INTRODUCTION. 



45 



tellects in science have condescended to write text-books 

 of their subjects, by which a great reform has been brought 

 about in the higher scientific literature. 1 At the same time 

 after fifty years of experimental research and accumula- 

 tion of material it has become necessary to review the 

 fundamental principles on which scientific reasoning rests : 

 a more philosophical, not to say metaphysical, spirit is 

 manifesting itself within the limits of science. 2 In the 

 abstract, and especially the mathematical, sciences, real 

 progress depends now mainly upon the discovery of 

 methods of simplification, on conciseness and elegance 

 of treatment, and on the discovery of unifying principles 

 and generalising aspects. 3 



22. 



Scientific 

 reasoning 

 more philo- 

 sophical. 



1 This remark refers mainly to 

 England and Germany. In France, 

 as a result of giving lectures at the 

 Ecole Polytechnique, the Bureau 

 des Longitudes, the Faculte" des 

 Sciences, &c., the great mathema- 

 ticians and physicists of the cen- 

 tury have frequently worked up their 

 researches in connected treatises. 

 For such we are indebted to Lame, 

 Cauchy, Poncelet, and many others. 

 But the two works which in Eng- 

 land and Germany created probably 

 the greatest reform in the teaching 

 of the principles of natural phil- 

 osophy were Thomson and Tait's 

 ' Natural Philosophy ' (first sketch, 

 1863, 1st ed., 1867) and Kirchhoffs 

 ' Vorlesungen iiber Mechanik ' (Leip- 

 zig, 1877). 



- I refer principally to the various 

 writings of Helmholtz, following 

 those of Riemann, and the many 

 hints thrown out in Gauss's pub- 

 lished papers, and in his correspond- 

 ence with Schumacher. Helmholtz 

 has of all purely scientific writers 

 paid most attention to the meta- 

 physical foundations of geometry 



and dynamics, and has critically 

 examined the earlier theories of 

 Kant, published a century ago. It 

 is interesting in this respect to note 

 what Kant is reported to have said 

 to Stagemann in 1797: "I have 

 come with my writings a century 

 too soon ; after a hundred years 

 people will begin to understand me 

 rightly, and will then study my 

 books anew and appreciate them." 

 (Vid. 'Tagebiicher,' von Varnhagen 

 von Ense, Leipzig, 1861, vol. i. p. 

 46.) Next to Helmholtz we are 

 most indebted to Emil du Bois- 

 Reymond and his brother Paul. 

 See Emil's 'Reden' (Leipzig, 1886- 

 87, 2 vols.), and the posthumous 

 work of his brother : ' Ueber die 

 Grundlagen der Erkeuntniss in den 

 exacten Wissenschaften ' (Tubingen, 

 1890). 



3 An authority on this subject 

 says : " Generality of aspects and 

 methods, precision and elegance 

 of exposition, have, since the time 

 of Lagrange, become the common 

 property of those who claim to 

 be scientific mathematicians. This 



