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AN ESSAY
THE RELATION OF THE SEVERAL PARTS
MATHEMATICAL SCIENCE
TO
THE FUNDAMENTAL IDEA THEREIN CONTAINED;
THE SUBSTANCE OF WHICH WAS READ BEFORE
THE ASHMOLEAN SOCIETY ON THE EVENING ;OF MAY 14, 1849.
BY
BARTHOLOMEW PRICE, M.A.
FELLOW AND TUTOR OF PEMBROKE COLLEGE, OXFORD.
OXFORD,
PRINTED BY THOMAS COMBE, PRINTER TO THE UNIVERSITY, FOR
THE ASHMOLEAN SOCIETY,
M.DCCC.XLJX,
A
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On the relation of the several parts of a Mathematical
Science to the fwidamental Idea therein contained.
J. HE inquirer into the principles and structure of a mathematical
science has two questions to discuss : of which, one has refereace
directly to the source whence its scientific truth springs, and inci-
dentally only to the history of the rise and growth of the particular
science ; and the other to the arrangement and sequence or con-
nexion with each other of the different portions of the subject-
matter of the science in the more perfect and developed form in
which it is offered to his contemplation : the former of these two
questions belongs to Metaphysic, and the latter to Logic ; but they
are so intimately related to each other, that it is impossible to con-
fiider "eitlitr, without introducing matter whidh more properly be-
longs to the other : and although the object in the following paper
is to elucidate the latter, viz. the logical question, yet the assump-
tion of certain propositions, the discussion of whose certitude and
truth belongs to metaphysic, will be necessary ; but it is to be
borne in mind, that they have been assumed in order to enable us
to follow out our train of argument in the particular subject w^hich
we are going to discuss.
The * vexata quaestio" of metaphysic, from the rise of philo-
sophy to the present time, is this, " Have we, or have we not, any
ideas which are necessarily and universally true }" and it resolves
itself into a question of another form, " Have we, or have we not,
any knowledge independent of experience ?"
On these questions two schools have been formed ; one declares
that all knowledge is given in experience, and that its materials are
derived from sensations^ which are impressions on the mind of ex-
ternal objects ; and that the mind, endowed as it is with the facul-
ties of attention and retention, is the repository of them ; and that
from reflection on these, combination of them, and generalizations
founded on them, knowledge, real and systematic, is formed : the
other school declares, that experience gives us only a portion of our
knowledge ; that there are elements of it which were never derived
A 2
from sensatiob, ari(f''w^ifcK,''n account of their necessity and
univers'pJijtV,;kl^soldtcfly, ^Tatscfend all experience; that the rough
materials aVe given 'by e'rripiricai observation, and that they may be
accumulated in any quantity and quality ; but that the form which
is imposed on them, and which constitutes them systematic know-
ledge, is supplied by the mind : thus, that systematic knowledge
consists of two elements, of which one is acquired from experience,
and the other from certain ideas natural to the mind, which perhaps
observation and experience may call into action ; but that the know-
ledge arising from these has such characteristics of necessity and
universality which no amount* of experience can possibly give.
While then, of these two contradictory schools, one says that all
knowledge is derived from experience, and passes through the
senses to the mind, which is the storehouse of it; the other as-
sumes that there are certain ideas in the mind previous to all con-
tact with the world of sense, which are sources and mainsprings of
knowledge, and that whatever we know of external objects, we
know only so far as they accord with these ideas, and as far as
these ideas combine them : one therefore says that all real and
systematic knowledge comes from within ; the other, that it comes
from without : one, that all knowledge is subjective ; the other, that
there are certain real objects existent independently of the human
mind, which the mind has the capacity of cognizing, and that when,
and only when, these are cognized, has it real and substantial know-
ledge : the one says that our knowledge starts with ideas, and ends
with facts that accord with these ideas ; the other, that it begins
with facts, and by a plastic faculty of the human mind, which com-
bines many facts in one idea, ends with the idea ; and while one
school would sum up its philosophy in an axiomatic dictum of the
scholastic doctors, " nihil est in intellectu, quod non fuerit in sensu,"
and almost ignore the restriction of Leibnitz, " nisi ipse intellectus,"
the other school would reply by such arguments as these ; '^' You
may take away from your experience-conception of a body all that
is empirical therein, colour, hardness, softness, weight, this or that
shape, still the space remains which is occupied, and that you cannot
take away : or you may omit from your empirical conception of any
corporeal or incorporeal object all that experience teaches, yet you
cannot take away from it that by which you think upon it as a sub-
stance ; and thus it would argue that these and such conceptions
must have their seat in the faculty of cognition a priori."
An examination of the logical processes and arrangement of some
of the mathematical and physico-mathematical sciences will involve
the adoption of one or the other of the two contradictory systems.
It is beside the object of the present paper to discuss their re-
spective arguments ; but the four following propositions contain the
substance of what is assumed and will be said ; and from these it is
apparent which of the two systems we have adopted, and the illus-
trations, as far as their value goes, will give evidence to the truth
of it.
I. There are in the human mind certain general ideas or con-
ceptions, capable of development, such as those of number, space,
time, motion, substance, and the like.
II. The pure and exact sciences 'consist of systematised series
and aggregates of the several processes by which these pregnant
ideas are called into action and subsequently developed.
III. That although we cannot a priori assert that these sciences
are applicable to objects of the external world, and to the explana-
tion of phaenomena which we observe, yet they may be so, and are
so, in so far as we can prove that the primary and axiomatic parts
of the idea have their counterparts in nature.
