The “curious” property noted by Mr. A. P. Rollett (Note 1261 in the Mathematical Gazette for December 1937), with regard to the centres of circles touching one another and inscribed in the space between two given circles, leads to an interesting general theorem on all such sets of circles.

Given any two circles, and a set of circles touching one another and the two given circles, then, if r1r2, r3 are the radii of three successive circles of the set, of which r2 is known, r1 and r3 are the roots of a quadratic equation of which the coefficients are simple functions of r2 and the radii of the given circles.

In the Gazette of October, 1928 (Vol. XIV, p. 233), Professor H. G. Forder gives a simple inductive proof of the theorem on the Euler Numbers :

He remarks, however, that his method is not successful in proving completely the allied theorem for the coefficients of the tangent series

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.

The purpose of this short article is to give a simple method for constructing a magic square of any given odd size. The method consists basically of two cyclic permutations perpendicular to each other.

The 5 × 5 square’s construction is illustrated, a proof outlined, and a 17 × 17 square quoted.