Given a, b and A, there are sometimes two distinct solutions, sometimes one, and sometimes none. The geometrical explanation is a commonplace, from which it appears that two solutions occur if b>a>bsinA. [But see correction below.]

To determine logarithms to the base 1·1.—If N = ax, x is the logarithm of N to the base a. Hence, if N = 1·1x, x is the logarithm of N to the base 1·1, and, by plotting different values of x with the corresponding values of N, a logarithmic curve may be drawn. This curve will furnish an approximate logarithm of any number to the base 1·1.

The following examples do not usually find a place in text-books dealing with involution, and so may be interesting to those teaching the subject. They are simply special cases of the two well-known theorems: (1) that a straight line cuts the six sides of a complete quadrilateral in six points that form an involution, (2) that if any point in the plane of the quadrilateral be joined to the six vertices, the six rays so formed form a pencil in involution.

It frequently happens in teaching that one wishes to draw a right-angled triangle with rational sides, and at the same time, for the sake of variety if nothing else, one desires to avoid the hackneyed one whose sides are 3, 4, 5.