IV. That thus every such science to be useful must consist of
two parts ; one in which the idea pregnant with consequences is
analysed and developed ; and a second, in which the phsenomena
of nature are observed with the view of determining, first, whether
at all, and, secondly, how far, the laws of the external world accord
with the axiomatic statement of the science.
Without entering into the question whence the mind gets its
ideas, whether they are innate, or whether they are acquired ;
whether they are the necessary results of a comparison of the
objects of the external world, or whether the mind does not of
necessity get them so far as to express them by words from the
circumstances in which itself grows and is a mind ; yet independent
of such an investigation, in the construction of a science which
arises from such an idea as I conceive to be at the foundation of
the purely mathematical sciences, there must be two logical pro-
cesses ; one, the deductive, by means of which parts and properties
of the idea are enunciated and defined, divisions and classifications
are made, what is arbitrary and accidental is separated from what
is necessary and germinal, by means of which propositions are formed
from the comparison of parts of the idea which are more and less
general, and the equivalents of such propositions are framed, and
from general propositions others less general are deduced ; that
process, in short, by which we resolve the pregnant idea into its
several constituent parts, and deduce particular propositions which
the general statement of the idea imports; but when our deductive
science has been thus constructed from an idea in the mind, our
next business is, to inquire, whether our scientific idea and 'pro-
positions have their counterparts in the external world of sense;
our science may be merely an intellectual discipHne, useful perhaps
for cultivating- the mental faculties, and indicating the import of
pregnant propositions, and no more than this. Or, since man is
not like one of Aristotle's deities airdpKrjs, happy in purely sub-
jective contemplation, but inasmuch as he has body as well as
soul, and lives in contact with a marvellous complication of external
phsenomena, to the examination of which he is driven by mental
impulses, it is likely that the external world corresponds to his
subjective ideas ; at all events it is worth while to inquire whether
it does : hence arises the necessity of examining facts and pheno-
mena, with the view of detecting general laws subject to which
they take place, in order that we may compare them with the pro-
positions, axiomatic or inferred, of our science ; we must therefore
analyse effects with the object of determining the laws of their
necessary antecedent causes: hence there must be another process,
which shall guide us in gathering from our experience of and inter-
course with the external world the general laws subject to whicll
its agencies are conducted; which shall give us rules to ascenJ^
from particular and observed facts to general propositions ; which
shall furnish us with criteria to discover the general law lying con'-'^
cealed in the particular instance ; by this process we shall inter-^"
rogate nature, and determine how far her laws are identical witM^^
those of our scientific idea ; this process is manifestly the reverb'-
one to the former, and is called the inductive; and although w^
might a priori (on account of the manner in which, as metaphyi''"
sicians say, the idea is arrived at) expect that our science would
correspond with the phaenomena of nature, yet we can apply it only
so far as the inductive propositions allow us ; this latter process then
will in a manner make real and substantial that science which
before was only subjective, being a creature of and existing in the
human mind ; in the complete discussion therefore of any .badswJbif
science which has an objective reality as well as subjective existen^^c.
these two processes must enter. "^'
If this be a true account of the constitution of a science, the divifj!
sion of sciences into "pure" and "applied" is founded on a wrong'
prmciple ; for the more correct division would be into sciences
"perfect" and "imperfect:" those, that is, which have reached a
state in which the above two elements are combined, and those
which want one or the other of them. By perfect sciences there-
fore I mean such as have a leading and pregnant idea, supphed by
the mind ; an idea distinctly apprehended, clearly defined, and the
limits of it accurately set out : that which renders the truths of the
science superior to all experimental evidence, and gives them an uni-
versality and a necessity which is characteristic of the science ; such
are, I conceive, the ideas of number, geometrical space, motion, and
such like, and therefore the several corresponding sciences of num-
ber, geometry, mechanics, ought to have the corresponding notes of
perfection ; these however have an application of greater or less
latitude in the phsenomena of the external world ; greater or less, I
say, because the idea has certain phases and the a priori science
certain parts, the counterparts to which nature does not exhibit :
though she therefore may illustrate and interpret much, yet there may
be and is much which goes beyond her^; although probably she has
first suggested the idea which otherwise would lie dormant, still the
mind, by a salient and plastic power of its own, has given to the idea
such powers, that it is no longer to be kept within the limits which
nature and experience impose, but going beyond these constructs a
science ; and such sciences have in many cases foretold, anticipated,^
and led the way to some of the most famous discoveries of natural
philosophy, results have been deductively inferred from them which
have taken the world by surprise, and an examination of nature has
subsequently shewn her exact accordance with them : assuming then
the principles of the science to be true, all consequences, those ana-
logous to nature's laws no more than others, follow with the same
logical cogency ; and the deductive processes of our sciences of
number, geometrical space, and motion, are exactly the same for all
abstract properties of number, for the most recondite investigations
of infinitesimal analysis, for all conceivable geometrical surfaces, and
for all laws of dynamical force, as they are for the applied branches
a A late eminent Professor of Moral Pliilosophy in this University, Mr. Mills, .
writes as follows in a lecture on the Origin of Ideas : " We may safely allow that
sensation gives the first impulse j we may agree with Bacon and Locke that
knowledge is built upon experience ; but it is the active and independent power
of the understanding which regulates and fashions anew the information commu-*'
nicated by the feeUngs of sensation, and which ascends from the first lessons of
experience to the general and immutable principles of virtue and science. By
this divine light of reason kindled in the soul, man vindicates his high original
and future destiny ; developes the faculties and energies with which his Creator
has endowed him; and, so far as a humble sense of his dependence on the Foun- '
tain of all intelligence will permit, feels a just pride as he contemplates the moral
and intellectual strength of Butler, Pascal, or Newton."
