522 



NATURE 



{March 28, 1889 



of his "Logic" (p. 25), he says that in geometry no results 

 are admitted by help of observation and testimony, but only 

 by reasoning from the definition, yet he afterwards (p. 55) states 

 that, as space and its properties appear undeniably to be learned 

 by sense, the argument seems to him to preponderate for naming 

 geometry a mixed science, and believing that its propositions 

 are real and not verbal truths. And Potts ^ says that geometry 

 seems to rest on the simplest inductions from experience and 

 observation, and that its principles are founded on facts 

 co?nizab]e by the senses. 



But it is to Reid - that the idea of a more precise mathematical 

 treatment of the subject is due, and his name ought to head the 

 roll on which will be inscribed the names of Lobatschewsky, 

 Riemann, and other investigators. Kant, indeed, disposes of 

 ■such questions summarily, by saying that it follows from his 

 premisses that the propositions of geometry are not the deter- 

 minations of a mere creature of our feigning fancy, but that they 

 necessarily hold of space, and consequently of all that may be met 

 within it, because space is nothing else than the form of all the 

 external phenomena, in which alone objects of sense can be 

 given ("Prolegomena," 3 p. 51). He adds (pp. 51 and 53) that 

 ■external phenomena must necessarily and precisely agree with 

 the propositions of the geometer. Whether Kant's allusion to 

 "superficial metaphysicians" points to the Pyrrhonists and 

 Epicureans ■* or to others, and, possibly, even to Reid, whom 

 he had mentioned before (Preface, p. viii.), does not appear. 

 Whatever opinions be formed of Kant's theory, or of the nature 

 of space, his view is impressive. Confine that view to two 

 ■dimensions, and suppose the surface of a sphere to be inhabited 

 "by a being destitute of any conception of a third dimension, and 

 whose senses are unaffected by any point not situated or any 

 motion not taking place on that surface. He could only estimate 

 direction and position by the tangent to the path of the visual ray 

 at the point where that path meets his visual organ, and would 

 think that all objects were situate in one plane. His geometry 

 would be Euclidian ; for, if he could form a notion of the actual 

 paths of rays, he would have a conception of the third dimension 

 in space.® Here Kant and Riemann would apparently be at 

 issue ; for, if a more general conception of space is to be rendered 

 special by actual measurements on the sphere, then, after an 

 enlarged experience, the Euclidian conception would have to be 

 expelled and replaced by some other. And all this would have 

 to be done without praying in aid the excluded third dimension. 



Aristotle •• notices that the nature of everything is best seen in 

 its smallest portions, and Kant " remarks that there was a time 

 when mathematicians, who were philosophers too, began to 

 doubt, not the truth of their geometrical propositions as far as 

 they regard space, but the objective validity and applicability of 

 the conception itself, and of all its determinations, to nature ; as 

 they were apprehensive that a line in nature might consist of 

 physical points, and that consequently true space in the object 

 might consist of simple parts, though space as conceived by the 

 geometer cannot so consist. Clifford* would have given due 

 weight to the doubts of the philosophical mathematicians. He 

 •even suggests that the properties of space may change with time. 

 Now, a number may be a function of an angle ; the very angle 

 itself determines those numbers (ratios of lines) which we call 

 sines and cosines. But, says De Morgan," in every case but this 

 it is impossible to conceive number a function of magnitude. It 

 seems almost equally difificult to entertain Clifford's conjecture, 

 -which, nevertheless, measurements might verify. The sentence. 

 Nam tempora et spatia sunt sui ipsoruni et rerum omnium quasi 

 Loca, in Newton's Scholium, though it may suggest that omni- 

 presence does not involve extension in space, implies no func- 

 tional relation between space and time. The words "then and 



' Potts (Robert), "Euclid's Elements of Geometry," &c. (Cambridge and 

 London, 1845); Notes to Book i., p. 41. 



'^ Thomas Reid, "An Inquiry into the Human Mind on the Principles of 

 Common Sense " (1764). My pagings refer to the Calcutta Reprint of 1869. 

 •Chapter vi. treats (pp. 94-277) of Seeing ; its Section vii. (pp. 120-24), of 

 Visible Figure and Extension; and its Section ix. (pp. 132-45), of the 

 ■Geometry of Visibles. In Section viii. (pp. 125-32), we have Some Queries 

 concerning Visible Figure answered. 



3 I cite from Richardson's Translation (London, 1819) ; and cannot now 

 give the corresponding paging in that of Prof Mahaffy. 



■* Montucla, •' Histoire" (2de edition, An. vii.), p. 21. 



5 See Cayley, Southport Address, pp. 11. 12. 



6 See Bacon, "Advancement of Learning," p. 108. 



7 Kant, " Prolegomena," p. 52. 



8 William Kingdon Cliff rd, " Mathematical Papers" (London, 1882). See 

 pp. xl. and xl i. of the Introduction, by H. J. S. Smith. 



9 De Morgan, "On the General Principles of which the Composition or 

 Aggregation of Forces is a Consequence " (Camb. Trans., vol. x., part 2, pp. 

 394, 295, footnote). 



there," accompanying every material allegation in indictments, 

 would suffice to show that the opinions of the world at large on 

 certain characteristics of time and space ^ were in accord with 

 that of the philosophers. Indeed, their isolation, as forms of 

 intuition, may no more be a peculiarity of Kant's system than is 

 his distinction between analytical and synthetical judgments. 

