It is well known that the six rotations of an equilateral triangle form a group; that is to say that, if an equilateral triangle ABC occupies a certain triangular space, there are six operations and six only which when performed either singly or in succession will leave the triangle still occupying the same space, but with vertices in different positions. The set of six possible orientations of the movable triangle are drawn below and numbered.

If on the sides of a triangle ABC equilateral triangles are constructed, inwardly and outwardly, and their circumcircles drawn, the 6 circles will in general have 11 points of intersection: A, B, C; 2 points on each of the 3 sides; a point F common to the circumcircles of the outer triangles, and a point N common to those of the inner triangles (Fig. 1).

This article uses complex numbers to investigate the figure, and has as its main object the establishment of an extremal property of N.

To introduce the proposed rule we make use of it to settle the signs of terms in the expansion of determinants such as :

where the symbol for the element has superscript r denoting rows and subscript s denoting columns.

The use of Mathematics we understand as not merely utilitarian applications, such as those by which astronomy is employed in navigation, “whereby our nautical logbooks can be better kept, and water transport of all kinds has grown more commodious”; flight too can be made in the air, but also the cultivation at the same time of the intellect, to think correctly, and arrive at the right conclusion from data which are certain ; and the enlargement generally of the interest in this life.

And so our subject received its name from MA⊖HMATA, meaning disciplined exact knowledge, and formed an important branch in the old University ideal, to teach how to live a life, not merely to make a living. To keep the soul alive, body and soul together, and to cure intellectual timidity, the worst form of cowardice.

Binal Fractions. The reciprocals (1/N) of any number (N) will be termed Binal Fractions, or shortly Binals, when expressed in the binary scale. The notation used is similar to that of ordinary decimals. Thus (see table at end):

1°. A point is used to mark the beginning of the fraction.

2°. The binal (1/N) consists of a certain sequence of ciphers and units (the only digits used) to the right of the point; partly non-recurrent and partly recurrent.

Ladies and Gentlemen,

There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.