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The Astronomical Observatories of Jai Singh c15.s1: Hindu Astronomy

The Astronomical Observatories of Jai Singh c15.s1

Language
English

Contents

Hindu Astronomy

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² The following notes attempt to give, very briefly, only a fair notion of Hindu astronomy. No attempt has been made at completeness. For further information reference should be made to the works enumerated in the annexed bibliography (p. 142 seq.).

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64. There is a certain amount of very interesting mythological astronomy recorded in the Vedas, but the earliest formal Hindu astronomical works are the *Jyotisha Vedānga* and the *Sūrya Prajñapti*, the latter of which exhibits a strange cosmography (with two suns, two moons, etc.³) while both have the crude elements of a scientific astronomy. These works are of considerable historical interest: they show little, if any, Greek influence.

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³ The astronomical notions of the early Christian writers were often far more absurd. See p. 83.

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Soon after the beginning of the Christian era the traditional astronomical system in India was largely discarded, and the system in vogue in the Greek schools was imported and assimilated. In the *Pañcha Siddhāntikā* of Varāha Mihira we, possibly, have summaries of two western books—the *Paulisa* and *Romaka Siddhāntas*, but, quite apart from this, there is abundant evidence to show, not only Greek influence, but, Greek domination. The representative Indian work, that exhibits the astronomy of this period, is the *Sūrya Siddhānta*. In its original form this work was probably composed about A.D. 400, and the recension now in use about A.D. 1100. Since then very little attempt at advance has been made. The orthodox still accept the *Sūrya Siddhānta* as authoritative, and other works are not essentially different.

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Such are the facts, but there has been an extraordinary amount of misconception current. According to Hindu tradition the *Sūrya Siddhānta* was composed some millions of years ago.⁴ Bailly, towards the end of the eighteen century, considered that Indian astronomy had been founded on accurate observations made thousands of years before the Christian era. Laplace, basing his arguments on figures given by Bailly, decided, that some 3,000 years B.C., the Indian astronomers had recorded observations of the planets correct to one second; Playfair⁵ supported Bailly's views; Sir William Jones argued that correct observations must have been made as early as 1181 B.C.; and so on; but, with the researches of Bentley, Colebrooke, Weber, Whitney, Thibaut and others, more correct views were introduced; and it has long been known that the figures used by Bailly are comparatively modern.

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⁴ *Sūrya Siddhānta*, i, 2-3.

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⁵ Afterwards both Laplace and Playfair recanted. See my *Hindu Astronomy*.

Vedic Astronomy

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65. Vedic Astronomy is more poetical than exact, and it is of interest, apart from its poetic value, chiefly as a subject of controversy. Certain scholars, *e.g.*, Dikshit, Tilak, Jacobi and others, argue, from rather vague astronomical premises, partly based on the texts, an extreme antiquity for the Vedic writings; others do not accept their views.

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The Vedic year was 12 months of 30 days each, with an occasional intercalary month, "the thirteenth month fabricated of days and nights, having thirty members." (A.V. XIII, 3, 8.) There is no indication of any definite cycle. (The five-year cycle appears later.) The year was also divided into two equal courses or *ayanas*, a northern course or *Uttarāyana* beginning at the winter solstice, and a southern course or *Dakshiṇāyana* beginning at the summer solstice.

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In the Rig Veda two asterisms only are mentioned, Maghā and Phalguni; but in later Vedic texts (*e.g.*, A.V., XIX, 7, 1-5) a complete list of the 28 *nakshatras* or asterisms is given. This list is headed by *Krittikās* or the Pleiades, which marked, it is believed, the vernal equinox of the Vedic year; and this is a foundation, although not a very secure one, for Vedic chronology.¹

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¹ According to Albirūni (II, 8), in his time, the year was commenced with Chitra, Bhādrapadā, Krittikā, or Mrigaśiras, according to locality or predilection. See also Fleet J.R.A.S., 1916, p. 570.

