As I can pretend to only limited practical experience in the matter of teaching the foundations of Mathematics, and of Geometry in particular, I have thought that the time allowed for my remarks may be advantageously occupied in a rapid survey of what the geometrical method has accomplished, and the estimation in which it has been held at different periods of the history of the development of Mathematical Science. Thus, avoiding expert questions relating to Education, we may begin by considering from a general standpoint what is implied in the science of Geometry, for the right ordering of which, in its educational relations, this Association has been working for a long series of years. It would, I presume, be against the instincts of our members to interpret the subject in any narrow sense. Geometry is usually based on the partly mathematical, partly logical, framework which has descended to us from Greek times, and which is chiefly associated with its earliest great systematization in the work of Euclid. But modern development would perhaps render any definition of its scope incomplete that did not include all methods in formal and physical Mathematics which proceed by direct contemplation of the relations of the subjects considered, instead of through representations of them that are merely quantitative or numerical.

The method usually given in text books on Algebra for finding the present value of an annuity to last for a given number of years, or the accumulated amount of an annuity left unpaid for a given time, consists in writing down the present values or amounts of the several payments, and summing them by the usual formula for a geometrical progression. The following method, by which the result is found from first principles without the use of formulae, is so simple and instructive that I am rather surprised that the idea does not appear to have suggested itself to teachers and writers of text books before now, or, if it has done so, that it has received so little notice.

The majority of text-books on Geometrical Conies do not give any proof of the fundamental property that one and only one conic passes through any five given points, no three of which lie on a straight line. Special attention is drawn to this omission by the Rev. Dr. C. Taylor in an article published in the Gazette for May, 1895. The following is a description of three methods which various attempts at obtaining an elementary proof of the theorem have suggested. It is, however, very desirable to have a simpler proof than any of these, if such can be given.