oduced system into the use of exponents to denote the powers of a
quantity, thus opening the way for the binomial theorem; he was the
first to throw clear light on the negative roots of equations; his is
the theorem by use of which the maximum number of positive or negative
roots of an equation can be ascertained. Analytical geometry originated
with his investigation of the nature and origin of curves.

His mathematical improvements opened the way for the reform of physical
science and for its immense modern advance. In his optical
investigations he established the law of refraction of light. His
ingenious theory of the vortices--tracing gravity, magnetism, light, and
heat, to the whirling or revolving movements of the molecules of matter
with which the universe is filled--was accepted as science for about a
quarter of a century.

In mental science Descartes's primary instrument for search of truth was
Doubt: everything was to be doubted until it had been proved. This was
provisional skepticism, merely to provide against foregone conclusions.
It was not to preclude belief, but to summon and assure belief as
distinct from the inane submission to authority, to prejudice, or to
impulse. In this process of doubting everything, the philosopher comes
at last to one fact which he cannot doubt--the fact that he exists; for
if he did not exist he could not be thinking his doubt. _Cogito, ergo
sum_ is one point of absolute knowledge; it is a clear and ultimate
perception.

The first principle of his philosophy is, that our consciousness is
truthful in its proper sphere, also that our thought is truthful and
trustworthy under these two conditions--when the thought is clear and
vivid, and when it is held to a theme utterly distinct from every other
theme; since it is impossible for us to believe that either man who
thinks, or the universe concerning which he thinks, is organized on the
basis of a lie. There are "necessary truths," and they are discoverable.

A second principle is, the inevitable ascent of our thought from the
fragmentary to the perfect, from the finite to the infinite. Thus the
thought of the infinite is an "innate idea," a part of man's potential
consciousness. This principle (set forth in one of the selections given
herewith) is the Cartesian form of the _a priori_ argument for the
Divine existence, which like other _a priori_ forms is viewed by critics
not as a proof in pure logic, but as a commanding and lum