premise, namely A, so that we get the
required result:
A = AX.
We should have reached the same result if the first premise had been of
the form
A = AB ꖌ AC ꖌ .... ꖌ AQ.
We can always prove a proposition, if we find it more convenient, by
proving its equivalent. To assert that all not-B’s are not-A’s, is
exactly the same as to assert that all A’s are B’s. Accordingly we may
ascertain that A = AB by first ascertaining that *b* = *ab*. If we
observe, for instance, that all substances which are not solids are
also not capable of double refraction, it follows necessarily that all
double refracting substances are solids. We may convince ourselves that
all electric substances are nonconductors of electricity, by reflecting
that all good conductors do not, and in fact cannot, retain electric
excitation. When we come to questions of probability it will be found
desirable to prove, as far as possible, both the original proposition
and its equivalent, as there is then an increased area of observation.
The number of alternatives which may arise in the division of a class
varies greatly, and may be any number from two upwards. Thus it is
probable that every substance is either magnetic or diamagnetic, and no
substance can be both at the same time. The division then must be made
in the form
A = AB*c* ꖌ A*b*C.
If now we can prove that all magnetic substances are capable of
polarity, say B = BD, and also that all diamagnetic substances are
capable of polarity, C = CD, it follows by substitution that all
substances are capable of polarity, or A = AD. We commonly divide
the class substance into the three subclasses, solid, liquid, and
gas; and if we can show that in each of these forms it obeys Carnot’s
thermodynamic law, it follows that all substances obey that law.
Similarly we may show that all vertebrate animals possess red blood,
if we can show separately that fish, reptiles, birds, marsupials, and
mammals possess red blood, there being, as far as is known, only five
principal subclasses of vertebrata.
Our inductions will often be embarrassed by exceptions, real or
apparent. We might affirm that all gems are incombustible were not
diamonds undoubtedly combustible. Nothing seems more evident than that
all the metals are opaque until we examine them in fine films, when
gold and silver are found to be transparent. All plants absorb carbonic
acid except certain fungi; all the bodies of the planetary system
have a progressive motion from west to east, except the satellites of
Uranus and Neptune. Even some of the profoundest laws of matter are not
quite universal; all solids expand by heat except india-rubber, and
possibly a few other substances; all liquids which have been tested
expand by heat except water below 4° C. and fused bismuth; all gases
have a coefficient of expansion increasing with the temperature, except
hydrogen. In a later chapter I shall consider how such anomalous cases
may be regarded and classified; here we have only to express them in a
consistent manner by our notation.
Let us take the case of the transparency of metals, and assign the
terms thus:--
A = metal D = iron
B = gold E, F, &c. = copper, lead, &c.
C = silver X = opaque.
Our premises will be
A = B ꖌ C ꖌ D ꖌ E, &c.
B = B*x*
C = C*x*
D = DX
E = EX,
and so on for the rest of the metals. Now evidently
A*bc* = (D ꖌ E ꖌ F ꖌ ...)*bc*,
and by substitution as before we shall obtain
A*bc* = A*bc*X,
or in words, “All metals not gold nor silver are opaque;” at the same
time we have
A(B ꖌ C) = AB ꖌ AC = AB*x* ꖌ AC*x* = A(B ꖌ C)*x*,
or “Metals which are either gold or silver are not opaque.”
In some cases the problem of induction assumes a much higher degree of
complexity. If we examine the properties of crystallized substances
we may find some properties which are common to all, as cleavage or
fracture in definite planes; but it would soon become requisite to
break up the class into several minor ones. We should divide crystals
according to the seven accepted systems--and we should then find
that crystals of each system possess many common properties. Thus
crystals of the Regular or Cubical system expand equally by heat,
conduct heat and electricity with uniform rapidity, and are of like
elasticity in all directions; they have but one index of refraction
for light; and every facet is repeated in like relation to each of
the three axes. Crystals of the system having one principal axis
will be found to possess the various physical powers of conduction,
refraction, elasticity, &c., uniformly in directions perpendicular
to the principal axis; in other directions their properties vary
according to complicated laws. The remaining systems in which the
crystals possess three unequal axes, or have inclined axes, exhibit
still more complicated results, the effects of the crystal upon light,
heat, electricity, &c., varying in all directions. But when we pursue
induction into the intricacies of its application to nature we really
enter upon the subject of classification, which we must take up again
in a later part of this work.
*Solution of the Inverse or Inductive Problem, involving Two Classes*.
It is now plain that Induction consists in passing back from a series
of combinations to the laws by which such combinations are governed.
The natural law that all metals are conductors of electricity really
means that in nature we find three classes of objects, namely--
1. Metals, conductors;
2. Not-metals, conductors;
3. Not-metals, not-conductors.
It comes to the same thing if we say that it excludes the existence
of the class, “metals not-conductors.” In the same way every other
law or group of laws will really mean the exclusion from existence
of certain combinations of the things, circumstances or phenomena
governed by those laws. Now in logic, strictly speaking, we treat not
the phenomena, nor the laws, but the general forms of the laws; and a
little consideration will show that for a finite number of things the
possible number of forms or kinds of law governing them must also be
finite. Using general terms, we know that A and B can be present or
absent in four ways and no more--thus:
AB, A*b*, *a*B, *ab*;
therefore every possible law which can exist concerning the relation
of A and B must be marked by the exclusion of one or more of the above
combinations. The number of possible laws then cannot exceed the
number of selections which we can make from these four combinations.
Since each combination may be present or absent, the number of cases
to be considered is 2 × 2 × 2 × 2, or sixteen; and these cases are all
shown in the following table, in which the sign 0 indicates absence or
non-existence of the combination shown at the left-hand column in the
same line, and the mark 1 its presence:--
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
* * * * * * *
AB 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
A*b* 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
*a*B 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
*ab* 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
Thus in column sixteen we find that all the conceivable combinations
are present, which means that there are no special laws in existence
in such a case, and that the combinations are governed only by the
universal Laws of Identity and Difference. The example of metals and
conductors of electricity would be represented by the twelfth column;
and every other mode in which two things or qualities might present
themselves is shown in one or other of the columns. More than half
the cases may indeed be at once rejected, because they involve the
entire absence of a term or its negative. It has been shown to be a
logical principle that every term must have its negative (p. 111),
and when this is not the case, inconsistency between the conditions
of combination must exist. Thus if we laid down the two following
propositions, “Graphite conducts electricity,” and “Graphite does not
conduct electricity,” it would amount to asserting the impossibility
of graphite existing at all; or in general terms, A is B and A is
not B result in destroying altogether the combinations containing A,
a case shown in the fourth column of the above table. We therefore
restrict our attention to those cases which may be represented in
natural phenomena when at least two combinations are present, and which
correspond to those columns of the table in which each of A, *a*,
B, *b* appears. These cases are shown in the columns marked with an
asterisk.
We find that seven cases remain for examination, thus characterised--
Four cases exhibiting three combinations,
Two cases exhibiting two combinations,
One case exhibiting four combinations.
It has already been pointed out that a proposition of the form A =
AB destroys one combination, A*b*, so that this is the form of law
applying to the twelfth column. But by changing one or more of the
terms in A = AB into its negative, or by interchanging A and B, *a* and
*b*, we obtain no less than eight different varieties of the one form;
thus--
12th case. 8th case. 15th case. 14th case.
A = AB A = A*b* *a* = *a*B *a* = *ab*
*b* = *ab* B = *a*B *b* = A*b* B = AB
The reader of the preceding sections will see that each proposition
in the lower line is logically equivalent to, and is in fact the
contrapositive of, that above it (p. 83). Thus the propositions
A = A*b* and B = *a*B both give the same combinations, shown in the
eighth column of the table, and trial shows that the twelfth, eighth,
fifteenth and fourteenth columns are thus accounted for. We come to
this conclusion then--*The general form of proposition* A = AB *admits
of four logically distinct varieties, each capable of expression in two
modes*.
In two columns of the table, namely the seventh and tenth, we observe
that two combinations are missing. Now a simple identity A = B renders
impossible both A*b* and *a*B, accounting for the tenth case; and if we
change B into *b* the identity A = *b* accounts for the seventh case.
There may indeed be two other varieties of the simple identity, namely
*a* = *b* and *a* = B; but it has already been shown repeatedly that
these are equivalent respectively to A = B and A = *b* (p. 115). As
the sixteenth column has already been accounted for as governed by no
special conditions, we come to the following general conclusion:--The
laws governing the combinations of two terms must be capable of
expression either in a partial identity or a simple identity; the
partial identity is capable of only four logically distinct varieties,
and the simple identity of two. Every logical relation between two
terms must be expressed in one of these six forms of law, or must be
logically equivalent to one of them.
In short, we may conclude that in treating of partial and complete
identity, we have exhaustively treated the modes in which two terms or
classes of objects can be related. Of any two classes it can be said
that one must either be included in the other, or must be identical
with it, or a like relation must exist between one class and the
negative of the other. We have thus completely solved the inverse
logical problem concerning two terms.[85]
[85] The contents of this and the following section nearly correspond
with those of a paper read before the Manchester Literary and
Philosophical Society on December 26th, 1871. See Proceedings of the
Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. pp.
119–130.
*The Inverse Logical Problem involving Three Classes.*
No sooner do we introduce into the problem a third term C, than the
investigation assumes a far more complex character, so that some
readers may prefer to pass over this section. Three terms and their
negatives may be combined, as we have frequently seen, in eight
different combinations, and the effect of laws or logical conditions
is to destroy any one or more of these combinations. Now we may make
selections from eight things in 2^{8} or 256 ways; so that we have no
less than 256 different cases to treat, and the complete solution is
at least fifty times as troublesome as with two terms. Many series of
combinations, indeed, are contradictory, as in the simpler problem,
and may be passed over, the test of consistency being that each of the
letters A, B, C, *a*, *b*, *c*, shall appear somewhere in the series of
combinations.
My mode of solving the problem was as follows:--Having written out the
whole of the 256 series of combinations, I examined them separately and
struck out such as did not fulfil the test of consistency. I then chose
some form of proposition involving two or three terms, and varied it
in every possible manner, both by the circular interchange of letters
(A, B, C into B, C, A and then into C, A, B), and by the substitution
for any one or more of the terms of the corresponding negative terms.
For instance, the proposition AB = ABC can be first varied by circular
interchange so as to give BC = BCA and then CA = CAB. Each of these
three can then be thrown into eight varieties by negative change. Thus
AB = ABC gives *a*B = *a*BC, A*b* = A*b*C, AB = AB*c*, *ab* = *ab*C,
and so on. Thus there may possibly exist no less than twenty-four
varieties of the law having the general form AB = ABC, meaning that
whatever has the properties of A and B has those also of C. It by no
means follows that some of the varieties may not be equivalent to
others; and trial shows, in fact, that AB = ABC is exactly the same
in meaning as A*c* = A*bc* or B*c* = B*ca*. Thus the law in question
has but eight varieties of distinct logical meaning. I now ascertain
by actual deductive reasoning which of the 256 series of combinations
result from each of these distinct laws, and mark them off as soon as
found. I then proceed to some other form of law, for instance A = ABC,
meaning that whatever has the qualities of A has those also of B and
C. I find that it admits of twenty-four variations, all of which are
found to be logically distinct; the combinations being worked out, I am
able to mark off twenty-four more of the list of 256 series. I proceed
in this way to work out the results of every form of law which I can
find or invent. If in the course of this work I obtain any series of
combinations which had been previously marked off, I learn at once that
the law giving these combinations is logically equivalent to some law
previously treated. It may be safely inferred that every variety of the
apparently new law will coincide in meaning with some variety of the
former expression of the same law. I have sufficiently verified this
assumption in some cases, and have never found it lead to error. Thus
as AB = ABC is equivalent to A*c* = A*bc*, so we find that *ab* = *ab*C
is equivalent to *ac* = *ac*B.
Among the laws treated were the two A = AB and A = B which involve only
two terms, because it may of course happen that among three things two
only are in special logical relation, and the third independent; and
the series of combinations representing such cases of relation are sure
to occur in the complete enumeration. All single propositions which
I could invent having been treated, pairs of propositions were next
investigated. Thus we have the relations, “All A’s are B’s, and all
B’s are C’s,” of which the old logical syllogism is the development.
We may also have “all A’s are all B’s, and all B’s are C’s,” or even
“all A’s are all B’s, and all B’s are all C’s.” All such premises admit
of variations, greater or less in number, the logical distinctness
of which can only be determined by trial in detail. Disjunctive
propositions either singly or in pairs were also treated, but were
often found to be equivalent to other propositions of a simpler form;
thus A = ABC ꖌ A*bc* is exactly the same in meaning as AB = AC.
This mode of exhaustive trial bears some analogy to that ancient
mathematical process called the Sieve of Eratosthenes. Having taken
a long series of the natural numbers, Eratosthenes is said to have
calculated out in succession all the multiples of every number, and
to have marked them off, so that at last the prime numbers alone
remained, and the factors of every number were exhaustively discovered.
My problem of 256 series of combinations is the logical analogue, the
chief points of difference being that there is a limit to the number of
cases, and that prime numbers have no analogue in logic, since every
series of combinations corresponds to a law or group of conditions.
But the analogy is perfect in the point that they are both inverse
processes. There is no mode of ascertaining that a number is prime but
by showing that it is not the product of any assignable factors. So
there is no mode of ascertaining what laws are embodied in any series
of combinations but trying exhaustively the laws which would give them.
Just as the results of Eratosthenes’ method have been worked out to
a great extent and registered in tables for the convenience of other
mathematicians, I have endeavoured to work out the inverse logical
problem to the utmost extent which is at present practicable or useful.
I have thus found that there are altogether fifteen conditions or
series of conditions which may govern the combinations of three
terms, forming the premises of fifteen essentially different kinds
of arguments. The following table contains a statement of these
conditions, together with the numbers of combinations which are
contradicted or destroyed by each, and the numbers of logically
distinct variations of which the law is capable. There might be
also added, as a sixteenth case, that case where no special logical
condition exists, so that all the eight combinations remain.
+---------+-------------------------------+-----------+------------+
| | | Number of | Number of |
|Reference| Propositions expressing the | distinct |combinations|
| Number. | general type of the logical | logical |contradicted|
| | conditions. |variations.| by each. |
+---------+-------------------------------+-----------+------------+
| I. | A = B | 6 | 4 |
| II. | A = AB | 12 | 2 |
| III. | A = B, B = C | 4 | 6 |
| IV. | A = B, B = BC | 24 | 5 |
| V. | A = AB, B = BC | 24 | 4 |
| VI. | A = BC | 24 | 4 |
| VII. | A = ABC | 24 | 3 |
| VIII. | AB = ABC | 8 | 1 |
| IX. | A = AB, *a*B = *a*B*c* | 24 | 3 |
| X. | A = ABC, *ab* = *ab*C | 8 | 4 |
| XI. | AB = ABC, *ab* = *abc* | 4 | 2 |
| XII. | AB = AC | 12 | 2 |
| XIII. | A = BC ꖌ A*bc* | 8 | 3 |
| XIV. | A = BC ꖌ *bc* | 2 | 4 |
| XV. | A = ABC, *a* = B*c* ꖌ *b*C | 8 | 5 |
+---------+-------------------------------+-----------+------------+
There are sixty-three series of combinations derived from
self-contradictory premises, which with 192, the sum of the numbers of
distinct logical variations stated in the third column of the table,
and with the one case where there are no conditions or laws at all,
make up the whole conceivable number of 256 series.
We learn from this table, for instance, that two propositions of
the form A = AB, B = BC, which are such as constitute the premises
of the old syllogism Barbara, exclude as impossible four of the
eight combinations in which three terms may be united, and that
these propositions are capable of taking twenty-four variations by
transpositions of the terms or the introduction of negatives. This
table then presents the results of a complete analysis of all the
possible logical relations arising in the case of three terms, and the
old syllogism forms but one out of fifteen typical forms. Generally
speaking, every form can be converted into apparently different
propositions; thus the fourth type A = B, B = BC may appear in the
form A = ABC, *a* = *ab*, or again in the form of three propositions
A = AB, B = BC, *a*B = *a*B*c*; but all these sets of premises yield
identically the same series of combinations, and are therefore of
equivalent logical meaning. The fifth type, or Barbara, can also be
thrown into the equivalent forms A = ABC, *a*B = *a*BC and A = AC,
B = A ꖌ *a*BC. In other cases I have obtained the very same logical
conditions in four modes of statements. As regards mere appearance and
form of statement, the number of possible premises would be very great,
and difficult to exhibit exhaustively.
The most remarkable of all the types of logical condition is the
fourteenth, namely, A = BC ꖌ *bc*. It is that which expresses the
division of a genus into two doubly marked species, and might be
illustrated by the example--“Component of the physical universe =
matter, gravitating, or not-matter (ether), not-gravitating.” It is
capable of only two distinct logical variations, namely, A = BC ꖌ *bc*
and A = B*c* ꖌ *b*C. By transposition or negative change of the letters
we can indeed obtain six different expressions of each of these
propositions; but when their meanings are analysed, by working out the
combinations, they are found to be logically equivalent to one or other
of the above two. Thus the proposition A = BC ꖌ *bc* can be written in
any of the following five other modes,
*a* = *b*C ꖌ B*c*, B = CA ꖌ *ca*, *b* = *c*A ꖌ C*a*,
C = AB ꖌ *ab*, *c* = *a*B ꖌ A*b*.
I do not think it needful to publish at present the complete table of
193 series of combinations and the premises corresponding to each. Such
a table enables us by mere inspection to learn the laws obeyed by any
set of combinations of three things, and is to logic what a table of
factors and prime numbers is to the theory of numbers, or a table of
integrals to the higher mathematics. The table already given (p. 140)
would enable a person with but little labour to discover the law of any
combinations. If there be seven combinations (one contradicted) the law
must be of the eighth type, and the proper variety will be apparent.
If there be six combinations (two contradicted), either the second,
eleventh, or twelfth type applies, and a certain number of trials will
disclose the proper type and variety. If there be but two combinations
the law must be of the third type, and so on.
The above investigations are complete as regards the possible logical
relations of two or three terms. But when we attempt to apply the
same kind of method to the relations of four or more terms, the labour
becomes impracticably great. Four terms give sixteen combinations
compatible with the laws of thought, and the number of possible
selections of combinations is no less than 2^{16} or 65,536. The
following table shows the extraordinary manner in which the number of
possible logical relations increases with the number of terms involved.
+---------+-------------+---------------------------------------+
|Number of| Number of |Number of possible selections of combi-|
| terms. | possible | nations corresponding to consistent |
| |combinations.| or inconsistent logical relations. |
+-----------------------+---------------------------------------+
| 2 | 4 | 16 |
| 3 | 8 | 256 |
| 4 | 16 | 65,536 |
| 5 | 32 | 4,294,967,296 |
| 6 | 64 | 18,446,744,073,709,551,616 |
+---------+-------------+---------------------------------------+
Some years of continuous labour would be required to ascertain the
types of laws which may govern the combinations of only four things,
and but a small part of such laws would be exemplified or capable of
practical application in science. The purely logical inverse problem,
whereby we pass from combinations to their laws, is solved in the
preceding pages, as far as it is likely to be for a long time to come;
and it is almost impossible that it should ever be carried more than a
single step further.
In the first edition, vol i. p. 158, I stated that I had not been
able to discover any mode of calculating the number of cases in which
inconsistency would be implied in the selection of combinations from
the Logical Alphabet. The logical complexity of the problem appeared
to be so great that the ordinary modes of calculating numbers of
combinations failed, in my opinion, to give any aid, and exhaustive
examination of the combinations in detail seemed to be the only method
applicable. This opinion, however, was mistaken, for both Mr. R. B.
Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers
of inconsistent cases both for three and for four terms, without much
difficulty. In the case of four terms they find that there are 1761
inconsistent selections and 63,774 consistent, which with one case
where no condition exists, make up the total of 65,536 possible
selections.
The inconsistent cases are distributed in the manner shown in the
following table:--
+--------------+---------------------------------------------------+
| Number of | |
| Combinations | 0 1 2 3 4 5 6 7 8 9 10, &c. |
| remaining. | |
+--------------+---------------------------------------------------+
| Number of | |
| Inconsistent | 1 16 112 352 536 448 224 64 8 0 0, &c. |
| Cases. | |
+--------------+---------------------------------------------------+
When more than eight combinations of the Logical Alphabet (p. 94,
column V.) remain unexcluded, there cannot be inconsistency. The whole
numbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are
given in the 17th line of the Arithmetical Triangle given further on in
the Chapter on Combinations and Permutations, the sum of the numbers in
that line being 65,536.
*Professor Clifford on the Types of Compound Statement involving Four
Classes.*
In the first edition (vol. i. p. 163), I asserted that some years of
labour would be required to ascertain even the precise number of types
of law governing the combinations of four classes of things. Though I
still believe that some years’ labour would be required to work out the
types themselves, it is clearly a mistake to suppose that the *numbers*
of such types cannot be calculated with a reasonable amount of labour,
Professor W. K. Clifford having actually accomplished the task. His
solution of the numerical problem involves the use of a complete new
system of nomenclature and is far too intricate to be fully described
here. I can only give a brief abstract of the results, and refer
readers, who wish to follow out the reasoning, to the Proceedings of
the Literary and Philosophical Society of Manchester, for the 9th
January, 1877, vol. xvi., p. 88, where Professor Clifford’s paper is
printed in full.
By a *simple statement* Professor Clifford means the denial of the
existence of any single combination or *cross-division*, of the
classes, as in ABCD = 0, or A*b*C*d* = 0. The denial of two or more
such combinations is called a *compound statement*, and is further said
to be *twofold*, *threefold*, &c., according to the number denied. Thus
ABC = 0 is a twofold compound statement in regard to four classes,
because it involves both ABCD = 0 and ABC*d* = 0. When two compound
statements can be converted into one another by interchange of the
classes, A, B, C, D, with each other or with their complementary
classes, *a*, *b*, *c*, *d*, they are called *similar*, and all similar
statements are said to belong to the same *type*.
Two statements are called *complementary* when they deny between them
all the sixteen combinations without both denying any one; or, which
is the same thing, when each denies just those combinations which
the other permits to exist. It is obvious that when two statements
are similar, the complementary statements will also be similar,
and consequently for every type of *n*-fold statement, there is a
complementary type of (16--*n*)-fold statement. It follows that we need
only enumerate the types as far as the eighth order; for the types
of more-than-eight-fold statement will already have been given as
complementary to types of lower orders.
One combination, ABCD, may be converted into another A*b*C*d* by
interchanging one or more of the classes with the complementary
classes. The number of such changes is called the *distance*, which in
the above case is 2. In two similar compound statements the distances
of the combinations denied must be the same; but it does not follow
that when all the distances are the same, the statements are similar.
There is, however, only one example of two dissimilar statements having
the same distances. When the distance is 4, the two combinations
are said to be *obverse* to one another, and the statements denying
them are called *obverse statements*, as in ABCD = 0 and *abcd* = 0
or again A*b*C*d* = 0 and *a*B*c*D = 0. When any one combination is
given, called the *origin*, all the others may be grouped in respect
of their relations to it as follows:--Four are at distance *one* from
it, and may be called *proximates*; six are at distance *two*, and may
be called *mediates*; four are at distance *three*, and may be called
*ultimates*; finally the obverse is at distance *four*.
Origin and Six Obverse and
four proximates. mediates. four ultimates.
*ab*CD
|
*a*BCD A*bc*D | A*b*C*d* A*bcd*
| \ | / |
| \ | / |
| \|/ |
ABC*d*--ABCD--A*b*CD + *abc*D--*abcd*--*a*B*cd*
| /|\ |
| / | \ |
| / | \ |
AB*c*D *a*B*c*D | *a*BC*d* *ab*C*d*.
|
AB*cd*
It will be seen that the four proximates are respectively obverse to
the four ultimates, and that the mediates form three pairs of obverses.
Every proximate or ultimate is distant 1 and 3 respectively from such a
pair of mediates.
Aided by this system of nomenclature Professor Clifford proceeds to an
exhaustive enumeration of types, in which it is impossible to follow
him. The results are as follows:--
1-fold statements 1 type }
2 " " 4 types}
3 " " 6 " }
4 " " 19 " } 159
5 " " 27 " }
6 " " 47 " }
7 " " 55 " }
8-fold statements 78 "
Now as each seven-fold or less-than-seven-fold statement is
complementary to a nine-fold or more-than-nine-fold statement, it
follows that the complete number of types will be 159 × 2 + 78 = 396.
It appears then that the types of statement concerning four classes
are only about 26 times as numerous as those concerning three classes,
fifteen in number, although the number of possible combinations is 256
times as great.
Professor Clifford informs me that the knowledge of the possible
groupings of subdivisions of classes which he obtained by this inquiry
has been of service to him in some applications of hyper-elliptic
functions to which he has subsequently been led. Professor Cayley has
since expressed his opinion that this line of investigation should
be followed out, owing to the bearing of the theory of compound
combinations upon the higher geometry.[86] It seems likely that many
unexpected points of connection will in time be disclosed between the
sciences of logic and mathematics.
[86] *Proceedings of the Manchester Literary and Philosophical
Society*, 6th February, 1877, vol. xvi., p. 113.
*Distinction between Perfect and Imperfect Induction.*
We cannot proceed with advantage before noticing the extreme difference
which exists between cases of perfect and those of imperfect induction.
We call an induction *perfect* when all the objects or events which
can possibly come under the class treated have been examined. But in
the majority of cases it is impossible to collect together, or in any
way to investigate, the properties of all portions of a substance or
of all the individuals of a race. The number of objects would often
be practically infinite, and the greater part of them might be beyond
our reach, in the interior of the earth, or in the most distant parts
of the Universe. In all such cases induction is *imperfect*, and is
affected by more or less uncertainty. As some writers have fallen into
much error concerning the functions and relative importance of these
two branches of reasoning, I shall have to point out that--
1. Perfect Induction is a process absolutely requisite, both in the
performance of imperfect induction and in the treatment of large
bodies of facts of which our knowledge is complete.
2. Imperfect Induction is founded on Perfect Induction, but involves
another process of inference of a widely different character.
It is certain that if I can draw any inference at all concerning
objects not examined, it must be done on the data afforded by the
objects which have been examined. If I judge that a distant star obeys
the law of gravity, it must be because all other material objects
sufficiently known to me obey that law. If I venture to assert that
all ruminant animals have cloven hoofs, it is because all ruminant
animals which have come under my notice have cloven hoofs. On the
other hand, I cannot safely say that all cryptogamous plants possess
a purely cellular structure, because some cryptogamous plants, which
have been examined by botanists, have a partially vascular structure.
The probability that a new cryptogam will be cellular only can be
estimated, if at all, on the ground of the comparative numbers of
known cryptogams which are and are not cellular. Thus the first step
in every induction will consist in accurately summing up the number
of instances of a particular phenomenon which have fallen under our
observation. Adams and Leverrier, for instance, must have inferred
that the undiscovered planet Neptune would obey Bode’s law, because
*all the planets known at that time obeyed it*. On what principles the
passage from the known to the apparently unknown is warranted, must be
carefully discussed in the next section, and in various parts of this
work.
It would be a great mistake, however, to suppose that Perfect Induction
is in itself useless. Even when the enumeration of objects belonging
to any class is complete, and admits of no inference to unexamined
objects, the statement of our knowledge in a general proposition is a
process of so much importance that we may consider it necessary. In
many cases we may render our investigations exhaustive; all the teeth
or bones of an animal; all the cells in a minute vegetable organ; all
the caves in a mountain side; all the strata in a geological section;
all the coins in a newly found hoard, may be so completely scrutinized
that we may make some general assertion concerning them without fear
of mistake. Every bone might be proved to contain phosphate of lime;
every cell to enclose a nucleus; every cave to hide remains of extinct
animals; every stratum to exhibit signs of marine origin; every coin
to be of Roman manufacture. These are cases where our investigation
is limited to a definite portion of matter, or a definite area on the
earth’s surface.
There is another class of cases where induction is naturally and
necessarily limited to a definite number of alternatives. Of the
regular solids we can say without the least doubt that no one has
more than twenty faces, thirty edges, and twenty corners; for by the
principles of geometry we learn that there cannot exist more than five
regular solids, of each of which we easily observe that the above
statements are true. In the theory of numbers, an endless variety of
perfect inductions might be made; we can show that no number less than
sixty possesses so many divisors, and the like is true of 360; for it
does not require a great amount of labour to ascertain and count all
the divisors of numbers up to sixty or 360. I can assert that between
60,041 and 60,077 no prime number occurs, because the exhaustive
examination of those who have constructed tables of prime numbers
proves it to be so.
In matters of human appointment or history, we can frequently have
a complete limitation of the number of instances to be included in
an induction. We might show that the propositions of the third book
of Euclid treat only of circles; that no part of the works of Galen
mentions the fourth figure of the syllogism; that none of the other
kings of England reigned so long as George III.; that Magna Charta has
not been repealed by any subsequent statute; that the price of corn in
England has never been so high since 1847 as it was in that year; that
the price of the English funds has never been lower than it was on the
23rd of January, 1798, when it fell to 47-1/4.
It has been urged against this process of Perfect Induction that it
gives no new information, and is merely a summing up in a brief form
of a multitude of particulars. But mere abbreviation of mental labour
is one of the most important aids we can enjoy in the acquisition
of knowledge. The powers of the human mind are so limited that
multiplicity of detail is alone sufficient to prevent its progress
in many directions. Thought would be practically impossible if every
separate fact had to be separately thought and treated. Economy of
mental power may be considered one of the main conditions on which our
elevated intellectual position depends. Mathematical processes are for
the most part but abbreviations of the simpler acts of addition and
subtraction. The invention of logarithms was one of the most striking
additions ever made to human power: yet it was a mere abbreviation of
operations which could have been done before had a sufficient amount
of labour been available. Similar additions to our power will, it
is hoped, be made from time to time; for the number of mathematical
problems hitherto solved is but an indefinitely small fraction of those
which await solution, because the labour they have hitherto demanded
renders them impracticable. So it is throughout all regions of thought.
The amount of our knowledge depends upon our power of bringing it
within practicable compass. Unless we arrange and classify facts and
condense them into general truths, they soon surpass our powers of
memory, and serve but to confuse. Hence Perfect Induction, even as a
process of abbreviation, is absolutely essential to any high degree of
mental achievement.
*Transition from Perfect to Imperfect Induction.*
It is a question of profound difficulty on what grounds we are
warranted in inferring the future from the present, or the nature
of undiscovered objects from those which we have examined with our
senses. We pass from Perfect to Imperfect Induction when once we
allow our conclusion to apply, at all events apparently, beyond the
data on which it was founded. In making such a step we seem to gain
a net addition to our knowledge; for we learn the nature of what was
unknown. We reap where we have never sown. We appear to possess the
divine power of creating knowledge, and reaching with our mental arms
far beyond the sphere of our own observation. I shall have, indeed, to
point out certain methods of reasoning in which we do pass altogether
beyond the sphere of the senses, and acquire accurate knowledge which
observation could never have given; but it is not imperfect induction
that accomplishes such a task. Of imperfect induction itself, I venture
to assert that it never makes any real addition to our knowledge, in
the meaning of the expression sometimes accepted. As in other cases
of inference, it merely unfolds the information contained in past
observations; it merely renders explicit what was implicit in previous
experience. It transmutes, but certainly does not create knowledge.
There is no fact which I shall more constantly keep before the reader’s
mind in the following pages than that the results of imperfect
induction, however well authenticated and verified, are never more than
probable. We never can be sure that the future will be as the present.
We hang ever upon the will of the Creator: and it is only so far as He
has created two things alike, or maintains the framework of the world
unchanged from moment to moment, that our most careful inferences can
be fulfilled. All predictions, all inferences which reach beyond their
data, are purely hypothetical, and proceed on the assumption that new
events will conform to the conditions detected in our observation of
past events. No experience of finite duration can give an exhaustive
knowledge of the forces which are in operation. There is thus a
double uncertainty; even supposing the Universe as a whole to proceed
unchanged, we do not really know the Universe as a whole. We know only
a point in its infinite extent, and a moment in its infinite duration.
We cannot be sure, then, that our observations have not escaped some
fact, which will cause the future to be apparently different from the
past; nor can we be sure that the future really will be the outcome of
the past. We proceed then in all our inferences to unexamined objects
and times on the assumptions--
1. That our past observation gives us a complete knowledge of what
exists.
2. That the conditions of things which did exist will continue to be
the conditions which will exist.
We shall often need to illustrate the character of our knowledge of
nature by the simile of a ballot-box, so often employed by mathematical
writers in the theory of probability. Nature is to us like an infinite
ballot-box, the contents of which are being continually drawn, ball
after ball, and exhibited to us. Science is but the careful observation
of the succession in which balls of various character present
themselves; we register the combinations, notice those which seem to
be excluded from occurrence, and from the proportional frequency of
those which appear we infer the probable character of future drawings.
But under such circumstances certainty of prediction depends on two
conditions:--
1. That we acquire a perfect knowledge of the comparative numbers of
balls of each kind within the box.
2. That the contents of the ballot-box remain unchanged.
Of the latter assumption, or rather that concerning the constitution
of the world which it illustrates, the logician or physicist can
have nothing to say. As the Creation of the Universe is necessarily
an act passing all experience and all conception, so any change in
that Universe, or, it may be, a termination of it, must likewise be
infinitely beyond the bounds of our mental faculties. No science
no reasoning upon the subject, can have any validity; for without
experience we are without the basis and materials of knowledge. It
is the fundamental postulate accordingly of all inference concerning
the future, that there shall be no arbitrary change in the subject
of inference; of the probability or improbability of such a change I
conceive that our faculties can give no estimate.
The other condition of inductive inference--that we acquire an
approximately complete knowledge of the combinations in which events
do occur, is in some degree within our power. There are branches
of science in which phenomena seem to be governed by conditions of
a most fixed and general character. We have ground in such cases
for believing that the future occurrence of such phenomena can be
calculated and predicted. But the whole question now becomes one
of probability and improbability. We seem to leave the region of
logic to enter one in which the number of events is the ground of
inference. We do not really leave the region of logic; we only leave
that where certainty, affirmative or negative, is the result, and the
agreement or disagreement of qualities the means of inference. For the
future, number and quantity will commonly enter into our processes of
reasoning; but then I hold that number and quantity are but portions
of the great logical domain. I venture to assert that number is wholly
logical, both in its fundamental nature and in its developments.
Quantity in all its forms is but a development of number. That which is
mathematical is not the less logical; if anything it is more logical,
in the sense that it presents logical results in a higher degree of
complexity and variety.
Before proceeding then from Perfect to Imperfect Induction I must
devote a portion of this work to treating the logical conditions
of number. I shall then employ number to estimate the variety of
combinations in which natural phenomena may present themselves, and
the probability or improbability of their occurrence under definite
circumstances. It is in later parts of the work that I must endeavour
to establish the notions which I have set forth upon the subject of
Imperfect Induction, as applied in the investigation of Nature, which
notions maybe thus briefly stated:--
1. Imperfect Induction entirely rests upon Perfect Induction for its
materials.
2. The logical process by which we seem to pass directly from
examined to unexamined cases consists in an inverse application of
deductive inference, so that all reasoning may be said to be either
directly or inversely deductive.
3. The result is always of a hypothetical character, and is never
more than probable.
4. No net addition is ever made to our knowledge by reasoning; what
we know of future events or unexamined objects is only the unfolded
contents of our previous knowledge, and it becomes less probable as
it is more boldly extended to remote cases.
BOOK II.
NUMBER, VARIETY, AND PROBABILITY.
CHAPTER VIII.
PRINCIPLES OF NUMBER.
Not without reason did Pythagoras represent the world as ruled by
number. Into almost all our acts of thought number enters, and in
proportion as we can define numerically we enjoy exact and useful
knowledge of the Universe. The science of numbers, too, has hitherto
presented the widest and most practicable training in logic. So free
and energetic has been the study of mathematical forms, compared
with the forms of logic, that mathematicians have passed far in
advance of pure logicians. Occasionally, in recent times, they have
condescended to apply their algebraic instrument to a reflex treatment
of the primary logical science. It is thus that we owe to profound
mathematicians, such as John Herschel, Whewell, De Morgan, or Boole,
the regeneration of logic in the present century. I entertain no
doubt that it is in maintaining a close alliance with quantitative
reasoning that we must look for further progress in our comprehension
of qualitative inference.
I cannot assent, indeed, to the common notion that certainty begins and
ends with numerical determination. Nothing is more certain than logical
truth. The laws of identity and difference are the tests of all that is
certain throughout the range of thought, and mathematical reasoning is
cogent only when it conforms to these conditions, of which logic is the
first development. And if it be erroneous to suppose that all certainty
is mathematical, it is equally an error to imagine that all which is
mathematical is certain. Many processes of mathematical reasoning are
of most doubtful validity. There are points of mathematical doctrine
which must long remain matter of opinion; for instance, the best form
of the definition and axiom concerning parallel lines, or the true
nature of a limit. In the use of symbolic reasoning questions occur on
which the best mathematicians may differ, as Bernoulli and Leibnitz
differed irreconcileably concerning the existence of the logarithms of
negative quantities.[87] In fact we no sooner leave the simple logical
conditions of number, than we find ourselves involved in a mazy and
mysterious science of symbols.
[87] Montucla. *Histoire des Mathématiques*, vol. iii. p. 373.
Mathematical science enjoys no monopoly, and not even a supremacy,
in certainty of results. It is the boundless extent and variety of
quantitative questions that delights the mathematical student. When
simple logic can give but a bare answer Yes or No, the algebraist
raises a score of subtle questions, and brings out a crowd of curious
results. The flower and the fruit, all that is attractive and
delightful, fall to the share of the mathematician, who too often
despises the plain but necessary stem from which all has arisen. In
no region of thought can a reasoner cast himself free from the prior
conditions of logical correctness. The mathematician is only strong and
true as long as he is logical, and if number rules the world, it is
logic which rules number.
Nearly all writers have hitherto been strangely content to look upon
numerical reasoning as something apart from logical inference. A long
divorce has existed between quality and quantity, and it has not
been uncommon to treat them as contrasted in nature and restricted
to independent branches of thought. For my own part, I believe that
all the sciences meet somewhere. No part of knowledge can stand
wholly disconnected from other parts of the universe of thought; it
is incredible, above all, that the two great branches of abstract
science, interlacing and co-operating in every discourse, should
rest upon totally distinct foundations. I assume that a connection
exists, and care only to inquire, What is its nature? Does the science
of quantity rest upon that of quality; or, *vice versâ*, does the
science of quality rest upon that of quantity? There might conceivably
be a third view, that they both rest upon some still deeper set of
principles.
It is generally supposed that Boole adopted the second view, and
treated logic as an application of algebra, a special case of
analytical reasoning which admits only two quantities, unity and zero.
It is not easy to ascertain clearly which of these views really was
accepted by Boole. In his interesting biographical sketch of Boole,[88]
the Rev. R. Harley protests against the statement that Boole’s logical
calculus imported the conditions of number and quantity into logic.
He says: “Logic is never identified or confounded with mathematics;
the two systems of thought are kept perfectly distinct, each being
subject to its own laws and conditions. The symbols are the same for
both systems, but they have not the same interpretation.” The Rev. J.
Venn, again, in his review of Boole’s logical system,[89] holds that
Boole’s processes are at bottom logical, not mathematical, though
stated in a highly generalized form and with a mathematical dress. But
it is quite likely that readers of Boole should be misled. Not only
have his logical works an entirely mathematical appearance, but I find
on p. 12 of his *Laws of Thought* the following unequivocal statement:
“That logic, as a science, is susceptible of very wide applications
is admitted; but it is equally certain that its ultimate forms and
processes are mathematical.” A few lines below he adds, “It is not of
the essence of mathematics to be conversant with the ideas of number
and quantity.”
[88] *British Quarterly Review*, No. lxxxvii, July 1866.
[89] *Mind*, October 1876, vol. i. p. 484.
The solution of the difficulty is that Boole used the term mathematics
in a wider sense than that usually attributed to it. He probably
adopted the third view, so that his mathematical *Laws of Thought* are
the common basis both of logic and of quantitative mathematics. But
I do not care to pursue the subject because I think that, in either
case Boole was wrong. In my opinion logic is the superior science, the
general basis of mathematics as well as of all other sciences. Number
is but logical discrimination, and algebra a highly developed logic.
Thus it is easy to understand the deep analogy which Boole pointed out
between the forms of algebraic and logical deduction. Logic resembles
algebra as the mould resembles that which is cast in it. Boole mistook
the cast for the mould. Considering that logic imposes its own laws
upon every branch of mathematical science, it is no wonder that we
constantly meet with the traces of logical laws in mathematical
processes.
*The Nature of Number.*
Number is but another name for *diversity*. Exact identity is unity,
and with difference arises plurality. An abstract notion, as was
pointed out (p. 28), possesses a certain *oneness*. The quality of
*justice*, for instance, is one and the same in whatever just acts it
is manifested. In justice itself there are no marks of difference by
which to discriminate justice from justice. But one just act can be
discriminated from another just act by circumstances of time and place,
and we can count many acts thus discriminated each from each. In like
manner pure gold is simply pure gold, and is so far one and the same
throughout. But besides its intrinsic qualities, gold occupies space
and must have shape and size. Portions of gold are always mutually
exclusive and capable of discrimination, in respect that they must be
each without the other. Hence they may be numbered.
Plurality arises when and only when we detect difference. For instance,
in counting a number of gold coins I must count each coin once, and not
more than once. Let C denote a coin, and the mark above it the order of
counting. Then I must count the coins
C′ + C″ + C‴ + C″″ + ....
If I were to count them as follows
C′ + C″ + C‴ + C‴ + C″″ + ...,
I should make the third coin into two, and should imply the existence
of difference where there is no difference.[90] C‴ and C‴ are but
the names of one coin named twice over. But according to one of the
conditions of logical symbols, which I have called the Law of Unity
(p. 72), the same name repeated has no effect, and
A ꖌ A = A.
[90] *Pure Logic*, Appendix, p. 82, § 192.
We must apply the Law of Unity, and must reduce all identical
alternatives before we can count with certainty and use the processes
of numerical calculation. Identical alternatives are harmless in
logic, but are wholly inadmissible in number. Thus logical science
ascertains the nature of the mathematical unit, and the definition may
be given in these terms--*A unit is any object of thought which can be
discriminated from every other object treated as a unit in the same
problem.*
It has often been said that units are units in respect of being
perfectly similar to each other; but though they may be perfectly
similar in some respects, they must be different in at least one point,
otherwise they would be incapable of plurality. If three coins were
so similar that they occupied the same space at the same time, they
would not be three coins, but one coin. It is a property of space that
every point is discriminable from every other point, and in time every
moment is necessarily distinct from any other moment before or after.
Hence we frequently count in space or time, and Locke, with some other
philosophers, has held that number arises from repetition in time.
Beats of a pendulum may be so perfectly similar that we can discover no
difference except that one beat is before and another after. Time alone
is here the ground of difference and is a sufficient foundation for the
discrimination of plurality; but it is by no means the only foundation.
Three coins are three coins, whether we count them successively or
regard them all simultaneously. In many cases neither time nor space
is the ground of difference, but pure quality alone enters. We can
discriminate the weight, inertia, and hardness of gold as three
qualities, though none of these is before nor after the other, neither
in space nor time. Every means of discrimination may be a source of
plurality.
Our logical notation may be used to express the rise of number.
The symbol A stands for one thing or one class, and in itself must
be regarded as a unit, because no difference is specified. But the
combinations AB and A*b* are necessarily *two*, because they cannot
logically coalesce, and there is a mark B which distinguishes one
from the other. A logical definition of the number *four* is given in
the combinations ABC, AB*c*, A*b*C, A*bc*, where there is a double
difference. As Puck says--
“Yet but three? Come one more;
Two of both kinds makes up four.”
I conceive that all numbers might be represented as arising out of
the combinations of the Logical Alphabet, more or less of each series
being struck out by various logical conditions. The number three, for
instance, arises from the condition that A must be either B or C, so
that the combinations are ABC, AB*c*, A*b*C.
*Of Numerical Abstraction.*
There will now be little difficulty in forming a clear notion of
the nature of numerical abstraction. It consists in abstracting the
character of the difference from which plurality arises, retaining
merely the fact. When I speak of *three men* I need not at once specify
the marks by which each may be known from each. Those marks must exist
if they are really three men and not one and the same, and in speaking
of them as many I imply the existence of the requisite differences.
Abstract number, then, is *the empty form of difference*; the abstract
number *three* asserts the existence of marks without specifying their
kind.
Numerical abstraction is thus seen to be a different process from
logical abstraction (p. 27), for in the latter process we drop out
of notice the very existence of difference and plurality. In forming
the abstract notion *hardness*, we ignore entirely the diverse
circumstances in which the quality may appear. It is the concrete
notion *three hard objects*, which asserts the existence of hardness
along with sufficient other undefined qualities, to mark out *three*
such objects. Numerical thought is indeed closely interwoven with
logical thought. We cannot use a concrete term in the plural, as
*men*, without implying that there are marks of difference. But when we
use an abstract term, we deal with unity.
The origin of the great generality of number is now apparent. Three
sounds differ from three colours, or three riders from three horses;
but they agree in respect of the variety of marks by which they can be
discriminated. The symbols 1 + 1 + 1 are thus the empty marks asserting
the existence of discrimination. But in dropping out of sight the
character of the differences we give rise to new agreements on which
mathematical reasoning is founded. Numerical abstraction is so far from
being incompatible with logical abstraction that it is the origin of
our widest acts of generalization.
*Concrete and Abstract Number.*
The common distinction between concrete and abstract number can now be
easily stated. In proportion as we specify the logical characters of
the things numbered, we render them concrete. In the abstract number
three there is no statement of the points in which the *three* objects
agree; but in *three coins*, *three men*, or *three horses*, not only
are the objects numbered but their nature is restricted. Concrete
number thus implies the same consciousness of difference as abstract
number, but it is mingled with a groundwork of similarity expressed in
the logical terms. There is identity so far as logical terms enter;
difference so far as the terms are merely numerical.
The reason of the important Law of Homogeneity will now be apparent.
This law asserts that in every arithmetical calculation the logical
nature of the things numbered must remain unaltered. The specified
logical agreement of the things must not be affected by the unspecified
numerical differences. A calculation would be palpably absurd which,
after commencing with length, gave a result in hours. It is equally
absurd, in a purely arithmetical point of view, to deduce areas from
the calculation of lengths, masses from the combination of volume
and density, or momenta from mass and velocity. It must remain for
subsequent consideration to decide in what sense we may truly say that
two linear feet multiplied by two linear feet give four superficial
feet; arithmetically it is absurd, because there is a change of unit.
As a general rule we treat in each calculation only objects of one
nature. We do not, and cannot properly add, in the same sum yards of
cloth and pounds of sugar. We cannot even conceive the result of adding
area to velocity, or length to density, or weight to value. The units
added must have a basis of homogeneity, or must be reducible to some
common denominator. Nevertheless it is possible, and in fact common, to
treat in one complex calculation the most heterogeneous quantities, on
the condition that each kind of object is kept distinct, and treated
numerically only in conjunction with its own kind. Different units,
so far as their logical differences are specified, must never be
substituted one for the other. Chemists continually use equations which
assert the equivalence of groups of atoms. Ordinary fermentation is
represented by the formula
C^{6} H^{12} O^{6} = 2C^{2} H^{6} O + 2CO^{2}.
Three kinds of units, the atoms respectively of carbon, hydrogen, and
oxygen, are here intermingled, but there is really a separate equation
in regard to each kind. Mathematicians also employ compound equations
of the same kind; for in, *a* + *b* √ - 1 = *c* + *d* √ - 1,
it is impossible by ordinary addition to add *a* to *b* √ - 1.
Hence we really have the separate equations *a* = *b*, and
*c* √ - 1 = *d* √ - 1. Similarly an equation between
two quaternions is equivalent to four equations between ordinary
quantities, whence indeed the name *quaternion*.
*Analogy of Logical and Numerical Terms.*
If my assertion is correct that number arises out of logical
conditions, we ought to find number obeying all the laws of logic.
It is almost superfluous to point out that this is the case with the
fundamental laws of identity and difference, and it only remains to
show that mathematical symbols do really obey the special conditions
of logical symbols which were formerly pointed out (p. 32). Thus the
Law of Commutativeness, is equally true of quality and quantity. As in
logic we have
AB = BA,
so in mathematics it is familiarly known that
2 × 3 = 3 × 2, or *x* × *y* = *y* × *x*.
The properties of space are as indifferent in multiplication as we
found them in pure logical thought.
Similarly, as in logic
triangle or square = square or triangle,
or generally A ꖌ B = B ꖌ A,
so in quantity 2 + 3 = 3 + 2,
or generally *x* + *y* = *y* + *x*.
The symbol ꖌ is not identical with +, but it is thus far analogous.
How far, now, is it true that mathematical symbols obey the Law of
Simplicity expressed in the form
AA = A,
or the example
Round round = round?
Apparently there are but two numbers which obey this law; for it is
certain that
*x* × *x* = *x*
is true only in the two cases when *x* = 1, or *x* = 0.
In reality all numbers obey the law, for 2 × 2 = 2 is not really
analogous to AA = A. According to the definition of a unit already
given, each unit is discriminated from each other in the same problem,
so that in 2′ × 2″, the first *two* involves a different discrimination
from the second *two*. I get four kinds of things, for instance, if I
first discriminate “heavy and light” and then “cubical and spherical,”
for we now have the following classes--
heavy, cubical. light, cubical.
heavy, spherical. light, spherical.
But suppose that my two classes are in both cases discriminated by the
same difference of light and heavy, then we have
heavy heavy = heavy,
heavy light = 0,
light heavy = 0,
light light = light.
Thus, (heavy or light) × (heavy or light) = (heavy or light).
In short, *twice two is two* unless we take care that the second two
has a different meaning from the first. But under similar circumstances
logical terms give the like result, and it is not true that A′A″ = A′,
when A″ is different in meaning from A′.
In a similar manner it may be shown that the Law of Unity
A ꖌ A = A.
holds true alike of logical and mathematical terms. It is absurd indeed
to say that
*x* + *x* = *x*
except in the one case when *x* = absolute zero. But this contradiction
*x* + *x* = *x* arises from the fact that we have already defined
the units in one x as differing from those in the other. Under such
circumstances the Law of Unity does not apply. For if in
A′ ꖌ A″ = A′
we mean that A″ is in any way different from A′ the assertion of
identity is evidently false.
The contrast then which seems to exist between logical and mathematical
symbols is only apparent. It is because the Laws of Simplicity and
Unity must always be observed in the operation of counting that those
laws seem no further to apply. This is the understood condition under
which we use all numerical symbols. Whenever I write the symbol 5 I
really mean
1 + 1 + 1 + 1 + 1,
and it is perfectly understood that each of these units is distinct
from each other. If requisite I might mark them thus
1′+ 1″ + 1‴ + 1″″ + 1″‴.
Were this not the case and were the units really
1′ + 1″ + 1″ + 1‴ + 1″″,
the Law of Unity would, as before remarked, apply, and
1″ + 1″ = 1″.
Mathematical symbols then obey all the laws of logical symbols, but
two of these laws seem to be inapplicable simply because they are
presupposed in the definition of the mathematical unit. Logic thus lays
down the conditions of number, and the science of arithmetic developed
as it is into all the wondrous branches of mathematical calculus is but
an outgrowth of logical discrimination.
*Principle of Mathematical Inference.*
The universal principle of all reasoning, as I have asserted, is that
which allows us to substitute like for like. I have now to point out
how in the mathematical sciences this principle is involved in each
step of reasoning. It is in these sciences indeed that we meet with the
clearest cases of substitution, and it is the simplicity with which the
principle can be applied which probably led to the comparatively early
perfection of the sciences of geometry and arithmetic. Euclid, and
the Greek mathematicians from the first, recognised *equality* as the
fundamental relation of quantitative thought, but Aristotle rejected
the exactly analogous, but far more general relation of identity, and
thus crippled the formal science of logic as it has descended to the
present day.
Geometrical reasoning starts from the axiom that “things equal to the
same thing are equal to each other.” Two equalities enable us to infer
a third equality; and this is true not only of lines and angles, but
of areas, volumes, numbers, intervals of time, forces, velocities,
degrees of intensity, or, in short, anything which is capable of being
equal or unequal. Two stars equally bright with the same star must be
equally bright with each other, and two forces equally intense with a
third force are equally intense with each other. It is remarkable that
Euclid has not explicitly stated two other axioms, the truth of which
is necessarily implied. The second axiom should be that “Two things of
which one is equal and the other unequal to a third common thing, are
unequal to each other.” An equality and inequality, in short, give an
inequality, and this is equally true with the first axiom of all kinds
of quantity. If Venus, for instance, agrees with Mars in density, but
Mars differs from Jupiter, then Venus differs from Jupiter. A third
axiom must exist to the effect that “Things unequal to the same thing
may or may not be equal to each other.” *Two inequalities give no
ground of inference whatever.* If we only know, for instance, that
Mercury and Jupiter differ in density from Mars, we cannot say whether
or not they agree between themselves. As a fact they do not agree;
but Venus and Mars on the other hand both differ from Jupiter and yet
closely agree with each other. The force of the axioms can be most
clearly illustrated by drawing equal and unequal lines.[91]
[91] *Elementary Lessons in Logic* (Macmillan), p. 123. It is pointed
out in the preface to this Second Edition, that the views here given
were partially stated by Leibnitz.
The general conclusion then must be that where there is equality there
may be inference, but where there is not equality there cannot be
inference. A plain induction will lead us to believe that *equality is
the condition of inference concerning quantity*. All the three axioms
may in fact be summed up in one, to the effect, that “*in whatever
relation one quantity stands to another, it stands in the same relation
to the equal of that other*.”
The active power is always the substitution of equals, and it is an
accident that in a pair of equalities we can make the substitution
in two ways. From *a* = *b* = *c* we can infer *a* = *c*, either by
substituting in *a* = *b* the value of *b* as given in *b* = *c*,
or else by substituting in *b* = *c* the value of *b* as given in
*a* = *b*. In *a* = *b* ~ *d* we can make but the one substitution of
*a* for *b*. In *e* ~ *f* ~ *g* we can make no substitution and get no
inference.
In mathematics the relations in which terms may stand to each other are
far more varied than in pure logic, yet our principle of substitution
always holds true. We may say in the most general manner that *In
whatever relation one quantity stands to another, it stands in the same
relation to the equal of that other.* In this axiom we sum up a number
of axioms which have been stated in more or less detail by algebraists.
Thus, “If equal quantities be added to equal quantities, the sums will
be equal.” To explain this, let
*a* = *b*, *c* = *d*.
Now *a* + *c*, whatever it means, must be identical with itself, so that
*a* + *c* = *a* + *c*.
In one side of this equation substitute for the quantities their
equivalents, and we have the axiom proved
*a* + *c* = *b* + *d*.
The similar axiom concerning subtraction is equally evident, for
whatever *a* - *c* may mean it is equal to *a* - *c*, and therefore by
substitution to *b* - *d*. Again, “if equal quantities be multiplied by
the same or equal quantities, the products will be equal,” For evidently
*ac* = *ac*,
and if for *c* in one side we substitute its equal *d*, we have
*ac* = *ad*,
and a second similar substitution gives us
*ac* = *bd*.
