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232311168 511 893 267 63 584 Mars, 2296828522 Jupiter, 364226455 Saturn, 146567298 Eqninoxes, 199669 Days, 1577916450000 .. .. •• .. •a •• .. .. The following numbers, which are of use in com- putation, are derived from those in the above table. Solar or Sanra months, in 4320000000 years, = 51840000000 Lonations in ditto, 57753300000—4320000000 s 53433300000 Intercalary lunations, = 53433300000 — 51840000000 sr 1593300000 Tithis, or lonar days, = 53433300000 x 30 ==1602999000000 Intercalary Tithis, 1,602,999,000,000. 1,577,916,450,000 = 25,082,550,000 Sideral days, 1577916450000 + 4320000000 =1682236450000 Having thus given a complete view of the manner in which the first and most ancient of the modern astronomical systems of the Hindm was framed, it Digitized by Google OF THE HINDUS. 93 must, I believe, be sufficiently obvious to any per- son acquainted with computation, that the positions of the planets given by such system, must necessa- rily be nearer the truth at the time it v^ras framed, than at any other distant period, either before or after. For, though entire numbers cannot be made to give exactly the positions of the planets according to observation, as maybe seen by the example above given respecting the number of revolutions of Venus, which gives the position of that planet about 6' 41" too great, yet still, the errors upon the whole, will be less then than at any other distant time : and this self-evident principle will be found to exist, not only in Hindu astronoliiical works, but also in all astronomical tables whatever ; for every astronomer, whether his system or tables be real or artificial, must necessarily endeavour to give the positions of the planets as correct as he can, at least in his own time ; for otherwise they would be of no use. Let us, therefore, now, apply this principle in de- termining the antiquity of the above system, in the same manner as if we met with it by accident, and did not know when it was framed. For this purpose there are two methods. The first is, to determine the error in the positions of the planets at some fixed point of time, and then to divide the errors so found, by the errors or difierences in the mean motions; the mean result will point out the time sufficiently near for our purpose. The second is, to determine the errors in the positions of the planets at different pe- riods, till one is found at which the sum of the errors is the least possible, and after which the errors again begin to increase. The period when the errors are least, is the time the system was framed, or very near it. Digitized by Google 04 THE MODERN ASTRONOMY To determine the antiquity of the system by the first method, let the errors in the positions of the planets at the beginning of the Kali Yuga, (18th February 1612 J. P. at 6 A.M. 75^50' East of Greenwich,) be determined. The positions of the planets by the system, at the beginning of the Kali Yuga, will be had by multi- plying the four last figures of the number of revolu- tions of each, as given in the Table, by 4567, re- serving in the product the four right-hand figures, which will express the position of the planet in de- cimal parts of a revolution, thus : — The Sod, 0000 x 4Se7 0000 Moon, 0000 X 4567 ^ 0000 Mercury, .... 8984 x 4667 = 9928 = 11 27° 24' 28",8 Venus, 9492 x 4607 9964 11 28 42 14 ,4 Mars, 8622 x 4667 ^ 9074 11 29 S 60 ,4 Jnpiter, 6466 x 4667 ^ 9985 = 11 29 27 36 ,0 Saturn, 7298 x 4667 9966 = 11 28 46 33,6 Mooq's apogee, 6868 x 4667 = 3486 4 6 29 46 ,6 .^-Node,... 1168 x 4667 == 4266 =.6 3 12 67 ,6 The rule above given, serving only to exhibit the positions of the planets, &c. at the beginning of the Kali Yuga, by the system, it may be proper to give here the general rule, which answers for all times. It is this ; As 4320000000 years is to the number of revolutions of the planet, &c. given in the Table; so the time elapsed firom the beginning of the Kalpa^ or modern creation, to the planet's, &c. mean place in the Hindu sphere at the end of that time.