March 28, 1889] 



NATURE 



521 



ON THE CONFLUENCES AND BIFURCATIONS 

 OF CERTAIN THEORIES} 



A XIOMS, says Proclus,- are common to all sciences, though 

 "'*■ each employs them in its peculiar subject-matter. A little 

 further on ^ he cites Aristotle * as saying that one science is 

 more certain than another, viz. that which emanates from more 

 simple suppositions than that which uses more various principles ; 

 and that which tells the why, than that which tells only the 

 simple existence of a thing ; and that which is conversant about 

 intelligibles, than that which touches and is employed about 

 sensibles. 



Proclus adds that, according to these definitions of certainty, 

 arithmetic is more certain than geometry, since its principles 

 excel by their simplicity. For the conception of unity has no 

 reference to position in space, while that of a point involves such 

 reference. In short, we may say that to count a number of 

 objects is a simpler operation than to measure the distances 

 between them. 



All this, and much more, shows how early the notion of what 

 is sometimes called a hierarchy of the sciences arose. Proclus's 

 order of precedence would seem to be this, viz. logic, ^ arithmetic, 

 geometry, mechanics, optics, dioptrics," and so on ; the progres- 

 sion being from the more to the less abstract, or from the 

 abstract to the concrete. 



Francis Bacon, mindful perhaps of Proclus,^ and duly appre- 

 ciating the power of mathematics as an instrument^ and its 

 value as a discipline," expressly takes the degree of abstractness 

 of a science as the mark for its classification. He places 

 mathematics, as the most abstract of sciences,'" at one end of the 

 scale and "policy" at the other. He does not graduate the 

 scale minutely, but it may be that, as in the case of the categories," 

 he attached no great value to such details. Distinguishing 

 philosophy from theology, Ic^ic, and mathematics,^- he assigns 

 to it the axioms which are common to several sciences and 

 the inquiry into essences, as quantity, similitude, diversity, 

 possibility, and the rest. Science he divides between meta- 

 physics, the science of the abstract and permanent, and physics, 

 that of matter and its changes.'^ Bacon, in one place, names the 

 one universal science by the name of philosophy, while in 

 another he treats philosophy and metaphysics as two distinct 

 things,''' He uses the word metaphysics in a sense different from 

 that in which it was then " received. Mathematics he places as 

 a branch of metaphysics, and as having determined or deter- 

 minate quantity for its subject. To the pure mathematics, he 

 says, belong geometry and arithmetic ; the one handling con- 

 tinuous, and the other discrete quantity.'" If he means continuous 

 quantity so far as it is immovable, he agrees with the Pytha- 

 goreans. "^ 



Quantity, time, and space are placed by Aristotle among his 

 categories, or are implied in them. With regard to spaCe, he 

 does not seem to have reached the Kantian view in any way, nor 

 to be very clear in his meaning, though he apparently feels that 

 to realize space we must have motion. His conception of time 

 as one of the elements required for measuring motion, and his 

 starting thf problem as to whether we could have time without 

 a mind to conceive, seem a more distinct approximation, though 

 only an approximation, to Kant's view of time as merely a 

 subjective condition of perception.'* 



■ Presidential Address delivered by Sir J.ames Cockle, F.R.S., to the 

 London Mathematical Society, on November 8, 1888. 



- Proclus, "Commentaiics on the First Book of Euclid's Elements" 

 (Taylor's Translation, London, 1792), p. 92. 

 ^^ Proclus. of>. cit., p. 93. 



■* Taylor {ib. p. 93) supplies the reference to the first Analytics, t. 42. 



■' Proclus, op. cit , p. 79. Hume {" Treatise," vol. i., London, 1739, Book i. 

 Part 3, p. 129. et vid. p. 128) says that geometry falls short of that perfect 

 precision and certainty which are peculiar to arithmetic and algebra. 



(• Proclus, op. cit., p- og; et rid. pp.78, 79. 



' Bacon, 'The Proficience and Advancement of Learning" (Oxford, 

 1633), pp. 49. 50- 



** Bacon, op. cit., pp. 151, 152 ; et vid. pp. 119, 120. 



'' Bacon, op. cit., pp. 152, 205, and 231. 



'" Bacon, op. ait., p. 218 ; etvid. pp. 150, 151. 



" Bacon, op. cit., pp. 130, 131, 140, and 201 ; et vid. p. 161. 



'2 Bacon, op. cit., pp. 49, 50 ; et vid. pp. 130, 131, and 143. 



'3 Bacon, op. cit., p. 141. 



'■• Bacon, op. cit., pp. 130, 140. 



'5 Bacon, op. cit., p. 138 ; con/, pp. 146, 147. 



■0 Bacon, op. cit., pp. 150, 151. 



'^ Proclus (Taylor's Translation), p. 74. 



'"' For this summary of Aristotle's views I am indebted to Mr. Reginald H. 

