Of The Remaining Laws Of Nature.
§ 1. In the First Book we found that all the assertions which can be
conveyed by language, express some one or more of five different things:
Existence; Order in Place; Order in Time; Causation; and Resemblance.(197)
Of these, Causation, in our view of the subject, not being fundamentally
different from Order in Time, the five species of possible assertions are
reduced to four. The propositions which affirm Order in Time in either of
its two modes, Co-existence and Succession, have formed, thus far, the
subject of the present Book. And we have now concluded the exposition, so
far as it falls within the limits assigned to this work, of the nature of
the evidence on which these propositions rest, and the processes of
investigation by which they are ascertained and proved. There remain three
classes of facts: Existence, Order in Place, and Resemblance; in regard to
which the same questions are now to be resolved.
Regarding the first of these, very little needs be said. Existence in
general, is a subject not for our science, but for metaphysics. To
determine what things can be recognized as really existing, independently
of our own sensible or other impressions, and in what meaning the term is,
in that case, predicated of them, belongs to the consideration of “Things
in themselves,” from which, throughout this work, we have as much as
possible kept aloof. Existence, so far as Logic is concerned about it, has
reference only to phenomena; to actual, or possible, states of external or
internal consciousness, in ourselves or others. Feelings of sensitive
beings, or possibilities of having such feelings, are the only things the
existence of which can be a subject of logical induction, because the only
things of which the existence in individual cases can be a subject of
experience.
It is true that a thing is said by us to exist, even when it is absent,
and therefore is not and can not be perceived. But even then, its
existence is to us only another word for our conviction that we should
perceive it on a certain supposition; namely, if we were in the needful
circumstances of time and place, and endowed with the needful perfection
of organs. My belief that the Emperor of China exists, is simply my belief
that if I were transported to the imperial palace or some other locality
in Pekin, I should see him. My belief that Julius Cæsar existed, is my
belief that I should have seen him if I had been present in the field of
Pharsalia, or in the senate-house at Rome. When I believe that stars exist
beyond the utmost range of my vision, though assisted by the most powerful
telescopes yet invented, my belief, philosophically expressed, is, that
with still better telescopes, if such existed, I could see them, or that
they may be perceived by beings less remote from them in space, or whose
capacities of perception are superior to mine.
The existence, therefore, of a phenomenon, is but another word for its
being perceived, or for the inferred possibility of perceiving it. When
the phenomenon is within the range of present observation, by present
observation we assure ourselves of its existence; when it is beyond that
range, and is therefore said to be absent, we infer its existence from
marks or evidences. But what can these evidences be? Other phenomena;
ascertained by induction to be connected with the given phenomenon, either
in the way of succession or of co-existence. The simple existence,
therefore, of an individual phenomenon, when not directly perceived, is
inferred from some inductive law of succession or co-existence; and is
consequently not amenable to any peculiar inductive principles. We prove
the existence of a thing, by proving that it is connected by succession or
co-existence with some known thing.
With respect to _general_ propositions of this class, that is, which
affirm the bare fact of existence, they have a peculiarity which renders
the logical treatment of them a very easy matter; they are generalizations
which are sufficiently proved by a single instance. That ghosts, or
unicorns, or sea-serpents exist, would be fully established if it could be
ascertained positively that such things had been even once seen. Whatever
has once happened, is capable of happening again; the only question
relates to the conditions under which it happens.
So far, therefore, as relates to simple existence, the Inductive Logic has
no knots to untie. And we may proceed to the remaining two of the great
classes into which facts have been divided; Resemblance, and Order in
Place.
§ 2. Resemblance and its opposite, except in the case in which they assume
the names of Equality and Inequality, are seldom regarded as subjects of
science; they are supposed to be perceived by simple apprehension; by
merely applying our senses or directing our attention to the two objects
at once, or in immediate succession. And this simultaneous, or virtually
simultaneous, application of our faculties to the two things which are to
be compared, does necessarily constitute the ultimate appeal, wherever
such application is practicable. But, in most cases, it is not
practicable: the objects can not be brought so close together that the
feeling of their resemblance (at least a complete feeling of it) directly
arises in the mind. We can only compare each of them with some third
object, capable of being transported from one to the other. And besides,
even when the objects can be brought into immediate juxtaposition, their
resemblance or difference is but imperfectly known to us, unless we have
compared them minutely, part by part. Until this has been done, things in
reality very dissimilar often appear undistinguishably alike. Two lines of
very unequal length will appear about equal when lying in different
directions; but place them parallel with their farther extremities even,
and if we look at the nearer extremities, their inequality becomes a
matter of direct perception.
