where as before, a, b, c, d, e are the elements of a circulant of order five, we have rs where 0, 45 45 45 = 23 34 41 12 3 12 12 12 12 15 15 [159] = 0, -[32]+[32]-[3]+[33]. , is the complementary of the minor of the elements in the intertu v section of the rth and sth rows, and the tth and uth columns, after the elements in the vth column of the circulant are replaced by units. Using this same method we may also obtain vanishing aggregates of minors of circulants of even order. Thus for n = 4 we have It is otherwise evident that we may remove from (5) and (6) the restriction that n must be odd, and have them true for any order of circulant. November, 1918. QUESTIONS AND DISCUSSIONS. EDITED BY W. A. HURWITZ, Cornell University, Ithaca, N. Y. DISCUSSIONS. We present this month only one discussion, by Professor Florence P. Lewis, dealing with Euclid's parallel postulate. Professor Lewis sketches the history of the controversy over this postulate, and indicates its bearing on the problems of teaching. The subject is of great interest and importance. While it would be rash to assert that a knowledge of the history of non-Euclidean geometry and an acquaintance with the modern views regarding the nature of a postulate are indispensable prerequisites to successful teaching of elementary geometry, nevertheless it can not be denied that such equipment must notably enrich and vivify a teacher's own appreciation of the subject and thus add materially to the effectiveness of teaching. The subject is an extensive one, not readily amenable to adequate treatment in a short article. It is unlikely that any two persons, in giving a short account of the history of the parallel postulate and the development of non-Euclidean geometry, would make exactly the same selection of names to be mentioned or the same interpretation of historical facts. Still less likely would be the agreement of independent writers on the pedagogic aspects of the question. Thus our readers will probably occasionally disagree with parts of Professor Lewis's treatment or dissent from some of her conclusions. But it is believed that the article as a whole represents a consistent and just account, which should be of value to many readers, especially those concerned with the teaching of geometry. Professor Lewis's contention that we should cease to fear redundancy in our list of assumptions seems irrefutable. A course in geometry for adolescents should not be planned in the same way as a course for graduate students in the university. Why "prove" to a high school student that circles with equal radii are congruent? To the poor student the proof brings no added conviction; while the good student wonders why, if this proof is logically necessary, it is not even more essential to show that no arc of a circle is a segment of a straight line. Both statements would be welcome as assumptions. Perhaps it is not easy to draw the line between what is to be assumed and what is to be proved. Probably every child would accept as assumptions the propositions on congruence of triangles; probably almost none would be satisfied to accept the angle sum of a triangle without proof. It may require some discrimination to decide just which theorems arouse in the student that demand for an answer which is the kernel of all successful teaching; but there can be little doubt that such decision should form the basis of our future treatment of elementary geometry. HISTORY OF THE PARALLEL POSTULATE.1 By FLORENCE P. LEWIS, Goucher College. Like the famous problems of construction, Euclid's postulate concerning parallels is a thought that links the ages. Its history is a long story with dramatic climax and far-reaching influence on modern mathematical and general scientific thought. I wish to recall briefly the salient features of the story, and to state what seem to me its suggestions in regard to the teaching of elementary geometry. Euclid's fifth postulate (called also the eleventh or twelfth axiom) states: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side on which are the angles less than two right angles." The earliest commentators found fault with this statement as being not self-evident. Concerning the meaning of axiom, Aristotle says: "That which it is necessary for anyone to hold who is to learn anything at all is an axioms;" and "It is ignorance alone that could lead anyone to try to prove the axiom." Without going into the difficult question of the precise distinction to the Greek mind between axiom and postulate, we may take it that the character of being indisputable pertained to each. Postulates stating that a straight line joining any two points can be drawn, that a circle can be drawn with given center and radius, or that all right angles are equal, were accepted, while the postulate of parallels was scrutinized and admitted at best with reluctance. Proclus, writing in the fifth century A. D., gives some of the reasons for this attitude, and we may surmise others. The postulate makes a positive statement about a region beyond the reach of possible observation or geometrical intuition. Proclus insinuates that those who "suppose they have ground for instantaneous belief" are "yielding to mere plausible imaginings"; the conclusion is "plausible but not necessary." 2 The converse is proved in Proposition 27, book I of Euclid's Elements, and there seems to be no reason why this proposition should be more or less self-evident than its converse. The fact that the two lines continually approach each other was not a convincing argument to the Greek geometer who was acquainted with the relation of the hyperbola to its asymptote. The form of statement of the postulate is long and awkward compared with that of the others, and its obviousness thereby lessened. There is evidence that Euclid himself endeavored to prove the statement before putting it down as a postulate; for in some manuscripts it appears not with the others but only just before Proposition 29, where it is indispensable to the proof. If the order is significant, it indicates that the author did not at first intend to include this among the postulates, and that he finally did so only when he found that he could neither prove it nor proceed without it. 1 Read before the Association of Teachers of Mathematics in New England, May 3, 1919. 2 Cf. Heath's Euclid, Vol. I, and Bonola's Non-Euclidean Geometry, Chicago, Open Court, 1912, to which reference is made throughout this paper. 