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.

During the last half of the nineteenth century there was a great revival of interest in the field of geometry. A large part of this interest had its origin in a simple problem submitted to a contemporary mathematical periodical by a French army officer. The problem was to find a point O within a triangle ABC such that the angles OAB, OBC and OCA would be equal. The name of the army captain who submitted the problem was Brocard—Pierre René Jean-Baptiste Henri Brocard.

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.

Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.

Consider two triads of points, a, b, c, a', b', c' in a plane ; then, using homogeneous coordinates and regarding the points as specified by the corresponding vectors, we may write

and writing b × c = α, c × a = β, a × b = γ then a × a' = m1γ − n1β, b × b' = n2α − l2γ, c × c' = l3β − m3α.

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 mid-points of the sides of any quadrilateral form a parallelogram. Repeated applications of this inscription process will lead to two families of similar and similarly situated figures. It is proved below that the general n-sided polygon lying in space of h dimensions gives rise to similar sequences in the limit. If the initial polygon is not re-entrant convergence is rapid, as will be seen from Figures 1 and 5. The parallelogram is replaced by the affine transformation of a regular polygon lying in a plane.

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.

Find the path taken by a ship which pursues, and always directs its course towards, another ship (supposed moving in a straight line), their speeds being constant and in fixed ratio, and their initial directions at right angles.

The purpose of this article is to outline a system of notation for dealing with some of the properties of the general triangle. By means of it results obtained may be concisely stated and, by symbolic manipulation, other results implied but not explicitly realised, may be found.

Suppose C is a convex curve bounding a plane domain of area A. l will denote the length of C and d its diameter ; that is, the upper bound of the distance between any pair of its points. It is clear that, for all convex curves C, 2d≤l≤πd. In this paper I consider classes of convex curves C with a given value of l satisfying one or more further conditions. The aim will be in each case to find the curves which enclose minimum or maximum area. The existence of a curve of a given class ∑ for which the area enclosed attains the maximum or minimum possible follows from compactness in the class of convex curves in a bounded part of the plane.

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.