To ascertain whether, and in what, two phenomena resemble or differ, is
not always, therefore, so easy a thing as it might at first appear. When
the two can not be brought into juxtaposition, or not so that the observer
is able to compare their several parts in detail, he must employ the
indirect means of reasoning and general propositions. When we can not
bring two straight lines together, to determine whether they are equal, we
do it by the physical aid of a foot-rule applied first to one and then to
the other, and the logical aid of the general proposition or formula,
“Things which are equal to the same thing are equal to one another.” The
comparison of two things through the intervention of a third thing, when
their direct comparison is impossible, is the appropriate scientific
process for ascertaining resemblances and dissimilarities, and is the sum
total of what Logic has to teach on the subject.
An undue extension of this remark induced Locke to consider reasoning
itself as nothing but the comparison of two ideas through the medium of a
third, and knowledge as the perception of the agreement or disagreement of
two ideas; doctrines which the Condillac school blindly adopted, without
the qualifications and distinctions with which they were studiously
guarded by their illustrious author. Where, indeed, the agreement or
disagreement (otherwise called resemblance or dissimilarity) of any two
things is the very matter to be determined, as is the case particularly in
the sciences of quantity and extension; there, the process by which a
solution, if not attainable by direct perception, must be indirectly
sought, consists in comparing these two things through the medium of a
third. But this is far from being true of all inquiries. The knowledge
that bodies fall to the ground is not a perception of agreement or
disagreement, but of a series of physical occurrences, a succession of
sensations. Locke’s definitions of knowledge and of reasoning required to
be limited to our knowledge of, and reasoning about, resemblances. Nor,
even when thus restricted, are the propositions strictly correct; since
the comparison is not made, as he represents, between the ideas of the two
phenomena, but between the phenomena themselves. This mistake has been
pointed out in an earlier part of our inquiry,(198) and we traced it to an
imperfect conception of what takes place in mathematics, where very often
the comparison is really made between the ideas, without any appeal to the
outward senses; only, however, because in mathematics a comparison of the
ideas is strictly equivalent to a comparison of the phenomena themselves.
Where, as in the case of numbers, lines, and figures, our idea of an
object is a complete picture of the object, so far as respects the matter
in hand; we can, of course, learn from the picture, whatever could be
learned from the object itself by mere contemplation of it as it exists at
the particular instant when the picture is taken. No mere contemplation of
gunpowder would ever teach us that a spark would make it explode, nor,
consequently, would the contemplation of the idea of gunpowder do so; but
the mere contemplation of a straight line shows that it can not inclose a
space; accordingly the contemplation of the idea of it will show the same.
What takes place in mathematics is thus no argument that the comparison is
between the ideas only. It is always, either indirectly or directly, a
comparison of the phenomena.
In cases in which we can not bring the phenomena to the test of direct
inspection at all, or not in a manner sufficiently precise, but must judge
of their resemblance by inference from other resemblances or
dissimilarities more accessible to observation, we of course require, as
in all cases of ratiocination, generalizations or formulæ applicable to
the subject. We must reason from laws of nature; from the uniformities
which are observable in the fact of likeness or unlikeness.
§ 3. Of these laws or uniformities, the most comprehensive are those
supplied by mathematics; the axioms relating to equality, inequality, and
proportionality, and the various theorems thereon founded. And these are
the only Laws of Resemblance which require to be, or which can be, treated
apart. It is true there are innumerable other theorems which affirm
resemblances among phenomena; as that the angle of the reflection of light
is _equal_ to its angle of incidence (equality being merely exact
resemblance in magnitude). Again, that the heavenly bodies describe
_equal_ areas in equal times; and that their periods of revolution are
_proportional_ (another species of resemblance) to the sesquiplicate
powers of their distances from the centre of force. These and similar
propositions affirm resemblances, of the same nature with those asserted
in the theorems of mathematics; but the distinction is, that the
propositions of mathematics are true of all phenomena whatever, or at
least without distinction of origin; while the truths in question are
affirmed only of special phenomena, which originate in a certain way; and
the equalities, proportionalities, or other resemblances, which exist
between such phenomena, must necessarily be either derived from, or
identical with, the law of their origin—the law of causation on which they
depend. The equality of the areas described in equal times by the planets,
is _derived_ from the laws of the causes; and, until its derivation was
shown, it was an empirical law. The equality of the angles of reflection
and incidence is _identical_ with the law of the cause; for the cause is
the incidence of a ray of light upon a reflecting surface, and the
equality in question is the very law according to which that cause
produces its effects. This class, therefore, of the uniformities of
resemblance between phenomena, are inseparable, in fact and in thought,
from the laws of the production of those phenomena; and the principles of
induction applicable to them are no other than those of which we have
treated in the preceding chapters of this Book.
