SECTIONAL TRANSACTIONS.— A.* 345 



Prof. E. H. Neville. — The influence of the University on school 

 geometry (10.10). 



In the anticipation of developments proper to a later stage, the balance 

 between technical skill and general ideas has not always been well held in 

 the school course. Sometimes, as in the case of the circular points, verbal 

 dexterity is acquired in a field of which no rational account is given. Some- 

 times, as in the case of curvature, we are content with trivial exercises when 

 we might have exciting glimpses into higher geometry. The problem 

 should be considered in relation both to the potential specialist and to the 

 boy whose mathematics is a small part of a general education. 



Mr. H. G. Green. — Infinity in Euclidean geometry (10.30). 



The term ' line at infinity ' as a device to cover a case of failure in the conical 

 projection from plane to plane. 



Discussions on the line at infinity and distant points regarded as dis- 

 cussions of a line and points near it in a projected figure. Circular points. 



The plane at infinity : properties of the conicoids developed from the 

 quadrangle formed by the intersection of two conies. 



Prof. W. H. McCrea. — The circular points and elementary geometry 

 (ii.o). 



Elementary geometry in schools is metrical euclidean geometry. The 

 ' point at infinity ' on a line is defined to give convenient expression to 

 consequences of Euclid's parallel postulate. The points at infinity on all 

 lines in the euclidean plane satisfy formally the collinearity conditions ; 

 so we get the ' line at infinity.' 



It is convenient to define the ' orthogonal involution ' on the line at 

 infinity. Then, for example, a necessary and sufficient condition for a 

 conic to be a circle is that all pairs of this involution should be conjugate with 

 respect to the conic. 



We might proceed to note the analogy with the conjugate pairs of points 

 on the radical axis of a coaxal system ; these form an involution, and, if it 

 possesses double points, all circles of the system pass through them. So 

 we might choose to say that all circles in the plane pass through the imaginary 

 double points of the orthogonal involution, and so define the ' circular points.' 

 But it is better not to do this, the concept of imaginary elements being 

 foreign to this geometry. Instead, we can continue to work with the ortho- 

 gonal involution. Example: foci of a conic. 



Prof. H. S. Ruse. — Differential geometry (11.35). 



' The theory of the curvature, etc., of plane curves is usually treated as 

 a mere by-product of the calculus ; but since it contains many of the 

 essential ideas of the generalised differential geometry that forms the basis 

 of relativity and other branches of modern physics, it is worthy of greater 

 attention. The interest of beginners (especially those intending to specialise 

 in mathematics at a university) might be stimulated by the use of vector 

 methods and by a general account of how the formula of plane differential 

 geometry extend naturally to the theory of curves and surfaces in higher 

 space. Such ideas are by no means beyond the understanding of the 

 average student. 



General Discussion (12.0). 



