Some years ago there was a vogue for a series of books dealing imaginatively with the future and bearing such titles as Daedalus, or the Future of Science, Icarus, or the Future of Flying, and the like. I do not recall whether arithmetic was among the topics considered by the ingenious authors, but a further title, Archimedes, or the Future of Calculation, might certainly have been suggested by the developments of modern arithmometers.

We have shown how the theory of the Fractional Calculus enables us to expand functions in various forms. In the present paper we shall obtain some further expansions.

A λth integral of f(z) along a simple curve l is defined by

where γ is the least integer greater than or equal to zero such that R(λ) + γ>0, D stands for d/dz, and the integration and differentiation are along l.

Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.

Several writers have studied algebras in which multiplication is non-associative, that is, x yz≠xy z. It is necessary in a non-associative algebra to distinguish the possible interpretations of a power xn In a non-commutative non-associative algebra x2 is unique, x3 can mean xx2 or x2x; x4 can mean x xx2, x x2x, x2x2, xx2x or x2x x, x5 has 14 interpretations; x6 has 42; and so on. In a commutative non-associative algebra, the possible interpretations are fewer x3 is unique, x4 can mean xx3 or x2x2, x5 can mean x xx3, x x2x2 or x2x3, x6 has 6 interpretations, and so on. The problem considered here is how many meanings are there for xn (A) in a general non-commutative non-associative algebra ? (B) in a general commutative non-associative algebra ? The answer to (A) is I am not able to find any such simple formula for (B).

It has been the fashion in some branches of mathematics to use a rather high sounding and mystifying terminology for comparatively simple ideas and I fear that Geometry has fallen to the lure. It is, of course, essential that new conceptions should have new titles. There is, however, the very serious danger that, through familiarity and, perhaps, moments of carelessness more may be read into such titles than was originally intended. I propose to discuss Infinity in Euclidean Geometry, as I find continually, especially with young students, that the phrases “point at infinity” and “line at infinity” have led to concepts of concrete existence which, whatever may be the case in other philosophies and branches of science, are in this one quite unjustified and which would be exceedingly awkward if they were correct.

The aim of this article is to convey the impressions received by a French Professor of Mathematics on reading the book referred to in the title, and thus to throw some light on similarities and differences between methods of teaching in the two countries.

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.

This paper is a sequel to a paper entitled “The Harmonic Conies”, Math. Gazette, Vol. XIX, page 98. How is it that text books on Algebraic Geometry deal with invariants and covariante of two conies yet never so much as hint at the existence of such forms for more than two conies?

Elementary geometry will be supposed to have its usual meaning as euclidean metrical real geometry of the plane. I propose to point out that, while “points at infinity” and “the line at infinity” may be properly introduced in conformity with the axioms, the “circular points” cannot be so introduced. They do not belong to this geometry; and those results which do properly belong to it, and are often made apparently to depend on the existence of the circular points, should therefore be expressed with no mention of these points.

From some viewpoints the beginnings of Analysis were marked by the publication, in 1656, by J. Wallis of the Arithmetica Infinitorum, in which he obtains the product ad inf.

He defines, very early in the book, a “limit” in this way (considering a series which tends to the value ⅓) “When the number of terms increases, the excess above ⅓ decreases, so that at length it becomes less than any assignable quantity; moreover, if the number of terms becomes infinite, the excess will have vanished completely”. We still use substantially the same definition.

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.

As a science becomes mature it usually relies more and more on mathematical ideas. It has followed that in so far as mathematics is not entirely a “pure” subject, its content has been dictated by the more mature, that is, by the physical sciences. Nevertheless, it is slowly being appreciated that the permeation of mathematics through to the non-physical sciences must imply sooner or later a partial reorientation of the existing attitude.