It is otherwise with the truths of mathematics. The laws of equality and
inequality between spaces, or between numbers, have no connection with
laws of causation. That the angle of reflection is equal to the angle of
incidence, is a statement of the mode of action of a particular cause; but
that when two straight lines intersect each other the opposite angles are
equal, is true of all such lines and angles, by whatever cause produced.
That the squares of the periodic times of the planets are proportional to
the cubes of their distances from the sun, is a uniformity derived from
the laws of the causes (or forces) which produce the planetary motions;
but that the square of any number is four times the square of half the
number, is true independently of any cause. The only laws of resemblance,
therefore, which we are called upon to consider independently of
causation, belong to the province of mathematics.
§ 4. The same thing is evident with respect to the only one remaining of
our five categories, Order in Place. The order in place, of the effects of
a cause, is (like every thing else belonging to the effects) a consequence
of the laws of that cause. The order in place, or, as we have termed it,
the collocation, of the primeval causes, is (as well as their resemblance)
in each instance an ultimate fact, in which no laws or uniformities are
traceable. The only remaining general propositions respecting order in
place, and the only ones which have nothing to do with causation, are some
of the truths of geometry; laws through which we are able, from the order
in place of certain points, lines, or spaces, to infer the order in place
of others which are connected with the former in some known mode; quite
independently of the particular nature of those points, lines, or spaces,
in any other respect than position or magnitude, as well as independently
of the physical cause from which in any particular case they happen to
derive their origin.
It thus appears that mathematics is the only department of science into
the methods of which it still remains to inquire. And there is the less
necessity that this inquiry should occupy us long, as we have already, in
the Second Book, made considerable progress in it. We there remarked, that
the directly inductive truths of mathematics are few in number; consisting
of the axioms, together with certain propositions concerning existence,
tacitly involved in most of the so-called definitions. And we gave what
appeared conclusive reasons for affirming that these original premises,
from which the remaining truths of the science are deduced, are,
notwithstanding all appearances to the contrary, results of observation
and experience; founded, in short, on the evidence of the senses. That
things equal to the same thing are equal to one another, and that two
straight lines which have once intersected one another continue to
diverge, are inductive truths; resting, indeed, like the law of universal
causation, only on induction _per enumerationem simplicem_; on the fact
that they have been perpetually perceived to be true, and never once found
to be false. But, as we have seen in a recent chapter that this evidence,
in the case of a law so completely universal as the law of causation,
amounts to the fullest proof, so is this even more evidently true of the
general propositions to which we are now adverting; because, as a
perception of their truth in any individual case whatever, requires only
the simple act of looking at the objects in a proper position, there never
could have been in their case (what, for a long period, there were in the
case of the law of causation) instances which were apparently, though not
really, exceptions to them. Their infallible truth was recognized from the
very dawn of speculation; and as their extreme familiarity made it
impossible for the mind to conceive the objects under any other law, they
were, and still are, generally considered as truths recognized by their
own evidence, or by instinct.
§ 5. There is something which seems to require explanation, in the fact
that the immense multitude of truths (a multitude still as far from being
exhausted as ever) comprised in the mathematical sciences, can be elicited
from so small a number of elementary laws. One sees not, at first, how it
is that there can be room for such an infinite variety of true
propositions, on subjects apparently so limited.