8
of these several sciences : for commercial arithmetic, for land-mea-
suring, for physical astronomy.
It is true that in a particular science one element may appear
more prominent than the other ; in our science of mechanics the ex-
perimental and apphed part at once arrests the attention, for the
whole world is in motion, matter's state is continually changing,
for momentum is ever being transferred from one body to another,
and thus in most cases we fail to discover the pure science of motion
or to realize its idea it is lost in its applied form ; yet not the less
really does it exist, and a science of motion may be drawn from it :
it is on this principle that D'Alembert has constructed his " Traits
de Dynamique;" so on the other hand, the science of number seems
to us now to be the necessary development of the abstract idea of
quantupHcity ; we do not seek for any confirmation of its truths in
experience, we scarcely look beyond the range of the abstract science
for illustration, so easily apprehensible do we think its truths to be ;
thus in common arithmetic, as soon as the idea of definite number
is apprehended, we do not even ordinarily think it necessary to state
such axioms as " If equal numbers be added to equals the wholes are
equal," " Numbers equal to the same number are equal to one an-
other," or to explain what equality of numbers is ; and the common
rules of arithmetic, those of addition, subtraction, proportion, &c. are
at once assented to ; and these become the first premises of our
science, which are immediately deduced from the idea, and from
which all the results of the science follow by a deductive process ;
yet there is an applied part of the science which renders our common
arithmetic useful : commercial arithmetic, our systems of weights
and measures, division of time, exchanges, are all applications of the
rules and processes of the exact science ; but before this can be lo-
gically done, we must have assured ourselves by an inductive pro-
cess, which however may be so rapid as to escape observation, that
the axioms and laws of our pure science have their counterparts in
the applied : as for instance, that matter is discretive and susceptible
of discontinuous, though of definite, division ; that it is additive
simply, and that when two units are added, one is not absorbed into
the other : these and such like properties o f the pure science must
also be true in the matter of the applied science ; doubtless these
two elements may be best traced in the history of the science's
growth, and in the case of common arithmetic we are under a disad-
vantage, for little or nothing is known of its history ; in the earliest
records it is found to be in its principles and rules nearly as perfect
as it is now ; and as its idea is so simple, and as it exists in only
one dimension, the mind apprehends it at once, and we are wont to
overlook the inductive and apphed element. In another branch
however of the same science, viz. in Infinitesimal Analysis, we have
more ample illustration, so that the applied element is more clearly
apprehended than the ideal; the conception of number is at the
foundation of it, but under other forms than those with which we
conceived of it in arithmetic, viz. under the forms of infinite dis-
cretiveness and of continuousness ; for whereas in arithmetic the
idea of number is of it in certain determined proportions, and in
algebra of certain undetermined, and in both of these in finite pro-
portions, so in infinitesimal analysis our conception is of number in
infinitely small, and therefore in infinitely large proportions ; and
whereas again in arithmetic and algebra we pass " per saltus" from
one number to another, and neglect all intermediate numbers, so
in infinitesimal analysis we conceive of number as increasing by
continuous growth ; discontinuousness and finiteness are properties
of number in arithmetic and algebra; but in the infinitesimal cal-
culus continuousness and infinity are parts of the pregnant idea,
and which are developed into the science ; and the conception of
it under such peculiar phases authorizes us to define infinitesimals
and their orders, to construct rules for determining them, or, in
other words, to create the subject materials of the science, to invent
a convenient nomenclature, to enunciate axiomatic laws respecting
them, such as the following : " a finite number of infinitesimals of
a given order makes no appreciable increase when added to finite
numbers, or to infinitesimals of a lower order;" and to deduce
from them all the consequences which necessarily follow. Such is
the method of infinitesimal analysis, and the assemblage of the pro-
cesses above described constitutes the Differential Calculus. In the
results we are led to such simple formulae, which are so immedi-
ately applicable to questions of geometiy and mechanics (the sub-
jects which originated the science) that the principles of the pure
science are lost sight of; and although the science is as perfect as
any, and the two elements distinctly are combined in it, yet the
ideal has scarcely been recognised ; but we know the history of it ;
how it presented itself in a confused and applied state, and under
two difi^erent phases to its founders : how Newton realized the
property of continuous growth, and Leibnitz that of infinity and
infinitesimals ; and how it has reached its present perfect state by
successive improvements; and we know now that it has a sure
and logical basis, on the strength of which it is not only applicable
to the explanation and prediction of geometrical and mechanical
10
truths, but also such a due and exact conception has been formed of
its idea, as has led to the most recondite results, as to the calculus
of variations, the theory of definite integrals, and so on. It is to
the ideal element that the labours of the most eminent analysts have
been directed in the latter years ; and if we may judge of the future
by past success, we may hope that within a few years the integral
calculus may be nearly as complete as the diflferential.
In geometry I may remark, that for the most part the ideal
element prevails over the experimental.
