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 envelop. "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."[1]
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 Rule" 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."[2] Here then Descartes declares that, as much as possible, he had studied mathematics all his life, as a preparation or propedeutic to philosophy. It appears that Descartes 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 abstract 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, entitled, "The World as Will and Idea," as it appeared in its second edi-
