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«The astronomy we find in texts from ancient India is similar to that we know from ancient Greco-Roman sources, so much so that the prevailing view is ...»

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Mean Motions and Longitudes in Indian Astronomy

Dennis W. Duke, Florida State University

The astronomy we find in texts from ancient India is similar to that we know from

ancient Greco-Roman sources, so much so that the prevailing view is that astronomy in

India was in large part adapted from Greco-Roman sources transmitted to India. 1

However, there are sometimes differences in the details of how fundamental ideas are

implemented. One such area is the technique for dealing with mean motions and longitudes. The Greek methods explained by Ptolemy are essentially identical to what we use today: one specifies a mean longitude λ0 for some specific day and time – the epoch t0 – and uses known mean motions ω to compute the mean longitude λ at any other time from the linear relation λ = λ0 + ω (t − t0 ).

Rather than specify a mean longitude λ0 for some epoch t0, the methods used in Indian texts from the 5th century and later instead assume the mean or true longitudes of all planets are zero at both the beginning and end of some very long time period of millions or billions of years, and specify the number of orbital rotations of the planets during those intervals, so that mean longitudes for any date may be computed using R λ= (t − t0 ) Y where R is the number of revolutions in some number of years Y, and t – t0 is the elapsed time in years since some epoch time at which all the longitudes were zero. Such ‘great year’ schemes may have been used by Greco-Roman astronomers, 2 but if so we have nothing surviving that explains in detail how such methods worked, or even if they were seriously used. Since the Indian texts give only computational algorithms, and those often cryptically, we never find any explanation of the underlying derivations of the techniques used in those texts. Hence it is of interest to understand is as much detail as possible the methods used to construct the revolution numbers R.

In Greek astronomy as exemplified in the Almagest, the units of ω are generally revolutions per year or degrees per day, and the revolutions can be either sidereal (returns to a specific fixed star) or tropical (returns to a specific cardinal point of the Sun’s orbit).

The mean motions ω are determined by dividing the total degrees traveled over many years by the number of days in the time interval or by specifying period relations giving the number of revolutions in longitude and anomaly in, typically, a few decades of years.

In the case of the planets, small fractional corrections are also specified. 3 These period relations are all seen first in Babylonian records and were certainly determined using empirical observations recorded over several centuries. 4 The positions in mean longitude and anomaly are deduced from involved and complicated comparisons of observed true

–  –  –

For all the planets there are relations between the mean motions in longitude ωp and anomaly ωa and the mean motion ωS of the Sun. In the Almagest the mean position on the epicycle is reckoned with respect to the apogee of the epicycle, which is the point on the epicycle on the extension of the line from the center of uniform rotation to the center of the epicycle, and of course that line is rotating with constant speed. For the outer planets Saturn, Jupiter, and Mars – those that can achieve any elongation from the Sun – these mean motions satisfy the relation

–  –  –

For the inner planets Venus and Mercury – those that achieve only limited elongations from the Sun – the conventional mean motion of the planet is exactly the mean motion of the Sun, so at some point it apparently became useful to specify the mean motion according to

–  –  –

so that the mean position on the epicycle is now reckoned with respect to a fixed direction in space, typically either a fixed star or a cardinal point. In modern terms, we of course recognize this as the heliocentric motion of the inner planets, and so in fact under this definition ωp is the heliocentric mean motion for both the inner and outer planets.

One consequence of these relations is that an author need specify only ωp for each planet, and ωa can be immediately derived. In the Almagest, perhaps for convenience, Ptolemy in fact specifies and tabulates both the mean motion in longitude and anomaly for all the planets, while in the Planetary Hypotheses he is more economical, specifying only the single planetary mean motions ωp as defined above for each planet. 6 All ancient Indian texts on astronomy follow exactly the same scheme of specifying just one mean motion per planet, the ωp mentioned above, although, just as in the Almagest, it is explicitly stated that for the outer planets the rotations of the epicycle equal the rotations of the Sun, and for the inner planets the mean motion of the planet equals the mean motion of the Sun. 7

The Literary Background

The Indian astronomy texts not only mention very long time intervals, but also give some information on the structure of those intervals. In the Paitamahasiddhanta (hereinafter Paita) and the Brahmasphutasiddhanta (hereinafter BSS), 8 the fundamental long time interval is a kalpa of Y = 4,320,000,000 years and the numbers of revolutions, which are the same in both documents, range from R = 146,567,298 for Saturn to R = 57,753,300,000 for the Moon. In this system the kalpa is constructed from Dennis Duke Page 2 1/30/2008 mahayugas of 4,320,000 years according to a rather elaborate scheme: a kalpa consists of 14 manvantaras, and each manvantara consists of 71 mahayugas, or 306,720,000 years.