3 Even the meaning of this phrase requires further elucidation. Most of the early geometers appear to have attacked the problem. Proclus quotes and criticizes several proofs, and gives one of his own. He instances one writer who even attempted to prove the falsity of the statement, the argument being similar to those used in Zeno's paradoxes. The common opinion, however, seems to have been that the postulate stated a truth, but that it ought to be proved. Euclid had proved two sides of a triangle greater than the third, which is far more obvious than this. If the statement was true it should be proved in order to convince the doubters; if false, it should be removed. In no case should it be retained among the fundamental presuppositions. Sir Henry Savile (1621) and the Italian Saccheri (1733) refer to it as a blot or blemish on a work that is otherwise perfect, and this expresses the common attitude of mathematicians until the first quarter of the nineteenth century. Early attempts at proof usually took the form of a change in the definition of parallels, or the substitution, conscious or unconscious, of a new assumption. Neither of these methods resulted in satisfaction to any but their inventors; for the definitions usually concealed an assumption, and the new postulates were no more obvious than the old. Posidonius, quoted by Proclus, defines parallels as lines everywhere equidistant. This begs the question; surely such parallels do not meet, but may there not be in the same plane other lines, not equidistant, which also do not meet? The definition involves also the assumption, that the locus of points in a plane at a given distance from a straight line is a straight line, and this was not self-evident.1 Ptolemy says that two lines on one side of a transversal are no more parallel than their extensions on the other side; hence if the two angles on one side are together less than two right angles, so also are the two angles on the other side, which is impossible since the sum of the four angles is four right angles. This is another way of saying that through a point but one parallel to a given line can be drawn, which is exactly Euclid's postulate. Proclus himself assumes (with some concealment) that if a line cuts one of two parallels it cuts the other, which is again postulate 5. Even as late as the close of the eighteenth century we find this argument advanced by one Thibault, and attributed also to Playfair: Let a line segment with one end A at a vertex of a triangle be rotated through the exterior angle. Translate it along the side until A comes to the next vertex and repeat the process. We finally arrive at the original position and must therefore have rotated through 360°. Hence the sum of the interior angles of a triangle is 180°; and, since Legendre had satisfactorily proved that this proposition entails Euclid's postulate of parallels, the latter is at last demonstrated. The fact that the same process could equally well be carried out with a spherical triangle, in which the angle-sum is not 180°, might have given him pause. The assumption that translation and rotation are independent operations is in fact equivalent to Euclid's postulate. 1 It should be noted that even the meaning of the criterion suggests several questions of logic. If two lines are so placed that perpendiculars to one of them from points on the other are equal, will the same statement hold when the rôles of the two lines are reversed? Will a perpendicular to one of two non-intersecting lines necessarily be perpendicular to the other? Of course the answers to these questions are closely bound up with the very postulate under discussion.-EDITOR. Heath gives (1. c.) a long and instructive list of these substitutes. In the course of centuries the minds of those interested became clear on one point: they did not wish merely to know whether it was possible to substitute some other assumption for Euclid's, though this question has its interest; they wished to know primarily whether exactly his form of the postulate was logically deducible from his other postulates and established theorems. To change the postulate was merely to re-state the problem. After certain Arabs and Persians had had their say in their day, the curtain rises on the Italian Renaissance of the sixteenth century, where the problem was attacked with great vigor. French and British assailants were not lacking. The first modern work devoted entirely to the subject was published by Cataldi in 1603. When the eighteenth century took up the unfinished business of proving the parallel postulate, we find most of the giants of those days attacking the enemy of geometers with an even keener sense that without victory there could be no peace. Yet d'Alembert toward the close of the century could still refer to the state of the theory of parallels as "the scandal of elementary geometry." Klügel in 1763 examined thirty demonstrations of the postulate. He was perhaps the first to express doubt of its demonstrability. Lagrange, according to De Morgan, in about 1800, when in the act of presenting to the French Academy a prepared memoir on parallels, interrupted his reading with the exclamation, "Il faut que j'y songe encore," and withdrew his manuscript. While the results of these investigations were on the whole negative, certain positive and valuable results were nevertheless obtained. The relation between the parallel postulate and the angle sum of a triangle was clearly brought out. Legendre proved that if in a single triangle the angle sum is two right angles, the postulate holds. Other equivalents are of interest. John Wallis and Laplace wished to assume: There exists a figure of arbitrary size similar to any given figure. Gauss could proceed rigorously provided he could prove the existence of a rectilinear triangle whose area is greater than any previously assigned area. W. Bolyai could have succeeded with the assumption that a circle can be passed through any three points not in a straight line. It must be borne in mind, moreover, that few mathematical questions have served so well as whetstones on which to sharpen the critical powers of mankind.. The work of the Italian priest Saccheri deserves notice because his method is that which finally brought the discussion to a close. Though published in 1733 his results did not become well known until after 1880, and therefore had little influence on other investigations. Legendre's Réflexions, published a hundred years later, covered much of the same ground without advancing quite so far. The title of Saccheri's work is Euclides ab omni Naevo Vindicatus, Euclid Vindicated of every Flaw. His plan was to prove the postulate by assuming its contradictory and showing that an inconsistency followed. He succeeded in proving that, according as in one triangle the angle sum is greater than, equal to, or less than two right angles, the same holds in every triangle, and that accordingly Euclid's postulate or one of its contradictories will hold. |