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If the vernal equinox was marked by *Krittikās*, then the period of fixing this was about 2350 years B.C., when the vernal equinox was approximately of the same longitude as Alcyone (η Tauri), the brightest of the Pleiades. But the only evidence we have is the occurrence of the list of *nakshatras* with *Krittikās* at the head and, if the assumption made is true, the only legitimate conclusion is that this list must have been prepared at some time after 2350 B.C. It may have been an exotic list; or it possibly might be a genuine record of Hindu observation at some time or other. There is another difficulty in the fact that, according to the Hindu records, *Krittikās*, apparently, marked the vernal equinox for a very considerable period.

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Other parts of the Vedic texts have also been used for the purpose of establishing their great antiquity: *e.g.*, Jacobi attempted to prove that the Vedic year commenced with the summer solstice. His arguments are based on the following very doubtful rendering of a verse of the ‘Frog Hymn’:-

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"Those leaders of rites observe the institutes of the gods, and disregard not the season of [the twelfth month]:² as the year revolves and the rains return, then scorched and heated they obtain Freedom."

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² Other translations give "the twelve months."

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Dikshit,³ from a passage of the Brāhmaṇas (Ś.B. II, 1, 2⁴), fixes the age of its composition at 3000 B.C. The words "They (the *Krittikās*) do not move from the eastern quarter while the other asterisms do move from the eastern quarter" he takes to mean, definitely, that the asterism Kṛittikas (Pleiades), and no other asterism, was on the equator; and he writes "In my opinion the statement conclusively proves that the passage was composed not later than 3000 B.C." Many other similar interpretations have been strived after.

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³ Indian Antiquary, 1895, XXIV, p. 245.

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In Vedic texts no definite mention is made of the planets, although much ingenuity has been exercised in interpreting the texts otherwise. There are possible references to eclipses, which Ludwig, with some skill, has attempted to identify.

Vedāṅga Astronomy

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66. (a) The Jyotisha Vedāṅga¹ and the Sūryaprajñapti contain the earliest formal astronomical statements. The former introduces the 5-year cycle of 1830 apparent solar days. The year was tropical in theory and contained 366 apparent solar days, and was, therefore, too long. The sidereal year was 367 sidereal days. The lunar day or tithi was 61/62 of an ordinary day, but was reckoned as equivalent to an ordinary day for calendar purposes, one tithi being omitted as occasion required. There were 27 nakshatras, each supposed to occupy 360/27 = 13 1/3 degrees of the ecliptic, and each nakshatra was considered to be divided into 124 equal divisions, or aṃśas. The sun, therefore, traversed 5 × 27 asterisms in the five-year cycle, or 5 × 27 × 124 / 1830 aṃśas a day; and to traverse one nakshatra it took 13 1/3 days = 13 days 335 kalās, since there are 603 kalās in one day. The moon traversed 67 × 27 / 1830 = 603/610 nakshatras in one day, and one nakshatra in 1 day 7 kalās.

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¹ Vedāṅga is the name of certain works, or classes of works, regarded as auxiliary to the Veda. They are generally considered as of a somewhat later date.

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The five-year cycle appears to have commenced with the winter solstice, and Śraviṣṭha is said to have marked the beginning of this cycle, and also the beginning of the sun's progress, and also the winter solstice—all of which are in agreement. If Śraviṣṭha is to be identified with β, α, γ and δ Delphini (as it usually is), then it marked the winter solstice about B.C. 1100. But a list of asterisms given in the text begins with Aśvini (β, and γ Arietis), which marked the vernal equinox about the beginning of the Christian era. The Vedāṅga states that, during the northern progress of the sun, the days increase in length at an even rate of 1·57 minutes a day, or 4 hours 48 minutes in six months of 183 days²: the northern and southern progress are considered equal.

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² The table of climates on page 131 shows that an increase of 4 1/2 hours corresponds to a latitude of 36° 38' N., the obliquity being taken as 23 1/2 degrees. For a greater obliquity it would be further north.

