The Teaching of Geometry

by David Eugene Smith

The other definitions of plane geometry need not be discussed, since all that have any historical interest have been considered. On the whole it may be said that our definitions to-day are not in general so carefully considered as those of Euclid, who weighed each word with greatest skill, but they are more teachable to beginners, and are, on the whole, more satisfactory from the educational standpoint. The greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small.

Since teachers are occasionally disturbed over the form in which definitions are stated, it is well to say a few words upon this subject. There are several standard types that may be used. (1) We may use the dictionary form, putting the word defined first, thus: "Right triangle. A triangle that has one of its angles a right angle." This is scientifically correct, but it is not a complete sentence, and hence it is not easily repeated when it has to be quoted as an authority. (2) We may put the word defined at the end, thus: "A triangle that has one of its angles a right angle is called a right triangle." This is more satisfactory. (3) We may combine (1) and (2), thus: "Right triangle. A triangle that has one of its angles a right angle is called a right triangle." This is still better, for it has the catchword at the beginning of the paragraph.

There is occasionally some mental agitation over the trivial things of a definition, such as the use of the words "is called." It would not be a very serious matter if they were omitted, but it is better to have them there. The reason is that they mark the statement at once as a definition. For example, suppose we say that "a triangle that has one of its angles a right angle is a right triangle." We have also the fact that "a triangle whose base is the diameter of a semicircle and whose vertex lies on the semicircle is a right triangle." The style of statement is the same, and we have nothing in the phraseology to show that the first is a definition and the second a theorem. This may happen with most of the definitions, and hence the most careful writers have not consented to omit the distinctive words in question.

Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter.

1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them.

2. Admit no obscure or equivocal terms without defining them.

3. Use in the definitions only terms that are perfectly understood or are there explained.

4. Omit no necessary principles without general agreement, however clear and evident they may be.

5. Set forth in the axioms only those things that are in themselves perfectly evident.