The Astronomical Observatories of Jai Singh c15.s1.t4: Hindu Astronomical Calculations

The Astronomical Observatories of Jai Singh c15.s1.t4

Language: English

Hindu Astronomical Calculations

c15.s1.t4.1

70. The important elements in the *Sūrya Siddhānta* are:—

| A | Years in the *yuga* or age | . . . . . | 4,320,000 | A. |
| B | Sidereal days | . . . . . | 1582,237,828 | B. |
| C | Natural or civil days | . . . . . | 1577,917,828 | C = B—A. |
| D | Solar months | . . . . . | 51,840,000 | D = 12 × A. |
| E | Sidereal months | . . . . . | 57,753,336 | E. |
| F | Synodic months | . . . . . | 53,433,336 | F = E—A. |
| G | Intercalary months (*adhimāsa*) | . . . . . | 1,593,336 | G = F—D. |
| H | Lunar days (*tithi*) | . . . . . | 1603,000,080 | H = 30 × F. |
| I | Omitted lunar days (*tithi kshaya*) | . . . . . | 25,082,252 | I = H—C. |

c15.s1.t4.3

(*a*) One of the most frequent of the Hindu calculations is concerned with finding the *ahargaṇa*, or ‘sum of days’ that have elapsed since the beginning of a *yuga*. Thus at the commencement of the Śaka year 953 (A.D. 1031, February 25th) 3,244,132 years of the *chaturyuga* had elapsed, and to find the number of civil days the calculation is as follows:— 12 × 3,244,132 × G/D gives 1,196,525 intercalary months,² whence the lunar days elapsed are 30 (12 × 3,244,132 + 1,196,525)=1,203,783,270. Again 1,203,783,270 × I/H gives 18,835,679 omitted lunar days, and the number of civil days that have elapsed is therefore 1,203,783,270—18,835,679=1,184,947,591.

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² An intercalary month only occurs when it is complete, hence the fractional part is omitted.

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(*b*) Another characteristic example of the calculations is concerned with the equation of the centre (Kendra). The mean position of a planet was calculated from its number of revolutions in a *yuga* (p. 74). This was corrected by hypothecating certain epicyclic motions. The mean motion in a circle (deferent) gave wrong positions, so the planet was supposed to move in a second circle (epicycle), whose centre lay on the circumference of the mean circle (deferent), and the corrected position was calculated on this theory. For all but the sun and moon (which required only one) two corrections were made, (1) the equation of the conjunction, (2) the equation of the apsis—by two separate epicycles; and by combining these two equations the ‘true place’ of the planet was determined. The calculation is complicated, and entailed very considerable skill of sorts. Here are calculations for the planet Venus for a particular position.

c15.s1.t4.6

CALCULATIONS FOR THE EQUATION OF THE CENTRE FOR VENUS. Given mean longitude, L = 8s 18° 13′ Longitude of conjunction, C = 10s 21° 50′ Longitude of apsis, A = 2s 19° 52′ Epicycle of apsis, E, varies from 11° to 12°. Difference, Δ = 1° Epicycle of conjunction, E, varies from 260° to 262° Difference, Δ = 2°

c15.s1.t4.7

| . | First calculation for equation of conjunction. | Second calculation for equation of apsis. | Third calculation for equation of apsis. | Fourth calculation for equation of conjunction. |
| Longitude . . | 8s 18° 13′ | + 26° 7′ / 2 = 9° 17′ | + 22′ / 2 = 9° 1′ 28″ | 8s 18° 13′ + 23′ = 8s 18° 36′ = 2s 3° 37′ - 23′ = 2s 3° 14′ |
| a. Mean commutation = C-L | 2s 3° 37′ | . . | . . | . . |
| a'. Equated anomaly = A-L | . . | 5 18° 35′ | - 22′ / 2 = 5° 18° 24′ | . . |
| b. Bhujajyā = Sin a or a' | 3080′ | 689′ | 691′ | 3069′ |
| c. Kotijyā = Cos a or a' | 1527′ | 3369′ | 3368′ | 1521′ |
| d. Correction for epicycle = Δ | 1° 47′ | 12′ | 12′ | . . |
| e. Corrected epicycle = E-d | 260° 13′ | 11° 48′ | 11° 48′ | 260° 13′ |
| f. = be / 360° | 2226′ | 22·3′ | 22·6′ | 2218′ |
| g. = ce / 360° | 1104′ | 110′ | 110·4′ | 1118′ |
| h. Chala karna = √(r±g)² + f² | 5058′ | 3548′ | 3458′ | 5067′ |
| i. = rf / h | 1515 | 21·9′ | 22·5′ | 1505′ |
| j. = sin⁻¹ i | + 26° 7′ | + 0° 22· | + 0° 23′ | + 25° 59′ |
| The true longitude is therefore | . . | 8s 18° 36′ | + 25° 59′ | = 9s 14° 35′ |

c15.s1.t4.8

The radius(r) is supposed to be divided into 3438 equal parts, which are, as a matter of convenience, termed minutes, and the sines, etc. (bhujajyā, kotijyā, chala karna, etc.), are expressed in terms of this radius (see my Indian Mathematics, p. 11).

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The corrections for epicycles are simple proportions: the difference for 90° is Δ, what is the difference for θ degrees? The dimensions of the epicycle are expressed in degrees, etc., in such a way that (r' being the radius of the epicycle) 2 π r' : 2 π :: the number of degrees : 360.

c15.s1.t4.10

71. A great many interesting topics must be omitted in this brief sketch of Hindu astronomical theory. Āryabhaṭa taught that the earth rotated upon its axis, and a proper explanation of eclipses, but was not approved. The works of Brahmagupta and Bhāskara have considerable interest in matters of detail, but do not differ fundamentally from the Sūrya Siddhānta. Indeed, since the time of composition of this work there has been been practically no alteration of fundamental importance in the Hindu theory.

c15.s1.t4.11

At the present time there are three schools of astronomers: (i) The Saura-paksha, (ii) the Ārya-paksha, (iii) the Brahma-paksha; and these only differ¹ in matters of detail. For example, a distinctive feature is the length of the year² employed. These are:—
(i) Saura-paksha year 365d 6h 12m 36·56s. (ii) Ārya-paksha year 365d 6h 12m 30s.
(iii) Brahma-paksha year 365d 6h 12m 30·915s.

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¹ Really the differentiation is a geographical one. The Sūrya Siddhānta is the standard authority in the greater part of India, but the first Ārya-Siddhānta is the authority in the Tamil and Malayāḷam countries of Southern India, while Brahmagupta is followed in Gujarāt, Rājputāna and North-West India.

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² Theoretically, at least, the year is a sidereal one, but there is some vagueness, and there are no records of the methods by which the results were attained. See R. Sewell, The Indian Calendar, pp. 7-10.

Linked texts (1)

Sūryasiddhānta — Maya, Sanskrit