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(b) The Sūryaprajñapti is a Jain treatise on astronomy, that is similar in many respects to the Vedāṅga. The Jainas held the old Indian idea of the heavenly bodies revolving round Meru, and, as a corollary to this, they conceived two suns, two moons and two sets of constellations. The five-year cycle began with the summer solstice, with the sun in Pushya, and Thibaut thought³ this was a correction, from actual observation, of the older Vedāṅga.

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³ Thibaut calculates a precession of 17° 3', or a difference of 1246 years between the Vedāṅga and the Sūryaprajñapti, but gives a caution as to the uncertainty of the deduction.

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Another point of difference was the employment of 28 nakshatras of unequal extent, and this altered, theoretically, the positions of the nakshatras—in some cases to a very considerable extent, and makes our estimations of the periods in which these works were composed very uncertain.

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(c) The characteristics of this period are:—

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(1) The five-year cycle.

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(2) The division of the sphere into 27 or 28 nakshatras.

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(3) Equal daily change in the length of the day.

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(4) Omission of any explicit reference to the planets.

Greek Astronomy in India

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67. Varāha Mihira and others, about A.D. 550, made popular new ideas borrowed from the west: they remodelled the Hindu astronomical system on Greek lines. Varāha Mihira’s astrological works contain numerous Greek technical terms and show, unmistakably, Greek influence. His great astronomical work, the Pañchasiddhāntikā, consists of summaries of the Paitāmaha, Vāsisṭha, Romaka, Pauliśa¹ and Saura Siddhāntas. “The Siddhānta made by Pauliśa is accurate, near to it is the Siddhānta proclaimed by Romaka, more accurate is Sāvitra and the two remaining ones are far from the truth.” The summary of the Paitāmaha Siddhānta exhibits the teaching of the Vedānga stage but adds the epoch of 2 Śaka (=A. D. 80). The Vāsisṭha Siddhānta appears to represent the transition stage. It alters the longest day rule and introduces shadow calculations, and the lagna or ‘rising sign’ notion; while the other three introduce, unequivocally, the Greek teaching. The main characteristics of the Romaka Siddhānta are—

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¹ Albirūni writes (i, 153): “Paulisa-siddhānta, so called from Paulisa the Greek, from the city of Samtra, which I suppose to be Alexandria.”

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(a) A cycle of 2850=19×150 years, perhaps based on the Metonic cycle.

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(b) A year of 365ᵈ 5ʰ 55ᵐ 12ˢ, which is exactly the tropical year of Hipparchus.

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(c) The epoch of 427 Śaka (=A.D. 505).

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(d) Omission of mention of epicycles.

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In the Pauliśa Siddhānta the following points are noteworthy:—

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(a) A year of 365ᵈ 6ʰ 12ᵐ.

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(b) A special rule for finding the place of the moon.

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(c) Very rough rules for eclipses.

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(d) Differences in longitudes between Avanti (Ujjain) and Vārāṇasī (Benares) and Yavana (Alexandria) are given.

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(e) A table of sines agreeing with Ptolemy’s table of chords.

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68. The Sūrya Siddhānta is probably the best known astronomical work of the Hindus. The several sections of the accepted text² relate to—

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² Varāha Mihira’s summary of this work differs in some details from the text now in use, but not essentially, e.g., the length of the sidereal year in the two works is: Old Sūrya Siddhānta 365ᵈ 6ʰ 12ᵐ 36ˢ. Modern Sūrya Siddhānta 365ᵈ 6ʰ 12ᵐ 36·56ˢ. The text now accepted possibly dates from about A.D. 1100, while the earlier limit for the original Sūrya Siddhānta is about A.D. 400.

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1. The mean motions of the planets.

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2. The true places of the planets.

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3. The gnomon.

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4. Eclipses.³ 5. Planetary conjunctions.
6. Asterisms.
7. Heliacal risings and settings.¹
8. Instruments.
9. Time. Cosmogony.
10. Astrology.

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³ That subject which is the greatest mystery, which perplexes the minds of writers of astronomical works.” (P.S. i, 5.)

