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6688 4 The sun at the given moment is supposed to be just entering the first sign in the Hindu sphere ; but its longitude at the same instant in the European sphere was 10* T 15' 48", the diflFerence V 28^ 44' 12' is the difference between the two spheres at the time ; which being added to the longitudes of the planets in the tropical sphere, reduces their places to the Hindu ; from which it may be easily seen that the planets were not in the position assumed, and that the errors in the assumption so made were, In the SWi place, 0® 0' 0^ Moon'sdo 8 8 48 — Mercnry'sdo. ,.'. 82 88 69 + VeniM*8do 82 62 61 ^ Mar'sdo 12 18 \7^ Jupiter'sdo 16 49 21-- Satmm'ido 21 18 20 + Digitized by Google 88 THE MODERN ASTRONOMY The marks or signs — + show, that. the assumed po- sition falls short of, or exceeds the real mean place, by the quantity to which they are annexed : thus the position assumed being 0% falls short of the moon's mean place at the time by 3° 8' 48", and exceeds the mean place of Mercury by 32° 38' 69", because Mer- cury was then only in 10" 27° 21' I" instead of 0' 0° 0' 0", the assumed position. From the circumstance here stated, it must be self-evident, that in deriving the mean annual motions of the planets from the assumed position at the given time, the mean motions of the moon, Venus and Jupiter, must come out greater, and those of Mer- cury, Mars, and Saturn, less, than the truth, — and that the errors, in such mean annual motions would, if nothing else operated to the contrary, be in pro- portion to the errors above exhibited in the position assumed. Having thus shown the principal cause of the dif- ference between the Europeans and modern Hindus in respect of the quantities of the mean annual motions of the planets, we may now proceed to determine the mean motions themselves, preparatory to our showing the manner in which the astronomical sys- tem was formed and connected with the system of years already mentioned. I have already stated, that the ancient astronomy of the Hindus terminated in March, A.D. 538, at which period the new system was introduced. Iti this year the vernal equinoctial point, the beginning of the Lunar Asterism Aswini, and the beginning of the month Vaisakha, were supposed to coincide, (See Plate VL) which point was, therefore, made the commencement of the year in this new system ; so Digitized by Google OF THE HINDUS. 89 that the sun was supposed to enter into the sign Aries of the tropical sphere, and into the first sign of the Hindu sphere at the same moment of time. Now the instant of the mean vernal equinox in that year, was the 21st March, about six in the morning, on the meridian of UjeiA, and the number of days elapsed from 6 A.M. 18th February, in the year 1612 of the Julian period, to the instant of the ver- nal equinox in A.D. 638, was 1329176 days, which being divided by the number of Hindu years, viz. 3639 then completed, we obtam 365** 6** 12°" 2 P 57'^ or 365* 15*** 30' 54" 64"' for the length of the Hindu year ; which, however, must undergo a correction, in order to adjust it to the new system, under the following conditions : — 1st, The Kalpa, or system, is to commence with Sunday at sunrise, as thejirst day of the week. — ^2d, The number of days in the Kalpa, or in 4,320,000,000 years, must be complete without a f relation. — 3rd, The number of days from the creation, or beginning of the Kalpa, to the 18th February, in the year 1612 of the Julian period, or in 1972944000 years, must be complete without a fraction.