________________
10
EPIGRAPHIA INDICA.
[VOL. XIV.
always 21600'. The formula then for all authorities, a being the angle of mean anomaly, is : Equation centrel = minutes in epicycle x sina
21600 252 A. The First Arya-Siddhānta gives for the dimension of the epioycle 13° 30' or 810'. Hence by that authority -
810 Equation centre = 21600 sin, a. = go sin. «. Since there are 3° 45' between each base-angle, the difference in minutes between each is 225', and the measure of first or average difference of equation for each intermediate minute of anomaly is the difference between two consecutive equations divided by 225. Taken in seconds, this difference is given in col. 6. Multiply the minutes of difference between the base-angle and the given anomaly-angle by the amount given in col. 6, and, taking the result in seconds, apply it to the base-equation, and you have the correct equation for the given anomaly-angle.
For an example take the 2nd and 3rd sines. The 2nd sine, i.e. of anomaly-angle 7° 30', is 449'. Multiply by 3 and divide by 80. Result 0° 16' 50"-25,
The 3rd sine, of anomaly 11° 18', is 671'. Multiply by 3 and divide by 80. Result 0° 25' 9"-75.
'The difference between the two results is 8' 1950. This is the total difference in 225'. which is the difference between the two anomaly-angles. 8 19.50 divided by 225 gives for each minute of angle the increment 2.22.
B. Equation of the centre by the Sürya-Siddhānta.-This calculation is made on the same fundamental principle.
The Surya-Siddhanta (cf. Jacobi, abore, I, 441) assumes a contraction of the epicyclo amounting to 20' at the end of each of the odd quadrants. If this contraction at any point is called q, we have q : 20' :: sin. a : sin. 90°. :.9=20 57
30...9=20 ans. sin. 90o=3438' (see Table XLVID.
20' Hence 9= 3438
sin, a, The Sürya-Siddhānta gives for the dimension of the epicycle 14o.
Hence the formula for the equation without the contraction would be 26sin. a. With the
146
contraction it is 26 sina 1: 14 20
?
sin. a 360 3 438' x 21600- sin.' a; or, finally
3713040 The best anthorities agree that this is the correct formula
Ench equation for the several base angles has been calculated by this formula and fully worked out for nine decimals of a second. The results are given in fall in Table XLVIIA, col. 7, and in abbreviated form in Table XLVII, col. 7. The difference in equation per minute of anomaly-arc has been calpulated by dividing the difference between consecutive base-equations in minutes by 225, and taking the result in seconds. This is tabulated in full in Table XLVIIA, col. 8, and in abbreviated form in Table XLVII, col. 8.
253 C. Equation of the centre by the Second Arya-Siddhanta and Siddhānta-Siromaņi.The same fundamental principle holds good. The epicycle is (Epig. Ind. I. 341) 13°
When an angle is very small, as is the case with even the greatest of the equation-angles, which is only about 2° 10', the sine is taken to be equal to the are. Hence the presumed equality in the text of "sin. equation and "equation centre." Table XLVII shews that the side of 8° 45' is 225', the mudem the are. The vine of 1 is 60', also the same w the arc.
