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COMPUTATION OF DATES.
The motion in seconds in one solar year, according to the Surya Siddhanta, is thus 01161; similarly for the 2nd Arya Siddhanta it is 0"1383, and for Brahma Siddhanta 0.144. Subtracting the amounts for 3000 years from the sun's mean anomaly for OK.Y., we find the same for 3000 K.Y., viz. (1) 282° 46′ 24"; (2) 282° 7' 29"; (3) 282° 7' 12"; as entered in Table XIII in the column headed O's an.
441
76. The tables for the equations of the centres of the sun and moon are calculated from the epicycles. Their dimensions are the following:
According to Arya &. 31° 30'
13 30
2nd Arya S. 31° 34'
13 40
Brahma S. 31° 36′
18 40
Epicycle of the moon Epicycle of the sun. Now according to Hindu astronomy, sin. eq.: sin. an. :: minutes in the epicycle: minutes
in the orbit.
In all these calculations the Hindu sines have to be used. Thus we find e.g. the eq. for 's an. =45° (sin 45°-2431), according to the first Arya Siddhanta, 212'71= 3° 32′ 43′′; according to the second Arya Siddh. 213'65-3° 33′ 39′′.
3438 20 x Bina
77. The epicycles of the moon and sun, according to the Surya Siddhánta, have circumferences of 32° and 14° respectively, and are assumed to contract at the odd quadrants by 20'. The amount of the contraction at any other point, say at anom. a, is 20x ; hence the equation of the sun's centre for anomaly a is sin a 3439 x 300 x Which formula will be found convenient for the calculation of the table. This has been done by Davies (48. Res. vol. II, p. 256); I have taken Davies' tables from Warren's Kala Sankalita, Tables XXII and XXIII.
78. The General Tables yield approximately correct results with the smallest amount of calculation; but they do not conform strictly to the data of any Siddhanta, but are based on the European tables of Largeteau" with this difference that while Largeteau expresses the mean distance of sun and moon, a, in 10,000th parts of the circle, these tables furnish the same element, called tithi, in 30th parts of the synodical revolution. But the mean anomaly of the moon is expressed in the same way in both. For 3200 K.Y.99 A.D. 18th March, Largeteau's tables give a-moon's age 2575, and 6-857, for mean midnight at Paris. Reducing this for mean sunrise at Lanka we must add the increments of a and b for 1h 6m, viz. 15 and 2, which give a 2590 and b 859. From a we subtract 200 (the sum of the equations of and at their maximum), multiply by 30, and divide by 10,000; which gives 7.17 the required tithi for 3200 K.Y. as in Table I. The value of b found above, 859," is transferred to column c's an. of Table I without further change. The same elements in Table II can easily be derived from Largeteau's Table for the years of the 9th century, attention being paid to the leap years.
47 Additions a la Connaissance des Temps, 1846, pp. 1-29, containing Tables pour le calcul des Syzygies écliptiques ou quelconques; par M. C. L. Largeteau. These short tables are founded on those of Delambre for the sun and of Damoiseau for the moon, and take only the larger equations into account. M. Largeteau uses six quantities in his tables, but does not explain what each indicates; they are,-a moon's age (or distance from the sun) in 10,000ths of a lunation-300 (sum of negative equations); b= moon's mean anomaly (Hansen's g); e2a-b; d= sun's mean anomaly (Hansen's g'); e= moon's distance from the Node or Hansen's g+w; and fun's distance from Moon's Node or 2e2a (that is Hansen's 2g+2). The last four quantities are given in 1000th parts of the circumference. Similar handy tables, but sexagesi mal, and with more equations were published in the seventh edition of the Encyclopædia Britannica, and others in Guznmere's Astronomy (Philadelphia 1858).-J. 13.
If the degrees in 'Distance O' Table XIII, &c., be multiplied by 30 we obtain this element a according to the different Sid. Antas; thus for K.Y. 3200 we have 89° 6 x 30 = 2688; or if we divide the same by 12, we have 7'47 tithi. Again for b, from Table XIII, 1320 10 x 100+ 36 367, and 367-500= 867, differing by about 3° from the European value. Hansen's Tables de la Lune give for the value of the tithi here, 7:1637 and for 's anom. 858-11.-J. B.
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