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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.XLIX. 
 
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 , .-rThe Aslimolean Society desire it to he understood ■. 
 that they are not answerable^ as a hody^ for any fact^^, 
 ¥easoninqSy or qnintons, advanced in paMert'm^^ 
 
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On the relation of the several parts of a Mathematical 
 Science to the fundamental Idea therein contained. 
 
 J. HE inquirer into the principles and structure of a mathematical 
 science has two questions to discuss : of which, one has reference 
 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 diiFerent 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 either, 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 which 
 we are going to discuss. 
 
 The ** vexata qusestio" 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 sensatioh, aiid'^w^ifcH, '-©n account of their necessity and 
 univers'pJit^J,; ^l^s^kt'dy, ^Tatscfend all experience; that the rough 
 materials aVe giVeh' 'by empirical 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 heside 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 phsenomena 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 
 
6 
 
 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 alrdpKrjs, 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 which 
 its agencies are conducted; which shall give us rules to ascenti"^ 
 from particular and observed facts to general propositions; whiiifr^ 
 shall furnish us with criteria to discover the general law lying coii-^ 
 cealed in the particular instance ; by this process we shall intei'i^^ 
 rogate nature, and determine how far her laws are identical wit^'^ 
 those of our scientific idea ; this process is manifestly the reversk^'^ 
 one to the former, and is called the inductive; and although w^ ' 
 might a priori (on account of the manner in which, as metaphy!.-'" 
 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 exact 
 science which has an objective rea% as-weli-as subjective existence- 
 these two processes must enter. ' ''-'^ .noiHiam- -■'u: 
 
 If this be a true account of the constitution of a science, the divif,'^ 
 sion of sciences into "pure" and "applied" is founded on a wrong * 
 pnnciple ; 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 
 
 J.t 
 
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.' 
 her, 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 Plulosophy in this University, Mr. Mills,, 
 writes as follows in a lecture on the Origin of Ideas : *' We may safely allow th^t 
 sensation gives the first impulse ; 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- i 
 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 
 quantuplicity ; 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 ; tbat 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 
 grpwth, 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 geometry 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 phsenomena 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 lequires a process difibrent 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 
 
13 
 
 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 diflference 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; still 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 lengthy 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" (Kara TTrjKKoTrjra) , 
 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 w^hich 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 different 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 hnes 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 \:ertex 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 application 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 sam€ 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 ; 
 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 follows : 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- 
 
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 rw' at rest^ but which subse- 
 quently moves with a velocity v , then we have 
 
 mv=m!v; 
 
 hence if we knew v, and v , (and our ideas of space and time enable 
 us to determine these,) we can compare m and rri; 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 a:*, 
 
 1= ^- 
 
 dz 
 dt~ 
 
 z; 
 
 dP'X d^v d^z 
 and therefore -n^, -r-r, -—7, are the several increments of the re- 
 dt^ dt^ dt'^ 
 
 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 y—b z—c 
 
 a p 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 ^. 
 
 ,s.^i b It will be observed that in the preceding sketch, nothing has been said of 
 
 •;2«r 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 
 
 c^onccption be the fundamental idea, the business of the science will be to 
 
1 
 
 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 view, 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 supphes, 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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 UNIVERSITY OF CALIFORNIA LIBRARY