 This distinction was present to the mind of Bacon,- as well as 

 to that of Locke, whom Kant cites (" Prolegomena," p. 25), and 

 who, elsewhere than in the place cited, adverts to the distinction. 

 That which Locke had styled a trifling proposition, Kant called 

 an analytical judgment ; and that which Locke (" Essay concern- 

 ing Human Understanding," book iv., chap, viii.. Sect. 8) styled 

 a real truth, Kant would would have called a synthetical judg- 

 ment. With Hume, too, Kant is in some respects in close 

 relation. Hume ("Treatise," vol. i., book i., part 2, pp. 

 53-124) treats specially of the ideas of space and time. Hume, 

 again (" Inquiry," p. 17 ; Essay iv., p. 50), distinguishes between 

 results attained by reasonings a priori a.\\A results arising entirely 

 from experience ("Inquiry," p. 17 ; Essay, p. 49). He seems 

 to allow conception a sufficiently wide range, for he urges 

 (" Inquiry," p. 13 ; Essay ii., pp. 26, 27) that, in one exceptional 

 instance, there may be an idea not arising from a corresponding 

 impression ; viz. in the case when from the impressions of two 

 distinct shades of a particular colour, a conception is formed of 

 an intermediate shade of the same colour. He asserts (" In- 

 quiry," p. 1 18) that the only objects of the abstract sciences or of 

 demonstration are quantity and number. 



If, as Clifford ^ seems to think, there are no sufficient grounds 

 for maintaining that, if our space has curvature, it must be con- 

 tained in a space of more dimensions and no curvature, one 

 difficulty is apparently removed. The one-dimensioned time is 

 something very different to space, from which the higher- 

 dimensioned entity might differ still more ; and if a solid be 

 treated as the shadow or projection in Euclid's space of, say, a 

 four-dimensioned body, that part of the body which lies outside 

 the shadow seems to have no quality analogous to impenetrability 

 or inertia, nor indeed any quality which affects the senses or 

 deranges the results of calculation. Prof Cayley says (Southport 

 Address, p. 11) that Riemann's idea reems to be that of modify- 

 ing the notion of distance, not that of treating it as a locus in four- 

 dimensional space. The suggestion (Cayley, ih. p. 10) of a rule 

 changing its length by an alteration of temperature facilitates 

 apprehension. Prof, von Helmholtz has considered the effect 

 of the changes in sensible phenomena which a transition to a 

 spherical or pseudo-spherical world, if such things be, would 

 produce ; and he has taken an independent view of the subject 

 in other respects.'' 



De Morgan ^ professed to have been puzzled to know on which 

 side the meeting of parallels took place, or whether on both. 

 He concludes that they never meet. This, however, does not 

 shake, nor is it to be supposed that he wished '' it to shake, the 

 belief in modern methods, for he apparently admits that inter- 

 pretation of forms may demand conclusions which can be reached 

 by reasoning on infinity, if increase without limit show approach. 

 He observes that it is clearly conceived by the logicians that all 

 division is reducible to simple dichotomy anr" its repetitions, and 

 that when the logician has once shown division, difference, he 

 does not trouble himself with the difficulty of repetitions. De 



' I .should have been glad to have given Locke's and Kant's descriptions 

 of space and time, and to have compared them with Newton's. But I cannot 

 omit to refer to a Smith's Prize paper, by Mr. Robert Franklin Muirhead, 

 printed in the Philosophical Magazine ioT June 1S87, S. 5, vol. x.xiii. pp. 

 473-89. 



= Bacon, " Advancement of Learning," p. 47. 



3 Clifford, "The Universal Statements of Arithmetic " Nineteenth Century 

 (1879, vol. v., pp. 513-22 ; vide p. 522). 



■* A paper in Mind, by Prof, von Helmholtz, elicited a criticism from Prof. 

 Land, which produced a reply ; and with a brief note appended to a paper on 

 another subject, by Prof Land, the discussion closed. See Mind, vol. i. 

 pp. 301-21 ; vol ii. pp. 38-46; vol. iii. pp. 212-25 and 551-55; and vol. iv. 

 pp. 591-96. 



5 De Morgan, " On Infinity," &c. (Camb. Trans., vol. xi. Part i, 1865, 

 pp. 145-89; vid. pp. 173,176, 180, 147). In connection with this paper, the 

 comments of Mr. W. S. B. Wool house in the Educational Times (Reprint, 

 vol. vi. pp. 49-52) should be considered. And in connection with a para- 

 graph at pp. 161, I^2, of De Morgan's paper, the leading paragraph of p. 424 

 of a previous paper of h s, '"On the Theory of Errors of Observation " (C. 

 T., vol. X. Pt. 2, 1862), should be read. In the last-mentioned passage he 

 distinguishes between the zero and the indivisible of probability. Hamilton, 

 of Edinburgh, following earlier authorities, expre.s.sly restricts the application 

 of logic to finite things. But it does not therefore follow that logicians in 

 general lurn a deaf ear to all reasoning upon infinities and infinitesimals, 

 and that they reject results stamped with authority and universally 

 accepted. 



6 This sufficiently appears from a statement at p. 15 of his paper, " On 

 the Rojt," &c. (Camb. Trans., vol. xi. Pt, 2). 



