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.

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 classical propositions of the elementary theory of numbers concerning divisibility, primes and linear indeterminate equations, are all based on the possibility of division with a remainder.