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.

The values of the principal radii of curvature ρ1; ρ2 at a point on an ellipsoid are α2/p, β2/p where α2, β2 are the squares of the semi-axes of the central section parallel to the tangent plane at the point, and p is the distance from the centre to the tangent plane.

(1) If the straight line joining two points subtends equal angles at two other points on the same side of it, the four points are con cyclic.

A fundamental result in the theory of conics is the following:

Two opposite vertices of a complete quadrilateral are conjugate points for any conic which passes through the four remaining vertices.

An interesting determinant occurs in the fifth volume of Muir's History1. It is

where n = ½ (p – 1), and ars is the smallest positive integer such that

rars = s (mod p), (1) p being any odd prime number. It is evident that each element ars is unique and non zero. For p = 5, 7, 11 the determinants are

(2) respectively, and their values are

– 5 , 72, 114.

A and B study an array of counters, placed in straight lines on a table, in separated groups (e.g. 28, 53, 1, 16). The counters are to be removed under the following conditions :–

Each “player” may take away

(1) one counter from anywhere (e.g. 27, 53, 1, 16) or (28, 20, 32, 1, 16) etc.

or (2) three counters from anywhere (e.g. 28, 53, 14) or (27, 52, 1, 15) or (28, 25, 17, 6, 2, 1, 16) etc.

or (3) two adjacent counters from one group (e.g. 28, 51, 1, 16) or (28, 53, 1, 8, 6) etc.

We consider an equation with constant coefficients

where a≠0 and f(x) is continuous in a suitable interval. Suppose that the symbolic polynomial P(D) has been fully decomposed into its (real or complex) linear factors, so that the equation may be written

where b1, …, bq are distinct, and m1+…+mq = n. The Complementary Function being now known, we may write down a particular integral of (1) by Cauchy's method.

The figure shews the dissection in the form

Let m, x denote AD, DE, where E is the projection of A on BC.

Quadrilateral 1 is constructed with sides ½ (c + x), ½ (c – x), ½ (m + ½a + x), ½ (m – ½a – x); quadrilateral 6 with sides ½ (b – x), ½ (b + x), ½ (m – ½a – x), ½ (m + ½a – x). The square on AB is then dissected into four quadrilaterals equal to 1, plus the shaded square (AD2); similarly for the square on AC. These quadrilaterals are reassembled in the squares on BE, EC respectively with the right-angled corners in the reversed positions, so as to enclose squares each equal to DE2. Finally two quadrilaterals are subdivided (4, 11) and (3, 5, 10, 12) to fit in as shewn in the figure.

The integrals referred to here are those given by the following

Theorem:

where the integration is extended over

and l > 0, m > 0, n > 0, a ≤ a ≤ b < ∞.

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression

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.