In these four rules we have the essential part of one half of Des Cartes's system, the other, which is equally important, is the attempt to solve metaphysical problems by mathematical aid. To mathematics he had devoted much of his time. He it was who, at the age of twenty three, made the grand discovery of the applicability of algebra to geometry. While deeply engaged in mathematical studies and investigations, he came to the conclusion that mathematics were capable of a still further simplification, and of much more extended application. Impressed with the certainty of the conclusions arrived at by the aid of mathematical reasoning, he began to apply mathematics to metaphysics. His ambition was to found a system which should be solid and convincing. Having searched for certitude, he had found _its basis_ in consciousness; he next wanted a _method_, and hoped he had found it in mathematics. He tells us that "those long chains of reasoning, all simple and easy, by which geometers used to arrive at their most difficult demonstrations, suggested to him that all things which came within human knowledge, must follow each other in a similar chain; and that provided we abstain from admitting anything as true which is not so, and that we always preserve in them the order necessary to deduce one from the other, there can be none so remote to which we cannot finally attain, nor so obscure but that we may discover them." Acting out this, he dealt with metaphysics as we should with a problem from Euclid, and expected by rigorous reasoning to discover the truth. He, like Archimedes, had wished for a standing place from which to use the lever, that should overturn the world; but, having a sure standing place in the indubitable fact of his own existence, he did not possess sufficient courage to put forth the mighty power--it was left for one who came after him to fairly attempt the over-throw of the world of error so long existent. Cartesianism was sufficiently obnoxious to the divines to provoke their wrath; and yet, from some of its peculiarities, it has found many opponents amongst the philosophical party. The Cartesian philosophy is founded on two great principles, the one metaphysical, the other physical. The metaphysical is Des Cartes's foundation-stone--the "I think, therefore I am." This has been warmly attacked as not being logical. Des Cartes said his existence was a fact--a fact above and beyond all logic; logic could neither prove nor disprove it. The _Cogito ergo Sum_ was not new itself, but it was the first stone of a new building--the first step in a new road: from this fact Des Cartes tried to reach another, and from that others.