The Case of Geometry

Abstract

For a long time mathematics has been synonymous with geometry. In fact there have always existed other branches too, algebra, trigonometry, calculus, which, however, were not much more than collections of haphazard, badly founded rules, whereas geometry was a perfect conceptual system, where things rigorously followed from each other, and finally everything from definitions and axioms. Though other techniques were more proficient, geometry was genuine truth. But the high esteem in which geometry stood faded away. By their axiomatic systems Pasch and Hilbert revealed many gaps in classical geometry. On the other hand the Pasch-Hilbert style axiomatics were so complicated that you could just read them or do foundational research about them, but you could not do geometry within them, and, in any case, could not teach geometry with them.

We see by experience that among equal minds and all other things being equal, he who possesses geometry, conquers and acquires an entirely new rigour.

The early study of Euclid made me a hater of Geometry, which I hope may plead my excuses if I have shocked the opinions of any in this room (and I know there are some who rank Euclid as second in sacredness to the Bible alone, and as one of the advanced outposts of the British Constitution) by the tone in which I have previously alluded to it as a schoolbook; and yet, in spite of this repugnance, which had become a second nature in me, whenever I went far enough into any mathematical question, I found I touched, at last, a geometrical bottom.

During the last few years some determined efforts have been made to displace Euclid’s Elements in our schools, but the majority of experienced teachers still regard it as the best foundation of geometrical thinking that has yet been published.

First of all do not get bewitched by the diabolic charm of geometry; nothing can more extinguish in you the internal spirit of grace, of meditation and of mortification.

Oeuvres Complètes de Fénélon, VIII, 519 (ed. Paris 1852)

B. Pascal, Pensées et opuscules, Hachette, p. 165, Note

J. J. Sylvester, The collected math, papers, II, 660

W. W. Rouse Ball, A Short History of Mathematics, 1888, p. 51

Parts of this chapter were printed in Educational Studies 3 (1971) 413–435.