It has been shown that the evolution of logarithms involves consideration of Napier’s knowledge of Greek; this in turn is connected with his “educational and cultural environment”. He certainly studied abroad for some years but where he spent the time is largely a matter for speculation. Now even speculation must be based on some form of reasoning, therefore let it be taken as a hypothesis that as his mathematical equipment was considerably in advance of that of his contemporaries, this equipment must have been the result of ideas planted by his teachers. Consequently an explanation of Napier’s travels may be made by considering his work in mathematics and by discovering similarities in the works of mathematicians who were in Europe during the years of his residence abroad. It must be remembered also that it was a time of acute religious differences, and that he belonged to a family that was ardently Protestant in its sympathies. His own Protestantism is quite openly expressed in his work on the Apocalypse. Therefore these speculations are based on the “historico-religious” background of the period and on Napier’s mathematical attainments.

The three-dimensional analogue of Ptolemy’s theorem (Simson’s Euclid, Prop. D) has been given by Cayley (Collected Works, I, 1). The following note contains the theory of such generalizations to space of any number of dimensions.

Taking Euclidean space of n dimensions we denote the Cartesian coordinates of a point As by (xs,a) where the Greek suffix is a dummy and takes the values 1, 2, 3, ..., n, n + 1. We make the convention that xs, n+1 = 1 for every point As.

The Algebra Report is silent on the teaching of indices. The writer has found that some graphical illustration on the lines (probably well known) suggested below gives the work more meaning for boys of various degrees of ability

The first step is to plot the values of 2x, 1·5x, 1·2x, 1x, and 0·5x for the values x = 1, 2 and 3, and, assuming that we may, to join up the sets of points by smooth curves. This, of course, is done rather tentatively, as three points do not give much definition to a curve, but the values of all except 2x when x = 4 can be added and make the curves more definite. To include 24 makes the scale of the graph undesirably small.

If a particle is constrained to move on a sphere, the projection of its acceleration at P upon the tangent plane at P is here called its tangential acceleration. The line of action of the tangential acceleration touches a great circle; if O is a point on this great circle, the tangential acceleration at P is said to be directed to O. A tangential acceleration which is always directed to a fixed point O is called a central tangential acceleration.

It is proposed to discuss in this paper some properties of the general Frégier-point and point out its connections with well known conics like the Apollonian hyperbola and the nine-point conic of a quadrangle. Incidentally, we discover a pedal conic of a quadrangle which has very intimate relations with the nine-point conic.

The method is best introduced by a simple example.

To work outas a recurring decimal.

Instead of long division by 19, proceed as follows

[Explanation. Divide 2 into 1·0 and write the quotient 5 one decimal place further on than usual.