Nothing can convince us more that “things are not what they seem” than a comparison of the descriptions by two men of what they remember of Cambridge in the seventies ! I should of course recognise my friend Professor Forsyths picture of the Cambridge of that time and am grateful for his calling to my mind so many memories of the past. But for me he has painted them in far too drab a colour, so that one wonders whether his undergraduate days could have been like mine some of the happiest in his life. The reading of his paper enforces upon me the truth of the principle that environment and personalities are not things in themselves, but are our own “constructs”—each of us giving our own “reality” to such evasive phenomena.

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible.

The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles.

1.1. Relation between mutual distances of 4 points in a plane.

Let O, A, B, C be the four given points.

In a previous article (October 1933) it was shown how the “projection method” could be applied to the drawing of bending moment diagrams for fixed concentrated loads. It is proposed to extend this method to cases where there are uniformly distributed loads in addition to the concentrated ones.