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
solution. The former (instrumental) solution is discussed by J. E. Hofmann in „Ueber die sog. Platonische Konstruktion von Kubikwurzeln", Sudhoffs Archiv, Band 58, Heft 1, 1974, pp. 60—63. But by means of our reconstruction of the arachne, square and cubic roots can be extracted. We submit, therefore, that the so-called Platonic solution to a :x = x :y = y :b and Eudoxus's solution are identical, albeit the arachne, in addition to illustrating the proof, also suggests the method of invention preceding the proof. It is true that Heath (A History of Greek Mathematics, i, pp. 255—258) concluded that the so-called Platonic solution was invented in the Academy by someone contemporaty with or later than Menaechmus, Eudoxus's pupil. But Heath's argument depends on the fact that in analytical terms the former solution, being found by means of a curve of the third degree, is more difficult than that of Menaechmus, which is found by means of the intersection of curves of the second degree (either two parabolas or a parabola and a hyperbola). It is not the Cartesian analytical treatment, however, but the use of the instrument, which must be considered here. For Plutarch (Marcell. 14, p. 59 f., Sint.) tells us that Eudoxus used ,,machines" constructed on the basis of geometrical theories, especially on the teory of the two mean proportionals (see Paulys RE, s.v. Eudoxos, §4). Besides, Plato (at tm. 32B) seems to refer to the problem of the two mean proportionals and to its solution as to something well known. In addition to the use of an instrument, two main points emerge from the tradition pertaining to Eudoxus's solution (D 24—29 in Lasserre). First, Eudoxus used „curved lines" {xxyaivhui) in the discovery of the solution, but did not refer to them in the proof. And second, Eutocius criticizes Eudoxus for having confused a discrete proportion (e.g. a :b = c :d) with a continuous one (e.g. a:b = b:c). Earlier commentators have never succeeded in combinig these two features (see Lasserre, op. cit., pp. 163—6, and Heath, op. cit., i. pp. 249—251). Yet both can be understood when the use of the arachne is considered. Leaving aside its more obvious use in the extraction of square roots, or the finding of one mean proportional, we will concentrate upon the cubic roots. The fact that even square roots can be extracted, this being a special case of the more general problem of finding two mean proportionals, should be remembered, however. For it is from the connection between the problems of one mean proportional and a ovvx/xia taken in the sense of a quadratic value of a rectangle (see Ärpad Szabö, Anfänge der Griechischen Mathematik, 1969) that we have continued our studies pertaining to the dynamic world-view in [8]). A mediating step is the reconstruction of Eucoxus's arachne. Let us outline first the proof of the so-called Platonic instrumental solution, which is possible as soon as the instrument has reached a desired position. If a = 1 and b = 2, say, the solution to a:x = x:y = y:b appears as in the attached figure (Fig. 10). Because the angles AOM, MON, NOB, AMN and BNM are right angles, the triangles AMO, MON and NOB are similar, and their respective sides proportional. 3 _ Hence it can be proved (without any „curved lines") that, in modern terms, 1:^2 == 3 3 _ 3 _ = ^2:^4 = ^4:2. In other words, one has obtained by means of the arachne two 3 _ 3 _ approximations: x %|/2 , y % /4 .
