The subject of this address is Technical Education. I wish to examine its essential nature and also its relation to a liberal education. Such an enquiry may help us to realise the conditions for the successful working of a national system of technical training. It is also a very burning question among mathematical teachers; for mathematics is included in most technical courses.

The following notes are a development of subject-matter which is treated in the writer’s book on Cross-Ratio Geometry, chiefly in Chap. XIV., and the reader is referred to articles in that work for the demonstration of any property that is given here without proof.

That the advantage of using vector analysis in mathematical physics has met with so little recognition by British applied mathematicians must appear rather strange to those of other countries. There are probably not more than half a dozen among us who habitually use vector methods and notation, at any rate in published work. The fact is still more surprising when we recall the work of Maxwell and Heaviside, seeing that the former gave a sort of authority to the curl and divergence, while the latter had so much to do with the systematic development of the vectorial calculus.

Perhaps the chief reason why vector analysis has not come into more general favour with us is that our leaders in applied mathematics have not felt the need of it. It has often been remarked, and perhaps with some degree of truth, that nothing can be accomplished by vector methods that cannot also be done by Cartesian analysis ; and therefore, it is argued, the change is unnecessary and useless. If we had to deal only with minds of special mathematical ability and analytical insight, this conclusion might be accepted. But with the average student so much of his attention is occupied in dealing with the complex array of symbols of partial differentiation to which he is often led in Cartesian analysis, that he is unable to grasp the inner meaning of the work. It is difficult for him in many cases even to see exactly what is expressed in the formulae obtained, involving as they do the three components of a vector quantity in combinations not easy to visualise. And even if the student succeeds in following the argument it is often almost impossible for him to remember either the train of reasoning or the result arrived at. He does not perhaps see why his equations should be differentiated partially with respect to x, y, z and added, or with respect to z, y and subtracted.

The Mathematical Association existed first of all as a society for the improvement of geometrical teaching in the schools, and had as its avowed intention the infusion of a spirit of sweet-reasonableness into abstract geometry. Whether the Association aimed in the first instance at the unconditional abdication of Euclid is difficult to say, but at all events such was the final result. So successful were the labours of the Association in the realm of geometry that it resolved to extend its activities into the domains of Arithmetic and Algebra. To these were quite recently added the Calculus and Mechanics; but, broadly speaking, it must be said that its activities have been almost entirely confined within the somewhat narrow limits of the Secondary Schools. The contention of this communication is that great and clamant opportunities of service lie outside the walls of our more academic Secondary Schools, and it is proposed to deal with some of those newer spheres of activity.

To find the probability that if two quantities chosen at random are measured correct to n figures, their product so obtained is correct to n figures.

Convergents to tan nθ.

If the numerator and denominator of the fraction are both convergent series, one value of tan nθ is