We might prove a like axiom concerning division in an exactly
similar manner. I might even extend the list of axioms and say that
“Equal powers of equal numbers are equal.” For certainly, whatever
*a* × *a* × *a* may mean, it is equal to *a* × *a* × *a*; hence by our
usual substitution it is equal to *b* × *b* × *b*. The same will be
true of roots of numbers and ^{c}√*a* = ^{d}√*b* provided that
the roots are so taken that the root of *a* shall really be related
to *a* as the root of *b* is to *b*. The ambiguity of meaning of an
operation thus fails in any way to shake the universality of the
principle. We may go further and assert that, not only the above common
relations, but all other known or conceivable mathematical relations
obey the same principle. Let Q*a* denote in the most general manner
that we do something with the quantity *a*; then if *a* = *b* it
follows that
Q*a* = Q*b*.
The reader will also remember that one of the most frequent operations
in mathematical reasoning is to substitute for a quantity its equal,
as known either by assumed, natural, or self-evident conditions.
Whenever a quantity appears twice over in a problem, we may apply
what we learn of its relations in one place to its relations in the
other. All reasoning in mathematics, as in other branches of science,
thus involves the principle of treating equals equally, or similars
similarly. In whatever way we employ quantitative reasoning in the
remaining parts of this work, we never can desert the simple principle
on which we first set out.
*Reasoning by Inequalities.*
I have stated that all the processes of mathematical reasoning may
be deduced from the principle of substitution. Exceptions to this
assertion may seem to exist in the use of inequalities. The greater of
a greater is undoubtedly a greater, and what is less than a less is
certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than
Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little
consideration discloses sufficient reason for believing that even in
such cases, where equality does not apparently enter, the force of the
reasoning entirely depends upon underlying and implied equalities.
In the first place, two statements of mere difference do not give
any ground of inference. We learn nothing concerning the comparative
heights of St. Paul’s and Westminster Abbey from the assertions that
they both differ in height from St. Peter’s at Rome. We need something
more than inequality; we require one identity in addition, namely the
identity in direction of the two differences. Thus we cannot employ
inequalities in the simple way in which we do equalities, and, when we
try to express what other conditions are requisite, we find ourselves
lapsing into the use of equalities or identities.
In the second place, every argument by inequalities may be represented
in the form of equalities. We express that *a* is greater than *b* by
the equation
*a* = *b* + *p*, (1)
where *p* is an intrinsically positive quantity, denoting the
difference of *a* and *b*. Similarly we express that *b* is greater
than *c* by the equation
*b* = *c* + *q*, (2)
and substituting for *b* in (1) its value in (2) we have
*a* = *c* + *q* + *p*. (3)
Now as *p* and *q* are both positive, it follows that *a* is greater
than *c*, and we have the exact amount of excess specified. It will be
easily seen that the reasoning concerning that which is less than a
less will result in an equation of the form
*c* = *a* - *r* - *s*.
Every argument by inequalities may then be thrown into the form of an
equality; but the converse is not true. We cannot possibly prove that
two quantities are equal by merely asserting that they are both greater
or both less than another quantity. From *e* > *f* and *g* > *f*, or
*e* < *f* and *g* < *f*, we can infer no relation between *e* and *g*.
And if the reader take the equations *x* = *y* = 3 and attempt to prove
that therefore *x* = 3, by throwing them into inequalities, he will
find it impossible to do so.
From these considerations I gather that reasoning in arithmetic or
algebra by so-called inequalities, is only an imperfectly expressed
reasoning by equalities, and when we want to exhibit exactly and
clearly the conditions of reasoning, we are obliged to use equalities
explicitly. Just as in pure logic a negative proposition, as expressing
mere difference, cannot be the means of inference, so inequality can
never really be the true ground of inference. I do not deny that
affirmation and negation, agreement and difference, equality and
inequality, are pairs of equally fundamental relations, but I assert
that inference is possible only where affirmation, agreement, or
equality, some species of identity in fact, is present, explicitly or
implicitly.
*Arithmetical Reasoning.*
It may seem somewhat inconsistent that I assert number to arise out of
difference or discrimination, and yet hold that no reasoning can be
grounded on difference. Number, of course, opens a most wide sphere
for inference, and a little consideration shows that this is due to
the unlimited series of identities which spring up out of numerical
abstraction. If six people are sitting on six chairs, there is no
resemblance between the chairs and the people in logical character.
But if we overlook all the qualities both of a chair and a person and
merely remember that there are marks by which each of six chairs may
be discriminated from the others, and similarly with the people, then
there arises a resemblance between the chairs and the people, and this
resemblance in number may be the ground of inference. If on another
occasion the chairs are filled by people again, we may infer that these
people resemble the others in number though they need not resemble them
in any other points.
Groups of units are what we really treat in arithmetic. The number
*five* is really 1 + 1 + 1 + 1 + 1, but for the sake of conciseness we
substitute the more compact sign 5, or the name *five*. These names
being arbitrarily imposed in any one manner, an infinite variety of
relations spring up between them which are not in the least arbitrary.
If we define *four* as 1 + 1 + 1 + 1, and *five* as 1 + 1 + 1 + 1 + 1,
then of course it follows that *five* = *four* + 1; but it would be
equally possible to take this latter equality as a definition, in
which case one of the former equalities would become an inference. It
is hardly requisite to decide how we define the names of numbers,
provided we remember that out of the infinitely numerous relations
of one number to others, some one relation expressed in an equality
must be a definition of the number in question and the other relations
immediately become necessary inferences.
In the science of number the variety of classes which can be formed is
altogether infinite, and statements of perfect generality may be made
subject only to difficulty or exception at the lower end of the scale.
Every existing number for instance belongs to the class *m* + 7; that
is, every number must be the sum of another number and seven, except of
course the first six or seven numbers, negative quantities not being
here taken into account. Every number is the half of some other, and so
on. The subject of generalization, as exhibited in mathematical truths,
is an infinitely wide one. In number we are only at the first step of
an extensive series of generalizations. As number is general compared
with the particular things numbered, so we have general symbols for
numbers, and general symbols for relations between undetermined
numbers. There is an unlimited hierarchy of successive generalizations.
*Numerically Definite Reasoning.*
It was first discovered by De Morgan that many arguments are valid
which combine logical and numerical reasoning, although they cannot be
included in the ancient logical formulas. He developed the doctrine of
the “Numerically Definite Syllogism,” fully explained in his *Formal
Logic* (pp. 141–170). Boole also devoted considerable attention to the
determination of what he called “Statistical Conditions,” meaning the
numerical conditions of logical classes. In a paper published among the
Memoirs of the Manchester Literary and Philosophical Society, Third
Series, vol. IV. p. 330 (Session 1869–70), I have pointed out that we
can apply arithmetical calculation to the Logical Alphabet. Having
given certain logical conditions and the numbers of objects in certain
classes, we can either determine the numbers of objects in other
classes governed by those conditions, or can show what further data
are required to determine them. As an example of the kind of questions
treated in numerical logic, and the mode of treatment, I give the
following problem suggested by De Morgan, with my mode of representing
its solution.
“For every man in the house there is a person who is aged; some of the
men are not aged. It follows that some persons in the house are not
men.”[92]
[92] *Syllabus of a Proposed System of Logic*, p. 29.
Now let A = person in house,
B = male,
C = aged.
By enclosing a logical symbol in brackets, let us denote the number of
objects belonging to the class indicated by the symbol. Thus let
(A) = number of persons in house,
(AB) = number of male persons in house,
(ABC) = number of aged male persons in house,
and so on. Now if we use *w* and *w*′ to denote unknown numbers,
the conditions of the problem may be thus stated according to my
interpretation of the words--
(AB) = (AC) - *w*, (1)
that is to say, the number of persons in the house who are aged is at
least equal to, and may exceed, the number of male persons in the house;
(AB*c*) = *w*′, (2)
that is to say, the number of male persons in the house who are not
aged is some unknown positive quantity.
If we develop the terms in (1) by the Law of Duality (pp. 74, 81, 89),
we obtain
(ABC) + (AB*c*) = (ABC) + (A*b*C) - *w*.
Subtracting the common term (ABC) from each side and substituting for
(AB*c*) its value as given in (2), we get at once
(A*b*C) = *w* + *w*′,
and adding (A*bc*) to each side, we have
(A*b*) = (A*bc*) + *w* + *w*′.
The meaning of this result is that the number of persons in the house
who are not men is at least equal to *w* + *w*′, and exceeds it by the
number of persons in the house who are neither men nor aged (A*bc*).
It should be understood that this solution applies only to the terms of
the example quoted above, and not to the general problem for which De
Morgan intended it to serve as an illustration.
As a second instance, let us take the following question:--The
whole number of voters in a borough is *a*; the number against whom
objections have been lodged by liberals is *b*; and the number against
whom objections have been lodged by conservatives is *c*; required the
number, if any, who have been objected to on both sides. Taking
A = voter,
B = objected to by liberals,
C = objected to by conservatives,
then we require the value of (ABC). Now the following equation is
identically true--
(ABC) = (AB) + (AC) + (A*bc*) - (A). (1)
For if we develop all the terms on the second side we obtain
(ABC) = (ABC) + (AB*c*) + (ABC) + (A*b*C) + (A*bc*)
- (ABC) - (AB*c*) - (A*b*C) - (A*bc*);
and striking out the corresponding positive and negative terms, we have
left only (ABC) = (ABC). Since then (1) is necessarily true, we have
only to insert the known values, and we have
(ABC) = *b* + *c* - *a* + (A*bc*).
Hence the number who have received objections from both sides is equal
to the excess, if any, of the whole number of objections over the
number of voters together with the number of voters who have received
no objection (A*bc*).
The following problem illustrates the expression for the common part of
any three classes:--The number of paupers who are blind males, is equal
to the excess, if any, of the sum of the whole number of blind persons,
added to the whole number of male persons, added to the number of those
who being paupers are neither blind nor males, above the sum of the
whole number of paupers added to the number of those who, not being
paupers, are blind, and to the number of those who, not being paupers,
are male.
The reader is requested to prove the truth of the above statement, (1)
by his own unaided common sense; (2) by the Aristotelian Logic; (3) by
the method of numerical logic just expounded; and then to decide which
method is most satisfactory.
*Numerical meaning of Logical Conditions.*
In many cases classes of objects may exist under special logical
conditions, and we must consider how these conditions can be
interpreted numerically. Every logical proposition gives rise to a
corresponding numerical equation. Sameness of qualities occasions
sameness of numbers. Hence if
A = B
denotes the identity of the qualities of A and B, we may conclude that
(A) = (B).
It is evident that exactly those objects, and those objects only, which
are comprehended under A must be comprehended under B. It follows that
wherever we can draw an equation of qualities, we can draw a similar
equation of numbers. Thus, from
A = B = C
we infer
A = C;
and similarly from
(A) = (B) = (C),
meaning that the numbers of A’s and C’s are equal to the number of B’s,
we can infer
(A) = (C).
But, curiously enough, this does not apply to negative propositions and
inequalities. For if
A = B ~ D
means that A is identical with B, which differs from D, it does not
follow that
(A) = (B) ~ (D).
Two classes of objects may differ in qualities, and yet they may agree
in number. This point strongly confirms me in the opinion which I have
already expressed, that all inference really depends upon equations,
not differences.
The Logical Alphabet thus enables us to make a complete analysis of any
numerical problem, and though the symbolical statement may sometimes
seem prolix, I conceive that it really represents the course which the
mind must follow in solving the question. Although thought may outstrip
the rapidity with which the symbols can be written down, yet the mind
does not really follow a different course from that indicated by the
symbols. For a fuller explanation of this natural system of Numerically
Definite Reasoning, with more abundant illustrations and an analysis
of De Morgan’s Numerically Definite Syllogism, I must refer the
reader to the paper[93] in the Memoirs of the Manchester Literary and
Philosophical Society, already mentioned, portions of which, however,
have been embodied in the present section.
[93] It has been pointed out to me by Mr. C. J. Monroe, that section
14 (p. 339) of this paper is erroneous, and ought to be cancelled.
The problem concerning the number of paupers illustrates the answer
which should have been obtained. Mr. A. J. Ellis, F.R.S., had
previously observed that my solution in the paper of De Morgan’s
problem about “men in the house” did not answer the conditions
intended by De Morgan, and I therefore give in the text a more
satisfactory solution.
The reader may be referred, also, to Boole’s writings upon the
subject in the *Laws of Thought*, chap. xix. p. 295, and in a paper
on “Propositions Numerically Definite,” communicated by De Morgan, in
1868, to the Cambridge Philosophical Society, and printed in their
*Transactions*, vol. xi. part ii.
CHAPTER IX.
THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.
Nature may be said to be evolved from the monotony of non-existence
by the creation of diversity. It is plausibly asserted that we are
conscious only so far as we experience difference. Life is change, and
perfectly uniform existence would be no better than non-existence.
Certain it is that life demands incessant novelty, and that nature,
though it probably never fails to obey the same fixed laws, yet
presents to us an apparently unlimited series of varied combinations
of events. It is the work of science to observe and record the kinds
and comparative numbers of such combinations of phenomena, occurring
spontaneously or produced by our interference. Patient and skilful
examination of the records may then disclose the laws imposed on matter
at its creation, and enable us more or less successfully to predict, or
even to regulate, the future occurrence of any particular combination.
The Laws of Thought are the first and most important of all the laws
which govern the combinations of phenomena, and, though they be binding
on the mind, they may also be regarded as verified in the external
world. The Logical Alphabet develops the utmost variety of things and
events which may occur, and it is evident that as each new quality is
introduced, the number of combinations is doubled. Thus four qualities
may occur in 16 combinations; five qualities in 32; six qualities in
64; and so on. In general language, if n be the number of qualities,
2^{n} is the number of varieties of things which may be formed from
them, if there be no conditions but those of logic. This number,
it need hardly be said, increases after the first few terms, in an
extraordinary manner, so that it would require 302 figures to express
the number of combinations in which 1,000 qualities might conceivably
present themselves.
If all the combinations allowed by the Laws of Thought occurred
indifferently in nature, then science would begin and end with those
laws. To observe nature would give us no additional knowledge, because
no two qualities would in the long run be oftener associated than any
other two. We could never predict events with more certainty than we
now predict the throws of dice, and experience would be without use.
But the universe, as actually created, presents a far different and
much more interesting problem. The most superficial observation shows
that some things are constantly associated with other things. The more
mature our examination, the more we become convinced that each event
depends upon the prior occurrence of some other series of events.
Action and reaction are gradually discovered to underlie the whole
scene, and an independent or casual occurrence does not exist except
in appearance. Even dice as they fall are surely determined in their
course by prior conditions and fixed laws. Thus the combinations of
events which can really occur are found to be comparatively restricted,
and it is the work of science to detect these restricting conditions.
In the English alphabet, for instance, we have twenty-six letters. Were
the combinations of such letters perfectly free, so that any letter
could be indifferently sounded with any other, the number of words
which could be formed without any repetition would be 2^{26} - 1, or
67,108,863, equal in number to the combinations of the twenty-seventh
column of the Logical Alphabet, excluding one for the case in which
all the letters would be absent. But the formation of our vocal organs
prevents us from using the far greater part of these conjunctions of
letters. At least one vowel must be present in each word; more than two
consonants cannot usually be brought together; and to produce words
capable of smooth utterance a number of other rules must be observed.
To determine exactly how many words might exist in the English language
under these circumstances, would be an exceedingly complex problem,
the solution of which has never been attempted. The number of existing
English words may perhaps be said not to exceed one hundred thousand,
and it is only by investigating the combinations presented in the
dictionary, that we can learn the Laws of Euphony or calculate the
possible number of words. In this example we have an epitome of the
work and method of science. The combinations of natural phenomena are
limited by a great number of conditions which are in no way brought to
our knowledge except so far as they are disclosed in the examination of
nature.
It is often a very difficult matter to determine the numbers
of permutations or combinations which may exist under various
restrictions. Many learned men puzzled themselves in former centuries
over what were called Protean verses, or verses admitting many
variations in accordance with the Laws of Metre. The most celebrated of
these verses was that invented by Bernard Bauhusius, as follows:[94]--
“Tot tibi sunt dotes, Virgo, quot sidera cœlo.”
[94] Montucla, *Histoire*, &c., vol. iii. p. 388.
One author, Ericius Puteanus, filled forty-eight pages of a work in
reckoning up its possible transpositions, making them only 1022. Other
calculators gave 2196, 3276, 2580 as their results. Wallis assigned
3096, but without much confidence in the accuracy of his result.[95]
It required the skill of James Bernoulli to decide that the number of
transpositions was 3312, under the condition that the sense and metre
of the verse shall be perfectly preserved.
[95] Wallis, *Of Combinations*, &c., p. 119.
In approaching the consideration of the great Inductive problem,
it is very necessary that we should acquire correct notions as
to the comparative numbers of combinations which may exist under
different circumstances. The doctrine of combinations is that part of
mathematical science which applies numerical calculation to determine
the numbers of combinations under various conditions. It is a part of
the science which really lies at the base not only of other sciences,
but of other branches of mathematics. The forms of algebraical
expressions are determined by the principles of combination, and
Hindenburg recognised this fact in his Combinatorial Analysis. The
greatest mathematicians have, during the last three centuries,
given their best powers to the treatment of this subject; it was
the favourite study of Pascal; it early attracted the attention of
Leibnitz, who wrote his curious essay, *De Arte Combinatoria*, at
twenty years of age; James Bernoulli, one of the very profoundest
mathematicians, devoted no small part of his life to the investigation
of the subject, as connected with that of Probability; and in his
celebrated work, *De Arte Conjectandi*, he has so finely described the
importance of the doctrine of combinations, that I need offer no excuse
for quoting his remarks at full length.
“It is easy to perceive that the prodigious variety which appears
both in the works of nature and in the actions of men, and which
constitutes the greatest part of the beauty of the universe, is owing
to the multitude of different ways in which its several parts are
mixed with, or placed near, each other. But, because the number of
causes that concur in producing a given event, or effect, is oftentimes
so immensely great, and the causes themselves are so different one
from another, that it is extremely difficult to reckon up all the
different ways in which they may be arranged or combined together, it
often happens that men, even of the best understandings and greatest
circumspection, are guilty of that fault in reasoning which the writers
on logic call *the insufficient or imperfect enumeration of parts or
cases*: insomuch that I will venture to assert, that this is the chief,
and almost the only, source of the vast number of erroneous opinions,
and those too very often in matters of great importance, which we are
apt to form on all the subjects we reflect upon, whether they relate to
the knowledge of nature, or the merits and motives of human actions.
“It must therefore be acknowledged, that that art which affords a cure
to this weakness, or defect, of our understandings, and teaches us so
to enumerate all the possible ways in which a given number of things
may be mixed and combined together, that we may be certain that we have
not omitted any one arrangement of them that can lead to the object
of our inquiry, deserves to be considered as most eminently useful and
worthy of our highest esteem and attention. And this is the business
of *the art or doctrine of combinations*. Nor is this art or doctrine
to be considered merely as a branch of the mathematical sciences. For
it has a relation to almost every species of useful knowledge that the
mind of man can be employed upon. It proceeds indeed upon mathematical
principles, in calculating the number of the combinations of the things
proposed: but by the conclusions that are obtained by it, the sagacity
of the natural philosopher, the exactness of the historian, the skill
and judgment of the physician, and the prudence and foresight of the
politician may be assisted; because the business of all these important
professions is but *to form reasonable conjectures* concerning the
several objects which engage their attention, and all wise conjectures
are the results of a just and careful examination of the several
different effects that may possibly arise from the causes that are
capable of producing them.”[96]
[96] James Bernoulli, *De Arte Conjectandi*, translated by Baron
Maseres. London, 1795, pp. 35, 36.
*Distinction of Combinations and Permutations.*
We must first consider the deep difference which exists between
Combinations and Permutations, a difference involving important logical
principles, and influencing the form of mathematical expressions.
In *permutation* we recognise varieties of order, treating AB as a
different group from BA. In *combination* we take notice only of the
presence or absence of a certain thing, and pay no regard to its place
in order of time or space. Thus the four letters *a*, *e*, *m*, *n*
can form but one combination, but they occur in language in several
permutations, as *name*, *amen*, *mean*, *mane*.
We have hitherto been dealing with purely logical questions, involving
only combination of qualities. I have fully pointed out in more than
one place that, though our symbols could not but be written in order
of place and read in order of time, the relations expressed had no
regard to place or time (pp. 33, 114). The Law of Commutativeness, in
fact, expresses the condition that in logic we deal with combinations,
and the same law is true of all the processes of algebra. In some
cases, order may be a matter of indifference; it makes no difference,
for instance, whether gunpowder is a mixture of sulphur, carbon, and
nitre, or carbon, nitre, and sulphur, or nitre, sulphur, and carbon,
provided that the substances are present in proper proportions and
well mixed. But this indifference of order does not usually extend to
the events of physical science or the operations of art. The change of
mechanical energy into heat is not exactly the same as the change from
heat into mechanical energy; thunder does not indifferently precede and
follow lightning; it is a matter of some importance that we load, cap,
present, and fire a rifle in this precise order. Time is the condition
of all our thoughts, space of all our actions, and therefore both in
art and science we are to a great extent concerned with permutations.
Language, for instance, treats different permutations of letters as
having different meanings.
Permutations of things are far more numerous than combinations of those
things, for the obvious reason that each distinct thing is regarded
differently according to its place. Thus the letters A, B, C, will
make different permutations according as A stands first, second, or
third; having decided the place of A, there are two places between
which we may choose for B; and then there remains but one place for
C. Accordingly the permutations of these letters will be altogether
3 × 2 × 1 or 6 in number. With four things or letters, A, B, C, D,
we shall have four choices of place for the first letter, three for
the second, two for the third, and one for the fourth, so that there
will be altogether, 4 × 3 × 2 × 1, or 24 permutations. The same simple
rule applies in all cases; beginning with the whole number of things
we multiply at each step by a number decreased by a unit. In general
language, if *n* be the number of things in a combination, the number
of permutations is
*n* (*n* - 1)(*n* - 2) .... 4 . 3 . 2 . 1.
If we were to re-arrange the names of the days of the week, the
possible arrangements out of which we should have to choose the new
order, would be no less than 7 . 6 . 5 . 4 . 3 . 2 . 1, or 5040, or,
excluding the existing order, 5039.
The reader will see that the numbers which we reach in questions
of permutation, increase in a more extraordinary manner even than
in combination. Each new object or term doubles the number of
combinations, but increases the permutations by a factor continually
growing. Instead of 2 × 2 × 2 × 2 × .... we have 2 × 3 × 4 × 5 × ....
and the products of the latter expression immensely exceed those of the
former. These products of increasing factors are frequently employed,
as we shall see, in questions both of permutation and combination. They
are technically called *factorials*, that is to say, the product of all
integer numbers, from unity up to any number *n* is the *factorial*
of *n*, and is often indicated symbolically by *n*!. I give below the
factorials up to that of twelve:--
24 = 1 . 2 . 3 . 4
120 = 1 . 2 ... 5
720 = 1 . 2 ... 6
5,040 = 7!
40,320 = 8!
362,880 = 9!
3,628,800 = 10!
39,916,800 = 11!
479,001,600 = 12!
The factorials up to 36! are given in Rees’s ‘Cyclopædia,’ art.
*Cipher*, and the logarithms of factorials up to 265! are to be
found at the end of the table of logarithms published under the
superintendence of the Society for the Diffusion of Useful Knowledge
(p. 215). To express the factorial 265! would require 529 places of
figures.
Many writers have from time to time remarked upon the extraordinary
magnitude of the numbers with which we deal in this subject. Tacquet
calculated[97] that the twenty-four [sic] letters of the alphabet may
be arranged in more than 620 thousand trillions of orders; and Schott
estimated[98] that if a thousand millions of men were employed for the
same number of years in writing out these arrangements, and each man
filled each day forty pages with forty arrangements in each, they would
not have accomplished the task, as they would have written only 584
thousand trillions instead of 620 thousand trillions.
[97] *Arithmeticæ Theoria.* Ed. Amsterd. 1704. p. 517.
[98] Rees’s *Cyclopædia*, art. *Cipher*.
In some questions the number of permutations may be restricted
and reduced by various conditions. Some things in a group may
be undistinguishable from others, so that change of order will
produce no difference. Thus if we were to permutate the letters of
the name *Ann*, according to our previous rule, we should obtain
3 × 2 × 1, or 6 orders; but half of these arrangements would be
identical with the other half, because the interchange of the two
*n*’s has no effect. The really different orders will therefore be
(3 . 2 . 1)/(1 . 2) or 3, namely *Ann*, *Nan*, *Nna*. In the word
*utility* there are two *i*’s and two *t*’s, in respect of both of
which pairs the numbers of permutations must be halved. Thus we obtain
(7 . 6 . 5 . 4 . 3 . 2 . 1)/(1 . 2 . 1 . 2) or 1260, as the number of
permutations. The simple rule evidently is--when some things or letters
are undistinguished, proceed in the first place to calculate all the
possible permutations as if all were different, and then divide by
the numbers of possible permutations of those series of things which
are not distinguished, and of which the permutations have therefore
been counted in excess. Thus since the word *Utilitarianism* contains
fourteen letters, of which four are *i*’s, two *a*’s, and two *t*’s,
the number of distinct arrangements will be found by dividing the
factorial of 14, by the factorials of 4, 2, and 2, the result being
908,107,200. From the letters of the word *Mississippi* we can get in
like manner 11!/(4! × 4! × 2!) or 34,650 permutations, which is not
the one-thousandth part of what we should obtain were all the letters
different.
*Calculation of Number of Combinations.*
Although in many questions both of art and science we need to calculate
the number of permutations on account of their own interest, it far
more frequently happens in scientific subjects that they possess but
an indirect interest. As I have already pointed out, we almost always
deal in the logical and mathematical sciences with *combinations*, and
variety of order enters only through the inherent imperfections of our
symbols and modes of calculation. Signs must be used in some order,
and we must withdraw our attention from this order before the signs
correctly represent the relations of things which exist neither before
nor after each other. Now, it often happens that we cannot choose all
the combinations of things, without first choosing them subject to the
accidental variety of order, and we must then divide by the number of
possible variations of order, that we may get to the true number of
pure combinations.
Suppose that we wish to determine the number of ways in which we can
select a group of three letters out of the alphabet, without allowing
the same letter to be repeated. At the first choice we can take any
one of 26 letters; at the next step there remain 25 letters, any one
of which may be joined with that already taken; at the third step
there will be 24 choices, so that apparently the whole number of ways
of choosing is 26 × 25 × 24. But the fact that one choice succeeded
another has caused us to obtain the same combinations of letters in
different orders; we should get, for instance, *a*, *p*, *r* at one
time, and *p*, *r*, *a* at another, and every three distinct letters
will appear six times over, because three things can be arranged in
six permutations. To get the number of combinations, then, we must
divide the whole number of ways of choosing, by six, the number of
permutations of three things, obtaining (26 × 25 × 24)/(1 × 2 × 3) or
2,600.
It is apparent that we need the doctrine of combinations in order
that we may in many questions counteract the exaggerating effect of
successive selection. If out of a senate of 30 persons we have to
choose a committee of 5, we may choose any of 30 first, any of 29 next,
and so on, in fact there will be 30 × 29 × 28 × 27 × 26 selections;
but as the actual character of the members of the committee will not
be affected by the accidental order of their selection, we divide by
1 × 2 × 3 × 4 × 5, and the possible number of different committees will
be 142,506. Similarly if we want to calculate the number of ways in
which the eight major planets may come into conjunction, it is evident
that they may meet either two at a time or three at a time, or four or
more at a time, and as nothing is said as to the relative order or
place in the conjunction, we require the number of combinations. Now
a selection of 2 out of 8 is possible in (8 . 7)/(1 . 2) or 28 ways;
of 3 out of 8 in (8 . 7 . 6)/(1 . 2 . 3) or 56 ways; of 4 out of 8 in
(8 . 7 . 6 . 5)/(1 . 2 . 3 . 4) or 70 ways; and it may be similarly
shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers
of ways are 56, 28, 8, and 1. Thus we have solved the whole question
of the variety of conjunctions of eight planets; and adding all the
numbers together, we find that 247 is the utmost possible number of
modes of meeting.
In general algebraic language, we may say that a group of *m* things
may be chosen out of a total number of *n* things, in a number of
combinations denoted by the formula
(*n* . (*n*-1)(*n*-2)(*n*-3) .... (*n* - *m* + 1))/(1 . 2 . 3 . 4 .... *m*)
The extreme importance and significance of this formula seems to have
been first adequately recognised by Pascal, although its discovery
is attributed by him to a friend, M. de Ganières.[99] We shall find
it perpetually recurring in questions both of combinations and
probability, and throughout the formulæ of mathematical analysis traces
of its influence may be noticed.
[99] *Œuvres Complètes de Pascal* (1865), vol. iii. p. 302. Montucla
states the name as De Gruières, *Histoire des Mathématiques*,
vol. iii. p. 389.
*The Arithmetical Triangle.*
The Arithmetical Triangle is a name long since given to a series
of remarkable numbers connected with the subject we are treating.
According to Montucla[100] “this triangle is in the theory of
combinations and changes of order, almost what the table of Pythagoras
is in ordinary arithmetic, that is to say, it places at once under the
eyes the numbers required in a multitude of cases of this theory.” As
early as 1544 Stifels had noticed the remarkable properties of these
numbers and the mode of their evolution. Briggs, the inventor of the
common system of logarithms, was so struck with their importance that
he called them the Abacus Panchrestus. Pascal, however, was the first
who wrote a distinct treatise on these numbers, and gave them the name
by which they are still known. But Pascal did not by any means exhaust
the subject, and it remained for James Bernoulli to demonstrate fully
the importance of the *figurate numbers*, as they are also called. In
his treatise *De Arte Conjectandi*, he points out their application
in the theory of combinations and probabilities, and remarks of the
Arithmetical Triangle, “It not only contains the clue to the mysterious
doctrine of combinations, but it is also the ground or foundation of
most of the important and abstruse discoveries that have been made in
the other branches of the mathematics.”[101]
[100] *Histoire des Mathématiques*, vol. iii. p. 378.
[101] Bernoulli, *De Arte Conjectandi*, translated by Francis
Maseres. London, 1795, p. 75.
The numbers of the triangle can be calculated in a very easy manner by
successive additions. We commence with unity at the apex; in the next
line we place a second unit to the right of this; to obtain the third
line of figures we move the previous line one place to the right, and
add them to the same figures as they were before removal; we can then
repeat the same process *ad infinitum*. The fourth line of figures,
for instance, contains 1, 3, 3, 1; moving them one place and adding as
directed we obtain:--
Fourth line ... 1 3 3 1
1 3 3 1
--------------
Fifth line .... 1 4 6 4 1
1 4 6 4 1
----------------
Sixth line .... 1 5 10 10 5 1
Carrying out this simple process through ten more steps we obtain the
first seventeen lines of the Arithmetical Triangle as printed on the
next page. Theoretically speaking the Triangle must be regarded as
infinite in extent, but the numbers increase so rapidly that it soon
becomes impracticable to continue the table. The longest table of the
numbers which I have found is in Fortia’s “Traité des Progressions”
(p. 80), where they are given up to the fortieth line and the ninth
column.
THE ARITHMETICAL TRIANGLE.
Line. First Column.
1 1 Second Column.
2 1 1 Third Column.
3 1 2 1 Fourth Column.
4 1 3 3 1 Fifth Column.
5 1 4 6 4 1 Sixth Column.
6 1 5 10 10 5 1 Seventh Column.
7 1 6 15 20 15 6 1 Eighth Column.
8 1 7 21 35 35 21 7 1 Ninth Column.
9 1 8 28 56 70 56 28 8 1 Tenth Column.
10 1 9 36 84 126 126 84 36 9 1 Eleventh Column.
11 1 10 45 120 210 252 210 120 45 10 1 Twelfth Column.
12 1 11 55 165 330 462 462 330 165 55 11 1 Thirteenth Column.
13 1 12 66 220 495 792 924 792 495 220 66 12 1 Fourteenth Column.
14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Fifteenth Column.
15 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 Sixteenth Column.
16 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 Seventeenth Col.
17 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
Examining these numbers, we find that they are connected by an
unlimited series of relations, a few of the more simple of which may be
noticed. Each vertical column of numbers exactly corresponds with an
oblique series descending from left to right, so that the triangle is
perfectly symmetrical in its contents. The first column contains only
*units*; the second column contains the *natural numbers*, 1, 2, 3,
&c.; the third column contains a remarkable series of numbers, 1, 3,
6, 10, 15, &c., which have long been called *the triangular numbers*,
because they correspond with the numbers of balls which may be arranged
in a triangular form, thus--
[Illustration]
The fourth column contains the *pyramidal numbers*, so called because
they correspond to the numbers of equal balls which can be piled in
regular triangular pyramids. Their differences are the triangular
numbers. The numbers of the fifth column have the pyramidal numbers
for their differences, but as there is no regular figure of which
they express the contents, they have been arbitrarily called the
*trianguli-triangular numbers*. The succeeding columns have, in a
similar manner, been said to contain the *trianguli-pyramidal*, the
*pyramidi-pyramidal* numbers, and so on.[102]
[102] Wallis’s *Algebra*, Discourse of Combinations, &c., p. 109.
From the mode of formation of the table, it follows that the
differences of the numbers in each column will be found in the
preceding column to the left. Hence the *second differences*, or the
*differences of differences*, will be in the second column to the left
of any given column, the third differences in the third column, and so
on. Thus we may say that unity which appears in the first column is the
*first difference* of the numbers in the second column; the *second
difference* of those in the third column; the *third difference* of
those in the fourth, and so on. The triangle is seen to be a complete
classification of all numbers according as they have unity for any of
their differences.
Since each line is formed by adding the previous line to itself, it
is evident that the sum of the numbers in each horizontal line must be
double the sum of the numbers in the line next above. Hence we know,
without making the additions, that the successive sums must be 1, 2,
4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the
Logical Alphabet. Speaking generally, the sum of the numbers in the
*n*th line will be 2^{*n* - 1}.
Again, if the whole of the numbers down to any line be added together,
we shall obtain a number less by unity than some power of 2; thus,
the first line gives 1 or 2^{1} - 1; the first two lines give 3 or
2^{2} - 1; the first three lines 7 or 2^{3} - 1; the first six lines
give 63 or 2^{6} - 1; or, speaking in general language, the sum of the
first *n* lines is 2^{*n*} - 1. It follows that the sum of the numbers
in any one line is equal to the sum of those in all the preceding
lines increased by a unit. For the sum of the *n*th line is, as
already shown, 2^{*n* - 1}, and the sum of the first *n* - 1 lines is
2^{*n* - 1} - 1, or less by a unit.
This account of the properties of the figurate numbers does not
approach completeness; a considerable, probably an unlimited, number of
less simple and obvious relations might be traced out. Pascal, after
giving many of the properties, exclaims[103]: “Mais j’en laisse bien
plus que je n’en donne; c’est une chose étrange combien il est fertile
en propriétés! Chacun peut s’y exercer.” The arithmetical triangle may
be considered a natural classification of numbers, exhibiting, in the
most complete manner, their evolution and relations in a certain point
of view. It is obvious that in an unlimited extension of the triangle,
each number, with the single exception of the number *two*, has at
least two places.
[103] *Œuvres Complètes*, vol. iii. p. 251.
Though the properties above explained are highly curious, the greatest
value of the triangle arises from the fact that it contains a complete
statement of the values of the formula (p. 182), for the numbers of
combinations of *m* things out of *n*, for all possible values of *m*
and *n*. Out of seven things one may be chosen in seven ways, and
seven occurs in the eighth line of the second column. The combinations
of two things chosen out of seven are (7 × 6)/(1 × 2) or 21, which
is the third number in the eighth line. The combinations of three
things out of seven are (7 × 6 × 5)/(1 × 2 × 3) or 35, which appears
fourth in the eighth line. In a similar manner, in the fifth, sixth,
seventh, and eighth columns of the eighth line I find it stated in
how many ways I can select combinations of 4, 5, 6, and 7 things out
of 7. Proceeding to the ninth line, I find in succession the number
of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out
of 8 things. In general language, if I wish to know in how many ways
*m* things can be selected in combinations out of *n* things, I must
look in the *n* + 1^{th} line, and take the *m* + 1^{th} number, as
the answer. In how many ways, for instance, can a subcommittee of
five be chosen out of a committee of nine. The answer is 126, and
is the sixth number in the tenth line; it will be found equal to
(9 . 8 . 7 . 6 . 5)/(1 . 2 . 3 . 4 . 5), which our formula (p. 182)
gives.
The full utility of the figurate numbers will be more apparent when
we reach the subject of probabilities, but I may give an illustration
or two in this place. In how many ways can we arrange four pennies as
regards head and tail? The question amounts to asking in how many ways
we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the *fifth*
line of the triangle gives us the complete answer, thus--
We can select No head and 4 tails in 1 way.
" 1 head and 3 tails in 4 ways.
" 2 heads and 2 tails in 6 ways.
" 3 heads and 1 tail in 4 ways.
" 4 heads and 0 tail in 1 way.
The total number of different cases is 16, or 2^{4}, and when we come
to the next chapter, it will be found that these numbers give us the
respective probabilities of all throws with four pennies.
I gave in p. 181 a calculation of the number of ways in which eight
planets can meet in conjunction; the reader will find all the numbers
detailed in the ninth line of the arithmetical triangle. The sum of the
whole line is 2^{8} or 256; but we must subtract a unit for the case
where no planet appears, and 8 for the 8 cases in which only one planet
appears; so that the total number of conjunctions is 2^{8} -1 - 8
or 247. If an organ has eleven stops we find in the twelfth line the
numbers of ways in which we can draw them, 1, 2, 3, or more at a time.
Thus there are 462 ways of drawing five stops at once, and as many of
drawing six stops. The total number of ways of varying the sound is
2048, including the single case in which no stop at all is drawn.
One of the most important scientific uses of the arithmetical triangle
consists in the information which it gives concerning the comparative
frequency of divergencies from an average. Suppose, for the sake of
argument, that all persons were naturally of the equal stature of five
feet, but enjoyed during youth seven independent chances of growing one
inch in addition. Of these seven chances, one, two, three, or more,
may happen favourably to any individual; but, as it does not matter
what the chances are, so that the inch is gained, the question really
turns upon the number of combinations of 0, 1, 2, 3, &c., things out of
seven. Hence the eighth line of the triangle gives us a complete answer
to the question, as follows:--
Out of every 128 people--
Feet Inches.
One person would have the stature of 5 0
7 persons " " 5 1
21 persons " " 5 2
35 persons " " 5 3
35 persons " " 5 4
21 persons " " 5 5
7 persons " " 5 6
1 person " " 5 7
By taking a proper line of the triangle, an answer may be had under
any more natural supposition. This theory of comparative frequency of
divergence from an average, was first adequately noticed by Quetelet,
and has lately been employed in a very interesting and bold manner by
Mr. Francis Galton,[104] in his remarkable work on “Hereditary Genius.”
We shall afterwards find that the theory of error, to which is made the
ultimate appeal in cases of quantitative investigation, is founded upon
the comparative numbers of combinations as displayed in the triangle.
[104] See also Galton’s Lecture at the Royal Institution, 27th
February, 1874; Catalogue of the Special Loan Collection of
Scientific Instruments, South Kensington, Nos. 48, 49; and Galton,
*Philosophical Magazine*, January 1875.
*Connection between the Arithmetical Triangle and the Logical Alphabet.*
There exists a close connection between the arithmetical triangle
described in the last section, and the series of combinations of
letters called the Logical Alphabet. The one is to mathematical science
what the other is to logical science. In fact the figurate numbers, or
those exhibited in the triangle, are obtained by summing up the logical
combinations. Accordingly, just as the total of the numbers in each
line of the triangle is twice as great as that for the preceding line
(p. 186), so each column of the Alphabet (p. 94) contains twice as many
combinations as the preceding one. The like correspondence also exists
between the sums of all the lines of figures down to any particular
line, and of the combinations down to any particular column.
By examining any column of the Logical Alphabet we find that the
combinations naturally group themselves according to the figurate
numbers. Take the combinations of the letters A, B, C, D; they consist
of all the ways in which I can choose four, three, two, one, or none of
the four letters, filling up the vacant spaces with negative terms.
There is one combination, ABCD, in which all the positive letters are
present; there are four combinations in each of which three positive
letters are present; six in which two are present; four in which only
one is present; and, finally, there is the single case, *abcd*, in
which all positive letters are absent. These numbers, 1, 4, 6, 4, 1,
are those of the fifth line of the arithmetical triangle, and a like
correspondence will be found to exist in each column of the Logical
Alphabet.
Numerical abstraction, it has been asserted, consists in overlooking
the kind of difference, and retaining only a consciousness of its
existence (p. 158). While in logic, then, we have to deal with each
combination as a separate kind of thing, in arithmetic we distinguish
only the classes which depend upon more or less positive terms being
present, and the numbers of these classes immediately produce the
numbers of the arithmetical triangle.
It may here be pointed out that there are two modes in which we
can calculate the whole number of combinations of certain things.
Either we may take the whole number at once as shown in the Logical
Alphabet, in which case the number will be some power of two, or else
we may calculate successively, by aid of permutations, the number of
combinations of none, one, two, three things, and so on. Hence we
arrive at a necessary identity between two series of numbers. In the
case of four things we shall have
2 = 1 + 4/1 + (4 . 3)/(1 . 2) + (4 . 3 . 2)/(1 . 2 . 3) +
(4 . 3 . 2 . 1)/(1 . 2 . 3 . 4).
In a general form of expression we shall have
2 = 1 + *n*/1 + (*n* . (*n* - 1))/(1 . 2) + (*n*
(*n* - 1)(*n* - 2))/(1 . 2 . 3) + &c.,
the terms being continued until they cease to have any value. Thus we
arrive at a proof of simple cases of the Binomial Theorem, of which
each column of the Logical Alphabet is an exemplification. It may be
shown that all other mathematical expansions likewise arise out of
simple processes of combination, but the more complete consideration of
this subject must be deferred to another work.
*Possible Variety of Nature and Art.*
We cannot adequately understand the difficulties which beset us in
certain branches of science, unless we have some clear idea of the vast
numbers of combinations or permutations which may be possible under
certain conditions. Thus only can we learn how hopeless it would be
to attempt to treat nature in detail, and exhaust the whole number of
events which might arise. It is instructive to consider, in the first
place, how immensely great are the numbers of combinations with which
we deal in many arts and amusements.
In dealing a pack of cards, the number of hands, of thirteen cards
each, which can be produced is evidently 52 × 51 × 50 × ... × 40
divided by 1 × 2 × 3 ... × 13. or 635,013,559,600. But in whist
four hands are simultaneously held, and the number of distinct
deals becomes so vast that it would require twenty-eight figures to
express it. If the whole population of the world, say one thousand
millions of persons, were to deal cards day and night, for a hundred
million of years, they would not in that time have exhausted one
hundred-thousandth part of the possible deals. Even with the same hands
of cards the play may be almost infinitely varied, so that the complete
variety of games at whist which may exist is almost incalculably great.
It is in the highest degree improbable that any one game of whist was
ever exactly like another, except it were intentionally so.
The end of novelty in art might well be dreaded, did we not find that
nature at least has placed no attainable limit, and that the deficiency
will lie in our inventive faculties. It would be a cheerless time
indeed when all possible varieties of melody were exhausted, but it
is readily shown that if a peal of twenty-four bells had been rung
continuously from the so-called beginning of the world to the present
day, no approach could have been made to the completion of the possible
changes. Nay, had every single minute been prolonged to 10,000 years,
still the task would have been unaccomplished.[105] As regards ordinary
melodies, the eight notes of a single octave give more than 40,000
permutations, and two octaves more than a million millions. If we were
to take into account the semitones, it would become apparent that it
is impossible to exhaust the variety of music. When the late Mr. J. S.
Mill, in a depressed state of mind, feared the approaching exhaustion
of musical melodies, he had certainly not bestowed sufficient study on
the subject of permutations.
[105] Wallis, *Of Combinations*, p. 116, quoting Vossius.
Similar considerations apply to the possible number of natural
substances, though we cannot always give precise numerical results. It
was recommended by Hatchett[106] that a systematic examination of all
alloys of metals should be carried out, proceeding from the binary ones
to more complicated ternary or quaternary ones. He can hardly have been
aware of the extent of his proposed inquiry. If we operate only upon
thirty of the known metals, the number of binary alloys would be 435,
of ternary alloys 4060, of quaternary 27,405, without paying regard
to the varying proportions of the metals, and only regarding the kind
of metal. If we varied all the ternary alloys by quantities not less
than one per cent., the number of these alloys would be 11,445,060.
An exhaustive investigation of the subject is therefore out of the
question, and unless some laws connecting the properties of the alloy
and its components can be discovered, it is not apparent how our
knowledge of them can ever be more than fragmentary.
[106] *Philosophical Transactions* (1803), vol. xciii. p. 193.
The possible variety of definite chemical compounds, again, is
enormously great. Chemists have already examined many thousands
of inorganic substances, and a still greater number of organic
compounds;[107] they have nevertheless made no appreciable impression
on the number which may exist. Taking the number of elements at
sixty-one, the number of compounds containing different selections of
four elements each would be more than half a million (521,855). As the
same elements often combine in many different proportions, and some of
them, especially carbon, have the power of forming an almost endless
number of compounds, it would hardly be possible to assign any limit
to the number of chemical compounds which may be formed. There are
branches of physical science, therefore, of which it is unlikely that
scientific men, with all their industry, can ever obtain a knowledge in
any appreciable degree approaching to completeness.
[107] Hofmann’s *Introduction to Chemistry*, p. 36.
*Higher Orders of Variety.*
The consideration of the facts already given in this chapter will
not produce an adequate notion of the possible variety of existence,
unless we consider the comparative numbers of combinations of different
orders. By a combination of a higher order, I mean a combination of
groups, which are themselves groups. The immense numbers of compounds
of carbon, hydrogen, and oxygen, described in organic chemistry, are
combinations of a second order, for the atoms are groups of groups.
The wave of sound produced by a musical instrument may be regarded as
a combination of motions; the body of sound proceeding from a large
orchestra is therefore a complex aggregate of sounds, each in itself
a complex combination of movements. All literature may be said to be
developed out of the difference of white paper and black ink. From the
unlimited number of marks which might be chosen we select twenty-six
conventional letters. The pronounceable combinations of letters are
probably some trillions in number. Now, as a sentence is a selection
of words, the possible sentences must be inconceivably more numerous
than the words of which it may be composed. A book is a combination
of sentences, and a library is a combination of books. A library,
therefore, may be regarded as a combination of the fifth order, and the
powers of numerical expression would be severely tasked in attempting
to express the number of distinct libraries which might be constructed.
The calculation, of course, would not be possible, because the union
of letters in words, of words in sentences, and of sentences in books,
is governed by conditions so complex as to defy analysis. I wish only
to point out that the infinite variety of literature, existing or
possible, is all developed out of one fundamental difference. Galileo
remarked that all truth is contained in the compass of the alphabet. He
ought to have said that it is all contained in the difference of ink
and paper.
One consequence of successive combination is that the simplest marks
will suffice to express any information. Francis Bacon proposed for
secret writing a biliteral cipher, which resolves all letters of the
alphabet into permutations of the two letters *a* and *b*. Thus A
was *aaaaa*, B *aaaab*, X *babab*, and so on.[108] In a similar way,
as Bacon clearly saw, any one difference can be made the ground of a
code of signals; we can express, as he says, *omnia per omnia*. The
Morse alphabet uses only a succession of long and short marks, and
other systems of telegraphic language employ right and left strokes.
A single lamp obscured at various intervals, long or short, may be
made to spell out any words, and with two lamps, distinguished by
colour, position, or any other circumstance, we could at once represent
Bacon’s biliteral alphabet. Babbage ingeniously suggested that every
lighthouse in the world should be made to spell out its own name or
number perpetually, by flashes or obscurations of various duration
and succession. A system like that of Babbage is now being applied
to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John
Hopkinson.
[108] *Works*, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s
*Encyclopædia*, art. *Cipher*.
Let us calculate the numbers of combinations of different orders which
may arise out of the presence or absence of a single mark, say A. In
these figures
+---+---+ +---+---+ +---+---+ +---+---+
| A | A | | A | | | | A | | | |
+---+---+ +---+---+ +---+---+ +---+---+
we have four distinct varieties. Form them into a group of a higher
order, and consider in how many ways we may vary that group by omitting
one or more of the component parts. Now, as there are four parts,
and any one may be present or absent, the possible varieties will
be 2 × 2 × 2 × 2, or 16 in number. Form these into a new whole, and
proceed again to create variety by omitting any one or more of the
sixteen. The number of possible changes will now be 2 . 2 . 2 . 2 .
2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2, or 2^{16}, and we can
repeat the process again and again. We are imagining the creation of
objects, whose numbers are represented by the successive orders of the
powers of *two*.
At the first step we have 2; at the next 2^{2}, or 4; at the third
(2^{2})^{2}, or 16, numbers of very moderate amount. Let the reader
calculate the next term, ((2^{2})^{2})^{2}, and he will be surprised
to find it leap up to 65,536. But at the next step he has to calculate
the value of 65,536 *two*’s multiplied together, and it is so great
that we could not possibly compute it, the mere expression of the
result requiring 19,729 places of figures. But go one step more and we
pass the bounds of all reason. The sixth order of the powers of *two*
becomes so great, that we could not even express the number of figures
required in writing it down, without using about 19,729 figures for
the purpose. The successive orders of the powers of two have then the
following values so far as we can succeed in describing them:--
First order 2
Second order 4
Third order 16
Fourth order 65,536
Fifth order, number expressed by 19,729 figures.
Sixth order, number expressed by
figures, to express the number
of which figures would require
about 19,729 figures.
It may give us some notion of infinity to remember that at this sixth
step`, having long surpassed all bounds of intuitive conception, we
make no approach to a limit. Nay, were we to make a hundred such steps,
we should be as far away as ever from actual infinity.
It is well worth observing that our powers of expression rapidly
overcome the possible multitude of finite objects which may exist in
any assignable space. Archimedes showed long ago, in one of the most
remarkable writings of antiquity, the *Liber de Arcnæ Numero*, that
the grains of sand in the world could be numbered, or rather, that
if numbered, the result could readily be expressed in arithmetical
notation. Let us extend his problem, and ascertain whether we could
express the number of atoms which could exist in the visible universe.
The most distant stars which can now be seen by telescopes--those of
the sixteenth magnitude--are supposed to have a distance of about
33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for
supposing that there do not exist more than from 3 × 10^{24} to 10^{26}
molecules in a cubic centimetre of a solid or liquid substance.[109]
Assuming these data to be true, for the sake of argument, a simple
calculation enables us to show that the almost inconceivably vast
sphere of our stellar system if entirely filled with solid matter,
would not contain more than about 68 × 10^{90} atoms, that is to say,
a number requiring for its expression 92 places of figures. Now, this
number would be immensely less than the fifth order of the powers of
two.
[109] *Nature*, vol. i. p. 553.
In the variety of logical relations, which may exist between a certain
number of logical terms, we also meet a case of higher combinations.
We have seen (p. 142) that with only six terms the number of possible
selections of combinations is 18,446,744,073,709,551,616. Considering
that it is the most common thing in the world to use an argument
involving six objects or terms, it may excite some surprise that the
complete investigation of the relations in which six such terms may
stand to each other, should involve an almost inconceivable number of
cases. Yet these numbers of possible logical relations belong only to
the second order of combinations.
CHAPTER X.
THE THEORY OF PROBABILITY.
The subject upon which we now enter must not be regarded as an isolated
and curious branch of speculation. It is the necessary basis of the
judgments we make in the prosecution of science, or the decisions we
come to in the conduct of ordinary affairs. As Butler truly said,
“Probability is the very guide of life.” Had the science of numbers
been studied for no other purpose, it must have been developed for
the calculation of probabilities. All our inferences concerning the
future are merely probable, and a due appreciation of the degree of
probability depends upon a comprehension of the principles of the
subject. I am convinced that it is impossible to expound the methods of
induction in a sound manner, without resting them upon the theory of
probability. Perfect knowledge alone can give certainty, and in nature
perfect knowledge would be infinite knowledge, which is clearly beyond
our capacities. We have, therefore, to content ourselves with partial
knowledge--knowledge mingled with ignorance, producing doubt.
A great difficulty in this subject consists in acquiring a precise
notion of the matter treated. What is it that we number, and measure,
and calculate in the theory of probabilities? Is it belief, or opinion,
or doubt, or knowledge, or chance, or necessity, or want of art? Does
probability exist in the things which are probable, or in the mind
which regards them as such? The etymology of the name lends us no
assistance: for, curiously enough, *probable* is ultimately the same
word as *provable*, a good instance of one word becoming differentiated
to two opposite meanings.
Chance cannot be the subject of the theory, because there is really
no such thing as chance, regarded as producing and governing events.
The word chance signifies *falling*, and the notion of falling is
continually used as a simile to express uncertainty, because we can
seldom predict how a die, a coin, or a leaf will fall, or when a bullet
will hit the mark. But everyone sees, after a little reflection, that
it is in our knowledge the deficiency lies, not in the certainty of
nature’s laws. There is no doubt in lightning as to the point it shall
strike; in the greatest storm there is nothing capricious; not a grain
of sand lies upon the beach, but infinite knowledge would account for
its lying there; and the course of every falling leaf is guided by the
principles of mechanics which rule the motions of the heavenly bodies.
Chance then exists not in nature, and cannot coexist with knowledge;
it is merely an expression, as Laplace remarked, for our ignorance
of the causes in action, and our consequent inability to predict the
result, or to bring it about infallibly. In nature the happening of
an event has been pre-determined from the first fashioning of the
universe. *Probability belongs wholly to the mind.* This is proved by
the fact that different minds may regard the very same event at the
same time with widely different degrees of probability. A steam-vessel,
for instance, is missing and some persons believe that she has sunk
in mid-ocean; others think differently. In the event itself there can
be no such uncertainty; the steam-vessel either has sunk or has not
sunk, and no subsequent discussion of the probable nature of the event
can alter the fact. Yet the probability of the event will really vary
from day to day, and from mind to mind, according as the slightest
information is gained regarding the vessels met at sea, the weather
prevailing there, the signs of wreck picked up, or the previous
condition of the vessel. Probability thus belongs to our mental
condition, to the light in which we regard events, the occurrence
or non-occurrence of which is certain in themselves. Many writers
accordingly have asserted that probability is concerned with degree or
quantity of belief. De Morgan says,[110] “By degree of probability
we really mean or ought to mean degree of belief.” The late Professor
Donkin expressed the meaning of probability as “quantity of belief;”
but I have never felt satisfied with such definitions of probability.
The nature of *belief* is not more clear to my mind than the notion
which it is used to define. But an all-sufficient objection is, that
*the theory does not measure what the belief is, but what it ought to
be*. Few minds think in close accordance with the theory, and there
are many cases of evidence in which the belief existing is habitually
different from what it ought to be. Even if the state of belief in any
mind could be measured and expressed in figures, the results would be
worthless. The value of the theory consists in correcting and guiding
our belief, and rendering our states of mind and consequent actions
harmonious with our knowledge of exterior conditions.
[110] *Formal Logic*, p. 172.
This objection has been clearly perceived by some of those who still
used quantity of belief as a definition of probability. Thus De
Morgan adds--“Belief is but another name for imperfect knowledge.”
Donkin has well said that the quantity of belief is “always relative
to a particular state of knowledge or ignorance; but it must be
observed that it is absolute in the sense of not being relative to
any individual mind; since, the same information being presupposed,
all minds *ought* to distribute their belief in the same way.”[111]
Boole seemed to entertain a like view, when he described the theory as
engaged with “the equal distribution of ignorance;”[112] but we may
just as well say that it is engaged with the equal distribution of
knowledge.
