Page:A History of Mathematics (1893).djvu/204

 1649 were passed by him in Holland in the study, principally, of physics and metaphysics. His residence in Holland was during the most brilliant days of the Dutch state. In 1637 he published his Discours de la Méthode, containing among others an essay of 106 pages on geometry. His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.

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 sometimes 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.

The Latin term for "ordinate," used by Descartes comes from the expression lineæ ordinatæ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623–1697), a professor of mathematics in Rome.[3] Descartes' geometry was called "analytical geometry," partly