## X19o 28o 9934 33o 3344 23o o56 12718

in close agreement with the given entry (127;25°).

In order to find the true longitude of Mars at t, one must add X' to the mean longitude of Mars at the time when the mean anomaly is o°, most recently before t.

In Figure 3, the observer is at O, the center of the deferent is at D, the equant point is at E, the center of the epicycle is at C, the vernal pointis in the direction OV, the mean position of Mars is in the direction OM, the mean Sun is in the direction OS, the mean epicyclic apogee is at Ae, the true epicyclic apogee is at A OS is parallel to CM, and the true position of Mars is at M. There are two equations: q1 is approximated by c3(k'), and q2 is approximated by c6(a) + c4(K')- c5(a).

The entry for the latitude, P(19o, 28o) = +o;51°, results from multiplying the northern latitude (a function of true anomaly) by the minutes of proportion (a function of the true center). The true anomaly at that time computed above was 91;12°. To obtain the true center it is necessary to correct the previously found mean center, k' = 19;34°, by the amount of the corresponding equation of center, -3;3o°; hence, the true center is 16;4° (= 19;34° - 3;3o°). In the tables for the latitude of Mars (Table 73) we find +o;53° for the northern latitude and 57;31 for the minutes of proportion, after interpolation. Their product is +o;5o,48° = +o;51°, in agreement with the text.

Finally, to recompute the entry for elongation, n(190, 280) = 58;46°, one needs to know the true longitudes of the Sun and Mars at time t. As stated above, the true longitude of Mars is the sum of À'(190, 280) = 127;25° and its mean longitude when a is 0° (56;27°). The result is 183;52°. As for the true longitude of the Sun at time t, one must first determine the mean solar longitude at time t. This results from adding the mean solar anomaly found above (87;42°) to the mean longitude of Mars at time t (156;1,23° = 56;27° + 190d ■ 0;31,26,39°/d), where 0;31,26,39°/d is the mean motion in longitude of Mars. The result is 243;43°. Hence, the mean solar anomaly at time t is 243;43° - 90;45° = 152;58°, where 90;45° is the longitude of the solar apogee at that time. The solar equation corresponding to 152;58° is -1;0,43° = -1;1°, and thus the true solar longitude at time t is 242;42° (= 243;43° - 1;1°). Finally, n(190, 280) = 242;42° - 183;52° = 58;50°, in agreement with the given entry.

Table 57. Mean motions for conjunction of the luminaries in collected years

Na, f. 94v: Tabula (. . .) luminarium Rc, f. 47r: Tabula radicum coniunctionium luminarium Ed. 1495, f. C6r-v: Tabula radicum coniunctionis luminarium Ed. 1526, f. 230r-v: Tabula radicum [symbol for conjunction] lumina-rium

In this table there are five columns: see Table 57. The first is for the argument: collected years from 40 to 2000 at intervals of 40y. Thus, all the entries correspond to leap years. The entries in the other columns represent the time (in days, hours, and minutes), the longitude of the luminaries at mean conjunction (in physical signs, degrees, and minutes), the mean lunar anomaly (in degrees and minutes), and the argument of lunar latitude (in physical signs, degrees, and minutes). The time displayed in column 2, headed superatio, is the excess over an integer number of mean synodic months; hence, the entries in this column do not exceed 29d 12;44h, which is the length of the mean synodic month used here. This value is mentioned in Chapter 22 and is explicitly given in Table 59, below. However, the entries in these tables were computed with a higher precision. Consider, for instance, the entry for 2000 years, 1d 18;35h. In 2000 years there are 2000y ■ 365;15d = 730,500 days and in 730,498d 5;25h (= 730,500d - 1d 18;35h) there is an integer number of mean synodic months, which happens to be 24,737. Dividing the time interval corresponding to 24,737 mean synodic months by that number, we find a length of the mean synodic month of 29d 12;44,3,3h, which is exactly the value generally used in Alfonsine astronomy. As was the case for previous tables, the entries for the solar longitude at conjunction, the lunar anomaly, and the argument for lunar latitude do not correspond to the exact number of years indicated in the first column; rather, they correspond to the nearest previous conjunction of the Sun and the Moon. This table presents data that are analogous to what we find in Table 60, and in the comments to that table we offer some sample computations of its entries.

Table 57: Mean motions for conjunction of the luminaries in collected years

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Table 57: Mean motions for conjunction of the luminaries in collected years

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