Let α1, β1, γ1, and α2, β2, γ2 be the complex numbers corresponding to the angular points A1, B1, C1 and A2, B2, C2 of two related triangles, T1, T2, in a plane, and let homologous vertices be denoted by the same letters.

As this is to be a purely scientific talk it is best to start from fundamental definitions and concrete propositions.

So first the definitions.

Mathematics is merely the high-sounding name designated to the efforts of a child in its attempt to count up to ten, which is crystallised in those immortal words, “One little piggy went to market, one little piggy stayed at home. .” If this is not acceptable to some of the unenlightened members, the alternative “addin’ up and takin’ away” may be substituted.

The curve formed by any flexible uniform chain supported at both ends, in equilibrium under the action of gravity, is the catenary

where a is the distance of the lowest point of the catenary above the horizontal x-axis. If the curve is required to pass through the origin, its equation becomes

and it is known that a = H/w, where H is the horizontal force applied at the ends of the catenary, and w is the weight per unit length of the chain.

Une note de M. L. Droussent, (M, t. LVI, 93), rappelle la propriété qu’ont les centres des cercles inscrits aux triangles ayant pour sommets ceux d’un quadrilatére convexe inscriptible, pris trois par trois, de coïncider avec les sommets d’un rectangle, dont l’origine paraît assez ancienne et qui a été étendue aux seize centres des cercles tritangents des mêmes triangles.

Les différents groupes de points collinéaires ou concycliques signalés dans la note précitée, qui sont aussi mis en évidence dans les articles dont nous venons de donner les références, suggèrent cependant d’autres remarques.

Let a1, a2, a3 denote the sides α1, α2, α3 the angles and A1, A2, A3 the vertices of a triangle in the Euclidean plane. A point P, whose distances from the sides α1, α2, α3 are in the ratio p1: p2: p3 will be denoted by P[p1]. p1, p2, p3 are called the normal coordinates of P. Thus the unit point E[l] is the incentro of the triangle, S[cosec α1] is the centroid, M[cos α1] is the circumcentre, and H[sec α1] is the orthocentre. Similarly, using normal line coordinates, we have l0 [l] is the unit line, l∞[sin α1] is the line at infinity and lH[cos α1] is the axis of the altitudes.

Direct and independent proof is given to the theorem, implicit in the possibility, and fundamental for the procedure of the simultaneous canonical reduction of a pair of real quadratic forms, one of which is positive definite. In this way a complete unification of the possibility and the procedure is effected.

Recently Ferrar (1) has considered this question of reduction by a method “free of difficult invariant factor arguments”, and Todd (2) the example of the orthogonal reduction of a single form in a direct and non-inductive fashion to remove, with reference to the traditional method for proving the possibility, “the aesthetic objection that it does not correspond to the practical method for obtaining the reduction”.

The concept of the direct product is widely used in the theory of groups, algebras, vector spaces and allied topics, but its use as a manipulative tool in matrix algebra is less common.

The present note shows how the direct product of matrices arises in elementary work, derives some of its more important properties, and gives some examples of its use.

This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.