This great work, generally attributed to Eudoxus, has attracted the attention of mathematicians of all time, but is now neglected both in the Schools and in the Universities.

The first person to discover a flaw in the logic of any of the propositions, as given in the text that has come down to us, was (I believe) Saccheri, who, in his Euclides ab omni naevo vindicatus, pointed out that in the demonstration of the 18th proposition it was assumed that if there be three magnitudes of which the first and second are of the same kind, then a fourth magnitude of the same kind as the third must exist such that the ratio of the first magnitude to the second was the same as that of the third to the fourth. This proposition he endeavoured unsuccessfully to prove, but it is now known that its validity depends on the Axiom of Continuity.

Much has been said about the danger of dividing the various branches of pure mathematics into “water-tight compartments,” and books have been written with a view to breaking down from the very beginning the barriers between the different subjects.

I feel, however, that to the average boy who has not yet reached Matriculation standard, the division of Mathematics into Arithmetic, Algebra and Geometry is very real and most helpful. A straight-forward question inmensuration, the study of the identity (a+b)2=a 2+2ab+b 2, and the theorem that the square on a line is equal to the sum of the squares on its two segments together with twice the rectangle contained by them, for example, will and should appear as separate questions—the first being linked with other similar exercises in Arithmetic, the second depending on algebraical multiplication and being a study in form rather than in area, and the third turning on a geometrical construction. Only after each has been treated in the appropriate way should we show the pupil the cross-connections existing in the three problems.

Principles. 1. If K be the centre of a conic and P a point not on it, the diameter conjugate to KP is parallel to the polar of P.

2. Conjugate diameters form a pencil in involution. Hence if S=0 and S' =0 be the equations of two known pairs of conjugate diameters, the general equation is S + λS' =0.

The method then is to find the diameters conjugate to KA, KB, KC, where ABC is the triangle of reference, by means of 1, and then to write down the general equation by means of 2.

The difference between any number and another formed by reversing its digits has certain interesting properties, which, I believe, have not been explicitly noticed before, and these properties determine the necessary and sufficient conditions that a number may be expressed as such a difference.

To begin with, let us take a number having an even number of digits and denote it by

where a p , b p are the middle digits, b 1 the units digit and a 1>b 1.