"Let A1 be any point on a between any given points A and B; suppose A2, A3, A4, ... so taken that A1 lies between A and A, A, between A1 and Ag, etc., and also such that the segments AA1, A12 are equal; then must there be in the ... series Ag, A3, A4 2, ... a point An such that B lies between A and A. - The denial of this axiom leads to the non-Archimedean geometry." Hilbert inserts the necessary definitions for understanding these postulates (axioms), and adds numerous corollaries showing the far-reaching effect of the statements; but this is not the place to enter this interesting field. Whether or not his postulates are sufficient, it is evident that tacitly or openly they are assumed in our elementary rectilinear geometry. Their consideration should convince the teacher that the question of the postulates is by no means the simple one which the text-books sometimes make us feel. Thus geometry is exact, not because its postulates necessarily agree with the facts of the external world; that is not of so much moment. It is exact because it postulates definitely at the outset certain few statements concerning figures in space, and then applies logic to see what other statements can be deduced therefrom. CHAPTER XII TYPICAL PARTS OF GEOMETRY The introduction to demonstrative geometry may well be made independent of the text-book, unless the book offers some special preparatory work. If the pupils have not a reasonable knowledge of geometric drawing, a few days may profitably be devoted to this subject exclusively. Professor Minchin has this to say of the English schools, and the same is almost as true of our own: "So far as I am able to learn by inquiry, Euclid is taught in all our schools without the aid of rule, compasses, protractor, or scale. This is quite in accordance with the traditional method- the classical method which, unfortunately, so greatly dominates English education and quite conducive to long-delayed knowledge of the subject. "Now the use of the above simple instruments for all beginners in geometry is the first change that I advocate, whether we continue to teach from Euclid's book or from one proceeding on simpler and better lines. Well-drawn figures possess an enormous teaching power, not merely in geometry, but in all branches of mathematics and mathematical physics." 1 1 The Teaching of Geometry, The School World, Vol. I, p. 161 (1899). Before undertaking the ordinary text-book demonstrations the teacher will also find it of great value to offer a few preliminary theorems which pave the way for the usual sequence of propositions, giving a notion of what is meant by a logical proof, and creating a habit of working out independent demonstrations. The following, for example, might be given in this way: (1) All right angles are equal (if the text-book postulates the demonstrable fact of the equality of straight angles); (2) At a point in a given line not more than one perpendicular can be drawn to that line in the same plane-not that one can be drawn, as so many text-books affirm but fail to prove; (3) The complements of equal angles are equal; the proposition concerning vertical angles, and several others of the simpler ones selected from the first "book." After a little work of this kind the pupil is prepared to understand the nature of a logical proof. Independence will begin to assert itself, confidence in his ability to handle a proposition without a slavish dependence upon his text-book, while mere memorizing will fail to secure the usual foothold at the start. These two points may now be impressed: (1) Every statement in a proof must be based upon a postulate, an axiom, a definition, or some proposition previously considered; (2) No statement is true simply because it appears from the figure to be true. With this preliminary treatment of a dozen or more simple propositions, and with some instruction concerning geometric drawing, the text-book sequence may be undertaken with much less danger of discouragement, of slovenly work, of groping in the dark, and of mere memorizing. Symbols-The contest between the opponents of all symbols and the advocates of mathematical shorthand in geometry, as in other branches of the science, is about over. In England Todhunter's Euclid is giving place to the Harpur, Hall and Stevens, McKay, Nixon, and others which make extensive use of symbols, while in America Chauvenet's excellent work has had to give place to less scholarly but more usable textbooks. In general one is practically bound by the symbols in the book in the hands of the class. A few notes upon the subject may, however, be suggestive. In the first place, only generally recognized mathematical symbols should have place; in a world-subject like mathematics, provincialism is especially to be condemned. We may think that || would be a better sign of equality than =, but the world does not think so, and we have no right to set up a new sign language. In this respect it is unfortunate that some of our American writers should continue to use the provincial symbol for equivalence (~), not only because it is difficult to make, but because it has no standing among mathematicians. Indeed, the distinction between equal and equivalent is so nearly obliterated T in our language that many teachers now use the more exact term "congruent" for what some English writers call "identically equal," even though the textbook in their classes has the word "equal." The symbol for congruence (), a combination of the symbols for similarity (~, an S laid on its side, from similis) and equality (=), is so full of meaning and is so generally recognized by the mathematical world that its more complete introduction in elementary work is desirable. It is certainly not open to the objection of novelty, for it dates from Leibnitz, nor of the provincialism and want of significance which characterize the American symbol for equivalence. The modern symbols for limit (=, still in its provincial stage), identity (=), and non-equality (#), in addition to the ordinary algebraic signs, are also convenient. There is also much advantage in following the modern method of reading angles and lines, and of lettering triangles. Among the ancients, when angles B A were always considered as less than 180°, it was a matter of little moment whether one should read the angle here illustrated AOB or BOA. But now that we recognize angles of any number of degrees, as when we turn a screw through 90°, 180°, 270°, 360°, 450°, ., it becomes necessary to distinguish the two conjugate angles in the figure. The |