an excellent mathematician of the first century B.C., the following remarks: 'The ancients, defining a cone as the revolution of a right-angled triangle about one of the sides containing the right angle, naturally supposed also that all conics are right and there is only one kind of section in each-in the right-angled cone the section which we now call a parabola, in the obtuse-angled a hyperbola, and in the acute-angled an ellipse. You will find the sections so named among the ancients. Hence just as they considered the theorem of the two right angles for each kind of triangle, the equilateral first, then the isosceles, and lastly the scalene, whereas the later writers stated the theorem in a general form as follows, 'In every triangle the three interior angles are equal to two right angles,' so also with the conic sections, they regarded the so-called 'section of a right-angled cone' in the right-angled cone only, supposed to be cut by a plane perpendicular to one side of the cone: and similarly the sections of the obtuse-angled and acute-angled cones they exhibited only in such cones respectively, applying to all cones cutting planes perpendicular to one side of the cone. . . . But afterwards Apollonius of Perga discovered the general theorem that in every cone, whether right or scalene, all the sections may be obtained according to the different directions in which the cutting plane meets the cone.'"'1 While the early treatment of conic sections shows the difficulties in reaching the conception of the general, it is the same treatment of the theorems in elementary geometry that is of interest to us here. We thus see the early difficulties of the race; the tendency to pass from the special to the general is not only characteristic of the individual in his development, but of the race also. As to the conciseness of geometric proofs, it is probable that compared with our practice to-day there was little reference to previous theorems, and hence a certain amount of tediousness was avoided. Even Euclid, who systematized the existing geometry, referred to previous propositions in only an indirect way. But to the learners of geometry in these early times, the method must have seemed far from being concise, where necessary steps in the proofs were insisted upon. Alexander the Great, who had Menæchmus for a teacher, complained of the length of the proofs. When he asked his teacher if the instruction could not 1 1 Gow, p. 137. be made somewhat shorter, Menæchmus replied, “O King, in the material world there are roads for common people and roads for kings, but there is only one road to geometry, and that is for all."1 A similar story is told of Euclid in reply to the question of King Ptolemy. The geometers before Euclid have given us plans of attack that are standard to-day. We shall refer in particular to the notion of locus, the method of exhaustion, reductio ad absurdum, and analysis. Allman2 contends that the conception of geometric locus is due to Thales. He bases his conclusion in part on the fact that Thales knew that any angle inscribed in a semi-circle is a right angle. Dr. Allman may be right, but his conclusions do not seem warranted. It seems probable though, that the notion of geometric locus was understood before the time of Archytas (cir. 400 B.C.), for Archytas not only employed this idea, but used the intersection of loci for the determination of a point.3 We learn also from the Eudemian Summary that Aristaus (cir. 320 B.C.) wrote on solid loci, and later Hermotimus of Colophon composed some propositions on loci. About this time curves of all kinds were called running loci, the straight line and the circle were called plane loci, and the conic sections solid loci.1 At least by the time of Plato, the conception of geometric locus was fully appreciated.5 Eudoxus is generally credited with the perfection of the method of exhaustion. Before him Antiphon and Bryson had employed the process of exhaustion in seeking the quadrature of the circle. This was the process of exhausting the area of a circle by means of inscribed and circumscribed polygons. Antiphon used only the series of inscribed polygons, while Bryson used both the inscribed and circumscribed. The latter, the more rigid, was adopted by Euclid, and we find it in use to-day in many of our texts. But the so-called method of exhaustion 1 Bretschneider, Die Geometrie und die Geometer vor Euklides, pp. 162-163 'Op. cit., p. 13. See Allman, pp. 111-114. Archytas effects the duplication of the cube by the intersections of a cylinder, cone, and hemisphere. 