Dating an Eclipse and its Battle

Reports of ancient observations of the Moon's position can be of interest to historians because these observations hold out hope of dating specific events and, more generally, of relating ancient chronological systems to our own.

We can, for example, calculate past eclipses, particularly large eclipses of the Sun (by the Moon) on a particular day visible at a particular place, and then identify the ret-rodicted eclipse with a recorded eclipse. This has been done regarding the story of a battle between the Medes and the Lydians in what is now northern Turkey. The battle was brought to an abrupt end by a total solar eclipse, which modern science informs us took place in that region on May 28 in 584 b.c.

ecliptic. In consequence, a straight line from the epicycle's center to the center of the ecliptic cut off equal arcs in equal times on the concentric circle, and a line from the apogee of the eccentric deferent to the ecliptic's center also cut off equal arcs in equal times on the concentric circle. But regular motion of the apogee of the eccentric deferent did not make regular that deferent's revolution about its own center. Furthermore, the regular motion of the epicycle's center with respect to the concentric circle did not translate into regular circular motion with respect to the deferent, the circle on which the epicycle's center actually moved.

Modifications oflunartheory necessary to bring it into agreement with observation violated Ptolemy's own requirement of regular circular motion with respect to the center of the orbit. The violation here, however, was obscure and easily ignored, at least relatively so, in contrast to his discussion later in the Almagest of planetary motions.

If all that weren't enough, the motion of the Moon still had to be linked with the motion of the Sun if eclipses were to be accounted for.

After the Moon, Ptolemy moved on to the stars, in Books VII and VIII of the Almagest. And after the stars, he moved on to the planets in Books IX through XIII, the last five books of the Almagest.

Ptolemy accepted the traditional order of the planets, starting with the Earth in the center of the universe and moving outward: the Moon, Venus, Mercury, the Sun, Mars, Jupiter, and Saturn, all surrounded by the outer sphere of the fixed stars.

Next, Ptolemy restated the basic problem of devising a system of regular circular motions: "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 as 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)" (Almagest, IX 2).

The demonstration would be difficult: Ptolemy warned: "The successful accomplishment of this aim as 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). Observations of the planets were available only over short periods, while the error in systematic observations becomes less over longer periods. In addition to uncertain observations, there was more than one anomaly, and the anomalies were intertwined: "In the case of research about the anomalies, the fact that there are two anomalies appearing for each of the planets, and that they are unequal in magnitude and in the times of their returns, works a good deal of confusion. For one of the anomalies is seen to have relation to the Sun, and the other to the parts of the zodiac, but both are mixed together so it is very hard to determine what belongs to each; and most of the old observations were thrown together carelessly and grossly" (Almagest, IX 2).

One of the two planetary anomalies (deviations from uniform circular motion) is a planet's seemingly irregular orbital speed, which is linked to its position relative to the ecliptic. The second anomaly is a planet's retrograde motion, which occurs when a superior planet (Mars, Jupiter, or Saturn, in orbit beyond the Earth) is near opposition (on the opposite side of the Earth from the Sun); thus the phenomenon is related to the Sun. The inferior planets (Mercury and Venus, in orbits between the Earth and the Sun) were observed to retrograde when they were in conjunction with the Sun (in approximately the same direction as the Sun, viewed from the Earth).

Note that in the heliocentric model, the inferior planets can never be in opposition. From the Earth, they are always seen near the Sun. In Ptolemy's

Figure 9.2: Opposition for Inferior Planets. E is the

In a geocentric (Earth-centered) planetary model, an inferior planet (any planet located between the Earth and the Sun) can be in opposition to the Sun (on the opposite side of the Earth from the Sun). In a heliocentric (Sun-centered) model, a minor planet can never be in opposition.

Figure 9.2: Opposition for Inferior Planets. E is the

In a geocentric (Earth-centered) planetary model, an inferior planet (any planet located between the Earth and the Sun) can be in opposition to the Sun (on the opposite side of the Earth from the Sun). In a heliocentric (Sun-centered) model, a minor planet can never be in opposition.

planetary model, however, there was nothing inherent in the geometry to prevent opposition for an inferior planet. He would have to adjust the speeds and starting positions of the inferior planets and the Sun to prevent his model from moving an inferior planet into opposition.

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