The Teaching of Geometry

by David Eugene Smith

In view of these efforts it is well to go back to Euclid and see what this great teacher of university men[46] had to suggest. The following are the five "common notions" that Euclid deemed sufficient for the purposes of elementary geometry.

1. Things equal to the same thing are also equal to each other. This axiom has persisted in all elementary textbooks. Of course it is a simple matter to attempt criticism,—to say that -2 is the square root of 4, and +2 is also the square root of 4, whence -2 = +2; but it is evident that the argument is not sound, and that it does not invalidate the axiom. Proclus tells us that Apollonius attempted to prove the axiom by saying, "Let a equal b, and b equal c. I say that a equals c. For, since a equals b, a occupies the same space as b. Therefore a occupies

the same space as c. Therefore a equals c." The proof is of no value, however, save as a curiosity.

2. And if to equals equals are added, the wholes are equal.

3. If equals are subtracted from equals, the remainders are equal.

Axioms 2 and 3 are older than Euclid's time, and are the only ones given by him relating to the solution of the equation. Certain other axioms were added by later writers, as, "Things which are double of the same thing are equal to one another," and "Things which are halves of the same thing are equal to one another." These two illustrate the ancient use of duplatio (doubling) and mediatio (halving), the primitive forms of multiplication and division. Euclid would not admit the multiplication axiom, since to him this meant merely repeated addition. The partition (halving) axiom he did not need, and if needed, he would have inferred its truth. There are also the axioms, "If equals are added to unequals, the wholes are unequal," and "If equals are subtracted from unequals, the remainders are unequal," neither of which Euclid would have used because he did not define "unequals." The modern arrangement of axioms, covering addition, subtraction, multiplication, division, powers, and roots, sometimes of unequals as well as equals, comes from the development of algebra. They are not all needed for geometry, but in so far as they show the relation of arithmetic, algebra, and geometry, they serve a useful purpose. There are also other axioms concerning unequals that are of advantage to beginners, even though unnecessary from the standpoint of strict logic.

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.