CHAPTER II THE WORK OF EUCLID AND HIS INFLUENCE ON THE SUBSEQUENT TEACHING OF GEOMETRY' THE ELEMENTS OF EUCLID As has been mentioned, Euclid (cir. 300 B.C.) compiled and arranged in an orderly manner the work of his predecessors. Proclus, after quoting from the Eudemian Summary, closes with these words," "Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theatetus's, and brought to irrefragable proof propositions which had been less strictly proved by his predecessors." Before considering the special features of this work, let us summarize the main contributions to Euclid according to the opinion of Dr. Allman. He holds that after the Pythagoreans, Euclid was indebted most to Eudoxus and Theætetus. To the Pythagoreans he attributes the substance of Books I, II, and IV, the doctrine of proportion and of similar figures "together with the discoveries respecting the application, excess, and defect of areas—the subject matter of the sixth book: the theorems arrived at, however, were proved for commensurable magnitudes only, and assumed to hold good for all." To Eudoxus is credited 1 For a readable account of the work of Euclid and his later influence, see Frankland, The Story of Euclid. The relation of Euclid's "Elements" to later texts and progress in geometry is set forth by Professor Gino Loria in his Della varia fortuna di Euclide. In lighter vein, one may read an entertaining controversy over the merits of Euclid by turning to Dodgson, Euclid and his Modern Rivals. To Theætetus is Books V and VI, and the bulk of Book XII. given a part of Book X and Book XIII. Euclid also is given the credit of contributing to Book X. We cannot take Allman's judgment as final, but his conclusions show how little Euclid contributed to the subject-matter of geometry. The fourteenth book, which has been assigned to Euclid, was written about 150 years later by Hypsicles. He has been credited with adding also the fifteenth book, but the opinion is gaining ground that this was written by Damascius of Damascus (cir. 490 A.D.).1 The original text which bears Euclid's name consisted of thirteen books. The sequence of subject-matter in these books is as follows:2 Books I and II are on the geometry of straight lines and areas. Book III treats of circles. Book IV, of regular figures. Book V, the theory of proportion for all kinds of magnitudes. Book VI, the application of this theory to plane figures. Books VI, VIII, and IX treat of the arithmetical theory of proportion. Book X is on arithmetical characteristics of the divisions of a straight line. Books XI and XII treat the geometry of solids, and Book XIII is on the five regular polyhedra. Book I is introduced by the definitions of point, line, etc. Here are given five postulates and five common notions (later called axioms). One of the postulates deserves our attention, for it represents the points of departure of so-called non-euclidean geometries. It is, "If two straight lines are cut by a third straight line so as to make the sum of the interior angles on the same side of the transversal less than two right angles, then these two lines, if produced, will meet on that side." Euclid, being unable to prove this, had to assume its truth. Playfair (1795) stated the equivalent of this in the form, "Two intersecting straight lines cannot both be parallel to the same straight line." The denial of this led to the non-euclidean geometries of Bolyai and Lobachevski. The lack of correspondence between the historic development of geometry and its sequence as shown in Euclid's "Elements" has already been referred to.3 It has been shown, for 1 Gow, pp. 272, 312-313. Also De Morgan's 3 See p. 16 above. 2 See Heiberg, Euclidis Elementa, Leipzig, 1883. article, Eucleides, in Smith's Dictionary of Greek and Roman Biography and Mythology. example, that the geometry of circles, the theory of proportion, and the study of the regular solids were developed in an order entirely different from their sequence in the "Elements." Euclid put solid geometry after plane geometry, although the former was partially developed before the latter had been systematized or even fully developed. The systematizing of definitions, axioms, and postulates was not accomplished until Plato and Aristotle turned their attention to this fundamental part of geometry.1 Euclid was not concerned with the subjectmatter of geometry from the standpoint of its growth, his task being to organize this material so as to get a minimum of logical friction. The sequence set by Euclid has to a large extent been studiously copied by his followers. Another thing peculiar to Euclid was the elimination of all practical work. We have observed that it was largely due to Plato that mechanics was divorced from geometry. Euclid simply reflected the traditions of the Greek people. Geometry was to be evolved from within itself, so there was not even mensuration in the "Elements." Even the quadrature of the circle was omitted. Furthermore, there were no original exercises such as we have to-day in our geometries. All the propositions were worked out in full. It was a book to read, not one to develop, as the term is