Modern scientists try to quantify scientific theories and models, seemingly automatically, without thinking about why they are quantifying or even if they should quantify. A folklorist or an anthropologist looking for and identifying recurring general themes in the preoccupations of scientists might liken their quantitative disposition to a knee-jerk reaction or to the salivating of Pavlov's psychologically conditioned dogs when a bell rang. But if every time they talk or write scientists don't praise explicitly the tremendous power of quantitative reasoning in science, and in many other fields of human thought as well, still they are fully conscious of what they are doing and why. In Greek geometrical astronomy we can find the birth and trace the early development of the quantitative scientific method.
There is no evidence that Apollonius tried to quantify the new eccentric and epicycle models for saving the phenomena. For all we know, he may have been perfectly content with his brilliant qualitative explanations. He may not even have imagined that his geometrical model could take on quantitative, predictive power. Four centuries separate Apollonius's qualitative eccentrics and epicycles from Ptolemy's comprehensive quantitative model of planetary motions. Hipparchus bridges this chasm.
Ptolemy credits Hipparchus with developing the quantitative solar model presented in the Almagest, "but under conditions which forced him [Hipparchus], as far as concerns the effect over a long period, to conjecture rather than to predict, since he had found very few observations of fixed stars before his own time" (Almagest, VII 1). Ptolemy had more observations, and he corrected Hip-parchus's theory. Uncertainty exists over what Ptolemy contributed and what Hipparchus contributed, and some historians attribute to Hipparchus everything not explicitly claimed by Ptolemy as his own.
The primary objective of any solar theory was to demonstrate the motion of the Sun using only uniform circular motions. Ptolemy wrote: "It is now
Figure 8.1: Ptolemy's Geometrical Demonstration of the Solar Eccentricity and Apogee. The ecliptic (for Ptolemy, the apparent path of the Sun among the stars; in modern theory, the plane of the Earth's orbit around the Sun marked against the stars) is ABCD. It is centered on the Earth at E.
The diameters of the ecliptic circle AC and BD are perpendicular to each other and pass through the tropic and equinoctial points. Spring equinox (one of two points where the ecliptic and the plane of the Earth's equator intersect, as marked on the sphere of the stars; about March 21) is at A. Summer solstice (one of two points on the ecliptic where the Sun is at its greatest distance above or below the plane of the Earth's equator; about June 22 in the northern hemisphere, the beginning of summer, and the longest day of the year) is at B. Autumnal equinox (about September 22) is at C. Winter solstice (about December 22; the shortest day of the year) is at D.
If the Sun traveled with uniform motion (constant velocity) around the ecliptic circle ABCD, as observed from its center E, the Sun would traverse arcs AB, BC, CD, and DA in equal times. The seasons (spring, summer, fall, and winter) would be equal in length, a quarter of a year each.
From observations, however, as observed from E, arc HK is 941/2 days; KL is 921/2; LM is 881/g; and MH is 901/8, with HKLM an eccentric circle centered on F. Uniform motion around HKLM appears nonuniform to an observer at E.
To make arc HK the longest of arcs HK, KL, LM, and MH, as seen from E, it is readily obvious that the center F of the eccentric circle HKLM must be placed somewhere in quadrant AB. Ptolemy worked out geometrically and quantitatively precisely where in quadrant AB.
necessary to take up the apparent irregularity or anomaly of the Sun . . . And this can be accomplished by either hypothesis: (1) by that of the epicycle . . . But (2) it would be more reasonable to stick to the hypothesis of eccentricity which is simpler and completely effected by one and not two movements" (Almagest, III 4).
Ptolemy further demanded that theory determine the precise position of the Sun for any given time: "The first question is that of finding the ratio of eccentricity of the Sun's circle . . . what ratio the line between the eccentric circle's center and the ecliptic's center . . . has to the radius of the eccentric circle; and next at what section of the ecliptic the apogee [the point farthest from the Earth] of the eccentric circle is to be found" (Almagest, III 4).
This Hipparchus had done: "For having supposed the time from the spring equinox to the summer tropic [solstice] to be 941/2 days, and the time from the summer tropic to the autumn equinox to be 92V2 days, he proves from these appearances alone that the straight line between the aforesaid centers is very nearly V24 ofthe radius ofthe eccentric circle; and that its apogee proceeds the summer tropic by very nearly 2472° of the ecliptic's 360°" (Almagest, III 4). Ptolemy's own numerical solutions, using both Hipparchus's data and his own new observations, agreed with Hipparchus's determinations.
For the Moon's motion, Ptolemy used observations available to Hipparchus, Hipparchus's mathematical procedures, and also new observations, made "very accurately" by himself. Ptolemy corrected and improved the lunar theory handed down from Hipparchus. Next, Ptolemy tackled the planets:
Now, since our problem is to demonstrate, in the case of the five planets as in the case of the Sun and Moon, all their apparent irregularities so produced by means of regular and circular motions (for these are proper to the nature of divine things which are strangers to disparities and disorders) the successful accomplishment of this aim so truly belonging to mathematical theory in philosophy is to be considered a great thing, very difficult and as yet unattained in a reasonable way by anyone. (Almagest, IX 2)
Hipparchus, who had demonstrated the hypotheses of the Sun and the Moon, had not, according to Ptolemy, succeeded with the planets:
I consider Hipparchus to have been most zealous after the truth, . . . especially because of his having left us more examples of accurate observations than he ever got from his predecessors. He sought out the hypotheses of the Sun and Moon, and demonstrated as far as possible and by every available means that they were accomplished through uniform circular movements, but he did not attempt to give the principle of the hypotheses of the five planets, as far as we can tell from those memoirs of his which have come down to us, but only arranged the observations in a more useful way and showed the appearance to be inconsistent with the hypotheses of the mathematicians of his time. (Almagest, IX 2)
Hipparchus had found the problem of the planets too complex:
For not only did he think it necessary as it seemed to declare that . . . the regressions of each [planet] are unique and of such and such a magnitude . . . but he also thought that these movements could not be effected either by eccentric circles, or by circles concentric with the ecliptic but bearing epicycles, or even by both together. (Almagest, IX 2)
Yet Hipparchus was not content to settle for less than a full quantitative solution, nor was Ptolemy:
Hipparchus reasoned that no one who has progressed through the whole of mathematics to such a point of accuracy and zeal for truth would be content to
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