The perfection of a science depends on the due adjustment of
these two elements : this is evident to the historical inquirer into
the growth of a science : for the most part, the experimental facts
are first observed ; but they exist singly, one by one ; they require
binding, or, in the apt language of Professor Whewell, there must
be a colligation of them ; a principle of unity is required, and such
is the idea ; but this lying hid under a mass of special applications
of it, is but dimly, if at all, recognised at first, and thus the induc-
tive process advances but slowly and often unsuccessively : observers
may toil, and the result of their labours will be but a series of dis-
connected facts. When however, at last, by the sagacity of a master
mind the idea is recognised, unity is given to all ; all become parts
and applications of the one idea ; the chaotic observations become
systematic knowledge, and thus a philosophical and arranged science
is formed ; and a science too which is not only commensurate with
the facts observed, but of much wider applicability : for the idea
which has been generally suggested by observation has been supplied
by the mind, and the mind has given it properties of universality
and necessity, with which no amount of experience could ever invest
it ; and thus it involves consequences of greater extent than those
observations which called it from its source ; and it becomes
the germ of a pure science : and when it has been distinctly appre-
hended, particular phases of it are enunciated in precise language,
accurate and distinct divisions of the several branches of it are
posited, definitions are framed, and the consequences of the idea are
traced by a deductive process from the axioms, which are the major
premises of the first syllogisms : and a science thus constructed is
ready henceforward to unravel complex phaenomena which may
occur under conditions consistent with its axiomatic laws ; and also
to foretel what will happen when such and such circumstances con-
cur. The determination however of the consistency of these circum-
stances with the axiomatic laws requires a process difi*erent to the
deductive of which an outline is given : for particular facts and cases
11
must be analysed; their accidents must be eliminated; their essential
qualities examined, to detect whether at all, and how far, they accord
with the philosophical axioms of the pure science ; and we must en-
deavour in these instances to discover the connexion of necessary
antecedent and consequent, of cause and effect, which lies hid in
them : thus, for example, to anticipate what will be presently said,
ere the exact science of mechanics can be applied to the explanation
of the facts of Physical Astronomy, we must examine the matter of
the universe in order to determine whether it has properties accord-
ant with the laws of motion, which are the axioms of mechanics ;
and subsequently, as the exact science of motion will include all laws
of force, we must by observation determine what the particular law
is which prevails in physical astronomy ; and the examination of
facts with this object in view has led to the grandest instances of
induction : and inasmuch as perhaps otherwise our science might be
useful as an intellectual exercise, it hereby becomes an applied sci-
ence ; and thus, chronologically, the examination of external phse-
nomena will enter twice into the science ; first, in the formation of
it, when the idea is suggested ; and again, when external nature is
examined in order to determine the agreement or disagreement of
her laws with those of our a priori conception. A complete treatise
on any perfect science ought therefore to consist of two parts, one
in which the principles of the pure science are explained and its
consequences deduced, and a second in which are discussed the in-
ductive arguments and the inductive laws for the application of the
pure science to the phsenomena of nature. If therefore Logic be that
science which treats of the laws of thought, and constructs rules for
educing from given truths other truths which they contain, and with
this object enables us to analyse conceptions, to examine their con-
sistency or inconsistency, and affords tests whereby to judge of the
cogency of an argument, and if the methods of investigation which
it provides be applicable to all sciences and all man's knowledge,
whatever be the source of it, logic, fully to discharge its office, must
consist of two parts, and provide two processes : one the reverse of
the other ; one, that is, by which the special truths with which a
more general proposition is pregnant are evolved, and the other by
which we are enabled to discover a general proposition or law con-
cealed in a particular instance ; and if its character be thus universal,
that is, if it be the process of educing truth from given truths, which
either observation, or experiment, or conscience, or necessary attri-
butes of the mind, give to us, we must not transform it into a kind
of Hermeneutic or superior grammar, but we must exclude from it
12
all such questions as belong to metaphysic^ rhetoric, or to psycho-
logy, and restrict it to its own province and subject matter, to the
evolution of truth from other truths, whether by deduction or by
induction, and to the consideration of conceptions, whether they be
a priori or empirical, whether they be only in thought or expressed
in words.
Whether any science has attained to the utmost perfection is a
question which I suppose all will answer in the negative ; but that
some have reached that state in which the two elements are recog-
nised and adjusted, so that they may fairly be called " perfect"
in contradistinction to others which are still far behind them, will,
I think, be allowed by all but the most bigoted empiricist : the
question then of perfectness or imperfectness is one of degree ; but
amongst those which have attained to a degree of comparative per-
fection, I should mention that of Number in its three branches of
common Arithmetic, Algebra, and Infinitesimal calculus ; that of
Geometrical Space, and of the particular form of the science called
algebraical geometry ; that of Motion " la Mecanique." Yet there
are others which are on their way to this perfect state ; the ideas
of which have not yet been clearly apprehended, and the principles
of classification not yet distinctly framed ; but the materials for
which in their applied and partial form exist in the external world ;
from which certain general propositions have been formed by an
inductive process, but which are at present little else than observed
uniformities, being general formulae including in their grasp many
particular cases, but yet are not statements of axiomatic properties
of a central idea, fitted to be the nucleus and germ of a science ;
and therefore such knowledge, though it be to a degree systema-
tised, does not reach that standard of perfection to which the three
sciences above mentioned conform ; the dictum " that fluids press
equally in all directions," the idea of polarity which Professor
Whewell makes to be the germ of the electrical and its kindred
sciences, that of resemblance and analogy which enters so largely
into the structure of systematic treatises on botany and natural
history as the principle of classification, though they do colligate
^'What would otherwise be unconnected facts, and thus give order
where would otherwise be confusion ; yet are not pregnant ideas
of the respective sciences in the same way that number is of arith-
metic, and space of geometry, and motion of mechanics ; proofs of
this assertion are evident from the difference of relation that the
facts of the science bear to the above ideas according as new dis-
coveries are successivelv made ; ever since similar effects have been
13
found to arise from electrical action, whether developed by a ma-
chine, by a battery, or by a revolving magnet, the idea of polarity
has entered with more distinctness of conception into an explana-
tion of the phsenomena of machine-electricity ; and the principle of
the systematic arrangement of plants has been lately founded on a
consideration of their physiological characters instead of on their
external flowers as is the case in Linnaeus's arrangement ; in some
of the still- imperfect sciences the idea doubtless exists in the mind,
though in a rough and indistinctly-conceived form ; such as the
idea of mechanics was before the age of Galileo ; and such as per-
haps the idea of the theory of undulations is at present ; the exist-
ence of an etherial medium is assumed, the constitution of which
however has not yet been so clearly conceived that its properties
can be accurately stated; stiU however the science of optics is on
its road towards perfection, and has already satisfied a searching
test ; that is, it has predicted phsenomena which observation has
subsequently shewn to exist ; an evidence of its truth of the same
nature as that which gives to the theory of gravitation its greatest
certainty ; viz. the discovery of Neptune, the calculation of solar
nutation, the lunar disturbance owing to the earth's oblateness ;
we may venture therefore to foretel that ere long such an imperfect
science will be ranked amongst the exact sciences, having an idea
adequately reahzed, and its results accurately detailed.
