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

information in the ancient sources directly pertinent to Eudoxus. Hence additional, even though perhaps indirect, information must be sought from related sources. To our delight we have discovered that Plato, Eudoxus's contemporary and associate, gives us two invaluable clues in exact numerical terms in the construction of the world-soul in the Timaeus. One points towards the methods of numerical analysis in Pythagorean mathematics, the other towards an exact value for the obliquity of the ecliptic by these methods. We need not overstress the alleged, though likely, collaboration of Eudoxus and Plato. It is enough to say that this additional information suffices for our reconstruction of Eudoxus's methods of computation in astronomy and geography. For more than one hundred main parameters, including Eudoxus's fictitious parameters, can be computed starting from the known planetary periods [5]. It may be added that as far as real observational parameters are concerned, Schiaparelli's conjectures eventually turn out to be quite good (in the region of 2°—3° from the computed ones). Although the possibility of a mere coincidence in solving such a great number of exact parameter values is negligible, we suspect that many modern commentators may still oppose our use of additional information drawn from the Timaeus. We are, however, quite convinced that Plato's "great harmonia", described in full earnest and most carefully worded, can be tapped for new astronomical information then current in the Academy. For it can be shown that in general, when any two line segments partly overlap so that both are equally divisible by the overlapping part, their end-points create a set of harmonious points the ratio of division being either 2:1 or 3:1 (depending on the point of view adopted). And Plato's "great harmonia'

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'' is constructed starting from just these basic "double and triple intervals". This highly interesting result is communicated in [1, 2] and constitutes a major step in our argument for the correctness of our Eudoxan reconstructions. We interpret this result in terms of a discovery of Plato's rationale of the "great harmonia", and shall include the use of harmonious points in the arsenal of ancient mathematical tools legitimately used in our reconstructions. In fact we make use of harmonious points both in the scales of the arachne and in the outline of Eudoxus's practical computations. Now it must be noted that harmonious points in the ratios of division mentioned (2:1 and 3:1) are obtained as soon as Plato's basic double and triple intervals are interpreted in terms of line segments added to one another (e.g. 1 + 2 + 4+8 and 3 + 9 + 27), and their harmonic and arithmetical means, together with the additional intervals of 9:8 (all of them clearly stated by Plato), are likewise interpreted and inserted between the basic intervals. And according to Plato, they are so interpreted and inserted, [4]. Here, a different arrangement of Plato's basic intervals in one row (e.g. 1, 2, 3, 4, 9, 8, 27) is advocated by Taylor and Cornford, who consequently see little or no mathematical significance in Plato's "great harmonia". But their arrangement is simply in contradiction to the text, for Plato explicitly says that when the insertions are executed, the intervals of 3:2, 4:3, and 9:8 are created, whereas the insertions into the series of the basic intervals arranged in one row produce an extra, unwarranted interval [4, p. 34]. Moreover, the better arrangement in two rows, in the form of a "lambda-like diagram", suggests a value for the obliquity of the ecliptic, which is a central feature in Plato's astronomical model, in terms of a Pythagorean, right-