680 SCIENTIFIC THOUGHT. 



quarters the one purely theoretical, the other practical. 

 Accordingly the doctrine of forms and arrangements has 

 during the last century been developed by mathematicians 

 in two distinct interests, which only quite lately seem to 

 approach and assist each other. 

 ss. The purely abstract or theoretical interest came from 



Theory of 



numbers, the side of the theory of numbers, a branch of research 

 which was revived by Legendre in France and by the 

 youthful genius of Gauss in Germany ; the more practical 

 one came from the theory of equations, notably in its 

 application to problems of geometry. The methods by 

 which these subjects were treated had in the early part 

 of the nineteenth century undergone a great change. 

 The older inductive method in both branches namely, 

 in the solution of equations and in the investigation of 

 the properties of numbers relied mainly on ingenious 

 devices which were mostly of special, not of general, 

 value. Theorems were found by induction, and had 

 afterwards to be proved by rigorous logical deduction. 

 Success depended on the degree of care with which the 

 mind operated with mathematical symbols, and rested 

 frequently on the intuition, if not the inspiration, of 

 genius. Two of the greatest mathematical minds 

 Fermat l in France and Newton 2 in England stood 



1 Pierre Fermat (1601-65) pre- 

 pared an edition of the Treatise of 

 Diophantus, and his marginal notes 

 contain many theorems referring 

 to the properties of numbers which 

 have been the subject of much 

 comment and examination by 

 mathematicians of the first rank 

 down to the present day. In 

 letters to contemporaries he re- 

 ferred to many of these dis- 

 coveries, and to his proofs, which 

 he did not communicate. Some 



of these proofs seem not to have 

 satisfied him, being deficient in 

 rigour. In spite of the labours of 

 Euler, Lagrange, Cauchy, Dirich- 

 let, Kummer, and others, one of 

 these theorems still awaits proof. 

 A full account of Fermat's theorems 

 is given in Cantor's ' Geschichte der 

 Mathematik,' vol. ii. 2nd ed., p. 

 773 sqq. Also in W. Rouse 

 Ball's ' History of Mathematics, ' 

 p. 260 sqq. 



2 Newton, in his 'Universal 



