4. _Things that coincide with one another are equal to one another._ This is no longer included in the list of axioms. It is rather a definition of "equal," or of "congruent," to take the modern term. If not a definition, it is certainly a postulate rather than an axiom, being purely geometric in character. It is probable that Euclid included it to show that superposition is to be considered a legitimate form of proof, but why it was not placed among the postulates is not easily seen. At any rate it is unfortunately worded, and modern writers generally insert the postulate of motion instead,--that a figure may be moved about in space without altering its size or shape. The German philosopher, Schopenhauer (1844), criticized Euclid's axiom as follows: "Coincidence is either mere tautology or something entirely empirical, which belongs not to pure intuition but to external sensuous experience. It presupposes, in fact, the mobility of figures." 5. _The whole is greater than the part._ To this Clavius (1574) added, "The whole is equal to the sum of its parts," which may be taken to be a definition of "whole," but which is helpful to beginners, even if not logically necessary. Some writers doubt the genuineness of this axiom. Having considered the axioms of Euclid, we shall now consider the axioms that are needed in the study of elementary geometry. The following are suggested, not from the standpoint of pure logic, but from that of the needs of the teacher and pupil. 1. _If equals are added to equals, the sums are equal._ Instead of this axiom, the one numbered 8 below is often given first. For convenience in memorizing, however, it is better to give the axioms in the following order: (1) addition, (2) subtraction, (3) multiplication, (4) division, (5) powers and roots,--all of equal quantities. 2. _If equals are subtracted from equals, the remainders are equal._ 3. _If equals are multiplied by equals, the products are equal._ 4. _If equals are divided by equals, the quotients are equal._ 5. _Like powers or like positive roots of equals are equal._ Formerly students of geometry knew nothing of algebra, and in particular nothing of negative quantities. Now, however, in American schools a pupil usually studies algebra a year before he studies demonstrative geometry. It is therefore better, in speaking of roots, to limit them to positive numbers, since the two square roots of 4 (+2 and -2), for example, are not equal. If the pupil had studied complex numbers before he began geometry, it would have been advisable to limit the roots still further to real roots, since the four fourth roots of 1 (+1, -1, +[sqrt](-1), -[sqrt](-1)), for example, are not equal save in absolute value. It is well, however, to eliminate these fine distinctions as far as possible, since their presence only clouds the vision of the beginner.