(xm - 1)/ m >(xn - 1)/n, m > n.— I have sometimes found the following geometrical method of attack useful in explaining this theorem to students of only average ability. The Lemmas introduced are almost self-evident, and admit of simple analytical demonstration.

Professor Chrystal remarks (Algebra, Chap. VII. §4) that “for tntative processes no general rule can be given.” The tentative processes consist in arranging the terms in groups in such a way as either to manifest a factor common to these groups or aggregates of terms, or to bring the expression under one of the Standard Forms of which the factors are already known, such as a2-b2, a3 - b3, a3 + b3.

(1) Let EA, EB be tangents to a circle with centre O. To prove EA = EB, the following is often served up:—

By subtraction from the equal angles EAO, EBO

and from either of (1), x= ± 3, y = ± 2.

For some questions in arithmetic it is convenient to be able to decimalise shillings and pence mentally to any required number of decimal places.

Let XOX′ = 1 be the diameter of a circle,

Then the lines representing the sines and cosines of the angles,

are marked in the figure.