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

the same value for the obliquity of the ecliptic is used as the one which we deduced from Plato's Timaeus, these two ratios correspond to the latitudes of Babylon and Egypt (the ruins of Babylon and Alexandria). The accuracy is even better than that obtained by the later Hellenistic method (the error being some 16'). The principles of the reconstructed Eudoxan method are illustrated in the attached figure (Fig. 6). In modern terms, if R = 12 = radius of the summer tropic, in the case K

M = 5:3, then P'Q = R cos /?, OP' = R tana, and tan <p = QP':0P' = R cos /S: R tana. Hence cos /? = tan <p tana. It is seen that the use of this method presupposes that the celestical pole P can be determined at the summer solstice. How this is accomplished by means of the arachne, is shown in the next figure (Fig. 7). The ratio K

ss can be read from an ordinary scale on the arm (a:b) because the instrument indicates both the horizontal plane, diameters of the tropics, the obliquity of the ecliptic, and the celestial pole at the same time. The value of this argument gains in strength because Hipparchus's remarks in his commentary on Aratus are independent of the other traditions pertaining to Eudoxus's theory of the homocentric spheres. Moreover, Hipparchus seems to be drawing on Eudoxus directly, repeating his results as they stand and not trying to reconcile them with later observations, for he obviously does not know Eudoxus's method. A similar case is Hipparchus's wrong conclusion from Pytheas's gnomon value G

ss — 120:41 4/5 for Massalia. Hipparchus's surprising miscalculations (error 2°) suggest that he had access to data computed in a way quite different from his. The Geometrical Score of the Harmony of the Spheres Owing to the scales based on harmonious points (the ratios of division being 2:1 and 3:1), of the measuring unit, the arache can also be used in determining proportional angular distances between stars. In this type of observation one tries to find quandruples of stars on a line corresponding to sets of harmonious points on the scales of the instrument. At least two motives can be suggested for this type of observation: (i) an astronomer, especially one associated with the Academy, may wish to show that the starry heavens are constructed according to principles of geometrical harmony (which does not exclude the musical harmony, as can be seen from the musical intervals used in Plato's ,,great harmony"), and (ii) an astronomer operating without proper starmaps, and hence obliged to refer to fairly inaccurate and changing descriptions of the constellations, may wish to specify and standardize these descriptions starting from principles of geometrical harmony. Looking at the extant material pertaining to Eudoxus's (lost) books Phaenomena and Enoptron (fragments 1—120 in Lasserre's Die Fragmente des Eudoxos von Knidos, 1966), one can hardly doubt that Eudoxus was occupied with problems of specification and standardization although, as is the case elsewhere too, only his results are known while his method must be reconstructed from these. In uncertain cases, though, for instance when the location of a special star in one or another constellation must be decided upon, it is quite natural to apply geometrical principles. Suppose that three stars unquestionably belong to a constellation and form three