Each manvantara is preceded and followed by one of 15 sandhis or twilight periods consisting of 4/10th of a mahayuga, or 1,728,000 years, and so a kalpa consists altogether of exactly 1,000 mahayugas. Each mahayuga is itself divided into four parts with lengths in the ratios 4:3:2:1, hence a krtayuga of 1,728,000 years (so that a sandhi is a krtayuga), a tretayuga of 1,296,000 years, a dvaparayuga of 864,000 years, and a kaliyuga of 432,000 years. Finally, to connect all this to a ‘modern’ date, it is said that what is effectively sunrise on –3101 Feb 18 occurred at the beginning of the kaliyuga of the 28th mahayuga of the seventh manvantara, hence after the lapse of six manvantaras and seven sandhis and 27.9 mahayugas, all of which together makes 4567 periods of 432,000 years, or 1,972,944,000 years in total, which is 0.4567 of a kalpa. As we shall see below, the number 4567 is intimately involved in the construction of the revolution numbers. The version of the Paita that survives does not give the connection to a modern date, but it is quite corrupted and from the great overlap with the BSS we can be sure that the Paita, or something closely related to it, was Brahmagupta’s source, and that the time details we find described by Brahmagupta were originally in the Paita or the closely related source.

In the Aryabhatiya Aryabhata uses a very similar scheme. He keeps the same mahayuga of 4,320,000 years and gives the revolutions of each planet in a mahayuga, with numbers ranging from 146,564 for Saturn to 57,753,336 for the Moon. However, he then declares that a kalpa is 14 manvantaras and each manvantara is 72 mahayugas, so for him a kalpa is 1,008 mahayugas or 4,354,560,000 years. He further divides the mahayuga into four equal kaliyugas, each of 1,080,000 years. Finally, in his sunrise scheme he says that the same date Brahmagupta used, sunrise on –3101 Feb 18, occurs at the beginning of the final kaliyuga of the 28th mahayuga of the seventh manvantara, hence after the lapse of six manvantaras and 27.75 mahayugas or 1,986,120,000 years, which is about 0.4561 of his kalpa, and which is about as close to Brahmagupta’s 0.4567 as Aryabhata could get, given his system. However, as we shall see below, the number 4561 plays no role whatsoever in Aryabhata's construction of the revolution numbers.

The parallels in all these astronomy texts – the final kaliyuga of the 28th mahayuga of the 7th manu of the current kalpa – are obviously striking, and in fact mirror well–established tradition in older Indian literary texts. 9 The Hindu epic poem Mahabharata, probably compiled in the second half of the first millennium B.C., describes the general scheme of yugas and kalpas in a section that is generally thought to be a late interpolation. 10 The entire scheme for constructing the kalpa and locating the current kaliyuga within it is described in the Vishnu Purana, 11 probably compiled sometime between the 1st century B.C. and the 4th century A.D., and everything except the position in the 28th mahayuga is described in the Law Code of Manu, 12 probably compiled in the first or second century A.D. What is missing from all the literary texts is any information tying the events in those texts, most importantly the date of the battle of Bharata, which occurred at the beginning of the current kaliyuga, with any modern date. However, as we shall see below, the 3600 year interval between the beginning of the current kaliyuga and 499 Mar 21 is given directly or indirectly in the Paita, the BSS, the Aryabhatiya, and in early

–  –  –

Aryabhatiya Revolution Numbers R Our task now is to understand the construction of the revolution numbers in both systems.