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¹ For some historical account of this topic see Bouché-Leclercq, *L'Astrologie grecque*, p. 111f.

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69(a). The topics dealt with in most of the later Hindu works are fundamentally the same as those of the Sūrya Siddhānta, and the following notes apply, fairly generally, to all these works. The earth is considered as a fixed unsupported sphere, round which the other heavenly bodies revolve.² Its diameter is given as 1,600 yojanas, and the distance of the moon as 51,570 yojanas, or roughly the same relative distance as Ptolemy gives (61 1/4 radii of the earth). The distances of the other planets are calculated on the assumption that they move with equal velocities. The equation of the centre of a planet is calculated by assuming epicycles, but an apparently indigenous notion is introduced by making the epicycle oval. Ptolemy's theory of the equant is omitted, and certain other improvements of Ptolemy, relating to the moon and Mercury, are also omitted. The precession of the equinoxes is explained as a sort of libration, within limits of 27 degrees east and west of a fixed position, at a rate of 54 seconds a year; and the obliquity is generally reckoned at 24 degrees.

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² Āryabhaṭa attempted to revive the theory of Heraclitus that it was the earth's own rotation that produced the apparent motions of the heavenly bodies; but he was condemned as unorthodox, and the view has never been accepted by Hindu astronomers generally.

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The Greek names of the signs of the zodiac were adapted, and the seven day week introduced; many Greek astrological terms and some Greek mathematical terms were adopted without change. Some of the old cosmological notions, that did not interfere with the new ideas, were retained. The Greek teaching was, indeed, accepted as a whole—but the evidence points to the curious fact, that the Greek astronomy introduced is that of a period preceding Ptolemy, although Ptolemy lived in the second century of our era, and the Hindu-Greek astronomical works did not appear earlier than A.D. 400.

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69(b). One of the most notable features of Hindu astronomy of this period is the employment of immense cycles. To express the planetary elements in integral numbers the astronomers assumed an artificial epoch of general conjunction, and a period of recurring conjunctions. The last general conjunction was supposed to be at 3102 B.C., and the cycle or yuga of recurring conjunction is supposed to consist of 4,320,000 years.³ The planetary elements are expressed in terms of this cycle as follows:—

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³ The number 4,320,000=20×60³ is suggestive of Babylonian influence. See also J. Adam, *The Nuptial Number of Plato*.

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| Planet. | Revolutions. | Planet. | Revolutions. |
| Sun | 4,320,000 | Saturn | 146,568 |
| Mercury . . . . . . | 17,937,060 | Moon : Sidereal revolutions | 57,753,336 |
| Venus . . . . . . | 7,022,376 | " Synodic " | 53,433,336 |
| Mars . . . . . . | 2,296,832 | " Revolution of apsis | 488,203 |
| Jupiter . . . . . . | 364,220 | " " , node | 232,233⁴ |

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⁴ As Whitney points out all these numbers except the last two are divisible by four and this seems to indicate that the last two were later additions.

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There was, of course, no general conjunction at 3102 B.C., which period was arrived at by calculating backwards, according to the rules. Bentley assumed that, at that particular time when the calculation was made, the positions of the planets were known with some accuracy: he then calculated, according to the *Sūrya Siddhānta* rules, the positions of the planets at several periods, and found them most correct for A.D. 1091, and concluded that the elements were fixed and the work composed towards the end of the eleventh century.¹

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¹ For further details see my *Hindu Astronomy*. Bentley's general argument is quite sound, but he must not be accepted as reliable on all points.

Hindu Astronomical Calculations

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

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| 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. |

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(*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.

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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°

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| . | 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′ |

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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.

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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.

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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.

Hindu Star Lists

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72. The determination of the position of the stars with exactitude does not seem to have interested the ancient Hindu astronomers.¹ In early works the brief lists of stars with celestial co-ordinates are given in connexion with the path of the sun and moon through the nakshatras. In each nakshatra the position of a junction star or *yogatārā* was determined. The *Pañcha Siddhāntikā* mentions seven of these while the *Sūrya Siddhānta* gives the position of 28, one for each nakshatra, and also of seven other stars. The *Siddhānta Śiromani* and *Brahma Siddhānta* mention only Canopus and Sirius. After these, such lists as Mahendra Suri's (given in Appendix A) sometimes occur.