To begin with the science of number. The elementary or ultimate truths of
this science are the common axioms concerning equality, namely, “Things
which are equal to the same thing are equal to one another,” and “Equals
added to equals make equal sums” (no other axioms are required),(199)
together with the definitions of the various numbers. Like other so-called
definitions, these are composed of two things, the explanation of a name,
and the assertion of a fact; of which the latter alone can form a first
principle or premise of a science. The fact asserted in the definition of
a number is a physical fact. Each of the numbers two, three, four, etc.,
denotes physical phenomena, and connotes a physical property of those
phenomena. Two, for instance, denotes all pairs of things, and twelve all
dozens of things, connoting what makes them pairs, or dozens; and that
which makes them so is something physical; since it can not be denied that
two apples are physically distinguishable from three apples, two horses
from one horse, and so forth; that they are a different visible and
tangible phenomenon. I am not undertaking to say what the difference is;
it is enough that there is a difference of which the senses can take
cognizance. And although a hundred and two horses are not so easily
distinguished from a hundred and three, as two horses are from
three—though in most positions the senses do not perceive any
difference—yet they may be so placed that a difference will be
perceptible, or else we should never have distinguished them, and given
them different names. Weight is confessedly a physical property of things;
yet small differences between great weights are as imperceptible to the
senses in most situations, as small differences between great numbers; and
are only put in evidence by placing the two objects in a peculiar
position—namely, in the opposite scales of a delicate balance.
What, then, is that which is connoted by a name of number? Of course, some
property belonging to the agglomeration of things which we call by the
name; and that property is, the characteristic manner in which the
agglomeration is made up of, and may be separated into, parts. I will
endeavor to make this more intelligible by a few explanations.
When we call a collection of objects _two_, _three_, or _four_, they are
not two, three, or four in the abstract; they are two, three, or four
things of some particular kind; pebbles, horses, inches, pounds’ weight.
What the name of number connotes is, the manner in which single objects of
the given kind must be put together, in order to produce that particular
aggregate. If the aggregate be of pebbles, and we call it _two_, the name
implies that, to compose the aggregate, one pebble must be joined to one
pebble. If we call it _three_, one and one and one pebble must be brought
together to produce it, or else one pebble must be joined to an aggregate
of the kind called _two_, already existing. The aggregate which we call
_four_, has a still greater number of characteristic modes of formation.
One and one and one and one pebble may be brought together; or two
aggregates of the kind called _two_ may be united; or one pebble may be
added to an aggregate of the kind called _three_. Every succeeding number
in the ascending series, may be formed by the junction of smaller numbers
in a progressively greater variety of ways. Even limiting the parts to
two, the number may be formed, and consequently may be divided, in as many
different ways as there are numbers smaller than itself; and, if we admit
of threes, fours, etc., in a still greater variety. Other modes of
arriving at the same aggregate present themselves, not by the union of
smaller, but by the dismemberment of larger aggregates. Thus, _three
pebbles_ may be formed by taking away one pebble from an aggregate of
four; _two pebbles_, by an equal division of a similar aggregate; and so
on.
Every arithmetical proposition; every statement of the result of an
arithmetical operation; is a statement of one of the modes of formation of
a given number. It affirms that a certain aggregate might have been formed
by putting together certain other aggregates, or by withdrawing certain
portions of some aggregate; and that, by consequence, we might reproduce
those aggregates from it, by reversing the process.
Thus, when we say that the cube of 12 is 1728, what we affirm is this:
that if, having a sufficient number of pebbles or of any other objects, we
put them together into the particular sort of parcels or aggregates called
twelves; and put together these twelves again into similar collections;
and, finally, make up twelve of these largest parcels; the aggregate thus
formed will be such a one as we call 1728; namely, that which (to take the
most familiar of its modes of formation) may be made by joining the parcel
called a thousand pebbles, the parcel called seven hundred pebbles, the
parcel called twenty pebbles, and the parcel called eight pebbles.
The converse proposition that the cube root of 1728 is 12, asserts that
this large aggregate may again be decomposed into the twelve twelves of
twelves of pebbles which it consists of.
The modes of formation of any number are innumerable; but when we know one
mode of formation of each, all the rest may be determined deductively. If
we know that _a_ is formed from _b_ and _c_, _b_ from _a_ and _e_, _c_
from _d_ and _f_, and so forth, until we have included all the numbers of
any scale we choose to select (taking care that for each number the mode
of formation be really a distinct one, not bringing us round again to the
former numbers, but introducing a new number), we have a set of
propositions from which we may reason to all the other modes of formation
of those numbers from one another. Having established a chain of inductive
truths connecting together all the numbers of the scale, we can ascertain
the formation of any one of those numbers from any other by merely
traveling from one to the other along the chain. Suppose that we know only
the following modes of formation: 6=4+2, 4=7–3, 7=5+2, 5=9–4. We could
determine how 6 may be formed from 9. For 6=4+2=7–3+2=5+2–3+2=9–4+2–3+2.