As so much has been said above on the science of number in
illustration of the relation to each other of the two elements of a
science, it is unnecessary to add more ; but I propose to consider
the sciences of Geometry and Motion at greater length, and to shew
how exactly the processes which I have attempted to describe hold
good in these particular sciences. The idea of the science of geome-
try is geometrical space ; the term cannot be defined, being too large
in all that it implies to admit of being trammelled with words ; and
for this very reason it is, that it is pregnant with consequences. In
the construction of the science, then, certain phases of the idea are
first enunciated, and certain properties of it are stated which are
necessary to an adequate conception of it ; and if which were not,
geometrical space would not be what we conceive it to be. These
statements are our axioms, and the major premises of the first
syllogisms from which we deduce the several specific propositions
which enunciate properties of the leading idea. The axioms are
such as, "Things equal to the same thing are equal to one another,"
and the following six axioms of the first book of Euclid. For all
these, though true, perhaps, of every thing capable of measurement.
14
are yet, I conceive, enunciated in Euclid as properties of geometrical
space only. So again are properties of space expressed in the fol-
lowing axioms : the conception of equal spaces in the axiom " Mag-
nitudes which coincide with one another, or exactly fill the same
space, are equal" "The whole is greater than its part;" and the
conception of ratio of spaces in the axiom in the fifth book "Ratio
is the relation of two quantities to each other in respect of (not
position or colour or hardness, but) quantuplicity" {Kma nrfKtKoTrjra) ,
These are true of all geometrical space, involved in any adequate
conception of it, and without which it would not be what it is : they
are truths too of such certainty as no amount of experience could
ever give to them ; experience may suggest them, but it will give
but a rough outline and a crude form of them ; it will present to us
a line nearly straight, and nearly without breadth, as for instance
the joint of two boards of different colours : but there are uneven-
nesses which the mind must abstract, and thus provide for itself the
conception of a line perfectly straight, and lying evenly between its
extreme points. In a similar manner are our conceptions of a per-
fect sphere and a perfect ellipse formed : the mind then acts on
these imperfect objects, and gives them a necessity which is peculiar
to its own constitution ; and the enunciation of such axioms is the
first step in the philosophical construction of a science. Again, the
idea is such that space is tridimensional ; hence arises a perfect divi-
sion of the subject into three parts, corresponding to space of one,
to space of two, and to space of three dimensions. Of space of one
dimension or lines we have a twofold division of lines, curved and
straight ; and of curved lines we have several species of circles,
ellipses, cycloids, curves of double curvature, and so on : and we
have figures formed of combinations of straight lines, as e. g. tri-
angles, parallelograms, trapeziums ; and we have parallel straight
lines ; and thus there is the most perfect division possible of the
several specific forms which the pregnant idea involves, and a most
perfect subordination of classes from even summum genus down to
infima species ; but these many divisions having been made, it be-
comes necessary to explain the meaning of the terms we employ ;
hence our need of definitions, which are definitions of the several
classes; not the same as the axioms, inasmuch as they are not
pregnant with consequences, and are not used as premises in any
syllogism by means of which we deduce new properties from old
ones ; but as many of the specific forms of the original idea involve
consequences peculiar to that particular species, it is necessary to
enunciate axioms of rather a diflferent kind, which shall state pro-
15
perties, of not the idea in all its generality, but of certain particular
species of it, such as the axiom about straight lines "Two
straight lines cannot enclose a space ;" that about right angles
" All right angles are equal ;" that about parallel straight lines, and
so on ; all these differ from the definitions, inasmuch as the defini-
tions are definitions only of the words, and do not involve any of
the properties of the things whose names they explain. But axioms,
specific though they be, are replete with all the consequences which
are involved in the particular forms of geometrical space correspond-
ing to them. By help of the axiom " Two straight hues cannot en-
close a space," the fourth proposition of the first book of Euclid is
proved ; but the proof of no proposition rests on the definition that
" straight lines are those which lie evenly between its extreme
points ;" for whereas the axiom enunciates an essential property of
such lines from which all other properties of lines may be educed ;
the definition gives little else than a synonym of a straight line.