 Roe, who referred me to Ueberweg's " Hist, of Phil.," p. 164, for a more 

 general statement, and to p. 165 for a list of the best books for its fuller 

 elucidation, adding that in Ritte and Preller's extracts, pp. 288 and 289, will 



Newton, in the Scholium to his definitions, distinguishes be- 

 tween absolute and relative time, the latter being time conceived 

 in its relation to phenomena. Of absolute time (otherwise called 

 duration) which has no relation to anything external, he says 

 that it flows equably, and that its rate of flow and the order of 

 its parts are immutable. In his "Fluxions " he uses the word 

 time in a somewhat different sense, viz. as meaning the inde- 

 pendent variable, characterized by an equable increase, fluxion, 

 or flow.^ Sir W. Rowan Hamilton treated algebra as the 

 science of pure time, but his doctrine is not entirely - assented to 

 by De Morgan, nor by Prof. Cayley, who indeed, in his South- 

 port Address (p. 19), intimates dissent from it. Proclus does not 

 connect arithmetic with time, and Prof. Cayley suggests [ib. p. 

 18) that, in any case, the notion of number or plurality is not 

 more dependent on time than on space. By the logicians, time 

 seems to be regarded as the more abstract of the pure intuitions. 

 In fact, time is implied in memory and in thought itself, and 

 Prof. Francis W. Newman observes that no man could get 

 through a syllogism if he forgot the first premiss while dwell- 

 ing on the second.^ Moreover, he has recourse to the idea of 

 time when he comes to discuss propositions,^ and Boole investi- 

 gates the nature of the connection of his own secondary pro- 

 positions with the idea of time.' The ancient Indians had their 

 cyclical periods, but not therefore necessarily any notion of a 

 uniform curvature (so to say) of time. 



Absolute space, says Newton, perpetually remains similar to 

 itself and immovable ; and, further on in the Scholium, he adds 

 that the order of its parts is immutable. In the preface to 

 the " Principia"he had observed that the description of straight 

 lines and circles, on which geometry is founded, belongs to 

 mechanics, and he follows up this train of thought. But, whether 

 he means to detach himself from Plato, I miist leave others to 

 say. It is said to be certain that he was familiar with Bacon's 

 works ; that he uses the word axiom, not in Euclid's sense, but 

 in Bacon's, thus giving the name of axioms to the laws of 

 motion, which, of course are ascertained by the scrutiny of 

 nature, and to those general experimental truths which form 

 the groundwork of optics." Now Bacon says that, in his judg- 

 ment, the senses are sufficient to certify and report truth, either 

 immediately or by way of comparison." Moreover, he suggests 

 that the rule Qu<e in eodem tertio conveniunt, et inter se con- 

 veniunt, a rule so potent in logic as that ail syllogisms are built 

 upon it, is taken from the mathematics.** In seeking an origin 

 for the more abstract in the less abstract, Bacon is not solitary. 

 Thomas Stephens Davies suggested " that the argument from 

 superposition had its origin in mechanical considerations, and 

 from the fitting together of material figures. Moreover, it is 

 conceivable that some observant person among the ancient 

 Egyptians, whose custom it was to stamp their bricks, noticing 

 the resemblances of the marks and the correspondence of the 

 impressions with the impressing tool, may have been led to a 

 recognition of the rule quoted by Bacon. The doctrine that 

 there enters into geometry an element derived from the senses 

 has, indeed, appeared in books designed for ordinary readers. 

 Thus, Prof. Newman, writing in 1836-38, although in one part 



be found all the important passages from Aristotle bearing on the question. 

 As to the views of Boole, see his " Laws of Thought " (London, 1854), pp. 

 162 et seq. ; see also p. 419. Boole treats of space at pp. 163, 175, and 418 ; 

 and at p. 17s he quotes Aristotle's statements respecting the ex istence of 

 space in three dimensions. 



I Newton, " Fluxions," pp. 26 and 38 of the small edition (London, 1737). 

 This is a genuine work of Newton's. As to its bibliography, see Notes and 

 Queries, 2nd S., vol. x. pp. 163, 232. 233 ; 3rd S., vol. xi. pp. 514, 515 ; 4th 

 S., vol. ii. p. 316 ; 5th S., vol. IV. p. 401 ; 6th S., vol. iv. pp. 129, 130; vol. 

 V. pp. 263, 264, 304, 305, and 426. This octavo edition is very scarce. 

 Indeed, I only know of two copies, viz. my own copy and one in the library 

 of the Royal Astronomical Society. 



^ De Morgan, " On the Foundation of Algebra," Cambridge Transac- 

 tions, vol. v.i. pp. 173-87; see pp. 175, 176. The remarks of Prof Cayley 

 on Whewell, at p. 18 of his Southport Address, are applicable to Rowan 

 Hamilton. . ,_ 



3 Newman, " Lectures on Logic, or on the Science of Evidence, ' &c. 

 (Oxford, 1838), p. 15. 



4 Newman, op. cit., pp. 32-34- 



5 Boole, "An Investigation of the Laws of Thought " (London, 1854), pp. 

 162 et set], 



6 See the account of the " Novum Organon" in the " Library of Useful 

 Knowledge," p. 10. 



^ Bacon, "Advancement of Learning" (cited supra), p. 193. 



8 Bacon, op. cit.. p. 132. 



» T. S. Davies, Ge0metric.1l Notes, Mechanics' Magazine, vol. liu. (1850), 

 pp. ito, 169, 262, 291, 442. Davies points out "the connection between 

 parallels and similar triangles" He thinks that Aristotle's secession from 

 the school of Plato arose from his enforcement of his own logical doctrines. 

 Davies rejects the notion of a geometry built upon definitions alone without 

 the assistance of axioms. 



