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

It is shown in [5] that if an arbitrary parameter n is introduced in accordance with Elementa V. 15, and (2) is considered as a generalized proportion (3) TjJ

ml>

T

i

I

°J: (T™

mb

±T

i

I

^) = nT°°f:n, and means are discovered for mastering (3), all Eudoxan parameters characterizing the theory of the homocentric spheres can be given an exact numerical value. The solutions have the following form in terms of the sides a, b, c of a general right-angled triangle, or in terms of two integers p, q generating a similar Pythagorean triangle (p =» q, relatively prime and not both odd) in the usual way: (4) for xy:(x + y) = nT

comb

:n (5) forxy:(x—y) = nT

comb

:n n = x + y = c + b = = > (p + q)

2 n = x—y = c—b = = =- (p—q)

2 x — (a + b + c):2 = => p(p + q) x = (a—b + c):2 = => p(p—q) y = (b + c—a):2 = = > q(p + q) y = (a + b—c):2 = = > q(p—q) x:y = (a + c):b = p:q x:y = b:(c—a) = p:q How these solutions are obtained by using the methods of analysis and synthesis (this being a direct contribution to a better understanding of Eudoxus's ideas of analysis) is explained in detail in [5, 7]. The solutions are perfectly satisfactory as regards ancient mathematical tools, and one can hardly imagine simpler solutions than ours. Moreover, the solutions x:y = p:q alone are really needed, and in [7] it is shown that these solutions, which are invariable with respect to the sign in (3), have a simple geometrical interpretation. For if in a Pythagorean right-angled triangle tan a = 2pq:(p

2

—q

2

), then tan (a/2) = q:p by Elementa VI. 3. This fact we have made use of in the construction of the measuring unit of the arachne. Despite the simple geometrical interpretation and its relatively easy instrumental realization, we are still speaking about the possible way of Eudoxus's invention, or, if you wish, about the mathematical background of the method of finding the correct solution to (3). This heuristic process is described in algebraic terms in the Appendix to [5], and it shows a most interesting feature mentioned above: the auxiliary parameter n disappears in the synthetic part of the method. Moreover, the general view on language as a painting corresponding point to point to reality is suggested—a view that seems to be shared by Plato, too, in his semantics of time. But the problem of Eudoxus's practical calculations remains to be tackled. Details are discussed in [8]; suffice it to say here that the problem can be solved within the general framework of ancient mathematics. For we have seen that Plato's "great harmonia" was constructed by means of harmonious points (the ratios of division being 2:1 and 3:1), and by their means also the practical calculations can be made. The following examples are intended to illustrate these calculations. [Moon] By means of the arachne, the Moon's maximum deviation from the Eudoxan ecliptic can be observed and measured in terms of tan (a/2) = q:p. Let p :q now stand for the ratio of division in harmonious points, and note that calendaric considerations also suggest the "idealized" observation, for (p + q):p = 30:29 is equal to the ratio of the "full" and "hollow" months, i.e. 30 days:29 days. Thus we have the following situation describing the combination of the second and third lunar motions (diagram not to scale), where two periods can be solved starting from