Analytic geometry of the plane

Abstract

The main idea of analytic geometry is that geometric investigations can be carried out by means of algebraic calculations. This method has proved extraordinarily fruitful. The fusion of geometric and algebraic thinking, together with functional thinking, provides an important help to man’s understanding of the exploration and comprehension of objective reality. At the same time the method is particularly attractive mathematically and gives rise to important elements in the training of the mind. The birth of the method of analytic geometry, and the consequent growth of the methods of the differential and integral calculus, characterize the transition to modern mathematics. The year of birth can be taken to be 1637, when Descartes (1596–1650) published his Discours de la Méthode anonymously, to avoid a dispute with the church. In this work, which is also significant for the history of philosophy, the third part, entitled La Géométrie, systematically expounds the fundamental principle of analytic geometry. Shortly before, Fermat (1601–1665) had also worked out the method of analytic geometry, but his treatise Ad locos pianos et solidos isagoge (Introduction to planar and spatial geometric loci) was not published until 1679. Since the ‘Geometry’ of Descartes had also the better notation, the development of the method of analytic geometry is usually attributed to Descartes. Its present form was, however, developed a long time after Descartes, particularly by Euler (1707–1783). For example, Descartes did not use two axes, and only since the time of Euler, to whom a large part of the modern notation is due, have far-reaching conclusions been drawn from the equations of geometric loci, while Descartes and Fermat generally regarded their investigations as ending when the equation had been set up.