The Motion of the Moon in the Romaka Siddhanta

GEORGE ABRAHAM

Communicated by B. L. VAN DER WAERDEN

Summary

Formulae are fitted to the different sets of values, given by the translators, for the equation of centre for the motion of the moon according to the Romaka Siddhhnta, as found in the Paficasiddhfmtikft of VARAHA Mining.

Introduction

In an earlier paper [1], formulae were fitted to the sets of values which describe the true motion of the sun according to the Romaka, Pauli~a and Vasi.st.ha Sid- dhfmtas, as found in the Paficasiddhfmtikfi, which gives an explicit formula only for the Sfiryasiddhfmta.

For the moon's motion, the Paficasiddhfintikft gives the formulae of the SQrya and Vasi~t.ha Siddhfintas. and a set of values of the equation of centre for the Romaka Siddh~nta. In this paper the Romaka numbers are analysed in the same way as for the sun, and likely formulae are suggested.

Translators of the Paficasiddh~ntik~t [2, 3, 4] agree on the numbers in verse 3 of chapter VIII, dealing with the true motion of the sun. But there are wide differences in the understanding of the moon's motion, in verse 6 of chapter VIII, with respect to the values of the equation of centre. The translation of THIBAUT and DVWEDI and that of PIN6REE have only degrees and minutes, while SASTRI has degrees, minutes and seconds, and all disagree on the numbers. SASTRI com- ments:

"'The intervals of the equation of centre are given in minutes and seconds, as in the case of the sun, with the special mention of degrees where there are full degrees. But the text here is so corrupt that we are not certain about the numbers. So we have to depend much on guessing."

326 G. ABRAHAM

The Translations

Verse 4 o f chap te r VI I I gives the fo rmu la for the mean longi tude l, and verse 5

the fo rmu la for the a n o m a l y x = i - - m, m being the longi tude o f the m o o n ' s apogee. In o rder to get the t rue long i tude l o f the moon , we need the co r rec t ion

t e rm y = T - I ca l led the equa t ion o f centre. Verse 6 gives the first differences Ay o f y, for the values x = 15 °, 30 °, 45 °, . . . 90 ° o f the anomaly .

THIBAUT and DVlVEDI'S t r ans la t ion o f verse 6 is:

One degree plus 14, 11 and 2 (minutes) ; four t imes eighteen (i.e. 72) lessened by e ight t imes three (24); five t imes six (30); and sixty minus eight t imes six (12). The las t two quant i t ies are to be lessened by one.

PINGREE t rans la tes this verse as :

A degree plus 1 4 ( = 1 ° 14'), 11 ( = 1 ° "11) and 2 ( = 1 ° 2 ' ) ; 4 t i m e s 18 minus 8 t imes 3 ( = 48 ' ) ; 52 ( = 25') and 6 t imes 16 minus 90 ( = 6 ' ) ; ( these are) used with the moon .

KUPPANNA SASTRI gives a very different ve rs ion :

F o r the half-signs o f a n o m a l y the intervals o f the equa t ion o f centre are 1 ° + 14' + 25", 1 ° + 11' + 48", 1 ° -k 2 ' - 9" , 48 ' - 15", 48' - 18' - 0" , 4 8 ' - 1 8 ' - 2 0 ' - 1" (i.e. 1 ° 1 4 ' 2 5 ' ' , 1 ° 1 1 ' 4 8 ' ' , 1 ° 1 ' 5 1 ' ' , 4 7 ' 4 5 " , 3 0 ' 0 " , 9' 59").

These three sets o f values o f Ay and the co r r e spond ing y are col lected in Table 1.

Table 1

x 15 ° 30 ° 45 ° 60 ° 75 ° 90 °

THIBAUT

PINGREE

SASTRI

Ay 1 ° 14' 1 ° 11' 1 ° 2' 48' 29' 11' y 1 ° 14' 2 ° 25' 3 ° 27' 4 ° 15' 4 ° 44' 4 ° 56'

Ay 1 ° 14' 1 ° 11' 1 ° 2' 48' 25' 6' y 1 ° 14' 2 ° 25' 3 ° 27' 4 ° 15' 4 ° 40' 4 ° 46'

Ay 1 ° 14' 25" 1 ° 11' 48" 1 ° 1' 51" 47' 45" 30' 9' 59" y 1 ° 14' 25" 2 ° 26' 13" 3 ° 28' 4" 4 ° 15 ~ 49" 4 ° 45' 49" 4 ° 55' 48"

The Moon in the Romaka Siddh~nta 327

The Formulae

I follow the same procedure as for the Romaka sun, and take for FOURIER Analysis the formula

5

(1) Y = ~a b2r+l sin {(2r + 1) x}, r = 0

where fl = 15 °, andf(kfl) are the given values of the equation of centre, Table 1. The results are shown in Table 2, in minutes of arc.

Table 2

bl ba b5 b7 b9 bll

THIBAUT 293.65 -- 1.71 --0.63 --0.24 +0.29 --0.41 PINGREE 290.69 +0.90 --2.64 + 1.08 --0.43 --0.03 SASTRI 295.00 -- 1.34 --0.59 --0.24 --0.21 --0.01

These coefficients suggest formulae of the type:

I. y = a s i n x ; II. y = a s i n x - b s i n 3 x

where a and b are integral numbers of minutes. For the PINGREE coefficients in Table 2, formula II would not be suitable because of the large values of b5 and b7. Table 3 gives the best values of the formula I, the last column being the standard errors.

Table 3

a s.e.

Tm~ArJT 294 1.5 PINGREE 290 2.5 SASTRI 295 1.1

For the formula II, the best values are

(2a) 294' sin x - 2' sin 3x, s.e. = 0.7',

(2b) 295 'sin x - 1' sin 3x, s.e. = 0.6'

the first for the TrlIBAUa" and DVIVEDI numbers, the second for those of SASTRL

328 G. ABRAHAM

AS in the case of the sun, I have also tested the SrQyasiddhfinta formula (verses 7, 8, chapter IX)

31 (3) sin y = ~ sin x .

The standard errors for the three sets of numbers are 2 . 2 ' , 5 • 3' and 1 • 4'.

Conclusion

The formulae (2a), (2b) provide the best fit. The agreement with the simple formulae tested in this paper is the best for the SASTRI set of numbers. I t is inter- esting to compare the best fit obtained by me [1] for the Romaka sun

(4) y = 141' sin x - 2' sin 3x,

similar to (2a), (2b) above.

References

1. G. ABRAHAM, Theories for the Motion of the Sun in the Paficasiddhfmtikfi, Arch. for History of Exact Sciences Vol. 34, Nr. 3, 1985.

2. G. THmAUT & M. S. DVIVEDI, The Paficasiddh~mtikfi of Varfihamihira, Benares, 1889. 3. O. NEU6EBAUER & D. PINGREE, The Paficasiddhgntikfi of Varfihamihira, Danske

Vidensk. Selskab. Histor.-Filos. Skrifter 6, 1, 1970 and 6, 2, 1971. 4. T. S. KUPPANNA SASSTRI, Translation with notes of the Paficasiddhfintikfi, to be

published by the Indian National Science Academy, Delhi.

Plot 1520, 12 th Main Road Annanagar, Madras 600040

India

(Received October 7, 1985)