Nearly a month of the summer term remains to be got through by the somewhat jaded boys and girls after the Higher School examination is over. For many years I have been casting round for stimulating topics to discuss with my more or less advanced mathematicians, and this paper is concerned with my latest find. Its title refers to the four tangents at the points of contact of the nine-points circle of a triangle with the four circles touching the sides. I am aware that a great many properties of the points of contact themselves have been investigated, and I cannot flatter myself that I have found out anything new about the figure formed by the tangents thereat. But the literature of any branch of mathematics is so vast that it seemed best for me, instead of hunting in books, to try to find out what I could in my own way. The effort to do so resulted in a delightful mathematical adventure, which has enabled me to devise something to bring out to our pupils what analytical geometry really is about.

A Striking thing about contemporary education is the increasing demand for mathematical training. In the eighteenth century the only departments of science which conspicuously called for it were astronomy and such branches of mechanics and optics as had arisen in connection with astronomical pursuits. Navigation was perhaps the only important profession for which any mathematical equipment was an indispensable prerequisite. Though an eighteenth century H. G. Wells might have anticipated minor avenues of future employment for professional mathematicians as teachers attached to artillery or actuarial work, he would scarcely have foreseen that chemistry, power production, genetics, psychology and social statistics would severally enlist the services of the mathematician. The educational problem which arises from the rapid mathematisation of science during the past half-century has found us unprepared, and is largely an unsolved one.

In the textbooks there are many examples concerning the leading features of the triangle whose sides are given by the equations ax2 + 2hxy + by2 = 0 and L ≡ lx + my + n=0. These are usually easy because of the use that can be made of the origin. But when we replace the first equation by the general equation