[111] *Philosophical Magazine*, 4th Series, vol. i. p. 355.
[112] *Transactions of the Royal Society of Edinburgh*, vol. xxi.
part 4.
I prefer to dispense altogether with this obscure word belief, and to
say that the theory of probability deals with *quantity of knowledge*,
an expression of which a precise explanation and measure can presently
be given. An event is only probable when our knowledge of it is diluted
with ignorance, and exact calculation is needed to discriminate how
much we do and do not know. The theory has been described by some
writers as professing *to evolve knowledge out of ignorance*; but as
Donkin admirably remarked, it is really “a method of avoiding the
erection of belief upon ignorance.” It defines rational expectation
by measuring the comparative amounts of knowledge and ignorance, and
teaches us to regulate our actions with regard to future events in a
way which will, in the long run, lead to the least disappointment. It
is, as Laplace happily said, *good sense reduced to calculation*. This
theory appears to me the noblest creation of intellect, and it passes
my conception how two such men as Auguste Comte and J. S. Mill could be
found depreciating it and vainly questioning its validity. To eulogise
the theory ought to be as needless as to eulogise reason itself.
*Fundamental Principles of the Theory.*
The calculation of probabilities is really founded, as I conceive, upon
the principle of reasoning set forth in preceding chapters. We must
treat equals equally, and what we know of one case may be affirmed of
every case resembling it in the necessary circumstances. The theory
consists in putting similar cases on a par, and distributing equally
among them whatever knowledge we possess. Throw a penny into the air,
and consider what we know with regard to its way of falling. We know
that it will certainly fall upon a side, so that either head or tail
will be uppermost; but as to whether it will be head or tail, our
knowledge is equally divided. Whatever we know concerning head, we know
also concerning tail, so that we have no reason for expecting one more
than the other. The least predominance of belief to either side would
be irrational; it would consist in treating unequally things of which
our knowledge is equal.
The theory does not require, as some writers have erroneously supposed,
that we should first ascertain by experiment the equal facility of
the events we are considering. So far as we can examine and measure
the causes in operation, events are removed out of the sphere of
probability. The theory comes into play where ignorance begins, and the
knowledge we possess requires to be distributed over many cases. Nor
does the theory show that the coin will fall as often on the one side
as the other. It is almost impossible that this should happen, because
some inequality in the form of the coin, or some uniform manner in
throwing it up, is almost sure to occasion a slight preponderance
in one direction. But as we do not previously know in which way a
preponderance will exist, we have no reason for expecting head more
than tail. Our state of knowledge will be changed should we throw up
the coin many times and register the results. Every throw gives us some
slight information as to the probable tendency of the coin, and in
subsequent calculations we must take this into account. In other cases
experience might show that we had been entirely mistaken; we might
expect that a die would fall as often on each of the six sides as on
each other side in the long run; trial might show that the die was a
loaded one, and falls most often on a particular face. The theory would
not have misled us: it treated correctly the information we had, which
is all that any theory can do.
It may be asked, as Mill asks, Why spend so much trouble in calculating
from imperfect data, when a little trouble would enable us to render a
conclusion certain by actual trial? Why calculate the probability of a
measurement being correct, when we can try whether it is correct? But I
shall fully point out in later parts of this work that in measurement
we never can attain perfect coincidence. Two measurements of the
same base line in a survey may show a difference of some inches, and
there may be no means of knowing which is the better result. A third
measurement would probably agree with neither. To select any one of the
measurements, would imply that we knew it to be the most nearly correct
one, which we do not. In this state of ignorance, the only guide is
the theory of probability, which proves that in the long run the mean
of divergent results will come most nearly to the truth. In all other
scientific operations whatsoever, perfect knowledge is impossible, and
when we have exhausted all our instrumental means in the attainment of
truth, there is a margin of error which can only be safely treated by
the principles of probability.
The method which we employ in the theory consists in calculating the
number of all the cases or events concerning which our knowledge is
equal. If we have the slightest reason for suspecting that one event
is more likely to occur than another, we should take this knowledge
into account. This being done, we must determine the whole number of
events which are, so far as we know, equally likely. Thus, if we have
no reason for supposing that a penny will fall more often one way than
another, there are two cases, head and tail, equally likely. But if
from trial or otherwise we know, or think we know, that of 100 throws
55 will give tail, then the probability is measured by the ratio of 55
to 100.
The mathematical formulæ of the theory are exactly the same as those
of the theory of combinations. In this latter theory we determine in
how many ways events may be joined together, and we now proceed to use
this knowledge in calculating the number of ways in which a certain
event may come about. It is the comparative numbers of ways in which
events can happen which measure their comparative probabilities. If
we throw three pennies into the air, what is the probability that two
of them will fall tail uppermost? This amounts to asking in how many
possible ways can we select two tails out of three, compared with the
whole number of ways in which the coins can be placed. Now, the fourth
line of the Arithmetical Triangle (p. 184) gives us the answer. The
whole number of ways in which we can select or leave three things is
eight, and the possible combinations of two things at a time is three;
hence the probability of two tails is the ratio of three to eight. From
the numbers in the triangle we may similarly draw all the following
probabilities:--
One combination gives 0 tail. Probability 1/8.
Three combinations gives 1 tail. Probability 3/8.
Three combinations give 2 tails. Probability 3/8.
One combination gives 3 tails. Probability 1/8.
We can apply the same considerations to the imaginary causes of the
difference of stature, the combinations of which were shown in p. 188.
There are altogether 128 ways in which seven causes can be present or
absent. Now, twenty-one of these combinations give an addition of two
inches, so that the probability of a person under the circumstances
being five feet two inches is 21/128. The probability of five feet
three inches is 35/128; of five feet one inch 7/128; of five feet
1/128, and so on. Thus the eighth line of the Arithmetical Triangle
gives all the probabilities arising out of the combinations of seven
causes.
*Rules for the Calculation of Probabilities.*
I will now explain as simply as possible the rules for calculating
probabilities. The principal rule is as follows:--
Calculate the number of events which may happen independently of each
other, and which, as far as is known, are equally probable. Make this
number the denominator of a fraction, and take for the numerator the
number of such events as imply or constitute the happening of the
event, whose probability is required.
Thus, if the letters of the word *Roma* be thrown down casually in a
row, what is the probability that they will form a significant Latin
word? The possible arrangements of four letters are 4 × 3 × 2 × 1,
or 24 in number (p. 178), and if all the arrangements be examined,
seven of these will be found to have meaning, namely *Roma*, *ramo*,
*oram*, *mora*, *maro*, *armo*, and *amor*. Hence the probability of a
significant result is 7/24.
We must distinguish comparative from absolute probabilities. In drawing
a card casually from a pack, there is no reason to expect any one card
more than any other. Now, there are four kings and four queens in a
pack, so that there are just as many ways of drawing one as the other,
and the probabilities are equal. But there are thirteen diamonds, so
that the probability of a king is to that of a diamond as four to
thirteen. Thus the probabilities of each are proportional to their
respective numbers of ways of happening. Again, I can draw a king in
four ways, and not draw one in forty-eight, so that the probabilities
are in this proportion, or, as is commonly said, the *odds* against
drawing a king are forty-eight to four. The odds are seven to seventeen
in favour, or seventeen to seven against the letters R,o,m,a,
accidentally forming a significant word. The odds are five to three
against two tails appearing in three throws of a penny. Conversely,
when the odds of an event are given, and the probability is required,
*take the odds in favour of the event for numerator, and the sum of the
odds for denominator*.
It is obvious that an event is certain when all the combinations of
causes which can take place produce that event. If we represent the
probability of such event according to our rule, it gives the ratio
of some number to itself, or unity. An event is certain not to happen
when no possible combination of causes gives the event, and the ratio
by the same rule becomes that of 0 to some number. Hence it follows
that in the theory of probability certainty is expressed by 1, and
impossibility by 0; but no mystical meaning should be attached to these
symbols, as they merely express the fact that *all* or *no* possible
combinations give the event.
By a *compound event*, we mean an event which may be decomposed into
two or more simpler events. Thus the firing of a gun may be decomposed
into pulling the trigger, the fall of the hammer, the explosion of
the cap, &c. In this example the simple events are not *independent*,
because if the trigger is pulled, the other events will under proper
conditions necessarily follow, and their probabilities are therefore
the same as that of the first event. Events are *independent* when
the happening of one does not render the other either more or less
probable than before. Thus the death of a person is neither more nor
less probable because the planet Mars happens to be visible. When
the component events are independent, a simple rule can be given for
calculating the probability of the compound event, thus--*Multiply
together the fractions expressing the probabilities of the independent
component events.*
The probability of throwing tail twice with a penny is 1/2 × 1/2,
or 1/4; the probability of throwing it three times running is
1/2 × 1/2 × 1/2, or 1/8; a result agreeing with that obtained in
an apparently different manner (p. 202). In fact, when we multiply
together the denominators, we get the whole number of ways of happening
of the compound event, and when we multiply the numerators, we get the
number of ways favourable to the required event.
Probabilities may be added to or subtracted from each other under the
important condition that the events in question are exclusive of each
other, so that not more than one of them can happen. It might be argued
that, since the probability of throwing head at the first trial is
1/2, and at the second trial also 1/2, the probability of throwing it
in the first two throws is 1/2 + 1/2, or certainty. Not only is this
result evidently absurd, but a repetition of the process would lead
us to a probability of 1-1/2 or of any greater number, results which
could have no meaning whatever. The probability we wish to calculate is
that of one head in two throws, but in our addition we have included
the case in which two heads appear. The true result is 1/2 + 1/2 × 1/2
or 3/4, or the probability of head at the first throw, added to the
exclusive probability that if it does not come at the first, it will
come at the second. The greatest difficulties of the theory arise
from the confusion of exclusive and unexclusive alternatives. I may
remind the reader that the possibility of unexclusive alternatives was
a point previously discussed (p. 68), and to the reasons then given
for considering alternation as logically unexclusive, may be added
the existence of these difficulties in the theory of probability. The
erroneous result explained above really arose from overlooking the
fact that the expression “head first throw or head second throw” might
include the case of head at both throws.
*The Logical Alphabet in questions of Probability.*
When the probabilities of certain simple events are given, and it is
required to deduce the probabilities of compound events, the Logical
Alphabet may give assistance, provided that there are no special
logical conditions so that all the combinations are possible. Thus,
if there be three events, A, B, C, of which the probabilities are, α,
β, γ, then the negatives of those events, expressing the absence of
the events, will have the probabilities 1 - α, 1 - β, 1 - γ. We have
only to insert these values for the letters of the combinations and
multiply, and we obtain the probability of each combination. Thus the
probability of ABC is αβγ; of A*bc*, α(1 - β)(1 - γ).
We can now clearly distinguish between the probabilities of exclusive
and unexclusive events. Thus, if A and B are events which may happen
together like rain and high tide, or an earthquake and a storm, the
probability of A or B happening is not the sum of their separate
probabilities. For by the Laws of Thought we develop A ꖌ B into
AB ꖌ A*b* ꖌ *a*B, and substituting α and β, the probabilities of A
and B respectively, we obtain α . β + α . (1 - β) + (1 - α) . β or
α + β - α . β. But if events are *incompossible* or incapable of
happening together, like a clear sky and rain, or a new moon and a full
moon, then the events are not really A or B, but A not-B, or B not-A,
or in symbols A*b* ꖌ *a*B. Now if we take μ = probability of A*b* and ν
= probability of *a*B, then we may add simply, and the probability of
A*b* ꖌ *a*B is μ + ν.
Let the reader carefully observe that if the combination AB cannot
exist, the probability of A*b* is not the product of the probabilities
of A and *b*. When certain combinations are logically impossible, it
is no longer allowable to substitute the probability of each term for
the term, because the multiplication of probabilities presupposes the
independence of the events. A large part of Boole’s Laws of Thought
is devoted to an attempt to overcome this difficulty and to produce
a General Method in Probabilities by which from certain logical
conditions and certain given probabilities it would be possible to
deduce the probability of any other combinations of events under those
conditions. Boole pursued his task with wonderful ingenuity and power,
but after spending much study on his work, I am compelled to adopt
the conclusion that his method is fundamentally erroneous. As pointed
out by Mr. Wilbraham,[113] Boole obtained his results by an arbitrary
assumption, which is only the most probable, and not the only possible
assumption. The answer obtained is therefore not the real probability,
which is usually indeterminate, but only, as it were, the most probable
probability. Certain problems solved by Boole are free from logical
conditions and therefore may admit of valid answers. These, as I have
shown,[114] may be solved by the combinations of the Logical Alphabet,
but the rest of the problems do not admit of a determinate answer, at
least by Boole’s method.
[113] *Philosophical Magazine*, 4th Series, vol. vii. p. 465;
vol. viii. p. 91.
[114] *Memoirs of the Manchester Literary and Philosophical Society*,
3rd Series, vol. iv. p. 347.
*Comparison of the Theory with Experience.*
The Laws of Probability rest upon the fundamental principles of
reasoning, and cannot be really negatived by any possible experience.
It might happen that a person should always throw a coin head
uppermost, and appear incapable of getting tail by chance. The theory
would not be falsified, because it contemplates the possibility of
the most extreme runs of luck. Our actual experience might be counter
to all that is probable; the whole course of events might seem to be
in complete contradiction to what we should expect, and yet a casual
conjunction of events might be the real explanation. It is just
possible that some regular coincidences, which we attribute to fixed
laws of nature, are due to the accidental conjunction of phenomena
in the cases to which our attention is directed. All that we can
learn from finite experience is capable, according to the theory of
probabilities, of misleading us, and it is only infinite experience
that could assure us of any inductive truths.
At the same time, the probability that any extreme runs of luck will
occur is so excessively slight, that it would be absurd seriously
to expect their occurrence. It is almost impossible, for instance,
that any whist player should have played in any two games where the
distribution of the cards was exactly the same, by pure accident
(p. 191). Such a thing as a person always losing at a game of
pure chance, is wholly unknown. Coincidences of this kind are not
impossible, as I have said, but they are so unlikely that the lifetime
of any person, or indeed the whole duration of history, does not give
any appreciable probability of their being encountered. Whenever we
make any extensive series of trials of chance results, as in throwing
a die or coin, the probability is great that the results will agree
nearly with the predictions yielded by theory. Precise agreement must
not be expected, for that, as the theory shows, is highly improbable.
Several attempts have been made to test, in this way, the accordance of
theory and experience. Buffon caused the first trial to be made by a
young child who threw a coin many times in succession, and he obtained
1992 tails to 2048 heads. A pupil of De Morgan repeated the trial for
his own satisfaction, and obtained 2044 tails to 2048 heads. In both
cases the coincidence with theory is as close as could be expected, and
the details may be found in De Morgan’s “Formal Logic,” p. 185.
Quetelet also tested the theory in a rather more complete manner, by
placing 20 black and 20 white balls in an urn and drawing a ball out
time after time in an indifferent manner, each ball being replaced
before a new drawing was made. He found, as might be expected, that the
greater the number of drawings made, the more nearly were the white
and black balls equal in number. At the termination of the experiment
he had registered 2066 white and 2030 black balls, the ratio being
1·02.[115]
[115] *Letters on the Theory of Probabilities*, translated by Downes,
1849, pp. 36, 37.
I have made a series of experiments in a third manner, which seemed to
me even more interesting, and capable of more extensive trial. Taking
a handful of ten coins, usually shillings, I threw them up time after
time, and registered the numbers of heads which appeared each time. Now
the probability of obtaining 10, 9, 8, 7, &c., heads is proportional
to the number of combinations of 10, 9, 8, 7, &c., things out of 10
things. Consequently the results ought to approximate to the numbers in
the eleventh line of the Arithmetical Triangle. I made altogether 2048
throws, in two sets of 1024 throws each, and the numbers obtained are
given in the following table:--
+-------------------+-----------+---------+---------+----------+-----------+
|Character of Throw.|Theoretical| First | Second | Average. |Divergence.|
| | Numbers. | Series. | Series. | | |
+-------------------+-----------+---------+---------+----------+-----------+
| 10 Heads 0 Tail | 1 | 3 | 1 | 2 | + 1 |
| 9 " 1 " | 10 | 12 | 23 | 17-1/2 | + 7-1/2 |
| 8 " 2 " | 45 | 57 | 73 | 65 | + 20 |
| 7 " 3 " | 120 | 129 | 123 | 126 | + 6 |
| 6 " 4 " | 210 | 181 | 190 | 185-1/2 | - 25 |
| 5 " 5 " | 252 | 257 | 232 | 244-1/2 | - 7-1/2 |
| 4 " 6 " | 210 | 201 | 197 | 199 | - 11 |
| 3 " 7 " | 120 | 111 | 119 | 115 | - 5 |
| 2 " 8 " | 45 | 52 | 50 | 51 | + 6 |
| 1 " 9 " | 10 | 21 | 15 | 18 | + 8 |
| 0 " 10 " | 1 | 0 | 1 | 1/2 | - 1/2 |
+-------------------+-----------+---------+---------+----------+-----------+
| Totals | 1024 | 1024 | 1024 | 1024 | - 1 |
+-------------------+-----------+---------+---------+----------+-----------+
The whole number of single throws of coins amounted to 10 × 2048, or
20,480 in all, one half of which or 10,240 should theoretically give
head. The total number of heads obtained was actually 10,353, or 5222
in the first series, and 5131 in the second. The coincidence with
theory is pretty close, but considering the large number of throws
there is some reason to suspect a tendency in favour of heads.
The special interest of this trial consists in the exhibition, in a
practical form, of the results of Bernoulli’s theorem, and the law
of error or divergence from the mean to be afterwards more fully
considered. It illustrates the connection between combinations and
permutations, which is exhibited in the Arithmetical Triangle, and
which underlies many important theorems of science.
*Probable Deductive Arguments*.
With the aid of the theory of probabilities, we may extend the sphere
of deductive argument. Hitherto we have treated propositions as
certain, and on the hypothesis of certainty have deduced conclusions
equally certain. But the information on which we reason in ordinary
life is seldom or never certain, and almost all reasoning is really a
question of probability. We ought therefore to be fully aware of the
mode and degree in which deductive reasoning is affected by the theory
of probability, and many persons may be surprised at the results which
must be admitted. Some controversial writers appear to consider, as De
Morgan remarked,[116] that an inference from several equally probable
premises is itself as probable as any of them, but the true result is
very different. If an argument involves many propositions, and each of
them is uncertain, the conclusion will be of very little force.
[116] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 396.
The validity of a conclusion may be regarded as a compound event,
depending upon the premises happening to be true; thus, to obtain the
probability of the conclusion, we must multiply together the fractions
expressing the probabilities of the premises. If the probability is
1/2 that A is B, and also 1/2 that B is C, the conclusion that A is
C, on the ground of these premises, is 1/2 × 1/2 or 1/4. Similarly
if there be any number of premises requisite to the establishment
of a conclusion and their probabilities be *p*, *q*, *r*, &c., the
probability of the conclusion on the ground of these premises is
*p* × *q* × *r* × ... This product has but a small value, unless each
of the quantities *p*, *q*, &c., be nearly unity.
But it is particularly to be noticed that the probability thus
calculated is not the whole probability of the conclusion, but that
only which it derives from the premises in question. Whately’s[117]
remarks on this subject might mislead the reader into supposing that
the calculation is completed by multiplying together the probabilities
of the premises. But it has been fully explained by De Morgan[118] that
we must take into account the antecedent probability of the conclusion;
A may be C for other reasons besides its being B, and as he remarks,
“It is difficult, if not impossible, to produce a chain of argument of
which the reasoner can rest the result on those arguments only.” The
failure of one argument does not, except under special circumstances,
disprove the truth of the conclusion it is intended to uphold,
otherwise there are few truths which could survive the ill-considered
arguments adduced in their favour. As a rope does not necessarily break
because one or two strands in it fail, so a conclusion may depend upon
an endless number of considerations besides those immediately in view.
Even when we have no other information we must not consider a statement
as devoid of all probability. The true expression of complete doubt is
a ratio of equality between the chances in favour of and against it,
and this ratio is expressed in the probability 1/2.
[117] *Elements of Logic*, Book III. sections 11 and 18.
[118] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 400.
Now if A and C are wholly unknown things, we have no reason to believe
that A is C rather than A is not C. The antecedent probability is then
1/2. If we also have the probabilities that A is B, 1/2 and that B is
C, 1/2 we have no right to suppose that the probability of A being C
is reduced by the argument in its favour. If the conclusion is true
on its own grounds, the failure of the argument does not affect it;
thus its total probability is its antecedent probability, added to the
probability that this failing, the new argument in question establishes
it. There is a probability 1/2 that we shall not require the special
argument; a probability 1/2 that we shall, and a probability 1/4
that the argument does in that case establish it. Thus the complete
result is 1/2 + 1/2 × 1/4, or 5/8. In general language, if *a* be the
probability founded on a particular argument, and *c* the antecedent
probability of the event, the general result is 1 - (1 - *a*)(1 - *c*),
or *a* + *c* - *ac*.
We may put it still more generally in this way:--Let *a*, *b*, *c*, &c.
be the probabilities of a conclusion grounded on various arguments. It
is only when all the arguments fail that our conclusion proves finally
untrue; the probabilities of each failing are respectively, 1 - *a*,
1 - *b*, 1 - *c*, &c.; the probability that they will all fail is
(1 - *a*)(1 - *b*)(1 - *c*) ...; therefore the probability that the
conclusion will not fail is 1 - (1 - *a*)(1 - *b*)(1 - *c*) ... &c. It
follows that every argument in favour of a conclusion, however flimsy
and slight, adds probability to it. When it is unknown whether an
overdue vessel has foundered or not, every slight indication of a lost
vessel will add some probability to the belief of its loss, and the
disproof of any particular evidence will not disprove the event.
We must apply these principles of evidence with great care, and
observe that in a great proportion of cases the adducing of a weak
argument does tend to the disproof of its conclusion. The assertion
may have in itself great inherent improbability as being opposed to
other evidence or to the supposed law of nature, and every reasoner
may be assumed to be dealing plainly, and putting forward the whole
force of evidence which he possesses in its favour. If he brings but
one argument, and its probability *a* is small, then in the formula
1 - (1 - *a*)(1 - *c*) both *a* and *c* are small, and the whole
expression has but little value. The whole effect of an argument thus
turns upon the question whether other arguments remain, so that we
can introduce other factors (1 - *b*), (1 - *d*), &c., into the above
expression. In a court of justice, in a publication having an express
purpose, and in many other cases, it is doubtless right to assume
that the whole evidence considered to have any value as regards the
conclusion asserted, is put forward.
To assign the antecedent probability of any proposition, may be a
matter of difficulty or impossibility, and one with which logic and
the theory of probability have little concern. From the general body of
science in our possession, we must in each case make the best judgment
we can. But in the absence of all knowledge the probability should
be considered = 1/2, for if we make it less than this we incline to
believe it false rather than true. Thus, before we possessed any means
of estimating the magnitudes of the fixed stars, the statement that
Sirius was greater than the sun had a probability of exactly 1/2; it
was as likely that it would be greater as that it would be smaller; and
so of any other star. This was the assumption which Michell made in
his admirable speculations.[119] It might seem, indeed, that as every
proposition expresses an agreement, and the agreements or resemblances
between phenomena are infinitely fewer than the differences (p. 44),
every proposition should in the absence of other information be
infinitely improbable. But in our logical system every term may be
indifferently positive or negative, so that we express under the same
form as many differences as agreements. It is impossible therefore
that we should have any reason to disbelieve rather than to believe a
statement about things of which we know nothing. We can hardly indeed
invent a proposition concerning the truth of which we are absolutely
ignorant, except when we are entirely ignorant of the terms used. If I
ask the reader to assign the odds that a “Platythliptic Coefficient is
positive” he will hardly see his way to doing so, unless he regard them
as even.
[119] *Philosophical Transactions* (1767). Abridg. vol. xii. p. 435.
The assumption that complete doubt is properly expressed by 1/2 has
been called in question by Bishop Terrot,[120] who proposes instead
the indefinite symbol 0/0; and he considers that “the *à priori*
probability derived from absolute ignorance has no effect upon the
force of a subsequently admitted probability.” But if we grant that the
probability may have any value between 0 and 1, and that every separate
value is equally likely, then *n* and 1 - *n* are equally likely, and
the average is always 1/2. Or we may take *p* . *dp* to express the
probability that our estimate concerning any proposition should lie
between *p* and *p* + *dp*. The complete probability of the proposition
is then the integral taken between the limits 1 and 0, or again 1/2.
[120] *Transactions of the Edinburgh Philosophical Society*,
vol. xxi. p. 375.
*Difficulties of the Theory.*
The theory of probability, though undoubtedly true, requires very
careful application. Not only is it a branch of mathematics in which
oversights are frequently committed, but it is a matter of great
difficulty in many cases, to be sure that the formula correctly
represents the data of the problem. These difficulties often arise from
the logical complexity of the conditions, which might be, perhaps, to
some extent cleared up by constantly bearing in mind the system of
combinations as developed in the Indirect Logical Method. In the study
of probabilities, mathematicians had unconsciously employed logical
processes far in advance of those in possession of logicians, and the
Indirect Method is but the full statement of these processes.
It is very curious how often the most acute and powerful intellects
have gone astray in the calculation of probabilities. Seldom was Pascal
mistaken, yet he inaugurated the science with a mistaken solution.[121]
Leibnitz fell into the extraordinary blunder of thinking that the
number twelve was as probable a result in the throwing of two dice as
the number eleven.[122] In not a few cases the false solution first
obtained seems more plausible to the present day than the correct
one since demonstrated. James Bernoulli candidly records two false
solutions of a problem which he at first thought self-evident; and he
adds a warning against the risk of error, especially when we attempt
to reason on this subject without a rigid adherence to methodical
rules and symbols. Montmort was not free from similar mistakes.
D’Alembert constantly fell into blunders, and could not perceive,
for instance, that the probabilities would be the same when coins
are thrown successively as when thrown simultaneously. Some men of
great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste
Comte,[123] Poinsot, and J. S. Mill,[124] have so far misapprehended
the theory, as to question its value or even to dispute its validity.
The erroneous statements about the theory given in the earlier editions
of Mill’s *System of Logic* were partially withdrawn in the later
editions.
[121] Montucla, *Histoire des Mathématiques*, vol. iii. p. 386.
[122] Leibnitz *Opera*, Dutens’ Edition, vol. vi. part i. p. 217.
Todhunter’s *History of the Theory of Probability*, p. 48. To the
latter work I am indebted for many of the statements in the text.
[123] *Positive Philosophy*, translated by Martineau, vol. ii. p. 120.
[124] *System of Logic*, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.
Many persons have a fallacious tendency to believe that when a chance
event has happened several times together in an unusual conjunction, it
is less likely to happen again. D’Alembert seriously held that if head
was thrown three times running with a coin, tail would more probably
appear at the next trial.[125] Bequelin adopted the same opinion, and
yet there is no reason for it whatever. If the event be really casual,
what has gone before cannot in the slightest degree influence it. As
a matter of fact, the more often a casual event takes place the more
likely it is to happen again; because there is some slight empirical
evidence of a tendency. The source of the fallacy is to be found
entirely in the feelings of surprise with which we witness an event
happening by chance, in a manner which seems to proceed from design.
[125] Montucla, *Histoire*, vol. iii. p. 405; Todhunter, p. 263.
Misapprehension may also arise from overlooking the difference between
permutations and combinations. To throw ten heads in succession with a
coin is no more unlikely than to throw any other particular succession
of heads and tails, but it is much less likely than five heads and five
tails without regard to their order, because there are no less than
252 different particular throws which will give this result, when we
abstract the difference of order.
Difficulties arise in the application of the theory from our habitual
disregard of slight probabilities. We are obliged practically to accept
truths as certain which are nearly so, because it ceases to be worth
while to calculate the difference. No punishment could be inflicted
if absolutely certain evidence of guilt were required, and as Locke
remarks, “He that will not stir till he infallibly knows the business
he goes about will succeed, will have but little else to do but to
sit still and perish.”[126] There is not a moment of our lives when
we do not lie under a slight danger of death, or some most terrible
fate. There is not a single action of eating, drinking, sitting down,
or standing up, which has not proved fatal to some person. Several
philosophers have tried to assign the limit of the probabilities
which we regard as zero; Buffon named 1/10,000, because it is the
probability, practically disregarded, that a man of 56 years of age
will die the next day. Pascal remarked that a man would be esteemed a
fool for hesitating to accept death when three dice gave sixes twenty
times running, if his reward in case of a different result was to be a
crown; but as the chance of death in question is only 1 ÷ 6^{60}, or
unity divided by a number of 47 places of figures, we may be said to
incur greater risks every day for less motives. There is far greater
risk of death, for instance, in a game of cricket or a visit to the
rink.
[126] *Essay concerning Human Understanding*, bk. iv. ch. 14. § 1.
Nothing is more requisite than to distinguish carefully between the
truth of a theory and the truthful application of the theory to actual
circumstances. As a general rule, events in nature and art will present
a complexity of relations exceeding our powers of treatment. The
intricate action of the mind often intervenes and renders complete
analysis hopeless. If, for instance, the probability that a marksman
shall hit the target in a single shot be 1 in 10, we might seem to
have no difficulty in calculating the probability of any succession of
hits; thus the probability of three successive hits would be one in a
thousand. But, in reality, the confidence and experience derived from
the first successful shot would render a second success more probable.
The events are not really independent, and there would generally be
a far greater preponderance of runs of apparent luck, than a simple
calculation of probabilities could account for. In some persons,
however, a remarkable series of successes will produce a degree of
excitement rendering continued success almost impossible.
Attempts to apply the theory of probability to the results of judicial
proceedings have proved of little value, simply because the conditions
are far too intricate. As Laplace said, “Tant de passions, d’intérêts
divers et de circonstances compliquent les questions relatives à ces
objets, qu’elles sont presque toujours insolubles.” Men acting on a
jury, or giving evidence before a court, are subject to so many complex
influences that no mathematical formulas can be framed to express the
real conditions. Jurymen or even judges on the bench cannot be regarded
as acting independently, with a definite probability in favour of each
delivering a correct judgment. Each man of the jury is more or less
influenced by the opinion of the others, and there are subtle effects
of character and manner and strength of mind which defy analysis. Even
in physical science we can in comparatively few cases apply the theory
in a definite manner, because the data required are too complicated and
difficult to obtain. But such failures in no way diminish the truth and
beauty of the theory itself; in reality there is no branch of science
in which our symbols can cope with the complexity of Nature. As Donkin
said,--
“I do not see on what ground it can be doubted that every definite
state of belief concerning a proposed hypothesis, is in itself capable
of being represented by a numerical expression, however difficult or
impracticable it may be to ascertain its actual value. It would be very
difficult to estimate in numbers the *vis viva* of all the particles of
a human body at any instant; but no one doubts that it is capable of
numerical expression.”[127]
[127] *Philosophical Magazine*, 4th Series, vol. i. p. 354.
The difficulty, in short, is merely relative to our knowledge and
skill, and is not absolute or inherent in the subject. We must
distinguish between what is theoretically conceivable and what is
practicable with our present mental resources. Provided that our
aspirations are pointed in a right direction, we must not allow them
to be damped by the consideration that they pass beyond what can now
be turned to immediate use. In spite of its immense difficulties of
application, and the aspersions which have been mistakenly cast upon
it, the theory of probabilities, I repeat, is the noblest, as it
will in course of time prove, perhaps the most fruitful branch of
mathematical science. It is the very guide of life, and hardly can
we take a step or make a decision of any kind without correctly or
incorrectly making an estimation of probabilities. In the next chapter
we proceed to consider how the whole cogency of inductive reasoning
rests upon probabilities. The truth or untruth of a natural law, when
carefully investigated, resolves itself into a high or low degree of
probability, and this is the case whether or not we are capable of
producing precise numerical data.
CHAPTER XI.
PHILOSOPHY OF INDUCTIVE INFERENCE.
We have inquired into the nature of perfect induction, whereby we pass
backwards from certain observed combinations of events, to the logical
conditions governing such combinations. We have also investigated the
grounds of that theory of probability, which must be our guide when we
leave certainty behind, and dilute knowledge with ignorance. There is
now before us the difficult task of endeavouring to decide how, by the
aid of that theory, we can ascend from the facts to the laws of nature;
and may then with more or less success anticipate the future course
of events. All our knowledge of natural objects must be ultimately
derived from observation, and the difficult question arises--How can
we ever know anything which we have not directly observed through one
of our senses, the apertures of the mind? The utility of reasoning is
to assure ourselves that, at a determinate time and place, or under
specified conditions, a certain phenomenon will be observed. When
we can use our senses and perceive that the phenomenon does occur,
reasoning is superfluous. If the senses cannot be used, because the
event is in the future, or out of reach, how can reasoning take their
place? Apparently, at least, we must infer the unknown from the known,
and the mind must itself create an addition to the sum of knowledge.
But I hold that it is quite impossible to make any real additions to
the contents of our knowledge, except through new impressions upon
the senses, or upon some seat of feeling. I shall attempt to show
that inference, whether inductive or deductive, is never more than
an unfolding of the contents of our experience, and that it always
proceeds upon the assumption that the future and the unperceived will
be governed by the same conditions as the past and the perceived, an
assumption which will often prove to be mistaken.
In inductive as in deductive reasoning the conclusion never passes
beyond the premises. Reasoning adds no more to the implicit contents of
our knowledge, than the arrangement of the specimens in a museum adds
to the number of those specimens. Arrangement adds to our knowledge
in a certain sense: it allows us to perceive the similarities and
peculiarities of the specimens, and on the assumption that the museum
is an adequate representation of nature, it enables us to judge of
the prevailing forms of natural objects. Bacon’s first aphorism holds
perfectly true, that man knows nothing but what he has observed,
provided that we include his whole sources of experience, and the whole
implicit contents of his knowledge. Inference but unfolds the hidden
meaning of our observations, and *the theory of probability shows how
far we go beyond our data in assuming that new specimens will resemble
the old ones*, or that the future may be regarded as proceeding
uniformly with the past.
*Various Classes of Inductive Truths.*
It will be desirable, in the first place, to distinguish between the
several kinds of truths which we endeavour to establish by induction.
Although there is a certain common and universal element in all our
processes of reasoning, yet diversity arises in their application.
Similarity of condition between the events from which we argue, and
those to which we argue, must always be the ground of inference; but
this similarity may have regard either to time or place, or the simple
logical combination of events, or to any conceivable junction of
circumstances involving quality, time, and place. Having met with many
pieces of substance possessing ductility and a bright yellow colour,
and having discovered, by perfect induction, that they all possess a
high specific gravity, and a freedom from the corrosive action of
acids, we are led to expect that every piece of substance, possessing
like ductility and a similar yellow colour, will have an equally high
specific gravity, and a like freedom from corrosion by acids. This
is a case of the coexistence of qualities; for the character of the
specimens examined alters not with time nor place.
In a second class of cases, time will enter as a principal ground of
similarity. When we hear a clock pendulum beat time after time, at
equal intervals, and with a uniform sound, we confidently expect that
the stroke will continue to be repeated uniformly. A comet having
appeared several times at nearly equal intervals, we infer that it will
probably appear again at the end of another like interval. A man who
has returned home evening after evening for many years, and found his
house standing, may, on like grounds, expect that it will be standing
the next evening, and on many succeeding evenings. Even the continuous
existence of an object in an unaltered state, or the finding again of
that which we have hidden, is but a matter of inference depending on
experience.
A still larger and more complex class of cases involves the relations
of space, in addition to those of time and quality. Having observed
that every triangle drawn upon the diameter of a circle, with its
apex upon the circumference, apparently contains a right angle,
we may ascertain that all triangles in similar circumstances will
contain right angles. This is a case of pure space reasoning, apart
from circumstances of time or quality, and it seems to be governed by
different principles of reasoning. I shall endeavour to show, however,
that geometrical reasoning differs but in degree from that which
applies to other natural relations.
*The Relation of Cause and Effect.*
In a very large part of the scientific investigations which must be
considered, we deal with events which follow from previous events, or
with existences which succeed existences. Science, indeed, might arise
even were material nature a fixed and changeless whole. Endow mind
with the power to travel about, and compare part with part, and it
could certainly draw inferences concerning the similarity of forms, the
coexistence of qualities, or the preponderance of a particular kind of
matter in a changeless world. A solid universe, in at least approximate
equilibrium, is not inconceivable, and then the relation of cause and
effect would evidently be no more than the relation of before and
after. As nature exists, however, it is a progressive existence, ever
moving and changing as time, the great independent variable, proceeds.
Hence it arises that we must continually compare what is happening now
with what happened a moment before, and a moment before that moment,
and so on, until we reach indefinite periods of past time. A comet
is seen moving in the sky, or its constituent particles illumine
the heavens with their tails of fire. We cannot explain the present
movements of such a body without supposing its prior existence, with
a definite amount of energy and a definite direction of motion; nor
can we validly suppose that our task is concluded when we find that it
came wandering to our solar system through the unmeasured vastness of
surrounding space. Every event must have a cause, and that cause again
a cause, until we are lost in the obscurity of the past, and are driven
to the belief in one First Cause, by whom the course of nature was
determined.
*Fallacious Use of the Term Cause.*
The words Cause and Causation have given rise to infinite trouble
and obscurity, and have in no slight degree retarded the progress of
science. From the time of Aristotle, the work of philosophy has been
described as the discovery of the causes of things, and Francis Bacon
adopted the notion when he said “*vere scire esse per causas scire*.”
Even now it is not uncommonly supposed that the knowledge of causes is
something different from other knowledge, and consists, as it were, in
getting possession of the keys of nature. A single word may thus act
as a spell, and throw the clearest intellect into confusion, as I have
often thought that Locke was thrown into confusion when endeavouring to
find a meaning for the word *power*.[128] In Mill’s *System of Logic*
the term *cause* seems to have re-asserted its old noxious power. Not
only does Mill treat the Laws of Causation as almost coextensive with
science, but he so uses the expression as to imply that when once we
pass within the circle of causation we deal with certainties.
[128] *Essay concerning Human Understanding*, bk. ii. chap. xxi.
The philosophical danger which attaches to the use of this word may
be thus described. A cause is defined as the necessary or invariable
antecedent of an event, so that when the cause exists the effect will
also exist or soon follow. If then we know the cause of an event, we
know what will certainly happen; and as it is implied that science,
by a proper experimental method, may attain to a knowledge of causes,
it follows that experience may give us a certain knowledge of future
events. But nothing is more unquestionable than that finite experience
can never give us certain knowledge of the future, so that either
a cause is not an invariable antecedent, or else we can never gain
certain knowledge of causes. The first horn of this dilemma is hardly
to be accepted. Doubtless there is in nature some invariably acting
mechanism, such that from certain fixed conditions an invariable result
always emerges. But we, with our finite minds and short experience, can
never penetrate the mystery of those existences which embody the Will
of the Creator, and evolve it throughout time. We are in the position
of spectators who witness the productions of a complicated machine, but
are not allowed to examine its intimate structure. We learn what does
happen and what does appear, but if we ask for the reason, the answer
would involve an infinite depth of mystery. The simplest bit of matter,
or the most trivial incident, such as the stroke of two billiard balls,
offers infinitely more to learn than ever the human intellect can
fathom. The word cause covers just as much untold meaning as any of the
words *substance*, *matter*, *thought*, *existence*.
*Confusion of Two Questions.*
The subject is much complicated, too, by the confusion of two distinct
questions. An event having happened, we may ask--
(1) Is there any cause for the event?
(2) Of what kind is that cause?
No one would assert that the mind possesses any faculty capable of
inferring, prior to experience, that the occurrence of a sudden noise
with flame and smoke indicates the combustion of a black powder, formed
by the mixture of black, white, and yellow powders. The greatest
upholder of *à priori* doctrines will allow that the particular aspect,
shape, size, colour, texture, and other qualities of a cause must be
gathered through the senses.
The question whether there is any cause at all for an event, is of
a totally different kind. If an explosion could happen without any
prior existing conditions, it must be a new creation--a distinct
addition to the universe. It may be plausibly held that we can imagine
neither the creation nor annihilation of anything. As regards matter,
this has long been held true; as regards force, it is now almost
universally assumed as an axiom that energy can neither come into
nor go out of existence without distinct acts of Creative Will. That
there exists any instinctive belief to this effect, indeed, seems
doubtful. We find Lucretius, a philosopher of the utmost intellectual
power and cultivation, gravely assuming that his raining atoms could
turn aside from their straight paths in a self-determining manner,
and by this spontaneous origination of energy determine the form of
the universe.[129] Sir George Airy, too, seriously discussed the
mathematical conditions under which a perpetual motion, that is, a
perpetual source of self-created energy, might exist.[130] The larger
part of the philosophic world has long held that in mental acts there
is free will--in short, self-causation. It is in vain to attempt to
reconcile this doctrine with that of an intuitive belief in causation,
as Sir W. Hamilton candidly allowed.
[129] *De Rerum Natura*, bk. ii. ll. 216–293.
[130] *Cambridge Philosophical Transactions* (1830), vol. iii. pp.
369–372.
It is obvious, moreover, that to assert the existence of a cause for
every event cannot do more than remove into the indefinite past the
inconceivable fact and mystery of creation. At any given moment matter
and energy were equal to what they are at present, or they were not;
if equal, we may make the same inquiry concerning any other moment,
however long prior, and we are thus obliged to accept one horn of the
dilemma--existence from infinity, or creation at some moment. This is
but one of the many cases in which we are compelled to believe in one
or other of two alternatives, both inconceivable. My present purpose,
however, is to point out that we must not confuse this supremely
difficult question with that into which inductive science inquires on
the foundation of facts. By induction we gain no certain knowledge;
but by observation, and the inverse use of deductive reasoning, we
estimate the probability that an event which has occurred was preceded
by conditions of specified character, or that such conditions will be
followed by the event.
*Definition of the Term Cause.*
Clear definitions of the word cause have been given by several
philosophers. Hobbes has said, “A cause is the sum or aggregate of all
such accidents, both in the agents and the patients, as concur in the
producing of the effect propounded; all which existing together, it
cannot be understood but that the effect existeth with them; or that
it can possibly exist if any of them be absent.” Brown, in his *Essay
on Causation*, gave a nearly corresponding statement. “A cause,” he
says,[131] “may be defined to be the object or event which immediately
precedes any change, and which existing again in similar circumstances
will be always immediately followed by a similar change.” Of the
kindred word *power*, he likewise says:[132] “Power is nothing more
than that invariableness of antecedence which is implied in the belief
of causation.”
[131] *Observations on the Nature and Tendency of the Doctrine of Mr.
Hume, concerning the Relation of Cause and Effect.* Second ed. p. 44.
[132] Ibid. p. 97.
These definitions may be accepted with the qualification that our
knowledge of causes in such a sense can be probable only. The work of
science consists in ascertaining the combinations in which phenomena
present themselves. Concerning every event we shall have to determine
its probable conditions, or the group of antecedents from which it
probably follows. An antecedent is anything which exists prior to
an event; a consequent is anything which exists subsequently to an
antecedent. It will not usually happen that there is any probable
connection between an antecedent and consequent. Thus nitrogen is an
antecedent to the lighting of a common fire; but it is so far from
being a cause of the lighting, that it renders the combustion less
active. Daylight is an antecedent to all fires lighted during the day,
but it probably has no appreciable effect upon their burning. But
in the case of any given event it is usually possible to discover a
certain number of antecedents which seem to be always present, and with
more or less probability we conclude that when they exist the event
will follow.
Let it be observed that the utmost latitude is at present enjoyed in
the use of the term *cause*. Not only may a cause be an existent thing
endowed with powers, as oxygen is the cause of combustion, gunpowder
the cause of explosion, but the very absence or removal of a thing may
also be a cause. It is quite correct to speak of the dryness of the
Egyptian atmosphere, or the absence of moisture, as being the cause of
the preservation of mummies, and other remains of antiquity. The cause
of a mountain elevation, Ingleborough for instance, is the excavation
of the surrounding valleys by denudation. It is not so usual to speak
of the existence of a thing at one moment as the cause of its existence
at the next, but to me it seems the commonest case of causation which
can occur. The cause of motion of a billiard ball may be the stroke of
another ball; and recent philosophy leads us to look upon all motions
and changes, as but so many manifestations of prior existing energy.
In all probability there is no creation of energy and no destruction,
so that as regards both mechanical and molecular changes, the cause is
really the manifestation of existing energy. In the same way I see not
why the prior existence of matter is not also a cause as regards its
subsequent existence. All science tends to show us that the existence
of the universe in a particular state at one moment, is the condition
of its existence at the next moment, in an apparently different
state. When we analyse the meaning which we can attribute to the word
*cause*, it amounts to the existence of suitable portions of matter
endowed with suitable quantities of energy. If we may accept Horne
Tooke’s assertion, *cause* has etymologically the meaning of *thing
before*. Though, indeed, the origin of the word is very obscure, its
derivatives, the Italian *cosa*, and the French *chose*, mean simply
*thing*. In the German equivalent *ursache*, we have plainly the
original meaning of *thing before*, the *sache* denoting “interesting
or important object,” the English *sake*, and *ur* being the equivalent
of the English *ere*, *before*. We abandon, then, both etymology and
philosophy, when we attribute to the *laws of causation* any meaning
beyond that of the *conditions* under which an event may be expected to
happen, according to our observation of the previous course of nature.
I have no objection to use the words cause and causation, provided they
are never allowed to lead us to imagine that our knowledge of nature
can attain to certainty. I repeat that if a cause is an invariable
and necessary condition of an event, we can never know certainly
whether the cause exists or not. To us, then, a cause is not to be
distinguished from the group of positive or negative conditions which,
with more or less probability, precede an event. In this sense, there
is no particular difference between knowledge of causes and our general
knowledge of the succession of combinations, in which the phenomena of
nature are presented to us, or found to occur in experimental inquiry.
*Distinction of Inductive and Deductive Results.*
We must carefully avoid confusing together inductive investigations
which terminate in the establishment of general laws, and those which
seem to lead directly to the knowledge of future particular events.
That process only can be called induction which gives general laws,
and it is by the subsequent employment of deduction that we anticipate
particular events. If the observation of a number of cases shows that
alloys of metals fuse at lower temperatures than their constituent
metals, I may with more or less probability draw a general inference
to that effect, and may thence deductively ascertain the probability
that the next alloy examined will fuse at a lower temperature than
its constituents. It has been asserted, indeed, by Mill,[133] and
partially admitted by Mr. Fowler,[134] that we can argue directly from
case to case, so that what is true of some alloys will be true of the
next. Professor Bain has adopted the same view of reasoning. He thinks
that Mill has extricated us from the dead lock of the syllogism and
effected a total revolution in logic. He holds that reasoning from
particulars to particulars is not only the usual, the most obvious and
the most ready method, but that it is the type of reasoning which best
discloses the real process.[135] Doubtless, this is the usual result of
our reasoning, regard being had to degrees of probability; but these
logicians fail entirely to give any explanation of the process by which
we get from case to case.
[133] *System of Logic*, bk. II. chap, iii.
[134] *Inductive Logic*, pp. 13, 14.
[135] Bain, *Deductive Logic*, pp. 208, 209.
It may be allowed that the knowledge of future particular events is
the main purpose of our investigations, and if there were any process
of thought by which we could pass directly from event to event without
ascending into general truths, this method would be sufficient, and
certainly the briefest. It is true, also, that the laws of mental
association lead the mind always to expect the like again in apparently
like circumstances, and even animals of very low intelligence must
have some trace of such powers of association, serving to guide them
more or less correctly, in the absence of true reasoning faculties.
But it is the purpose of logic, according to Mill, to ascertain
whether inferences have been correctly drawn, rather than to discover
them.[136] Even if we can, then, by habit, association, or any rude
process of inference, infer the future directly from the past, it is
the work of logic to analyse the conditions on which the correctness
of this inference depends. Even Mill would admit that such analysis
involves the consideration of general truths,[137] and in this, as in
several other important points, we might controvert Mill’s own views
by his own statements. It seems to me undesirable in a systematic work
like this to enter into controversy at any length, or to attempt to
refute the views of other logicians. But I shall feel bound to state,
in a separate publication, my very deliberate opinion that many of
Mill’s innovations in logical science, and especially his doctrine of
reasoning from particulars to particulars, are entirely groundless and
false.
[136] *System of Logic.* Introduction, § 4. Fifth ed. pp. 8, 9.
[137] Ibid. bk. II. chap. iii. § 5, pp. 225, &c.
*The Grounds of Inductive Inference.*
I hold that in all cases of inductive inference we must invent
hypotheses, until we fall upon some hypothesis which yields deductive
results in accordance with experience. Such accordance renders the
chosen hypothesis more or less probable, and we may then deduce, with
some degree of likelihood, the nature of our future experience, on the
assumption that no arbitrary change takes place in the conditions of
nature. We can only argue from the past to the future, on the general
principle set forth in this work, that what is true of a thing will
be true of the like. So far then as one object or event differs from
another, all inference is impossible, particulars as particulars can
no more make an inference than grains of sand can make a rope. We
must always rise to something which is general or same in the cases,
and assuming that sameness to be extended to new cases we learn their
nature. Hearing a clock tick five thousand times without exception or
variation, we adopt the very probable hypothesis that there is some
invariably acting machine which produces those uniform sounds, and
which will, in the absence of change, go on producing them. Meeting
twenty times with a bright yellow ductile substance, and finding it
always to be very heavy and incorrodible, I infer that there was some
natural condition which tended in the creation of things to associate
these properties together, and I expect to find them associated in the
next instance. But there always is the possibility that some unknown
change may take place between past and future cases. The clock may run
down, or be subject to a hundred accidents altering its condition.
There is no reason in the nature of things, so far as known to us, why
yellow colour, ductility, high specific gravity, and incorrodibility,
should always be associated together, and in other cases, if not in
this, men’s expectations have been deceived. Our inferences, therefore,
always retain more or less of a hypothetical character, and are so far
open to doubt. Only in proportion as our induction approximates to the
character of perfect induction, does it approximate to certainty. The
amount of uncertainty corresponds to the probability that other objects
than those examined may exist and falsity our inferences; the amount
of probability corresponds to the amount of information yielded by our
examination; and the theory of probability will be needed to prevent us
from over-estimating or under-estimating the knowledge we possess.
*Illustrations of the Inductive Process.*
To illustrate the passage from the known to the apparently unknown, let
us suppose that the phenomena under investigation consist of numbers,
and that the following six numbers being exhibited to us, we are
required to infer the character of the next in the series:--
5, 15, 35, 45, 65, 95.
The question first of all arises, How may we describe this series of
numbers? What is uniformly true of them? The reader cannot fail to
perceive at the first glance that they all end in five, and the problem
is, from the properties of these six numbers, to infer the properties
of the next number ending in five. If we test their properties by the
process of perfect induction, we soon perceive that they have another
common property, namely that of being *divisible by five without
remainder*. May we then assert that the next number ending in five is
also divisible by five, and, if so, upon what grounds? Or extending
the question, Is every number ending in five divisible by five? Does
it follow that because six numbers obey a supposed law, therefore
376,685,975 or any other number, however large, obeys the law? I answer
*certainly not*. The law in question is undoubtedly true; but its truth
is not proved by any finite number of examples. All that these six
numbers can do is to suggest to my mind the possible existence of such
a law; and I then ascertain its truth, by proving deductively from the
rules of decimal numeration, that any number ending in five must be
made up of multiples of five, and must therefore be itself a multiple.
To make this more plain, let the reader now examine the numbers--
7, 17, 37, 47, 67, 97.
They all end in 7 instead of 5, and though not at equal intervals, the
intervals are the same as in the previous case. After consideration,
the reader will perceive that these numbers all agree in being *prime
numbers*, or multiples of unity only. May we then infer that the next,
or any other number ending in 7, is a prime number? Clearly not, for
on trial we find that 27, 57, 117 are not primes. Six instances,
then, treated empirically, lead us to a true and universal law in one
case, and mislead us in another case. We ought, in fact, to have no
confidence in any law until we have treated it deductively, and have
shown that from the conditions supposed the results expected must
ensue. No one can show from the principles of number, that numbers
ending in 7 should be primes.
From the history of the theory of numbers some good examples of false
induction can be adduced. Taking the following series of prime numbers,
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c.,
it will be found that they all agree in being values of the general
expression *x*^{2} + *x* + 41, putting for *x* in succession the
values, 0, 1, 2, 3, 4, &c. We seem always to obtain a prime number, and
the induction is apparently strong, to the effect that this expression
always will give primes. Yet a few more trials disprove this false
conclusion. Put *x* = 40, and we obtain 40 × 40 + 40 + 41, or 41 × 41.
Such a failure could never have happened, had we shown any deductive
reason why *x*^{2} + *x* + 41 should give primes.
There can be no doubt that what here happens with forty instances,
might happen with forty thousand or forty million instances. An
apparent law never once failing up to a certain point may then suddenly
break down, so that inductive reasoning, as it has been described by
some writers, can give no sure knowledge of what is to come. Babbage
pointed out, in his Ninth Bridgewater Treatise, that a machine could
be constructed to give a perfectly regular series of numbers through
a vast series of steps, and yet to break the law of progression
suddenly at any required point. No number of particular cases as
particulars enables us to pass by inference to any new case. It is
hardly needful to inquire here what can be inferred from an infinite
series of facts, because they are never practically within our power;
but we may unhesitatingly accept the conclusion, that no finite number
of instances can ever prove a general law, or can give us certain
knowledge of even one other instance.
General mathematical theorems have indeed been discovered by the
observation of particular cases, and may again be so discovered. We
have Newton’s own statement, to the effect that he was thus led to
the all-important Binomial Theorem, the basis of the whole structure
of mathematical analysis. Speaking of a certain series of terms,
expressing the area of a circle or hyperbola, he says: “I reflected
that the denominators were in arithmetical progression; so that
only the numerical co-efficients of the numerators remained to be
investigated. But these, in the alternate areas, were the figures of
the powers of the number eleven, namely 11^{0}, 11^{1}, 11^{2}, 11^{3},
11^{4}; that is, in the first 1; in the second 1, 1; in the third 1,
2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] I
inquired, therefore, in what manner all the remaining figures could
be found from the first two; and I found that if the first figure be
called *m*, all the rest could be found by the continual multiplication
of the terms of the formula
((*m* - 0)/1) × ((*m* - 1)/2) × ((*m* - 2)/3) ×
((*m* - 3)/4) × &c.”[139]
[138] These are the figurate numbers considered in pages 183, 187, &c.
[139] *Commercium Epistolicum.* *Epistola ad Oldenburgum*, Oct. 24,
1676. Horsley’s *Works of Newton*, vol. iv. p. 541. See De Morgan in
*Penny Cyclopædia*, art. “Binomial Theorem,” p. 412.
It is pretty evident, from this most interesting statement, that
Newton, having simply observed the succession of the numbers, tried
various formulæ until he found one which agreed with them all. He
was so little satisfied with this process, however, that he verified
particular results of his new theorem by comparison with the results
of common multiplication, and the rule for the extraction of the
square root. Newton, in fact, gave no demonstration of his theorem;
and the greatest mathematicians of the last century, James Bernoulli,
Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with
discovering a conclusive method of deductive proof.
There can be no doubt that in geometry also discoveries have been
suggested by direct observation. Many of the now trivial propositions
of Euclid’s Elements were probably thus discovered, by the ancient
Greek geometers; and we have pretty clear evidence of this in the
Commentaries of Proclus.[140] Galileo was the first to examine the
remarkable properties of the cycloid, the curve described by a point in
the circumference of a wheel rolling on a plane. By direct observation
he ascertained that the area of the curve is apparently three times
that of the generating circle or wheel, but he was unable to prove this
exactly, or to verify it by strict geometrical reasoning. Sir George
Airy has recorded a curious case, in which he fell accidentally by
trial on a new geometrical property of the sphere.[141] But discovery
in such cases means nothing more than suggestion, and it is always by
pure deduction that the general law is really established. As Proclus
puts it, *we must pass from sense to consideration*.
[140] Bk. ii. chap. iv.
[141] *Philosophical Transactions* (1866), vol. 146, p. 334.
[Illustration]
Given, for instance, the series of figures in the accompanying diagram,
measurement will show that the curved lines approximate to semicircles,
and the rectilinear figures to right-angled triangles. These figures
may seem to suggest to the mind the general law that angles inscribed
in semicircles are right angles; but no number of instances, and no
possible accuracy of measurement would really establish the truth of
that general law. Availing ourselves of the suggestion furnished by
the figures, we can only investigate deductively the consequences
which flow from the definition of a circle, until we discover among
them the property of containing right angles. Persons have thought
that they had discovered a method of trisecting angles by plane
geometrical construction, because a certain complex arrangement of
lines and circles had appeared to trisect an angle in every case tried
by them, and they inferred, by a supposed act of induction, that it
would succeed in all other cases. De Morgan has recorded a proposed
mode of trisecting the angle which could not be discriminated by the
senses from a true general solution, except when it was applied to
very obtuse angles.[142] In all such cases, it has always turned out
either that the angle was not trisected at all, or that only certain
particular angles could be thus trisected. The trisectors were misled
by some apparent or special coincidence, and only deductive proof could
establish the truth and generality of the result. In this particular
case, deductive proof shows that the problem attempted is impossible,
and that angles generally cannot be trisected by common geometrical
methods.
[142] *Budget of Paradoxes*, p. 257.
*Geometrical Reasoning.*
This view of the matter is strongly supported by the further
consideration of geometrical reasoning. No skill and care could ever
enable us to verify absolutely any one geometrical proposition.
Rousseau, in his *Emile*, tells us that we should teach a child
geometry by causing him to measure and compare figures by
superposition. While a child was yet incapable of general reasoning,
this would doubtless be an instructive exercise; but it never could
teach geometry, nor prove the truth of any one proposition. All our
figures are rude approximations, and they may happen to seem unequal
when they should be equal, and equal when they should be unequal.
Moreover figures may from chance be equal in case after case, and yet
there may be no general reason why they should be so. The results of
deductive geometrical reasoning are absolutely certain, and are either
exactly true or capable of being carried to any required degree of
approximation. In a perfect triangle, the angles must be equal to one
half-revolution precisely; even an infinitesimal divergence would be
impossible; and I believe with equal confidence, that however many are
the angles of a figure, provided there are no re-entrant angles, the
sum of the angles will be precisely and absolutely equal to twice as
many right-angles as the figure has sides, less by four right-angles.
In such cases, the deductive proof is absolute and complete; empirical
verification can at the most guard against accidental oversights.
There is a second class of geometrical truths which can only be
proved by approximation; but, as the mind sees no reason why that
approximation should not always go on, we arrive at complete
conviction. We thus learn that the surface of a sphere is equal exactly
to two-thirds of the whole surface of the circumscribing cylinder, or
to four times the area of the generating circle. The area of a parabola
is exactly two-thirds of that of the circumscribing parallelogram.
The area of the cycloid is exactly three times that of the generating
circle. These are truths that we could never ascertain, nor even verify
by observation; for any finite amount of difference, less than what the
senses can discern, would falsify them.
There are geometrical relations again which we cannot assign
exactly, but can carry to any desirable degree of approximation. The
ratio of the circumference to the diameter of a circle is that of
3·14159265358979323846.... to 1, and the approximation may be carried
to any extent by the expenditure of sufficient labour. Mr. W. Shanks
has given the value of this natural constant, known as π, to the extent
of 707 places of decimals.[143] Some years since, I amused myself
by trying how near I could get to this ratio, by the careful use of
compasses, and I did not come nearer than 1 part in 540. We might
imagine measurements so accurately executed as to give us eight or ten
places correctly. But the power of the hands and senses must soon
stop, whereas the mental powers of deductive reasoning can proceed
to an unlimited degree of approximation. Geometrical truths, then,
are incapable of verification; and, if so, they cannot even be learnt
by observation. How can I have learnt by observation a proposition
of which I cannot even prove the truth by observation, when I am in
possession of it? All that observation or empirical trial can do is
to suggest propositions, of which the truth may afterwards be proved
deductively.
[143] *Proceedings of the Royal Society* (1872–3), vol. xxi. p. 319.
If Viviani’s story is to be believed, Galileo endeavoured to satisfy
himself about the area of the cycloid by cutting out several large
cycloids in pasteboard, and then comparing the areas of the curve and
the generating circle by weighing them. In every trial the curve seemed
to be rather less than three times the circle, so that Galileo, we are
told, began to suspect that the ratio was not precisely 3 to 1. It is
quite clear, however, that no process of weighing or measuring could
ever prove truths like these, and it remained for Torricelli to show
what his master Galileo had only guessed at.[144]
[144] *Life of Galileo*, Society for the Diffusion of Useful
Knowledge, p. 102.
Much has been said about the peculiar certainty of mathematical
reasoning, but it is only certainty of deductive reasoning, and equal
certainty attaches to all correct logical deduction. If a triangle be
right-angled, the square on the hypothenuse will undoubtedly equal the
sum of the two squares on the other sides; but I can never be sure that
a triangle is right-angled: so I can be certain that nitric acid will
not dissolve gold, provided I know that the substances employed really
correspond to those on which I tried the experiment previously. Here is
like certainty of inference, and like doubt as to the facts.
*Discrimination of Certainty and Probability.*
We can never recur too often to the truth that our knowledge of the
laws and future events of the external world is only probable. The mind
itself is quite capable of possessing certain knowledge, and it is well
to discriminate carefully between what we can and cannot know with
certainty. In the first place, whatever feeling is actually present to
the mind is certainly known to that mind. If I see blue sky, I may be
quite sure that I do experience the sensation of blueness. Whatever
I do feel, I do feel beyond all doubt. We are indeed very likely to
confuse what we really feel with what we are inclined to associate with
it, or infer inductively from it; but the whole of our consciousness,
as far as it is the result of pure intuition and free from inference,
is certain knowledge beyond all doubt.
In the second place, we may have certainty of inference; the
fundamental laws of thought, and the rule of substitution (p. 9),
are certainly true; and if my senses could inform me that A was
indistinguishable in colour from B, and B from C, then I should be
equally certain that A was indistinguishable from C. In short, whatever
truth there is in the premises, I can certainly embody in their correct
logical result. But the certainty generally assumes a hypothetical
character. I never can be quite sure that two colours are exactly
alike, that two magnitudes are exactly equal, or that two bodies
whatsoever are identical even in their apparent qualities. Almost all
our judgments involve quantitative relations, and, as will be shown in
succeeding chapters, we can never attain exactness and certainty where
continuous quantity enters. Judgments concerning discontinuous quantity
or numbers, however, allow of certainty; I may establish beyond doubt,
for instance, that the difference of the squares of 17 and 13 is the
product of 17 + 13 and 17 - 13, and is therefore 30 × 4, or 120.
Inferences which we draw concerning natural objects are never certain
except in a hypothetical point of view. It might seem to be certain
that iron is magnetic, or that gold is incapable of solution in
nitric acid; but, if we carefully investigate the meanings of these
statements, they will be found to involve no certainty but that of
consciousness and that of hypothetical inference. For what do I mean
by iron or gold? If I choose a remarkable piece of yellow substance,
call it gold, and then immerse it in a liquid which I call nitric acid,
and find that there is no change called solution, then consciousness
has certainly informed me that, with my meaning of the terms, “Gold is
insoluble in nitric acid.” I may further be certain of something else;
for if this gold and nitric acid remain what they were, I may be sure
there will be no solution on again trying the experiment. If I take
other portions of gold and nitric acid, and am sure that they really
are identical in properties with the former portions, I can be certain
that there will be no solution. But at this point my knowledge becomes
purely hypothetical; for how can I be sure without trial that the gold
and acid are really identical in nature with what I formerly called
gold and nitric acid. How do I know gold when I see it? If I judge by
the apparent qualities--colour, ductility, specific gravity, &c., I
may be misled, because there may always exist a substance which to the
colour, ductility, specific gravity, and other specified qualities,
joins others which we do not expect. Similarly, if iron is magnetic,
as shown by an experiment with objects answering to those names, then
all iron is magnetic, meaning all pieces of matter identical with my
assumed piece. But in trying to identify iron, I am always open to
mistake. Nor is this liability to mistake a matter of speculation
only.[145]
[145] Professor Bowen has excellently stated this view. *Treatise on
Logic.* Cambridge, U.S.A., 1866, p. 354.
The history of chemistry shows that the most confident inferences may
have been falsified by the confusion of one substance with another.
Thus strontia was never discriminated from baryta until Klaproth
and Haüy detected differences between some of their properties.
Accordingly chemists must often have inferred concerning strontia what
was only true of baryta, and *vice versâ*. There is now no doubt that
the recently discovered substances, cæsium and rubidium, were long
mistaken for potassium.[146] Other elements have often been confused
together--for instance, tantalum and niobium; sulphur and selenium;
cerium, lanthanum, and didymium; yttrium and erbium.
[146] Roscoe’s *Spectrum Analysis*, 1st edit., p. 98.
Even the best known laws of physical science do not exclude false
inference. No law of nature has been better established than that of
universal gravitation, and we believe with the utmost confidence that
any body capable of affecting the senses will attract other bodies,
and fall to the earth if not prevented. Euler remarks that, although
he had never made trial of the stones which compose the church of
Magdeburg, yet he had not the least doubt that all of them were heavy,
and would fall if unsupported. But he adds, that it would be extremely
difficult to give any satisfactory explanation of this confident
belief.[147] The fact is, that the belief ought not to amount to
certainty until the experiment has been tried, and in the meantime a
slight amount of uncertainty enters, because we cannot be sure that
the stones of the Magdeburg Church resemble other stones in all their
properties.
[147] Euler’s *Letters to a German Princess*, translated by Hunter.
2nd ed., vol. ii. pp. 17, 18.
In like manner, not one of the inductive truths which men have
established, or think they have established, is really safe from
exception or reversal. Lavoisier, when laying the foundations of
chemistry, met with so many instances tending to show the existence
of oxygen in all acids, that he adopted a general conclusion to that
effect, and devised the name oxygen accordingly. He entertained no
appreciable doubt that the acid existing in sea salt also contained
oxygen;[148] yet subsequent experience falsified his expectations. This
instance refers to a science in its infancy, speaking relatively to
the possible achievements of men. But all sciences are and ever will
remain in their infancy, relatively to the extent and complexity of
the universe which they undertake to investigate. Euler expresses no
more than the truth when he says that it would be impossible to fix
on any one thing really existing, of which we could have so perfect
a knowledge as to put us beyond the reach of mistake.[149] We may be
quite certain that a comet will go on moving in a similar path *if*
all circumstances remain the same as before; but if we leave out this
extensive qualification, our predictions will always be subject to the
chance of falsification by some unexpected event, such as the division
of Biela’s comet or the interference of an unknown gravitating body.
[148] Lavoisier’s *Chemistry*, translated by Kerr. 3rd ed., pp. 114,
121, 123.
[149] Euler’s *Letters*, vol. ii. p. 21.
Inductive inference might attain to certainty if our knowledge of the
agents existing throughout the universe were complete, and if we were
at the same time certain that the same Power which created the universe
would allow it to proceed without arbitrary change. There is always
a possibility of causes being in existence without our knowledge,
and these may at any moment produce an unexpected effect. Even when
by the theory of probabilities we succeed in forming some notion of
the comparative confidence with which we should receive inductive
results, it yet appears to me that we must make an assumption. Events
come out like balls from the vast ballot-box of nature, and close
observation will enable us to form some notion, as we shall see in the
next chapter, of the contents of that ballot-box. But we must still
assume that, between the time of an observation and that to which our
inferences relate, no change in the ballot-box has been made.
CHAPTER XII.
THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.
We have hitherto considered the theory of probability only in its
simple deductive employment, in which it enables us to determine
from given conditions the probable character of events happening
under those conditions. But as deductive reasoning when inversely
applied constitutes the process of induction, so the calculation of
probabilities may be inversely applied; from the known character
of certain events we may argue backwards to the probability of a
certain law or condition governing those events. Having satisfactorily
accomplished this work, we may indeed calculate forwards to the
probable character of future events happening under the same
conditions; but this part of the process is a direct use of deductive
reasoning (p. 226).
Now it is highly instructive to find that whether the theory of
probability be deductively or inductively applied, the calculation is
always performed according to the principles and rules of deduction.
The probability that an event has a particular condition entirely
depends upon the probability that if the condition existed the event
would follow. If we take up a pack of common playing cards, and observe
that they are arranged in perfect numerical order, we conclude beyond
all reasonable doubt that they have been thus intentionally arranged
by some person acquainted with the usual order of sequence. This
conclusion is quite irresistible, and rightly so; for there are but
two suppositions which we can make as to the reason of the cards being
in that particular order:--
1. They may have been intentionally arranged by some one who would
probably prefer the numerical order.
2. They may have fallen into that order by chance, that is, by some
series of conditions which, being unknown to us, cannot be known to
lead by preference to the particular order in question.
The latter supposition is by no means absurd, for any one order is as
likely as any other when there is no preponderating tendency. But we
can readily calculate by the doctrine of permutations the probability
that fifty-two objects would fall by chance into any one particular
order. Fifty-two objects can be arranged in 52 × 51 × ... × 3 × 2 × 1
or about 8066 × (10)^{64} possible orders, the number obtained
requiring 68 places of figures for its full expression. Hence it is
excessively unlikely that anyone should ever meet with a pack of cards
arranged in perfect order by accident. If we do meet with a pack so
arranged, we inevitably adopt the other supposition, that some person,
having reasons for preferring that special order, has thus put them
together.
We know that of the immense number of possible orders the numerical
order is the most remarkable; it is useful as proving the perfect
constitution of the pack, and it is the intentional result of certain
games. At any rate, the probability that intention should produce that
order is incomparably greater than the probability that chance should
produce it; and as a certain pack exists in that order, we rightly
prefer the supposition which most probably leads to the observed result.
By a similar mode of reasoning we every day arrive, and validly arrive,
at conclusions approximating to certainty. Whenever we observe a
perfect resemblance between two objects, as, for instance, two printed
pages, two engravings, two coins, two foot-prints, we are warranted in
asserting that they proceed from the same type, the same plate, the
same pair of dies, or the same boot. And why? Because it is almost
impossible that with different types, plates, dies, or boots some
apparent distinction of form should not be produced. It is impossible
for the hand of the most skilful artist to make two objects alike, so
that mechanical repetition is the only probable explanation of exact
similarity.
We can often establish with extreme probability that one document
is copied from another. Suppose that each document contains 10,000
words, and that the same word is incorrectly spelt in each. There is
then a probability of less than 1 in 10,000 that the same mistake
should be made in each. If we meet with a second error occurring in
each document, the probability is less than 1 in 10,000 × 9999, that
two such coincidences should occur by chance, and the numbers grow
with extreme rapidity for more numerous coincidences. We cannot make
any precise calculations without taking into account the character of
the errors committed, concerning the conditions of which we have no
accurate means of estimating probabilities. Nevertheless, abundant
evidence may thus be obtained as to the derivation of documents from
each other. In the examination of many sets of logarithmic tables, six
remarkable errors were found to be present in all but two, and it was
proved that tables printed at Paris, Berlin, Florence, Avignon, and
even in China, besides thirteen sets printed in England between the
years 1633 and 1822, were derived directly or indirectly from some
common source.[150] With a certain amount of labour, it is possible
to establish beyond reasonable doubt the relationship or genealogy of
any number of copies of one document, proceeding possibly from parent
copies now lost. The relations between the manuscripts of the New
Testament have been elaborately investigated in this manner, and the
same work has been performed for many classical writings, especially by
German scholars.
[150] Lardner, *Edinburgh Review*, July 1834, p. 277.
*Principle of the Inverse Method.*
The inverse application of the rules of probability entirely depends
upon a proposition which may be thus stated, nearly in the words of
Laplace.[151] *If an event can be produced by any one of a certain
number of different causes, all equally probable à priori, the
probabilities of the existence of these causes as inferred from the
event, are proportional to the probabilities of the event as derived
from these causes.* In other words, the most probable cause of an
event which has happened is that which would most probably lead to the
event supposing the cause to exist; but all other possible causes are
also to be taken into account with probabilities proportional to the
probability that the event would happen if the cause existed. Suppose,
to fix our ideas clearly, that E is the event, and C_{1} C_{2} C_{3}
are the three only conceivable causes. If C_{1} exist, the probability
is *p*_{1} that E would follow; if C_{2} or C_{3} exist, the like
probabilities are respectively *p*_{2} and *p*_{3}. Then as *p*_{1}
is to *p*_{2}, so is the probability of C_{1} being the actual cause
to the probability of C_{2} being it; and, similarly, as *p*_{2} is
to *p*_{3}, so is the probability of C_{2} being the actual cause to
the probability of C_{3} being it. By a simple mathematical process we
arrive at the conclusion that the actual probability of C_{1} being the
cause is
*p*_{1}/(*p*_{1} + *p*_{2} + *p*_{3});
[151] *Mémoires par divers Savans*, tom. vi.; quoted by Todhunter in
his *History of the Theory of Probability*, p. 458.
and the similar probabilities of the existence of C_{2} and C_{3} are,
*p*_{2}/(*p*_{1} + *p*_{2} + *p*_{3}) and
*p*_{3}/(*p*_{1} + *p*_{2} + *p*_{3}).
The sum of these three fractions amounts to unity, which correctly
expresses the certainty that one cause or other must be in operation.
We may thus state the result in general language. *If it is certain
that one or other of the supposed causes exists, the probability that
any one does exist is the probability that if it exists the event
happens, divided by the sum of all the similar probabilities.* There
may seem to be an intricacy in this subject which may prove distasteful
to some readers; but this intricacy is essential to the subject in
hand. No one can possibly understand the principles of inductive
reasoning, unless he will take the trouble to master the meaning of
this rule, by which we recede from an event to the probability of each
of its possible causes.
This rule or principle of the indirect method is that which common
sense leads us to adopt almost instinctively, before we have any
comprehension of the principle in its general form. It is easy to see,
too, that it is the rule which will, out of a great multitude of cases,
lead us most often to the truth, since the most probable cause of an
event really means that cause which in the greatest number of cases
produces the event. Donkin and Boole have given demonstrations of this
principle, but the one most easy to comprehend is that of Poisson.
He imagines each possible cause of an event to be represented by a
distinct ballot-box, containing black and white balls, in such a ratio
that the probability of a white ball being drawn is equal to that of
the event happening. He further supposes that each box, as is possible,
contains the same total number of balls, black and white; then, mixing
all the contents of the boxes together, he shows that if a white ball
be drawn from the aggregate ballot-box thus formed, the probability
that it proceeded from any particular ballot-box is represented by the
number of white balls in that particular box, divided by the total
number of white balls in all the boxes. This result corresponds to that
given by the principle in question.[152]
[152] Poisson, *Recherches sur la Probabilité des Jugements*, Paris,
1837, pp. 82, 83.
Thus, if there be three boxes, each containing ten balls in all, and
respectively containing seven, four, and three white balls, then on
mixing all the balls together we have fourteen white ones; and if
we draw a white ball, that is if the event happens, the probability
that it came out of the first box is 7/14; which is exactly equal to
(7/10)/(7/10 + 4/10 + 3/10), the fraction given by the rule of the
Inverse Method.
*Simple Applications of the Inverse Method.*
In many cases of scientific induction we may apply the principle of the
inverse method in a simple manner. If only two, or at the most a few
hypotheses, may be made as to the origin of certain phenomena, we may
sometimes easily calculate the respective probabilities. It was thus
that Bunsen and Kirchhoff established, with a probability little short
of certainty, that iron exists in the sun. On comparing the spectra
of sunlight and of the light proceeding from the incandescent vapour
of iron, it became apparent that at least sixty bright lines in the
spectrum of iron coincided with dark lines in the sun’s spectrum. Such
coincidences could never be observed with certainty, because, even if
the lines only closely approached, the instrumental imperfections of
the spectroscope would make them apparently coincident, and if one line
came within half a millimetre of another, on the map of the spectra,
they could not be pronounced distinct. Now the average distance of the
solar lines on Kirchhoff’s map is 2 mm., and if we throw down a line,
as it were, by pure chance on such a map, the probability is about
one-half that the new line will fall within 1/2 mm. on one side or the
other of some one of the solar lines. To put it in another way, we may
suppose that each solar line, either on account of its real breadth, or
the defects of the instrument, possesses a breadth of 1/2 mm., and that
each line in the iron spectrum has a like breadth. The probability then
is just one-half that the centre of each iron line will come by chance
within 1 mm. of the centre of a solar line, so as to appear to coincide
with it. The probability of casual coincidence of each iron line with
a solar line is in like manner 1/2. Coincidence in the case of each of
the sixty iron lines is a very unlikely event if it arises casually,
for it would have a probability of only (1/2)^{60} or less than 1
in a trillion. The odds, in short, are more than a million million
millions to unity against such casual coincidence.[153] But on the
other hypothesis, that iron exists in the sun, it is highly probable
that such coincidences would be observed; it is immensely more probable
that sixty coincidences would be observed if iron existed in the sun,
than that they should arise from chance. Hence by our principle it is
immensely probable that iron does exist in the sun.
[153] Kirchhoff’s *Researches on the Solar Spectrum*. First part,
translated by Roscoe, pp. 18, 19.
All the other interesting results, given by the comparison of spectra,
rest upon the same principle of probability. The almost complete
coincidence between the spectra of solar, lunar, and planetary light
renders it practically certain that the light is all of solar origin,
and is reflected from the surfaces of the moon and planets, suffering
only slight alteration from the atmospheres of some of the planets.
A fresh confirmation of the truth of the Copernican theory is thus
furnished.
Herschel proved in this way the connection between the direction of the
oblique faces of quartz crystals, and the direction in which the same
crystals rotate the plane of polarisation of light. For if it is found
in a second crystal that the relation is the same as in the first, the
probability of this happening by chance is 1/2; the probability that
in another crystal also the direction will be the same is 1/4, and so
on. The probability that in *n* + 1 crystals there would be casual
agreement of direction is the nth power of 1/2. Thus, if in examining
fourteen crystals the same relation of the two phenomena is discovered
in each, the odds that it proceeds from uniform conditions are more
than 8000 to 1.[154] Since the first observations on this subject were
made in 1820, no exceptions have been observed, so that the probability
of invariable connection is incalculably great.
[154] *Edinburgh Review*, No. 185, vol. xcii. July 1850, p. 32;
Herschel’s *Essays*, p. 421; *Transactions of the Cambridge
Philosophical Society*, vol. i. p. 43.
It is exceedingly probable that the ancient Egyptians had exactly
recorded the eclipses occurring during long periods of time, for
Diogenes Laertius mentions that 373 solar and 832 lunar eclipses had
been observed, and the ratio between these numbers exactly expresses
that which would hold true of the eclipses of any long period, of say
1200 or 1300 years, as estimated on astronomical grounds. It is evident
that an agreement between small numbers, or customary numbers, such
as seven, one hundred, a myriad, &c., is much more likely to happen
from chance, and therefore gives much less presumption of dependence.
If two ancient writers spoke of the sacrifice of oxen, they would in
all probability describe it as a hecatomb, and there would be nothing
remarkable in the coincidence. But it is impossible to point out any
special reason why an old writer should select such numbers as 373 and
832, unless they had been the results of observation.
On similar grounds, we must inevitably believe in the human origin
of the flint flakes so copiously discovered of late years. For
though the accidental stroke of one stone against another may often
produce flakes, such as are occasionally found on the sea-shore, yet
when several flakes are found in close company, and each one bears
evidence, not of a single blow only, but of several successive blows,
all conducing to form a symmetrical knife-like form, the probability
of a natural and accidental origin becomes incredibly small, and the
contrary supposition, that they are the work of intelligent beings,
approximately certain.[155]
[155] Evans’ *Ancient Stone Implements of Great Britain*. London,
1872 (Longmans).
*The Theory of Probability in Astronomy.*
The science of astronomy, occupied with the simple relations of
distance, magnitude, and motion of the heavenly bodies, admits more
easily than almost any other science of interesting conclusions founded
on the theory of probability. More than a century ago, in 1767, Michell
showed the extreme probability of bonds connecting together systems
of stars. He was struck by the unexpected number of fixed stars
which have companions close to them. Such a conjunction might happen
casually by one star, although possibly at a great distance from the
other, happening to lie on a straight line passing near the earth.
But the probabilities are so greatly against such an optical union
happening often in the expanse of the heavens, that Michell asserted
the existence of some connection between most of the double stars.
It has since been estimated by Struve, that the odds are 9570 to 1
against any two stars of not less than the seventh magnitude falling
within the apparent distance of four seconds of each other by chance,
and yet ninety-one such cases were known when the estimation was made,
and many more cases have since been discovered. There were also four
known triple stars, and yet the odds against the appearance of any one
such conjunction are 173,524 to 1.[156] The conclusions of Michell have
been entirely verified by the discovery that many double stars are
connected by gravitation.
[156] Herschel, *Outlines of Astronomy*, 1849, p. 565; but Todhunter,
in his *History of the Theory of Probability*, p. 335, states that
the calculations do not agree with those published by Struve.
Michell also investigated the probability that the six brightest stars
in the Pleiades should have come by accidents into such striking
proximity. Estimating the number of stars of equal or greater
brightness at 1500, be found the odds to be nearly 500,000 to 1 against
casual conjunction. Extending the same kind of argument to other
clusters, such as that of Præsepe, the nebula in the hilt of Perseus’
sword, he says:[157] “We may with the highest probability conclude,
the odds against the contrary opinion being many million millions
to one, that the stars are really collected together in clusters
in some places, where they form a kind of system, while in others
there are either few or none of them, to whatever cause this may be
owing, whether to their mutual gravitation, or to some other law or
appointment of the Creator.”
[157] *Philosophical Transactions*, 1767, vol. lvii. p. 431.
The calculations of Michell have been called in question by the late
James D. Forbes,[158] and Mr. Todhunter vaguely countenances his
objections,[159] otherwise I should not have thought them of much
weight. Certainly Laplace accepts Michell’s views,[160] and if Michell
be in error it is in the methods of calculation, not in the general
validity of his reasoning and conclusions.
[158] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 401,
December 1850; also August 1849.
[159] *History*, &c., p. 334.
[160] *Essai Philosophique*, p. 57.
Similar calculations might no doubt be applied to the peculiar drifting
motions which have been detected by Mr. R A. Proctor in some of the
constellations.[161] The odds are very greatly against any numerous
group of stars moving together in any one direction by chance. On like
grounds, there can be no doubt that the sun has a considerable proper
motion because on the average the fixed stars show a tendency to move
apparently from one point of the heavens towards that diametrically
opposite. The sun’s motion in the contrary direction would explain
this tendency, otherwise we must believe that thousands of stars
accidentally agree in their direction of motion, or are urged by some
common force from which the sun is exempt. It may be said that the
rotation of the earth is proved in like manner, because it is immensely
more probable that one body would revolve than that the sun, moon,
planets, comets, and the whole of the stars of the heavens should be
whirled round the earth daily, with a uniform motion superadded to
their own peculiar motions. This appears to be mainly the reason which
led Gilbert, one of the earliest English Copernicans, and in every
way an admirable physicist, to admit the rotation of the earth, while
Francis Bacon denied it.
[161] *Proceedings of the Royal Society*; 20 January, 1870;
*Philosophical Magazine*, 4th Series, vol. xxxix. p. 381.
In contemplating the planetary system, we are struck with the
similarity in direction of nearly all its movements. Newton remarked
upon the regularity and uniformity of these motions, and contrasted
them with the eccentricity and irregularity of the cometary
orbits.[162] Could we, in fact, look down upon the system from the
northern side, we should see all the planets moving round from west
to east, the satellites moving round their primaries, and the sun,
planets, and satellites rotating in the same direction, with some
exceptions on the verge of the system. In the time of Laplace eleven
planets were known, and the directions of rotation were known for the
sun, six planets, the satellites of Jupiter, Saturn’s ring, and one of
his satellites. Thus there were altogether 43 motions all concurring,
namely:--
Orbital motions of eleven planets 11
Orbital motions of eighteen satellites 18
Axial rotations 14
--
43
[162] *Principia*, bk. ii. General scholium.
The probability that 43 motions independent of each other would
coincide by chance is the 42nd power of 1/2, so that the odds are
about 4,400,000,000,000 to 1 in favour of some common cause for the
uniformity of direction. This probability, as Laplace observes,[163]
is higher than that of many historical events which we undoubtingly
believe. In the present day, the probability is much increased by the
discovery of additional planets, and the rotation of other satellites,
and it is only slightly weakened by the fact that some of the outlying
satellites are exceptional in direction, there being considerable
evidence of an accidental disturbance in the more distant parts of the
system.
[163] *Essai Philosophique*, p. 55. Laplace appears to count the
rings of Saturn as giving two independent movements.
Hardly less remarkable than the uniform direction of motion is the
near approximation of the orbits of the planets to a common plane.
Daniel Bernoulli roughly estimated the probability of such an agreement
arising from accident as 1 ÷ (12)^{6} the greatest inclination of any
orbit to the sun’s equator being 1-12th part of a quadrant. Laplace
devoted to this subject some of his most ingenious investigations. He
found the probability that the sum of the inclinations of the planetary
orbits would not exceed by accident the actual amount (·914187
of a right angle for the ten planets known in 1801) to be (1/10)!
(·914187)^{10} or about ·00000011235. This probability may be combined
with that derived from the direction of motion, and it then becomes
immensely probable that the constitution of the planetary system arose
out of uniform conditions, or, as we say, from some common cause.[164]
[164] Lubbock, *Essay on Probability*, p. 14. De Morgan, *Encyc.
Metrop.* art. *Probability*, p. 412. Todhunter’s *History of the
Theory of Probability*, p. 543. Concerning the objections raised to
these conclusions by Boole, see the *Philosophical Magazine*, 4th
Series, vol. ii. p. 98. Boole’s *Laws of Thought*, pp. 364–375.
If the same kind of calculation be applied to the orbits of comets,
the result is very different.[165] Of the orbits which have been
determined 48·9 per cent. only are direct or in the same direction as
the planetary motions.[166] Hence it becomes apparent that comets do
not properly belong to the solar system, and it is probable that they
are stray portions of nebulous matter which have accidentally become
attached to the system by the attractive powers of the sun or Jupiter.
[165] Laplace, *Essai Philosophique*, pp. 55, 56.
[166] Chambers’ *Astronomy*, 2nd ed. pp. 346–49.
*The General Inverse Problem.*
In the instances described in the preceding sections, we have been
occupied in receding from the occurrence of certain similar events to
the probability that there must have been a condition or cause for
such events. We have found that the theory of probability, although
never yielding a certain result, often enables us to establish an
hypothesis beyond the reach of reasonable doubt. There is, however,
another method of applying the theory, which possesses for us even
greater interest, because it illustrates, in the most complete manner,
the theory of inference adopted in this work, which theory indeed it
suggested. The problem to be solved is as follows:--
*An event having happened a certain number of times, and failed a
certain number of times, required the probability that it will happen
any given number of times in the future under the same circumstances.*
All the *larger* planets hitherto discovered move in one direction
round the sun; what is the probability that, if a new planet exterior
to Neptune be discovered, it will move in the same direction? All
known permanent gases, except chlorine, are colourless; what is the
probability that, if some new permanent gas should be discovered, it
will be colourless? In the general solution of this problem, we wish to
infer the future happening of any event from the number of times that
it has already been observed to happen. Now, it is very instructive to
find that there is no known process by which we can pass directly from
the data to the conclusion. It is always requisite to recede from the
data to the probability of some hypothesis, and to make that hypothesis
the ground of our inference concerning future events. Mathematicians,
in fact, make every hypothesis which is applicable to the question in
hand; they then calculate, by the inverse method, the probability of
every such hypothesis according to the data, and the probability that
if each hypothesis be true, the required future event will happen. The
total probability that the event will happen is the sum of the separate
probabilities contributed by each distinct hypothesis.
To illustrate more precisely the method of solving the problem, it
is desirable to adopt some concrete mode of representation, and the
ballot-box, so often employed by mathematicians, will best serve
our purpose. Let the happening of any event be represented by the
drawing of a white ball from a ballot-box, while the failure of an
event is represented by the drawing of a black ball. Now, in the
inductive problem we are supposed to be ignorant of the contents of
the ballot-box, and are required to ground all our inferences on our
experience of those contents as shown in successive drawings. Rude
common sense would guide us nearly to a true conclusion. Thus, if we
had drawn twenty balls one after another, replacing the ball after each
drawing, and the ball had in each case proved to be white, we should
believe that there was a considerable preponderance of white balls in
the urn, and a probability in favour of drawing a white ball on the
next occasion. Though we had drawn white balls for thousands of times
without fail, it would still be possible that some black balls lurked
in the urn and would at last appear, so that our inferences could never
be certain. On the other hand, if black balls came at intervals, we
should expect that after a certain number of trials the black balls
would appear again from time to time with somewhat the same frequency.
The mathematical solution of the question consists in little more
than a close analysis of the mode in which our common sense proceeds.
If twenty white balls have been drawn and no black ball, my common
sense tells me that any hypothesis which makes the black balls in
the urn considerable compared with the white ones is improbable; a
preponderance of white balls is a more probable hypothesis, and as a
deduction from this more probable hypothesis, I expect a recurrence
of white balls. The mathematician merely reduces this process of
thought to exact numbers. Taking, for instance, the hypothesis that
there are 99 white and one black ball in the urn, he can calculate the
probability that 20 white balls would be drawn in succession in those
circumstances; he thus forms a definite estimate of the probability
of this hypothesis, and knowing at the same time the probability of a
white ball reappearing if such be the contents of the urn, he combines
these probabilities, and obtains an exact estimate that a white ball
will recur in consequence of this hypothesis. But as this hypothesis
is only one out of many possible ones, since the ratio of white and
black balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has
to repeat the estimate for every such possible hypothesis. To make the
method of solving the problem perfectly evident, I will describe in the
next section a very simple case of the problem, originally devised for
the purpose by Condorcet, which was also adopted by Lacroix,[167] and
has passed into the works of De Morgan, Lubbock, and others.
[167] *Traité élémentaire du Calcul des Probabilités*, 3rd ed.
(1833), p. 148.
*Simple Illustration of the Inverse Problem.*
Suppose it to be known that a ballot-box contains only four black or
white balls, the ratio of black and white balls being unknown. Four
drawings having been made with replacement, and a white ball having
appeared on each occasion but one, it is required to determine the
probability that a white ball will appear next time. Now the hypotheses
which can be made as to the contents of the urn are very limited in
number, and are at most the following five:--
4 white and 0 black balls
3 " " 1 " "
2 " " 2 " "
1 " " 3 " "
0 " " 4 " "
The actual occurrence of black and white balls in the drawings puts the
first and last hypothesis out of the question, so that we have only
three left to consider.
If the box contains three white and one black, the probability of
drawing a white each time is 3/4, and a black 1/4; so that the compound
event observed, namely, three white and one black, has the probability
3/4 × 3/4 × 3/4 × 1/4, by the rule already given (p. 204). But as it is
indifferent in what order the balls are drawn, and the black ball might
come first, second, third, or fourth, we must multiply by four, to
obtain the probability of three white and one black in any order, thus
getting 27/64.
Taking the next hypothesis of two white and two black balls
in the urn, we obtain for the same probability the quantity
1/2 × 1/2 × 1/2 × 1/2 × 4, or 16/64, and from the third hypothesis of
one white and three black we deduce likewise 1/4 × 1/4 × 1/4 × 3/4 × 4,
or 3/64. According, then, as we adopt the first, second, or third
hypothesis, the probability that the result actually noticed would
follow is 27/64, 16/64, and 3/64. Now it is certain that one or
other of these hypotheses must be the true one, and their absolute
probabilities are proportional to the probabilities that the observed
events would follow from them (pp. 242, 243). All we have to do, then,
in order to obtain the absolute probability of each hypothesis, is to
alter these fractions in a uniform ratio, so that their sum shall be
unity, the expression of certainty. Now, since 27 + 16 + 3 = 46, this
will be effected by dividing each fraction by 46, and multiplying by
64. Thus the probabilities of the first, second, and third hypotheses
are respectively--
27/46, 16/46, 3/46.
The inductive part of the problem is completed, since we have found
that the urn most likely contains three white and one black ball, and
have assigned the exact probability of each possible supposition. But
we are now in a position to resume deductive reasoning, and infer the
probability that the next drawing will yield, say a white ball. For if
the box contains three white and one black ball, the probability of
drawing a white one is certainly 3/4; and as the probability of the box
being so constituted is 27/46, the compound probability that the box
will be so filled and will give a white ball at the next trial, is
27/46 × 3/4 or 81/184.
Again, the probability is 16/46 that the box contains two white and two
black, and under those conditions the probability is 1/2 that a white
ball will appear; hence the probability that a white ball will appear
in consequence of that condition, is
16/46 × 1/2 or 32/184.
From the third supposition we get in like manner the probability
3/46 × 1/4 or 3/184.
Since one and not more than one hypothesis can be true, we may add
together these separate probabilities, and we find that
81/184 + 32/184 + 3/184 or 116/184
is the complete probability that a white ball will be next drawn under
the conditions and data supposed.
*General Solution of the Inverse Problem.*
In the instance of the inverse method described in the last section,
the balls supposed to be in the ballot-box were few, for the purpose of
simplifying the calculation. In order that our solution may apply to
natural phenomena, we must render our hypotheses as little arbitrary
as possible. Having no *à priori* knowledge of the conditions of the
phenomena in question, there is no limit to the variety of hypotheses
which might be suggested. Mathematicians have therefore had recourse
to the most extensive suppositions which can be made, namely, that the
ballot-box contains an infinite number of balls; they have then varied
the proportion of white to black balls continuously, from the smallest
to the greatest possible proportion, and estimated the aggregate
probability which results from this comprehensive supposition.
To explain their procedure, let us imagine that, instead of an infinite
number, the ballot-box contains a large finite number of balls, say
1000. Then the number of white balls might be 1 or 2 or 3 or 4, and so
on, up to 999. Supposing that three white and one black ball have been
drawn from the urn as before, there is a certain very small probability
that this would have occurred in the case of a box containing one white
and 999 black balls; there is also a small probability that from such
a box the next ball would be white. Compound these probabilities, and
we have the probability that the next ball really will be white, in
consequence of the existence of that proportion of balls. If there be
two white and 998 black balls in the box, the probability is greater
and will increase until the balls are supposed to be in the proportion
of those drawn. Now 999 different hypotheses are possible, and the
calculation is to be made for each of these, and their aggregate taken
as the final result. It is apparent that as the number of balls in
the box is increased, the absolute probability of any one hypothesis
concerning the exact proportion of balls is decreased, but the
aggregate results of all the hypotheses will assume the character of a
wider average.
When we take the step of supposing the balls within the urn to be
infinite in number, the possible proportions of white and black balls
also become infinite, and the probability of any one proportion
actually existing is infinitely small. Hence the final result that
the next ball drawn will be white is really the sum of an infinite
number of infinitely small quantities. It might seem impossible to
calculate out a problem having an infinite number of hypotheses,
but the wonderful resources of the integral calculus enable this
to be done with far greater facility than if we supposed any large
finite number of balls, and then actually computed the results. I
will not attempt to describe the processes by which Laplace finally
accomplished the complete solution of the problem. They are to be found
described in several English works, especially De Morgan’s *Treatise
on Probabilities*, in the *Encyclopædia Metropolitana*, and Mr.
Todhunter’s *History of the Theory of Probability*. The abbreviating
power of mathematical analysis was never more strikingly shown. But
I may add that though the integral calculus is employed as a means
of summing infinitely numerous results, we in no way abandon the
principles of combinations already treated. We calculate the values of
infinitely numerous factorials, not, however, obtaining their actual
products, which would lead to an infinite number of figures, but
obtaining the final answer to the problem by devices which can only be
comprehended after study of the integral calculus.
It must be allowed that the hypothesis adopted by Laplace is in some
degree arbitrary, so that there was some opening for the doubt which
Boole has cast upon it.[168] But it may be replied, (1) that the
supposition of an infinite number of balls treated in the manner of
Laplace is less arbitrary and more comprehensive than any other that
can be suggested. (2) The result does not differ much from that
which would be obtained on the hypothesis of any large finite number
of balls. (3) The supposition leads to a series of simple formulas
which can be applied with ease in many cases, and which bear all the
appearance of truth so far as it can be independently judged by a sound
and practiced understanding.
[168] *Laws of Thought*, pp. 368–375.
*Rules of the Inverse Method.*
By the solution of the problem, as described in the last section, we
obtain the following series of simple rules.
1. *To find the probability that an event which has not hitherto been
observed to fail will happen once more, divide the number of times the
event has been observed increased by one, by the same number increased
by two.*
If there have been *m* occasions on which a certain event might have
been observed to happen, and it has happened on all those occasions,
then the probability that it will happen on the next occasion of the
same kind (*m* + 1)/(*m* + 2). For instance, we may say that there are
nine places in the planetary system where planets might exist obeying
Bode’s law of distance, and in every place there is a planet obeying
the law more or less exactly, although no reason is known for the
coincidence. Hence the probability that the next planet beyond Neptune
will conform to the law is 10/11.
2. *To find the, probability that an event which has not hitherto
failed will not fail for a certain number of new occasions, divide the
number of times the event has happened increased by one, by the same
number increased by one and the number of times it is to happen.*
An event having happened *m* times without fail, the probability that
it will happen *n* more times is (*m* + 1)/(*m* + *n* + 1). Thus the
probability that three new planets would obey Bode’s law is 10/13; but
it must be allowed that this, as well as the previous result, would be
much weakened by the fact that Neptune can barely be said to obey the
law.
*3. An event having happened and failed a certain number of times, to
find the probability that it will happen the next time, divide the
number of times the event has happened increased by one, by the whole
number of times the event has happened or failed increased by two.*
If an event has happened *m* times and failed *n* times,
the probability that it will happen on the next occasion is
(*m* + 1)/(*m* + *n* + 2). Thus, if we assume that of the elements
discovered up to the year 1873, 50 are metallic and 14 non-metallic,
then the probability that the next element discovered will be metallic
is 51/66. Again, since of 37 metals which have been sufficiently
examined only four, namely, sodium, potassium, lanthanum, and
lithium, are of less density than water, the probability that the
next metal examined or discovered will be less dense than water is
(4 + 1)/(37 + 2) or 5/39.
We may state the results of the method in a more general manner
thus,[169]--If under given circumstances certain events A, B, C, &c.,
have happened respectively *m*, *n*, *p*, &c., times, and one or other
of these events must happen, then the probabilities of these events are
proportional to *m* + 1, *n* + 1, *p* + 1, &c., so that the probability
of A will be (*m* + 1)/(*m* + 1 + *n* + 1 + *p* + 1 + &c.) But if new
events may happen in addition to those which have been observed, we
must assign unity for the probability of such new event. The odds then
become 1 for a new event, *m* + 1 for A, *n* + 1 for B, and so on, and
the absolute probability of A is (*m* + 1)/(1 + *m* + 1 + *n* + 1 + &c.)
[169] De Morgan’s *Essay on Probabilities*, Cabinet Cyclopædia, p. 67.
It is interesting to trace out the variations of probability according
to these rules. The first time a casual event happens it is 2 to 1
that it will happen again; if it does happen it is 3 to 1 that it
will happen a third time; and on successive occasions of the like
kind the odds become 4, 5, 6, &c., to 1. The odds of course will be
discriminated from the probabilities which are successively 2/3, 3/4,
4/5, &c. Thus on the first occasion on which a person sees a shark,
and notices that it is accompanied by a little pilot fish, the odds
are 2 to 1, or the probability 2/3, that the next shark will be so
accompanied.
When an event has happened a very great number of times, its
happening once again approaches nearly to certainty. If we suppose
the sun to have risen one thousand million times, the probability
that it will rise again, on the ground of this knowledge merely, is
(1,000,000,000 + 1)/(1,000,000,000 + 1 + 1). But then the probability
that it will continue to rise for as long a period in the future is
only (1,000,000,000 + 1)/(2,000,000,000 + 1), or almost exactly 1/2.
The probability that it will continue so rising a thousand times
as long is only about 1/1001. The lesson which we may draw from
these figures is quite that which we should adopt on other grounds,
namely, that experience never affords certain knowledge, and that
it is exceedingly improbable that events will always happen as we
observe them. Inferences pushed far beyond their data soon lose
any considerable probability. De Morgan has said,[170] “No finite
experience whatsoever can justify us in saying that the future shall
coincide with the past in all time to come, or that there is any
probability for such a conclusion.” On the other hand, we gain the
assurance that experience sufficiently extended and prolonged will
give us the knowledge of future events with an unlimited degree of
probability, provided indeed that those events are not subject to
arbitrary interference.
[170] *Essay on Probabilities*, p. 128.
It must be clearly understood that these probabilities are only such
as arise from the mere happening of the events, irrespective of any
knowledge derived from other sources concerning those events or the
general laws of nature. All our knowledge of nature is indeed founded
in like manner upon observation, and is therefore only probable. The
law of gravitation itself is only probably true. But when a number of
different facts, observed under the most diverse circumstances, are
found to be harmonized under a supposed law of nature, the probability
of the law approximates closely to certainty. Each science rests upon
so many observed facts, and derives so much support from analogies or
connections with other sciences, that there are comparatively few cases
where our judgment of the probability of an event depends entirely
upon a few antecedent events, disconnected from the general body of
physical science.
Events, again, may often exhibit a regularity of succession or
preponderance of character, which the simple formula will not take into
account. For instance, the majority of the elements recently discovered
are metals, so that the probability of the next discovery being that
of a metal, is doubtless greater than we calculated (p. 258). At
the more distant parts of the planetary system, there are symptoms
of disturbance which would prevent our placing much reliance on any
inference from the prevailing order of the known planets to those
undiscovered ones which may possibly exist at great distances. These
and all like complications in no way invalidate the theoretic truth of
the formulas, but render their sound application much more difficult.
Erroneous objections have been raised to the theory of probability, on
the ground that we ought not to trust to our *à priori* conceptions
of what is likely to happen, but should always endeavour to obtain
precise experimental data to guide us.[171] This course, however,
is perfectly in accordance with the theory, which is our best and
only guide, whatever data we possess. We ought to be always applying
the inverse method of probabilities so as to take into account all
additional information. When we throw up a coin for the first time, we
are probably quite ignorant whether it tends more to fall head or tail
upwards, and we must therefore assume the probability of each event as
1/2. But if it shows head in the first throw, we now have very slight
experimental evidence in favour of a tendency to show head. The chance
of two heads is now slightly greater than 1/4, which it appeared to
be at first,[172] and as we go on throwing the coin time after time,
the probability of head appearing next time constantly varies in a
slight degree according to the character of our previous experience. As
Laplace remarks, we ought always to have regard to such considerations
in common life. Events when closely scrutinized will hardly ever prove
to be quite independent, and the slightest preponderance one way or
the other is some evidence of connection, and in the absence of better
evidence should be taken into account.
[171] J. S. Mill, *System of Logic*, 5th edition, bk. iii. chap.
xviii. § 3.
[172] Todhunter’s *History*, pp. 472, 598.
The grand object of seeking to estimate the probability of future
events from past experience, seems to have been entertained by James
Bernoulli and De Moivre, at least such was the opinion of Condorcet;
and Bernoulli may be said to have solved one case of the problem.[173]
The English writers Bayes and Price are, however, undoubtedly the first
who put forward any distinct rules on the subject.[174] Condorcet and
several other eminent mathematicians advanced the mathematical theory
of the subject; but it was reserved to the immortal Laplace to bring
to the subject the full power of his genius, and carry the solution of
the problem almost to perfection. It is instructive to observe that a
theory which arose from petty games of chance, the rules and the very
names of which are forgotten, gradually advanced, until it embraced the
most sublime problems of science, and finally undertook to measure the
value and certainty of all our inductions.
[173] Todhunter’s *History*, pp. 378, 379.
[174] *Philosophical Transactions*, [1763], vol. liii. p. 370, and
[1764], vol. liv. p. 296. Todhunter, pp. 294–300.
*Fortuitous Coincidences.*
We should have studied the theory of probability to very little
purpose, if we thought that it would furnish us with an infallible
guide. The theory itself points out the approximate certainty, that we
shall sometimes be deceived by extraordinary fortuitous coincidences.
There is no run of luck so extreme that it may not happen, and it
may happen to us, or in our time, as well as to other persons or in
other times. We may be forced by correct calculation to refer such
coincidences to a necessary cause, and yet we may be deceived. All
that the calculus of probability pretends to give, is *the result in
the long run*, as it is called, and this really means in *an infinity
of cases*. During any finite experience, however long, chances may be
against us. Nevertheless the theory is the best guide we can have. If
we always think and act according to its well-interpreted indications,
we shall have the best chance of escaping error; and if all persons,
throughout all time to come, obey the theory in like manner, they will
undoubtedly thereby reap the greatest advantage.
No rule can be given for discriminating between coincidences which
are casual and those which are the effects of law. By a fortuitous
or casual coincidence, we mean an agreement between events, which
nevertheless arise from wholly independent and different causes or
conditions, and which will not always so agree. It is a fortuitous
coincidence, if a penny thrown up repeatedly in various ways always
falls on the same side; but it would not be fortuitous if there were
any similarity in the motions of the hand, and the height of the
throw, so as to cause or tend to cause a uniform result. Now among the
infinitely numerous events, objects, or relations in the universe, it
is quite likely that we shall occasionally notice casual coincidences.
There are seven intervals in the octave, and there is nothing very
improbable in the colours of the spectrum happening to be apparently
divisible into the same or similar series of seven intervals. It is
hardly yet decided whether this apparent coincidence, with which Newton
was much struck, is well founded or not,[175] but the question will
probably be decided in the negative.
[175] Newton’s *Opticks*, Bk. I., Part ii. Prop. 3; *Nature*, vol. i.
p. 286.
It is certainly a casual coincidence which the ancients noticed between
the seven vowels, the seven strings of the lyre, the seven Pleiades,
and the seven chiefs at Thebes.[176] The accidents connected with the
number seven have misled the human intellect throughout the historical
period. Pythagoras imagined a connection between the seven planets and
the seven intervals of the monochord. The alchemists were never tired
of drawing inferences from the coincidence in numbers of the seven
planets and the seven metals, not to speak of the seven days of the
week.
[176] Aristotle’s *Metaphysics*, xiii. 6. 3.
A singular circumstance was pointed out concerning the dimensions
of the earth, sun, and moon; the sun’s diameter was almost exactly
110 times as great as the earth’s diameter, while in almost exactly
the same ratio the mean distance of the earth was greater than the
sun’s diameter, and the mean distance of the moon from the earth was
greater than the moon’s diameter. The agreement was so close that it
might have proved more than casual, but its fortuitous character is
now sufficiently shown by the fact, that the coincidence ceases to be
remarkable when we adopt the amended dimensions of the planetary system.
A considerable number of the elements have atomic weights, which are
apparently exact multiples of that of hydrogen. If this be not a law to
be ultimately extended to all the elements, as supposed by Prout, it
is a most remarkable coincidence. But, as I have observed, we have no
means of absolutely discriminating accidental coincidences from those
which imply a deep producing cause. A coincidence must either be very
strong in itself, or it must be corroborated by some explanation or
connection with other laws of nature. Little attention was ever given
to the coincidence concerning the dimensions of the sun, earth, and
moon, because it was not very strong in itself, and had no apparent
connection with the principles of physical astronomy. Prout’s Law
bears more probability because it would bring the constitution of
the elements themselves in close connection with the atomic theory,
representing them as built up out of a simpler substance.
In historical and social matters, coincidences are frequently pointed
out which are due to chance, although there is always a strong popular
tendency to regard them as the work of design, or as having some hidden
meaning. If to 1794, the number of the year in which Robespierre fell,
we add the sum of its digits, the result is 1815, the year in which
Napoleon fell; the repetition of the process gives 1830 the year
in which Charles the Tenth abdicated. Again, the French Chamber of
Deputies, in 1830, consisted of 402 members, of whom 221 formed the
party called “La queue de Robespierre,” while the remainder, 181 in
number, were named “Les honnêtes gens.” If we give to each letter a
numerical value corresponding to its place in the alphabet, it will be
found that the sum of the values of the letters in each name exactly
indicates the number of the party.
A number of such coincidences, often of a very curious character,
might be adduced, and the probability against the occurrence of each
is enormously great. They must be attributed to chance, because they
cannot be shown to have the slightest connection with the general
laws of nature; but persons are often found to be greatly influenced
by such coincidences, regarding them as evidence of fatality, that
is of a system of causation governing human affairs independently of
the ordinary laws of nature. Let it be remembered that there are an
infinite number of opportunities in life for some strange coincidence
to present itself, so that it is quite to be expected that remarkable
conjunctions will sometimes happen.
In all matters of judicial evidence, we must bear in mind the probable
occurrence from time to time of unaccountable coincidences. The Roman
jurists refused for this reason to invalidate a testamentary deed, the
witnesses of which had sealed it with the same seal. For witnesses
independently using their own seals might be found to possess identical
ones by accident.[177] It is well known that circumstantial evidence of
apparently overwhelming completeness will sometimes lead to a mistaken
judgment, and as absolute certainty is never really attainable, every
court must act upon probabilities of a high amount, and in a certain
small proportion of cases they must almost of necessity condemn the
innocent victims of a remarkable conjuncture of circumstances.[178]
Popular judgments usually turn upon probabilities of far less amount,
as when the palace of Nicomedia, and even the bedchamber of Diocletian,
having been on fire twice within fifteen days, the people entirely
refused to believe that it could be the result of accident. The Romans
believed that there was fatality connected with the name of Sextus.
“Semper sub Sextis perdita Roma fuit.”
[177] Possunt autem omnes testes et uno annulo signare testamentum
Quid enim si septem annuli una sculptura fuerint, secundum quod
Pomponio visum est?--*Justinian*, ii. tit. x. 5.
[178] See Wills on *Circumstantial Evidence*, p. 148.
The utmost precautions will not provide against all contingencies.
To avoid errors in important calculations, it is usual to have them
repeated by different computers; but a case is on record in which three
computers made exactly the same calculations of the place of a star,
and yet all did it wrong in precisely the same manner, for no apparent
reason.[179]
[179] *Memoirs of the Royal Astronomical Society*, vol. iv. p. 290,
quoted by Lardner, *Edinburgh Review*, July 1834, p. 278.
*Summary of the Theory of Inductive Inference.*
The theory of inductive inference stated in this and the previous
chapters, was suggested by the study of the Inverse Method of
Probability, but it also bears much resemblance to the so-called
Deductive Method described by Mill, in his celebrated *System of
Logic*. Mill’s views concerning the Deductive Method, probably form
the most original and valuable part of his treatise, and I should
have ascribed the doctrine entirely to him, had I not found that
the opinions put forward in other parts of his work are entirely
inconsistent with the theory here upheld. As this subject is the most
important and difficult one with which we have to deal, I will try to
remedy the imperfect manner in which I have treated it, by giving a
recapitulation of the views adopted.