5 See Montucla, Histoire de mathématiques, Tome I, p. 183; Chasles, Aperçu historique des methodes en géometrié, p. 5. was established later by Eudoxus. It embodies the two propositions mentioned below.1 In establishing the method of exhaustion, Eudoxus, as we can see in the propositions cited, made use of the reductio ad absurdum, which is a most powerful instrument of attack in mathematical theory. In this connection we should mention the use of geometric reduction. This idea is so common to-day that we do not dignify it with a definition. Proclus,' discussing the recasting of the duplication problem by Hippocrates, credits him with the invention of this method, which he defines as a transition from one problem or theorem to another, which, being solved or proved, the thing proposed necessarily follows. This method then was employed Y 2 Y3 3, 1I. If from A more than its half be taken, and from the remainder more than its half, and so on, the remainder will at last become less than B, where B is any magnitude named at the outset (and of the same kind as A), however small. II. Let there be two magnitudes, P and Q, both of the same kind; and let a succession of other magnitudes, called X1, X2, X, ... be nearer and nearer to P, so that any one, X1, shall differ from P less than half as much as its predecessor differed. Let Y1, be a succession of quantities similarly related to Q; and let the ratios X, to Y1, X2 to Y2, and so on, be all the same with each other, and the same with that of A to B. Then it must be that P is to as A to B. (It is obvious, from the conditions, that if X, be greater than P, Y, is greater than 2, etc.) Suppose X1, X2, etc., less than P, and therefore Y1, Y2, etc., less than Q. Then if A is not to B as P to Q, A is to Bas P to some other quantity S greater or less than Q; say, less than Q. Then (by hyp. and I) we can find some one of the series Y1, Y.,, (say Y) which is nearer to Q than S is to Q and which is therefore greater than S. Then since X, is to Y, as A to B, or as P to S, we have X is to Yn as P to S, or X n n n n n to P as Y to S; from which, since X, is less than P, But Y is also greater than S, which is absurd; therefore, A is not to B as P to less than Q. Neither is A to B as P'to more than Q (which call S), for in that case S is to P as B to A; let S be to P as Q to T, then S is to Q as P to T; from which, S being greater than Q, P is greater than T. But B is to A as S to P, that is, as Q to less than P, which is proved to be impossible by the reasoning of the last case. Consequently, A is not to B as P to more than Q, or to less than Q; that is, A is to B as P to Q, which was to be shown. Let P and Q be two circles, A and B the squares on their diameters, X, and Y1 inscribed squares, X, and Y, inscribed regular octagons, X, and Y, inscribed regular figures of sixteen sides, etc., the preceding process gives the proof that circles are to one another as the squares of their diameters. See De Morgan's article, Geometry of the Greeks, in the Penny Encyclopedia, and Gow, pp. 171, 172. 2 Proclus, ed. Friedlein, p. 212. 2 3 in the discovery of new theorems or problems. It sets up a chain of steps by which certain conclusions are reached. This we see prepares the way for the reductio ad absurdum which in turn embodies the idea of analysis. As we know, the method of analysis is employed by supposing for the time being that the desired theorem is true or the sought problem is effected so as to find the necessary underlying conditions. In reductio ad absurdum the contrary theorem is proved to be not true by analysis. We must also then credit Eudoxus with employing this method, for the method of exhaustion involves the reductio ad absurdum, and it in turn employs the method of analysis. "Both Proclus (ed. Friedlein, p. 211), and Diogenes Laertius (III, 24), state that Plato invented the method of proof by analysis." Plato and Eudoxus were contemporaries. We have positive evidence of Eudoxus employing this method, as shown above, but Proclus and Diogenes Laertius give the credit to Plato. It is most probable that Plato systematized the method and gave it a definite form. The discussion of geometric problems owes its origin to the Greeks. We are accustomed to-day to attack a problem by analysis, then, after the necessary conditions are found, the construction is made, the deductive proof is given, and then is added the discussion of the conditions under which the problem is or is not solvable. The Greeks called this the diorismus. The Eudemian Summary2 credits Leon, a student of the Platonic school, with its invention. While there are divided opinions as to whom belongs the credit of these various methods, it is certain that they were systematized and practiced in Athens during the time of Plato. Regarding the general characteristics of the Greek elementary geometry, two stand out prominently: (1) It was essentially deductive. Undoubtedly this was to a large extent the cause of mechanics being expelled from geometry. (2) The restriction as to the use of compasses and rule necessarily divided the subject-matter of geometry; so the conic sections did not find a place in common with the geometry of the straight line and the circle. Hence in Euclid's "Elements" we find only 1 Gow, p. 176; also see Cantor, I, p. 207. 'See above, p. 14. a treatment of the latter geometry. This division exists to-day in the elementary field, although it has been the custom for some authors to put a synthetic treatment of the conics at the end of the text after the geometry of solids. As for methods of attack in elementary geometry, the Greeks invented all those that are in common use to-day. We should specially mention the method of analysis and the reductio ad absurdum. |