History Online - Astronomy

Example 2. Consider eclipse 25 (see pp. 189-190) described in the History of Rome by Livy (Bk. XXXVII, 4.4). The characteristics extracted from Livy's text are as follows.

(1) The eclipse was solar.
(2) It occurred 5 days earlier than the ides of July, i.e., on July 10.
(3) The approximate coordinates of its observation zone were 30° ^ lat. N. < 45° and 10° <, long. E. <, 25°.
(4) In the observation zone, the moon's trajectory passed below the centre of the sun during the eclipse if the moon and sun were projected on the celestial sphere.
The traditional solution suggested in Ginzel's canon was March 14, 190 B.C. However, since condition 2 was not fulfilled, the astronomers also offered other astronomical solutions, e.g., July 17, 188 B.C. But the conditions of the problem posed were not satisfied in this case either (p. 190). Owing to the absence of other astronomical solutions for the time interval 300-100 B.C., determined beforehand due to the a priori requirements of tradition, and which would satisfy conditions 1-4 better, the traditional one of 190 B.C. was retained in the canon.

Meanwhile, the application of the formal astronomical dating method and extension of the interval in which an exact solution was being sought to the periods from 600 B.C. to A.D. 1600 permits us to reach the following conclusion .

(1) As it turns out, there is an eclipse fully satisfying all the conditions of the problem.
(2) This exact solution is unique for the interval from 600 B.C. to A.D. 1600.
(3) It is July 10, A.D. 967.
(4) It is stable with respect to a small perturbation of the initial data, namely a perturbation of the principal condition 2, which means that it remains unique in extending the search interval from July 10 to July 9 and 11, i.e., by one day. This exact solution was found in assuming, naturally, that the Julian names of the months correspond to the Julian calendar.

An example: the eclipse described by Livy and Plutarch

Example 3. Consider the list of descriptive characteristics of eclipse 27 (see p. 190) also described in Livy's History of Rome, Bk. LIV, 36.1. See also Plutarch's Vitae Aemilius Paulus, 17.

(1) The eclipse was lunar.
(2) It occurred on the night of September 4 to September 5.
(3) The observation zone was bounded by lat. 40° and 50° N, and long. 10° and 25° E.
(4) It occurred from 2 to 4 A.M. local time.
(5) Its phase was close to 12', and possibly exceeded 12'. Remark. The phase of a solar eclipse is found by the formula $ = 12A, where A is the ratio of the part of the sun's diameter, covered by the moon at the eclipse's maximum, to the whole diameter. However, in the case of a supertotal lunar eclipse, a quantity proportional to its duration is added to the phase of 12' (the moon stays in the shadow of the earth for a long time). Hence, the phase of a lunar eclipse can reach 22'7.
(6) This lunar eclipse occurred after the summer solstice. The traditional solution given in the canon (p. 190) is June 21, 168 B.C. This does not satisfy conditions 2 and 6 of our problem. Attempts of many authors to find a better astronomical solution for the interval from 300 to 100 B.C., determined a priori from the requirements of tradition, did not lead to positive results. Omitting the details, we should note that the situation is perfectly similar to the one described in Examples 1 and 2.

Application of the formal astronomical dating method and extension of the search time interval from 600 B.C. to A.D. 1600 permit us to draw the following conclusions .

(1) There exist exact astronomical solutions fully satisfying all conditions 1-6.
(2) There are only three exact solutions for the time interval from 600 B.C. to A.D. 1600.
(3) These solutions are (a) the night of September 4 to September 5, A.D. 415, (b) the night of September 4 to September 5, A.D. 955, and (c) the night of September 4 to September 5, A.D 1020 (Vol. 5, pp. 266-272).
(4) For a small perturbation of the initial data, i.e., while considering lunar eclipses occurring not only at night but also at sunset, there arises only one more possible solution, (d) the night of September 4 to Septemer 5, A.D. 434.

Yet another solution is theoretically possible, namely, the lunar eclipse of September 4 to September 5, 106 B.C.; however, it possesses the phase 5'9, which is far too small. If we select those with greatest phases from the above astronomical solutions (see condition 5 of the problem), then two of them are ideally suitable, namely the eclipse of A.D. 955 with the phase of 16'1 and that of A.D. 1020 with the phase of 18'7, the latter being still more adequate than the former.

Upon further perturbation of the initial data of the problem, namely, also considering the night of September 3 to September 4, four other new astronomical solutions present themselves; however, they all relate to the medieval period, occurring in A.D. 453, 936, 1457, and 1476. A perturbation of the initial data in the other direction, namely, considering the night of September 5 to September 6, is impossible, which follows from Livy's text.