certain; nay, had even flatly contradicted each other. This led him to the gigantic resolution of taking nothing whatever on authority, but of subjecting everything to scrutinous examination, according to new methods of inquiry. The certainty of the conclusions in geometry and arithmetic brought out in his mind the contrast between the true and false ways of seeking the truth. He thereupon attempted to apply mathematical reasoning to all sciences. "Comparing the mysteries of nature with the laws of mathematics, he dared to hope that the secrets of both could be unlocked with the same key." Thus he built up a system of philosophy called Cartesianism.
Great as was Descartes' celebrity as a metaphysician, it may be fairly questioned whether his claim to be remembered by posterity as a mathematician is not greater. His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever. At the age of twenty-one, Descartes enlisted in the army of Prince Maurice of Orange. His years of soldiering were years of leisure, in which he had time to pursue his studies. At that time mathematics was his favourite science. But in 1625 he ceased to devote himself to pure mathematics. Sir William Hamilton is in error when he states that Descartes considered mathematical studies absolutely pernicious as a means of internal culture. In a letter to Mersenne, Descartes says: "M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature.… You know that all my physics is nothing else than geometry." The years between 1629 and
