In a period of anomaly the planet, initially at the apogee on its epicycle, returns to it and travels 6 physical signs; the intermediary tabulated values of A/(t - t0, k) represent the increment of the true longitude at the beginning of each of the days listed. This increment is not counted from Aries 0°; rather, it is an increment over the mean longitude at the time when the mean anomaly was 0°.
By inspection of the table we see that the three equations were derived in the same way according to the expressions:
qL(t - to, k) = [X'(t - to, k) - X'(t - to, k + 10°)]/10, qB(t - to, k) = [p(t - to, k) - p(t - to, k + 10°)]/10, qE(t - to, K) = [n(t - to, K + 1o°) - n(t - to, k)]/1o.
Similarly, the tabulated daily velocity results from the following expression:
vD(t - to, k) = [X'(t - to + 1od, k) - X'(t - to, k)]/1o, whereas the tabulated hourly velocity results from:
The previous expression for vD applies for all planets, except Mercury; since the entries for this planet are given at intervals of 3 days, the above expression turns into:
vD(t - to, k) = [X'(t - to + 3d, k) - X'(t - to, k)]/3.
These quantities are used for interpolation purposes. Chapter 14 provides a succinct explanation of the use of the tables for the planets, and in Chapter 18 we are given a worked example for Mars for November 17, 1447 at 18;40h. Although not stated explicitly, the presentation of the double argument tables is that of a set of templates for computing the planet's true longitude at a given time. The general method to compute the true longitude, the latitude, and the true elongation of a planet by means of Bianchini's tables consists, first, in computing from the tables for mean motions the mean anomaly and the mean longitude of the planet at a given time. Then one has to find the time when, most recently, the mean anomaly was 0° and the argument of center corresponding to that time. Next, one enters the double argument table whose heading is closest to that mean argument of center and looks for the entries for the longitude, the latitude, and the true elongation closest to the time since the moment when the mean anomaly was, most recently, equal to 0° (i.e., subtracting the time corresponding to the number of completed cycles of anomaly). Then one interpolates in the double argument table to take into account the excesses of time and argument of center. The interpolation is greatly facilitated by the tabulated quantities. After these interpolations, the true longitude of the planet is found by adding the resulting quantity to the mean longitude of the planet when its mean anomaly was, most recently, equal to 0°. As an illustration of this method, let us consider the following quantities, t, a(t), V(t), t0, a(to), K(t0), where t is the given time, and t0 is the time when the anomaly was most_recently equal to 0°. From the mean motion tables one finds a(t) and V(t). Then we seek t0: for this we have to subtract from t the time corresponding to a number of complete cycles of the planet's anomaly such that a(t - t0) is the excess of mean anomaly since the time, t0, when the mean anomaly was most recently 0°. In other words, t - t0 cannot exceed the period of the planet's anomaly. To compute the mean argument of center for t0, K(t0), we subtract the longitude of the apogee from V(t0). We then enter the double argument table with t - t0 and K(t0). After suitable interpolations, we find the quantity V, and the true longitude at t, V(t), is V(t0) + V (t - t0, K(t0)). The quantity, V, is the sum of the mean argument of center in the heading (i.e., at time t0), the increment in the argument of center from t0 to t, and the appropriate corrections for anomaly and argument of center. Thus
V(t) = V(t0) + V = V(t0) + K(t0) + AK(t - t0) + c(a, K), where
K(t) = K(t0) + AK(t - t0), and c(a, k) = c(a(t), k) = c(a(t - t0), k).
In the worked example for Mars, we are first told to find in Tables 32 and 33 the entries for 1440y and 6y for the two quantities: time (superatio) and increment in longitude (motus). We are also told to add the number of days in October and 17d 18;40h to the time. Again, we note that Bianchini uses complete days. After adding the corresponding radices for the Incarnation found in Table 37, the resulting values are:
time = 1751d 17;47h = 285d 14;12h (1440y) + 631d 3;15h (6y) + 304d (October) + 17d 18;40h + 513d 5;40h (Incar.)
increment in longitude = 5,18;59° = 1,29;5° (1440y) + 1,37;29° (6y) + 2,12;25° (Incar.).
We are then told to subtract from the time shown above the entry corresponding to 2 periods of anomaly in Table 38, and to add to the increment in longitude the corresponding entry for 2 periods of anomaly. Again, we note that the modus operandi in Bianchini's tables, in order to use the tables as templates, is: subtract the periods in time, and add the corresponding entries for the other quantities. The resulting values are:
time = 191d 21;1h = 1751d 17;47h - 1559d 20,46h (2 periods of anom.)
increment in longitude = 56;27° = 5,18;59° - 1,37;28° (2 periods of anom.).
