The Teaching of Geometry

by David Eugene Smith

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, +√(-1), -√(-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.

It should also be noted that these five axioms might be combined in one, namely, If equals are operated on by equals in the same way, the results are equal. In Axiom 1 this operation is addition, in Axiom 2 it is subtraction, and so on. Indeed, in order to reduce the number of axioms two are already combined in Axiom 5. But there is a good reason for not combining the first four with the fifth, and there is also a good reason for combining two in Axiom 5. The reason is that these are the axioms continually used in equations, and to combine them all in one would be to encourage laxness of thought on the part of the pupil. He would always say "by Axiom 1" whatever he did to an equation, and the teacher would not be certain whether the pupil was thinking definitely of dividing equals by equals, or had a hazy idea that he was manipulating an equation in some other way that led to an answer. On the other hand, Axiom 5 is not used as often as the preceding four, and the interchange of integral and fractional exponents is relatively common, so that the joining of these two axioms in one for the purpose of reducing the total number is justifiable.

6. If unequals are operated on by positive equals in the same way, the results are unequal in the same order. This includes in a single statement the six operations mentioned in the preceding axioms; that is, if a > b and if x = y, then a + x > b + y, a - x > b - y, ax > by, etc. The reason for thus combining six axioms in one in the case of inequalities is apparent. They are rarely used in geometry, and if a teacher is in doubt as to the pupil's knowledge, he can easily inquire in the few cases that arise, whereas it would consume a great deal of time to do this for the many equations that are met. The axiom is stated in such a way as to exclude multiplying or dividing by negative numbers, this case not being needed.

7. If unequals are added to unequals in the same order, the sums are unequal in the same order; if unequals are subtracted from equals, the remainders are unequal in the reverse order. These are the only cases in which unequals are necessarily combined with unequals, or operate upon equals in geometry, and the axiom is easily explained to the class by the use of numbers.

8. Quantities that are equal to the same quantity or to equal quantities are equal to each other. In this axiom the word "quantity" is used, in the common manner of the present time, to include number and all geometric magnitudes (length, area, volume).

9. A quantity may be substituted for its equal in an equation or in an inequality. This axiom is tacitly assumed by all writers, and is very useful in the proofs of geometry. It is really the basis of several other axioms, and if we were seeking the "irreducible minimum," it would replace them. Since, however, we are seeking only a reasonably abridged list of convenient assumptions that beginners will understand and use, this axiom has much to commend it. If we consider the equations (1) a = x and (2) b = x, we see that for x in equation (1) we may substitute b from equation (2) and have a = b; in other words, that Axiom 8 is included in Axiom 9. Furthermore, if (1) a = b and (2) x = y, then since a + x is the same as a + x, we may, by substituting, say that a + x = a + x = b + x = b + y. In other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes others has a legitimate place, because a beginner would be too much confused by seeing its entire scope, and because he will make frequent use of it in his mathematical work.