It may therefore be formed by taking away 4 and 3, and adding 2 and 2. If
we know besides that 2+2=4, we obtain 6 from 9 in a simpler mode, by
merely taking away 3.
It is sufficient, therefore, to select one of the various modes of
formation of each number, as a means of ascertaining all the rest. And
since things which are uniform, and therefore simple, are most easily
received and retained by the understanding, there is an obvious advantage
in selecting a mode of formation which shall be alike for all; in fixing
the connotation of names of number on one uniform principle. The mode in
which our existing numerical nomenclature is contrived possesses this
advantage, with the additional one, that it happily conveys to the mind
two of the modes of formation of every number. Each number is considered
as formed by the addition of a unit to the number next below it in
magnitude, and this mode of formation is conveyed by the place which it
occupies in the series. And each is also considered as formed by the
addition of a number of units less than ten, and a number of aggregates
each equal to one of the successive powers of ten; and this mode of its
formation is expressed by its spoken name, and by its numerical character.
What renders arithmetic the type of a deductive science, is the fortunate
applicability to it of a law so comprehensive as “The sums of equals are
equals:” or (to express the same principle in less familiar but more
characteristic language), Whatever is made up of parts, is made up of the
parts of those parts. This truth, obvious to the senses in all cases which
can be fairly referred to their decision, and so general as to be
co-extensive with nature itself, being true of all sorts of phenomena (for
all admit of being numbered), must be considered an inductive truth, or
law of nature, of the highest order. And every arithmetical operation is
an application of this law, or of other laws capable of being deduced from
it. This is our warrant for all calculations. We believe that five and two
are equal to seven, on the evidence of this inductive law, combined with
the definitions of those numbers. We arrive at that conclusion (as all
know who remember how they first learned it) by adding a single unit at a
time: 5 + 1=6, therefore 5+1+1=6+1=7; and again 2=1+1, therefore
5+2=5+1+1=7.
§ 6. Innumerable as are the true propositions which can be formed
concerning particular numbers, no adequate conception could be gained,
from these alone, of the extent of the truths composing the science of
number. Such propositions as we have spoken of are the least general of
all numerical truths. It is true that even these are co-extensive with all
nature; the properties of the number four are true of all objects that are
divisible into four equal parts, and all objects are either actually or
ideally so divisible. But the propositions which compose the science of
algebra are true, not of a particular number, but of all numbers; not of
all things under the condition of being divided in a particular way, but
of all things under the condition of being divided in any way—of being
designated by a number at all.
Since it is impossible for different numbers to have any of their modes of
formation completely in common, it is a kind of paradox to say, that all
propositions which can be made concerning numbers relate to their modes of
formation from other numbers, and yet that there are propositions which
are true of all numbers. But this very paradox leads to the real principle
of generalization concerning the properties of numbers. Two different
numbers can not be formed in the same manner from the same numbers; but
they may be formed in the same manner from different numbers; as nine is
formed from three by multiplying it into itself, and sixteen is formed
from four by the same process. Thus there arises a classification of modes
of formation, or in the language commonly used by mathematicians, a
classification of Functions. Any number, considered as formed from any
other number, is called a function of it; and there are as many kinds of
functions as there are modes of formation. The simple functions are by no
means numerous, most functions being formed by the combination of several
of the operations which form simple functions, or by successive
repetitions of some one of those operations. The simple functions of any
number _x_ are all reducible to the following forms: _x_+_a_, _x_–_a_,
_ax_, _x_/_a_, log. _x_ (to the base _a_), and the same expressions varied
by putting _x_ for _a_ and _a_ for _x_, wherever that substitution would
alter the value: to which, perhaps, ought to be added sin _x_, and arc
(sin=_x_). All other functions of _x_ are formed by putting some one or
more of the simple functions in the place of _x_ or _a_, and subjecting
them to the same elementary operations.
