Technikatörténeti szemle 10. (1978)

A MÉRÉS ÉS A MÉRTÉKEK AZ EMBER MŰVELŐDÉSÉBEN című konferencián Budapesten, 1976. április 27–30-án elhangzott előadások II. - Maula, E.: A szögmérés kezdetei

points of a set of harmonious points. A fourth star corresponding to the fourth harmonious point on the scale, may then be grouped together with the previous ones. If one consults modern star-maps and the list of stars mentioned by Eudoxus, this procedure suggests itself. It is worthwhile, therefore, to show that harmonious points can indeed be defined in terms of Eudoxus's contribution to Euclid's Elementa, Book v. Let us consider similar triangles of two different kinds (Fig. 8). Suppose that the line segments EC, CF are equal. Then on the one hand the triangles ACE and ADG, and on the other hand the triangles CFB and BGD, are similar. Hence the points A, B, C, D form a set of harmonious points (A, B; C, D). In order to derive the condition for harmonious points, we now take any equimultiples of the line segments AD and EC, and any equimultiples of the line segments AC and GD. Let the former equimultiples be, for instance, halves of, and the latter equimultiples double, the corresponding line segments. Then the equimultiple of AD falls short of that of EC, and the equimultiple of AC falls short of that of GD. Now, by Elem. v, Def. 5, we obtain the proportion (1) AC:AD = EG:GD and the proportion (2) CF:GD = CB:BD. Since EC = CF, we obtain further by Elem. v. 7 the proportion (3) EC:GD = CF:GD. From (1) and (3) we obtain by Elem. v. 11 the proportion (4) AC:AD = CF:GD, and from (2) and (4) by the same proposition the proportion (5) AC:AD = CB:BD. Finally, from (5) we obtain by Elem. v. 16 the proportion (6) AC :CB = AD :BD. This is the condition under which the points A, B, C, D form a set of harmonious points (A, B; C, D). As for the interrelations of the line segments partly overlapping on the scales of the measuring unit and the angles, these are best represented by means of the line segments and the tangents of the angles (for tan a = the ratio of the gnomon, to its shadow). We obtain several interrelations which are illustrated by the attached figure (Fig. 9). We list some of them below, (i) tan (a + /S):tana = a:(a—d), where the coefficient (a—d)

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/ad gives the ratio of division, (ii) tan (<x + /3+y):tan (a + /3) = = (a + b—d):a, where the coefficient is a:(b—d), (iii) tan (a + /J+y):tan a = = (a + b—d):(a—d), where the coefficient is (a—d):(b—d), (iv) tan (<p + y>) :tan <p = = b:(b—d), where the coefficient is (a + b—-d):b, (v) tan («5 + <p + y>) :tan (<p + y>) = = (a+b—d):b, where the coefficient is, contrary to the previous case, b:(b—d), and finally (vi) tan (<5 + <p + y):tan ip =s (a + b—d):(b—d) or the ratio of division of the line segments, the endpoints of which create the set of harmonious points (A, B; C, D). Irrationality and Invention It is well known that the ancients had discovered not only theoretical geometrical solutions but also practical instrumental solutions to the problem of the two mean proportionals, which is a:x = x:y = y:b. The main sources are Pappus's Gollectio, Book iii, ed. F. Hultsch (Berlin, 1876—8, Band I, pp. 56—64) and Eutocius's commentary on Archimedes's De sphaera et cylindro, Book ii, Prop. 1, appearing in J. L. Heiberg's critical edition of Archimedes's Opera (second edition, Band III, Leipzig 1915, pp. 54—106). The whole tradition seems to derive from Eratosthenes's Platonicus, and through him from Eudemus (see Lasserre, op. cit., p. 163 ff.). We are especially interested in the so-called Platonic solution and Eudoxus's