352 



SCIENTIFIC THOUGHT. 



29. 

 Geometrical 

 axioms. 



I may, in passing, mention here that in the course of 

 our century certain views have been put forward in pure 

 mathematics, or rather in geometry, which make it con- 

 ceivable, if not probable, that our ideas of space might 

 not apply to immeasurably small or to immeasurably 

 large dimensions.^ Should the future progress of thought 



Miething, ' L. Euler's Lehre voin 

 Aether,' p. 30). In quite recent 

 times a similar position has again 

 been taken up by Paul du Bois- 

 Reymond in his essay " Ueber die 

 Unbegreiflichkeit der F'ernkraft," 

 ill the ' Xaturwissenschaftliche 

 Rundschau' (vol. iii. No. 14), and 

 in his posthumous work, ' Ueber 

 die Grundlagen der Erkenntniss in 

 den exacten Wissenschaften ' (Tii- 

 bingen, 1890), in which he adds 

 action at a distance as a third 

 " ignorabimus " or unknowable pro- 

 blem to the two given in his 

 brother Emil's address, " Ueber die 

 Grenzen des Naturerkennens " 

 (1872, reprinted in 'Reden,' vol. 

 i. p. 105). On the Continent, 

 about thirty years ago, the fruit- 

 lessness of pursuing this problem 

 seemed generally admitted. Helm- 

 holtz in 1847 speaks of the initial 

 assumption "that all actions in na- 

 ture are to be reduced to attracting 

 and repelling forces, whose inten- 

 sitj' depends merely on the distance 

 of points mutuallj' acting on each 

 other" {actio in distans), and Du 

 Bois-Reymond repeats this in 1871 

 in liis address. But it is significant 

 that Helmholtz, who (through his 

 memoir on vortex motion in 1858) 

 gave such an impetus to the me- 

 chanical explanations of molecular 

 forces, modified his views on this 

 point (see his address on Magnus, 

 1871, ' Vortriige und Reden,' vol. 

 ii.) ; accordingly in the reprint of 

 his memoir of 1847 he has accom- 

 panied it with .some significant re- 

 marks on the necessity of that in- 

 itial assumption (18S1, ' Wissen- 



schaftliche Abhandlungen,' vol. i. 

 p. 68). 



^ lliemann was probably the first 

 to give expression to this line of 

 thought. His memoir on this sub- 

 ject, " On the Hypotheses which lie 

 at the Foundation of Geometry," 

 bears the date 1854. It was read 

 before the Philosophical Faculty of 

 Gottingen in the presence and at the 

 request of Gauss, on whom it made 

 a profound impression (see the bio- 

 graphical notice on Riemann by 

 Dedekind, attached to Riemann's 

 'Gesammelte Werke,' Leipzig, 1876). 

 The memoir was not published till 

 after Riemann's death in 1867. In 

 England the late Prof. Clifford in- 

 troduced the subject to the Cam- 

 bridge Philosophical Society in 1 870 : 

 " The axioms of plane geometry are 

 true within the limits of experiment 

 on the surface of a sheet of paper, 

 and yet we know that the sheet is 

 really covered with a number of 

 small ridges and furrows, upon 

 which these axioms are not true. 

 Similarly although the axioms of 

 solid geometry are true within the 

 limits of experiment for finite por- 

 tions of our space, yet we have no 

 reason to conclude that they are 

 true for verj' small portions ; and 

 if any help can be got thereby for 

 the explanation of physical pheno- 

 mena, we may have reason to con- 

 clude that they are not true for 

 very small portions of space " (see 

 Clifford's ' Mathematical Papers,' 

 p. 21 . Compare also his lectures 

 on "The Philosophy of the Pure 

 Sciences " in ' Lectures and Essays,' 

 vol. i. p. 295 sqq.) 



