Section III., 1906. [ HI ] Trans. R. S. C. 



VII. — The Foundations of Geometry — Presidential Address to 



Section III. 



By Professor Alfred Baker^ M.A. 



(Read May 22nd, 1906). 



It is satisfactory to know that the advances that are being made 

 in the purely intellectual domain of mathematics are, in a sense, com- 

 parable with the remarkable achievements of the ph3'Sicist and of the 

 chemist, though necessarily appealing to a more limited circle, and lesa 

 encouraged by the stimulating influence of popular applause. In 

 mathematics activity shows itself in two directions — the boundaries 

 of the science are being enlarged, and its foundations are being subjected 

 to the most searching examination. In analysis the examination of 

 the number concept has produced many remarkable results ; in geometry 

 the search-light of a penetrating logic has revealed the base of the 

 subject with remarkable clearness. It is of this latter field of enquiry — 

 the foundations of geometry — I wish to speak. 



In Euclid each proposition rests on preceding propositions, and the 

 reasoning is unassailable. But when we go down to the lowest stones 

 of the structure — to the axioms — we find ourselves in serious diffi-' 

 culties. Every intelligent schoolboy has had his trouble with the 

 eleventh axiom, respecting parallel lines, and it has puzzled many a 

 philosopher. It has been claimed that Euclid reckoned it amongst 

 his postulates ; and certainly, if its self-evidence had never been asserted, 

 and if the assumption implied in it had been asked as a concession, 

 the nature of the foundation of the science of geometry would have 

 been much more clearly revealed, and much useless labour would have 

 been saved. Everyone is more or less acquainted with the struggles 

 to prove this axiom. Perhaps nothing reveals the subtlety of thfe 

 subject better than the well-known story told of the great Lagrange. 

 Observing that the formulae of spherical trigonometry did not depend 

 on the eleventh axiom, Lagrange thought to develop a proof of the 

 axiom based on this fact. He prepared his paper and actually began 

 to read it before the Academy. Suddenly stopping, he said, " II faut 

 que j'y songe encore," put the paper in his pocket, and never after- 

 wards referred to the matter, at least in public. 



The search for the unattainable was closed by the labours of Gauss, 

 Bolyai, Lobachevski and Kiemann, the pioneers undoubtedly being 



