the controversy with obstinacy. He had a controversy also with Roberval on the cycloid. This curve has been called
the "Helen of geometers," on account of its beautiful properties
and the controversies which their discovery occasioned. Its quadrature by Roberval was generally considered a brilliant achievement, but Descartes commented on it by saying that any one moderately well versed in geometry might have done this. He then sent a short demonstration of his own. On Roberval's intimating that he had been assisted by a knowledge of the solution, Descartes constructed the tangent to the curve, and challenged Roberval and Fermat to do the same. Fermat accomplished it, but Roberval never succeeded in solving this problem, which had cost the genius of Descartes but a moderate degree of attention.
He studied some new curves, now called "ovals of Descartes,"
which were intended by him to serve in the construction of converging lenses, but which yielded no results of practical value.
The application of algebra to the doctrine of curved lines
reacted favourably upon algebra. As an abstract science, Descartes improved it by the systematic use of exponents and
by the full interpretation and construction of negative quantities.
Descartes also established some theorems on the theory of equations. Celebrated is his "rule of signs" for determining
the number of positive and negative roots; viz. an equation may have as many roots as there are variations of signs, and as many roots as there are permanencies of signs. Descartes was charged by Wallis with availing himself, without acknowledgment,
of Harriot's theory of equations, particularly his mode
of generating equations; but there seems to be no good ground for the charge. Wallis also claimed that Descartes failed to observe that the above rule of signs is not true whenever the
equation has imaginary roots; but Descartes does not say that the equation always has, but that it may have so many roots. It is true that Descartes does not consider the case of imaginaries directly, but further on in his Geometry he gives incontestable evidence of being able to handle this case also.
In mechanics, Descartes can hardly be said to have advanced
beyond Galileo. The latter had overthrown the ideas of
Aristotle on this subject, and Descartes simply "threw himself upon the enemy" that had already been "put to the rout." His statement of the first and second laws of motion was an
improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens.