The Motion of the Sun in the Vasis..tha SiddhSnta

GEORGE ABRAHAM

Communicated by B. L. VAN DER WAERDEN

KUPPANNA SASTRI1 translated from the Paficasiddh~ntik~ 2 II,1 a table of times t the sun needs to cover the 12 signs of the zodiac, in quarter-days, namely 126 diminished by

1, 0, 0, 0, 2, 4, 7, 9, 9, 8, 6, 5

according to Vasi.st.ha-Siddh~nta. He thinks these values would have been obtained empirically, most probably by an analysis of eclipses. However the usual method in Babylonian, Greek, Hindu and Arab astronomy is to start from a theoretical model (piecewise linear or epicyclic motion), to determine the constants empirically (e.g. by means of eclipses.) and finally to compute a table from the model. VASlS.THA himself in calculating the motion of the moon, uses a model, namely a quadratic function (SP-665)P /63 , according to KUPPANNA SASTRI (page 26).

Assuming a theoretical model for the sun also, we shall try in this paper to find out the model used.

Summing up the times t, we obtain the time T in quarter-days the sun takes to reach the initial points of the signs.

Let T o be the time in quarter-days for the sun in uniform motion to reach the same initial points. Then To= 14612/360, (2=longitude) because the mean sun moves 360 ° in 1461 quarter-days, according to the text.

Let D = 4 ( T - T o ) . Then the values of D at the beginning of each sign are

2 0 30 60 90 120 150 180 210 240 270 300 330 360 D 0 13 30 47 64 73 74 63 44 25 10 3 0

Plotting D as a function of 2, we see that the points lie on a smooth curve with one exception at sign 11 with deviations of the order of magnitude 1 unit due to the rounding of the given data to multiples of a quarter-day. We should expect larger deviations in empirical data because an error of ½ degree in 2 at the time of an eclipse would cause an error of 8 units in D.

The symmetry of the curve suggests that the value 3 of D for 2 = 330 should be - 1, i.e. the corresponding t should be 7 instead of 6. Then the maximum of the curve is 75, the minimum - 1 ; hence the mean is 37. A line drawn at height

Archive for History of Exact Sciences, Volume 22. © by Springer-Verlag 1980

2 G. ABRAHAM

37 intersects the curve in two poin ts exact ly 180 ° apar t , which again suggests der iva t ion from a model . The po in ts of in tersect ion at 71 ° and 251 ° are obvious ly the assumed apogee and perigee.

The true a n o m a l y is x = / l - 71. Let u be the mean anomaly , w the difference u - x , and T, the t ime at which the sun reaches apogee. Since u is the m o t i o n of the mean sun, we have

(1)

360 ( 2 - 71) = 1~601T- 2 + C w=u- x = l - - ~ ( r - T°)-

360 360 D - - - + C 1461 ( T - To)+ C = 1461 4

where C is a constant . At apogee, w = 0 and D = 3 7 , so (1) becomes

360 37 (2) 0 = ~ C.

1461 4

Sub t rac t ing (2) f rom (1) we ob ta in

( D - 37) 360 90 (3) w = 4 1461 - 1461 (D - 37).

If we p lo t w as a funct ion of x or of u, we get someth ing l ike a sine function, with a l i t t le a symmet ry abou t the m a x i m u m and min imum. W e have therefore fi t ted a curve of the form

(4) W' = (a + b u') sin u,

where u' is the angle in rad ians co r r e spond ing to u - 9 0 ° for 0 < u < 1 8 0 ° and

270 ° - u for 180 ° < u < 360 °. This would co r re spond to an epicycle with var iab le radius

p=a+bu'.

If w are the given values from (3) co r re spond ing to the values w' of the curve (4), 12

we find a and b by min imiz ing the sum of the squares of the devia t ions ~ (w~ - w' , ) 2 . , = 1

W e ob ta in

a = 2 . 2 8 °, b - 0 . 1 2 °.

The values of w and w' are given in the fol lowing table.

x - 7 1 -41 - 11 19 49 79 109 139 169 199 229 259

w - 2.28 - 1.48 - 0.43 0.62 1.66 2.22 2.28 1-60 0.43 -0 .74 -1 .66 -2 .10

u -73 .3 -42 .5 -11 .4 19.6 50.7 81.2 111.3 140.6 169.4 198.3 227.3 256.9

w' - 2.16 - 1.47 - 0.42 0.71 1-70 2.23 2.16 1.52 0.45 -0 ,76 -1 .74 -2-25

The Motion of the Sun in the Vasis.t.ha Siddhgnta 3

Compar ing w' with w, we see that the mot ion of the sun in the Vasist.ha follows the formula (4) very closely. However there is no explicit use of t r igonometr ic methods elsewhere in the Vasistha.

I am indebted to Professor B. L.VAN DER WAERDEN for his guidance.

References

1. KUPPANNASASTRI, T.S., The Vasistha Sun and Moon in Var~hamihira's Pafica- siddh~ntik~, Journal of Oriental Research, 25, 1957, p 19-41.

2. NEUGEBAUER, O., & D. PINGREE, The Pafichasiddh~ntik~ of Var~hamihira, Kongl. Danske Videnske. Selsk. Hist.-Fil. Skrifter, 6, 1, Part I, p. 35, Part II, pp. 15-17.

$66 Annanagar Madras 600040

(Received February 13, 1980)