ColinMaclaurin, of West Highland ancestry, a son of the manse, was born at Kilmodan, between the Kyles of Bute and upper Loch Fyne, in February 1698. His father, who was minister there, died soon after, and his mother died when he was 9; the care of Colin and his brothers devolved on their uncle Daniel, who was minister of the neighbouring parish of Kilfinnan. At the age of 11 Colin, already proficient in Latin and Greek, entered Glasgow University, and graduated M.A. in his fifteenth year. It was here that he discovered his mathematical bent, and he is said to have mastered the first six books of Euclid in a few days after finding them accidentally in a friend's rooms.

That the determinant

where hr is the rth complete homogeneous symmetric function in a set of n arguments, is equal to the quotient of a particular pair of alternants was shown essentially by Jacobi in 1841 and by Trudi in 1864. The present note exhinits this well-know relation, (3), as the immediate consequence of a simple matrix equality.

1. This celebrated problem is treated in nearly all the textbooks on probability; for example in Bertrand's Calcul des Probabilités, 1889, pp. 15–17, in Poincare's of the same title, 1896, pp. 36–38, and in most of the recent textbooks. The problem may be stated in abstract terms as follows: Among the n! permutations (α1α2α3… αn) of the natural order (123…n), how many have no αi equal to j? The problem has been clothed in many picturesque (and highly unlikely) “representations”; for example, by imagining n letters placed at random in n addressed envelopes, and inquiring what is the chance that no letter is in its correct envelope; or by imagining n gentlemen returning at random to their n houses; and so on, ad risum. Various derivations have also been given of the probability in question, namely the first n + 1 terms of the expansion of e-1, to which function the probability converges with rapidity as n increases.

The familiar Lemma introduced by Goursat in his proof of Cauchy's theorem suggests the following necessary and sufficient condition for differentiability of a complex function f(z).

The object of this note is to outline a rigorous evaluation of Planck's integral by methods which presuppose no more than an elementary knowledge of the Calculus. The proof has been divided into four theorems each of which is of some interest in itself.

Consider three sequences αn, Dn, kn (n = 1, 2, 3,…), such that Dnan→0 and, for n > 1,

Then Hence, writing Σ for ,

if either series converges. This will be applied to derive various series for π from the two known results1

,

where Nn = 3. 5. ….(2n + 1).

Much interest has always been aroused by this theorem, which asserts that a triangle is isosceles when two of the internal bisectors of its angles are equal. Recently McBride has given a proof, together with a selection from the numerous others which have been published. The following proof, based mainly on Euclid, Book III, differs from any I have come across, and establishes a slightly wider theorem.

Theorem. If a circle cut all the sides (produced if necessary) of an equilateral polygon, the algebraic sum of the intercepts between the vertices and the circle is zero; i.e., if any side AB of the polygon be cut by the circle in P and Q, then Σ(AP + BQ) = 0, the intercepts being signed by fixing a positive direction round the contour of the polygon.

My attention was drawn by an Art teacher to the following approximate construction for inscribing a regular polygon of n sides in a given circle, having diameter AB, centre O. Find C in AB so that AC: AB = 2: n, and construct the equilateral triangle ABD. If DC produced meets the circle in E, then AE is approximately a side of the required polygon.

1948 is the bicentenary of Euler's discovery that

This note gives a brief account of the subsequent work on these relations and a proof of the equivalence of limit and series which appears to involve new features.