In order to carry on general reasonings on the subject of Functions, we
require a nomenclature enabling us to express any two numbers by names
which, without specifying what particular numbers they are, shall show
what function each is of the other; or, in other words, shall put in
evidence their mode of formation from one another. The system of general
language called algebraical notation does this. The expressions _a_ and
a2+3a denote, the one any number, the other the number formed from it in a
particular manner. The expressions _a_, _b_, _n_, and (_a+b_)n, denote any
three numbers, and a fourth which is formed from them in a certain mode.
The following may be stated as the general problem of the algebraical
calculus: F being a certain function of a given number, to find what
function F will be of any function of that number. For example, a binomial
_a + b_ is a function of its two parts _a_ and _b_, and the parts are, in
their turn, functions of _a + b_: now (_a + b_)n is a certain function of
the binomial; what function will this be of _a_ and _b_, the two parts?
The answer to this question is the binomial theorem. The formula _(a +
b)__n__ = a__n__ + n/1 a__n–1__ b + n.n–1/1.2 a__n–2__ b__2_, etc., shows
in what manner the number which is formed by multiplying _a + b_ into
itself _n_ times, might be formed without that process, directly from _a,
b_, and _n_. And of this nature are all the theorems of the science of
number. They assert the identity of the result of different modes of
formation. They affirm that some mode of formation from _x_, and some mode
of formation from a certain function of _x_, produce the same number.
Such, as above described, is the aim and end of the calculus. As for its
processes, every one knows that they are simply deductive. In
demonstrating an algebraical theorem, or in resolving an equation, we
travel from the _datum_ to the _quæsitum_ by pure ratiocination; in which
the only premises introduced, besides the original hypotheses, are the
fundamental axioms already mentioned—that things equal to the same thing
are equal to one another, and that the sums of equal things are equal. At
each step in the demonstration or in the calculation, we apply one or
other of these truths, or truths deducible from them, as, that the
differences, products, etc., of equal numbers are equal.
It would be inconsistent with the scale of this work, and not necessary to
its design, to carry the analysis of the truths and processes of algebra
any further; which is also the less needful, as the task has been, to a
very great extent, performed by other writers. Peacock’s Algebra, and Dr.
Whewell’s _Doctrine of Limits_, are full of instruction on the subject.
The profound treatises of a truly philosophical mathematician, Professor
De Morgan, should be studied by every one who desires to comprehend the
evidence of mathematical truths, and the meaning of the obscurer processes
of the calculus, and the speculations of M. Comte, in his _Cours de
Philosophie Positive_, on the philosophy of the higher branches of
mathematics, are among the many valuable gifts for which philosophy is
indebted to that eminent thinker.
§ 7. If the extreme generality, and remoteness not so much from sense as
from the visual and tactual imagination, of the laws of number, renders it
a somewhat difficult effort of abstraction to conceive those laws as being
in reality physical truths obtained by observation; the same difficulty
does not exist with regard to the laws of extension. The facts of which
those laws are expressions, are of a kind peculiarly accessible to the
senses, and suggesting eminently distinct images to the fancy. That
geometry is a strictly physical science would doubtless have been
recognized in all ages, had it not been for the illusions produced by two
circumstances. One of these is the characteristic property, already
noticed, of the facts of geometry, that they may be collected from our
ideas or mental pictures of objects as effectually as from the objects
themselves. The other is, the demonstrative character of geometrical
truths; which was at one time supposed to constitute a radical distinction
between them and physical truths; the latter, as resting on merely
probable evidence, being deemed essentially uncertain and unprecise. The
advance of knowledge has, however, made it manifest that physical science,
in its better understood branches, is quite as demonstrative as geometry.
The task of deducing its details from a few comparatively simple
principles is found to be any thing but the impossibility it was once
supposed to be; and the notion of the superior certainty of geometry is an
illusion, arising from the ancient prejudice which, in that science,
mistakes the ideal data from which we reason, for a peculiar class of
realities, while the corresponding ideal data of any deductive physical
science are recognized as what they really are, hypotheses.
Every theorem in geometry is a law of external nature, and might have been
ascertained by generalizing from observation and experiment, which in this
case resolve themselves into comparison and measurement. But it was found
practicable, and, being practicable, was desirable, to deduce these truths
by ratiocination from a small number of general laws of nature, the
certainty and universality of which are obvious to the most careless
observer, and which compose the first principles and ultimate premises of
the science. Among these general laws must be included the same two which
we have noticed as ultimate principles of the Science of Number also, and
which are applicable to every description of quantity; viz., The sums of
equals are equal, and Things which are equal to the same thing are equal
to one another; the latter of which may be expressed in a manner more
suggestive of the inexhaustible multitude of its consequences, by the
following terms: Whatever is equal to any one of a number of equal
magnitudes, is equal to any other of them. To these two must be added, in
geometry, a third law of equality, namely, that lines, surfaces, or solid
spaces, which can be so applied to one another as to coincide, are equal.