All inductive reasoning is but the inverse application of deductive
reasoning. Being in possession of certain particular facts or events
expressed in propositions, we imagine some more general proposition
expressing the existence of a law or cause; and, deducing the
particular results of that supposed general proposition, we observe
whether they agree with the facts in question. Hypothesis is thus
always employed, consciously or unconsciously. The sole conditions to
which we need conform in framing any hypothesis is, that we both have
and exercise the power of inferring deductively from the hypothesis to
the particular results, which are to be compared with the known facts.
Thus there are but three steps in the process of induction:--
(1) Framing some hypothesis as to the character of the general law.
(2) Deducing consequences from that law.
(3) Observing whether the consequences agree with the particular facts
under consideration.
In very simple cases of inverse reasoning, hypothesis may seem
altogether needless. To take numbers again as a convenient
illustration, I have only to look at the series,
1, 2, 4, 8, 16, 32, &c.,
to know at once that the general law is that of geometrical
progression; I need no successive trial of various hypotheses, because
I am familiar with the series, and have long since learnt from what
general formula it proceeds. In the same way a mathematician becomes
acquainted with the integrals of a number of common formulas, so
that he need not go through any process of discovery. But it is none
the less true that whenever previous reasoning does not furnish the
knowledge, hypotheses must be framed and tried (p. 124).
There naturally arise two cases, according as the nature of the
subject admits of certain or only probable deductive reasoning.
Certainty, indeed, is but a singular case of probability, and the
general principles of procedure are always the same. Nevertheless,
when certainty of inference is possible, the process is simplified.
Of several mutually inconsistent hypotheses, the results of which can
be certainly compared with fact, but one hypothesis can ultimately
be entertained. Thus in the inverse logical problem, two logically
distinct conditions could not yield the same series of possible
combinations. Accordingly, in the case of two terms we had to choose
one of six different kinds of propositions (p. 136), and in the case
of three terms, our choice lay among 192 possible distinct hypotheses
(p. 140). Natural laws, however, are often quantitative in character,
and the possible hypotheses are then infinite in variety.
When deduction is certain, comparison with fact is needed only to
assure ourselves that we have rightly selected the hypothetical
conditions. The law establishes itself, and no number of particular
verifications can add to its probability. Having once deduced from
the principles of algebra that the difference of the squares of two
numbers is equal to the product of their sum and difference, no number
of particular trials of its truth will render it more certain. On the
other hand, no finite number of particular verifications of a supposed
law will render that law certain. In short, certainty belongs only to
the deductive process, and to the teachings of direct intuition; and
as the conditions of nature are not given by intuition, we can only be
certain that we have got a correct hypothesis when, out of a limited
number conceivably possible, we select that one which alone agrees with
the facts to be explained.
In geometry and kindred branches of mathematics, deductive reasoning
is conspicuously certain, and it would often seem as if the
consideration of a single diagram yields us certain knowledge of a
general proposition. But in reality all this certainty is of a purely
hypothetical character. Doubtless if we could ascertain that a supposed
circle was a true and perfect circle, we could be certain concerning a
multitude of its geometrical properties. But geometrical figures are
physical objects, and the senses can never assure us as to their exact
forms. The figures really treated in Euclid’s *Elements* are imaginary,
and we never can verify in practice the conclusions which we draw with
certainty in inference; questions of degree and probability enter.
Passing now to subjects in which deduction is only probable, it ceases
to be possible to adopt one hypothesis to the exclusion of the others.
We must entertain at the same time all conceivable hypotheses, and
regard each with the degree of esteem proportionate to its probability.
We go through the same steps as before.
(1) We frame an hypothesis.
(2) We deduce the probability of various series of possible
consequences.
(3) We compare the consequences with the particular facts, and observe
the probability that such facts would happen under the hypothesis.
The above processes must be performed for every conceivable hypothesis,
and then the absolute probability of each will be yielded by the
principle of the inverse method (p. 242). As in the case of certainty
we accept that hypothesis which certainly gives the required results,
so now we accept as most probable that hypothesis which most probably
gives the results; but we are obliged to entertain at the same time
all other hypotheses with degrees of probability proportionate to the
probabilities that they would give the same results.
So far we have treated only of the process by which we pass from
special facts to general laws, that inverse application of deduction
which constitutes induction. But the direct employment of deduction
is often combined with the inverse. No sooner have we established
a general law, than the mind rapidly draws particular consequences
from it. In geometry we may almost seem to infer that *because* one
equilateral triangle is equiangular, therefore another is so. In
reality it is not because one is that another is, but because all
are. The geometrical conditions are perfectly general, and by what
is sometimes called *parity of reasoning* whatever is true of one
equilateral triangle, so far as it is equilateral, is true of all
equilateral triangles.
Similarly, in all other cases of inductive inference, where we seem to
pass from some particular instances to a new instance, we go through
the same process. We form an hypothesis as to the logical conditions
under which the given instances might occur; we calculate inversely
the probability of that hypothesis, and compounding this with the
probability that a new instance would proceed from the same conditions,
we gain the absolute probability of occurrence of the new instance in
virtue of this hypothesis. But as several, or many, or even an infinite
number of mutually inconsistent hypotheses may be possible, we must
repeat the calculation for each such conceivable hypothesis, and then
the complete probability of the future instance will be the sum of the
separate probabilities. The complication of this process is often very
much reduced in practice, owing to the fact that one hypothesis may be
almost certainly true, and other hypotheses, though conceivable, may be
so improbable as to be neglected without appreciable error.
When we possess no knowledge whatever of the conditions from which
the events proceed, we may be unable to form any probable hypotheses
as to their mode of origin. We have now to fall back upon the general
solution of the problem effected by Laplace, which consists in
admitting on an equal footing every conceivable ratio of favourable
and unfavourable chances for the production of the event, and then
accepting the aggregate result as the best which can be obtained. This
solution is only to be accepted in the absence of all better means,
but like other results of the calculus of probability, it comes to our
aid where knowledge is at an end and ignorance begins, and it prevents
us from over-estimating the knowledge we possess. The general results
of the solution are in accordance with common sense, namely, that
the more often an event has happened the more probable, as a general
rule, is its subsequent recurrence. With the extension of experience
this probability increases, but at the same time the probability is
slight that events will long continue to happen as they have previously
happened.
We have now pursued the theory of inductive inference, as far as
can be done with regard to simple logical or numerical relations.
The laws of nature deal with time and space, which are infinitely
divisible. As we passed from pure logic to numerical logic, so we must
now pass from questions of discontinuous, to questions of continuous
quantity, encountering fresh considerations of much difficulty. Before,
therefore, we consider how the great inductions and generalisations
of physical science illustrate the views of inductive reasoning just
explained, we must break off for a time, and review the means which we
possess of measuring and comparing magnitudes of time, space, mass,
force, momentum, energy, and the various manifestations of energy in
motion, heat, electricity, chemical change, and the other phenomena of
nature.
BOOK III.
METHODS OF MEASUREMENT.
CHAPTER XIII.
THE EXACT MEASUREMENT OF PHENOMENA.
As physical science advances, it becomes more and more accurately
quantitative. Questions of simple logical fact after a time resolve
themselves into questions of degree, time, distance, or weight. Forces
hardly suspected to exist by one generation, are clearly recognised
by the next, and precisely measured by the third generation. But
one condition of this rapid advance is the invention of suitable
instruments of measurement. We need what Francis Bacon called
*Instantiæ citantes*, or *evocantes*, methods of rendering minute
phenomena perceptible to the senses; and we also require *Instantiæ
radii* or *curriculi*, that is measuring instruments. Accordingly,
the introduction of a new instrument often forms an epoch in the
history of science. As Davy said, “Nothing tends so much to the
advancement of knowledge as the application of a new instrument. The
native intellectual powers of men in different times are not so much
the causes of the different success of their labours, as the peculiar
nature of the means and artificial resources in their possession.”
In the absence indeed of advanced theory and analytical power, a
very precise instrument would be useless. Measuring apparatus and
mathematical theory should advance *pari passu*, and with just such
precision as the theorist can anticipate results, the experimentalist
should be able to compare them with experience. The scrupulously
accurate observations of Flamsteed were the proper complement to the
intense mathematical powers of Newton.
Every branch of knowledge commences with quantitative notions of a
very rude character. After we have far progressed, it is often amusing
to look back into the infancy of the science, and contrast present
with past methods. At Greenwich Observatory in the present day, the
hundredth part of a second is not thought an inconsiderable portion
of time. The ancient Chaldæans recorded an eclipse to the nearest
hour, and the early Alexandrian astronomers thought it superfluous to
distinguish between the edge and centre of the sun. By the introduction
of the astrolabe, Ptolemy and the later Alexandrian astronomers could
determine the places of the heavenly bodies within about ten minutes
of arc. Little progress then ensued for thirteen centuries, until
Tycho Brahe made the first great step towards accuracy, not only by
employing better instruments, but even more by ceasing to regard an
instrument as correct. Tycho, in fact, determined the errors of his
instruments, and corrected his observations. He also took notice of
the effects of atmospheric refraction, and succeeded in attaining an
accuracy often sixty times as great as that of Ptolemy. Yet Tycho and
Hevelius often erred several minutes in the determination of a star’s
place, and it was a great achievement of Rœmer and Flamsteed to reduce
this error to seconds. Bradley, the modern Hipparchus, carried on the
improvement, his errors in right ascension, according to Bessel, being
under one second of time, and those of declination under four seconds
of arc. In the present day the average error of a single observation
is probably reduced to the half or quarter of what it was in Bradley’s
time; and further extreme accuracy is attained by the multiplication
of observations, and their skilful combination according to the theory
of error. Some of the more important constants, for instance that of
nutation, have been determined within the tenth part of a second of
space.[180]
[180] Baily, *British Association Catalogue of Stars*, pp. 7, 23.
It would be a matter of great interest to trace out the dependence of
this progress upon the introduction of new instruments. The astrolabe
of Ptolemy, the telescope of Galileo, the pendulum of Galileo and
Huyghens, the micrometer of Horrocks, and the telescopic sights and
micrometer of Gascoygne and Picard, Rœmer’s transit instrument,
Newton’s and Hadley’s quadrant, Dollond’s achromatic lenses, Harrison’s
chronometer, and Ramsden’s dividing engine--such were some of the
principal additions to astronomical apparatus. The result is, that we
now take note of quantities, 300,000 or 400,000 times as small as in
the time of the Chaldæans.
It would be interesting again to compare the scrupulous accuracy of a
modern trigonometrical survey with Eratosthenes’ rude but ingenious
guess at the difference of latitude between Alexandria and Syene--or
with Norwood’s measurement of a degree of latitude in 1635. “Sometimes
I measured, sometimes I paced,” said Norwood; “and I believe I am
within a scantling of the truth.” Such was the germ of those elaborate
geodesical measurements which have made the dimensions of the globe
known to us within a few hundred yards.
In other branches of science, the invention of an instrument has
usually marked, if it has not made, an epoch. The science of heat might
be said to commence with the construction of the thermometer, and it
has recently been advanced by the introduction of the thermo-electric
pile. Chemistry has been created chiefly by the careful use of the
balance, which forms a unique instance of an instrument remaining
substantially in the form in which it was first applied to scientific
purposes by Archimedes. The balance never has been and probably never
can be improved, except in details of construction. The torsion
balance, introduced by Coulomb towards the end of last century, has
rapidly become essential in many branches of investigation. In the
hands of Cavendish and Baily, it gave a determination of the earth’s
density; applied in the galvanometer, it gave a delicate measure of
electrical forces, and is indispensable in the thermo-electric pile.
This balance is made by simply suspending any light rod by a thin wire
or thread attached to the middle point. And we owe to it almost all the
more delicate investigations in the theories of heat, electricity, and
magnetism.
Though we can now take note of the millionth of an inch in space,
and the millionth of a second in time, we must not overlook the fact
that in other operations of science we are yet in the position of the
Chaldæans. Not many years have elapsed since the magnitudes of the
stars, meaning the amounts of light they send to the observer’s eye,
were guessed at in the rudest manner, and the astronomer adjudged a
star to this or that order of magnitude by a rough comparison with
other stars of the same order. To Sir John Herschel we owe an attempt
to introduce a uniform method of measurement and expression, bearing
some relation to the real photometric magnitudes of the stars.[181]
Previous to the researches of Bunsen and Roscoe on the chemical action
of light, we were devoid of any mode of measuring the energy of light;
even now the methods are tedious, and it is not clear that they give
the energy of light so much as one of its special effects. Many natural
phenomena have hardly yet been made the subject of measurement at all,
such as the intensity of sound, the phenomena of taste and smell, the
magnitude of atoms, the temperature of the electric spark or of the
sun’s photosphere.
[181] *Outlines of Astronomy*, 4th ed. sect. 781, p. 522. *Results of
Observations at the Cape of Good Hope*, &c., p. 37.
To suppose, then, that quantitative science treats only of exactly
measurable quantities, is a gross if it be a common mistake. Whenever
we are treating of an event which either happens altogether or does
not happen at all, we are engaged with a non-quantitative phenomenon,
a matter of fact, not of degree; but whenever a thing may be greater
or less, or twice or thrice as great as another, whenever, in short,
ratio enters even in the rudest manner, there science will have a
quantitative character. There can be little doubt, indeed, that
every science as it progresses will become gradually more and more
quantitative. Numerical precision is the soul of science, as Herschel
said, and as all natural objects exist in space, and involve molecular
movements, measurable in velocity and extent, there is no apparent
limit to the ultimate extension of quantitative science. But the reader
must not for a moment suppose that, because we depend more and more
upon mathematical methods, we leave logical methods behind us. Number,
as I have endeavoured to show, is logical in its origin, and quantity
is but a development of number, or analogous thereto.
*Division of the Subject.*
The general subject of quantitative investigation will have to be
divided into several parts. We shall firstly consider the means at
our disposal for measuring phenomena, and thus rendering them more or
less amenable to mathematical treatment. This task will involve an
analysis of the principles on which accurate methods of measurement are
founded, forming the subject of the remainder of the present chapter.
As measurement, however, only yields ratios, we have in the next
chapter to consider the establishment of unit magnitudes, in terms of
which our results may be expressed. As every phenomenon is usually the
sum of several distinct quantities depending upon different causes,
we have next to investigate in Chapter XV. the methods by which we
may disentangle complicated effects, and refer each part of the joint
effect to its separate cause.
It yet remains for us in subsequent chapters to treat of quantitative
induction, properly so called. We must follow out the inverse logical
method, as it presents itself in problems of a far higher degree of
difficulty than those which treat of objects related in a simple
logical manner, and incapable of merging into each other by addition
and subtraction.
*Continuous Quantity.*
The phenomena of nature are for the most part manifested in quantities
which increase or decrease continuously. When we inquire into the
precise meaning of continuous quantity, we find that it can only be
described as that which is divisible without limit. We can divide
a millimetre into ten, or a hundred, or a thousand, or ten thousand
parts, and mentally at any rate we can carry on the division *ad
infinitum*. Any finite space, then, must be conceived as made up of an
infinite number of parts each infinitely small. We cannot entertain the
simplest geometrical notions without allowing this. The conception of a
square involves the conception of a side and diagonal, which, as Euclid
beautifully proves in the 117th proposition of his tenth book, have no
common measure,[182] meaning no finite common measure. Incommensurable
quantities are, in fact, those which have for their only common measure
an infinitely small quantity. It is somewhat startling to find, too,
that in theory incommensurable quantities will be infinitely more
frequent than commensurable. Let any two lines be drawn haphazard;
it is infinitely unlikely that they will be commensurable, so that
the commensurable quantities, which we are supposed to deal with in
practice, are but singular cases among an infinitely greater number of
incommensurable cases.
[182] See De Morgan, *Study of Mathematics*, in U.K.S. Library, p. 81.
Practically, however, we treat all quantities as made up of the
least quantities which our senses, assisted by the best measuring
instruments, can perceive. So long as microscopes were uninvented, it
was sufficient to regard an inch as made up of a thousand thousandths
of an inch; now we must treat it as composed of a million millionths.
We might apparently avoid all mention of infinitely small quantities,
by never carrying our approximations beyond quantities which the
senses can appreciate. In geometry, as thus treated, we should never
assert two quantities to be equal, but only to be *apparently* equal.
Legendre really adopts this mode of treatment in the twentieth
proposition of the first book of his Geometry; and it is practically
adopted throughout the physical sciences, as we shall afterwards
see. But though our fingers, and senses, and instruments must stop
somewhere, there is no reason why the mind should not go on. We can
see that a proof which is only carried through a few steps in fact,
might be carried on without limit, and it is this consciousness of no
stopping-place, which renders Euclid’s proof of his 117th proposition
so impressive. Try how we will to circumvent the matter, we cannot
really avoid the consideration of the infinitely small and the
infinitely great. The same methods of approximation which seem confined
to the finite, mentally extend themselves to the infinite.
One result of these considerations is, that we cannot possibly adjust
two quantities in absolute equality. The suspension of Mahomet’s coffin
between two precisely equal magnets is theoretically conceivable but
practically impossible. The story of the *Merchant of Venice* turns
upon the infinite improbability that an exact quantity of flesh could
be cut. Unstable equilibrium cannot exist in nature, for it is that
which is destroyed by an infinitely small displacement. It might be
possible to balance an egg on its end practically, because no egg has
a surface of perfect curvature. Suppose the egg shell to be perfectly
smooth, and the feat would become impossible.
*The Fallacious Indications of the Senses.*
I may briefly remind the reader how little we can trust to our
unassisted senses in estimating the degree or magnitude of any
phenomenon. The eye cannot correctly estimate the comparative
brightness of two luminous bodies which differ much in brilliancy;
for we know that the iris is constantly adjusting itself to the
intensity of the light received, and thus admits more or less light
according to circumstances. The moon which shines with almost dazzling
brightness by night, is pale and nearly imperceptible while the eye is
yet affected by the vastly more powerful light of day. Much has been
recorded concerning the comparative brightness of the zodiacal light at
different times, but it would be difficult to prove that these changes
are not due to the varying darkness at the time, or the different
acuteness of the observer’s eye. For a like reason it is exceedingly
difficult to establish the existence of any change in the form or
comparative brightness of nebulæ; the appearance of a nebula greatly
depends upon the keenness of sight of the observer, or the accidental
condition of freshness or fatigue of his eye. The same is true of
lunar observations; and even the use of the best telescope fails to
remove this difficulty. In judging of colours, again, we must remember
that light of any given colour tends to dull the sensibility of the eye
for light of the same colour.
Nor is the eye when unassisted by instruments a much better judge
of magnitude. Our estimates of the size of minute bright points,
such as the fixed stars, are completely falsified by the effects of
irradiation. Tycho calculated from the apparent size of the star-discs,
that no one of the principal fixed stars could be contained within the
area of the earth’s orbit. Apart, however, from irradiation or other
distinct causes of error our visual estimates of sizes and shapes are
often astonishingly incorrect. Artists almost invariably draw distant
mountains in ludicrous disproportion to nearer objects, as a comparison
of a sketch with a photograph at once shows. The extraordinary apparent
difference of size of the sun or moon, according as it is high in the
heavens or near the horizon, should be sufficient to make us cautious
in accepting the plainest indications of our senses, unassisted by
instrumental measurement. As to statements concerning the height of the
aurora and the distance of meteors, they are to be utterly distrusted.
When Captain Parry says that a ray of the aurora shot suddenly
downwards between him and the land which was only 3,000 yards distant,
we must consider him subject to an illusion of sense.[183]
[183] Loomis, *On the Aurora Borealis*. Smithsonian Transactions,
quoting Parry’s Third Voyage, p. 61.
It is true that errors of observation are more often errors of judgment
than of sense. That which is actually seen must be so far truly seen;
and if we correctly interpret the meaning of the phenomenon, there
would be no error at all. But the weakness of the bare senses as
measuring instruments, arises from the fact that they import varying
conditions of unknown amount, and we cannot make the requisite
corrections and allowances as in the case of a solid and invariable
instrument.
Bacon has excellently stated the insufficiency of the senses for
estimating the magnitudes of objects, or detecting the degrees in which
phenomena present themselves. “Things escape the senses,” he says,
“because the object is not sufficient in quantity to strike the sense:
as all minute bodies; because the percussion of the object is too
great to be endured by the senses: as the form of the sun when looking
directly at it in mid-day; because the time is not proportionate to
actuate the sense: as the motion of a bullet in the air, or the quick
circular motion of a firebrand, which are too fast, or the hour-hand of
a common clock, which is too slow; from the distance of the object as
to place: as the size of the celestial bodies, and the size and nature
of all distant bodies; from prepossession by another object: as one
powerful smell renders other smells in the same room imperceptible;
from the interruption of interposing bodies: as the internal parts of
animals; and because the object is unfit to make an impression upon
the sense: as the air or the invisible and untangible spirit which is
included in every living body.”
*Complexity of Quantitative Questions.*
One remark which we may well make in entering upon quantitative
questions, has regard to the great variety and extent of phenomena
presented to our notice. So long as we deal only with a simply logical
question, that question is merely, Does a certain event happen? or,
Does a certain object exist? No sooner do we regard the event or object
as capable of more and less, than the question branches out into many.
We must now ask, How much is it compared with its cause? Does it change
when the amount of the cause changes? If so, does it change in the same
or opposite direction? Is the change in simple proportion to that of
the cause? If not, what more complex law of connection holds true? This
law determined satisfactorily in one series of circumstances may be
varied under new conditions, and the most complex relations of several
quantities may ultimately be established.
In every question of physical science there is thus a series of steps
the first one or two of which are usually made with ease while the
succeeding ones demand more and more careful measurement. We cannot
lay down any invariable series of questions which must be asked from
nature. The exact character of the questions will vary according
to the nature of the case, but they will usually be of an evident
kind, and we may readily illustrate them by examples. Suppose that
we are investigating the solution of some salt in water. The first
is a purely logical question: Is there solution, or is there not?
Assuming the answer to be in the affirmative, we next inquire, Does
the solubility vary with the temperature, or not? In all probability
some variation will exist, and we must have an answer to the further
question, Does the quantity dissolved increase, or does it diminish
with the temperature? In by far the greatest number of cases salts and
substances of all kinds dissolve more freely the higher the temperature
of the water; but there are a few salts, such as calcium sulphate,
which follow the opposite rule. A considerable number of salts resemble
sodium sulphate in becoming more soluble up to a certain temperature,
and then varying in the opposite direction. We next require to assign
the amount of variation as compared with that of the temperature,
assuming at first that the increase of solubility is proportional to
the increase of temperature. Common salt is an instance of very slight
variation, and potassium nitrate of very considerable increase with
temperature. Accurate observations will probably show, however, that
the simple law of proportionate variation is only approximately true,
and some more complicated law involving the second, third, or higher
powers of the temperature may ultimately be established. All these
investigations have to be carried out for each salt separately, since
no distinct principles by which we may infer from one substance to
another have yet been detected. There is still an indefinite field
for further research open; for the solubility of salts will probably
vary with the pressure under which the medium is placed; the presence
of other salts already dissolved may have effects yet unknown. The
researches already effected as regards the solvent power of water must
be repeated with alcohol, ether, carbon bisulphide, and other media,
so that unless general laws can be detected, this one phenomenon of
solution can never be exhaustively treated. The same kind of questions
recur as regards the solution or absorption of gases in liquids, the
pressure as well as the temperature having then a most decided effect,
and Professor Roscoe’s researches on the subject present an excellent
example of the successive determination of various complicated
laws.[184]
[184] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790.
There is hardly a branch of physical science in which similar
complications are not ultimately encountered. In the case of gravity,
indeed, we arrive at the final law, that the force is the same for
all kinds of matter, and varies only with the distance of action.
But in other subjects the laws, if simple in their ultimate nature,
are disguised and complicated in their apparent results. Thus the
effect of heat in expanding solids, and the reverse effect of forcible
extension or compression upon the temperature of a body, will vary
from one substance to another, will vary as the temperature is already
higher or lower, and, will probably follow a highly complex law, which
in some cases gives negative or exceptional results. In crystalline
substances the same researches have to be repeated in each distinct
axial direction.
In the sciences of pure observation, such as those of astronomy,
meteorology, and terrestrial magnetism, we meet with many interesting
series of quantitative determinations. The so-called fixed stars, as
Giordano Bruno divined, are not really fixed, and may be more truly
described as vast wandering orbs, each pursuing its own path through
space. We must then determine separately for each star the following
questions:--
1. Does it move?
2. In what direction?
3. At what velocity?
4. Is this velocity variable or uniform?
5. If variable, according to what law?
6. Is the direction uniform?
7. If not, what is the form of the apparent path?
8. Does it approach or recede? 9. What is the form of the real path?
The successive answers to such questions in the case of certain binary
stars, have afforded a proof that the motions are due to a central
force coinciding in law with gravity, and doubtless identical with it.
In other cases the motions are usually so small that it is exceedingly
difficult to distinguish them with certainty. And the time is yet
far off when any general results as regards stellar motions can be
established.
The variation in the brightness of stars opens an unlimited field for
curious observation. There is not a star in the heavens concerning
which we might not have to determine:--
1. Does it vary in brightness?
2. Is the brightness increasing or decreasing?
3. Is the variation uniform?
4. If not, according to what law does it vary?
In a majority of cases the change will probably be found to have a
periodic character, in which case several other questions will arise,
such as--
5. What is the length of the period?
6. Are there minor periods?
7. What is the law of variation within the period?
8. Is there any change in the amount of variation?
9. If so, is it a secular, *i.e.* a continually growing change, or does
it give evidence of a greater period?
Already the periodic changes of a certain number of stars have been
determined with accuracy, and the lengths of the periods vary from less
than three days up to intervals of time at least 250 times as great.
Periods within periods have also been detected.
There is, perhaps, no subject in which more complicated quantitative
conditions have to be determined than terrestrial magnetism. Since
the time when the declination of the compass was first noticed, as
some suppose by Columbus, we have had successive discoveries from
time to time of the progressive change of declination from century to
century; of the periodic character of this change; of the difference
of the declination in various parts of the earth’s surface; of the
varying laws of the change of declination; of the dip or inclination
of the needle, and the corresponding laws of its periodic changes; the
horizontal and perpendicular intensities have also been the subject of
exact measurement, and have been found to vary with place and time,
like the directions of the needle; daily and yearly periodic changes
have also been detected, and all the elements are found to be subject
to occasional storms or abnormal perturbations, in which the eleven
year period, now known to be common to many planetary relations,
is apparent. The complete solution of these motions of the compass
needle involves nothing less than a determination of its position
and oscillations in every part of the world at any epoch, the like
determination for another epoch, and so on, time after time, until the
periods of all changes are ascertained. This one subject offers to men
of science an almost inexhaustible field for interesting quantitative
research, in which we shall doubtless at some future time discover the
operation of causes now most mysterious and unaccountable.
*The Methods of Accurate Measurement.*
In studying the modes by which physicists have accomplished very exact
measurements, we find that they are very various, but that they may
perhaps be reduced under the following three classes:--
1. The increase or decrease, in some determinate ratio, of the quantity
to be measured, so as to bring it within the scope of our senses, and
to equate it with the standard unit, or some determinate multiple or
sub-multiple of this unit.
2. The discovery of some natural conjunction of events which will
enable us to compare directly the multiples of the quantity with those
of the unit, or a quantity related in a definite ratio to that unit.
3. Indirect measurement, which gives us not the quantity itself, but
some other quantity connected with it by known mathematical relations.
*Conditions of Accurate Measurement.*
Several conditions are requisite in order that a measurement may be
made with great accuracy, and that the results may be closely accordant
when several independent measurements are made.
In the first place the magnitude must be exactly defined by sharp
terminations, or precise marks of inconsiderable thickness. When a
boundary is vague and graduated, like the penumbra in a lunar eclipse,
it is impossible to say where the end really is, and different people
will come to different results. We may sometimes overcome this
difficulty to a certain extent, by observations repeated in a special
manner, as we shall afterwards see; but when possible, we should choose
opportunities for measurement when precise definition is easy. The
moment of occultation of a star by the moon can be observed with great
accuracy, because the star disappears with perfect suddenness; but
there are other astronomical conjunctions, eclipses, transits, &c.,
which occupy a certain length of time in happening, and thus open the
way to differences of opinion. It would be impossible to observe with
precision the movements of a body possessing no definite points of
reference. The colours of the complete spectrum shade into each other
so continuously that exact determinations of refractive indices would
have been impossible, had we not the dark lines of the solar spectrum
as precise points for measurement, or various kinds of homogeneous
light, such as that of sodium, possessing a nearly uniform length of
vibration.
In the second place, we cannot measure accurately unless we have the
means of multiplying or dividing a quantity without considerable error,
so that we may correctly equate one magnitude with the multiple or
submultiple of the other. In some cases we operate upon the quantity
to be measured, and bring it into accurate coincidence with the actual
standard, as when in photometry we vary the distance of our luminous
body, until its illuminating power at a certain point is equal to
that of a standard lamp. In other cases we repeat the unit until it
equals the object, as in surveying land, or determining a weight by
the balance. The requisites of accuracy now are:--(1) That we can
repeat unit after unit of exactly equal magnitude; (2) That these
can be joined together so that the aggregate shall really be the sum
of the parts. The same conditions apply to subdivision, which may be
regarded as a multiplication of subordinate units. In order to measure
to the thousandth of an inch, we must be able to add thousandth after
thousandth without error in the magnitude of these spaces, or in their
conjunction.
*Measuring Instruments.*
To consider the mechanical construction of scientific instruments, is
no part of my purpose in this book. I wish to point out merely the
general purpose of such instruments, and the methods adopted to carry
out that purpose with great precision. In the first place we must
distinguish between the instrument which effects a comparison between
two quantities, and the standard magnitude which often forms one of
the quantities compared. The astronomer’s clock, for instance, is
no standard of the efflux of time; it serves but to subdivide, with
approximate accuracy, the interval of successive passages of a star
across the meridian, which it may effect perhaps to the tenth part of
a second, or 1/864000 part of the whole. The moving globe itself is
the real standard clock, and the transit instrument the finger of the
clock, while the stars are the hour, minute, and second marks, none
the less accurate because they are disposed at unequal intervals. The
photometer is a simple instrument, by which we compare the relative
intensity of rays of light falling upon a given spot. The galvanometer
shows the comparative intensity of electric currents passing through a
wire. The calorimeter gauges the quantity of heat passing from a given
object. But no such instruments furnish the standard unit in terms of
which our results are to be expressed. In one peculiar case alone does
the same instrument combine the unit of measurement and the means of
comparison. A theodolite, mural circle, sextant, or other instrument
for the measurement of angular magnitudes has no need of an additional
physical unit; for the circle itself, or complete revolution, is
the natural unit to which all greater or lesser amounts of angular
magnitude are referred.
The result of every measurement is to make known the purely numerical
ratio existing between the magnitude to be measured, and a certain
other magnitude, which should, when possible, be a fixed unit or
standard magnitude, or at least an intermediate unit of which the
value can be ascertained in terms of the ultimate standard. But though
a ratio is the required result, an equation is the mode in which the
ratio is determined and expressed. In every measurement we equate
some multiple or submultiple of one quantity, with some multiple or
submultiple of another, and equality is always the fact which we
ascertain by the senses. By the eye, the ear, or the touch, we judge
whether there is a discrepancy or not between two lights, two sounds,
two intervals of time, two bars of metal. Often indeed we substitute
one sense for the other, as when the efflux of time is judged by
the marks upon a moving slip of paper, so that equal intervals of
time are represented by equal lengths. There is a tendency to reduce
all comparisons to the comparison of space magnitudes, but in every
case one of the senses must be the ultimate judge of coincidence or
non-coincidence.
Since the equation to be established may exist between any multiples or
submultiples of the quantities compared, there naturally arise several
different modes of comparison adapted to different cases. Let *p* be
the magnitude to be measured, and *q* that in terms of which it is to
be expressed. Then we wish to find such numbers *x* and *y*, that the
equation *p = (x/y)q* may be true. This equation may be presented in
four forms, namely:--
First Form. Second Form. Third Form. Fourth Form.
*p = (x/y)q* *p(y/x) = q* *py = qx* *p/x = q/y*
Each of these modes of expressing the same equation corresponds to one
mode of effecting a measurement.
When the standard quantity is greater than that to be measured, we
often adopt the first mode, and subdivide the unit until we get a
magnitude equal to that measured. The angles observed in surveying,
in astronomy, or in goniometry are usually smaller than a whole
revolution, and the measuring circle is divided by the use of the
screw and microscope, until we obtain an angle undistinguishable from
that observed. The dimensions of minute objects are determined by
subdividing the inch or centimetre, the screw micrometer being the most
accurate means of subdivision. Ordinary temperatures are estimated by
division of the standard interval between the freezing and boiling
points of water, as marked on a thermometer tube.
In a still greater number of cases, perhaps, we multiply the standard
unit until we get a magnitude equal to that to be measured. Ordinary
measurement by a foot rule, a surveyor’s chain, or the excessively
careful measurements of the base line of a trigonometrical survey by
standard bars, are sufficient instances of this procedure.
In the second case, where *p(y/x) = q*, we multiply or divide a
magnitude until we get what is equal to the unit, or to some magnitude
easily comparable with it. As a general rule the quantities which we
desire to measure in physical science are too small rather than too
great for easy determination, and the problem consists in multiplying
them without introducing error. Thus the expansion of a metallic bar
when heated from 0°C to 100° may be multiplied by a train of levers or
cog wheels. In the common thermometer the expansion of the mercury,
though slight, is rendered very apparent, and easily measurable by the
fineness of the tube, and many other cases might be quoted. There are
some phenomena, on the contrary, which are too great or rapid to come
within the easy range of our senses, and our task is then the opposite
one of diminution. Galileo found it difficult to measure the velocity
of a falling body, owing to the considerable velocity acquired in a
single second. He adopted the elegant device, therefore, of lessening
the rapidity by letting the body roll down an inclined plane, which
enables us to reduce the accelerating force in any required ratio.
The same purpose is effected in the well-known experiments performed
on Attwood’s machine, and the measurement of gravity by the pendulum
really depends on the same principle applied in a far more advantageous
manner. Wheatstone invented a beautiful method of galvanometry for
strong currents, which consists in drawing off from the main current a
certain determinate portion, which is equated by the galvanometer to a
standard current. In short, he measures not the current itself but a
known fraction of it.
In many electrical and other experiments, we wish to measure the
movements of a needle or other body, which are not only very slight
in themselves, but the manifestations of exceedingly small forces. We
cannot even approach a delicately balanced needle without disturbing
it. Under these circumstances the only mode of proceeding with
accuracy, is to attach a very small mirror to the moving body, and
employ a ray of light reflected from the mirror as an index of its
movements. The ray may be considered quite incapable of affecting the
body, and yet by allowing the ray to pass to a sufficient distance, the
motions of the mirror may be increased to almost any extent. A ray of
light is in fact a perfectly weightless finger or index of indefinite
length, with the additional advantage that the angular deviation is
by the law of reflection double that of the mirror. This method was
introduced by Gauss, and is now of great importance; but in Wollaston’s
reflecting goniometer a ray of light had previously been employed as an
index. Lavoisier and Laplace had also used a telescope in connection
with the pyrometer.
It is a great advantage in some instruments that they can be readily
made to manifest a phenomenon in a greater or less degree, by a very
slight change in the construction. Thus either by enlarging the bulb
or contracting the tube of the thermometer, we can make it give
more conspicuous indications of change of temperature. The ordinary
barometer, on the other hand, always gives the variations of pressure
on one scale. The torsion balance is remarkable for the extreme
delicacy which may be attained by increasing the length and lightness
of the rod, and the length and thinness of the supporting thread.
Forces so minute as the attraction of gravitation between two balls, or
the magnetic and diamagnetic attraction of common liquids and gases,
may thus be made apparent, and even measured. The common chemical
balance, too, is capable theoretically of unlimited sensibility.
The third mode of measurement, which may be called the Method of
Repetition, is of such great importance and interest that we must
consider it in a separate section. It consists in multiplying both
magnitudes to be compared until some multiple of the first is found
to coincide very nearly with some multiple of the second. If the
multiplication can be effected to an unlimited extent, without the
introduction of countervailing errors, the accuracy with which the
required ratio can be determined is unlimited, and we thus account for
the extraordinary precision with which intervals of time in astronomy
are compared together.
The fourth mode of measurement, in which we equate submultiples of
two magnitudes, is comparatively seldom employed, because it does not
conduce to accuracy. In the photometer, perhaps, we may be said to use
it; we compare the intensity of two sources of light, by placing them
both at such distances from a given surface, that the light falling
on the surface is tolerable to the eye, and equally intense from each
source. Since the intensity of light varies inversely as the square
of the distance, the relative intensities of the luminous bodies are
proportional to the squares of their distances. The equal intensity of
two rays of similarly coloured light may be most accurately ascertained
in the mode suggested by Arago, namely, by causing the rays to pass in
opposite directions through two nearly flat lenses pressed together.
There is an exact equation between the intensities of the beams when
Newton’s rings disappear, the ring created by one ray being exactly the
complement of that created by the other.
*The Method of Repetition.*
The ratio of two quantities can be determined with unlimited accuracy,
if we can multiply both the object of measurement and the standard unit
without error, and then observe what multiple of the one coincides or
nearly coincides with some multiple of the other. Although perfect
coincidence can never be really attained, the error thus arising
may be indefinitely reduced. For if the equation *py* = *qx* be
uncertain to the amount *e*, so that *py* = *qx* ± *e*, then we have
*p* = *q(x/y)* ± *e/y* , and as we are supposed to be able to make *y*
as great as we like without increasing the error *e*, it follows that
we can make *e* ÷ *y* as small as we like, and thus approximate within
an inconsiderable quantity to the required ratio *x* ÷ *y*.
This method of repetition is naturally employed whenever quantities
can be repeated, or repeat themselves without error of juxtaposition,
which is especially the case with the motions of the earth and heavenly
bodies. In determining the length of the sidereal day, we determine the
ratio between the earth’s revolution round the sun, and its rotation on
its own axis. We might ascertain the ratio by observing the successive
passages of a star across the zenith, and comparing the interval by a
good clock with that between two passages of the sun, the difference
being due to the angular movement of the earth round the sun. In such
observations we should have an error of a considerable part of a second
at each observation, in addition to the irregularities of the clock.
But the revolutions of the earth repeat themselves day after day, and
year after year, without the slightest interval between the end of one
period and the beginning of another. The operation of multiplication
is perfectly performed for us by nature. If, then, we can find an
observation of the passage of a star across the meridian a hundred
years ago, that is of the interval of time between the passage of the
sun and the star, the instrumental errors in measuring this interval by
a clock and telescope may be greater than in the present day, but will
be divided by about 36,524 days, and rendered excessively small. It is
thus that astronomers have been able to ascertain the ratio of the mean
solar to the sidereal day to the 8th place of decimals (1·00273791 to
1), or to the hundred millionth part, probably the most accurate result
of measurement in the whole range of science.
The antiquity of this mode of comparison is almost as great as that of
astronomy itself. Hipparchus made the first clear application of it,
when he compared his own observations with those of Aristarchus, made
145 years previously, and thus ascertained the length of the year.
This calculation may in fact be regarded as the earliest attempt at
an exact determination of the constants of nature. The method is the
main resource of astronomers; Tycho, for instance, detected the slow
diminution of the obliquity of the earth’s axis, by the comparison of
observations at long intervals. Living astronomers use the method as
much as earlier ones; but so superior in accuracy are all observations
taken during the last hundred years to all previous ones, that it is
often found preferable to take a shorter interval, rather than incur
the risk of greater instrumental errors in the earlier observations.
It is obvious that many of the slower changes of the heavenly bodies
must require the lapse of large intervals of time to render their
amount perceptible. Hipparchus could not possibly have discovered the
smaller inequalities of the heavenly motions, because there were no
previous observations of sufficient age or exactness to exhibit them.
And just as the observations of Hipparchus formed the starting-point
for subsequent comparisons, so a large part of the labour of present
astronomers is directed to recording the present state of the heavens
so exactly, that future generations of astronomers may detect changes,
which cannot possibly become known in the present age.
The principle of repetition was very ingeniously employed in an
instrument first proposed by Mayer in 1767, and carried into practice
in the Repeating Circle of Borda. The exact measurement of angles
is indispensable, not only in astronomy but also in trigonometrical
surveys, and the highest skill in the mechanical execution of the
graduated circle and telescope will not prevent terminal errors of
considerable amount. If instead of one telescope, the circle be
provided with two similar telescopes, these may be alternately directed
to two distant points, say the marks in a trigonometrical survey, so
that the circle shall be turned through any multiple of the angle
subtended by those marks, before the amount of the angular revolution
is read off upon the graduated circle. Theoretically speaking, all
error arising from imperfect graduation might thus be indefinitely
reduced, being divided by the number of repetitions. In practice, the
advantage of the invention is not found to be very great, probably
because a certain error is introduced at each observation in the
changing and fixing of the telescopes. It is moreover inapplicable to
moving objects like the heavenly bodies, so that its use is confined to
important trigonometrical surveys.
The pendulum is the most perfect of all instruments, chiefly because
it admits of almost endless repetition. Since the force of gravity
never ceases, one swing of the pendulum is no sooner ended than the
other is begun, so that the juxtaposition of successive units is
absolutely perfect. Provided that the oscillations be equal, one
thousand oscillations will occupy exactly one thousand times as great
an interval of time as one oscillation. Not only is the subdivision of
time entirely dependent on this fact, but in the accurate measurement
of gravity, and many other important determinations, it is of the
greatest service. In the deepest mine, we could not observe the
rapidity of fall of a body for more than a quarter of a minute, and
the measurement of its velocity would be difficult, and subject to
uncertain errors from resistance of air, &c. In the pendulum, we have a
body which can be kept rising and falling for many hours, in a medium
entirely under our command or if desirable in a vacuum. Moreover, the
comparative force of gravity at different points, at the top and bottom
of a mine for instance, can be determined with wonderful precision, by
comparing the oscillations of two exactly similar pendulums, with the
aid of electric clock signals.
To ascertain the comparative times of vibration of two pendulums, it
is only requisite to swing them one in front of the other, to record
by a clock the moment when they coincide in swing, so that one hides
the other, and then count the number of vibrations until they again
come to coincidence. If one pendulum makes *m* vibrations and the other
*n*, we at once have our equation *pn* = *qm*; which gives the length
of vibration of either pendulum in terms of the other. This method of
coincidence, embodying the principle of repetition in perfection, was
employed with wonderful skill by Sir George Airy, in his experiments on
the Density of the Earth at the Harton Colliery, the pendulums above
and below being compared with clocks, which again were compared with
each other by electric signals. So exceedingly accurate was this method
of observation, as carried out by Sir George Airy, that he was able to
measure a total difference in the vibrations at the top and bottom of
the shaft, amounting to only 2·24 seconds in the twenty-four hours,
with an error of less than one hundredth part of a second, or one part
in 8,640,000 of the whole day.[185]
[185] *Philosophical Transactions*, (1856) vol. 146, Part i. p. 297.
The principle of repetition has been elegantly applied in observing
the motion of waves in water. If the canal in which the experiments are
made be short, say twenty feet long, the waves will pass through it
so rapidly that an observation of one length, as practised by Walker,
will be subject to much terminal error, even when the observer is very
skilful. But it is a result of the undulatory theory that a wave is
unaltered, and loses no time by complete reflection, so that it may be
allowed to travel backwards and forwards in the same canal, and its
motion, say through sixty lengths, or 1200 feet, may be observed with
the same accuracy as in a canal 1200 feet long, with the advantage of
greater uniformity in the condition of the canal and water.[186] It
is always desirable, if possible, to bring an experiment into a small
compass, so that it may be well under command, and yet we may often by
repetition enjoy at the same time the advantage of extensive trial.
[186] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 345.
Scott Russell, *British Association Report*, 1837, p. 432.
One reason of the great accuracy of weighing with a good balance is
the fact, that weights placed in the same scale are naturally added
together without the slightest error. There is no difficulty in the
precise juxtaposition of two grams, but the juxtaposition of two metre
measures can only be effected with tolerable accuracy, by the use of
microscopes and many precautions. Hence, the extreme trouble and cost
attaching to the exact measurement of a base line for a survey, the
risk of error entering at every juxtaposition of the measuring bars,
and indefatigable attention to all the requisite precautions being
necessary throughout the operation.
*Measurements by Natural Coincidence.*
In certain cases a peculiar conjunction of circumstances enables us to
dispense more or less with instrumental aids, and to obtain very exact
numerical results in the simplest manner. The mere fact, for instance,
that no human being has ever seen a different face of the moon from
that familiar to us, conclusively proves that the period of rotation
of the moon on its own axis is equal to that of its revolution round
the earth. Not only have we the repetition of these movements during
1000 or 2000 years at least, but we have observations made for us
at very remote periods, free from instrumental error, no instrument
being needed. We learn that the seventh satellite of Saturn is subject
to a similar law, because its light undergoes a variation in each
revolution, owing to the existence of some dark tract of land; now
this failure of light always occurs while it is in the same position
relative to Saturn, clearly proving the equality of the axial and
revolutional periods, as Huygens perceived.[187] A like peculiarity in
the motions of Jupiter’s fourth satellite was similarly detected by
Maraldi in 1713.
[187] *Hugenii Cosmotheoros*, pp. 117, 118. Laplace’s *Système*,
translated, vol. i. p. 67.
Remarkable conjunctions of the planets may sometimes allow us to
compare their periods of revolution, through great intervals of time,
with much accuracy. Laplace in explaining the long inequality in the
motions of Jupiter and Saturn, was assisted by a conjunction of these
planets, observed at Cairo, towards the close of the eleventh century.
Laplace calculated that such a conjunction must have happened on the
31st of October, A.D. 1087; and the discordance between the distances
of the planets as recorded, and as assigned by theory, was less than
one-fifth part of the apparent diameter of the sun. This difference
being less than the probable error of the early record, the theory was
confirmed as far as facts were available.[188]
[188] Grant’s *History of Physical Astronomy*, p. 129.
Ancient astronomers often showed the highest ingenuity in turning
any opportunities of measurement which occurred to good account.
Eratosthenes, as early as 250 B.C., happening to hear that the sun at
Syene, in Upper Egypt, was visible at the summer solstice at the bottom
of a well, proving that it was in the zenith, proposed to determine
the dimensions of the earth, by measuring the length of the shadow of
a rod at Alexandria on the same day of the year. He thus learnt in a
rude manner the difference of latitude between Alexandria and Syene and
finding it to be about one fiftieth part of the whole circumference, he
ascertained the dimensions of the earth within about one sixth part
of the truth. The use of wells in astronomical observation appears to
have been occasionally practised in comparatively recent times as by
Flamsteed in 1679.[189] The Alexandrian astronomers employed the moon
as an instrument of measurement in several sagacious modes. When the
moon is exactly half full, the moon, sun, and earth, are at the angles
of a right-angled triangle. Aristarchus measured at such a time the
moon’s elongation from the sun, which gave him the two other angles of
the triangle, and enabled him to judge of the comparative distances
of the moon and sun from the earth. His result, though very rude, was
far more accurate than any notions previously entertained, and enabled
him to form some estimate of the comparative magnitudes of the bodies.
Eclipses of the moon were very useful to Hipparchus in ascertaining
the longitude of the stars, which are invisible when the sun is above
the horizon. For the moon when eclipsed must be 180° distant from the
sun; hence it is only requisite to measure the distance of a fixed star
in longitude from the eclipsed moon to obtain with ease its angular
distance from the sun.
[189] Baily’s *Account of Flamsteed*, p. lix.
In later times the eclipses of Jupiter have served to measure an angle;
for at the middle moment of the eclipse the satellite must be in the
same straight line with the planet and sun, so that we can learn from
the known laws of movement of the satellite the longitude of Jupiter
as seen from the sun. If at the same time we measure the elongation or
apparent angular distance of Jupiter from the sun, as seen from the
earth, we have all the angles of the triangle between Jupiter, the sun,
and the earth, and can calculate the comparative magnitudes of the
sides of the triangle by trigonometry.
The transits of Venus over the sun’s face are other natural events
which give most accurate measurements of the sun’s parallax, or
apparent difference of position as seen from distant points of the
earth’s surface. The sun forms a kind of background on which the place
of the planet is marked, and serves as a measuring instrument free
from all the errors of construction which affect human instruments.
The rotation of the earth, too, by variously affecting the apparent
velocity of ingress or egress of Venus, as seen from different places,
discloses the amount of the parallax. It has been sufficiently shown
that by rightly choosing the moments of observation, the planetary
bodies may often be made to reveal their relative distance, to measure
their own position, to record their own movements with a high degree
of accuracy. With the improvement of astronomical instruments, such
conjunctions become less necessary to the progress of the science,
but it will always remain advantageous to choose those moments for
observation when instrumental errors enter with the least effect.
In other sciences, exact quantitative laws can occasionally be obtained
without instrumental measurement, as when we learn the exactly equal
velocity of sounds of different pitch, by observing that a peal of
bells or a musical performance is heard harmoniously at any distance
to which the sound penetrates; this could not be the case, as Newton
remarked, if one sound overtook the other. One of the most important
principles of the atomic theory, was proved by implication before the
use of the balance was introduced into chemistry. Wenzel observed,
before 1777, that when two neutral substances decompose each other,
the resulting salts are also neutral. In mixing sodium sulphate and
barium nitrate, we obtain insoluble barium sulphate and neutral sodium
nitrate. This result could not follow unless the nitric acid, requisite
to saturate one atom of sodium, were exactly equal to that required
by one atom of barium, so that an exchange could take place without
leaving either acid or base in excess.
An important principle of mechanics may also be established by a simple
acoustical observation. When a rod or tongue of metal fixed at one
end is set in vibration, the pitch of the sound may be observed to
be exactly the same, whether the vibrations be small or great; hence
the oscillations are isochronous, or equally rapid, independently of
their magnitude. On the ground of theory, it can be shown that such a
result only happens when the flexure is proportional to the deflecting
force. Thus the simple observation that the pitch of the sound of a
harmonium, for instance, does not change with its loudness establishes
an exact law of nature.[190]
[190] Jamin, *Cours de Physique*, vol. i. p. 152.
A closely similar instance is found in the proof that the intensity
of light or heat rays varies inversely as the square of the distance
increases. For the apparent magnitude certainly varies according to
this law; hence, if the intensity of light varied according to any
other law, the brightness of an object would be different at different
distances, which is not observed to be the case. Melloni applied the
same kind of reasoning, in a somewhat different form, to the radiation
of heat-rays.
*Modes of Indirect Measurement.*
Some of the most conspicuously beautiful experiments in the whole range
of science, have been devised for the purpose of indirectly measuring
quantities, which in their extreme greatness or smallness surpass the
powers of sense. All that we need to do, is to discover some other
conveniently measurable phenomenon, which is related in a known ratio
or according to a known law, however complicated, with that to be
measured. Having once obtained experimental data, there is no further
difficulty beyond that of arithmetic or algebraic calculation.
Gold is reduced by the gold-beater to leaves so thin, that the most
powerful microscope would not detect any measurable thickness. If we
laid several hundred leaves upon each other to multiply the thickness,
we should still have no more than 1/100th of an inch at the most to
measure, and the errors arising in the superposition and measurement
would be considerable. But we can readily obtain an exact result
through the connected amount of weight. Faraday weighed 2000 leaves of
gold, each 3-3/8 inch square, and found them equal to 384 grains. From
the known specific gravity of gold it was easy to calculate that the
average thickness of the leaves was 1/282,000 of an inch.[191]
[191] Faraday, *Chemical Researches*, p. 393.
We must ascribe to Newton the honour of leading the way in methods of
minute measurement. He did not call waves of light by their right name,
and did not understand their nature; yet he measured their length,
though it did not exceed the 2,000,000th part of a metre or the one
fifty-thousandth part of an inch. He pressed together two lenses of
large but known radii. It was easy to calculate the interval between
the lenses at any point, by measuring the distance from the central
point of contact. Now, with homogeneous rays the successive rings of
light and darkness mark the points at which the interval between the
lenses is equal to one half, or any multiple of half a vibration of
the light, so that the length of the vibration became known. In a
similar manner many phenomena of interference of rays of light admit
of the measurement of the wave lengths. Fringes of interference arise
from rays of light which cross each other at a small angle, and an
excessively minute difference in the lengths of the waves makes a very
perceptible difference in the position of the point at which two rays
will interfere and produce darkness.
Fizeau has recently employed Newton’s rings to measure small amounts of
motion. By merely counting the number of rings of sodium monochromatic
light passing a certain point where two glass plates are in close
proximity, he is able to ascertain with the greatest accuracy and ease
the change of distance between these glasses, produced, for instance,
by the expansion of a metallic bar, connected with one of the glass
plates.[192]
[192] *Proceedings of the Royal Society*, 30th November, 1866.
Nothing excites more admiration than the mode in which scientific
observers can occasionally measure quantities, which seem beyond
the bounds of human observation. We know the *average* depth of the
Pacific Ocean to be 14,190 feet, not by actual sounding, which would
be impracticable in sufficient detail, but by noticing the rate of
transmission of earthquake waves from the South American to the
opposite coasts, the rate of movement being connected by theory with
the depth of the water.[193] In the same way the average depth of
the Atlantic Ocean is inferred to be no less than 22,157 feet, from
the velocity of the ordinary tidal waves. A tidal wave again gives
beautiful evidence of an effect of the law of gravity, which we could
never in any other way detect. Newton estimated that the moon’s force
in moving the ocean is only one part in 2,871,400 of the whole force of
gravity, so that even the pendulum, used with the utmost skill, would
fail to render it apparent. Yet, the immense extent of the ocean allows
the accumulation of the effect into a very palpable amount; and from
the comparative heights of the lunar and solar tides, Newton roughly
estimated the comparative forces of the moon’s and sun’s gravity at the
earth.[194]
[193] Herschel, *Physical Geography*, § 40.
[194] *Principia*, bk. iii. Prop. 37, *Corollaries*, 2 and 3. Motte’s
translation, vol. ii. p. 310.
A few years ago it might have seemed impossible that we should ever
measure the velocity with which a star approaches or recedes from the
earth, since the apparent position of the star is thereby unaltered.
But the spectroscope now enables us to detect and even measure such
motions with considerable accuracy, by the alteration which it
causes in the apparent rapidity of vibration, and consequently in
the refrangibility of rays of light of definite colour. And while
our estimates of the lateral movements of stars depend upon our very
uncertain knowledge of their distances, the spectroscope gives the
motions of approach and recess irrespective of other motions excepting
that of the earth. It gives in short the motions of approach and recess
of the stars relatively to the earth.[195]
[195] Roscoe’s *Spectrum Analysis*, 1st ed. p. 296.
The rapidity of vibration for each musical tone, having been accurately
determined by comparison with the Syren (p. 10), we can use sounds as
indirect indications of rapid vibrations. It is now known that the
contraction of a muscle arises from the periodical contractions of each
separate fibre, and from a faint sound or susurrus which accompanies
the action of a muscle, it is inferred that each contraction lasts for
about one 300th part of a second. Minute quantities of radiant heat are
now always measured indirectly by the electricity which they produce
when falling upon a thermopile. The extreme delicacy of the method
seems to be due to the power of multiplication at several points in the
apparatus. The number of elements or junctions of different metals in
the thermopile can be increased so that the tension of the electric
current derived from the same intensity of radiation is multiplied;
the effect of the current upon the magnetic needle can be multiplied
within certain bounds, by passing the current many times round it in
a coil; the excursions of the needle can be increased by rendering it
astatic and increasing the delicacy of its suspension; lastly, the
angular divergence can be observed, with any required accuracy, by the
use of an attached mirror and distant scale viewed through a telescope
(p. 287). Such is the delicacy of this method of measuring heat, that
Dr. Joule succeeded in making a thermopile which would indicate a
difference of 0°·000114 Cent.[196]
[196] *Philosophical Transactions* (1859), vol. cxlix. p. 94.
