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
angled triangle. For an acute angle in the triangle with the hypotenuse of 39 = = 3 + 9 + 27, and the shorter side of 15 = 1 + 2+4+8 units of (relative) length, is obtainable from tan e = 5/12 or sin e = 5/13. And the same result is also obtained if one begins with two integers p = 3, q = 2 generating Pythagorean triples, i.e. with the same integers that figure in the formation of harmonious points by overlapping line segments, for in this case c = p 2 + q 2 = 13, a = p 2 —q 2 = 5, and b = = 2pq = 12, where a 2 + b 2 = c 2 and p > q; p, q relatively prime and not both odd [4, 5]. In this way we have deduced the obliquity of the ecliptic in Plato's cosmological system in the Timaeus, and since this must have been a central parameter value generally known in the Academy, we further assume that even Eudoxus used precisely this value. Our final results make it clear that the assumption is correct. To the possible objection that s P i at0 % 22° 37' obtained in this manner is less than the real value in Platos' and Euduxus's time e rea i = 23° 44' (see e.g. D. R. Dicks, Early Greek Astronomy to Aristotle, p. 154, n. 240), we answer that so it indeed should be in Eudoxus's system, for Eudoxus is known to have credited the Sun with a fictitious deviation from his ecliptic. This implies that the loxotes adopted by Eudoxus was somewhat too small as compared with actual observation. Moreover, even the ficitious additional deviation can be obtained by exactly the same method [5]. There are, of course, other interpretations of details based on the additional information we have drawn from Plato, and other smaller discoveries about the numerical parameter values of Eudoxus (including a computational connection between the synodic periods), but this may suffice to illustrate our use of the sources and some decisions made in their interpretation. More details will emerge from our description of the different usages of the arachne (below). Eudoxus's Method of Celestial Computations In the teory of the homocentric spheres, the basic problem of explanation concerns the computations needed in the combination of two spherical motions. If these motions are characterized by angular velocities, the combination assumes the following form. (1) +_a) LNA ( WORE >J;eo ind ( WorE >= +_G) OOMB ( W OR B > Here the indices (I, II) refer to two spheres, the indices (ind, comb) to the individual and resulting motions, and (W, E) to their directions. These are some of the Eudoxan parameters qualitatively described in the tradition. Some of them can easily be given even an exact value. In fact the problems are not difficult as far as Eudoxus's first and second spheres for the seven planets are concerned. The real problems appear in the combination of his second and third, or second and fourth spheres. Hence we may take these as our examples. Systematical study of all alternatives is undertaken in [4]. Since co = 1/T, or the inverse of rotation time, (1) may be restated as, say, (2) X oomb = X eomb T ln( ' • (T oomb +_ T ind )