-^Ath, The days so elapsed, must, when divided by seven, leave a remainder of five, to indicate that the 18th February 1612 J. P. fell on Friday. To comply with all these conditions, the length of the Hindu year, when cor- rected, comes out 365* 6** 12°* 9% or 365* 15** 30' 22' 30"', diflfering a few seconds from the former. Com- mencing, therefore, this year on the 18th February 1612 J, P. at 6 A.M. the termination of the 3639th year falls on the 20th of March A.D. 538, at 53' 51" past 4 P.M. at which instant the following were the mean positions of the planets : — N Digitized by Google 90 THE MODERN ASTRONOMY Sao, EaropetDiphere, ll' 29<^ 25' S3" Hioda sphere, 0$ O'' 0' 0^ Moon, 1 IT 46 32 1 18 20 59 Mercury, 5 5 44 45 5 6 19 12 Venus, 4 17 3T 54 4 18 12 1 Mw^, 9 1 42 31 9 2 16 68 Jupiter, 9 21 19 26 9 21 53 53 SttQm, 5 15 12 35 5 15 47 02 On the 18th February 1612 J. P. at 6 A.M. they were assumed to be in 0% or the beginning of the Hindu sphere ; therefore, to get their mean annual motions from this assumption, we must get their re- volutions for the time elapsed, (3639 years com- plete,) and add them to the above positions, which will then give us the following : Sun, 8639'^^- 0 00° 00' 0" Moon, 48649 Mercury, 15109 Venas, 5915 Mars, 1934 Japiter, 306 Saturn, 123 These quantities, being now divided by the time, 3639 years, we shall get the mean annual motions of each, as follow : Son, I'^ev. o 0° 0' 0" Moon, 13 Mercury, ...•••..• 4 Venus, 1 Mars, 0 Jupiter, •• 0 Saturn, 0 0 12 12 40 The mean annual motions thus deduced, as from two actual observations, would of course give the positions of the planets on the 20th March A. D. 538 at 53' 51" past 4 P.M. reckoning such motions as commencing at the epoch of mean conjunction. But, conformably to the nature of the system to be constructed, it is requisite that the planetary mo- tions should commence with the KalpUy or modem 1 18 20 59 5 6 19 12 4 18 12 21 9 2 16 58 9 21 53 53 5 15 47 02 4 12 46 30 1 24 45 1 7 15 11 45 6 11 24 5 1 0 21 8 Digitized by Google OF THE HINDUS. 91 creation ; therefore the motions just found will re* quire a correction to adjust them to the system of years already mentioned. For this purpose, it is necessary, in the first place, that the number of re- volutions of each planet in the period of 4320000000 years should be cmnplete, and entire without a frac- tion ; and, secondly, that the number to be assigned shall, when reckoned as commencing from the crea- tion, or beginning of the Kalpa, give the mean place of the planet in A. D. 538, when the system was framed, as near the truth as the nature of integral numbers will admit. The method of doing this I will now show ; — a single example will be sufficient for this purpose. Let it be required to find from the mean motions above determined, the number of revolutions of Venus in the period of 4320000000 years, so that the same being reckoned from the creation, (1972947639 years before A.D. 538,) it shall give the mean place of the planet, sufficiently correct to answer all Hindu purposes. First step. — The mean motion of Venus for 3639 years is found above to be 5915'*^- 4* 18^ 12' 21". Therefore, as 3639 years give this quantity, so 4320000000 years will give 7022384850 revolutions nearly : then say. As 4320000000 years to 7022384850 revolutions, so the years elapsed from the creation in A. D. 538 = 1972947639, to (3207129076"^) 4' 16° 24' 20*; com- paring this with 4* 18° 12' 21", the mean longitude of Venus in A.D. 538, it will be found too little by 1° 48' 1"; to make up this deficiency, we must find what diflFerence one revolution will make, thus : — As 4320000000^^ : 1'*^:: 1972947639^^ : 5* 14° 24' 44' 17"' 30^^- T^ 12^ Having thus found the difier- Digitized by Google 92 THE MODERN ASTRONOMY ence that one revolution would make, we must find by trials what number of revolutions will jnake up for the deficiency, 1° 48' 1"; — this will be found to be 4642 ; for, if we multiply 5" 14° 24' 44", &c. by this number, we shall get, rejecting the revolutions as of no use, T 54' 42", which exceeds 1° 48' 1" by only 6' 41": therefore, adding 4642 to 7022384850, we have the corrected number, equal to 7022389492, which is the veiy same that is given by the inventor of the system, in the following table; and in this manner all the rest were formed and adjusted to the number of years above given. Table of the revolutions of the planets, apsides, and nodes in a Kalpa, or 43200000Q0 yeais, formed in the manner above explained by the author of the system in A.D. 538. Planets. Apsides. Nodes Retrograde. Son's rerolntions, 4320000000 Moon, 5TT53300000 Mercnry, 17936998984 Venas, 7022389492 480 .. .. 488105858 332 653 292 855 41 .. ..