198 A History ofMathematics
Fig. 6Classical descriptive geometry as it is still practised today. The three projections are united to give a general view, using the
algebra of vectors.world, at least for some geometers.^4 Of course, in a sense this has always been so; but the idea
that the study of geometry was derived from knowledge of the world, whether innate (part of our
mental structure) or empirical (derived from observation) became a dominant one for the next two
centuries. The contrast with Plato’s view that geometrical ideas were in some way above practice is
striking.
The main developments in geometry from Newton’s time on—and this is important when we
come to consider the relative importance of the ‘new’ geometries—concerned, naturally, the
increasing introduction of coordinates and of calculus as tools. To study curves and surfaces
meant to study their equations, even if diagrams were used as aids to understanding. In the late
eighteenth century, the French geometer Gaspard Monge developed what was called ‘descript-
ive geometry’ a key subject in the immensely influential École Polytechnique. Very fashionable
throughout the nineteenth century, and still surviving as an essential part of practical training
although unknown in most mathematics departments, descriptive geometry was the study of
three-dimensional figures via their projections—plans, elevations, and so on; the breaking down
of a figure into its projections, and its reconstitution from them (Fig. 6); and it leaned heavily
on calculus in its more sophisticated parts. Partly because the use of coordinates was central,
partly because of the importance of practical application, it was not concerned with the ques-
tion of the world. In 1837 we find Monge’s follower Michel Chasles praising him precisely for
avoiding those diagrams (‘figures’) which were an essential starting point in thinking, say, about
parallels.[A]lthough descriptive geometry...by its nature makes an essential use of figures, it is only in its practical and
mechanical applications, where it plays an instrumental part, that it needs them: no one more than Monge thought
of and practised geometry without figures. There is a tradition in the École Polytechnique that Monge knew to an
amazing degree how to make his audience imagine the most complicated forms of extension in space, and their most
hidden properties, without any other aid but his hands, whose movements followed his words admirably...(Chasles
1837, p. 209)- In an earlier draft, I used the phrase ‘confused with’ rather than ‘identical with’; but the confusion is more that of the historian,
who has to try to understand, from Newton on, whether a geometer is describing an abstract construction or the empirical universe.
Sometimes, but not always, the geometer will explain.