In the Mathematical Gazette of October 1928, page 234, there appears a quotation from a letter of Chief Baron Pollock to his son, when an undergraduate at Trinity, in which, speaking of Euclid, he says: “I consider (I. 4) to be much more of a ‘pons asinorum’ than the 5th. Indeed I regard it as the test of a geometric spirit. If it be fairly conquered, it proves the existence of that logical accuracy which is the soul of mathematics, and to elicit and cultivate which is the great benefit which such studies confer as a branch of education.”

If ab is the numerical value of the product of two factors a and b, and if h is added to a (for the purpose of bringing the product up to an assigned value P, or for any other reason), then bh must be added to ab. If, now, k be added to b (for the same or any purpose), the product must be increased by k (a + h) where (a + h) is the new value of the factor previously a. These facts are so elementary that they merit quoting only on account of the curious neglect of their systematic alternate use by the arithmetician.

Voting, in some form or other, must have been one of the activities of primitive man: and one can easily picture the cave-dweller expounding, perhaps with illustrations on the wall of his dwelling, some of the difficulties of theory and practice quite as earnestly as we shall consider the subject to-day, and, I am inclined to think, with no less success. The announcement of the results of any system was probably no less exciting than the clash of spear on shield that declared the verdict in Athens, or the public vote by a show of hands. Secret voting was only permitted when individual persons were directly concerned. It will be remembered that John Stuart Mill’s judgment is in exact accord with the ancient practice: his illustration of the exceptional case being that of club membership, where the “blackball” of the old-fashioned ballot-box still represents the pebble of ancient Greece. In Rome voting “by classes” was frequently used, until corruption led to the domination of the wealthy; and a form of ballot by the tabella was used about B.C. 120.