Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.

With his unaided faculties, a man is comparatively limited in performance. His vision, hearing and legs operate only over small distances, he cannot lift very heavy weights nor weave much cloth in an hour. But by making use of devices such as the aeroplane, the telephone, and the powered loom, he can exploit natural properties and forces, and thereby attain what is literally a superhuman standard of performance in all kinds of activities.

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process. Then certain standard forms are evaluated, the rules for application to differential equations are laid down and a selection of worked examples follows. But such a treatment can mystify a student who wants to know why, in following these rules, he should, in effect, firstly, multiply each term of the equation by e-px, secondly, integrate the resultant products from 0 to ∞, and, thirdly, obtain in this way the complete solution.

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.

If on the sides of a triangle ABC equilateral triangles are constructed, inwardly and outwardly, and their circumcircles drawn, the 6 circles will in general have 11 points of intersection: A, B, C; 2 points on each of the 3 sides; a point F common to the circumcircles of the outer triangles, and a point N common to those of the inner triangles (Fig. 1).

This article uses complex numbers to investigate the figure, and has as its main object the establishment of an extremal property of N.

This number arises naturally, of course, as the hypotenuse of a right triangle with legs of length 1. Notoriously, √2 was found to be irrational by the Pythagoreans in around 500 BC. Their mathematics and philosophy was based upon natural numbers and small number ratios, and thus this discovery would have been very troubling: the Pythagoreans wouldn’t be the only ones to react badly to the imposition of the irrational.