The short tradition, exemplified in Aryabhata’s sunrise and midnight systems, is to give ω, the mean motion in sidereal longitude, as the ratio of two large integers,

–  –  –

where R is the integral number of sidereal revolutions in each mahayuga of length Y years. Aryabhata stipulates that all the mean longitudes are zero at the beginning and end of each mahayuga, and in addition, at the beginning of the 4th of the four equal kaliyugas that make up each mahayuga. This condition requires that R must be divisible by four.

The beginning of the fourth kaliyuga is at the date

–  –  –

either midnight or sunrise, at approximately 76° east longitude. Thus the mean longitude in revolutions after an interval Δt = t – t0 will be of the form

–  –  –

where R´ is the integral number of revolutions accomplished and r′ is the fraction of a revolution accomplished over and above R´, and so λ in degrees is just 360 × r ′. As explained above, for Mercury and Venus the mean longitude is just the mean longitude of the Sun, so instead λ refers to the longitude of the epicycle radius, referred to a fixed direction in space (i.e. not the epicycle apogee, as we see in the Almagest).

Now Aryabhata asserts that in the sunrise system 4,320,000 years contains 1,577,917,500 days, while the same number of years in the midnight system contains 1,577,917,800 days. Thus the sidereal year length in the sunrise system is 365;15,31,15 days, while in the midnight system it is 365;15,31,30 days, and so in both cases adding 3,600 of the respective years brings us to the same modern date: 499 Mar 21 at noon. Both year lengths are longer than the real sidereal year of 365;15,22,54 days.

We can therefore determine the mean longitudes of the planets in Aryabhata’s systems on 499 Mar 21 at noon by computing λ = 3, 600 × ω. Since 3,600 is sexagesimally expressed as 1,0,0 the calculation is most transparent if we also express ω in a sexagesimal base.

Since in both systems R must be an integer divisible by 4, ω must be of the form

–  –  –

Since δ is a multiple of 12, λ will be an integral multiple of 1;12°, and if R changes by ±4 revolutions, then λ changes correspondingly by ±1;12°. Note also that if Aryabhata had kept the traditional 4:3:2:1 ratios his kaliyuga would have been 432,000 years instead of 1,080,000 years, and hence his rotation numbers must have been divisible by 10 and his granularity would be 3°, which he might have regarded as too large for useful work.

There is some evidence that such a scheme was at some point related to the midnight system and was influential in the creation of the zij al-Arkand, which is one of the earliest transmissions of Indian astronomy into Islam. 13 Using the R values from Aryabhata’s sunrise system, one finds for Saturn

–  –  –

so the mean longitude is 6 ×1;12 = 7;12. The leading two digits 31,54 are consistent with

the common period relations for Mars:

79 − 37 284 − 133 = 0;31,53,55..., = 0;31,54,5...

For Venus,

–  –  –

For the Sun, there is one revolution per year, and so the longitude of the Sun is zero on both –3101 Feb 18 at sunrise and 499 Mar 21 at noon, the first by construction, the second as a consequence of being exactly 3,600 years later.

For the Moon the short systems specify 57,753,336 rotations in longitude, 488,219 rotations of the lunar apogee, and 232,226 rotations of the lunar node. The node rotates in the clockwise direction, so the rotations will be counted as negative. While the longitudes of the Sun and Moon are, like the five planets, zero on –3101 Feb 18, the Moon’s apogee is at 90° and the ascending node is at 180°. Thus the rotations in longitude are divisible by four as usual, while the rotations of the apogee are three plus some multiple of four, and the rotations of the node are two plus some multiple of four.

–  –  –

For the Moon’s apogee, the revolutions are three plus some multiple of four in order that the longitude of the apogee be 90° on –3101 Feb 18. Hence

–  –  –

These deconstructions of the rotation numbers R for the five planets clearly suggest how the numbers might have been originally constructed: from the period relations one would compute n;α,β, from the mean longitudes, rounded to the nearest multiple of 1;12°, one would get γ,δ, and thus compute R as

R = 4,320, 000 × n;α, β, γ, δ = 20, 0, 0 × n;α, β, γ, δ.

Clearly the revolution numbers R contain exactly two components: (1) knowledge of widely known planetary period relations, and (2) rounded mean longitudes at noon on 499 Mar 21. The perfectly clean separation of these components in the sexagesimal construction certainly suggests a strong Greek influence. Also note that because of the

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