ATTACKS UPON THE STUDY OF MATHEMATICS 365 



where progress has been hitherto arrested. To apply it to arithmetic 

 and geometry alone would be to occupy himself with " trifles," not only 

 because of the narrow field of application, but also because what was 

 practically his method had been thus applied long ago by the Greeks. 

 He wished to direct his method to unsolved problems. But he is free 

 to acknowledge that his method is found in mathematics as in an en- 

 velop. " Now I say that the mathematics are the envelope of this 

 method, not that I wish to conceal and envelop it, in order to keep the 

 vulgar away from it; on the contrary, I wish to dress and adorn it, in 

 such manner that it may be more easily grasped by the mind."-- 



In all this there is no attack whatever upon the culture value of 

 mathematics. Instead of hostility he shows friendliness to mathematics 

 as a gymnast of the mind. In his discussion of the " Fourth Eule " 

 there is a passage, not quoted by Hamilton, which bears directly upon 

 the question at issue : " This is why I have cultivated even to this day, 

 as much as I have been able, that universal mathematical science, so 

 that I believe I may hereafter devote myself to other sciences, without 

 fearing that my efforts may be premature."-^ Here then Descartes de- 

 clares that, as much as possible, he had studied mathematics all his life, 

 as a preparation or propedeutic to philosophy. It appears that Des- 

 cartes looked upon mathematical study as a desirable preparation to 

 philosophy, just as Plato had done nearly 2,000 years earlier. Looking 

 at the testimony contained in Descartes's writings, as a whole, there is 

 nothing in it to disturb in the least the belief in the educational value 

 of mathematical study. 



As a side issue we touch upon Hamilton's assertion that Descartes 

 in 1623 renounced mathematics for good. Hamilton does not say that 

 the work which is memorable in the history of mathematics as the 

 creation of analytical geometry was published by Descartes 14 years 

 later, in 1637. Did Descartes renounce mathematics for good? The 

 life of Descartes which was prepared by M. Thomas, a biography which 

 captured the prize offered by the French Academy in 1765, a biography 

 which is placed first in Cousin's edition of the works of Descartes, says 

 this about Descartes's renunciation (p. 89): "He attempted at least 

 five or six times to renounce them, but he always returned to them 

 again." M. Thomas adds : " He wished to occupy himself henceforth 

 only with morals ; but on the first occasion he returned to the study of 

 nature. Borne away in spite of himself, he plunged anew into the ab- 

 stract sciences" (p. 92). 



I proceed now to a review of a second attack upon mathematical 

 study, made by Schopenhauer, the pessimistic sage of Frankfort-on-the- 

 Main. His thoughts on mathematics are expressed in his work, en- 

 titled, " The World as Will and Idea," as it appeared in its second edi- 



" Southern Beview, Vol. 22, p. 270. 

 =3 Southern Beview, Vol. 22, p. 271. 