Some writers have asserted that this law of nature is a mere verbal
definition; that the expression “equal magnitudes” _means_ nothing but
magnitudes which can be so applied to one another as to coincide. But in
this opinion I can not agree. The equality of two geometrical magnitudes
can not differ fundamentally in its nature from the equality of two
weights, two degrees of heat, or two portions of duration, to none of
which would this definition of equality be suitable. None of these things
can be so applied to one another as to coincide, yet we perfectly
understand what we mean when we call them equal. Things are equal in
magnitude, as things are equal in weight, when they are felt to be exactly
similar in respect of the attribute in which we compare them: and the
application of the objects to each other in the one case, like the
balancing them with a pair of scales in the other, is but a mode of
bringing them into a position in which our senses can recognize
deficiencies of exact resemblance that would otherwise escape our notice.
Along with these three general principles or axioms, the remainder of the
premises of geometry consists of the so-called definitions: that is to
say, propositions asserting the real existence of the various objects
therein designated, together with some one property of each. In some cases
more than one property is commonly assumed, but in no case is more than
one necessary. It is assumed that there are such things in nature as
straight lines, and that any two of them setting out from the same point,
diverge more and more without limit. This assumption (which includes and
goes beyond Euclid’s axiom that two straight lines can not inclose a
space) is as indispensable in geometry, and as evident, resting on as
simple, familiar, and universal observation, as any of the other axioms.
It is also assumed that straight lines diverge from one another in
different degrees; in other words, that there are such things as angles,
and that they are capable of being equal or unequal. It is assumed that
there is such a thing as a circle, and that all its radii are equal; such
things as ellipses, and that the sums of the focal distances are equal for
every point in an ellipse; such things as parallel lines, and that those
lines are everywhere equally distant.(200)
§ 8. It is a matter of more than curiosity to consider, to what
peculiarity of the physical truths which are the subject of geometry, it
is owing that they can all be deduced from so small a number of original
premises; why it is that we can set out from only one characteristic
property of each kind of phenomenon, and with that and two or three
general truths relating to equality, can travel from mark to mark until we
obtain a vast body of derivative truths, to all appearance extremely
unlike those elementary ones.
The explanation of this remarkable fact seems to lie in the following
circumstances. In the first place, all questions of position and figure
may be resolved into questions of magnitude. The position and figure of
any object are determined by determining the position of a sufficient
number of points in it; and the position of any point may be determined by
the magnitude of three rectangular co-ordinates, that is, of the
perpendiculars drawn from the point to three planes at right angles to one
another, arbitrarily selected. By this transformation of all questions of
quality into questions only of quantity, geometry is reduced to the single
problem of the measurement of magnitudes, that is, the ascertainment of
the equalities which exist between them. Now when we consider that by one
of the general axioms, any equality, when ascertained, is proof of as many
other equalities as there are other things equal to either of the two
equals; and that by another of those axioms, any ascertained equality is
proof of the equality of as many pairs of magnitudes as can be formed by
the numerous operations which resolve themselves into the addition of the
equals to themselves or to other equals; we cease to wonder that in
proportion as a science is conversant about equality, it should afford a
more copious supply of marks of marks; and that the sciences of number and
extension, which are conversant with little else than equality, should be
the most deductive of all the sciences.
There are also two or three of the principal laws of space or extension
which are unusually fitted for rendering one position or magnitude a mark
of another, and thereby contributing to render the science largely
deductive. First, the magnitudes of inclosed spaces, whether superficial
or solid, are completely determined by the magnitudes of the lines and
angles which bound them. Secondly, the length of any line, whether
straight or curve, is measured (certain other things being given) by the
angle which it subtends, and _vicè versa_. Lastly, the angle which any two
straight lines make with each other at an inaccessible point, is measured
by the angles they severally make with any third line we choose to select.
