The angles which satisfy the equation

x=cosx,

occur in the solution of certain problems in analytical geometry.

It is easy to show that one value of the angle is coscoscos…cos 1, when the cosine is taken successively to infinity.

In what follows I propose to give the history of two theorenis and to state some of the consequences that have been developed from them.

The first theorem is:

If a triangle be inscribed in a circle, and from any point in the circumference perpendiculars be drawn to the sides, the feet of these perpendiculars lie in a straight line.

It can be shown by means of relative motion that if two bodies A and B move with velocities u and v in the same straight line, and a third body C move with velocity u + v also in the same straight line, the space passed over by C is equal to the sum of the spaces passed over by A and by B in the same time.