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¹ Albirūni writes (II, 83): "The Hindus are very little informed regarding the fixed stars. I never came across any one of them who knew the single stars of the lunar stations from eyesight, and was able to point them out to me with his fingers."

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2. The *Pañcha Siddhāntikā* record is as follows:—

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"The *yogatārā* of Kṛittikā is at the end of the sixth degree and three and a half *hastas* to the north of the ecliptic; that of Rohiṇī is at the end of the eighth degree, and five and a half *hastas* to the south of the ecliptic." (XIV, 34)
"The two stars of Punarvasu are at the eighth degree, and to the north and south of the ecliptic at an interval of eight *hastas*. The star of Pushya is at the fourth degree, three and a half *hastas* to the north." (35)
"Of Āśleṣhā the southern star is at the first degree one *hasta* (south); so also the northern star of Maghā the conjunction takes place in its own field,² at the sixth degree. Of Chitrā at seven and a half degrees, three *hastas* to the south."³

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² i.e., on the ecliptic.

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³ Since 24 *aṅgulas*=1 *hasta* and the diameter of the moon was reckoned as 15 *aṅgulas*, and its mean diameter as 32 minutes (SS. IV, 1) we have approximately 1 *aṅgula*=2' 8" and 1 *hasta*=51' 12" roughly; but possibly 1 *aṅgula* was meant to measure 2 minutes. Also 27 *nakshatras* occupy 360° and one therefore occupies 13⅓ degrees.

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The *Sūrya Siddhānta* gives the positions of the chief stars of the nakshatras in terms of polar latitude and longitude.⁴ The *Sūrya Siddhānta* stars and their position are given in appendix A.

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⁴ See also p. 8.

Hindu Astronomical Instruments

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73. The only instruments of practical utility for astronomical purposes described in ancient Hindu works are the sun-dial and clepsydra. An armillary sphere is also described as an instrument for purposes of demonstration. The only Hindu instrument of any antiquity actually found is the clepsydra, consisting of a metal bowl floating in a vessel of water.⁵

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⁵ It is the only instrument described in the *Ain-i-Akbari* (Ed. Jarret III, 16).

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The following is a summary of those parts of the early Hindu texts that deal with astronomical instruments. (i) The Clepsydra or Water clock is referred to in the *Jyotisha Vedānga*, where the amount of water that measures a *nāḍikā* (=24 minutes) is given. The more ancient form of water clock appears to have been simply a vessel with a small orifice at the bottom, through which the water flowed in a *nāḍikā*,⁶ but later on there came into use the form described in the *Sūrya Siddhānta* (XIII, 23): "A copper vessel, with a hole in the bottom, set in a vessel of pure water, sinks sixty times in a day and night, and is an accurate hemispherical instrument." The Pañcha siddhāntikā description ¹ (XIV, 32) is similar, but adds "Or else a nāḍikā may be measured by the time in which sixty slokas, each consisting of sixty long syllables, can be read out."

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⁶ J. F. Fleet, *The Ancient Indian Water Clock*, J.R.A.S., 1915, pp. 213-230.

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¹ Lala Chhotte Lāl's Jyotiṣa Vedāṅga, p. 12.

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A later description of the clepsydra is as follows: "A copper vessel, weighing 10 palas, six aṅgulas in height and twice as much in breadth at the mouth—this vessel of the capacity of 60 palas of water, and hemispherical in form, is called a ghaṭi. The aforesaid copper vessel, bored with a needle made of 3½ māshas ² of gold and 4 aṅgulas long, gets filled in one nāḍikā."

