While Professor Nunn’s treatment of Asymptotes (Math. Gazette, May 1926) gives the elementary student a method for finding the equations of linear asymptotes, the present writer feels that there is some weakness in the method, as the geometrical consequences of the various types of asymptote on the general form of a curve are not emphasised. There is, moreover, a serious difficulty for the advanced student. Professor Nunn regards two parallel asymptotes as a degenerate case of the asymptotic parabola, thus throwing over entirely the ideas of multiple points at “infinity”. This must lead to considerable modifications in the values and properties of Pliicker’s numbers. In fact, the change would affect the whole theory of the deficiency of curves.

The following lesson in Theoretical Arithmetic was originally intended as an Appendix to a Higher Arithmetic, and is now published in the hope that it may be of interest to the more select band of young scholars who may wish to know more of the subject than its application to the calculations of Stocks and Shares, et hoc omne genus.

Mr. W. C. Fletcher opened the discussion by urging practical teachers who would appreciate the difficulties of the subject to express their views. He then asked, Did the Pappus theorem depend on parallelism? The proof which Professor Baker happened to give made it so depend.

As the outcome of a short but occasionally painful experience of mathematical papers, I would suggest that such a paper, if it is to be of any interest at all to the majority of those to whom it is addressed, must necessarily be boring to some of them. It is therefore with this explanation that I am trying to put before you results which are not in any sense new, which are, indeed, the stock-in-trade of the present-day analyst, and therefore well known to many here, but which may interest some whose studies have not lain in this particular direction.

Many teachers know that Euclid’s proofs by superposition have been much criticised, but to most of us, perhaps, even after the publication of the Report on Geometry, the matter has remained somewhat obscure. As for purposes of my own I have had to try to clear my mind on the subject, possibly what I have so learnt may be of use to others.