When my previous paper on the “Dissection of Rectilineal Figures” was published (Mathematical Gazette, October, 1914, vol. vii. p. 381), I had not seen Mr. H. M. Taylor’s paper in the Messenger of Mathematics, 1905, vol. xxxv p. 81. He gives the four-part dissection of a pair of triangles, and two varieties of this; the three dissections being all of one type. He also gives the three-part and other dissections of a pair of parallelograms. He also gives a four-part dissection of a triangle and a parallelogram (equivalent, by projection, to that of a triangle and a square) which, for the case of an equilateral triangle and a square, was published in 1905 by Mr. H. E. Dudeney, This dissection, which I did not know before, is also applied by Mr. Taylor to the case of a quadrilateral and a triangle with a side equal to a diagonal of the quadrilateral. As I have already given two types of dissection of a triangle and a square, with four parts in their fundamental case, which I have called the first and second types, I will call this, which is shown in Fig. 17, the third type.

1. Let ABC be a triangle, H its orthocentre, P a point on the circumcircle, and D, E, F the feet of the perpendiculars from P on BC, CA, AB respectively. Then it is known that D, E, F are on a straight line which bisects PH. Through P draw lines making the angle ½π–α in the same sense with PD, PE, PF, PH to meet the sides in D′, E′, F′ and a line through H perpendicular to PH in H′. Then D′, E′, F′ are also on a line which bisects PH′. Call this line the Simson line (α) of P with respect to ABC

1. The stimulus that provoked this article was the following proof of the fact that the middle points of the diagonals of a complete quadrilateral are collinear.

For the 3 x 3 matrix A = [a ij], the trace t of A is the sum of its main-diagonal entries:

The characteristic equation of A is

a cubic equation in λ which can be written as

where Δ is the determinant of A and