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² About 56 grains troy. Fleet quotes another rule, which gives the weight as a suvarṇa (= 16 māshas), and length 4 aṅgulas, drawn out round or square. Bhāskara simply says (XI, 8) it "should have a hole bored in its bottom."

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In practice, no doubt, the dimensions of the bowl and the orifice were determined by experiment. Bhāskara (XI, 8) indeed says: "See how often it is filled and falls to the bottom of the pail of water in which it is placed. Divide 60 ghaṭis of day and night by the quotient, and it will give the measure of the clepsydra."

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(ii) The Gnomon.—The sun-dial described in the early treatises is of the simplest kind, consisting of a vertical rod, or gnomon, divided into 12 divisions. The descriptions are of a theoretical nature, and do not apply so much to the construction of instruments as to theoretical calculations. The Pañchasiddhāntikā (XIV, 14-16) instructions are: "Mark from the centre three times the end of the gnomon's shadow, and then describe two fish figures. Thereupon describe a circle, taking for radius a string that is fastened to the point in which the two strings issuing from the heads of the fish figures intersect, and that is so long as to reach the three points marked. On the given day the shadow of the gnomon moves in that circle, and the base of the gnomon is the south-north line; and the interval, in the north direction, is the midday shadow." (III, 1-7) This means, mark on any particular day the extremity of the shadow at three different times—and these three points are supposed to lie on a circle, the centre of which is found (in the usual way) by the so-called fish figures.³

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³ The 'fish figure' is the common part of two intersecting circles.

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The Sūrya Siddhānta directions (III, 1-7) are more elaborate but relate to exactly the same type of dial. They are as follows:—

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"(1) On a stony surface, made water level, or upon hard plaster, made level, there draw an even circle of a radius equal to any required number of digits of the gnomon. (2) At its centre set up the gnomon of twelve digits of the measure fixed upon; and, where the extremity of its shadow touches the circle in the former and after parts of the day, (3) fixing two points upon the circle, and calling them the forenoon and afternoon points, draw midway between them, by means of a fish figure, a north and south line. (4) Midway between the north and south directions draw, by a fish figure, an east and west line: and, in like manner, also by the fish figures, between the four cardinal directions, draw the intermediate directions. (5) Draw a circumscribing square, by meansof the lines going out from the centre: by the digits of its base lines projected upon that, is any given shadow reckoned. (6) The east and west line is called the prime vertical (*sama-maṇḍala*): it is likewise denominated the east and west hour circle (*unmaṇḍala*), and the equinoctial circle (*viṣuvan maṇḍala*). (7) Draw likewise an east and west line through the equinoctial shadow (*viṣuvad-bhā*); the interval between any given shadow and the line of the equinoctial shadow is denominated the measures of the amplitude¹ (*agrā*).”

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¹ Distance of the sun at rising or setting from east or west point of the horizon.

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(iii) Armillary Sphere.—The Sūrya Siddhānta (XIII, 1-6) gives instructions for the making of an elaborate armillary sphere:—