A striking case of indirect measurement is furnished by the revolving
mirror of Wheatstone and Foucault, whereby a minute interval of time
is estimated in the form of an angular deviation. Wheatstone viewed an
electric spark in a mirror rotating so rapidly, that if the duration
of the spark had been more than one 72,000th part of a second, the
point of light would have appeared elongated to an angular extent
of one-half degree. In the spark, as drawn directly from a Leyden
jar, no elongation was apparent, so that the duration of the spark
was immeasurably small; but when the discharge took place through
a bad conductor, the elongation of the spark denoted a sensible
duration.[197] In the hands of Foucault the rotating mirror gave a
measure of the time occupied by light in passing through a few metres
of space.
[197] Watts’ *Dictionary of Chemistry*, vol. ii. p. 393.
*Comparative Use of Measuring Instruments.*
In almost every case a measuring instrument serves, and should serve
only as a means of comparison between two or more magnitudes. As a
general rule, we should not attempt to make the divisions of the
measuring scale exact multiples or submultiples of the unit, but,
regarding them as arbitrary marks, should determine their values by
comparison with the standard itself. The perpendicular wires in the
field of a transit telescope, are fixed at nearly equal but arbitrary
distances, and those distances are afterwards determined, as first
suggested by Malvasia, by watching the passage of star after star
across them, and noting the intervals of time by the clock. Owing
to the perfectly regular motion of the earth, these time intervals
give exact determinations of the angular intervals. In the same way,
the angular value of each turn of the screw micrometer attached to a
telescope, can be easily and accurately ascertained.
When a thermopile is used to observe radiant heat, it would be almost
impossible to calculate on *à priori* grounds what is the value of
each division of the galvanometer circle, and still more difficult
to construct a galvanometer, so that each division should have a
given value. But this is quite unnecessary, because by placing the
thermopile before a body of known dimensions, at a known distance, with
a known temperature and radiating power, we measure a known amount
of radiant heat, and inversely measure the value of the indications
of the thermopile. In a similar way Dr. Joule ascertained the actual
temperature produced by the compression of bars of metal. For having
inserted a small thermopile composed of a single junction of copper and
iron wire, and noted the deflections of the galvanometer, he had only
to dip the bars into water of different temperatures, until he produced
a like deflection, in order to ascertain the temperature developed by
pressure.[198]
[198] *Philosophical Transactions* (1859), vol. cxlix. p. 119, &c.
In some cases we are obliged to accept a very carefully constructed
instrument as a standard, as in the case of a standard barometer or
thermometer. But it is then best to treat all inferior instruments
comparatively only, and determine the values of their scales by
comparison with the assumed standard.
*Systematic Performance of Measurements.*
When a large number of accurate measurements have to be effected, it
is usually desirable to make a certain number of determinations with
scrupulous care, and afterwards use them as points of reference for the
remaining determinations. In the trigonometrical survey of a country,
the principal triangulation fixes the relative positions and distances
of a few points with rigid accuracy. A minor triangulation refers every
prominent hill or village to one of the principal points, and then the
details are filled in by reference to the secondary points. The survey
of the heavens is effected in a like manner. The ancient astronomers
compared the right ascensions of a few principal stars with the moon,
and thus ascertained their positions with regard to the sun; the minor
stars were afterwards referred to the principal stars. Tycho followed
the same method, except that he used the more slowly moving planet
Venus instead of the moon. Flamsteed was in the habit of using about
seven stars, favourably situated at points all round the heavens. In
his early observations the distances of the other stars from these
standard points were determined by the use of the quadrant.[199] Even
since the introduction of the transit telescope and the mural circle,
tables of standard stars are formed at Greenwich, the positions being
determined with all possible accuracy, so that they can be employed for
purposes of reference by astronomers.
[199] Baily’s *Account of Flamsteed*, pp. 378–380.
In ascertaining the specific gravities of substances, all gases are
referred to atmospheric air at a given temperature and pressure;
all liquids and solids are referred to water. We require to compare
the densities of water and air with great care, and the comparative
densities of any two substances whatever can then be ascertained.
In comparing a very great with a very small magnitude, it is
usually desirable to break up the process into several steps, using
intermediate terms of comparison. We should never think of measuring
the distance from London to Edinburgh by laying down measuring rods,
throughout the whole length. A base of several miles is selected on
level ground, and compared on the one hand with the standard yard,
and on the other with the distance of London and Edinburgh, or any
other two points, by trigonometrical survey. Again, it would be
exceedingly difficult to compare the light of a star with that of the
sun, which would be about thirty thousand million times greater; but
Herschel[200] effected the comparison by using the full moon as an
intermediate unit. Wollaston ascertained that the sun gave 801,072
times as much light as the full moon, and Herschel determined that the
light of the latter exceeded that of α Centauri 27,408 times, so that
we find the ratio between the light of the sun and star to be that of
about 22,000,000,000 to 1.
[200] Herschel’s *Astronomy*, § 817, 4th. ed. p. 553.
*The Pendulum.*
By far the most perfect and beautiful of all instruments of measurement
is the pendulum. Consisting merely of a heavy body suspended freely
at an invariable distance from a fixed point, it is most simple in
construction; yet all the highest problems of physical measurement
depend upon its careful use. Its excessive value arises from two
circumstances.
(1) The method of repetition is eminently applicable to it, as already
described (p. 290).
(2) Unlike other instruments, it connects together three different
quantities, those of space, time, and force.
In most works on natural philosophy it is shown, that when the
oscillations of the pendulum are infinitely small, the square of the
time occupied by an oscillation is directly proportional to the length
of the pendulum, and indirectly proportional to the force affecting it,
of whatever kind. The whole theory of the pendulum is contained in the
formula, first given by Huygens in his *Horologium Oscillatorium*.
Time of oscillation = 3·14159 × √(length of pendulum/force).
The quantity 3·14159 is the constant ratio of the circumference and
radius of a circle, and is of course known with accuracy. Hence, any
two of the three quantities concerned being given, the third may be
found; or any two being maintained invariable, the third will be
invariable. Thus a pendulum of invariable length suspended at the
same place, where the force of gravity may be considered constant,
furnishes a measure of time. The same invariable pendulum being made
to vibrate at different points of the earth’s surface, and the times
of vibration being astronomically determined, the force of gravity
becomes accurately known. Finally, with a known force of gravity, and
time of vibration ascertained by reference to the stars, the length is
determinate.
All astronomical observations depend upon the first manner of using the
pendulum, namely, in the astronomical clock. In the second employment
it has been almost equally indispensable. The primary principle that
gravity is equal in all matter was proved by Newton’s and Gauss’
pendulum experiments. The torsion pendulum of Michell, Cavendish, and
Baily, depending upon exactly the same principles as the ordinary
pendulum, gave the density of the earth, one of the foremost natural
constants. Kater and Sabine, by pendulum observations in different
parts of the earth, ascertained the variation of gravity, whence comes
a determination of the earth’s ellipticity. The laws of electric
and magnetic attraction have also been determined by the method
of vibrations, which is in constant use in the measurement of the
horizontal force of terrestrial magnetism.
We must not confuse with the ordinary use of the pendulum its
application by Newton, to show the absence of internal friction against
space,[201] or to ascertain the laws of motion and elasticity.[202] In
these cases the extent of vibration is the quantity measured, and the
principles of the instrument are different.
[201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte’s Translation,
vol. ii. p. 107.
[202] Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i.
p. 33.
*Attainable Accuracy of Measurement.*
It is a matter of some interest to compare the degrees of accuracy
which can be attained in the measurement of different kinds of
magnitude. Few measurements of any kind are exact to more than six
significant figures,[203] but it is seldom that such accuracy can be
hoped for. Time is the magnitude which seems to be capable of the most
exact estimation, owing to the properties of the pendulum, and the
principle of repetition described in previous sections. As regards
short intervals of time, it has already been stated that Sir George
Airy was able to estimate one part in 8,640,000, an exactness, as he
truly remarks, “almost beyond conception.”[204] The ratio between the
mean solar and the sidereal day is known to be about one part in one
hundred millions, or to the eighth place of decimals, (p. 289).
[203] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 333.
[204] *Philosophical Transactions*, (1856), vol. cxlvi. pp. 330, 331.
Determinations of weight seem to come next in exactness, owing to the
fact that repetition without error is applicable to them. An ordinary
good balance should show about one part in 500,000 of the load. The
finest balance employed by M. Stas, turned with one part in 825,000 of
the load.[205] But balances have certainly been constructed to show
one part in a million,[206] and Ramsden is said to have constructed a
balance for the Royal Society, to indicate one part in seven millions,
though this is hardly credible. Professor Clerk Maxwell takes it for
granted that one part in five millions can be detected, but we ought to
discriminate between what a balance can do when first constructed, and
when in continuous use.
[205] *First Annual Report of the Mint*, p. 106.
[206] Jevons, in Watts’ *Dictionary of Chemistry*, vol. i. p. 483.
Determinations of length, unless performed with extraordinary care,
are open to much error in the junction of the measuring bars. Even
in measuring the base line of a trigonometrical survey, the accuracy
generally attained is only that of about one part in 60,000, or an
inch in the mile; but it is said that in four measurements of a base
line carried out very recently at Cape Comorin, the greatest error was
0·077 inch in 1·68 mile, or one part in 1,382,400, an almost incredible
degree of accuracy. Sir J. Whitworth has shown that touch is even a
more delicate mode of measuring lengths than sight, and by means of
a splendidly executed screw, and a small cube of iron placed between
two flat-ended iron bars, so as to be suspended when touching them, he
can detect a change of dimension in a bar, amounting to no more than
one-millionth of an inch.[207]
[207] British Association, Glasgow, 1856. *Address of the President
of the Mechanical Section*.
CHAPTER XIV.
UNITS AND STANDARDS OF MEASUREMENT.
As we have seen, instruments of measurement are only means of
comparison between one magnitude and another, and as a general rule we
must assume some one arbitrary magnitude, in terms of which all results
of measurement are to be expressed. Mere ratios between any series of
objects will never tell us their absolute magnitudes; we must have at
least one ratio for each, and we must have one absolute magnitude.
The number of ratios *n* are expressible in *n* equations, which will
contain at least *n* + 1 quantities, so that if we employ them to make
known *n* magnitudes, we must have one magnitude known. Hence, whether
we are measuring time, space, density, mass, weight, energy, or any
other physical quantity, we must refer to some concrete standard, some
actual object, which if once lost and irrecoverable, all our measures
lose their absolute meaning. This concrete standard is in all cases
arbitrary in point of theory, and its selection a question of practical
convenience.
There are two kinds of magnitude, indeed, which do not need to be
expressed in terms of arbitrary concrete units, since they pre-suppose
the existence of natural standard units. One case is that of abstract
number itself, which needs no special unit, because any object which
exists or is thought of as separate from other objects (p. 157)
furnishes us with a unit, and is the only standard required.
Angular magnitude is the second case in which we have a natural unit
of reference, namely the whole revolution or *perigon*, as it has
been called by Mr. Sandeman.[208] It is a necessary result of the
uniform properties of space, that all complete revolutions are equal
to each other, so that we need not select any one revolution, but can
always refer anew to space itself. Whether we take the whole perigon,
its half, or its quarter, is really immaterial; Euclid took the right
angle, because the Greek geometers had never generalised their notions
of angular magnitude sufficiently to treat angles of all magnitudes,
or of unlimited *quantity of revolution*. Euclid defines a right angle
as half that made by a line with its own continuation, which is of
course equal to half a revolution, but which was not treated as an
angle by him. In mathematical analysis a different fraction of the
perigon is taken, namely, such a fraction that the arc or portion of
the circumference included within it is equal to the radius of the
circle. In this point of view angular magnitude is an abstract ratio,
namely, the ratio between the length of arc subtended and the length
of the radius. The geometrical unit is then necessarily the angle
corresponding to the ratio unity. This angle is equal to about 57°,
17′, 44″·8, or decimally 57°·295779513... .[209] It was called by De
Morgan the *arcual unit*, but a more convenient name for common use
would be *radian*, as suggested by Professor Everett. Though this
standard angle is naturally employed in mathematical analysis, and any
other unit would introduce great complexity, we must not look upon it
as a distinct unit, since its amount is connected with that of the half
perigon, by the natural constant 3·14159... usually denoted by the
letter π.
[208] *Pelicotetics, or the Science of Quantity; an Elementary
Treatise on Algebra, and its groundwork Arithmetic.* By Archibald
Sandeman, M. A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304.
[209] De Morgan’s *Trigonometry and Double Algebra*, p. 5.
When we pass to other species of quantity, the choice of unit is found
to be entirely arbitrary. There is absolutely no mode of defining a
length, but by selecting some physical object exhibiting that length
between certain obvious points--as, for instance, the extremities of a
bar, or marks made upon its surface.
*Standard Unit of Time.*
Time is the great independent variable of all change--that which itself
flows on uninterruptedly, and brings the variety which we call motion
and life. When we reflect upon its intimate nature, Time, like every
other element of existence, proves to be an inscrutable mystery. We
can only say with St. Augustin, to one who asks us what is time, “I
know when you do not ask me.” The mind of man will ask what can never
be answered, but one result of a true and rigorous logical philosophy
must be to convince us that scientific explanation can only take place
between phenomena which have something in common, and that when we get
down to primary notions, like those of time and space, the mind must
meet a point of mystery beyond which it cannot penetrate. A definition
of time must not be looked for; if we say with Hobbes,[210] that it is
“the phantasm of before and after in motion,” or with Aristotle that it
is “the number of motion according to former and latter,” we obviously
gain nothing, because the notion of time is involved in the expressions
*before and after*, *former and latter*. Time is undoubtedly one of
those primary notions which can only be defined physically, or by
observation of phenomena which proceed in time.
[210] *English Works of Thos. Hobbes*, Edit. by Molesworth, vol. i.
p. 95.
If we have not advanced a step beyond Augustin’s acute reflections on
this subject,[211] it is curious to observe the wonderful advances
which have been made in the practical measurement of its efflux. In
earlier centuries the rude sun-dial or the rising of a conspicuous star
gave points of reference, while the flow of water from the clepsydra,
the burning of a candle, or, in the monastic ages, even the continuous
chanting of psalms, were the means of roughly subdividing periods, and
marking the hours of the day and night.[212] The sun and stars still
furnish the standard of time, but means of accurate subdivision have
become requisite, and this has been furnished by the pendulum and the
chronograph. By the pendulum we can accurately divide the day into
seconds of time. By the chronograph we can subdivide the second into
a hundred, a thousand, or even a million parts. Wheatstone measured
the duration of an electric spark, and found it to be no more than one
115,200th part of a second, while more recently Captain Noble has been
able to appreciate intervals of time not exceeding the millionth part
of a second.
[211] *Confessions*, bk. xi. chapters 20–28.
[212] Sir G. C. Lewis gives many curious particulars concerning the
measurement of time in his *Astronomy of the Ancients*, pp. 241, &c.
When we come to inquire precisely what phenomenon it is that we thus
so minutely measure, we meet insurmountable difficulties. Newton
distinguished time according as it was *absolute* or *apparent* time,
in the following words:--“Absolute, true, and mathematical time,
of itself and from its own nature, flows equably without regard to
anything external, and by another name is called *duration*; relative,
apparent and common time, is some sensible and external measure of
duration by the means of motion.”[213] Though we are perhaps obliged to
assume the existence of a uniformly increasing quantity which we call
time, yet we cannot feel or know abstract and absolute time. Duration
must be made manifest to us by the recurrence of some phenomenon. The
succession of our own thoughts is no doubt the first and simplest
measure of time, but a very rude one, because in some persons and
circumstances the thoughts evidently flow with much greater rapidity
than in other persons and circumstances. In the absence of all
other phenomena, the interval between one thought and another would
necessarily become the unit of time, but the most cursory observations
show that there are changes in the outward world much better fitted by
their constancy to measure time than the change of thoughts within us.
[213] *Principia*, bk. i. *Scholium to Definitions*. Translated by
Motte, vol. i. p. 9. See also p. 11.
The earth, as I have already said, is the real clock of the astronomer,
and is practically assumed as invariable in its movements. But on
what ground is it so assumed? According to the first law of motion,
every body perseveres in its state of rest or of uniform motion in
a right line, unless it is compelled to change that state by forces
impressed thereon. Rotatory motion is subject to a like condition,
namely, that it perseveres uniformly unless disturbed by extrinsic
forces. Now uniform motion means motion through equal spaces in equal
times, so that if we have a body entirely free from all resistance
or perturbation, and can measure equal spaces of its path, we have a
perfect measure of time. But let it be remembered that this law has
never been absolutely proved by experience; for we cannot point to any
body, and say that it is wholly unresisted or undisturbed; and even if
we had such a body, we should need some independent standard of time
to ascertain whether its motion was really uniform. As it is in moving
bodies that we find the best standard of time, we cannot use them to
prove the uniformity of their own movements, which would amount to a
*petitio principii*. Our experience comes to this, that when we examine
and compare the movements of bodies which seem to us nearly free from
disturbance, we find them giving nearly harmonious measures of time.
If any one body which seems to us to move uniformly is not doing so,
but is subject to fits and starts unknown to us, because we have no
absolute standard of time, then all other bodies must be subject to the
same arbitrary fits and starts, otherwise there would be discrepancy
disclosing the irregularities. Just as in comparing together a number
of chronometers, we should soon detect bad ones by their going
irregularly, as compared with the others, so in nature we detect
disturbed movement by its discrepancy from that of other bodies which
we believe to be undisturbed, and which agree nearly among themselves.
But inasmuch as the measure of motion involves time, and the measure
of time involves motion, there must be ultimately an assumption. We
may define equal times, as times during which a moving body under the
influence of no force describes equal spaces;[214] but all we can
say in support of this definition is, that it leads us into no known
difficulties, and that to the best of our experience one freely moving
body gives the same results as any other.
[214] Rankine, *Philosophical Magazine*, Feb. 1867, vol. xxxiii.
p. 91.
When we inquire where the freely moving body is, no perfectly
satisfactory answer can be given. Practically the rotating globe is
sufficiently accurate, and Thomson and Tait say: “Equal times are
times during which the earth turns through equal angles.”[215] No long
time has passed since astronomers thought it impossible to detect any
inequality in its movement. Poisson was supposed to have proved that a
change in the length of the sidereal day amounting to one ten-millionth
part in 2,500 years was incompatible with an ancient eclipse recorded
by the Chaldæans, and similar calculations were made by Laplace. But
it is now known that these calculations were somewhat in error, and
that the dissipation of energy arising out of the friction of tidal
waves, and the radiation of the heat into space, has slightly decreased
the rapidity of the earth’s rotatory motion. The sidereal day is now
longer by one part in 2,700,000, than it was in 720 B.C. Even before
this discovery, it was known that invariability of rotation depended
upon the perfect maintenance of the earth’s internal heat, which is
requisite in order that the earth’s dimensions shall be unaltered. Now
the earth being superior in temperature to empty space, must cool more
or less rapidly, so that it cannot furnish an absolute measure of time.
Similar objections could be raised to all other rotating bodies within
our cognisance.
[215] *Treatise on Natural Philosophy*, vol. i. p. 179.
The moon’s motion round the earth, and the earth’s motion round the
sun, form the next best measure of time. They are subject, indeed,
to disturbance from other planets, but it is believed that these
perturbations must in the course of time run through their rhythmical
courses, leaving the mean distances unaffected, and consequently, by
the third Law of Kepler, the periodic times unchanged. But there is
more reason than not to believe that the earth encounters a slight
resistance in passing through space, like that which is so apparent
in Encke’s comet. There may also be dissipation of energy in the
electrical relations of the earth to the sun, possibly identical with
that which is manifested in the retardation of comets.[216] It is
probably an untrue assumption then, that the earth’s orbit remains
quite invariable. It is just possible that some other body may be found
in the course of time to furnish a better standard of time than the
earth in its annual motion. The greatly superior mass of Jupiter and
its satellites, and their greater distance from the sun, may render the
electrical dissipation of energy less considerable than in the case of
the earth. But the choice of the best measure will always be an open
one, and whatever moving body we choose may ultimately be shown to be
subject to disturbing forces.
[216] *Proceedings of the Manchester Philosophical Society*, 28th
Nov. 1871, vol. xi. p. 33.
The pendulum, although so admirable an instrument for subdivision of
time, fails as a standard; for though the same pendulum affected by the
same force of gravity performs equal vibrations in equal times, yet
the slightest change in the form or weight of the pendulum, the least
corrosion of any part, or the most minute displacement of the point of
suspension, falsifies the results, and there enter many other difficult
questions of temperature, friction, resistance, length of vibration, &c.
Thomson and Tait are of opinion[217] that the ultimate standard of
chronometry must be founded on the physical properties of some body
of more constant character than the earth; for instance, a carefully
arranged metallic spring, hermetically sealed in an exhausted glass
vessel. But it is hard to see how we can be sure that the dimensions
and elasticity of a piece of wrought metal will remain perfectly
unchanged for the few millions of years contemplated by them. A nearly
perfect gas, like hydrogen, is perhaps the only kind of substance in
the unchanged elasticity of which we could have confidence. Moreover,
it is difficult to perceive how the undulations of such a spring could
be observed with the requisite accuracy. More recently Professor Clerk
Maxwell has made the novel suggestion, discussed in a subsequent
section, that undulations of light *in vacuo* would form the most
universal standard of reference, both as regards time and space.
According to this system the unit of time would be the time occupied
by one vibration of the particular kind of light whose wave length is
taken as the unit of length.
[217] *The Elements of Natural Philosophy*, part i. p. 119.
*The Unit of Space and the Bar Standard.*
Next in importance after the measurement of time is that of space.
Time comes first in theory, because phenomena, our internal thoughts
for instance, may change in time without regard to space. As to the
phenomena of outward nature, they tend more and more to resolve
themselves into motions of molecules, and motion cannot be conceived or
measured without reference both to time and space.
Turning now to space measurement, we find it almost equally difficult
to fix and define once and for ever, a unit magnitude. There are
three different modes in which it has been proposed to attempt the
perpetuation of a standard length.
(1) By constructing an actual specimen of the standard yard or metre,
in the form of a bar.
(2) By assuming the globe itself to be the ultimate standard of
magnitude, the practical unit being a submultiple of some dimension of
the globe.
(3) By adopting the length of the simple seconds pendulum, as a
standard of reference.
At first sight it might seem that there was no great difficulty in this
matter, and that any one of these methods might serve well enough;
but the more minutely we inquire into the details, the more hopeless
appears to be the attempt to establish an invariable standard. We must
in the first place point out a principle not of an obvious character,
namely, that *the standard length must be defined by one single
object*.[218] To make two bars of exactly the same length, or even two
bars bearing a perfectly defined ratio to each other, is beyond the
power of human art. If two copies of the standard metre be made and
declared equally correct, future investigators will certainly discover
some discrepancy between them, proving of course that they cannot both
be the standard, and giving cause for dispute as to what magnitude
should then be taken as correct.
[218] See Harris’ *Essay upon Money and Coins*, part. ii. [1758]
p. 127.
If one invariable bar could be constructed and maintained as the
absolute standard, no such inconvenience could arise. Each successive
generation as it acquired higher powers of measurement, would detect
errors in the copies of the standard, but the standard itself would be
unimpeached, and would, as it were, become by degrees more and more
accurately known. Unfortunately to construct and preserve a metre or
yard is also a task which is either impossible, or what comes nearly
to the same thing, cannot be shown to be possible. Passing over the
practical difficulty of defining the ends of the standard length
with complete accuracy, whether by dots or lines on the surface, or
by the terminal points of the bar, we have no means of proving that
substances remain of invariable dimensions. Just as we cannot tell
whether the rotation of the earth is uniform, except by comparing it
with other moving bodies, believed to be more uniform in motion, so
we cannot detect the change of length in a bar, except by comparing
it with some other bar supposed to be invariable. But how are we to
know which is the invariable bar? It is certain that many rigid and
apparently invariable substances do change in dimensions. The bulb of
a thermometer certainly contracts by age, besides undergoing rapid
changes of dimensions when warmed or cooled through 100° Cent. Can
we be sure that even the most solid metallic bars do not slightly
contract by age, or undergo variations in their structure by change
of temperature. Fizeau was induced to try whether a quartz crystal,
subjected to several hundred alternations of temperature, would be
modified in its physical properties, and he was unable to detect any
change in the coefficient of expansion.[219] It does not follow,
however, that, because no apparent change was discovered in a quartz
crystal, newly-constructed bars of metal would undergo no change.
[219] *Philosophical Magazine*, (1868), 4th Series, vol. xxxvi. p. 32.
The best principle, as it seems to me, upon which the perpetuation of
a standard of length can be rested, is that, if a variation of length
occurs, it will in all probability be of different amount in different
substances. If then a great number of standard metres were constructed
of all kinds of different metals and alloys; hard rocks, such as
granite, serpentine, slate, quartz, limestone; artificial substances,
such as porcelain, glass, &c., &c., careful comparison would show from
time to time the comparative variations of length of these different
substances. The most variable substances would be the most divergent,
and the standard would be furnished by the mean length of those which
agreed most closely with each other just as uniform motion is that of
those bodies which agree most closely in indicating the efflux of time.
*The Terrestrial Standard.*
The second method assumes that the globe itself is a body of invariable
dimensions and the founders of the metrical system selected the
ten-millionth part of the distance from the equator to the pole as
the definition of the metre. The first imperfection in such a method
is that the earth is certainly not invariable in size; for we know
that it is superior in temperature to surrounding space, and must be
slowly cooling and contracting. There is much reason to believe that
all earthquakes, volcanoes, mountain elevations, and changes of sea
level are evidences of this contraction as asserted by Mr. Mallet.[220]
But such is the vast bulk of the earth and the duration of its past
existence, that this contraction is perhaps less rapid in proportion
than that of any bar or other material standard which we can construct.
[220] *Proceedings of the Royal Society*, 20th June, 1872, vol. xx.
p. 438.
The second and chief difficulty of this method arises from the vast
size of the earth, which prevents us from making any comparison with
the ultimate standard, except by a trigonometrical survey of a most
elaborate and costly kind. The French physicists, who first proposed
the method, attempted to obviate this inconvenience by carrying out
the survey once for all, and then constructing a standard metre, which
should be exactly the one ten millionth part of the distance from the
pole to the equator. But since all measuring operations are merely
approximate, it was impossible that this operation could be perfectly
achieved. Accordingly, it was shown in 1838 that the supposed French
metre was erroneous to the considerable extent of one part in 5527. It
then became necessary either to alter the length of the assumed metre,
or to abandon its supposed relation to the earth’s dimensions. The
French Government and the International Metrical Commission have for
obvious reasons decided in favour of the latter course, and have thus
reverted to the first method of defining the metre by a given bar. As
from time to time the ratio between this assumed standard metre and the
quadrant of the earth becomes more accurately known, we have better
means of restoring that metre by reference to the globe if required.
But until lost, destroyed, or for some clear reason discredited,
the bar metre and not the globe is the standard. Thomson and Tait
remark that any of the more accurate measurements of the English
trigonometrical survey might in like manner be employed to restore our
standard yard, in terms of which the results are recorded.
*The Pendulum Standard.*
The third method of defining a standard length, by reference to the
seconds pendulum, was first proposed by Huyghens, and was at one time
adopted by the English Government. From the principle of the pendulum
(p. 302) it clearly appears that if the time of oscillation and the
force actuating the pendulum be the same, the length of the pendulum
must be the same. We do not get rid of theoretical difficulties, for
we must assume the attraction of gravity at some point of the earth’s
surface, say London, to be unchanged from time to time, and the
sidereal day to be invariable, neither assumption being absolutely
correct so far as we can judge. The pendulum, in short, is only an
indirect means of making one physical quantity of space depend upon two
other physical quantities of time and force.
The practical difficulties are, however, of a far more serious
character than the theoretical ones. The length of a pendulum is not
the ordinary length of the instrument, which might be greatly varied
without affecting the duration of a vibration, but the distance from
the centre of suspension to the centre of oscillation. There are no
direct means of determining this latter centre, which depends upon
the average momentum of all the particles of the pendulum as regards
the centre of suspension. Huyghens discovered that the centres of
suspension and oscillation are interchangeable, and Kater pointed
out that if a pendulum vibrates with exactly the same rapidity when
suspended from two different points, the distance between these points
is the true length of the equivalent simple pendulum.[221] But the
practical difficulties in employing Kater’s reversible pendulum are
considerable, and questions regarding the disturbance of the air, the
force of gravity, or even the interference of electrical attractions
have to be entertained. It has been shown that all the experiments made
under the authority of Government for determining the ratio between
the standard yard and the seconds pendulum, were vitiated by an error
in the corrections for the resisting, adherent, or buoyant power of
the air in which the pendulums were swung. Even if such corrections
were rendered unnecessary by operating in a vacuum, other difficult
questions remain.[222] Gauss’ mode of comparing the vibrations of a
wire pendulum when suspended at two different lengths is open to equal
or greater practical difficulties. Thus it is found that the pendulum
standard cannot compete in accuracy and certainty with the simple bar
standard, and the method would only be useful as an accessory mode of
restoring the bar standard if at any time again destroyed.
[221] Kater’s *Treatise on Mechanics*, Cabinet Cyclopædia, p. 154.
[222] Grant’s *History of Physical Astronomy*, p. 156.
*Unit of Density.*
Before we can measure the phenomena of nature, we require a third
independent unit, which shall enable us to define the quantity of
matter occupying any given space. All the changes of nature, as we
shall see, are probably so many manifestations of energy; but energy
requires some substratum or material machinery of molecules, in and by
which it may be manifested. Observation shows that, as regards force,
there may be two modes of variation of matter. As Newton says in the
first definition of the Principia, “the quantity of matter is the
measure of the same, arising from its density and bulk conjunctly.”
Thus the force required to set a body in motion varies both according
to the bulk of the matter, and also according to its quality. Two cubic
inches of iron of uniform quality, will require twice as much force
as one cubic inch to produce a certain velocity in a given time; but
one cubic inch of gold will require more force than one cubic inch of
iron. There is then some new measurable quality in matter apart from
its bulk, which we may call *density*, and which is, strictly speaking,
indicated by its capacity to resist and absorb the action of force.
For the unit of density we may assume that of any substance which is
uniform in quality, and can readily be referred to from time to time.
Pure water at any definite temperature, for instance that of snow
melting under inappreciable pressure, furnishes an invariable standard
of density, and by comparing equal bulks of various substances with
a like bulk of ice-cold water, as regards the velocity produced in a
unit of time by the same force, we should ascertain the densities of
those substances as expressed in that of water. Practically the force
of gravity is used to measure density; for a beautiful experiment with
the pendulum, performed by Newton and repeated by Gauss, shows that all
kinds of matter gravitate equally. Two portions of matter then which
are in equilibrium in the balance, may be assumed to possess equal
inertia, and their densities will therefore be inversely as their cubic
dimensions.
*Unit of Mass.*
Multiplying the number of units of density of a portion of matter,
by the number of units of space occupied by it, we arrive at the
quantity of matter, or, as it is usually called, the *unit of mass*, as
indicated by the inertia and gravity it possesses. To proceed in the
most simple manner, the unit of mass ought to be that of a cubic unit
of matter of the standard density; but the founders of the metrical
system took as their unit of mass, the cubic centimetre of water, at
the temperature of maximum density (about 4° Cent.). They called this
unit of mass the *gramme*, and constructed standard specimens of the
kilogram, which might be readily referred to by all who required to
employ accurate weights. Unfortunately the determination of the bulk
of a given weight of water at a certain temperature is an operation
involving many difficulties, and it cannot be performed in the present
day with a greater exactness than that of about one part in 5000, the
results of careful observers being sometimes found to differ as much as
one part in 1000.[223]
[223] Clerk Maxwell’s *Theory of Heat*, p. 79.
Weights, on the other hand, can be compared with each other to at least
one part in a million. Hence if different specimens of the kilogram be
prepared by direct weighing against water, they will not agree closely
with each other; the two principal standard kilograms agree neither
with each other, nor with their definition. According to Professor
Miller the so-called Kilogramme des Archives weighs 15432·34874 grains,
while the kilogram deposited at the Ministry of the Interior in Paris,
as the standard for commercial purposes, weighs 15432·344 grains. Since
a standard weight constructed of platinum, or platinum and iridium, can
be preserved free from any appreciable alteration, and since it can be
very accurately compared with other weights, we shall ultimately attain
the greatest exactness in our measurements of mass, by assuming some
single kilogram as a *provisional standard*, leaving the determination
of its actual mass in units of space and density for future
investigation. This is what is practically done at the present day,
and thus a unit of mass takes the place of the unit of density, both
in the French and English systems. The English pound is defined by a
certain lump of platinum, preserved at Westminster, and is an arbitrary
mass, chosen merely that it may agree as nearly as possible with old
English pounds. The gallon, the old English unit of cubic measurement,
is defined by the condition that it shall contain exactly ten pounds
weight of water at 62° Fahr.; and although it is stated that it has the
capacity of about 277·274 cubic inches, this ratio between the cubic
and linear systems of measurement is not legally enacted, but left open
to investigation. While the French metric system as originally designed
was theoretically perfect, it does not differ practically in this point
from the English system.
*Natural System of Standards.*
Quite recently Professor Clerk Maxwell has suggested that the
vibrations of light and the atoms of matter might conceivably be
employed as the ultimate standards of length, time, and mass. We
should thus arrive at a *natural system of standards*, which,
though possessing no present practical importance, has considerable
theoretical interest. “In the present state of science,” he says, “the
most universal standard of length which we could assume would be the
wave-length in vacuum of a particular kind of light, emitted by some
widely diffused substance such as sodium, which has well-defined lines
in its spectrum. Such a standard would be independent of any changes in
the dimensions of the earth, and should be adopted by those who expect
their writings to be more permanent than that body.”[224] In the same
way we should get a universal standard unit of time, independent of
all questions about the motion of material bodies, by taking as the
unit the periodic time of vibration of that particular kind of light
whose wave-length is the unit of length. It would follow that with
these units of length and time the unit of velocity would coincide with
the velocity of light in empty space. As regards the unit of mass,
Professor Maxwell, humorously as I should think, remarks that if we
expect soon to be able to determine the mass of a single molecule of
some standard substance, we may wait for this determination before
fixing a universal standard of mass.
[224] *Treatise on Electricity and Magnetism*, vol. i. p. 3.
In a theoretical point of view there can be no reasonable doubt that
vibrations of light are, as far as we can tell, the most fixed in
magnitude of all phenomena. There is as usual no certainty in the
matter, for the properties of the basis of light may vary to some
extent in different parts of space. But no differences could ever
be established in the velocity of light in different parts of the
solar system, and the spectra of the stars show that the times of
vibration there do not differ perceptibly from those in this part
of the universe. Thus all presumption is in favour of the absolute
constancy of the vibrations of light--absolute, that is, so far as
regards any means of investigation we are likely to possess. Nearly
the same considerations apply to the atomic weight as the standard of
mass. It is impossible to prove that all atoms of the same substance
are of equal mass, and some physicists think that they differ, so that
the fixity of combining proportions may be due only to the approximate
constancy of the mean of countless millions of discrepant weights. But
in any case the detection of difference is probably beyond our powers.
In a theoretical point of view, then, the magnitudes suggested by
Professor Maxwell seem to be the most fixed ones of which we have any
knowledge, so that they necessarily become the natural units.
In a practical point of view, as Professor Maxwell would be the first
to point out, they are of little or no value, because in the present
state of science we cannot measure a vibration or weigh an atom with
any approach to the accuracy which is attainable in the comparison
of standard metres and kilograms. The velocity of light is not known
probably within a thousandth part, and as we progress in the knowledge
of light, so we shall progress in the accurate fixation of other
standards. All that can be said then, is that it is very desirable
to determine the wave-lengths and periods of the principal lines of
the solar spectrum, and the absolute atomic weights of the elements,
with all attainable accuracy, in terms of our existing standards. The
numbers thus obtained would admit of the reproduction of our standards
in some future age of the world to a corresponding degree of accuracy,
were there need of such reference; but so far as we can see at present,
there is no considerable probability that this mode of reproduction
would ever be the best mode.
*Subsidiary Units.*
Having once established the standard units of time, space, and density
or mass, we might employ them for the expression of all quantities
of such nature. But it is often convenient in particular branches of
science to use multiples or submultiples of the original units, for the
expression of quantities in a simple manner. We use the mile rather
than the yard when treating of the magnitude of the globe, and the
mean distance of the earth and sun is not too large a unit when we
have to describe the distances of the stars. On the other hand, when
we are occupied with microscopic objects, the inch, the line or the
millimetre, become the most convenient terms of expression.
It is allowable for a scientific man to introduce a new unit in any
branch of knowledge, provided that it assists precise expression,
and is carefully brought into relation with the primary units. Thus
Professor A. W. Williamson has proposed as a convenient unit of
volume in chemical science, an absolute volume equal to about 11·2
litres representing the bulk of one gram of hydrogen gas at standard
temperature and pressure, or the *equivalent* weight of any other
gas, such as 16 grams of oxygen, 14 grams of nitrogen, &c.; in short,
the bulk of that quantity of any one of those gases which weighs as
many grams as there are units in the number expressing its atomic
weight.[225] Hofmann has proposed a new unit of weight for chemists,
called a *crith*, to be defined by the weight of one litre of hydrogen
gas at 0° C. and 0°·76 mm., weighing about 0·0896 gram.[226] Both of
these units must be regarded as purely subordinate units, ultimately
defined by reference to the primary units, and not involving any new
assumption.
[225] *Chemistry for Students*, by A. W. Williamson. Clarendon Press
Series, 2nd ed. Preface p. vi.
[226] *Introduction to Chemistry*, p. 131.
*Derived Units.*
The standard units of time, space, and mass having been once fixed,
many kinds of magnitude are naturally measured by units derived from
them. From the metre, the unit of linear magnitude follows in the most
obvious manner the centiare or square metre, the unit of superficial
magnitude, and the litre that is the cube of the tenth part of a metre,
the unit of capacity or volume. Velocity of motion is expressed by the
ratio of the space passed over, when the motion is uniform, to the time
occupied; hence the unit of velocity is that of a body which passes
over a unit of space in a unit of time. In physical science the unit of
velocity might be taken as one metre per second. Momentum is measured
by the mass moving, regard being paid both to the amount of matter and
the velocity at which it is moving. Hence the unit of momentum will be
that of a unit volume of matter of the unit density moving with the
unit velocity, or in the French system, a cubic centimetre of water of
the maximum density moving one metre per second.
An accelerating force is measured by the ratio of the momentum
generated to the time occupied, the force being supposed to act
uniformly. The unit of force will therefore be that which generates
a unit of momentum in a unit of time, or which causes, in the French
system, one cubic centimetre of water at maximum density to acquire in
one second a velocity of one metre per second. The force of gravity is
the most familiar kind of force, and as, when acting unimpeded upon any
substance, it produces in a second a velocity of 9·80868 . . metres per
second in Paris, it follows that the absolute unit of force is about
the tenth part of the force of gravity. If we employ British weights
and measures, the absolute unit of force is represented by the gravity
of about half an ounce, since the force of gravity of any portion of
matter acting upon that matter during one second, produces a final
velocity of 32·1889 feet per second or about 32 units of velocity.
Although from its perpetual action and approximate uniformity we find
in gravity the most convenient force for reference, and thus habitually
employ it to estimate quantities of matter, we must remember that it
is only one of many instances of force. Strictly speaking, we should
express weight in terms of force, but practically we express other
forces in terms of weight.
We still require the unit of energy, a more complex notion. The
momentum of a body expresses the quantity of motion which belongs or
would belong to the aggregate of the particles; but when we consider
how this motion is related to the action of a force producing or
removing it, we find that the effect of a force is proportional to the
mass multiplied by the square of the velocity and it is convenient to
take half this product as the expression required. But it is shown in
books upon dynamics that it will be exactly the same thing if we define
energy by a force acting through a space. The natural unit of energy
will then be that which overcomes a unit of force acting through a unit
of space; when we lift one kilogram through one metre, against gravity,
we therefore accomplish 9·80868 . . units of work, that is, we turn so
many units of potential energy existing in the muscles, into potential
energy of gravitation. In lifting one pound through one foot there is
in like manner a conversion of 32·1889 units of energy. Accordingly the
unit of energy will be in the English system, that required to lift
one pound through about the thirty-second part of a foot; in terms of
metric units, it will be that required to lift a kilogram through about
one tenth part of a metre.
Every person is at liberty to measure and record quantities in terms of
any unit which he likes. He may use the yard for linear measurement and
the litre for cubic measurement, only there will then be a complicated
relation between his different results. The system of derived units
which we have been briefly considering, is that which gives the most
simple and natural relations between quantitative expressions of
different kinds, and therefore conduces to ease of comprehension and
saving of laborious calculation.
It would evidently be a source of great convenience if scientific men
could agree upon some single system of units, original and derived, in
terms of which all quantities could be expressed. Statements would thus
be rendered easily comparable, a large part of scientific literature
would be made intelligible to all, and the saving of mental labour
would be immense. It seems to be generally allowed, too, that the
metric system of weights and measures presents the best basis for the
ultimate system; it is thoroughly established in Western Europe; it is
legalised in England; it is already commonly employed by scientific
men; it is in itself the most simple and scientific of systems. There
is every reason then why the metric system should be accepted at least
in its main features.
*Provisional Units.*
Ultimately, as we can hardly doubt, all phenomena will be recognised
as so many manifestations of energy; and, being expressed in terms of
the unit of energy, will be referable to the primary units of space,
time, and density. To effect this reduction, however, in any particular
case, we must not only be able to compare different quantities of
the phenomenon, but to trace the whole series of steps by which it
is connected with the primary notions. We can readily observe that
the intensity of one source of light is greater than that of another;
and, knowing that the intensity of light decreases as the square of
the distance increases, we can easily determine their comparative
brilliance. Hence we can express the intensity of light falling upon
any surface, if we have a unit in which to make the expression. Light
is undoubtedly one form of energy, and the unit ought therefore to be
the unit of energy. But at present it is quite impossible to say how
much energy there is in any particular amount of light. The question
then arises,--Are we to defer the measurement of light until we can
assign its relation to other forms of energy? If we answer Yes, it
is equivalent to saying that the science of light must stand still
perhaps for a generation; and not only this science but many others.
The true course evidently is to select, as the provisional unit of
light, some light of convenient intensity, which can be reproduced from
time to time in the same intensity, and which is defined by physical
circumstances. All the phenomena of light may be experimentally
investigated relatively to this unit, for instance that obtained after
much labour by Bunsen and Roscoe.[227] In after years it will become a
matter of inquiry what is the energy exerted in such unit of light; but
it may be long before the relation is exactly determined.
[227] *Philosophical Transactions* (1859), vol. cxlix. p. 884, &c.
A provisional unit, then, means one which is assumed and physically
defined in a safe and reproducible manner, in order that particular
quantities may be compared *inter se* more accurately than they can
yet be referred to the primary units. In reality the great majority
of our measurements are expressed in terms of such provisionally
independent units, and even the unit of mass, as we have seen, ought to
be considered as provisional.
The unit of heat ought to be simply the unit of energy, already
described. But a weight can be measured to the one-millionth part, and
temperature to less than the thousandth part of a degree Fahrenheit,
and to less therefore than the five-hundred thousandth part of the
absolute temperature, whereas the mechanical equivalent of heat
is probably not known to the thousandth part. Hence the need of a
provisional unit of heat, which is often taken as that requisite to
raise one gram of water through one degree Centigrade, that is from
0° to 1°. This quantity of heat is capable of approximate expression
in terms of time, space, and mass; for by the natural constant,
determined by Dr. Joule, and called the mechanical equivalent of heat,
we know that the assumed unit of heat is equal to the energy of 423·55
gram-metres, or that energy which will raise the mass of 423·55 grams
through one metre against 9·8... absolute units of force. Heat may also
be expressed in terms of the quantity of ice at 0° Cent., which it is
capable of converting into water under inappreciable pressure.
*Theory of Dimensions.*
In order to understand the relations between the quantities dealt with
in physical science, it is necessary to pay attention to the Theory of
Dimensions, first clearly stated by Joseph Fourier,[228] but in later
years developed by several physicists. This theory investigates the
manner in which each derived unit depends upon or involves one or more
of the fundamental units. The number of units in a rectangular area
is found by multiplying together the numbers of units in the sides;
thus the unit of length enters twice into the unit of area, which is
therefore said to have two dimensions with respect to length. Denoting
length by *L*, we may say that the dimensions of area are *L* × *L* or
*L*^{2}. It is obvious in the same way that the dimensions of volume or
bulk will be *L*^{3}.
[228] *Théorie Analytique de la Chaleur*, Paris; 1822, §§ 157–162.
The number of units of mass in a body is found by multiplying the
number of units of volume, by those of density. Hence mass is of
three dimensions as regards length, and one as regards density.
Calling density *D*, the dimensions of mass are *L*^{3}*D*. As already
explained, however, it is usual to substitute an arbitrary provisional
unit of mass, symbolised by *M*; according to the view here taken we
may say that the dimensions of *M* are *L*^{3}*D*.
Introducing time, denoted by *T*, it is easy to see that the dimensions
of velocity will be *L/T* or *LT*^{-1}, because the number of units
in the velocity of a body is found by *dividing* the units of length
passed over by the units of time occupied in passing. The acceleration
of a body is measured by the increase of velocity in relation to the
time, that is, we must divide the units of velocity gained by the units
of time occupied in gaining it; hence its dimensions will be *LT*^{-2}.
Momentum is the product of mass and velocity, so that its dimensions
are *MLT*^{-1}. The effect of a force is measured by the acceleration
produced in a unit of mass in a unit of time; hence the dimensions of
force are *MLT*^{-2}. Work done is proportional to the force acting and
to the space through which it acts; so that it has the dimensions of
force with that of length added, giving *ML*^{2}*T*^{-2}.
It should be particularly noticed that angular magnitude has no
dimensions at all, being measured by the ratio of the arc to the radius
(p. 305). Thus we have the dimensions *LL*^{-1} or *L*^{0}. This
agrees with the statement previously made, that no arbitrary unit of
angular magnitude is needed. Similarly, all pure numbers expressing
ratios only, such as sines and other trigonometrical functions,
logarithms, exponents, &c., are devoid of dimensions. They are absolute
numbers necessarily expressed in terms of unity itself, and are quite
unaffected by the selection of the arbitrary physical units. Angular
magnitude, however, enters into other quantities, such as angular
velocity, which has the dimensions 1/*T* or *T*^{-1}, the units of
angle being divided by the units of time occupied. The dimensions of
angular acceleration are denoted by *T*^{-2}.
The quantities treated in the theories of heat and electricity
are numerous and complicated as regards their dimensions. Thermal
capacity has the dimensions *ML*^{-3}, thermal conductivity,
*ML*^{-1}*T*^{-1}. In Magnetism the dimensions of the strength
of pole are *M*^{1/2}*L*^{3/2}*T*^{-1}, the dimensions of
field-intensity are *M*^{1/2}*L*^{-1/2}*T*^{-1}, and the intensity
of magnetisation has the same dimensions. In the science of
electricity physicists have to deal with numerous kinds of quantity,
and their dimensions are different too in the electro-static
and the electro-magnetic systems. Thus electro-motive force has
the dimensions *M*^{1/2}*L*^{1/2}*T*^{-1}, in the former, and
*M*^{1/2}*L*^{3/2}*T*^{-2} in the latter system. Capacity simply
depends upon length in electro-statics, but upon *L*^{-1}*T*^{2} in
electro-magnetics. It is worthy of particular notice that electrical
quantities have simple dimensions when expressed in terms of density
instead of mass. The instances now given are sufficient to show the
difficulty of conceiving and following out the relations of the
quantities treated in physical science without a systematic method of
calculating and exhibiting their dimensions. It is only in quite recent
years that clear ideas about these quantities have been attained. Half
a century ago probably no one but Fourier could have explained what
he meant by temperature or capacity for heat. The notion of measuring
electricity had hardly been entertained.
Besides affording us a clear view of the complex relations of physical
quantities, this theory is specially useful in two ways. Firstly, it
affords a test of the correctness of mathematical reasoning. According
to the *Principle of Homogeneity*, all the quantities *added* together,
and equated in any equation, must have the same dimensions. Hence if,
on estimating the dimensions of the terms in any equation, they be not
homogeneous, some blunder must have been committed. It is impossible
to add a force to a velocity, or a mass to a momentum. Even if the
numerical values of the two members of a non-homogeneous equation were
equal, this would be accidental, and any alteration in the physical
units would produce inequality and disclose the falsity of the law
expressed in the equation.
Secondly, the theory of units enables us readily and infallibly to
deduce the change in the numerical expression of any physical quantity,
produced by a change in the fundamental units. It is of course obvious
that in order to represent the same absolute quantity, a number must
vary inversely as the magnitude of the units which are numbered. The
yard expressed in feet is 3; taking the inch as the unit instead of
the foot it becomes 36. Every quantity into which the dimension length
enters positively must be altered in like manner. Changing the unit
from the foot to the inch, numerical expressions of volume must be
multiplied by 12 × 12 × 12. When a dimension enters negatively the
opposite rule will hold. If for the minute we substitute the second
as unit of time, then we must divide all numbers expressing angular
velocities by 60, and numbers expressing angular acceleration by
60 × 60. The rule is that a numerical expression varies inversely as
the magnitude of the unit as regards each whole dimension entering
positively, and it varies directly as the magnitude of the unit for
each whole dimension entering negatively. In the case of fractional
exponents, the proper root of the ratio of change has to be taken.
The study of this subject may be continued in Professor J. D. Everett’s
“Illustrations of the Centimetre-gramme-second System of Units,”
published by Taylor and Francis, 1875; in Professor Maxwell’s “Theory
of Heat;” or Professor Fleeming Jenkin’s “Text Book of Electricity.”
*Natural Constants.*
Having acquired accurate measuring instruments, and decided upon the
units in which the results shall be expressed, there remains the
question, What use shall be made of our powers of measurement? Our
principal object must be to discover general quantitative laws of
nature; but a very large amount of preliminary labour is employed in
the accurate determination of the dimensions of existing objects, and
the numerical relations between diverse forces and phenomena. Step
by step every part of the material universe is surveyed and brought
into known relations with other parts. Each manifestation of energy is
correlated with each other kind of manifestation. Professor Tyndall has
described the care with which such operations are conducted.[229]
[229] Tyndall’s *Sound*, 1st ed. p. 26.
“Those who are unacquainted with the details of scientific
investigation, have no idea of the amount of labour expended on
the determination of those numbers on which important calculations
or inferences depend. They have no idea of the patience shown by a
Berzelius in determining atomic weights; by a Regnault in determining
coefficients of expansion; or by a Joule in determining the mechanical
equivalent of heat. There is a morality brought to bear upon such
matters which, in point of severity, is probably without a parallel in
any other domain of intellectual action.”
Every new natural constant which is recorded brings many fresh
inferences within our power. For if *n* be the number of such constants
known, then 1/2 (*n*^{2}--*n*) is the number of ratios which are within
our powers of calculation, and this increases with the square of *n*.
We thus gradually piece together a map of nature, in which the lines of
inference from one phenomenon to another rapidly grow in complexity,
and the powers of scientific prediction are correspondingly augmented.
Babbage[230] proposed the formation of a collection of the constant
numbers of nature, a work which has at last been taken in hand by the
Smithsonian Institution.[231] It is true that a complete collection of
such numbers would be almost co-extensive with scientific literature,
since almost all the numbers occurring in works on chemistry,
mineralogy, physics, astronomy, &c., would have to be included.
Still a handy volume giving all the more important numbers and their
logarithms, referred when requisite to the different units in common
use, would be very useful. A small collection of constant numbers will
be found at the end of Babbage’s, Hutton’s, and many other tables of
logarithms, and a somewhat larger collection is given in Templeton’s
*Millwright and Engineer’s Pocket Companion*.
[230] British Association, Cambridge, 1833. Report, pp. 484–490.
[231] *Smithsonian Miscellaneous Collections*, vol. xii., the
Constants of Nature, part. i. Specific gravities compiled by F. W.
Clarke, 8vo. Washington, 1873.
Our present object will be to classify these constant numbers roughly,
according to their comparative generality and importance, under the
following heads:--
(1) Mathematical constants.
(2) Physical constants.
(3) Astronomical constants.
(4) Terrestrial numbers.
(5) Organic numbers.
(6) Social numbers.
*Mathematical Constants.*
At the head of the list of natural constants must come those which
express the necessary relations of numbers to each other. The ordinary
Multiplication Table is the most familiar and the most important of
such series of constants, and is, theoretically speaking, infinite in
extent. Next we must place the Arithmetical Triangle, the significance
of which has already been pointed out (p. 182). Tables of logarithms
also contain vast series of natural constants, arising out of the
relations of pure numbers. At the base of all logarithmic theory is
the mysterious natural constant commonly denoted by *e*, or ε, being
equal to the infinite series 1 + 1/1 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4
+...., and thus consisting of the sum of the ratios between the numbers
of permutations and combinations of 0, 1, 2, 3, 4, &c. things. Tables
of prime numbers and of the factors of composite numbers must not be
forgotten.
Another vast and in fact infinite series of numerical constants
contains those connected with the measurement of angles, and embodied
in trigonometrical tables, whether as natural or logarithmic sines,
cosines, and tangents. It should never be forgotten that though these
numbers find their chief employment in connection with trigonometry,
or the measurement of the sides of a right-angled triangle, yet the
numbers themselves arise out of numerical relations bearing no special
relation to space. Foremost among trigonometrical constants is the
well known number π, usually employed as expressing the ratio of the
circumference and the diameter of a circle; from π follows the value of
the arcual or natural unit of angular value as expressed in ordinary
degrees (p. 306).
Among other mathematical constants not uncommonly used may be mentioned
tables of factorials (p. 179), tables of Bernoulli’s numbers, tables of
the error function,[232] which latter are indispensable not only in the
theory of probability but also in several other branches of science.
[232] J. W. L. Glaisher, *Philosophical Magazine*, 4th Series,
vol. xlii. p. 421.
It should be clearly understood that the mathematical constants and
tables of reference already in our possession, although very extensive,
are only an infinitely small part of what might be formed. With the
progress of science the tabulation of new functions will be continually
demanded, and it is worthy of consideration whether public money
should not be available to reward the severe, long continued, and
generally thankless labour which must be gone through in calculating
tables. Such labours are a benefit to the whole human race as long as
it shall exist, though there are few who can appreciate the extent
of this benefit. A most interesting and excellent description of
many mathematical tables will be found in De Morgan’s article on
*Tables*, in the *English Cyclopædia*, Division of Arts and Sciences,
vol. vii. p. 976. An almost exhaustive critical catalogue of extant
tables is being published by a Committee of the British Association,
two portions, drawn up chiefly by Mr. J. W. L. Glaisher and Professor
Cayley, having appeared in the Reports of the Association for 1873 and
1875.
