The solution of the problem of drawing a circle to touch a given line BC at a given point X and to touch also a given circle v is very familiar; it is obvious that this solution must lead to an elementary verification of Eeuerbach’s theorem, but the extreme simplicity of this verification is commonly overlooked.

It is sometimes fascinating to return to the old geometry and try to find new properties of the triangle and quadrilateral. One of the most interesting ways of discovering such properties is to start with some well-known result and generalise it. The following notes are the result of an attempt to find circles in a triangle which satisfy the same kind of conditions as the Brocard circle.

Introduction. Many students interest themselves in finding a function which is equal to factorial n for integral values of n. There is a certain fascination about this function : it is so different from the transcendental functions of applied mathematics.

The work of writers like Rouse Ball in England, and Cajori and D. E. Smith in America, in revealing the fascinating story of the history of- arithmetic, has shown the subject developing slowly but surely along the corridor of time. This development, which culminates in present-day arithmetic, proceeds not only by increase of subject-matter, but involves suppression and modification both of matter and methods. It is not the haphazard growth of the wild plant, but the steady and systematic development which results from the pruning and shaping of the skilled and sympathetic gardener.