Hence also arises the necessity of a formal statement of some pro-
perty of parallel straight lines besides the definition of them. Here
also, did the limits of the paper admit of such extension, we might
shew how most of the properties of space of two dimensions depend
on the axiom concerning equality of magnitudes filling the same
space ; how exact the division is of figures, according to their
bounding lines, whether straight or curved ; and of surfaces into
plane and curved : and so we might proceed to discuss as parts of
the same science of space those results, which are involved in our
conception of it as a quantity infinitely discretive and continuous,
whereby we shall be supplied with principles for defining a point as
the inferior limit of space, a straight line as the superior limit of a
circle's arc (when the radius is infinitely large), and a plane as the
superior limit of a spherical surface, and parallel straight lines as the
sides of a triangle whose base is finite and viertex at an infinite dis-
tance. All these properties and innumerable others are involved in the
general idea of space, from which, according to the rules of deduc-
tive logic, they are to be deduced : the general idea is thus to be sepa-
rated into its constituent elements ; and the same logical process is
followed in all cases ; that is, it is indifferent to the positive science,
whether the curves and surfaces have their counterpart in nature or
not ; whether we are discussing the properties of the more compli-
cated curves, of cycloids, of lemniscates, surfaces of elasticity, or of
ellipses, triangles and spheres, in all cases the same rules are fol-
lowed, and the same axioms are the first major premises. But, on
the other hand, if we intend to apply our geometrical figures and
16
their properties to the explanation of cosmical phaenomena, that
is, if we intend to make our pure science an applied science, then
we must shew from observation and experiment that the fundamental
principles and axioms are true in the matter to which we are to
apj)ly their consequences : here then an inductive examination of
facts enters into the science ; general laws to which they are subject
must be discovered : as for instance, in the appKcation of geometry
to questions of astronomy, we must first assure ourselves that such
a motion as the planets are assumed to have is consistent with the
axioms of geometry ; and with this object we have to connect our
conception of a continuously-moving particle with that of geometri-
cal space capable of infinite discretiveness and of continuousness ; and
again by an accurate observation of the planet's position night after
night, and noting down its successive places^ we shall conclude that
its orbit is an ellipse ; and when this has been done, we shall be
authorized to apply to the planet's motion all the properties of the
ellipse ; our pure geometry will then come in, and by virtue of it
we can enunciate certain properties of such elliptic orbits, which
will go far to test the truth of our observations, and to foretel
certain results of such a motion. Such has been the course of astro-
nomy ; for had not the Greek geometers clearly apprehended, and
by their deductive processes accurately analysed the idea of geome-
trical space, and thereby fully discussed the properties of the conic
sections, the conception of an ellipse would not have been ready to
Kepler to simplify the complicated theory of epicyclical motion, and
Newton might not have been able by his mechanics to deduce from
the ellipse his law of gravitation ; and had not a positive science of
algebraical geometry been previously constructed, Fresnel might not
have been able to express his surface of elasticity and wave surface ;
and sir W. Hamilton might not have been able to discover the cuspal
points whereby he was led to the phaenomena of conical refraction.
The other science, the principles of the structure of which I
propose to consider, is Mechanics, " la M^canique," which I should
prefer to call the science of Motion ; for motion is its pregnant idea.
' Le mouvem.ent et ses proprietes gen^rales," D'Alembert writes
in the Discours Pr^liminaires, " sont le premier et le principal objet
de la mecanique ;" what motion is in itself is a metaphysical ques-
tion, the consideration of which does not fall within the scope of
the present paper : sufficient for us that we are able to state such
affections of it as may be the axioms of the science, and the basis
of our reasoning ; in the first place, we do not conceive of motion,
except of something moving, and that something we call matter ;
17
wiotiou must be clothed, and matter is that wherein it is clothed,,
and in which it consists. " In mechanics," says Professor Whewell,
*' we know of matter only as the subject on which force acts :" this
matter however of the abstract science is not necessarily what is
sensible and what gravitates and is heavy, for by matter we mean
none other than that which moves, and in which motion may reside:
thus the moving molecules of the etherial medium are such matter
as the science of motion recognises. The ultimate facts of the
science are motion and matter. Motion of matter also involves two
conditions ; it takes place in space and during time ; matter, we say,
exists in space, and such existence is called extensiveness ; and inas-
much as we do not conceive of two different particles of matter oc-
cupying the same space at the same time, we say that it has also the
quality of impenetrability. Time again enters into our conception
of matter in motion in two ways : we do not conceive a particle of
matter to be in two different places at the same time, and we do not
conceive of it passing from one position to another instantaneously 4
time must be occupied in the passage. Such are the four ideal ele-
ments on which and on their relation to each other the science of
motion is raised ; motion, matter, space, time. We treat of matter
in motion, and of motion as an affection of matter ; and motion
takes place in space and during time. From these elements arises
our conception of velocity, which is the degree of swiftness or slow-
ness with which matter changes its position. Velocity, we say, is
greater, the greater the space passed over in a given time, and the
less the time spent in passing over a given space ; and thus we may
define velocity, after the manner of most English mechanicians, to
be, according to the laws of variation, the ratio of the space passed
over to the time to which it is due, or after the manner of most
continental writers, to be the space passed through in an unit of
time. In the case however of variable velocity, the circumstances
adapt themselves to the principles of the infinitesimal calculus ; and
under the different aspects in which it is considered we should de-
fine velocity to the ratio of the increment of the space to the incre-
ment of the time to which it is due ; or we should reduce the unit
of time, so that it should become an infinitesimal. In either case
the determination of the finite velocity generated in a finite time