*Physical Constants.*
The second class of constants contains those which refer to the
actual constitution of matter. For the most part they depend upon
the peculiarities of the chemical substance in question, but we may
begin with those which are of the most general character. In a first
sub-class we may place the velocity of light or heat undulations, the
numbers expressing the relation between the lengths of the undulations,
and the rapidity of the undulations, these numbers depending only on
the properties of the ethereal medium, and being probably the same in
all parts of the universe. The theory of heat gives rise to several
numbers of the highest importance, especially Joule’s mechanical
equivalent of heat, the absolute zero of temperature, the mean
temperature of empty space, &c.
Taking into account the diverse properties of the elements we must
have tables of the atomic weights, the specific heats, the specific
gravities, the refractive powers, not only of the elements, but their
almost infinitely numerous compounds. The properties of hardness,
elasticity, viscosity, expansion by heat, conducting powers for heat
and electricity, must also be determined in immense detail. There are,
however, certain of these numbers which stand out prominently because
they serve as intermediate units or terms of comparison. Such are, for
instance, the absolute coefficients of expansion of air, water and
mercury, the temperature of the maximum density of water, the latent
heats of water and steam, the boiling-point of water under standard
pressure, the melting and boiling-points of mercury, and so forth.
*Astronomical Constants.*
The third great class consists of numbers possessing far less
generality because they refer not to the properties of matter, but to
the special forms and distances in which matter has been disposed in
the part of the universe open to our examination. We have, first of
all, to define the magnitude and form of the earth, its mean density,
the constant of aberration of light expressing the relation between
the earth’s mean velocity in space and the velocity of light. From
the earth, as our observatory, we then proceed to lay down the mean
distances of the sun, and of the planets from the same centre; all the
elements of the planetary orbits, the magnitudes, densities, masses,
periods of axial rotation of the several planets are by degrees
determined with growing accuracy. The same labours must be gone through
for the satellites. Catalogues of comets with the elements of their
orbits, as far as ascertainable, must not be omitted.
From the earth’s orbit as a new base of observations, we next proceed
to survey the heavens and lay down the apparent positions, magnitudes,
motions, distances, periods of variation, &c. of the stars. All
catalogues of stars from those of Hipparchus and Tycho, are full of
numbers expressing rudely the conformation of the visible universe.
But there is obviously no limit to the labours of astronomers; not
only are millions of distant stars awaiting their first measurements,
but those already registered require endless scrutiny as regards
their movements in the three dimensions of space, their periods of
revolution, their changes of brilliance and colour. It is obvious that
though astronomical numbers are conventionally called *constant*, they
are probably in all cases subject to more or less rapid variation.
*Terrestrial Numbers.*
Our knowledge of the globe we inhabit involves many numerical
determinations, which have little or no connection with astronomical
theory. The extreme heights of the principal mountains, the mean
elevations of continents, the mean or extreme depths of the oceans,
the specific gravities of rocks, the temperature of mines, the host of
numbers expressing the meteorological or magnetic conditions of every
part of the surface, must fall into this class. Many such numbers
are not to be called constant, being subject to periodic or secular
changes, but they are hardly more variable in fact than some which in
astronomical science are set down as constant. In many cases quantities
which seem most variable may go through rhythmical changes resulting
in a nearly uniform average, and it is only in the long progress of
physical investigation that we can hope to discriminate successfully
between those elemental numbers which are fixed and those which vary.
In the latter case the law of variation becomes the constant relation
which is the object of our search.
*Organic Numbers.*
The forms and properties of brute nature having been sufficiently
defined by the previous classes of numbers, the organic world, both
vegetable and animal, remains outstanding, and offers a higher series
of phenomena for our investigation. All exact knowledge relating to
the forms and sizes of living things, their numbers, the quantities
of various compounds which they consume, contain, or excrete, their
muscular or nervous energy, &c. must be placed apart in a class by
themselves. All such numbers are doubtless more or less subject to
variation, and but in a minor degree capable of exact determination.
Man, so far as he is an animal, and as regards his physical form, must
also be treated in this class.
*Social Numbers.*
Little allusion need be made in this work to the fact that man in
his economic, sanitary, intellectual, æsthetic, or moral relations
may become the subject of sciences, the highest and most useful of
all sciences. Every one who is engaged in statistical inquiry must
acknowledge the possibility of natural laws governing such statistical
facts. Hence we must allot a distinct place to numerical information
relating to the numbers, ages, physical and sanitary condition,
mortality, &c., of different peoples, in short, to vital statistics.
Economic statistics, comprehending the quantities of commodities
produced, existing, exchanged and consumed, constitute another
extensive body of science. In the progress of time exact investigation
may possibly subdue regions of phenomena which at present defy all
scientific treatment. That scientific method can ever exhaust the
phenomena of the human mind is incredible.
CHAPTER XV.
ANALYSIS OF QUANTITATIVE PHENOMENA.
In the two preceding chapters we have been engaged in considering how
a phenomenon may be accurately measured and expressed. So delicate
and complex an operation is a measurement which pretends to any
considerable degree of exactness, that no small part of the skill
and patience of physicists is usually spent upon this work. Much
of this difficulty arises from the fact that it is scarcely ever
possible to measure a single effect at a time. The ultimate object
must be to discover the mathematical equation or law connecting a
quantitative cause with its quantitative effect; this purpose usually
involves, as we shall see, the varying of one condition at a time,
the other conditions being maintained constant. The labours of the
experimentalist would be comparatively light if he could carry out
this rule of varying one circumstance at a time. He would then obtain
a series of corresponding values of the variable quantities concerned,
from which he might by proper hypothetical treatment obtain the
required law of connection. But in reality it is seldom possible to
carry out this direction except in an approximate manner. Before then
we proceed to the consideration of the actual process of quantitative
induction, it is necessary to review the several devices by which a
complicated series of effects can be disentangled. Every phenomenon
measured will usually be the sum, difference, or it may be the product
or quotient, of two or more different effects, and these must be in
some way analysed and separately measured before we possess the
materials for inductive treatment.
*Illustrations of the Complication of Effects.*
It is easy to bring forward a multitude of instances to show that a
phenomenon is seldom to be observed simple and alone. A more or less
elaborate process of analysis is almost always necessary. Thus if an
experimentalist wishes to observe and measure the expansion of a liquid
by heat, he places it in a thermometer tube and registers the rise of
the column of liquid in the narrow tube. But he cannot heat the liquid
without also heating the glass, so that the change observed is really
the difference between the expansions of the liquid and the glass. More
minute investigation will show the necessity perhaps of allowing for
further minute effects, namely the compression of the liquid and the
expansion of the bulb due to the increased pressure of the column as it
becomes lengthened.
In a great many cases an observed effect will be apparently at least
the simple sum of two separate and independent effects. The heat
evolved in the combustion of oil is partly due to the carbon and partly
to the hydrogen. A measurement of the heat yielded by the two jointly,
cannot inform us how much proceeds from the one and how much from the
other. If by some separate determination we can ascertain how much the
hydrogen yields, then by mere subtraction we learn what is due to the
carbon; and *vice versâ*. The heat conveyed by a liquid, may be partly
conveyed by true conduction, partly by convection. The light dispersed
in the interior of a liquid consists both of what is reflected by
floating particles and what is due to true fluorescence;[233] and we
must find some mode of determining one portion before we can learn the
other. The apparent motion of the spots on the sun, is the algebraic
sum of the sun’s axial rotation, and of the proper motion of the spots
upon the sun’s surface; hence the difficulty of ascertaining by direct
observations the period of the sun’s rotation.
[233] Stokes, *Philosophical Transactions* (1852), vol. cxlii. p. 529.
We cannot obtain the weight of a portion of liquid in a chemical
balance without weighing it with the containing vessel. Hence to have
the real weight of the liquid operated upon in an experiment, we must
make a separate weighing of the vessel, with or without the adhering
film of liquid according to circumstances. This is likewise the mode
in which a cart and its load are weighed together, the *tare* of the
cart previously ascertained being deducted. The variation in the height
of the barometer is a joint effect, partly due to the real variation
of the atmospheric pressure, partly to the expansion of the mercurial
column by heat. The effects may be discriminated, if, instead of one
barometer tube we have two tubes containing mercury placed closely side
by side, so as to have the same temperature. If one of them be closed
at the bottom so as to be unaffected by the atmospheric pressure, it
will show the changes due to temperature only, and, by subtracting
these changes from those shown in the other tube, employed as a
barometer, we get the real oscillations of atmospheric pressure. But
this correction, as it is called, of the barometric reading, is better
effected by calculation from the readings of an ordinary thermometer.
In other cases a quantitative effect will be the difference of
two causes acting in opposite directions. Sir John Herschel
invented an instrument like a large thermometer, which he called
the Actinometer,[234] and Pouillet constructed a somewhat similar
instrument called the Pyrheliometer, for ascertaining the heating power
of the sun’s rays. In both instruments the heat of the sun was absorbed
by a reservoir containing water, and the rise of temperature of the
water was exactly observed, either by its own expansion, or by the
readings of a delicate thermometer immersed in it. But in exposing the
actinometer to the sun, we do not obtain the full effect of the heat
absorbed, because the receiving surface is at the same time radiating
heat into empty space. The observed increment of temperature is in
short the difference between what is received from the sun and lost
by radiation. The latter quantity is capable of ready determination;
we have only to shade the instrument from the direct rays of the sun,
leaving it exposed to the sky, and we can observe how much it cools
in a certain time. The total effect of the sun’s rays will obviously
be the apparent effect *plus* the cooling effect in an equal time. By
alternate exposure in sun and shade during equal intervals the desired
result may be obtained with considerable accuracy.[235]
[234] *Admiralty Manual of Scientific Enquiry*, 2nd ed. p. 299.
[235] Pouillet, *Taylor’s Scientific Memoirs*, vol. iv. p. 45.
Two quantitative effects were beautifully distinguished in an
experiment of John Canton, devised in 1761 for the purpose of
demonstrating the compressibility of water. He constructed a
thermometer with a large bulb full of water and a short capillary
tube, the part of which above the water was freed from air. Under
these circumstances the water was relieved from the pressure of the
atmosphere, but the glass bulb in bearing that pressure was somewhat
contracted. He next placed the instrument under the receiver of an
air-pump, and on exhausting the air, the water sank in the tube. Having
thus obtained a measure of the effect of atmospheric pressure on the
bulb, he opened the top of the thermometer tube and admitted the air.
The level of the water now sank still more, partly from the pressure
on the bulb being now compensated, and partly from the compression of
the water by the atmospheric pressure. It is obvious that the amount of
the latter effect was approximately the difference of the two observed
depressions.
Not uncommonly the actual phenomenon which we wish to measure is
considerably less than various disturbing effects which enter into
the question. Thus the compressibility of mercury is considerably
less than the expansion of the vessels in which it is measured under
pressure, so that the attention of the experimentalist has chiefly
to be concentrated on the change of magnitude of the vessels. Many
astronomical phenomena, such as the parallax or the proper motions of
the fixed stars, are far less than the errors caused by instrumental
imperfections, or motions arising from precession, nutation, and
aberration. We need not be surprised that astronomers have from time to
time mistaken one phenomenon for another, as when Flamsteed imagined
that he had discovered the parallax of the Pole star.[236]
[236] Baily’s *Account of the Rev. John Flamsteed*, p. 58.
*Methods of Eliminating Error.*
In any particular experiment it is the object of the experimentalist to
measure a single effect only, and he endeavours to obtain that effect
free from interfering effects. If this cannot be, as it seldom or
never can really be, he makes the effect as considerable as possible
compared with the other effects, which he reduces to a minimum, and
treats as noxious errors. Those quantities, which are called *errors*
in one case, may really be most important and interesting phenomena in
another investigation. When we speak of eliminating error we really
mean disentangling the complicated phenomena of nature. The physicist
rightly wishes to treat one thing at a time, but as this object can
seldom be rigorously carried into practice, he has to seek some mode of
counteracting the irrelevant and interfering causes.
The general principle is that a single observation can render known
only a single quantity. Hence, if several different quantitative
effects are known to enter into any investigation, we must have at
least as many distinct observations as there are quantities to be
determined. Every complete experiment will therefore consist in general
of several operations. Guided if possible by previous knowledge of the
causes in action, we must arrange the determinations, so that by a
simple mathematical process we may distinguish the separate quantities.
There appear to be five principal methods by which we may accomplish
this object; these methods are specified below and illustrated in the
succeeding sections.
(1) *The Method of Avoidance.* The physicist may seek for some special
mode of experiment or opportunity of observation, in which the error is
non-existent or inappreciable.
(2) *The Differential Method.* He may find opportunities of observation
when all interfering phenomena remain constant, and only the subject
of observation is at one time present and another time absent; the
difference between two observations then gives its amount.
(3) *The Method of Correction.* He may endeavour to estimate the amount
of the interfering effect by the best available mode, and then make a
corresponding correction in the results of observation.
(4) *The Method of Compensation.* He may invent some mode of
neutralising the interfering cause by balancing against it an exactly
equal and opposite cause of unknown amount.
(5) *The Method of Reversal.* He may so conduct the experiment that
the interfering cause may act in opposite directions, in alternate
observations, the mean result being free from interference.
I. *Method of Avoidance of Error.*
Astronomers seek opportunities of observation when errors will
be as small as possible. In spite of elaborate observations and
long-continued theoretical investigation, it is not practicable to
assign any satisfactory law to the refractive power of the atmosphere.
Although the apparent change of place of a heavenly body produced by
refraction may be more or less accurately calculated yet the error
depends upon the temperature and pressure of the atmosphere, and, when
a ray is highly inclined to the perpendicular, the uncertainty in the
refraction becomes very considerable. Hence astronomers always make
their observations, if possible, when the object is at the highest
point of its daily course, *i.e.* on the meridian. In some kinds of
investigation, as, for instance, in the determination of the latitude
of an observatory, the astronomer is at liberty to select one or
more stars out of the countless number visible. There is an evident
advantage in such a case, in selecting a star which passes close to
the zenith, so that it may be observed almost entirely free from
atmospheric refraction, as was done by Hooke.
Astronomers endeavour to render their clocks as accurate as
possible, by removing the source of variation. The pendulum is
perfectly isochronous so long as its length remains invariable, and
the vibrations are exactly of equal length. They render it nearly
invariable in length, that is in the distance between the centres of
suspension and oscillation, by a compensatory arrangement for the
change of temperature. But as this compensation may not be perfectly
accomplished, some astronomers place their chief controlling clock in
a cellar, or other apartment, where the changes of temperature may
be as slight as possible. At the Paris Observatory a clock has been
placed in the caves beneath the building, where there is no appreciable
difference between the summer and winter temperature.
To avoid the effect of unequal oscillations Huyghens made his beautiful
investigations, which resulted in the discovery that a pendulum, of
which the centre of oscillation moved upon a cycloidal path, would
be perfectly isochronous, whatever the variation in the length of
oscillations. But though a pendulum may be easily rendered in some
degree cycloidal by the use of a steel suspension spring, it is found
that the mechanical arrangements requisite to produce a truly cycloidal
motion introduce more error than they remove. Hence astronomers seek
to reduce the error to the smallest amount by maintaining their clock
pendulums in uniform movement; in fact, while a clock is in good order
and has the same weights, there need be little change in the length
of oscillation. When a pendulum cannot be made to swing uniformly,
as in experiments upon the force of gravity, it becomes requisite to
resort to the third method, and a correction is introduced, calculated
on theoretical grounds from the amount of the observed change in the
length of vibration.
It has been mentioned that the apparent expansion of a liquid by heat,
when contained in a thermometer tube or other vessel, is the difference
between the real expansion of the liquid and that of the containing
vessel. The effects can be accurately distinguished provided that we
can learn the real expansion by heat of any one convenient liquid;
for by observing the apparent expansion of the same liquid in any
required vessel we can by difference learn the amount of expansion of
the vessel due to any given change of temperature. When we once know
the change of dimensions of the vessel, we can of course determine the
absolute expansion of any other liquid tested in it. Thus it became an
all-important object in scientific research to measure with accuracy
the absolute dilatation by heat of some one liquid, and mercury owing
to several circumstances was by far the most suitable. Dulong and
Petit devised a beautiful mode of effecting this by simply avoiding
altogether the effect of the change of size of the vessel. Two upright
tubes full of mercury were connected by a fine tube at the bottom, and
were maintained at two different temperatures. As mercury was free to
flow from one tube to the other by the connecting tube, the two columns
necessarily exerted equal pressures by the principles of hydrostatics.
Hence it was only necessary to measure very accurately by a
cathetometer the difference of level of the surfaces of the two columns
of mercury, to learn the difference of length of columns of equal
hydrostatic pressure, which at once gives the difference of density of
the mercury, and the dilatation by heat. The changes of dimension in
the containing tubes became a matter of entire indifference, and the
length of a column of mercury at different temperatures was measured
as easily as if it had formed a solid bar. The experiment was carried
out by Regnault with many improvements of detail, and the absolute
dilatation of mercury, at temperatures between 0° Cent. and 350°, was
determined almost as accurately as was needful.[237]
[237] Jamin, *Cours de Physique*, vol. ii. pp. 15–28.
The presence of a large and uncertain amount of error may render a
method of experiment valueless. Foucault devised a beautiful experiment
with the pendulum for demonstrating popularly the rotation of the
earth, but it could be of no use for measuring the rotation exactly. It
is impossible to make the pendulum swing in a perfect plane, and the
slightest lateral motion gives it an elliptic path with a progressive
motion of the axis of the ellipse, which disguises and often entirely
overpowers that due to the rotation of the earth.[238]
[238] *Philosophical Magazine*, 1851, 4th Series, vol. ii. *passim*.
Faraday’s laborious experiments on the relation of gravity and
electricity were much obstructed by the fact that it is impossible
to move a large weight of metal without generating currents of
electricity, either by friction or induction. To distinguish the
electricity, if any, directly due to the action of gravity from the
greater quantities indirectly produced was a problem of excessive
difficulty. Baily in his experiments on the density of the earth was
aware of the existence of inexplicable disturbances which have since
been referred with much probability to the action of electricity.[239]
The skill and ingenuity of the experimentalist are often exhausted
in trying to devise a form of apparatus in which such causes of error
shall be reduced to a minimum.
[239] Hearn, *Philosophical Transactions*, 1847, vol. cxxxvii.
pp. 217–221.
In some rudimentary experiments we wish merely to establish the
existence of a quantitative effect without precisely measuring its
amount; if there exist causes of error of which we can neither
render the amount known or inappreciable, the best way is to make
them all negative so that the quantitative effects will be less than
the truth rather than greater. Grove, for instance, in proving that
the magnetisation or demagnetisation of a piece of iron raises its
temperature, took care to maintain the electro-magnet by which the iron
was magnetised at a lower temperature than the iron, so that it would
cool rather than warm the iron by radiation or conduction.[240]
[240] *The Correlation of Physical Forces*, 3rd ed. p. 159.
Rumford’s celebrated experiment to prove that heat was generated out
of mechanical force in the boring of a cannon was subject to the
difficulty that heat might be brought to the cannon by conduction
from neighbouring bodies. It was an ingenious device of Davy to
produce friction by a piece of clock-work resting upon a block of
ice in an exhausted receiver; as the machine rose in temperature
above 32°, it was certain that no heat was received by conduction
from the support.[241] In many other experiments ice may be employed
to prevent the access of heat by conduction, and this device, first
put in practice by Murray,[242] is beautifully employed in Bunsen’s
calorimeter.
[241] *Collected Works of Sir H. Davy*, vol. ii. pp. 12–14. *Elements
of Chemical Philosophy*, p. 94.
[242] *Nicholson’s Journal*, vol. i. p. 241; quoted in *Treatise on
Heat*, Useful Knowledge Society, p. 24.
To observe the true temperature of the air, though apparently so easy,
is really a very difficult matter, because the thermometer is sure to
be affected either by the sun’s rays, the radiation from neighbouring
objects, or the escape of heat into space. These sources of error are
too fluctuating to allow of correction, so that the only accurate
mode of procedure is that devised by Dr. Joule, of surrounding the
thermometer with a copper cylinder ingeniously adjusted to the
temperature of the air, as described by him, so that the effect of
radiation shall be nullified.[243]
[243] Clerk Maxwell, *Theory of Heat*, p. 228. *Proceedings of the
Manchester Philosophical Society*, Nov. 26, 1867, vol. vii. p. 35.
When the avoidance of error is not practicable, it will yet be
desirable to reduce the absolute amount of the interfering error as
much as possible before employing the succeeding methods to correct
the result. As a general rule we can determine a quantity with less
inaccuracy as it is smaller, so that if the error itself be small
the error in determining that error will be of a still lower order
of magnitude. But in some cases the absolute amount of an error is
of no consequence, as in the index error of a divided circle, or the
difference between a chronometer and astronomical time. Even the rate
at which a clock gains or loses is a matter of little importance
provided it remain constant, so that a sure calculation of its amount
can be made.
2. *Differential Method.*
When we cannot avoid the existence of error, we can often resort
with success to the second mode by measuring phenomena under such
circumstances that the error shall remain very nearly the same in all
the observations, and neutralise itself as regards the purposes in
view. This mode is available whenever we want a difference between
quantities and not the absolute quantity of either. The determination
of the parallax of the fixed stars is exceedingly difficult, because
the amount of parallax is far less than most of the corrections for
atmospheric refraction, nutation, aberration, precession, instrumental
irregularities, &c., and can with difficulty be detected among these
phenomena of various magnitude. But, as Galileo long ago suggested,
all such difficulties would be avoided by the differential observation
of stars, which, though apparently close together, are really far
separated on the line of sight. Two such stars in close apparent
proximity will be subject to almost exactly equal errors, so that all
we need do is to observe the apparent change of place of the nearer
star as referred to the more distant one. A good telescope furnished
with an accurate micrometer is alone needed for the application of
the method. Huyghens appears to have been the first observer who
actually tried to employ the method practically, but it was not until
1835 that the improvement of telescopes and micrometers enabled
Struve to detect in this way the parallax of the star α Lyræ. It is
one of the many advantages of the observation of transits of Venus
for the determination of the solar parallax that the refraction of
the atmosphere affects in an exactly equal degree the planet and
the portion of the sun’s face over which it is passing. Thus the
observations are strictly of a differential nature.
By the process of substitutive weighing it is possible to ascertain
the equality or inequality of two weights with almost perfect freedom
from error. If two weights A and B be placed in the scales of the
best balance we cannot be sure that the equilibrium of the beam
indicates exact equality, because the arms of the beam may be unequal
or unbalanced. But if we take B out and put another weight C in, and
equilibrium still exists, it is apparent that the same causes of
erroneous weighing exist in both cases, supposing that the balance has
not been disarranged; B then must be exactly equal to C, since it has
exactly the same effect under the same circumstances. In like manner it
is a general rule that, if by any uniform mechanical process we get a
copy of an object, it is unlikely that this copy will be precisely the
same as the original in magnitude and form, but two copies will equally
diverge from the original, and will therefore almost exactly resemble
each other.
Leslie’s Differential Thermometer[244] was well adapted to the
experiments for which it was invented. Having two equal bulbs any
alteration in the temperature of the air will act equally by conduction
on each and produce no change in the indications of the instrument.
Only that radiant heat which is purposely thrown upon one of the bulbs
will produce any effect. This thermometer in short carries out the
principle of the differential method in a mechanical manner.
[244] Leslie, *Inquiry into the Nature of Heat*, p. 10.
3. *Method of Correction.*
Whenever the result of an experiment is affected by an interfering
cause to a calculable amount, it is sufficient to add or subtract this
amount. We are said to correct observations when we thus eliminate
what is due to extraneous causes, although of course we are only
separating the correct effects of several agents. The variation in the
height of the barometer is partly due to the change of temperature,
but since the coefficient of absolute dilatation of mercury has been
exactly determined, as already described (p. 341), we have only to make
calculations of a simple character, or, what is better still, tabulate
a series of such calculations for general use, and the correction for
temperature can be made with all desired accuracy. The height of the
mercury in the barometer is also affected by capillary attraction,
which depresses it by a constant amount depending mainly on the
diameter of the tube. The requisite corrections can be estimated with
accuracy sufficient for most purposes, more especially as we can check
the correctness of the reading of a barometer by comparison with a
standard barometer, and introduce if need be an index error including
both the error in the affixing of the scale and the effect due to
capillarity. But in constructing the standard barometer itself we must
take greater precautions; the capillary depression depends somewhat
upon the quality of the glass, the absence of air, and the perfect
cleanliness of the mercury, so that we cannot assign the exact amount
of the effect. Hence a standard barometer is constructed with a wide
tube, sometimes even an inch in diameter, so that the capillary effect
may be rendered almost zero.[245] Gay-Lussac made barometers in the
form of a uniform siphon tube, so that the capillary forces acting at
the upper and lower surfaces should balance and destroy each other;
but the method fails in practice because the lower surface, being
open to the air, becomes sullied and subject to a different force of
capillarity.
[245] Jevons, Watts’ *Dictionary of Chemistry*, vol. i. pp. 513–515.
In mechanical experiments friction is an interfering condition, and
drains away a portion of the energy intended to be operated upon in a
definite manner. We should of course reduce the friction in the first
place to the lowest possible amount, but as it cannot be altogether
prevented, and is not calculable with certainty from any general
laws, we must determine it separately for each apparatus by suitable
experiments. Thus Smeaton, in his admirable but almost forgotten
researches concerning water-wheels, eliminated friction in the most
simple manner by determining by trial what weight, acting by a cord
and roller upon his model water-wheel, would make it turn without
water as rapidly as the water made it turn. In short, he ascertained
what weight concurring with the water would exactly compensate for the
friction.[246] In Dr. Joule’s experiments to determine the mechanical
equivalent of heat by the condensation of air, a considerable amount
of heat was produced by friction of the condensing pump, and a small
portion by stirring the water employed to absorb the heat. This heat of
friction was measured by simply repeating the experiment in an exactly
similar manner except that no condensation was effected, and observing
the change of temperature then produced.[247]
[246] *Philosophical Transactions*, vol. li. p. 100.
[247] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 372.
We may describe as *test experiments* any in which we perform
operations not intended to give the quantity of the principal
phenomenon, but some quantity which would otherwise remain as an
error in the result. Thus in astronomical observations almost every
instrumental error may be avoided by increasing the number of
observations and distributing them in such a manner as to produce in
the final mean as much error in one way as in the other. But there
is one source of error, first discovered by Maskelyne, which cannot
be thus avoided, because it affects all observations in the same
direction and to the same average amount, namely the Personal Error of
the observer or the inclination to record the passage of a star across
the wires of the telescope a little too soon or a little too late.
This personal error was first carefully described in the *Edinburgh
Journal of Science*, vol. i. p. 178. The difference between the
judgment of observers at the Greenwich Observatory usually varies from
1/100 to 1/3 of a second, and remains pretty constant for the same
observers.[248] One practised observer in Sir George Airy’s pendulum
experiments recorded all his time observations half a second too early
on the average as compared with the chief observer.[249] In some
observers it has amounted to seven or eight-tenths of a second.[250]
De Morgan appears to have entertained the opinion that this source of
error was essentially incapable of elimination or correction.[251]
But it seems clear, as I suggested without knowing what had been
done,[252] that this personal error might be determined absolutely with
any desirable degree of accuracy by test experiments, consisting in
making an artificial star move at a considerable distance and recording
by electricity the exact moment of its passage over the wire. This
method has in fact been successfully employed in Leyden, Paris, and
Neuchatel.[253] More recently, observers were trained for the Transit
of Venus Expeditions by means of a mechanical model representing the
motion of Venus over the sun, this model being placed at a little
distance and viewed through a telescope, so that differences in the
judgments of different observers would become apparent. It seems likely
that tests of this nature might be employed with advantage in other
cases.
[248] *Greenwich Observations for* 1866, p. xlix.
[249] *Philosophical Transactions*, 1856, p. 309.
[250] Penny *Cyclopædia*, art. *Transit*, vol. xxv. pp. 129, 130.
[251] Ibid. art. *Observation*, p. 390.
[252] *Nature*, vol. i. p. 85.
[253] *Nature*, vol. i. p 337. See references to the Memoirs
describing the method.
Newton employed the pendulum for making experiments on the impact of
balls. Two balls were hung in contact, and one of them, being drawn
aside through a measured arc, was then allowed to strike the other,
the arcs of vibration giving sufficient data for calculating the
distribution of energy at the moment of impact. The resistance of the
air was an interfering cause which he estimated very simply by causing
one of the balls to make several complete vibrations without impact
and then marking the reduction in the lengths of the arcs, a proper
fraction of which reduction was added to each of the other arcs of
vibration when impact took place.[254]
[254] *Principia*, Book I. Law III. Corollary VI. Scholium. Motte’s
translation, vol. i. p. 33.
The exact definition of the standard of length is one of the most
important, as it is one of the most difficult questions in physical
science, and the different practice of different nations introduces
needless confusion. Were all standards constructed so as to give
the true length at a fixed uniform temperature, for instance the
freezing-point, then any two standards could be compared without the
interference of temperature by bringing them both to exactly the same
fixed temperature. Unfortunately the French metre was defined by a
bar of platinum at 0°C, while our yard was defined by a bronze bar at
62°F. It is quite impossible, then, to make a comparison of the yard
and metre without the introduction of a correction, either for the
expansion of platinum or bronze, or both. Bars of metal differ too so
much in their rates of expansion according to their molecular condition
that it is dangerous to infer from one bar to another.
When we come to use instruments with great accuracy there are many
minute sources of error which must be guarded against. If a thermometer
has been graduated when perpendicular, it will read somewhat
differently when laid flat, as the pressure of a column of mercury is
removed from the bulb. The reading may also be somewhat altered if it
has recently been raised to a higher temperature than usual, if it be
placed under a vacuous receiver, or if the tube be unequally heated
as compared with the bulb. For these minute causes of error we may
have to introduce troublesome corrections, unless we adopt the simple
precaution of using the thermometer in circumstances of position, &c.,
exactly similar to those in which it was graduated. There is no end to
the number of minute corrections which may ultimately be required. A
large number of experiments on gases, standard weights and measures,
&c., depend upon the height of the barometer; but when experiments
in different parts of the world are compared together we ought as a
further refinement to take into account the varying force of gravity,
which even between London and Paris makes a difference of ·008 inch of
mercury.
The measurement of quantities of heat is a matter of great difficulty,
because there is no known substance impervious to heat, and the problem
is therefore as difficult as to measure liquids in porous vessels.
To determine the latent heat of steam we must condense a certain
amount of the steam in a known weight of water, and then observe the
rise of temperature of the water. But while we are carrying out the
experiment, part of the heat will escape by radiation and conduction
from the condensing vessel or calorimeter. We may indeed reduce the
loss of heat by using vessels with double sides and bright surfaces,
surrounded with swans-down wool or other non-conducting materials; and
we may also avoid raising the temperature of the water much above that
of the surrounding air. Yet we cannot by any such means render the
loss of heat inconsiderable. Rumford ingeniously proposed to reduce
the loss to zero by commencing the experiment when the temperature
of the calorimeter is as much below that of the air as it is at the
end of the experiment above it. Thus the vessel will first gain and
then lose by radiation and conduction, and these opposite errors will
approximately balance each other. But Regnault has shown that the loss
and gain do not proceed by exactly the same laws, so that in very
accurate investigations Rumford’s method is not sufficient. There
remains the method of correction which was beautifully carried out by
Regnault in his determination of the latent heat of steam. He employed
two calorimeters, made in exactly the same way and alternately used to
condense a certain amount of steam, so that while one was measuring
the latent heat, the other calorimeter was engaged in determining
the corrections to be applied, whether on account of radiation and
conduction from the vessel or on account of heat reaching the vessel by
means of the connecting pipes.[255]
[255] Graham’s *Chemical Reports and Memoirs*, Cavendish Society, pp.
247, 268, &c.
4. *Method of Compensation.*
There are many cases in which a cause of error cannot conveniently be
rendered null, and is yet beyond the reach of the third method, that
of calculating the requisite correction from independent observations.
The magnitude of an error may be subject to continual variations, on
account of change of weather, or other fickle circumstances beyond
our control. It may either be impracticable to observe the variation
of those circumstances in sufficient detail, or, if observed, the
calculation of the amount of error may be subject to doubt. In these
cases, and only in these cases, it will be desirable to invent some
artificial mode of counterpoising the variable error against an equal
error subject to exactly the same variation.
We cannot weigh an object with great accuracy unless we make a
correction for the weight of the air displaced by the object, and add
this to the apparent weight. In very accurate investigations relating
to standard weights, it is usual to note the barometer and thermometer
at the time of making a weighing, and, from the measured bulks of
the objects compared, to calculate the weight of air displaced; the
third method in fact is adopted. To make these calculations in the
frequent weighings requisite in chemical analysis would be exceedingly
laborious, hence the correction is usually neglected. But when the
chemist wishes to weigh gas contained in a large glass globe for the
purpose of determining its specific gravity, the correction becomes of
much importance. Hence chemists avoid at once the error, and the labour
of correcting it, by attaching to the opposite scale of the balance a
dummy sealed glass globe of equal capacity to that containing the gas
to be weighed, noting only the difference of weight when the operating
globe is full and empty. The correction, being the same for both
globes, may be entirely neglected.[256]
[256] Regnault’s *Cours Elémentaire de Chimie*, 1851, vol i. p. 141.
A device of nearly the same kind is employed in the construction of
galvanometers which measure the force of an electric current by the
deflection of a suspended magnetic needle. The resistance of the needle
is partly due to the directive influence of the earth’s magnetism, and
partly to the torsion of the thread. But the former force may often be
inconveniently great as well as troublesome to determine for different
inclinations. Hence it is customary to connect together two equally
magnetised needles, with their poles pointing in opposite directions,
one needle being within and another without the coil of wire. As
regards the earth’s magnetism, the needles are now *astatic* or
indifferent, the tendency of one needle towards the pole being balanced
by that of the other.
An elegant instance of the elimination of a disturbing force by
compensation is found in Faraday’s researches upon the magnetism of
gases. To observe the magnetic attraction or repulsion of a gas seems
impossible unless we enclose the gas in an envelope, probably best made
of glass. But any such envelope is sure to be more or less affected
by the magnet, so that it becomes difficult to distinguish between
three forces which enter into the problem, namely, the magnetism of
the gas in question, that of the envelope, and that of the surrounding
atmospheric air. Faraday avoided all difficulties by employing two
equal and similar glass tubes connected together, and so suspended from
the arm of a torsion balance that the tubes were in similar parts of
the magnetic field. One tube being filled with nitrogen and the other
with oxygen, it was found that the oxygen seemed to be attracted and
the nitrogen repelled. The suspending thread of the balance was then
turned until the force of torsion restored the tubes to their original
places, where the magnetism of the tubes as well as that of the
surrounding air, being the same and in the opposite directions upon the
two tubes, could not produce any interference. The force required to
restore the tubes was measured by the amount of torsion of the thread,
and it indicated correctly the difference between the attractive powers
of oxygen and nitrogen. The oxygen was then withdrawn from one of the
tubes, and a second experiment made, so as to compare a vacuum with
nitrogen. No force was now required to maintain the tubes in their
places, so that nitrogen was found to be, approximately speaking,
indifferent to the magnet, that is, neither magnetic nor diamagnetic,
while oxygen was proved to be positively magnetic.[257] It required
the highest experimental skill on the part of Faraday and Tyndall, to
distinguish between what is apparent and real in magnetic attraction
and repulsion.
[257] Tyndall’s *Faraday*, pp. 114, 115.
Experience alone can finally decide when a compensating arrangement
is conducive to accuracy. As a general rule mechanical compensation
is the last resource, and in the more accurate observations it is
likely to introduce more uncertainty than it removes. A multitude
of instruments involving mechanical compensation have been devised,
but they are usually of an unscientific character,[258] because the
errors compensated can be more accurately determined and allowed for.
But there are exceptions to this rule, and it seems to be proved that
in the delicate and tiresome operation of measuring a base line,
invariable bars, compensated for expansion by heat, give the most
accurate results. This arises from the fact that it is very difficult
to determine accurately the temperature of the measuring bars under
varying conditions of weather and manipulation.[259] Again, the last
refinement in the measurement of time at Greenwich Observatory depends
upon mechanical compensation. Sir George Airy, observing that the
standard clock increased its losing rate 0·30 second for an increase of
one inch in atmospheric pressure, placed a magnet moved by a barometer
in such a position below the pendulum, as almost entirely to neutralise
this cause of irregularity. The thorough remedy, however, would be to
remove the cause of error altogether by placing the clock in a vacuous
case.
[258] See, for instance, the Compensated Sympiesometer,
*Philosophical Magazine*, 4th Series, vol. xxxix. p. 371.
[259] Grant, *History of Physical Astronomy*, pp. 146, 147.
We thus see that the choice of one or other mode of eliminating an
error depends entirely upon circumstances and the object in view; but
we may safely lay down the following conclusions. First of all, seek
to avoid the source of error altogether if it can be conveniently
done; if not, make the experiment so that the error may be as small,
but more especially as constant, as possible. If the means are at hand
for determining its amount by calculation from other experiments and
principles of science, allow the error to exist and make a correction
in the result. If this cannot be accurately done or involves too
much labour for the purposes in view, then throw in a counteracting
error which shall as nearly as possible be of equal amount in all
circumstances with that to be eliminated. There yet remains, however,
one important method, that of Reversal, which will form an appropriate
transition to the succeeding chapters on the Method of Mean Results and
the Law of Error.
5. *Method of Reversal.*
The fifth method of eliminating error is most potent and satisfactory
when it can be applied, but it requires that we shall be able to
reverse the apparatus and mode of procedure, so as to make the
interfering cause act alternately in opposite directions. If we can
get two experimental results, one of which is as much too great as
the other is too small, the error is equal to half the difference,
and the true result is the mean of the two apparent results. It is an
unavoidable defect of the chemical balance, for instance, that the
points of suspension of the pans cannot be fixed at exactly equal
distances from the centre of suspension of the beam. Hence two weights
which seem to balance each other will never be quite equal in reality.
The difference is detected by reversing the weights, and it may be
estimated by adding small weights to the deficient side to restore
equilibrium, and then taking as the true weight the geometric mean
of the two apparent weights of the same object. If the difference is
small, the arithmetic mean, that is half the sum, may be substituted
for the geometric mean, from which it will not appreciably differ.
This method of reversal is most extensively employed in practical
astronomy. The apparent elevation of a heavenly body is observed by a
telescope moving upon a divided circle, upon which the inclination of
the telescope is read off. Now this reading will be erroneous if the
circle and the telescope have not accurately the same centre. But if we
read off at the same time both ends of the telescope, the one reading
will be about as much too small as the other is too great, and the
mean will be nearly free from error. In practice the observation is
differently conducted, but the principle is the same; the telescope is
fixed to the circle, which moves with it, and the angle through which
it moves is read off at three, six, or more points, disposed at equal
intervals round the circle. The older astronomers, down even to the
time of Flamsteed, were accustomed to use portions only of a divided
circle, generally quadrants, and Römer made a vast improvement when he
introduced the complete circle.
The transit circle, employed to determine the meridian passage of
heavenly bodies, is so constructed that the telescope and the axis
bearing it, in fact the whole moving part of the instrument, can be
taken out of the bearing sockets and turned over, so that what was
formerly the western pivot becomes the eastern one, and *vice versâ*.
It is impossible that the instrument could have been so perfectly
constructed, mounted, and adjusted that the telescope should point
exactly to the meridian, but the effect of the reversal is that it will
point as much to the west in one position as it does to the east in the
other, and the mean result of observations in the two positions must be
free from such cause of error.
The accuracy with which the inclination of the compass needle can be
determined depends almost entirely on the method of reversal. The
dip needle consists of a bar of magnetised steel, suspended somewhat
like the beam of a delicate balance on a slender axis passing through
the centre of gravity of the bar, so that it is at liberty to rest
in that exact degree of inclination in the magnetic meridian which
the magnetism of the earth induces. The inclination is read off
upon a vertical divided circle, but to avoid error arising from the
centring of the needle and circle, both ends are read, and the mean
of the results is taken. The whole instrument is now turned carefully
round through 180°, which causes the needle to assume a new position
relatively to the circle and gives two new readings, in which any error
due to the wrong position of the zero of the division will be reversed.
As the axis of the needle may not be exactly horizontal, it is now
reversed in the same manner as the transit instrument, the end of the
axis which formerly pointed east being made to point west, and a new
set of four readings is taken.
Finally, error may arise from the axis not passing accurately through
the centre of gravity of the bar, and this error can only be detected
and eliminated on changing the magnetic poles of the bar by the
application of a strong magnet. The error is thus made to act in
opposite directions. To ensure all possible accuracy each reversal
ought to be combined with each other reversal, so that the needle will
be observed in eight different positions by sixteen readings, the mean
of the whole of which will give the required inclination free from all
eliminable errors.[260]
[260] Quetelet, *Sur la Physique du Globe*, p. 174. Jamin, *Cours de
Physique*, vol. i. p. 504.
There are certain cases in which a disturbing cause can with ease
be made to act in opposite directions, in alternate observations,
so that the mean of the results will be free from disturbance. Thus
in direct experiments upon the velocity of sound in passing through
the air between stations two or three miles apart, the wind is a
cause of error. It will be well, in the first place, to choose a
time for the experiment when the air is very nearly at rest, and the
disturbance slight, but if at the same moment signal sounds be made
at each station and observed at the other, two sounds will be passing
in opposite directions through the same body of air and the wind will
accelerate one sound almost exactly as it retards the other. Again, in
trigonometrical surveys the apparent height of a point will be affected
by atmospheric refraction and the curvature of the earth. But if in
the case of two points the apparent elevation of each as seen from
the other be observed, the corrections will be the same in amount,
but reversed in direction, and the mean between the two apparent
differences of altitude will give the true difference of level.
In the next two chapters we really pursue the Method of Reversal into
more complicated applications.
CHAPTER XVI.
THE METHOD OF MEANS.
All results of the measurement of continuous quantity can be only
approximately true. Were this assertion doubted, it could readily be
proved by direct experience. If any person, using an instrument of
the greatest precision, makes and registers successive observations
in an unbiassed manner, it will almost invariably be found that the
results differ from each other. When we operate with sufficient care
we cannot perform so simple an experiment as weighing an object in a
good balance without getting discrepant numbers. Only the rough and
careless experimenter will think that his observations agree, but
in reality he will be found to overlook the differences. The most
elaborate researches, such as those undertaken in connection with
standard weights and measures, always render it apparent that complete
coincidence is out of the question, and that the more accurate our
modes of observation are rendered, the more numerous are the sources
of minute error which become apparent. We may look upon the existence
of error in all measurements as the normal state of things. It is
absolutely impossible to eliminate separately the multitude of small
disturbing influences, except by balancing them off against each other.
Even in drawing a mean it is to be expected that we shall come near
the truth rather than exactly to it. In the measurement of continuous
quantity, absolute coincidence, if it seems to occur, must be only
apparent, and is no indication of precision. It is one of the most
embarrassing things we can meet when experimental results agree
too closely. Such coincidences should raise our suspicion that the
apparatus in use is in some way restricted in its operation, so as
not really to give the true result at all, or that the actual results
have not been faithfully recorded by the assistant in charge of the
apparatus.
If then we cannot get twice over exactly the same result, the question
arises, How can we ever attain the truth or select the result which may
be supposed to approach most nearly to it? The quantity of a certain
phenomenon is expressed in several numbers which differ from each
other; no more than one of them at the most can be true, and it is more
probable that they are all false. It may be suggested, perhaps, that
the observer should select the one observation which he judged to be
the best made, and there will often doubtless be a feeling that one
or more results were satisfactory, and the others less trustworthy.
This seems to have been the course adopted by the early astronomers.
Flamsteed, when he had made several observations of a star, probably
chose in an arbitrary manner that which seemed to him nearest to the
truth.[261]
[261] Baily’s *Account of Flamsteed*, p. 376.
When Horrocks selected for his estimate of the sun’s semi-diameter a
mean between the results of Kepler and Tycho, he professed not to do
it from any regard to the idle adage, “Medio tutissimus ibis,” but
because he thought it from his own observations to be correct.[262] But
this method will not apply at all when the observer has made a number
of measurements which are equally good in his opinion, and it is quite
apparent that in using an instrument or apparatus of considerable
complication the observer will not necessarily be able to judge whether
slight causes have affected its operation or not.
[262] *The Transit of Venus across the Sun*, by Horrocks, London,
1859, p. 146.
In this question, as indeed throughout inductive logic, we deal only
with probabilities. There is no infallible mode of arriving at the
absolute truth, which lies beyond the reach of human intellect, and
can only be the distant object of our long-continued and painful
approximations. Nevertheless there is a mode pointed out alike by
common sense and the highest mathematical reasoning, which is more
likely than any other, as a general rule, to bring us near the truth.
The ἄριστον μέτρον, or the *aurea mediocritas*, was highly esteemed
in the ancient philosophy of Greece and Rome; but it is not probable
that any of the ancients should have been able clearly to analyse and
express the reasons why they advocated the *mean* as the safest course.
But in the last two centuries this apparently simple question of the
mean has been found to afford a field for the exercise of the utmost
mathematical skill. Roger Cotes, the editor of the *Principia*, appears
to have had some insight into the value of the mean; but profound
mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange,
Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and
others, have hardly exhausted the subject.
*Several uses of the Mean Result.*
The elimination of errors of unknown sources, is almost always
accomplished by the simple arithmetical process of taking the *mean*,
or, as it is often called, the *average* of several discrepant numbers.
To take an average is to add the several quantities together, and
divide by the number of quantities thus added, which gives a quotient
lying among, or in the *middle* of, the several quantities. Before
however inquiring fully into the grounds of this procedure, it is
essential to observe that this one arithmetical process is really
applied in at least three different cases, for different purposes, and
upon different principles, and we must take great care not to confuse
one application of the process with another. A *mean result*, then, may
have any one of the following significations.
(1) It may give a merely representative number, expressing the general
magnitude of a series of quantities, and serving as a convenient mode
of comparing them with other series of quantities. Such a number is
properly called *The fictitious mean* or *The average result*.
(2) It may give a result approximately free from disturbing quantities,
which are known to affect some results in one direction, and other
results equally in the opposite direction. We may say that in this case
we get a *Precise mean result*.
(3) It may give a result more or less free from unknown and uncertain
errors; this we may call the *Probable mean result*.
Of these three uses of the mean the first is entirely different in
nature from the two last, since it does not yield an approximation
to any natural quantity, but furnishes us with an arithmetic result
comparing the aggregate of certain quantities with their number. The
third use of the mean rests entirely upon the theory of probability,
and will be more fully considered in a later part of this chapter. The
second use is closely connected, or even identical with, the Method of
Reversal already described, but it will be desirable to enter somewhat
fully into all the three employments of the same arithmetical process.
*The Mean and the Average.*
Much confusion exists in the popular, or even the scientific employment
of the terms *mean* and *average*, and they are commonly taken as
synonymous. It is necessary to ascertain carefully what significations
we ought to attach to them. The English word *mean* is equivalent to
*medium*, being derived, perhaps through the French *moyen*, from the
Latin *medius*, which again is undoubtedly kindred with the Greek
μεσος. Etymologists believe, too, that this Greek word is connected
with the preposition μετα, the German *mitte*, and the true English
*mid* or *middle*; so that after all the *mean* is a technical term
identical in its root with the more popular equivalent *middle*.
If we inquire what is the mean in a mathematical point of view, the
true answer is that there are several or many kinds of means. The old
arithmeticians recognised ten kinds, which are stated by Boethius, and
an eleventh was added by Jordanus.[263]
[263] De Morgan, Supplement to the *Penny Cyclopædia*, art. *Old
Appellations of Numbers*.
The *arithmetic mean* is the one by far the most commonly denoted by
the term, and that which we may understand it to signify in the absence
of any qualification. It is the sum of a series of quantities divided
by their number, and may be represented by the formula 1/2(*a* + *b*).
But there is also the *geometric mean*, which is the square root of the
product, √(*a* × *b*), or that quantity the logarithm of which
is the arithmetic mean of the logarithms of the quantities. There is
also the *harmonic mean*, which is the reciprocal of the arithmetic
mean of the reciprocals of the quantities. Thus if *a* and *b* be the
quantities, as before, their reciprocals are 1/*a* and 1/*b*, the
mean of which is 1/2 (1/*a* + 1/*b*), and the reciprocal again is
(2*ab*)/(*a* + *b*), which is the harmonic mean. Other kinds of means
might no doubt be invented for particular purposes, and we might apply
the term, as De Morgan pointed out,[264] to any quantity a function of
which is equal to a function of two or more other quantities, and is
such that the interchange of these latter quantities among themselves
will make no alteration in the value of the function. Symbolically, if
Φ(*y*, *y*, *y* ....) = Φ(*x*_{1}, *x*_{2}, *x*_{3} ....), then *y* is
a kind of mean of the quantities, *x*_{1}, *x*_{2}, &c.
[264] *Penny Cyclopædia*, art. *Mean*.
The geometric mean is necessarily adopted in certain cases. When we
estimate the work done against a force which varies inversely as
the square of the distance from a fixed point, the mean force is
the geometric mean between the forces at the beginning and end of
the path. When in an imperfect balance, we reverse the weights to
eliminate error, the true weight will be the geometric mean of the two
apparent weights. In almost all the calculations of statistics and
commerce the geometric mean ought, strictly speaking, to be used. If a
commodity rises in price 100 per cent. and another remains unaltered,
the mean rise of a price is not 50 per cent. because the ratio
150 : 200 is not the same as 100 : 150. The mean ratio is as unity to
√(1·00 × 2·00) or 1 to 1·41. The difference between the three
kinds of means in such a case[265] is very considerable; while the
rise of price estimated by the Arithmetic mean would be 50 per cent.
it would be only 41 and 33 per cent. respectively according to the
Geometric and Harmonic means.
[265] Jevons, *Journal of the Statistical Society*, June 1865,
vol. xxviii, p. 296.
In all calculations concerning the average rate of progress of a
community, or any of its operations, the geometric mean should be
employed. For if a quantity increases 100 per cent. in 100 years, it
would not on the average increase 10 per cent. in each ten years, as
the 10 per cent. would at the end of each decade be calculated upon
larger and larger quantities, and give at the end of 100 years much
more than 100 per cent., in fact as much as 159 per cent. The true mean
rate in each decade would be ^{10}√2 or about 1·07, that is, the
increase would be about 7 per cent. in each ten years. But when the
quantities differ very little, the arithmetic and geometric means are
approximately the same. Thus the arithmetic mean of 1·000 and 1·001
is 1·0005, and the geometric mean is about 1·0004998, the difference
being of an order inappreciable in almost all scientific and practical
matters. Even in the comparison of standard weights by Gauss’ method
of reversal, the arithmetic mean may usually be substituted for the
geometric mean which is the true result.
Regarding the mean in the absence of express qualification to the
contrary as the common arithmetic mean, we must still distinguish
between its two uses where it gives with more or less accuracy and
probability a really existing quantity, and where it acts as a mere
representative of other quantities. If I make many experiments to
determine the atomic weight of an element, there is a certain number
which I wish to approximate to, and the mean of my separate results
will, in the absence of any reasons to the contrary, be the most
probable approximate result. When we determine the mean density of
the earth, it is not because any part of the earth is of that exact
density; there may be no part exactly corresponding to the mean
density, and as the crust of the earth has only about half the mean
density, the internal matter of the globe must of course be above the
mean. Even the density of a homogeneous substance like carbon or gold
must be regarded as a mean between the real density of its atoms, and
the zero density of the intervening vacuous space.
The very different signification of the word “mean” in these two uses
was fully explained by Quetelet,[266] and the importance of the
distinction was pointed out by Sir John Herschel in reviewing his
work.[267] It is much to be desired that scientific men would mark the
difference by using the word *mean* only in the former sense when it
denotes approximation to a definite existing quantity; and *average*,
when the mean is only a fictitious quantity, used for convenience
of thought and expression. The etymology of this word “average” is
somewhat obscure; but according to De Morgan[268] it comes from
*averia*, “havings or possessions,” especially applied to farm stock.
By the accidents of language *averagium* came to mean the labour of
farm horses to which the lord was entitled, and it probably acquired in
this manner the notion of distributing a whole into parts, a sense in
which it was early applied to maritime averages or contributions of the
other owners of cargo to those whose goods have been thrown overboard
or used for the safety of the vessel.
[266] *Letters on the Theory of Probabilities*, transl. by Downes,
Part ii.
[267] Herschel’s *Essays*, &c. pp. 404, 405.
[268] *On the Theory of Errors of Observations, Cambridge
Philosophical Transactions*, vol. x. Part ii. 416.
*On the Average or Fictitious Mean.*
Although the average when employed in its proper sense of a fictitious
mean, represents no really existing quantity, it is yet of the
highest scientific importance, as enabling us to conceive in a single
result a multitude of details. It enables us to make a hypothetical
simplification of a problem, and avoid complexity without committing
error. The weight of a body is the sum of the weights of infinitely
small particles, each acting at a different place, so that a mechanical
problem resolves itself, strictly speaking, into an infinite number
of distinct problems. We owe to Archimedes the first introduction of
the beautiful idea that one point may be discovered in a gravitating
body such that the weight of all the particles may be regarded as
concentrated in that point, and yet the behaviour of the whole body
will be exactly represented by the behaviour of this heavy point. This
Centre of Gravity may be within the body, as in the case of a sphere,
or it may be in empty space, as in the case of a ring. Any two bodies,
whether connected or separate, may be conceived as having a centre of
gravity, that of the sun and earth lying within the sun and only 267
miles from its centre.
Although we most commonly use the notion of a centre or average point
with regard to gravity, the same notion is applicable to other cases.
Terrestrial gravity is a case of approximately parallel forces, and
the centre of gravity is but a special case of the more general Centre
of Parallel Forces. Wherever a number of forces of whatever amount
act in parallel lines, it is possible to discover a point at which
the algebraic sum of the forces may be imagined to act with exactly
the same effect. Water in a cistern presses against the side with a
pressure varying according to the depth, but always in a direction
perpendicular to the side. We may then conceive the whole pressure
as exerted on one point, which will be one-third from the bottom of
the cistern, and may be called the Centre of Pressure. The Centre
of Oscillation of a pendulum, discovered by Huyghens, is that point
at which the whole weight of the pendulum may be considered as
concentrated, without altering the time of oscillation (p. 315). When
one body strikes another the Centre of Percussion is that point in
the striking body at which all its mass might be concentrated without
altering the effect of the stroke. In position the Centre of Percussion
does not differ from the Centre of Oscillation. Mathematicians have
also described the Centre of Gyration, the Centre of Conversion, the
Centre of Friction, &c.
We ought carefully to distinguish between those cases in which an
*invariable* centre can be assigned, and those in which it cannot.
In perfect strictness, there is no such thing as a true invariable
centre of gravity. As a general rule a body is capable of possessing an
invariable centre only for perfectly parallel forces, and gravity never
does act in absolutely parallel lines. Thus, as usual, we find that our
conceptions are only hypothetically correct, and only approximately
applicable to real circumstances. There are indeed certain geometrical
forms called *Centrobaric*,[269] such that a body of that shape would
attract another exactly as if the mass were concentrated at the centre
of gravity, whether the forces act in a parallel manner or not.
Newton showed that uniform spheres of matter have this property,
and this truth proved of the greatest importance in simplifying his
calculations. But it is after all a purely hypothetical truth, because
we can nowhere meet with, nor can we construct, a perfectly spherical
and homogeneous body. The slightest irregularity or protrusion from
the surface will destroy the rigorous correctness of the assumption.
