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

is leaned upon, a discrete and a continuous analógia are equalized. Fourth, the whole method of invention with its increasingly more accurate approximations from above and from below, is remarkably akin to the known Pythagorean method of approximation to surds by algorithmic means. The discovery of this relationship seems to prove beyond reasonable doubt that Eudoxus's method indeed was of the type adumbrated here. Its implications for mathematical heuristics and theory-formation are farreaching. They are discussed in [8]. In the second method the arachne turns about the pivot on the enoptron, and the auxiliary trunk is useful (although perhaps not quite necessary). Suppose that we are trying to find the two mean proportionals between two line segments a, b. Now a is kept in all positions as the distance from the horizontal axis of the enoptron, measured at right angles from the main trunk of the measuring unit, and b is fixed on the scale of the main trunk, measured from the pivot onwards. (Fig. 14). Thus' the endpoint of 6 describes a circle, and the end-point of a describes a ,,curved line' (Fig. 15), while the other end-point of a follows the horizontal axis of the enoptron. In this case the desired position is obtained more ,,mechanically", i.e. as soon as a rectangle (completed by means of the auxiliary trunk, say) is obtained (the rectangle is MNCB). If one makes a = 1, b = 2, the condition for the „curved line" is x:y= = (/x

2 + y

2

):l, whence the equation (of the fourth degree) can readily be obtained. The second method adumbrated exhibits some similarity to Eudoxus's way of representing the apparent planetary motions by means of combined spherical rotations. It is not obvious, however, how Eutocius's remark on discrete and continuous proportions could be accounted for. But Plutarch's fits nicely here. And we cannot be very far from the birth-place of Menaechmus's discovery of the conic sections either. The Eclipse and the Corona The accuracy of Eudoxus's observations was eclipsed by Hipparchus's, his ,,curved lines" by Menaechmus's discovery of the conic sections, his methods of invention by Archimedes's kindred methods, and his arachne by the astrolabe of the Alexandrian astronomers. But owing to these lost achievements' solid foundation on mathematical principles, we still discern their corona in Eudoxus's extant mathematical, astronomical and geographical results. It is from these results that our reconstruction of Eudoxus's methods and observational techniques has been built. Hence our reconstruction of the arachne is a tribute to Eudoxus's praxis deeply ingrained in theory yet capable of giving rise to a concrete, instrumental aid to invention and observation. Eudoxus's Protophysik is oriented towards geometry and his logic of discovery towards approximative methods. On the other hand, his observations are theory-informed and the most characteristic feature of his geometry and methods of proof (including the method of exhaustion) is their orientation towards motion. True, we could say that one function of the arachne alone, its use as a computing machine in the extraction of approximations to square and cubie roots, admits of a fairly safe dating. The other functions discussed belong to the desiderata. They