becomes a problem of the integral calculus. When matter however
moves with an increased or diminished velocity^ a question arises
whether this change is due to an external cause acting on it, or to
an intrinsic power of its own to affect its own state. The old
Ari&totelian philosophy concluded, that cosmical matter at least had
c
18
the latter property. Galileo first enabled mechanicians to raise their
science on an axiom embodying the former alternative : matter, he
said, is inert, it has no intrinsic power to change its own state, whe-
ther that be of rest or motion : if it be at rest, it will remain at rest,
and if it be in motion, it will continue to move ; and to move with-
out any increase or diminution of velocity ; whatever action matter
may exert on other matter, it has no power of acting on itself ;
hence when matter's state changes we are authorized to seek for
the cause of the change in some source external to it ; and in an
adequate source, inasmuch as it will neither absorb into itself, nor
generate of its own resources any change of its state. This axiom
then of matters inertia is of the greatest importance to the construc-
tion of the science, for by it the equations of motion, which are the
first propositions of the science, are formed, and as follows, when
matter's velocity is changed, the expressed or developed change of
velocity is exactly equal to that impressed on or communicated to
it under the following conditions : motion is conceived to reside
in and to be of, matter ; and matter is conceived to be measur-
able, that is, according to the principles of number, one quan*
tity of matter is any number of times another quantity : and
thus if we take a certain quantity of matter as the unit, any
other may be expressed by a certain number of this unit ; but
this requires further explanation : we have spoken of matters ex-
isting in and occupying space, and this property we have called
extensiveness ; but a greater or less quantity of matter may be con-
tained in the same space, according as it is packed more closely
or rarely together ; it may be more or less dense ; such a property
we call intension, and we speak of matter as more or less intense ac-
cording as a greater or less quantity is contained in a given space :
hence the quantity of matter varies both as to the space it occupies,
and as to its density ; and therefore according to the principles of
number, any given quantity of matter will contain a certain number
of particles of matter, by being compared with a given unit of given
extension and given intension ; the mode of comparison will be evi-
dent from what foUows : when therefore any quantity of matter, or a
given mass, as it is called, is moving with a given velocity, each unit
particle moves with this velocity, and therefore if the particle had
been at rest, and afterwards is moving with this velocity, (both of
which suppositions are in accordance with the principles of the
science,) the velocity must have been communicated to each unit
particle from some external source, and therefore, by the law of
inertia, the whole transferred velocity will be as many times the ve-
k
19
locity of each unit particle, as there are particles in the body ; hence,
therefore, if the velocity of one particle be symbolised by v, the
whole velocity which has been transferred to the whole body will be
(if m represents the number of particles) 7nv : this expression, which
enters so largely into our science, we call momentum, or quantity of
motion, and we consider it transferable from one mass to another,
and to be such that none is lost in the transfer; so that we may now
modify what was said conditionally of quantities of matter being
equal both intensively and extensively, and unconditionally enunciate
the following axiom : when matter's momentum is changed the de-
veloped or expressed momentum is exactly equal to the communicated
or impressed momentum : this axiom when mathematically expressed
forms the equation of motion, and being capable of expansion, is the
major premiss from which all the results of the science deductively
follow ; if therefore there be a known mass m moving with a certain
velocity v, its momentum is equal to mv, and if the whole of its
momentum be transferred to a mass m' at rest, but which subse-
quently moves with a velocity v, then we have
hence if we knew v, and v , (and our ideas of space and time enable
us to determine these,) we can compare m and rn; and it is to be
observed that this is the mode we adopt for weighing masses.
In the case in which momentum is not instantaneously but gradu-
ally transferred from one mass to another, the successively transferred
elements of momentum are equal to the successively developed ones ;
and as the mass is unaffected by lapse of time, it remains the same,
and the equation of motion in this case becomes
Mass X impressed element of velocity
=mass X expressed element of velocity.
The limits of the paper do not allow me to enter into detail on the
method of resolving and compounding velocities ; but I may remark,
that as soon as we have determined that velocity is to be expressed by
ds
, the subject resolves itself into a question of number ; and the
laws of resolution and composition follow from the equation
ds^ = dx^ -f- dy^ + dz^.
As an example of the mode in which this equation is pregnant with
consequences, I will take the following: since the velocity expressed
!20
ds
along the path of the moving particle is represented by , we have
according to Maclaurin's method of resolution
= expressed velocity along the axis of ^,
1= -
dz
df" ''''
- . . ^x d^ d'^z , , . , e ^^
and therefore -r-s, -r-^, -77, are the several mcrements 01 the re-
dt^ dt^ dfi
solved expressed velocities due to an unit of time ; if therefore there
is no impressed velocity, each of these quantities is equal to zero ;
and therefore integrating twice, and adding constants
x~a yb z^_
a /3 y *
the first three of which equalities are the equations to a straight line,
whence we conclude, that if a particle moves, and without any velo-
city being impressed on it, it moves in a straight line.
The general idea of the science being that of motion^ it is of
motion either in act or in power, either actual or virtual : in the
latter case the science becomes that of statics, and its principle, by
the equality of impressed and expressed momenta, that of virtual
velocities; from which result the six equations of equilibrium ; hence
the problem of motion of rigid bodies is brought within the range of
the science by means of D'Alembert's principle, and likewise the
assemblage of propositions which forms the science. The primary
equations of motion then having been formed, all the results follow
by a deductive process from them ; they involve all the consequences
of the science, and however particular the last propositions be, and
however far removed from the first axiom, yet the axioms exist in
them, more or less near to the surface, and are capable of being
drawn out ; and the axioms are so general that they include all laws
of transferred momenta, and all results follow with a logical cogency
equally valid ^.