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(2) “Let the teacher, for the instruction of the pupil ……(3) prepare the wonder working fabric of the terrestrial and stellar sphere (*bhūbha gola*). Having fashioned an earth-globe of wood, of the desired size, (4) fix a staff, passing through the midst of it and protruding at either side for Meru; and likewise a couple of sustaining bands and the equinoctial circle; (5) these are to be made with graduated divisions (*aṅgula*) of degrees of the circle (*bhagaṇa*). Further, by means of the several day-radii, as adapted to the scale established for those other circles, (6) and, by means of the degrees of declination and latitude marked off upon the latter, at their own respective distances in declination, according to the declination of Aries, etc., (7) three bands are to be prepared and fastened: these answer also inversely for Cancer, etc. In the same manner, three for Libra, etc., answering also inversely for Capricorn, etc.: (8) and, situated in the southern hemisphere, are to be made and fastened to the two band-supporters. Those, likewise, of the asterisms situated in the southern and northern hemispheres, of Abhijit, (9) of the Seven Sages (*Saptarṣayas*) of Agastya, of Brahma, etc., are to be fixed. Just in the midst of all the equinoctial band is fixed. (10) Above the points of intersection of that and the supporting bands are the two solstices (*ayana*) and the two equinoxes (*viṣuvat*). From the place of the equinox, with the exact number of degrees, as proportioned to the whole circle, (11) fix by oblique chords, the spaces (*kṣetra*) of Aries and the rest; and so, likewise, another band, running obliquely from solstice to solstice, (12) and called the circle of declination (*krānti*): upon that the sun constantly revolves giving light: the moon and the other planets, also by their own nodes, which are situated in the ecliptic (*apa maṇḍala*), (13) being drawn away from it, are beheld at the limit of their removal in latitude (*vikṣepa*) from the corresponding point of declination. The orient ecliptic point (*lagna*) is that of the orient horizon; the occident point (*astamgachhat*) is similarly determined. (14) The meridian ecliptic point (*madhyama*) is as calculated by the equivalents in right ascension (*laṅkodayās*), for mid heaven (*hamadhya*) above. The sine which is between the meridian and the horizon (*kṣitija*) is styled the day measure (*antyā*), (15) and the sine of the sun’s ascensional distance (*charadala*) is to be recognised as the interval between the equator and the horizon. Having turned upward one’s own place, the circle of the horizon is midway of the sphere. (16) As covered with a casing (*vastra*) and as left uncovered, it is the sphere surrounded by Lokāloka. By the application of water is made ascertainment of the revolution of time. (17) One may construct a sphere instrument combined with quicksilver; this is a mystery, if plainly described it would be generally intelligible to the world. (18) Therefore let the supreme sphere be constructed according to the instruction of the preceptor. In each successive age, this construction, having become lost, (19) is by the sun's favour again revealed to some one or other at his pleasure."

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(iv) Other Instruments—" So also," the text continues, "one should construct instruments (yantras) for the ascertainment of time. (20) When quite alone, one should apply quicksilver to the wonder causing instrument.¹ By the gnomon (sanku), staff (yashti) arc (dhanus), circle (chakra), instruments for taking the shadow of various kinds .......... (21), By water instruments, the hemisphere (kapāla), etc., by the peacock, man, monkey, and by stringed sand receptacles, one may determine time accurately. (22) Quicksilver-holes, water, and cords, ropes and oil and water, mercury, and sand are used in these: these applications, too, are difficult ..........

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¹ See the Siddhānta Siromani xi 50-51. The instrument appears to be a perpetual motion machine, which consists of a wheel with hollow (tangential) spokes which are filled with mercury. "The wheel thus filled will, when placed on an axis supported by two posts, revolve of itself." This is an old friend.

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(24) So also a dial (narayantra) is good in daytime, and when the sun is clear."

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Such is the orthodox Hindu text relating to instruments. Nothing of material value appears to have been added to these instructions until the methods of the Yavanas were introduced, by Mahendra Sūri and others; but Bhāskara (Siddhānta Siromani XI, 16) claims to have invented an instrument called Phalaka Yantra,² which, he says, is an "excellent instrument, calculated to remove always the darkness of ignorance and is the delight of clever astronomers." This instrument is simply a board divided by horizontals into 90 equal parts. At the centre of the 30th graduation from the top a pin, or style, is placed perpendicular to the board, and round it a circle is drawn of radius=30 divisions, which is graduated in ghaṭis and degrees, and attached to the pin is an index arm (paṭṭika). The instrument is suspended by a chain, and is used for observational purposes. It is in fact part of a very simple astrolabe. Bhāskara did not seem very pleased with his instrument, for, he concludes (XI, 40) by saying "But what does a man of genius want with instruments, about which numerous works have treated? Let him only take a staff in his hand, and look at any object along it, casting his eye from its end to the top. There is nothing of which he will not then tell: its altitude, dimensions, etc." This sums up, very well indeed, the attitude of Hindu astronomers.

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² Compare the 'Balance Khorarie ou Fézaire' described by Delambre Astronomie du Moyen Age, p. 521

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