WHEN your chairman, Mr. Jackson, asked me to supply a title for the Presidential Address, I could think of no name short enough and suitable for these random remarks, à bâtons rompus for half an hour.

Professor Turner gave the Mathematical Association an account of some new methods in calculation, found successful for Halley’s Comet; and Ray Lankester held forth in the afternoon on his perennial subject of the place of Greek in education, about which Turner has given you his say on a previous occasion.

I must avow myself at the outset in complete disagreement with their opinion, as in Greek we come to the bedrock of all our knowledge in history, literature, science, and art. Greek is the lighthouse to guide us out of barbarism to the human ideal.

22. To obtain a Fagnano pair of points on an ellipse whose axes are given, it may be considered sufficient to construct the corresponding Fagnano quadrilateral. That is, having drawn the line hk (Fig. 14) with the points p and z marked, and having described a circle on hk as diameter, it is only necessary to draw through z some chord c1c2 and to join the sides of the quadrilateral c1hc2k.

The time has long passed when the teacher of elementary geometry, however elementary, can efficiently exercise his individuality in devising improved methods of exposition without taking some cognisance, however small, of the study of non-Euclidean geometry which has grown up out of the failure to prove what may be described as “Euclid’s Parallel Postulate.”

A paper has recently been published in the Proceedings of the Edinburgh Mathematical Society by Prof. Carslaw, of Sydney, N.S.W., containing an elementary interpretation of the Bolyai-Lobatschewsky geometry, which is so simple and so easily intelligible that I strongly recommend every teacher of mathematics or science to read it. It does away altogether with the necessity of forming conceptions of space which appear contradictory to experience, and shows the possibility of constructing a non-Euclidean geometry in ordinary Euclidean space.

Many instruments have been devised for mechanically recording the area of a closed curve when a tracing point is carried round its boundary, and, of these, probably the best known is the Amsler Planimeter invented in 1854 by Professor J. Amsler of Schaffhausen.

It belongs to the class known as polar planimeters, so called because the tracing arm turns about a fixed centre or pole.

The theory of the instrument has been given in various ways by different writers, but perhaps the neatest explanation is that given by, Professor Henrici in the British Association Report, 1894. As this is probably pretty well known, only the briefest resumé will be given here in order to make the extension of it and the present article readily intelligible. The explanation depends on the general theorem that the area, reckoned algebraically, swept out by a line of fixed length in a complete cycle, is equal to the difference of the areas described by its ends. In the Amsler Planimeter, one end of the tracing arm moves over the circumference of the circle described by the end of the pole arm while the tracing point at the other end describes the boundary of the area.