'%?' *> It will be observed that in the preceding sketch, nothing has been said of
S*^ force," the idea of which enters so largely into most of our text books, that
"'On it the whole science is raised ; the word has been purposely omitted, because,
first, force seems to be synonymous with mechanical cause, and therefore if its
conception be the fundamental idea, the biisiness of the science will be to
21
But after the pure science of motion has been evolved from the
pregnant idea, can we apply it to the explanation of phsenomena of
external Nature ? Not until we have established that the matter of
Nature has properties which are the counterparts of the axioms of
the science, and that the matter moves according to a law which the
principles of the science include. With the object of determining
these two points, we must examine and analyse Nature and her
matter, and by an inductive process conclude what her laws are:
first then we must inquire whether Nature's matter is moveable,
impenetrable, and inert ; whether it is in accordance with the laws
of composition of velocities, and of momentum. The first four of
these properties are proved inductively by the instances and experi-
ments cited in the ordinary text books, and into an examination of
which I need not therefore enter : the law of momentum is demon-
strated by Attwood's machine, whence we learn that the increment
of momentum due to any short time dt is equal to the product of
the mass moved, and the increment of the velocity due to the same
time dt. Such experiments prove the truth of the laws, axioms in
the case of the matter of the earth ; and then, by an inductive ex-
tension of such properties, we assume them to be true of the matter
of the whole material universe ; that is, we extend to the sun and
to the other members of the solar system properties of matter which
we have proved to be true of only one of the secondaries, viz. the
earth : and our extension too goes farther than this, inasmuch as
the law, subject to which momentum is transferred from one of the
discuss the relation between cause and effect ; and thus those writers who have
adopted this \dew, and stated its principle philosophically, enunciate as axioms
the mechanical translations of such propositions, as, " there is no effect without
a cause," " effects are produced by adequate causes," and speak of the com-
position of causes ; but the equations on which the whole science is raised state
the equality of momentum transferred and momentum developed which are
exactly the same thing under two different aspects, and cannot with justness of
language be called cause and effect ; the source of the communicated motion,
whether it be the muscular action of a cricketer's arm, the earth's attraction,
electrical action, and so on, may more properly be called the causes of the
motion, and into these the science of mechanics does not enter ; in its investi-
gations it assumes the laws according to which such momenta are impressed,
and equates such communicated motion to that which is expressed in the moving
mass : secondly, the word force has been used in so many different meanings,
that it is well to avoid an ambiguous expression which is constantly a stumbling-
block to students ; we mean by it roughly what produces or tends to produce
motion, and then we apply it to statical force, dynamical force, impulsive force,
accelerating force, moving force, labouring force, &c., &c., and by each term
intend a different thing.
22
bodies to another, can be tested only in its limit at the earth's sur-
face ; but all these extensions are made in accordance with canons
which an inductive logic supplies, and of the truth of them thence-
forward the mind has no doubt, and these become the "laws of
motion," or the axioms of the science in its appHed form : but I
would remark that such an inductive analysis of facts can never give
to such axioms that necessity and universality which are the charac-
teristics of the idea of the pure science ; it may fit them to the pur-
poses of physical astronomy, but cannot constitute them primary
laws of the exact science. Such an examination however fulfils its
object ; though it does not render the science of observation an exact
science, yet it enables us to apply our pure science, as far as it exhi-
bits nature's laws the counterpart of our axioms, and thus to deduce
from them the consequences which they involve. Again, the pure
science includes all laws of communicated momentum, but as there is
no a priori reason why nature's law should be one rather than
another, observation and experiment are required to discover her
particular law : here again the inductive process enters ; and in the
hands of Kepler it led to the three laws of equable description of
areas, elliptic orbits, and sesquiplicate ratio of periodic times and
mean distances; and these experimental laws having been ready for
Newton to apply his pure science to, were by it translated into
their mathematical equivalents, viz. the motion of planets under the
action of a central force and in one plane, the law of the inverse
square of the distance, and the action of the same central force on
all the planets. Had not Kepler lived and enunciated his three
laws, Newton might have constructed his fluxional calculus, and
applied it to mechanics, and yet might not have enunciated the law
of gravitation ; a science of dynamics might have been formed, but
it might have remained for some future observer to found the
science of physical astronomy ; numberless other instances have
occurred in the science of mechanics of a similar kind; but what
is above stated is enough to my present purpose to shew how in-
sufficient experiment and induction is to establish the axiomatic
laws of the pure science, and yet how necessary an examination of
nature's facts is to enable us to apply our pure science to the ex-
planation of phsenomena of the external world.
An examination of the processes and methods of the still imper-
fect sciences, such as those of physical optics, heat, electricity, leads
to the same results as that of the sciences of number, space, and
motion.
In conclusion, I may remark that the preceding investigation
23
suggests the course to be pursued with the best prospect of success
in an inquiry into the laws of nature ; not only must the phseno-
mena be examined to discover the necessary antecedent causes, and
must she be subjected to crucial experiments, but the idea of colli-
gation, which such an analysis has suggested, must be deductively
traced and its results compared with the corresponding facts of
observation ; the two processes will thus mutually illustrate and
strengthen each other, and finally a science will be formed with
two phases, such as has been described in the preceding pages ; it
will have those notes of universality and necessity, which nothing
short of the constitution of the human intellect can give to it, and
it will also in part be useful as an applied science in unravelling
the complicated wonders of the material world in which we are
placed.
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qNIVERSITY OF CALIFORNIA LIBRARY