By means of these general laws, the measurement of all lines, angles, and
spaces whatsoever might be accomplished by measuring a single straight
line and a sufficient number of angles; which is the plan actually pursued
in the trigonometrical survey of a country; and fortunate it is that this
is practicable, the exact measurement of long straight lines being always
difficult, and often impossible, but that of angles very easy. Three such
generalizations as the foregoing afford such facilities for the indirect
measurement of magnitudes (by supplying us with known lines or angles
which are marks of the magnitude of unknown ones, and thereby of the
spaces which they inclose), that it is easily intelligible how from a few
data we can go on to ascertain the magnitude of an indefinite multitude of
lines, angles, and spaces, which we could not easily, or could not at all,
measure by any more direct process.
§ 9. Such are the remarks which it seems necessary to make in this place,
respecting the laws of nature which are the peculiar subject of the
sciences of number and extension. The immense part which those laws take
in giving a deductive character to the other departments of physical
science, is well known; and is not surprising, when we consider that all
causes operate according to mathematical laws. The effect is always
dependent on, or is a function of, the quantity of the agent; and
generally of its position also. We can not, therefore, reason respecting
causation, without introducing considerations of quantity and extension at
every step; and if the nature of the phenomena admits of our obtaining
numerical data of sufficient accuracy, the laws of quantity become the
grand instrument for calculating forward to an effect, or backward to a
cause. That in all other sciences, as well as in geometry, questions of
quality are scarcely ever independent of questions of quantity, may be
seen from the most familiar phenomena. Even when several colors are mixed
on a painter’s palette, the comparative quantity of each entirely
determines the color of the mixture.
With this mere suggestion of the general causes which render mathematical
principles and processes so predominant in those deductive sciences which
afford precise numerical data, I must, on the present occasion, content
myself; referring the reader who desires a more thorough acquaintance with
the subject, to the first two volumes of M. Comte’s systematic work.
In the same work, and more particularly in the third volume, are also
fully discussed the limits of the applicability of mathematical principles
to the improvement of other sciences. Such principles are manifestly
inapplicable, where the causes on which any class of phenomena depend are
so imperfectly accessible to our observation, that we can not ascertain,
by a proper induction, their numerical laws; or where the causes are so
numerous, and intermixed in so complex a manner with one another, that
even supposing their laws known, the computation of the aggregate effect
transcends the powers of the calculus as it is, or is likely to be; or,
lastly, where the causes themselves are in a state of perpetual
fluctuation; as in physiology, and still more, if possible, in the social
science. The mathematical solutions of physical questions become
progressively more difficult and imperfect, in proportion as the questions
divest themselves of their abstract and hypothetical character, and
approach nearer to the degree of complication actually existing in nature;
insomuch that beyond the limits of astronomical phenomena, and of those
most nearly analogous to them, mathematical accuracy is generally obtained
“at the expense of the reality of the inquiry:” while even in astronomical
questions, “notwithstanding the admirable simplicity of their mathematical
elements, our feeble intelligence becomes incapable of following out
effectually the logical combinations of the laws on which the phenomena
are dependent, as soon as we attempt to take into simultaneous
consideration more than two or three essential influences.”(201) Of this,
the problem of the Three Bodies has already been cited, more than once, as
a remarkable instance; the complete solution of so comparatively simple a
question having vainly tried the skill of the most profound
mathematicians. We may conceive, then, how chimerical would be the hope
that mathematical principles could be advantageously applied to phenomena
dependent on the mutual action of the innumerable minute particles of
bodies, as those of chemistry, and still more, of physiology; and for
similar reasons those principles remain inapplicable to the still more
complex inquiries, the subjects of which are phenomena of society and
government.
The value of mathematical instruction as a preparation for those more
difficult investigations, consists in the applicability not of its
doctrines, but of its method. Mathematics will ever remain the most
perfect type of the Deductive Method in general; and the applications of
mathematics to the deductive branches of physics, furnish the only school
in which philosophers can effectually learn the most difficult and
important portion of their art, the employment of the laws of simpler
phenomena for explaining and predicting those of the more complex. These
grounds are quite sufficient for deeming mathematical training an
indispensable basis of real scientific education, and regarding (according
to the _dictum_ which an old but unauthentic tradition ascribes to Plato)
one who is ἀγεωμέτρητος, as wanting in one of the most essential
qualifications for the successful cultivation of the higher branches of
philosophy.