The method of introducing logarithms given in the Algebra Report consists in taking the powers of 1·1, which are easily calculated and are so closely packed that common logarithms can be deduced by proportional parts correct to three decimal places. It forms an interesting set of lessons for post-certificate pupils to discuss why this is so, and to extend the method with a view to obtaining greater accuracy Furthermore, one of the most important principles in pure mathematics is amply illustrated—that of sandwiching a function between two bounds. It is assumed that the expansion of loge(l + x) is known by the pupils and also its differential coefficient.

Nowadays every Fifth Form pupil uses logarithm tables. A discussion of the construction of the difference columns, with graphical illustration, would prepare him for the methods to be described below, of an iterative process for the approximate solution of any equation. Incidentally it would help him to realise more fully than he usually does to what degree of accuracy his logarithmic work is reliable. The method of obtaining logarithms by using powers of 1·1, as described in the Algebra Beport, uses the same principle of proportional parts, and, properly treated, helps to establish the general principle that, when the differences for unit increments in the entries used are constant, results obtained by proportional interpolation are accurate to the same number of figures as the entries.

If a tetrahedron with its opposite edges perpendicular is inscribed in a conicoid, its orthocentre (the intersection of the perpendiculars from the vertices to the opposite faces) will lie on the conicoid if the asymptotic cone of the conicoid has three perpendicular generators.

Let O, A, B, C be any four points not necessarily in one plane. Let H and K divide OA and BC proportionally, while M and N divide OB and AC proportionally. Then we shall prove that HK and MN meet at T such that OA, MN, BC are proportionally divided at H, T, K, while OB, HK, AC are proportionally divided at M, T, N, the geometrical relations being conveniently, and diagrammatically, expressed thus.

The idea of the representation of a linear system of plane curves by means of a flat space is of fundamental importance in algebraic geometry. In this paper some details of the representation of circles are worked out, and an attempt is made to give a three-dimensional picture of certain parts of circle geometry The theorems given in italics are thought to be new, although in a subject with such an enormous literature it might almost be said that no theorem can be a new one.

We continue here our work on fractional integration and differentiation of functions of a real variable which we began in a previous paper. All quantities in the present paper are real.

We define a λth integral, or a (−λ)th differential coefficient, of f(x) over an interval (a, x)by

where D stands for and γ is the least integer greater than or equal to zero such that λ + γ>0, f(x) is bounded on the path of integration, and is continuous on this path except possibly for a finite number of ordinary discontinuities.

The object of this address is to pass in review various theories of complex numbers and to scrutinise them from the standpoint of the teacher engaged in initiating his pupils into this subject. The material of the address is therefore very elementary and well known and roughly a century old, but it is arranged and combined in a way which may perhaps throw some new light on an old problem. It is important to realise at the outset the fundamental issues which are involved; and, although it is undesirable to intrude explicit philosophy into a first lesson on complex numbers, it is well for the teacher to have these metamathematical considerations clear in his own mind.