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No. 1.] THE TRUE LONGITUDE OF THE SUN IN HINDU ASTRONOMY.
256. In calculating the true sun's correct longitude and equation for each day for the preparation of Tables XLVIIIA and B I have obtained the equation by using the first or average difference in seconds as given in Table XLVII, cols. 6, 8, for each minute of anomalyangle between the base-angle of the Table and the given angle, in the belief that this represents the practice of the Hindus in bygone centuries. It is possible to calculate with still greater minuteness. We might perhaps be able, by use of some complicated. formula, to find out a more exact value of the difference in seconds applicable to the anomaly-angle under consideration; but this system would be so troublesome that it may be reasonably assumed to have never been adopted.
256 a. An example will best illustrate how each calculation for the 24-hour periods given in Tables XLVIII aud XLVIIIA was made. The value of the equation is based on the angle of mean anomaly, c, given in col. 2. The base-equation used is that for the base-angle next lower in the sine-table (XLVII, col. 5 or 7), the increment in the equation for the difference in angle between the base-angle and the given angle of anomaly being found by multiplying that difference in minutes and decimals by the amount given (col. 6 or 8) in seconds (this being the equation-difference per minute of anomaly-difference). The increment is added to or subtracted from the base-equations according as the consecutive base-equations are increasing or diminishing. The result is the exact equation for the given anomaly-angle, and this is entered in Table XLVIIIA or B, cols. 6, 7. This equation is added to or subtracted from the meau longitude of the sun (Table XLVIIIA or B, cols. 4, 5), and the result is the sun's true longitude, s (cols. 8, 9). The heading of the sine-Table (cols. 2, 11) shews whether the equation is plus or minus.
For an example I take Day 27 and work by the Arya-Siddhanta, using only the number of decimals given in my Tables.
Mean anomaly (Table XLVIIIA, col. 2). Next-lower base-anomaly (Table XLVIL, col. 2)
Difference
2° 44' 164'. The multiplier per minute of difference is (col. 6) 1".31. 164'-72124 x 1"-31-215-7848244. 215" 3' 35". Hence
Base equation for anomaly 123° 45' (Table XLVII, col. 5) Difference in equation above found, deducted since the values in col. 5 are diminishing
Exact equation for given anomaly
Sun's mean longitude (Table XLVIIIA, col. 4) Equation found (for sign column-heading)
126° 29'72124 -123 45
•
13
2° 44'72124
1° 47' 12"-75
3 35-7848244
1° 43′ 36"-9651756
24° 29′ 43′′-27
+1 43 36 97
Exact value of sun's true longitude, s
26° 13' 20"-24
This is converted into 10,000ths of the circle by Table XLVA, and both values are entered in cols. 8, 9, of Table XLVIII. Work by the other Siddhantas is precisely the same, the base. equations and multipliers being used, each set for its own authority.
In this way every figure of equation and true longitude has been worked out for every ay of the year.
