Rene Descartes

is responsible for having attempted to apply mathematical reasoning to all sciences after recognizing the certainty of the conclusions in geometry and arithmetic, which brought out in his mind the contrast between the true and false ways of seeking the truth.

Among the earliest thinkers of the seventeenth and eighteenth centuries, who employed their mental powers toward the destruction of old ideas and the up-building of new ones, ranks Rene Descartes. Though he professed orthodoxy in faith all his life, yet in science he was a profound sceptic. He found that the world's brightest thinkers had been long exercised in metaphysics, yet they had discovered nothing certain; nay, they had even flatly contradicted each other. ...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. His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever. His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.

is not responsible for having first applied algebra to geometry (as is often stated). Rather, he introduced into geometry an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations.

It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs some times used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree.

Florian CajoriThinkers

Florian Cajori