The operations of elementary algebra are of finite character; that is to say, an algebraic expression depends, explicitly or implicitly, only on a finite number of variables. The step from algebra to analysis is taken when we begin to consider expressions depending on an infinite number of variables; for instance, the limit of a sequence (an) depends on the infinite set of variables a1, a2, …. Such an infinite set of variables is most conveniently thought of as the set of values of a function; thus the sequence (an) is a function of the positive integral variable n, the single term a3, for example, being the value of the function when n is given the particular value 3. Thus the characteristic features of analysis are (i) the systematic use of functions of a variable capable of an infinity of distinct values, and (ii) the consideration of expressions that involve the values of such a function for an infinity of values of the independent variable.

The four star polyhedra discussed on pp. 171-173 of Lines’ Solid Geometry form an interesting series of constructions based on the angle of 72° The finished products form fine decorations for the mathematics laboratory, as well as visual aids.

The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.

One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.

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.

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.

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.

Methods of approximation, and successive approximation, are often used in more advanced mathematics, but it is not widely realised that they can with advantage be introduced at quite an early stage in the teaching of school mathematics. This note is a plea that these methods should be taught earlier; indeed it is written in the belief that a wider use of these methods would have not only a salutary effect on teaching and learning alike, but would help to eliminate some of the widespread troubles with which teachers of science are generally confronted and with which they often ask us to help. How often are we told by science colleagues that pupils calculate a physical quantity, for instance, a specific heat, to four decimal places when it is likely to be in error in the secorld place! Similarly those of us who teach practical mechanics, know full well how the resources of four- or five-figure logarithmic tables are invariably used to the full, when the data are correct only to two significant figures.