* By this rule the above positions may be verified, the number of years elapsed at the beginning of the Kali Yuga being 1972944000. By the same rule the mean an- nual motions are also obtained, by making the state- ment for one year. * If the time be days, you may make the days in the Kalpa year first number, the revolutions the second, and the given number of days the third. Digitized by Google OF THE HINDUS. 95 Having the positions of the planets at the begin- ning of the Kali Vuga, we compare them with their positions for the same instant by La Landers Tables, in order to find the errors or differences at that time, thus : — Mean places of the planets at the beginning of the Kali Yuga. By tke system as lMMr. By La Laode's TaWes. Knors. Son, .*. ^ fp or ff % ^ nr or ©ofO^ Moon, ........ 0 0 0 0 0 S 8 47,7 . S 8 47,7 Mercury, 11 27 24 28,8 10 27 21 0,9 ^ SO t 27,0 Vemis, 11 28 42 144 I 2 S2 S0,9 . 84 10 36,5 Man, 11 29 2 604 H 17 47 22,9 ^ 11 16 27,6 Jupiter, 11 29 27 S6,0 0 16 40 20,7 -17 21 44,7 Satan, 11 28 46 82,6 11 8 46 40, + 19 69 6S,6 Moon's apogee, . 4 5 29 46,6 S 29 68 19,2 + 6 81 26,4 Nodesopt. 5 8 12 67,6 5 6 S8 8,8 - S 25 6,2 Here, at the beginning of the Kali Vuga, we have the error in the place of Venus, equal to 34° 10' 36' 5, whereas in A.D. 538, it amounted to only 6' 41', which alone would be a convincing proof of the time when the system was framed ; for all the errors must incontrovertibly diminish, as we approach the time at which the observations were made, on which the system is founded. But to proceed : we must now compare the mean annual motions by the sjrstem, with La Lande's Ta- bles for the same space of time, 365^ &"' 12"- 9'-, re- duced to the Hindu sphere, in order to find the dif- ferences, or errors. By the System. 9y La Lande's Tables. Bmrs. Son, 0 0< O' 0^ 0 0< O' 0^ 0^ Moon, 4 12 46 SO 4 12 46 26,6140 8,3860 + Mercmy, 1 24 44 69,69(^2 1 24 46 33,8660 33^6708 - Venus, 7 15 11 56,8476 7 15 11 22,9260 33,9208 + Mars, 6 11 24 8,6666 6 11 24 19,6790 11,1224- Jiqiter, 1 0 21- 7,9365 1 0 20 51,5178 16,4187 + 8ahim, 0 12 12 60,1894 0 12 13 10,4427 20,2523- Moon's apogee, 1 10 40 31,7574 1 10 40 36,5950 4,8376 - Node, 0 19 21 38,8604 0 19 21 29,8975 3,4529 + Digitized by Google 96 T^lE MOD£RN ASTRONOMY Having now obtained the errors or differences in the mean annual motions, let the errors in position at the beginning of the Kali Yuga be divided by these, and we shall have the time, according to each, when there was no error, thus : — Moon, 30 8' 48' Sec. equat. + Mercury, •••... 80 Venus, 84 Mars, 11 16 27 Jupiter, 17 21 44 Saturn, 19 Moon's apogee, 6 — — . Node supt 3 J®' ^ \ divided by 3,386 gives 8591 9 ditto by 33,66 — 3213 6. 38,92 ~ 8627 6 11,12 — 3649 7 16,42 — 3806 68 6 20,26 — 3666 4 4,84 — 4109 2 3,46 — 8667 8 27 10 86 69 81 26 26 6 Their sum is 29117 Which, divided by 8, gives for a mean result 3639 years from the beginning of the Kali Yuga, or A. D. 538, the real epoch of the system. The following Table will now explain the second method, by which the en-ors in the system, at dif- ferent periods, are shown gradually diminishing, from the beginning of the Kali Yuga, down to the epoch at which it was framed, A.D. 538. Planets, KaUYoga, errors at. Kali Yoga 600. Kali Yaga 1900. Kali Yaga 2000. KaUYaga 3000. KaU Yoga 3890, or A. 0.538. Moon,* Mercury, Venus, Mars, Jupiter, Saturn, 8066'33"_ 30 328 + 84 1037 - 11 1628 + 17 2145 - 19 6964 + 7< 8'14''- 24 2646 + 28 8124 - 9 2615 ^ 14 8784 - 16 2928 + 6<^7'37''- 18 60 3 + 22 6211 - 7 34 1 + 11 6323 - 18 661 4 11 21 6 + 16 19 63 - 6 543 + 8 1428 - 8 3660 + in5'28"- 1 6966 + 6 5432 ~ 2 021 + 3 4049 - 2 6918 + OO r 20* + 3 58 40 - 0 06 41 + 0 1 64 + 0 46 58 - 0 36 24 - Moon's -i apogee J Moon's^ NodeS 6 8126 + 4 48 4 + 8 6441 + 2 6011 12934 + 0 88 2 + 8 26 6 - 2 6084 - 2 16 2 . 130 0 - 0 8227 - 0 4 19 + supt. 3 By this Table it may be seen, at one view, the /ast difference there is between the errors at the be- * Including the secular equation.