THE first Mathematical Colloquium held in Edinburgh met during the first week of August, and proved a great success. It was organised by the office-bearers of the Edinburgh Mathematical Society, A. G. Burgess, M.A., F.R.S.E., and Peter Comrie, M.A., B.Sc, F.R.S.E., respectively the President and Secretary of that Society, being also President and Secretary of the Colloquium. The idea of holding such a colloquium was an outcome of Professor Whittaker’s announcement that he purposed organising, as part of the Mathematical Honours curriculum in the University of Edinburgh, a mathematical laboratory for systematic numerical discussion of functions and methods of calculation. Several correspondents had expressed the hope that vacation courses in this line of study might be established; and it was decided to make a first experiment. It was resolved, however, not to limit the colloquium to a discussion of one branch of mathematics, but to enlarge its scope by the inclusion of two other domains of mathematical thought. The broad features of the programme we owe to Professor Whittaker; and its variety was such as to appeal to all types of mathematical mind.

Area Of a Rectangle.—Consider a rectangle A of height h and base b (Fig. 18). By moving the rectangle so that its base slides along the x-axis through a distance b, it takes up the second position A. Thus, on a length 26 of the x-axis, commencing at O, we have a rectangle of height h and area 2A, and so on.

A somewhat obvious simplification of Briggs’ logarithmic process was-discovered and given as one of three methods by Robert Flower in a rare small quarto tract, The ‘Radix a new way of making Logarithms, published in London by J Beecroft in 1771. Several tables of radices are given, the largest extending from r = 1 to 9 and n from 1 to 12 to twenty-three places.

Flower divides the given number, if necessary, by a power of ten and a single digit, so as to reduce the first figure to *9, and then multiplies by a succession of radices until all the digits become nines. The complement of the sum of the logarithms of the radices to the logarithm of the divisor gives the required logarithm. In some cases it is more convenient to multiply than to divide by a digit, the logarithm of the digit is then added to those of the radices and the complement to 1 taken.

The current elementary discussion of the Simple Pendulum is unsatisfactory, in that the problem is not reduced with sufficient directness to a case of S.H.M. It has to be artificially prefaced by consideration of a curvilinear motion in which the tangential resolute of the acceleration is proportional in magnitude to the arcual distance from a point of the path; and the discussion of this motion raises new points of difficulty altogether out of proportion to its significance in this connection. The net result is that the student’s appreciation of this first of the important applications of S.H.M. to “small oscillations” is lost, in dismay at finding that, even after he has mastered the essential difficulties of the S.H.M. itself, the first good application of it brings yet another awkward hurdle. And the physicist or engineer may well make this another case for railing impatiently at the devices for “dodging the Calculus” by which elementary theory is so apt to be obscured. Nevertheless, the teacher of Mathematics knows how important it is to postpone such Calculus difficulties as, e.g., to pave the way to the complete primitive of the S.H.M. differential equation and its uses (one of which gives the only clear-cut way of handling the ordinary discussion of the Simple Pendulum). But the postponement must not be obtained at the disproportionate cost of artificial complications which, for the sake of “elementary” treatment, cast a fog round the important features of the argument, without contributing anything of independent value to the student’s store of knowledge.

[71. vol. v. p. 330] Mr. Shovelton’s proof (vol. vii. p. 153) may be shortened by using central-difference operators. The theorem itself may be made clearer by splitting it up into two: