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.[36] 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, Coexistence 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 recognised 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 cannot 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 coexistence. The simple existence,
therefore, of an individual phenomenon, when not directly perceived, is
inferred from some inductive law of succession or coexistence: 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 coexistence 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 Space.
§ 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 cannot 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 cannot 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
cannot 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,[37] 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 learnt 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 cannot
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 cannot 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 reflexion 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 connexion with
laws of causation. That the angle of reflexion 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 an
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 everything 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 recognised 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 recognised 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,[38]) 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, &c., 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 cannot 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
endeavour 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, &c., 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 travelling 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 an 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 coextensive 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 coextensive 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 cannot 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_, _a x_, _x/a_, _x^a_, _a [root
of] x_, 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
_a^2 + 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 + &c._, 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.
Besides these general theorems of formulæ, what remains in the
algebraical calculus is the resolution of equations. But the resolution
of an equation is also a theorem. If the equation be _x^2 + ax = b_,
the resolution of this equation, viz. _x = -(1/2) a ± [root of]((1/4)
a^2 + b)_, is a general proposition, which may be regarded as an
answer to the question, If _b_ is a certain function of _x_ and _a_
(namely _x^2 + ax_), what function is _x_ of _b_ and _a_? The
resolution of equations is, therefore, a mere variety of the general
problem as above stated. The problem is--Given a function, what function
is it of some other function? And in the resolution of an equation, the
question is, to find what function of one of its own functions the
number itself is.
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, &c., 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 farther; 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 recognised 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 anything 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 recognised as what they really are,
mere 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 cannot agree. The
equality of two geometrical magnitudes cannot 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 pretended 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 recognise 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 cannot
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.[39]
§ 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 _vice versâ_. 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 few remarks which it seemed 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 cannot, 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 colours are mixed on a painter's palette, the
comparative quantity of each entirely determines the colour 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 cannot 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."[40] 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 [Greek: ageômetrêtos], as
wanting in one of the most essential qualifications for the successful
cultivation of the higher branches of philosophy.