SCIENTIFIC THOUGHT. 



4. 



Use of 

 mathe- 

 matics. 



of senate-house examiners and examinees, without for a 

 moment considering the question whether mathematical 

 thought as distinguished from mathematical problems is 

 capable of and has undergone any radical and funda- 

 mental change or development. 



Closely allied with this is the further question as to 

 the use of mathematics. Two extreme views have always 

 existed on this point. 1 To some, mathematics is only a 

 measuring and calculating instrument, 2 and their interest 



1 Of the two greatest mathemati- 

 cians of modern times, Newton and 

 Gauss, the former can be considered 

 as a representative of the first, the 

 latter of the second class ; neither of 

 them was exclusively so, and New- 

 ton's inventions in the pure science 

 of mathematics were probably equal 

 to Gauss's work in applied mathe- 

 matics. Newton's reluctance to 

 publish the method of fluxions in- 

 vented and used by him may per- 

 haps be attributed to the fact that 

 he was not satisfied with the logical 

 foundations of the calculus ; and 

 Gauss is known to have abandoned 

 his electro-dynamic speculations, as 

 he could not find a satisfactory 

 physical basis (see supra, p. 67). 

 Others who were not troubled by 

 similar logical or practical scruples 

 stepped in and did the work, to the 

 great benefit of scientific progress. 

 Newton's greatest work, the ' Prin- 

 cipia,' laid the foundation of mathe- 

 matical physics ; Gauss's greatest 

 work, the ' Disquisitiones Arith- 

 metical,' that of higher arithmetic 

 as distinguished from algebra. 

 Both works, written in the syn- 

 thetic style of the ancients, are 

 difficult, if not deterrent, in their 

 form, neither of them leading the 

 reader by easy steps to the 

 results. It took twenty or more 

 years before either of these works 

 received due recognition ; neither 



found favour at once before that 

 great tribunal of mathematical 

 thought, the Paris Academy of 

 Sciences. Newton's early reputa- 

 tion was established by other 

 researches and inventions, notably 

 in optics ; Gauss became known 

 through his theoretical rediscovery 

 of Ceres, the first of the minor 

 planets (see above, vol. i. p. 182). 

 The country of Newton is still pre- 

 eminent for its culture of mathe- 

 matical physics, that of Gauss for 

 the most abstract work in mathe- 

 matics. Not to speak of living 

 authorities, I need only mention 

 Stokes and Clerk-Maxwell on the 

 one side, Grassmann, Weierstrass, 

 and Georg Cantor on the other. 



2 Huxley said : " Mathematics- 

 may be compared to a mill of 

 exquisite workmanship which grinds 

 you stuff of any degree of fineness : 

 but, nevertheless, what you get out 

 depends on what you put in ; and 

 as the grandest mill in the world 

 will not extract wheat- flour from 

 peas-cods, so pages of formulae will 

 not get a definite result out of 

 loose data" ; and on another occa- 

 sion he said that mathematics "is 

 that study which knows nothing of 

 observation, nothing of induction, 

 nothing of experiment, nothing of 

 causation." The former statement 

 was endorsed by Lord Kelvin 

 (' Pop. Lectures,' &c., vol. ii. p. 



