that can only be overcome by the teacher setting a definite but reasonable time limit, and holding the pupil responsible if the work is not done in the time specified. If this matter is taken in hand the first day, and special effort made in the early weeks of the year, much of the difficulty can be overcome.

As to the nature of the recitation to bé expected from the pupil, no definite rule can be laid down, since it varies so much with the work of the day. In general, however, a pupil should state the theorem quickly, state exactly what is given and what is to be proved, with respect to the figure, and then give the proof. At first it is desirable that he should give the authorities in full, and later give only the essential part in a few words. It is better to avoid the expression "by previous proposition," for it soon comes to be abused, and of course the learning of section numbers in a book is a barbarism. It is only by continually stating the propositions used that a pupil comes to have well fixed in his memory the basal theorems of geometry, and without these he cannot make progress in his subsequent mathematics. In general, it is better to allow a pupil to finish his proof before asking him any questions, the constant interruptions indulged in by some teachers being the cause of no little confusion and hesitancy on the part of pupils. Sometimes it is well to have a figure drawn differently from the one in the book, or lettered differently, so as to make sure that the pupil has not memorized the proof, but in general such devices are unnecessary, for a teacher can easily discover whether the proof is thoroughly understood, either by the manner of the pupil or by some slight questioning. A good textbook has the figures systematically lettered in some helpful way that is easily followed by the class

that is listening to the recitation, and it is not advisable to abandon this for a random set of letters arranged in no proper order.

It is good educational policy for the teacher to commend at least as often as he finds fault when criticizing a recitation at the blackboard and when discussing the pupils' papers. Optimism, encouragement, sympathy, the genuine desire to help, the putting of one's self in the pupil's place, the doing to the pupil as the teacher would that he should do in return,- these are educational policies that make for better geometry as they make for better life.

The prime failure in teaching geometry lies unquestionably in the lack of interest on the part of the pupil, and this has been brought about by the ancient plan of simply reading and memorizing proofs. It is to get away from this that teachers resort to some such development of the lesson in advance, as has been suggested above. It is usually a good plan to give the easier propositions as exercises before they are reached in the text, where this can be done. An English writer has recently contributed this further idea:

It might be more stimulating to encourage investigation than to demand proofs of stated facts; that is to say, "Here is a figure drawn in this way, find out anything you can about it.” Some such exercises having been performed jointly by teachers and pupils, the lust of investigation and healthy competition which is present in every normal boy or girl might be awakened so far as to make such little researches really attractive; moreover, the training thus given is of far more value than that obtained by proving facts which are stated in advance, for it is seldom, if ever, that the problems of adult life present themselves in this manner. The spirit of the question, "What is true?" is positive and constructive, but that involved in "Is this true?” is negative and destructive.1

1 Carson, loc. cit., p. 12.

When the question is asked, "How shall I teach ?" or “What is the Method? there is no answer such as the questioner expects. A Japanese writer, Motowori, a great authority upon the Shinto faith of his people, once wrote these words: "To have learned that there is no way to be learned and practiced is really to have learned the way of the gods."

CHAPTER XI

THE AXIOMS AND POSTULATES

The interest as well as the value of geometry lies chiefly in the fact that from a small number of assumptions it is possible to deduce an unlimited number of conclusions. With the truth of these assumptions we are not so much concerned as with the reasoning by which we draw the conclusions, although it is manifestly desirable that the assumptions should not be false, and that they should be as few as possible.

It would be natural, and in some respects desirable, to call these foundations of geometry by the name "assumptions," since they are simply statements that are assumed to be true. The real foundation principles cannot be proved; they are the means by which we prove other statements. But as with most names of men or things, they have received certain titles that are time-honored, and that it is not worth the while to attempt to change. In English we call them axioms and postulates, and there is no more reason for attempting to change these terms than there is for attempting to change the names of geometry and of algebra.2

1 From the Greek yn, ge (earth), + μeтpeîv, metrein (to measure), although the science has not had to do directly with the measure of the earth for over two thousand years.

2 From the Arabic al (the) + jabr (restoration), referring to taking a quantity from one side of an equation and then restoring the balance by taking it from the other side (see page 37).

Since these terms are likely to continue, it is necessary to distinguish between them more carefully than is often done, and to consider what assumptions we are justified in including under each. In the first place, these names do not go back to Euclid, as is ordinarily supposed, although the ideas and the statements are his. "Postulate" is a Latin form of the Greek alτnua (aitema), and appears only in late translations. Euclid stated in substance, "Let the following be assumed." "Axiom" (ağíwpa, axioma) dates perhaps only from Proclus (fifth century A.D.), Euclid using the words "common notions (koivaì évvoiai, koinai ennoiai) for "axioms," as Aristotle before him had used "common things,' common principles," and "common opinions."

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The distinction between axiom and postulate was not clearly made by ancient writers. Probably what was in Euclid's mind was the Aristotelian distinction that an axiom was a principle common to all sciences, self-evident but incapable of proof, while the postulates were the assumptions necessary for building up the particular science under consideration, in this case geometry.1

We thus come to the modern distinction between axiom and postulate, and say that a general statement admitted to be true without proof is an axiom, while a postulate in geometry is a geometric statement admitted to be true, without proof. For example, when we say "If equals are added to equals, the sums are equal," we state an assumption that is taken also as true in arithmetic, in algebra, and in elementary mathematics in general. This is therefore an axiom. At one time such a

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1 One of the clearest discussions of the subject is in W. B. Frankland, "The First Book of Euclid's Elements,"" p. 26, Cambridge, 1905.