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* Ed. 1526 has 0. The typesetter omitted one entry in this column, causing an upwards shift of all subsequent entries and leaving nothing for the last row; hence he entered a '0'. The other entries in this column have silently been corrected to agree with the entries in the manuscripts.

* Ed. 1526 has 0. The typesetter omitted one entry in this column, causing an upwards shift of all subsequent entries and leaving nothing for the last row; hence he entered a '0'. The other entries in this column have silently been corrected to agree with the entries in the manuscripts.

Bianchini's canons give no hint of the way this table was constructed. The following method to recompute the entries is entirely based on Bianchini's tables but, although yielding good results, we cannot claim that this was the method used by the author. First, we should keep in mind that, as can be seen in Table 8, the yearly progress of the solar apogee ranges from 0;0,55° (around epoch) to 0;0,20° (around year 2000), and it is about 0;0,34° in Bianchini's time. Whatever the yearly increment, AAa, it implies a variation of the solar equation in a year, Ae, which in turn depends on the solar position within the year. To determine Ae corresponding to AAa, let us consider two instants approximately 1 year apart, when the true solar longitude returns to the same value, A. Between these two instants the true anomalies differ in AAa. If we take the mean anomalies to differ also in AAa, then Ae can be derived by interpolation from a table for the solar equation for the value of the mean anomaly, k, corresponding to a true anomaly, k = A - Aa. As can be seen in Table 8, in Bianchini's time Aa was about 90;40°. The value obtained for Ae then has to be converted into hours by dividing it by the hourly solar velocity taken from Table 11. Thus, for example, for A = 60° and for years 1444 and 1445, when the solar apogee was at 1,30;39,8° and 1,30;39,42°, as indicated in Table 8, AAa = 0;0,34h. The corresponding true solar anomalies were 329;20,52° and 329;20,18°. For these values of anomaly approximately 1 year apart, the increment in solar equation, Ae, is 0;0,1,3°, as derived from the table for the solar equation. Now, the solar velocity, v, for a solar anomaly of 329° is 0;2,23°/h, as indicated in Table 11. Thus, Ae/v is = 0;0,27h (text: 0;0,28h). It is entirely possible that the author only computed a few selected entries and filled in the rest by interpolation.

Table 92. Excess of revolution for the Sun in collected years

Nu, f. 96v: Tabula revolucionum secunda Rc, f. 79v: Tabula prima solis

Ed. 1526, ff. 387r: Tabula prima [symbol for the Sun] in revolutionibus annorum

This table gives the number of days, hours, minutes, and seconds that exceeds a number of years of 365 days, for collected years from 40y to 2000y at intervals of 40y. The first entry, for 40y, is 0d 7;9,21h, and differs by 1 second from that in Table 90. Curiously, the heading of this table in Regiomontanus's copy reads 'in annis expansis' instead of 'in annis collectis'.

Table 93. Excess of revolution for the Sun due to solar anomaly

Nu, f. 96v: Tabula revolucionum secunda Rc, f. 79v: Tabula 2 in revolutionibus

Ed. 1526, ff. 387v-388r: Tabula secunda [symbol for the Sun] in revo-lutionibus annorum

This table gives the number of days, hours, minutes, and seconds as a function of the solar anomaly, from 1° to 180°, and from 180° to 359°, at intervals of 1°, where each argument and its complement in 360° are given in the same row: see Table 93A. The maximum, 2d 4;46,17h, occurs at anomaly 92°-94°. We have recomputed the entries in this table by finding the quotient of the solar correction in the Parisian Alfonsine Table divided by the mean motion of the Sun, 0;59,8,19,37°/d, the value in these tables. The differences between text and computation as shown in Table 93A are less than 0;1h in all but one case, and differences of this magnitude have no significance here, although we are aware that the column for the differences, T(ext) - C(omputation), exhibits some sort of a sinusoidal component. We note that the computer of this table did not consider a variable motion for the Sun. The entry is the time it takes the Sun to travel the distance in longitude between its mean position and its true position as a function of the solar anomaly.

In the second worked example presented in Chapter 40 we are asked to consider as epoch Aug. 25, year 141 at 4h. We are told that the mean longitude of the Sun at that time was 2,33;5,1°, and that the solar apogee, the solar anomaly, and the true longitude of the Sun were 1,13;27,40°, 1,19;37,21°, and 2,30;58,35°, respectively. The problem here is to find the time of year 1450 when the Sun was in the same position as at the epoch. These two dates are separated by 1309 years. In Table 92 we find the excess for 1280y (9d 12;59,37h) and in Table 90 that for 29y (0;48,44h). The entry for 1280y has the letter M, for minue (subtract). The entry for 29y is to be found in the column headed 1, corresponding to the first year after a leap year, as is the case for year 141, and has the letter A, for adde (add). Subtracting the first entry from the epoch and adding the second entry, we obtain Aug. 15 at 15;49,7h. This time has now to be corrected because of the change in solar anomaly. In order to do this, we are asked to determine in Table 93 the excesses corresponding to two values of the solar anomaly. The first is the anomaly at epoch (1,19;37,21°) for which the entry in Table 93 is 2d 3;22,32h, after interpolation (2d 3;22,25h according to the worked example). The second is the anomaly that results from subtracting the mean longitude of the Sun at epoch (2,33;5,1° = 1,13;27,40° + 1,19;37,21°) from the value of the solar apogee for 1450 (1,30;42,33°, as deduced from Table 8), and it

Table 93A: Excess of revolution for the Sun due to solar anomaly

Anomaly

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