1. This is a sequel to a paper with similar title (Gazette, Vol. XXXI, pp. 270-8), which will be referred to as C.T. (I). It concerns Case I (§§ 8-11). For clearness of diagrams, a new shape of triangle has been used; and, to avoid confusion with previous results, and to suit the present development, different letters and suffixes are introduced.

The determination of centres of curvature and evolutes of well-known curves still forms an item of interest in the customary first course in Calculus. This note calls attention to an elegant construction for the centre of curvature of the conic based upon a property apparently forgotten [1] and offers an excuse for the consideration of evolutes.

As H. G. Wells pointed out a long time ago, in the country of the blind the two-eyed man tends to misjudge his neighbours. Relying almost entirely upon his power of vision, he ignores the other senses, and thinks that to be blind is to be mentally defective. In fact, a blind person sometimes has a more balanced mind than the man with keen edges, ten thumbs and ingrowing ears!

It is tempting to state the problems of the foundations of mathematics in general logical. terms, with words like “truth”, “proposition”, “necessary conclusions” In this way a sometimes useful, and always at any rate verbal, connection is made with old problems of logic; and for this reason also mathematical investigations into the foundations of mathematics have found a good number of popular exponents. But it seems that at times these expositions suffer considerably from rather ill-chosen slogans.

The method usually employed for the extraction of square roots involves a series of successive approximations, one for each significant decimal figure of the root. A binomial series may be found which converges quicker than Σ10-n, but this involves a number of separate divisions. Iteration by dividing an approximate value of the root into its square and taking the mean nearly doubles the number of significant figures at each operation. The methods here given converge rather faster than this type of iteration, and enables a root to be expressed as a simple infinite series. They arose from a question put to me by Mr. K. A. Kermack, as to how an ancient Roman could have obtained a good approximation to a square root as a rational fraction. I believe them to be novel, at least in part. It will clearly be sufficient to give methods for finding the square root of a positive integer k.

Most textbooks in dynamics deal with the problem of simple harmonic motion when there is damping present proportional to the velocity; the solution in this case is exact. If, on the other hand, we prescribe damping proportional to the square of the velocity the solutions, in general, can not be evaluated in finite terms, and so we have to resort to approximations. Since approximations play such a leading role in many current problems in applied mathematics, I feel that familiarity with approximations is an essential part of the early training of a good applied mathematician, whether he wishes to be a theorist or a practical engineer. The subject should be broached early.

Consider a thin thread (preferably nylon, which is strong, has low loss, and is resilient) attached to an electromagnetically-driven reed or tuning fork at one end, the other passing over a pulley, and tensioned by a weight W Suppose the free length of the thread is about 1 metre, and that the reed vibrates transversely to the thread. By adjusting W to get the correct tension, the first overtone of the thread may be obtained, there being a node at the middle. The frequency of the thread is the same as that of the reed. If the latter is rotated through 90° (the length of the thread baing unaltered), vibration occurs along the thread, thereby causing its tension to vary periodically The main frequency of tho thread is now half that of the reed, so a subharmonic of order two occurs, there being no node at the middle. This was demonstrated by F. Melde ninety years ago, using a tuning fork excited by a violin bow.