In elementary plane geometry there is a very well known theorem attributed to Wallace (1806) which runs:

Theorem I. If four lines are given in a plane, the circumscribing circles of the triangles which they form three by three are concurrent.

(1) Apparently it was in 1854 that Hermite first drew attention to the special determinant which now bears his name. It may be defined as being such that every two of its elements that are conjugate in position are conjugate-complex in form: and as a consequence its matrix is the sum of two matrices one of which is axisymmetric and the other zero-axial skew.

A lamina of mass m can be replaced by an equimomental system of three equal particles m placed at the vertices of a maximum inscribed triangle of a momental ellipse for the centre of mass. Similarly any rigid body can be replaced by an equimomental system of four equal particles (cf. Routh, Elementary Rigid Dynamics, 7th ed. (1905), Art. 44 and Note, p. 423).

The greatest number of straight lines that can lie upon a surface of order n (not being a ruled surface) is unknown, except if n is three. Salmon and Clebsch have shown that the points of contact of lines which have a four-point contact with the surface lie upon a locus of order n (11n – 24), the intersection of the surface of order n with another of order 11n – 24. Since a straight line lying wholly on the former surface must form a part of this locus, the number n (11n – 24) is an upper limit to the number of lines; if n is three, this gives 27, the correct number. But for values of n > 3, it is improbable that this limit1 can be reached.

1. In the treatment of the geometry of the ellipse, projection from the circle offers an easy and direct means of obtaining the properties of conjugate diameters of the ellipse. In text-books generally, little further use is made of the method of projection, except in the case of a few exercises. The following note shows how it can be employed to derive most of the well-known properties of the ellipse.

§ 1. The craze for extensive π-calculation which was so strange a feature of the last century was probably brought to an end not so much by the famous 707 decimals of W. Shanks in 1873 as by the demonstrations of Hermite and Lindemann, about the same time, regarding the transcendental nature of both e and π. Sporadic minor outbreaks of the disease still occur, of course, —Ramanujan in his earlier days was not entirely immune—and the series of the present note may seem symptomatic. It is hoped, however, that they will not be devoid of interest from the point of view of elementary trigonometry.