This means that 191d 21;1h before the time in question (Nov. 17 at 18;40h) the mean anomaly was 0°, and the mean longitude of the planet, X0 = 56;27°. This is exactly what follows from recomputing the longitude of Mars for May 9, 1447, a date for which the recomputed mean anomaly is 0°.
Then we must subtract the apogee of Mars for that time (2,14;31°) from the increment of longitude, shown above, to obtain 4,41;56°. The text refers to this quantity as centrum. We note that k, the argument of center, is 4,41;56° when the mean anomaly, a, is 0°, not that of November
17, 1447. We are then instructed to enter Table 41 with t - t0 = 190d and K = 4,40° = 280°, the closest values to the computed data found in the table. The corresponding entries in the double argument table are:
V (190, 280) = 2,7;25° qL(190, 280) = -0;7,12° vD(190, 280) = 0;34,36°/d vH(190, 280) = 0;1,26°/h P(190, 280) = +0;51° qB(190, 280) = -0;0,6° n(190, 280) = 58;46° qE(190, 280) = +0;9,18°.
Next come rules for interpolation: we are told to consider the excess of argument of center over 4s 40° (1;56° = 2°) and to multiply it by qL(190, 280), qB(190, 280), and qE(190, 280), to obtain the corresponding corrections to the other quantities:
A' V = -0;14° = -0;7,12° ■ 2 A' p = 0° = -0;0,6° ■ 2 A'n = 0;18° = +0;9,18° ■ 2, where A ' V, A 'P, and A 'n, are the excesses of longitude, latitude, and elongation, respectively. These corrections have to be added to V ' (190, 280), P(190, 280), and n(190, 280), respectively. We also have to consider the excesses of time over 190d, in days (1d) and hours (21;1h), and to multiply them by vD(190, 280) and vH(190, 280), respectively, to obtain:
correction due to the excess in days = 0;34,36° = 0;34,36°/d ■ 1d correction due to the excess in hours = 0;30° = 0;1,26°/h ■ 21;1h.
These two corrections add up to a correction due to the excess in time, q(A 't) = 1;5°, and it has to be added to V(190, 280), p(190, 280), and n(190, 280), respectively. Finally, the true longitude of Mars (on the 9th sphere) results from adding V' (190, 280), the excess of longitude (-0;14°), and the correction due to the excess in time (1;5°) to the longitude of the planet when the mean anomaly was 0° (56;27°):
3,4;43° = 56;27° + 2,7;25° - 0;14° + 1;5°, or, in general,
V = V + V (t - t0, K) + A' V + c1(A't), where t - t0 is the argument in the rows to the nearest day less than the time deduced from the time in question and the time when the mean anomaly was most recently 0°, and K is the argument of center for the columns which is the nearest multiple of 10° less than the argument of center at the time when the anomaly was most recently 0°.
Similarly, we find the true latitude of Mars by interpolating from P(190, 280) both in time (mean anomaly) and in argument of center. The excess in latitude due to the excess of argument of center is 0°, as mentioned above, and the excess in latitude due to the excess of time (1d 21;1h), c2(A't), is also 0°, according to Bianchini (the increment in latitude between 190d and 200d amounts to 0;6° = 0;57° - 0;51°). Hence, the true latitude is
or, in general, p = P(t - t0, K) +A'P + c2(A't), where t - t0 and K are defined as before.
In order to find the true elongation, that is, the difference between the true longitude of the Sun and that of the planet, we are told to find the excess of elongation due to the excess of time (1d 21;1h), which according to Bianchini is c3(A't) = 0;50,9° (the increment in elongation between 190d and 200d amounts to 4;27° = 1,3;13° - 58;46°), and to add it to the excess of elongation due to the argument of center, A'n = 0;18°, as mentioned above. Then, the true elongation of Mars at the given time results from adding these two excesses to n(190, 280), and the result is
or, in general, n = n(t - t0, K) + A'n + c3(A't), where t - t0 and K are defined as before.
Using the standard Alfonsine Tables, we have recomputed the entries for t - t0 = 190d and K = 4,40° = 280° in this table, namely, V(190, 280) = 2,7;25°, P(190, 280) = +0;51°, and n(190, 280) = 58;46°. In 190 days the increment in longitude is
and we define K' such that:
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