The spheroid, on the other hand, has no invariable centre at which its
mass may always be regarded as concentrated. The point from which its
resultant attraction acts will move about according to the distance
and position of the other attracting body, and it will only coincide
with the centre as regards an infinitely distant body whose attractive
forces may be considered as acting in parallel lines.
[269] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i.
p. 394.
Physicists speak familiarly of the poles of a magnet, and the term
may be used with convenience. But, if we attach any definite meaning
to the word, the poles are not the ends of the magnet, nor any fixed
points within, but the variable points from which the resultants of
all the forces exerted by the particles in the bar upon exterior
magnetic particles may be considered as acting. The poles are, in
short, Centres of Magnetic Forces; but as those forces are never
really parallel, these centres will vary in position according to
the relative place of the object attracted. Only when we regard the
magnet as attracting a very distant, or, strictly speaking, infinitely
distant particle, do its centres become fixed points, situated in short
magnets approximately at one-sixth of the whole length from each end of
the bar. We have in the above instances of centres or poles of force
sufficient examples of the mode in which the Fictitious Mean or Average
is employed in physical science.
*The Precise Mean Result.*
We now turn to that mode of employing the mean result which is
analogous to the method of reversal, but which is brought into practice
in a most extensive manner throughout many branches of physical
science. We find the simplest possible case in the determination of
the latitude of a place by observations of the Pole-star. Tycho Brahe
suggested that if the elevation of any circumpolar star were observed
at its higher and lower passages across the meridian, half the sum of
the elevations would be the latitude of the place, which is equal to
the height of the pole. Such a star is as much above the pole at its
highest passage, as it is below at its lowest, so that the mean must
necessarily give the height of the pole itself free from doubt, except
as regards incidental errors. The Pole-star is usually selected for the
purpose of such observations because it describes the smallest circle,
and is thus on the whole least affected by atmospheric refraction.
Whenever several causes are in action, each of which at one time
increases and at another time decreases the joint effect by equal
quantities, we may apply this method and disentangle the effects. Thus
the solar and lunar tides roll on in almost complete independence of
each other. When the moon is new or full the solar tide coincides, or
nearly so, with that caused by the moon, and the joint effect is the
sum of the separate effects. When the moon is in quadrature, or half
full, the two tides are acting in opposition, one raising and the other
depressing the water, so that we observe only the difference of the
effects. We have in fact--
Spring tide = lunar tide + solar tide;
Neap tide = lunar tide - solar tide.
We have only then to add together the heights of the maximum spring
tide and the minimum neap tide, and half the sum is the true height of
the lunar tide. Half the difference of the spring and neap tides on the
other hand gives the solar tide.
Effects of very small amount may be detected with great approach to
certainty among much greater fluctuations, provided that we have
a series of observations sufficiently numerous and long continued
to enable us to balance all the larger effects against each other.
For this purpose the observations should be continued over at least
one complete cycle, in which the effects run through all their
variations, and return exactly to the same relative positions as at the
commencement. If casual or irregular disturbing causes exist, we should
probably require many such cycles of results to render their effect
inappreciable. We obtain the desired result by taking the mean of all
the observations in which a cause acts positively, and the mean of all
in which it acts negatively. Half the difference of these means will
give the effect of the cause in question, provided that no other effect
happens to vary in the same period or nearly so.
Since the moon causes a movement of the ocean, it is evident that
its attraction must have some effect upon the atmosphere. The laws
of atmospheric tides were investigated by Laplace, but as it would
be impracticable by theory to calculate their amounts we can only
determine them by observation, as Laplace predicted that they would
one day be determined.[270] But the oscillations of the barometer thus
caused are far smaller than the oscillations due to several other
causes. Storms, hurricanes, or changes of weather produce movements
of the barometer sometimes as much as a thousand times as great as
the tides in question. There are also regular daily, yearly, or other
fluctuations, all greater than the desired quantity. To detect and
measure the atmospheric tide it was desirable that observations should
be made in a place as free as possible from irregular disturbances.
On this account several long series of observations were made at St.
Helena, where the barometer is far more regular in its movements than
in a continental climate. The effect of the moon’s attraction was
then detected by taking the mean of all the readings when the moon
was on the meridian and the similar mean when she was on the horizon.
The difference of these means was found to be only ·00365, yet it was
possible to discover even the variation of this tide according as the
moon was nearer to or further from the earth, though this difference
was only ·00056 inch.[271] It is quite evident that such minute effects
could never be discovered in a purely empirical manner. Having no
information but the series of observations before us, we could have
no clue as to the mode of grouping them which would give so small a
difference. In applying this method of means in an extensive manner
we must generally then have *à priori* knowledge as to the periods at
which a cause will act in one direction or the other.
[270] *Essai Philosophique sur les Probabilités*, pp. 49, 50.
[271] Grant, *History of Physical Astronomy*, p. 163.
We are sometimes able to eliminate fluctuations and take a mean result
by purely mechanical arrangements. The daily variations of temperature,
for instance, become imperceptible one or two feet below the surface
of the earth, so that a thermometer placed with its bulb at that
depth gives very nearly the true daily mean temperature. At a depth
of twenty feet even the yearly fluctuations are nearly effaced, and
the thermometer stands a little above the true mean temperature of the
locality. In registering the rise and fall of the tide by a tide-gauge,
it is desirable to avoid the oscillations arising from surface waves,
which is very readily accomplished by placing the float in a cistern
communicating by a small hole with the sea. Only a general rise or
fall of the level is then perceptible, just as in the marine barometer
the narrow tube prevents any casual fluctuations and allows only a
continued change of pressure to manifest itself.
*Determination of the Zero point.*
In many important observations the chief difficulty consists in
defining exactly the zero point from which we are to measure. We can
point a telescope with great precision to a star and can measure to
a second of arc the angle through which the telescope is raised or
lowered; but all this precision will be useless unless we know exactly
the centre point of the heavens from which we measure, or, what comes
to the same thing, the horizontal line 90° distant from it. Since the
true horizon has reference to the figure of the earth at the place of
observation, we can only determine it by the direction of gravity,
as marked either by the plumb-line or the surface of a liquid. The
question resolves itself then into the most accurate mode of observing
the direction of gravity, and as the plumb-line has long been found
hopelessly inaccurate, astronomers generally employ the surface of
mercury in repose as the criterion of horizontality. They ingeniously
observe the direction of the surface by making a star the index. From
the laws of reflection it follows that the angle between the direct
ray from a star and that reflected from a surface of mercury will
be exactly double the angle between the surface and the direct ray
from the star. Hence the horizontal or zero point is the mean between
the apparent place of any star or other very distant object and its
reflection in mercury.
A plumb-line is perpendicular, or a liquid surface is horizontal only
in an approximate sense; for any irregularity of the surface of the
earth, a mountain, or even a house must cause some deviation by its
attracting power. To detect such deviation might seem very difficult,
because every other plumb-line or liquid surface would be equally
affected by gravity. Nevertheless it can be detected; for if we place
one plumb-line to the north of a mountain, and another to the south,
they will be about equally deflected in opposite directions, and if
by observations of the same star we can measure the angle between the
plumb-lines, half the inclination will be the deviation of either,
after allowance has been made for the inclination due to the difference
of latitude of the two places of observation. By this mode of
observation applied to the mountain Schiehallion the deviation of the
plumb-line was accurately measured by Maskelyne, and thus a comparison
instituted between the attractive forces of the mountain and the whole
globe, which led to a probable estimate of the earth’s density.
In some cases it is actually better to determine the zero point by the
average of equally diverging quantities than by direct observation. In
delicate weighings by a chemical balance it is requisite to ascertain
exactly the point at which the beam comes to rest, and when standard
weights are being compared the position of the beam is ascertained
by a carefully divided scale viewed through a microscope. But when
the beam is just coming to rest, friction, small impediments or other
accidental causes may readily obstruct it, because it is near the
point at which the force of stability becomes infinitely small. Hence
it is found better to let the beam vibrate and observe the terminal
points of the vibrations. The mean between two extreme points will
nearly indicate the position of rest. Friction and the resistance of
air tend to reduce the vibrations, so that this mean will be erroneous
by half the amount of this effect during a half vibration. But by
taking several observations we may determine this retardation and
allow for it. Thus if *a*, *b*, *c* be the readings of the terminal
points of three excursions of the beam from the zero of the scale,
then 1/2(*a* + *b*) will be about as much erroneous in one direction
as 1/2(*b* + *c*) in the other, so that the mean of these two means,
or 1/4(*a* + 2*b* + *c*), will be exceedingly near to the point of
rest.[272] A still closer approximation may be made by taking four
readings and reducing them by the formula 1/6(*a* + 2*b* + 2*c* + *d*).
[272] Gauss, Taylor’s *Scientific Memoirs*, vol. ii. p. 43, &c.
The accuracy of Baily’s experiments, directed to determine the
density of the earth, entirely depended upon this mode of observing
oscillations. The balls whose gravitation was measured were so
delicately suspended by a torsion balance that they never came to
rest. The extreme points of the oscillations were observed both when
the heavy leaden attracting ball was on one side and on the other. The
difference of the mean points when the leaden ball was on the right
hand and that when it was on the left hand gave double the amount of
the deflection.
A beautiful instance of avoiding the use of a zero point is found in
Mr. E. J. Stone’s observations on the radiant heat of the fixed stars.
The difficulty of these observations arose from the comparatively great
amounts of heat which were sent into the telescope from the atmosphere,
and which were sufficient to disguise almost entirely the feeble heat
rays of a star. But Mr. Stone fixed at the focus of his telescope a
double thermo-electric pile of which the two parts were reversed in
order. Now any disturbance of temperature which acted uniformly upon
both piles produced no effect upon the galvanometer needle, and when
the rays of the star were made to fall alternately upon one pile and
the other, the total amount of the deflection represented double the
heating power of the star. Thus Mr. Stone was able to detect with
much certainty a heating effect of the star Arcturus, which even when
concentrated by the telescope amounted only to 0°·02 Fahr., and which
represents a heating effect of the direct ray of only about 0°·00000137
Fahr., equivalent to the heat which would be received from a three-inch
cubic vessel full of boiling water at the distance of 400 yards.[273]
It is probable that Mr. Stone’s arrangement of the pile might be
usefully employed in other delicate thermometric experiments subject to
considerable disturbing influences.
[273] *Proceedings of the Royal Society*, vol. xviii. p. 159 (Jan.
13, 1870). *Philosophical Magazine* (4th Series), vol. xxxix. p. 376.
*Determination of Maximum Points.*
We employ the method of means in a certain number of observations
directed to determine the moment at which a phenomenon reaches its
highest point in quantity. In noting the place of a fixed star at a
given time there is no difficulty in ascertaining the point to be
observed, for a star in a good telescope presents an exceedingly small
disc. In observing a nebulous body which from a bright centre fades
gradually away on all sides, it will not be possible to select with
certainty the middle point. In many such cases the best method is not
to select arbitrarily the supposed middle point, but points of equal
brightness on either side, and then take the mean of the observations
of these two points for the centre. As a general rule, a variable
quantity in reaching its maximum increases at a less and less rate,
and after passing the highest point begins to decrease by insensible
degrees. The maximum may indeed be defined as that point at which
the increase or decrease is null. Hence it will usually be the most
indefinite point, and if we can accurately measure the phenomenon we
shall best determine the place of the maximum by determining points on
either side at which the ordinates are equal. There is moreover this
advantage in the method that several points may be determined with the
corresponding ones on the other side, and the mean of the whole taken
as the true place of the maximum. But this method entirely depends upon
the existence of symmetry in the curve, so that of two equal ordinates
one shall be as far on one side of the maximum as the other is on the
other side. The method fails when other laws of variation prevail.
In tidal observations great difficulty is encountered in fixing
the moment of high water, because the rate at which the water is
then rising or falling, is almost imperceptible. Whewell proposed,
therefore, to note the time at which the water passes a fixed point
somewhat below the maximum both in rising and falling, and take
the mean time as that of high water. But this mode of proceeding
unfortunately does not give a correct result, because the tide follows
different laws in rising and in falling. There is a difficulty again in
selecting the highest spring tide, another object of much importance in
tidology. Laplace discovered that the tide of the second day preceding
the conjunction of the sun and moon is nearly equal to that of the
fifth day following; and, believing that the increase and decrease of
the tides proceeded in a nearly symmetrical manner, he decided that the
highest tide would occur about thirty-six hours after the conjunction,
that is half-way between the second day before and the fifth day
after.[274]
[274] Airy *On Tides and Waves*, Encycl. Metrop. pp. 364*-366*.
This method is also employed in determining the time of passage of the
middle or densest point of a stream of meteors. The earth takes two
or three days in passing completely through the November stream; but
astronomers need for their calculations to have some definite point
fixed within a few minutes if possible. When near to the middle they
observe the numbers of meteors which come within the sphere of vision
in each half hour, or quarter hour, and then, assuming that the law of
variation is symmetrical, they select a moment for the passage of the
centre equidistant between times of equal frequency.
The eclipses of Jupiter’s satellites are not only of great interest as
regards the motions of the satellites themselves, but were, and perhaps
still are, of use in determining longitudes, because they are events
occurring at fixed moments of absolute time, and visible in all parts
of the planetary system at the same time, allowance being made for the
interval occupied by the light in travelling. But, as is explained by
Herschel,[275] the moment of the event is wanting in definiteness,
partly because the long cone of Jupiter’s shadow is surrounded by
a penumbra, and partly because the satellite has itself a sensible
disc, and takes time in entering the shadow. Different observers using
different telescopes would usually select different moments for that
of the eclipse. But the increase of light in the emersion will proceed
according to a law the reverse of that observed in the immersion,
so that if an observer notes the time of both events with the same
telescope, he will be as much too soon in one observation as he is too
late in the other, and the mean moment of the two observations will
represent with considerable accuracy the time when the satellite is in
the middle of the shadow. Error of judgment of the observer is thus
eliminated, provided that he takes care to act at the emersion as he
did at the immersion.
[275] *Outlines of Astronomy*, 4th edition, § 538.
CHAPTER XVII.
THE LAW OF ERROR.
To bring error itself under law might seem beyond human power. He who
errs surely diverges from law, and it might be deemed hopeless out of
error to draw truth. One of the most remarkable achievements of the
human intellect is the establishment of a general theory which not
only enables us among discrepant results to approximate to the truth,
but to assign the degree of probability which fairly attaches to this
conclusion. It would be a mistake indeed to suppose that this law is
necessarily the best guide under all circumstances. Every measuring
instrument and every form of experiment may have its own special law
of error; there may in one instrument be a tendency in one direction
and in another in the opposite direction. Every process has its
peculiar liabilities to disturbance, and we are never relieved from the
necessity of providing against special difficulties. The general Law of
Error is the best guide only when we have exhausted all other means of
approximation, and still find discrepancies, which are due to unknown
causes. We must treat such residual differences in some way or other,
since they will occur in all accurate experiments, and as their origin
is assumed to be unknown, there is no reason why we should treat them
differently in different cases. Accordingly the ultimate Law of Error
must be a uniform and general one.
It is perfectly recognised by mathematicians that in each case a
special Law of Error may exist, and should be discovered if possible.
“Nothing can be more unlikely than that the errors committed in all
classes of observations should follow the same law,”[276] and the
special Laws of Error which will apply to certain instruments, as for
instance the repeating circle, have been investigated by Bravais.[277]
He concludes that every distinct cause of error gives rise to a curve
of possibility of errors, which may have any form,--a curve which
we may either be able or unable to discover, and which in the first
case may be determined by *à priori* considerations on the peculiar
nature of this cause, or which may be determined *à posteriori* by
observation. Whenever it is practicable and worth the labour, we ought
to investigate these special conditions of error; nevertheless, when
there are a great number of different sources of minute error, the
general resultant will always tend to obey that general law which we
are about to consider.
[276] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 324.
[277] *Letters on the Theory of Probabilities*, by Quetelet,
translated by O. G. Downes, Notes to Letter XXVI. pp. 286–295.
*Establishment of the Law of Error.*
Mathematicians agree far better as to the form of the Law of Error
than they do as to the manner in which it can be deduced and proved.
They agree that among a number of discrepant results of observation,
that mean quantity is probably the best approximation to the truth
which makes the sum of the squares of the errors as small as possible.
But there are three principal ways in which this law has been arrived
at respectively by Gauss, by Laplace and Quetelet, and by Sir John
Herschel. Gauss proceeds much upon assumption; Herschel rests upon
geometrical considerations; while Laplace and Quetelet regard the Law
of Error as a development of the doctrine of combinations. A number
of other mathematicians, such as Adrain of New Brunswick, Bessel,
Ivory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted
independent proofs or have modified or commented on those here to be
described. For full accounts of the literature of the subject the
reader should refer either to Mr. Todhunter’s *History of the Theory of
Probability* or to the able memoir of Mr. J. W. L. Glaisher.[278]
[278] *On the Law of Facility of Errors of Observations, and on the
Method of Least Squares*, Memoirs of the Royal Astronomical Society,
vol. xxxix. p. 75.
According to Gauss the Law of Error expresses the comparative
probability of errors of various magnitude, and partly from experience,
partly from *à priori* considerations, we may readily lay down certain
conditions to which the law will certainly conform. It may fairly
be assumed as a first principle to guide us in the selection of the
law, that large errors will be far less frequent and probable than
small ones. We know that very large errors are almost impossible, so
that the probability must rapidly decrease as the amount of the error
increases. A second principle is that positive and negative errors
shall be equally probable, which may certainly be assumed, because we
are supposed to be devoid of any knowledge as to the causes of the
residual errors. It follows that the probability of the error must be a
function of an even power of the magnitude, that is of the square, or
the fourth power, or the sixth power, otherwise the probability of the
same amount of error would vary according as the error was positive or
negative. The even powers *x*^{2}, *x*^{4}, *x*^{6}, &c., are always
intrinsically positive, whether *x* be positive or negative. There is
no *à priori* reason why one rather than another of these even powers
should be selected. Gauss himself allows that the fourth or sixth power
would fulfil the conditions as well as the second;[279] but in the
absence of any theoretical reasons we should prefer the second power,
because it leads to formulæ of great comparative simplicity. Did the
Law of Error necessitate the use of the higher powers of the error, the
complexity of the necessary calculations would much reduce the utility
of the theory.
[279] *Méthode des Moindres Carrés. Mémoires sur la Combinaison
des Observations, par Ch. Fr. Gauss. Traduit en Français par J.
Bertrand*, Paris, 1855, pp. 6, 133, &c.
By mathematical reasoning which it would be undesirable to attempt
to follow in this book, it is shown that under these conditions, the
facility of occurrence, or in other, words, the probability of error
is expressed by a function of the general form ε^{–*h*^{2} *x*^{2}}, in
which *x* represents the variable amount of errors. From this law, to
be more fully described in the following sections, it at once follows
that the most probable result of any observations is that which makes
the sum of the squares of the consequent errors the least possible.
Let *a*, *b*, *c*, &c., be the results of observation, and *x* the
quantity selected as the most probable, that is the most free from
unknown errors: then we must determine *x* so that (*a* - *x*)^{2} +
(*b* - *x*)^{2} + (*c* - *x*)^{2} + ... shall be the least possible
quantity. Thus we arrive at the celebrated *Method of Least Squares*,
as it is usually called, which appears to have been first distinctly
put in practice by Gauss in 1795, while Legendre first published in
1806 an account of the process in his work, entitled, *Nouvelles
Méthodes pour la Détermination des Orbites des Comètes*. It is worthy
of notice, however, that Roger Cotes had long previously recommended a
method of equivalent nature in his tract, “Estimatio Erroris in Mixta
Mathesi.”[280]
[280] De Morgan, *Penny Cyclopædia*, art. *Least Squares*.
*Herschel’s Geometrical Proof.*
A second way of arriving at the Law of Error was proposed by Herschel,
and although only applicable to geometrical cases, it is remarkable as
showing that from whatever point of view we regard the subject, the
same principle will be detected. After assuming that some general law
must exist, and that it is subject to the principles of probability,
he supposes that a ball is dropped from a high point with the
intention that it shall strike a given mark on a horizontal plane. In
the absence of any known causes of deviation it will either strike
that mark, or, as is infinitely more probable, diverge from it by an
amount which we must regard as error of unknown origin. Now, to quote
the words of Herschel,[281] “the probability of that error is the
unknown function of its square, *i.e.* of the sum of the squares of
its deviations in any two rectangular directions. Now, the probability
of any deviation depending solely on its magnitude, and not on its
direction, it follows that the probability of each of these rectangular
deviations must be the same function of *its* square. And since the
observed oblique deviation is equivalent to the two rectangular ones,
supposed concurrent, and which are essentially independent of one
another, and is, therefore, a compound event of which they are the
simple independent constituents, therefore its probability will be the
product of their separate probabilities. Thus the form of our unknown
function comes to be determined from this condition, viz., that the
product of such functions of two independent elements is equal to the
same function of their sum. But it is shown in every work on algebra
that this property is the peculiar characteristic of, and belongs only
to, the exponential or antilogarithmic function. This, then, is the
function of the square of the error, which expresses the probability
of committing that error. That probability decreases, therefore, in
geometrical progression, as the square of the error increases in
arithmetical.”
[281] *Edinburgh Review*, July 1850, vol. xcii. p. 17. Reprinted
*Essays*, p. 399. This method of demonstration is discussed by Boole,
*Transactions of Royal Society of Edinburgh*, vol. xxi. pp. 627–630.
*Laplace’s and Quetelet’s Proof of the Law.*
However much presumption the modes of determining the Law of Error,
already described, may give in favour of the law usually adopted, it is
difficult to feel that the arguments are satisfactory. The law adopted
is chosen rather on the grounds of convenience and plausibility, than
because it can be seen to be the necessary law. We can however approach
the subject from an entirely different point of view, and yet get to
the same result.
Let us assume that a particular observation is subject to four chances
of error, each of which will increase the result one inch if it occurs.
Each of these errors is to be regarded as an event independent of the
rest and we can therefore assign, by the theory of probability, the
comparative probability and frequency of each conjunction of errors.
From the Arithmetical Triangle (pp. 182–188) we learn that no error
at all can happen only in one way; an error of one inch can happen
in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches
respectively, will be 6, 4 and 1 in number.
We may infer that the error of two inches is the most likely to occur,
and will occur in the long run in six cases out of sixteen. Errors
of one and three inches will be equally likely, but will occur less
frequently; while no error at all, or one of four inches will be a
comparatively rare occurrence. If we now suppose the errors to act
as often in one direction as the other, the effect will be to alter
the average error by the amount of two inches, and we shall have the
following results:--
Negative error of 2 inches 1 way.
Negative error of 1 inch 4 ways.
No error at all 6 ways.
Positive error of 1 inch 4 ways.
Positive error of 2 inches 1 way.
We may now imagine the number of causes of error increased and the
amount of each error decreased, and the arithmetical triangle will
give us the frequency of the resulting errors. Thus if there be five
positive causes of error and five negative causes, the following
table shows the numbers of errors of various amount which will be the
result:--
+----------------------+-------------------+---+-------------------+
| Direction of Error. | Positive Error. | | Negative Error. |
+----------------------+-------------------+---+-------------------+
| Amount of Error. |5, 4, 3, 2, 1| 0 | 1, 2, 3, 4, 5|
+----------------------+-------------------+---+-------------------+
|Number of such Errors.|1, 10, 45, 120, 210|252|210, 120, 45, 10, 1|
+----------------------+-------------------+---+-------------------+
It is plain that from such numbers I can ascertain the probability
of any particular amount of error under the conditions supposed. The
probability of a positive error of exactly one inch is 210/1024, in
which fraction the numerator is the number of combinations giving
one inch positive error, and the denominator the whole number of
possible errors of all magnitudes. I can also, by adding together the
appropriate numbers get the probability of an error not exceeding a
certain amount. Thus the probability of an error of three inches or
less, positive or negative, is a fraction whose numerator is the sum of
45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, as before,
giving the result 1002/1024. We may see at once that, according to
these principles, the probability of small errors is far greater than
of large ones: the odds are 1002 to 22, or more than 45 to 1, that the
error will not exceed three inches; and the odds are 1022 to 2 against
the occurrence of the greatest possible error of five inches.
If any case should arise in which the observer knows the number and
magnitude of the chief errors which may occur, he ought certainly to
calculate from the Arithmetical Triangle the special Law of Error
which would apply. But the general law, of which we are in search,
is to be used in the dark, when we have no knowledge whatever of the
sources of error. To assume any special number of causes of error
is then an arbitrary proceeding, and mathematicians have chosen the
least arbitrary course of imagining the existence of an infinite
number of infinitely small errors, just as, in the inverse method of
probabilities, an infinite number of infinitely improbable hypotheses
were submitted to calculation (p. 255).
The reasons in favour of this choice are of several different kinds.
1. It cannot be denied that there may exist infinitely numerous causes
of error in any act of observation.
2. The law resulting from the hypothesis of a moderate number of causes
of error, does not appreciably differ from that given by the hypothesis
of an infinite number of causes of error.
3. We gain by the hypothesis of infinity a general law capable of ready
calculation, and applicable by uniform rules to all problems.
4. This law, when tested by comparison with extensive series of
observations, is strikingly verified, as will be shown in a later
section.
When we imagine the existence of any large number of causes of
error, for instance one hundred, the numbers of combinations become
impracticably large, as may be seen to be the case from a glance at
the Arithmetical Triangle, which proceeds only up to the seventeenth
line. Quetelet, by suitable abbreviating processes, calculated out
a table of probability of errors on the hypothesis of one thousand
distinct causes;[282] but mathematicians have generally proceeded on
the hypothesis of infinity, and then, by the devices of analysis,
have substituted a general law of easy treatment. In mathematical
works upon the subject, it is shown that the standard Law of Error is
expressed in the formula
*y* = *Y*ε^{-*cx*^{2}},
[282] *Letters on the Theory of Probabilities*, Letter XV. and
Appendix, note pp. 256–266.
in which *x* is the amount of the error, *Y* the maximum ordinate
of the curve of error, and *c* a number constant for each series of
observations, and expressing the amount of the tendency to error,
varying between one series of observations and another. The letter ε
is the mathematical constant, the sum of ratios between the numbers of
permutations and combinations, previously referred to (p. 330).
[Illustration]
To show the close correspondence of this general law with the special
law which might be derived from the supposition of a moderate number
of causes of error, I have in the accompanying figure drawn a curved
line representing accurately the variation of *y* when *x* in the above
formula is taken equal 0, 1/2, 1, 3/2, 2, &c., positive or negative,
the arbitrary quantities *Y* and *c* being each assumed equal to unity,
in order to simplify the calculations. In the same figure are inserted
eleven dots, whose heights above the base line are proportional to
the numbers in the eleventh line of the Arithmetical Triangle, thus
representing the comparative probabilities of errors of various amounts
arising from ten equal causes of error. The correspondence of the
general and the special Law of Error is almost as close as can be
exhibited in the figure, and the assumption of a greater number of
equal causes of error would render the correspondence far more close.
It may be explained that the ordinates NM, *nm*, *n′m′*, represent
values of *y* in the equation expressing the Law of Error. The
occurrence of any one definite amount of error is infinitely
improbable, because an infinite number of such ordinates might be
drawn. But the probability of an error occurring between certain limits
is finite, and is represented by a portion of the *area* of the curve.
Thus the probability that an error, positive or negative, not exceeding
unity will occur, is represented by the area M*mnn′m′*, in short, by
the area standing upon the line *nn′*. Since every observation must
either have some definite error or none at all, it follows that the
whole area of the curve should be considered as the unit expressing
certainty, and the probability of an error falling between particular
limits will then be expressed by the ratio which the area of the curve
between those limits bears to the whole area of the curve.
The mere fact that the Law of Error allows of the possible existence of
errors of every assignable amount shows that it is only approximately
true. We may fairly say that in measuring a mile it would be impossible
to commit an error of a hundred miles, and the length of life
would never allow of our committing an error of one million miles.
Nevertheless the general Law of Error would assign a probability for
an error of that amount or more, but so small a probability as to be
utterly inconsiderable and almost inconceivable. All that can, or in
fact need, be said in defence of the law is, that it may be made to
represent the errors in any special case to a very close approximation,
and that the probability of large and practically impossible errors, as
given by the law, will be so small as to be entirely inconsiderable.
And as we are dealing with error itself, and our results pretend to
nothing more than approximation and probability, an indefinitely small
error in our process of approximation is of no importance whatever.
*Logical Origin of the Law of Error.*
It is worthy of notice that this Law of Error, abstruse though the
subject may seem, is really founded upon the simplest principles.
It arises entirely out of the difference between permutations and
combinations, a subject upon which I may seem to have dwelt with
unnecessary prolixity in previous pages (pp. 170, 189). The order in
which we add quantities together does not affect the amount of the sum,
so that if there be three positive and five negative causes of error
in operation, it does not matter in which order they are considered as
acting. They may be intermixed in any arrangement, and yet the result
will be the same. The reader should not fail to notice how laws or
principles which appeared to be absurdly simple and evident when first
noticed, reappear in the most complicated and mysterious processes of
scientific method. The fundamental Laws of Identity and Difference gave
rise to the Logical Alphabet which, after abstracting the character of
the differences, led to the Arithmetical Triangle. The Law of Error is
defined by an infinitely high line of that triangle, and the law proves
that the mean is the most probable result, and that divergencies from
the mean become much less probable as they increase in amount. Now
the comparative greatness of the numbers towards the middle of each
line of the Arithmetical Triangle is entirely due to the indifference
of order in space or time, which was first prominently pointed out
as a condition of logical relations, and the symbols indicating them
(pp. 32–35), and which was afterwards shown to attach equally to
numerical symbols, the derivatives of logical terms (p. 160).
*Verification of the Law of Error.*
The theory of error which we have been considering rests entirely
upon an assumption, namely that when known sources of disturbances
are allowed for, there yet remain an indefinite, possibly an infinite
number of other minute sources of error, which will as often produce
excess as deficiency. Granting this assumption, the Law of Error must
be as it is usually taken to be, and there is no more need to verify
it empirically than to test the truth of one of Euclid’s propositions
mechanically. Nevertheless, it is an interesting occupation to verify
even the propositions of geometry, and it is still more instructive to
try whether a large number of observations will justify our assumption
of the Law of Error.
Encke has given an excellent instance of the correspondence of theory
with experience, in the case of observations of the differences of
Right Ascension of the sun and two stars, namely α Aquilæ and α Canis
minoris. The observations were 470 in number, and were made by Bradley
and reduced by Bessel, who found the probable error of the final result
to be only about one-fourth part of a second (0·2637). He then compared
the numbers of errors of each magnitude from 0·1 second upwards, as
actually given by the observations, with what should occur according to
the Law of Error.
The results were as follow:--[283]
+-------------------------+--------------------------+
| | Number of errors of each |
| Magnitude of the errors | magnitude according to |
| in parts of a second. +-------------+------------+
| | Observation.| Theory. |
+-------------------------+-------------+------------+
| 0·0 to 0·1 | 94 | 95 |
| ·1 " ·2 | 88 | 89 |
| ·2 " ·3 | 78 | 78 |
| ·3 " ·4 | 58 | 64 |
| ·4 " ·5 | 51 | 50 |
| ·5 " ·6 | 36 | 36 |
| ·6 " ·7 | 26 | 24 |
| ·7 " ·8 | 14 | 15 |
| ·8 " ·9 | 10 | 9 |
| ·9 " 1·0 | 7 | 5 |
| above 1·0 | 8 | 5 |
+-------------------------+-------------+------------+
[283] Encke, *On the Method of Least Squares*, Taylor’s *Scientific
Memoirs*, vol. ii. pp. 338, 339.
The reader will remark that the correspondence is very close, except
as regards larger errors, which are excessive in practice. It is one
objection, indeed, to the theory of error, that, being expressed in
a continuous mathematical function, it contemplates the existence of
errors of every magnitude, such as could not practically occur; yet
in this case the theory seems to under-estimate the number of large
errors.
Another comparison of the law with observation was made by Quetelet,
who investigated the errors of 487 determinations in time of the Right
Ascension of the Pole-Star made at Greenwich during the four years
1836–39. These observations, although carefully corrected for all known
causes of error, as well as for nutation, precession, &c., are yet of
course found to differ, and being classified as regards intervals of
one-half second of time, and then proportionately increased in number,
so that their sum may be one thousand, give the following results as
compared with what Quetelet’s theory would lead us to expect:--[284]
+------------+--------------------+------------+--------------------+
|Magnitude of| Number of Errors |Magnitude of| Number of Errors |
| error +------------+-------+ error +------------+-------+
| in tenths | by | by | in tenths | by | by |
|of a second.|Observation.|Theory.|of a second.|Observation.|Theory.|
+------------+------------+-------+------------+------------+-------+
| 0·0 | 168 | 163 | -- | -- | -- |
| +0·5 | 148 | 147 | -0·5 | 150 | 152 |
| +1·0 | 129 | 112 | -1·0 | 126 | 121 |
| +1·5 | 78 | 72 | -1·5 | 74 | 82 |
| +2·0 | 33 | 40 | -2·0 | 43 | 46 |
| +2·5 | 10 | 19 | -2·5 | 25 | 22 |
| +3·0 | 2 | 10 | -3·0 | 12 | 10 |
| -- | -- | -- | -3·5 | 2 | 4 |
+------------+------------+-------+------------+------------+-------+
[284] Quetelet, *Letters on the Theory of Probabilities*, translated
by Downes, Letter XIX. p. 88. See also Galton’s *Hereditary Genius*,
p. 379.
In this instance also the correspondence is satisfactory, but the
divergence between theory and fact is in the opposite direction to
that discovered in the former comparison, the larger errors being less
frequent than theory would indicate. It will be noticed that Quetelet’s
theoretical results are not symmetrical.
*The Probable Mean Result.*
One immediate result of the Law of Error, as thus stated, is that the
mean result is the most probable one; and when there is only a single
variable this mean is found by the familiar arithmetical process. An
unfortunate error has crept into several works which allude to this
subject. Mill, in treating of the “Elimination of Chance,” remarks in
a note[285] that “the mean is spoken of as if it were exactly the
same thing as the average. But the mean, for purposes of inductive
inquiry, is not the average, or arithmetical mean, though in a familiar
illustration of the theory the difference may be disregarded.” He goes
on to say that, according to mathematical principles, the most probable
result is that for which the sums of the squares of the deviations is
the least possible. It seems probable that Mill and other writers were
misled by Whewell, who says[286] that “The method of least squares is
in fact a method of means, but with some peculiar characters.... The
method proceeds upon this supposition: that all errors are not equally
probable, but that small errors are more probable than large ones.” He
adds that this method “removes much that is arbitrary in the method
of means.” It is strange to find a mathematician like Whewell making
such remarks, when there is no doubt whatever that the Method of Means
is only an application of the Method of Least Squares. They are, in
fact, the same method, except that the latter method may be applied to
cases where two or more quantities have to be determined at the same
time. Lubbock and Drinkwater say,[287] “If only one quantity has to
be determined, this method evidently resolves itself into taking the
mean of all the values given by observation.” Encke says,[288] that the
expression for the probability of an error “not only contains in itself
the principle of the arithmetical mean, but depends so immediately upon
it, that for all those magnitudes for which the arithmetical mean holds
good in the simple cases in which it is principally applied, no other
law of probability can be assumed than that which is expressed by this
formula.”
[285] *System of Logic*, bk. iii. chap. 17, § 3. 5th ed. vol. ii.
p. 56.
[286] *Philosophy of the Inductive Sciences*, 2nd ed. vol. ii.
pp. 408, 409.
[287] *Essay on Probability*, Useful Knowledge Society, 1833, p. 41.
[288] Taylor’s *Scientific Memoirs*, vol. ii. p. 333.
*The Probable Error of Results.*
When we draw a conclusion from the numerical results of observations we
ought not to consider it sufficient, in cases of importance, to content
ourselves with finding the simple mean and treating it as true. We
ought also to ascertain what is the degree of confidence we may place
in this mean, and our confidence should be measured by the degree of
concurrence of the observations from which it is derived. In some cases
the mean may be approximately certain and accurate. In other cases it
may really be worth little or nothing. The Law of Error enables us to
give exact expression to the degree of confidence proper in any case;
for it shows how to calculate the probability of a divergence of any
amount from the mean, and we can thence ascertain the probability that
the mean in question is within a certain distance from the true number.
The *probable error* is taken by mathematicians to mean the limits
within which it is as likely as not that the truth will fall. Thus
if 5·45 be the mean of all the determinations of the density of the
earth, and ·20 be approximately the probable error, the meaning is that
the probability of the real density of the earth falling between 5·25
and 5·65 is 1/2. Any other limits might have been selected at will.
We might calculate the limits within which it was one hundred or one
thousand to one that the truth would fall; but there is a convention to
take the even odds one to one, as the quantity of probability of which
the limits are to be estimated.
Many books on probability give rules for making the calculations,
but as, in the progress of science, persons ought to become more
familiar with these processes, I propose to repeat the rules here and
illustrate their use. The calculations, when made in accordance with
the directions, involve none but arithmetic or logarithmic operations.
The following are the rules for treating a mean result, so as
thoroughly to ascertain its trustworthiness.
1. Draw the mean of all the observed results.
2. Find the excess or defect, that is, the error of each result from
the mean.
3. Square each of these reputed errors.
4. Add together all these squares of the errors, which are of course
all positive.
5. Divide by one less than the number of observations. This gives the
*square of the mean error*.
6. Take the square root of the last result; it is the *mean error of a
single observation*.
7. Divide now by the square root of the number of observations, and we
get the *mean error of the mean result*.
8. Lastly, multiply by the natural constant 0·6745 (or approximately
by 0·674, or even by 2/3), and we arrive at the *probable error of the
mean result*.
Suppose, for instance, that five measurements of the height of a
hill, by the barometer or otherwise, have given the numbers of feet
as 293, 301, 306, 307, 313; we want to know the probable error of the
mean, namely 304. Now the differences between this mean and the above
numbers, *paying no regard to direction*, are 11, 3, 2, 3, 9; their
squares are 121, 9, 4, 9, 81, and the sum of the squares of the errors
consequently 224. The number of observations being 5, we divide by
1 less, or 4, getting 56. This is the square of the mean error, and
taking its square root we have 7·48 (say 7-1/2), the mean error of
a single observation. Dividing by 2·236, the square root of 5, the
number of observations, we find the mean error of the *mean* result to
be 3·35, or say 3-1/3, and lastly, multiplying by ·6745, we arrive at
the *probable error of the mean result*, which is found to be 2·259,
or say 2-1/4. The meaning of this is that the probability is one half,
or the odds are even that the true height of the mountain lies between
301-3/4 and 306-1/4 feet. We have thus an exact measure of the degree
of credibility of our mean result, which mean indicates the most likely
point for the truth to fall upon.
The reader should observe that as the object in these calculations
is only to gain a notion of the degree of confidence with which we
view the mean, there is no real use in carrying the calculations to
any great degree of precision; and whenever the neglect of decimal
fractions, or even the slight alteration of a number, will much
abbreviate the computations, it may be fearlessly done, except in
cases of high importance and precision. Brodie has shown how the law
of error may be usefully applied in chemical investigations, and some
illustrations of its employment may be found in his paper.[289]
[289] *Philosophical Transactions*, 1873, p. 83.
The experiments of Benzenberg to detect the revolution of the earth, by
the deviation of a ball from the perpendicular line in falling down a
deep pit, have been cited by Encke[290] as an interesting illustration
of the Law of Error. The mean deviation was 5·086 lines, and its
probable error was calculated by Encke to be not more than ·950 line,
that is, the odds were even that the true result lay between 4·136 and
6·036. As the deviation, according to astronomical theory, should be
4·6 lines, which lies well within the limits, we may consider that the
experiments are consistent with the Copernican system of the universe.
[290] Taylor’s *Scientific Memoirs*, vol. ii. pp. 330, 347, &c.
It will of course be understood that the probable error has regard only
to those causes of errors which in the long run act as much in one
direction as another; it takes no account of constant errors. The true
result accordingly will often fall far beyond the limits of probable
error, owing to some considerable constant error or errors, of the
existence of which we are unaware.
*Rejection of the Mean Result.*
We ought always to bear in mind that the mean of any series of
observations is the best, that is, the most probable approximation
to the truth, only in the absence of knowledge to the contrary. The
selection of the mean rests entirely upon the probability that unknown
causes of error will in the long run fall as often in one direction as
the opposite, so that in drawing the mean they will balance each other.
If we have any reason to suppose that there exists a tendency to error
in one direction rather than the other, then to choose the mean would
be to ignore that tendency. We may certainly approximate to the length
of the circumference of a circle, by taking the mean of the perimeters
of inscribed and circumscribed polygons of an equal and large number
of sides. The length of the circular line undoubtedly lies between the
lengths of the two perimeters, but it does not follow that the mean is
the best approximation. It may in fact be shown that the circumference
of the circle is *very nearly* equal to the perimeter of the inscribed
polygon, together with one-third part of the difference between the
inscribed and circumscribed polygons of the same number of sides.
Having this knowledge, we ought of course to act upon it, instead of
trusting to probability.
We may often perceive that a series of measurements tends towards an
extreme limit rather than towards a mean. In endeavouring to obtain
a correct estimate of the apparent diameter of the brightest fixed
stars, we find a continuous diminution in estimates as the powers of
observation increased. Kepler assigned to Sirius an apparent diameter
of 240 seconds; Tycho Brahe made it 126; Gassendi 10 seconds; Galileo,
Hevelius, and J. Cassini, 5 or 6 seconds. Halley, Michell, and
subsequently Sir W. Herschel came to the conclusion that the brightest
stars in the heavens could not have real discs of a second, and were
probably much less in diameter. It would of course be absurd to take
the mean of quantities which differ more than 240 times; and as the
tendency has always been to smaller estimates, there is a considerable
presumption in favour of the smallest.[291]
[291] Quetelet, *Letters*, &c. p. 116.
In many experiments and measurements we know that there is a
preponderating tendency to error in one direction. The readings of a
thermometer tend to rise as the age of the instrument increases, and
no drawing of means will correct this result. Barometers, on the other
hand, are likely to read too low instead of too high, owing to the
imperfection of the vacuum and the action of capillary attraction. If
the mercury be perfectly pure and no appreciable error be due to the
measuring apparatus, the best barometer will be that which gives the
highest result. In determining the specific gravity of a solid body the
chief danger of error arises from bubbles of air adhering to the body,
which would tend to make the specific gravity too small. Much attention
must always be given to one-sided errors of this kind, since the
multiplication of experiments does not remove the error. In such cases
one very careful experiment is better than any number of careless ones.
When we have reasonable grounds for supposing that certain experimental
results are liable to grave errors, we should exclude them in drawing
a mean. If we want to find the most probable approximation to the
velocity of sound in air, it would be absurd to go back to the old
experiments which made the velocity from 1200 to 1474 feet per second;
for we know that the old observers did not guard against errors arising
from wind and other causes. Old chemical experiments are valueless as
regards quantitative results. The old chemists found the atmosphere
in different places to differ in composition nearly ten per cent.,
whereas modern accurate experimenters find very slight variations.
Any method of measurement which we know to avoid a source of error is
far to be preferred to others which trust to probabilities for the
elimination of the error. As Flamsteed says,[292] “One good instrument
is of as much worth as a hundred indifferent ones.” But an instrument
is good or bad only in a comparative sense, and no instrument gives
invariable and truthful results. Hence we must always ultimately fall
back upon probabilities for the selection of the final mean, when other
precautions are exhausted.
[292] Baily, *Account of Flamsteed*, p. 56.
Legendre, the discoverer of the method of Least Squares,
recommended that observations differing very much from the results
of his method should be rejected. The subject has been carefully
investigated by Professor Pierce, who has proposed a criterion
for the rejection of doubtful observations based on the following
principle:[293]′“--observations should be rejected when the probability
of the system of errors obtained by retaining them is less than that
of the system of errors obtained by their rejection multiplied by the
probability of making so many and no more abnormal observations.”
Professor Pierce’s investigation is given nearly in his own words in
Professor W. Chauvenet’s “Manual of Spherical and Practical Astronomy,”
which contains a full and excellent discussion of the methods of
treating numerical observations.[294]
[293] Gould’s *Astronomical Journal*, Cambridge, Mass., vol. ii.
p. 161.
[294] Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558.
Very difficult questions sometimes arise when one or more results of
a method of experiment diverge widely from the mean of the rest. Are
we or are we not to exclude them in adopting the supposed true mean
result of the method? The drawing of a mean result rests, as I have
frequently explained, upon the assumption that every error acting in
one direction will probably be balanced by other errors acting in an
opposite direction. If then we know or can possibly discover any causes
of error not agreeing with this assumption, we shall be justified in
excluding results which seem to be affected by this cause.
In reducing large series of astronomical observations, it is not
uncommon to meet with numbers differing from others by a whole degree
or half a degree, or some considerable integral quantity. These are
errors which could hardly arise in the act of observation or in
instrumental irregularity; but they might readily be accounted for
by misreading of figures or mistaking of division marks. It would be
absurd to trust to chance that such mistakes would balance each other
in the long run, and it is therefore better to correct arbitrarily
the supposed mistake, or better still, if new observations can be
made, to strike out the divergent numbers altogether. When results
come sometimes too great or too small in a regular manner, we should
suspect that some part of the instrument slips through a definite
space, or that a definite cause of error enters at times, and not at
others. We should then make it a point of prime importance to discover
the exact nature and amount of such an error, and either prevent its
occurrence for the future or else introduce a corresponding correction.
In many researches the whole difficulty will consist in this detection
and avoidance of sources of error. Professor Roscoe found that the
presence of phosphorus caused serious and almost unavoidable errors in
the determination of the atomic weight of vanadium.[295] Herschel, in
reducing his observations of double stars at the Cape of Good Hope,
was perplexed by an unaccountable difference of the angles of position
as measured by the seven-feet equatorial and the twenty-feet reflector
telescopes, and after a careful investigation was obliged to be
contented with introducing a correction experimentally determined.[296]
[295] Bakerian Lecture, *Philosophical Transactions* (1868),
vol. clviii. p. 6.
[296] *Results of Observations at the Cape of Good Hope*, p. 283.
When observations are sufficiently numerous it seems desirable to
project the apparent errors into a curve, and then to observe whether
this curve exhibits the symmetrical and characteristic form of the
curve of error. If so, it may be inferred that the errors arise from
many minute independent sources, and probably compensate each other
in the mean result. Any considerable irregularity will indicate the
existence of one-sided or large causes of error, which should be made
the subject of investigation.
Even the most patient and exhaustive investigations will sometimes
fail to disclose any reason why some results diverge from others.
The question again recurs--Are we arbitrarily to exclude them? The
answer should be in the negative as a general rule. The mere fact
of divergence ought not to be taken as conclusive against a result,
and the exertion of arbitrary choice would open the way to the fatal
influence of bias, and what is commonly known as the “cooking” of
figures. It would amount to judging fact by theory instead of theory
by fact. The apparently divergent number may prove in time to be the
true one. It may be an exception of that valuable kind which upsets our
false theories, a real exception, exploding apparent coincidences, and
opening a way to a new view of the subject. To establish this position
for the divergent fact will require additional research; but in the
meantime we should give it some weight in our mean conclusions, and
should bear in mind the discrepancy as one demanding attention. To
neglect a divergent result is to neglect the possible clue to a great
discovery.
*Method of Least Squares.*
When two or more unknown quantities are so involved that they cannot
be separately determined by the Simple Method of Means, we can yet
obtain their most probable values by the Method of Least Squares,
without more difficulty than arises from the length of the arithmetical
computations. If the result of each observation gives an equation
between two unknown quantities of the form
*ax* + *by* = *c*
then, if the observations were free from error, we should need only two
observations giving two equations; but for the attainment of greater
accuracy, we may take many observations, and reduce the equations
so as to give only a pair with mean coefficients. This reduction is
effected by (1.), multiplying the coefficients of each equation by the
first coefficient, and adding together all the similar coefficients
thus resulting for the coefficients of a new equation; and (2.), by
repeating this process, and multiplying the coefficients of each
equation by the coefficient of the second term. Meaning by (sum of
*a*^{2}) the sum of all quantities of the same kind, and having the
same place in the equations as *a*^{2}, we may briefly describe the two
resulting mean equations as follows:--
(sum of *a*^{2}) . *x* + (sum of *ab*) . *y* = (sum of *ac*),
(sum of *ab*) . *x* + (sum of *b*^{2}) . *y* = (sum of *bc*).
When there are three or more unknown quantities the process is exactly
the same in nature, and we get additional mean equations by multiplying
by the third, fourth, &c., coefficients. As the numbers are in any
case approximate, it is usually unnecessary to make the computations
with accuracy, and places of decimals may be freely cut off to save
arithmetical work. The mean equations having been computed, their
solution by the ordinary methods of algebra gives the most probable
values of the unknown quantities.
*Works upon the Theory of Probability.*
Regarding the Theory of Probability and the Law of Error as most
important subjects of study for any one who desires to obtain a
complete comprehension of scientific method as actually applied in
physical investigations, I will briefly indicate the works in one or
other of which the reader will best pursue the study.
The best popular, and at the same time profound English work on the
subject is De Morgan’s “Essay on Probabilities and on their Application
to Life Contingencies and Insurance Offices,” published in the *Cabinet
Cyclopædia*, and to be obtained (in print) from Messrs. Longman. Mr.
Venn’s work on *The Logic of Chance* can now be procured in a greatly
enlarged second edition;[297] it contains a most interesting and able
discussion of the metaphysical basis of probability and of related
questions concerning causation, belief, design, testimony, &c.; but I
cannot always agree with Mr. Venn’s opinions. No mathematical knowledge
beyond that of common arithmetic is required in reading these works.
Quetelet’s *Letters* form a good introduction to the subject, and the
mathematical notes are of value. Sir George Airy’s brief treatise *On
the Algebraical and Numerical Theory of Errors of Observations and
the Combination of Observations*, contains a complete explanation of
the Law of Error and its practical applications. De Morgan’s treatise
“On the Theory of Probabilities” in the *Encyclopædia Metropolitana*,
presents an abstract of the more abstruse investigations of
Laplace, together with a multitude of profound and original remarks
concerning the theory generally. In Lubbock and Drinkwater’s work on
*Probability*, in the Library of Useful Knowledge, we have a concise
but good statement of a number of important problems. The Rev.
W. A. Whitworth has given, in a work entitled *Choice and Chance*,
a number of good illustrations of calculations both in combinations
and probabilities. In Mr. Todhunter’s admirable History we have an
exhaustive critical account of almost all writings upon the subject
of probability down to the culmination of the theory in Laplace’s
works. The Memoir of Mr. J. W. L. Glaisher has already been mentioned
(p. 375). In spite of the existence of these and some other good
English works, there seems to be a want of an easy and yet pretty
complete mathematical introduction to the study of the theory.
[297] *The Logic of Chance*, an Essay on the Foundations and Province
of the Theory of Probability, with especial reference to its
Logical Bearings and its Application to Moral and Social Science.
(Macmillan), 1876.
Among French works the Traité *Élémentaire du Calcul des Probabilités*,
by S. E. Lacroix, of which several editions have been published, and
which is not difficult to obtain, forms probably the best elementary
treatise. Poisson’s *Recherches sur la Probabilité des Jugements*
(Paris 1837), commence with an admirable investigation of the grounds
and methods of the theory. While Laplace’s great *Théorie Analytique
des Probabilités* is of course the “Principia” of the subject; his
*Essai Philosophique sur les Probabilités* is a popular discourse, and
is one of the most profound and interesting essays ever published. It
should be familiar to every student of logical method, and has lost
little or none of its importance by lapse of time.
*Detection of Constant Errors.*
The Method of Means is absolutely incapable of eliminating any error
which is always the same, or which always lies in one direction. We
sometimes require to be roused from a false feeling of security, and
to be urged to take suitable precautions against such occult errors.
“It is to the observer,” says Gauss,[298] “that belongs the task of
carefully removing the causes of constant errors,” and this is quite
true when the error is absolutely constant. When we have made a number
of determinations with a certain apparatus or method of measurement,
there is a great advantage in altering the arrangement, or even
devising some entirely different method of getting estimates of the
same quantity. The reason obviously consists in the improbability that
the same error will affect two or more different methods of experiment.
If a discrepancy is found to exist, we shall at least be aware of the
existence of error, and can take measures for finding in which way it
lies. If we can try a considerable number of methods, the probability
becomes great that errors constant in one method will be balanced or
nearly so by errors of an opposite effect in the others. Suppose that
there be three different methods each affected by an error of equal
amount. The probability that this error will in all fall in the same
direction is only 1/4; and with four methods similarly 1/8. If each
method be affected, as is always the case, by several independent
sources of error, the probability becomes much greater that in the mean
result of all the methods some of the errors will partially compensate
the others. In this case as in all others, when human vigilance has
exhausted itself, we must trust the theory of probability.
[298] Gauss, translated by Bertrand, p. 25.
In the determination of a zero point, of the magnitude of the
fundamental standards of time and space, in the personal equation of
an astronomical observer, we have instances of fixed errors; but as a
general rule a change of procedure is likely to reverse the character
of the error, and many instances may be given of the value of this
precaution. If we measure over and over again the same angular
magnitude by the same divided circle, maintained in exactly the same
position, it is evident that the same mark in the circle will be the
criterion in each case, and any error in the position of that mark will
equally affect all our results. But if in each measurement we use a
different part of the circle, a new mark will come into use, and as the
error of each mark cannot be in the same direction, the average result
will be nearly free from errors of division. It will be better still to
use more than one divided circle.
Even when we have no perception of the points at which error is
likely to enter, we may with advantage vary the construction of our
apparatus in the hope that we shall accidentally detect some latent
cause of error. Baily’s purpose in repeating the experiments of
Michell and Cavendish on the density of the earth was not merely to
follow the same course and verify the previous numbers, but to try
whether variations in the size and substance of the attracting balls,
the mode of suspension, the temperature of the surrounding air, &c.,
would yield different results. He performed no less than 62 distinct
series, comprising 2153 experiments, and he carefully classified and
discussed the results so as to disclose the utmost differences. Again,
in experimenting upon the resistance of the air to the motion of a
pendulum, Baily employed no less than 80 pendulums of various forms
and materials, in order to ascertain exactly upon what conditions
the resistance depends. Regnault, in his exact researches upon the
dilatation of gases, made arbitrary changes in the magnitude of parts
of his apparatus. He thinks that if, in spite of such modification,
the results are unchanged, the errors are probably of inconsiderable
amount;[299] but in reality it is always possible, and usually likely,
that we overlook sources of error which a future generation will
detect. Thus the pendulum experiments of Baily and Sabine were directed
to ascertain the nature and amount of a correction for air resistance,
which had been entirely misunderstood in the experiments by means of
the seconds pendulum, upon which was founded the definition of the
standard yard, in the Act of 5th George IV. c. 74. It has already been
mentioned that a considerable error was discovered in the determination
of the standard metre as the ten-millionth part of the distance from
the pole to the equator (p. 314).
[299] Jamin, *Cours de Physique*, vol. ii. p. 60.
We shall return in Chapter XXV. to the further consideration of the
methods by which we may as far as possible secure ourselves against
permanent and undetected sources of error. In the meantime, having
completed the consideration of the special methods requisite for
treating quantitative phenomena, we must pursue our principal subject,
and endeavour to trace out the course by which the physicist, from
observation and experiment, collects the materials of knowledge, and
then proceeds by hypothesis and inverse calculation to induce from them
the laws of nature.
Book IV.
INDUCTIVE INVESTIGATION.