used to-day. In short it was a philosophical treatise intended for mature minds. One further feature of Euclid should be mentioned. Hypothetical constructions were not permitted. Thus the theorem, "If a triangle have two sides equal, the angles opposite these sides are equal," could only be proved after it was shown how to construct an isosceles triangle. Hence Euclid begins his first book with constructions. This in itself is of pedagogical value, but the method was carried to the extreme. Modern usage allows these hypothetical constructions. Regarding the nature of theorems and problems, Euclid makes no distinction in his naming. He calls them all propositions. Euclid's predecessors had developed the various methods of at 1 Gow, p. 176. For an instructive article on the mathematical contributions of Aristotle, see Heiberg, Mathematisches zu Aristoteles, in Abhandlungen der Geschichte der Mathematischen Wissenschaften, v. 18, pp. 1-49. tack already mentioned, but had organized no systematic plan by which all propositions were subjected to an orderly method of proof. The "Elements" bears evidence of such a plan, and to Euclid we are thus indebted for: 1. The general enunciation of the proposition. 2. The particular statement. 3. The construction. 4. The proof. 5. The conclusion. 6. The affixing at the end the Q.E.D. or the Q.E.F.1 2 Some features of Euclid are adversely commented upon by De Morgan. The substance of three of his criticisms, which seem to be well founded, are: (1) Euclid makes no distinction between propositions which require demonstration and those which a logician would see to be nothing but different modes of stating a preceding proposition. Thus the statement, "Everything not A is not B" is equivalent to "Every B is A," but Euclid does not recognize this equivalence. (2) He fails to employ generalizing notions in certain cases. Thus in defining an angle as the sharp corner between two lines that meet, he does not consider the straight angle and the reflex angle. (3) He neglects the formal accuracy with which translators have endeavored to invest the "Elements." He refers to theorems in an indirect manner, either reasserting without reference, or saying "it has been demonstrated." Also, he places theorems among definitions, makes assumptions that are not in the postulates, and omits some necessary proofs. According to De Morgan, Euclid has been considered so perfect that later writers, thinking that they were restoring the original perfection of the book, were, as a matter of fact, improving on the "Elements." Simson is one of these to whom De Morgan refers. To summarize briefly, Euclid is to be considered the compiler and not the composer of the "Elements." Very little that was original is attributed to him. His great work was to systematize the logic of geometry. This applies not only to the sequence of propositions, but to the orderly arrangement of proof in the propositions. As we shall refer repeatedly to Euclid 1 Proclus, ed. Friedlein, pp. 203, 210. Compare Heiberg's Euclid is Elementa. See Gow, p. 199. 2 Op. cit., p. 71. and Euclidean geometry, three characteristic features of the book should be re-emphasized. They are: (1) Hypothetical constructions are not admitted. (2) All practical work is excluded. (3) All constructions are by means of straight edge and compasses only. This bars out the conic sections. THE LATER INFLUENCE OF EUCLID After Euclid, mathematics became more practical at Alexandria, due principally to the efforts of Archimedes, Hipparchus, and Heron of Alexandria. But the study of theoretical elementary geometry was kept alive although its subject-matter was little increased. Beginning with Theon of Alexandria, the father of Hypatia, the "Elements" of Euclid began a long series of editions. Theon, who lived in the fourth century A.D., lectured on mathematics at the University of Alexandria, where Euclid had been the first professor. There can be but little doubt that during the interim between Euclid and Theon the study of the "Elements" had been carried on, for the interest was such 675 years after Euclid that Theon prepared an edition of the "Elements" for his classes. This edition made some slight changes in the original, and added some commentaries, but the work as a whole was kept intact.1 Before the time of Theon, the "Elements" had become known throughout Greece, including Asia Minor and the Italian colonies. Indeed, in Italy, although geometry there was essentially practical, Euclid was not unknown, for Boethius (cir. 500 A.D.) incorporated in a work a statement of the propositions of Euclid I plus some others from the second, third, and fourth books, giving at the last the proofs of the first three propositions of Book I. 2 When Alexandria was destroyed by the Arabs, in 640 A.D., Greek learning found a home in the Syrian cities on the east coast of the Mediterranean. From these schools the Arabs of Bagdad gained something of Greek learning, and the works of Euclid, Archimedes, Apollonius, and Ptolemy were translated into the Arabic. Euclid was partially translated in the time of Harun al 1 De Morgan, op. cit., p. 68. 2 Ibid |