OH, wad some power the giftie gi’e us
To see oursels as others see us,
It wad frae mony a blunder free us,
And foolish notion;

in which case it is probable that wars would cease. During the last few years we have had glimpses of ourselves through the spectacles of the Central Powers; occasionally through the telescopes of the French, the Italians, or the Russians, and in the near future we are likely to get a good many pictures of ourselves as we appear to the last-named people. We are apt to think that our own view of ourselves is the only true view, and all other images are in error just as they differ from our own standard. Naturally other nations adopt the same course, and we are free to believe that in many cases their vision of themselves is far more at fault than our own self image. In this respect a study of Alice and her standards might be not without profit to the nations. When we want to see ourselves, or, at least, our faces, we usually employ a mirror, and what we see depends very much on the mirror we use. Sir George Greenhill was the first professor of Applied Mathematics at the Eoyal Indian Civil Engineering College, Coopers’ Hill. He told us that at the end of the first session a reception was held and the science departments were on view. A young lady entering the physical laboratory and seeing an inverted image of herself in a large concave mirror naively remarked to her companion, “They have hung that looking glass upside down.” But those who fixed the mirror were not to blame, as the lady would have discovered had she advanced past the centre of the curvature. The reply of modern mathematics to the prayer of Eobert Burns is the Theory of Relativity I have some doubt whether Burns would have appreciated the answer. As a rule, we use a plane mirror to inspect ourselves. This reverses certain spatial relations which can be expressed best, perhaps, by stating that it presents a right-handed screw as a left-handed screw and vice versû, or changes a wine screw into a beer screw, as Clerk Maxwell would say, referring to the tendrils of the vine and the hop.

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.

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.

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