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A Concise History of Science in India c6.s3: The Period of the Composition of the Astronomical Siddhāntas—the Five Siddhāntas, the Sūrya-siddhānta, the Astronomical Works and Commentaries of the Period

A Concise History of Science in India c6.s3

Language
English

Contents

The Period of the Composition of the Astronomical Siddhāntas—the Five Siddhāntas, the Sūrya-siddhānta, the Astronomical Works and Commentaries of the Period

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The first few centuries of the Christian era are very important in the history of Hindu astronomy. The inaccuracies and the insufficiency of the Vedic and the Sūtraic astronomy must have been felt for a long time. The crude calendrical astronomy based on uncorrected motions of the sun and the moon against a stellar zodiac had outlived its function and utility.

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a Das, pp. 30–42.

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b Kaye (2), p. 21.

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Accurate study of heavenly bodies, in which the planets were gradually to be incorporated, involved the application of more sophisticated mathematics, adoption of a system of co-ordinates, and more reliable determination of their periods of revolution, the sizes of the earth, the relative sizes of the sun and the moon, and so on. Although interest in the eclipse phenomena had been very ancient and recorded at several places in the earliest *Saṃhitās* and the *Brāhmaṇas*, their calculations and predictions were beyond the scope of the kind of astronomy so far practised and taught. All this must have occupied the serious attention of the astronomers of this period, for by about A.D. 400 or even earlier we for the first time come across an altogether new class of astronomical works called *siddhāntas* dealing with matters and methods unknown in the works of any previous period. This new class of works is characterized, among others, by the replacement of the *nakṣatra* system by the 12 signs of the zodiac, the correct length of the year, the correct rules for calculating day-lengths and oblique ascensions, the determination of mean longitudes by the *ahargaṇa* method, the study of planetary motions by reference to a celestial sphere, the system of great and small circles and the geometric models of eccentric circles and epicycles, the ideas of parallax and its calculation, and the calculation of solar and lunar eclipses. The emphasis on computations also opened the way to new methods of analysis. Integral solutions of indeterminate equations, rudiments of plane and spherical trigonometry, besides common geometrical, arithmetical and algebraic practices, became integral parts of the new science. Increasing mathematization very soon necessitated the incorporation of a few chapters on mathematics into most treatises on astronomy, and the practice was carried to such an extent that astronomy and mathematics became inseparable for several centuries to come.

The Five Siddhāntas

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The word *siddhānta* means ‘final conclusion’ or ‘solution’. Its application to the reformed astronomical texts of the period, seeking to present the correct solutions of astronomical problems, was no doubt quite appropriate. Tradition says that there were 18 original *siddhāntas*, as Sudhākara Dvivedi, in his *Gaṇakataraṅgiṇī*, mentions; these 18 *siddhāntas* were by Sūrya, Pitāmaha, Vyāsa, Vasiṣṭha, Atri, Parāśara, Kāśyapa, Nārada, Garga, Marīci, Manu, Aṅgirā, Lomaśa (Romaka), Paulīśa, Cyavana, Yavana, Bhṛgu and Śaunaka. Most of them are lost and several of them which invoked the name of some ancient sage to gain popularity cannot certainly claim to belong to this class of astronomy. Of the above list, the *Sūrya-siddhānta*, both in its old version as summarized by Varāhamihira and its later improved form through revisions by several authors, has come down to us. We are also indebted to Varāhamihira for preserving for us the main features of four others, namely the *Paitāmaha*, the *Vāsiṣṭha*, the *Paulīśa* and the *Romaka*. The teachings of the five *siddhāntas* as summarized by Varāha in his *Pañcasiddhāntikā* will now be briefly discussed.

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*The Paitāmaha-siddhānta*

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This is the most inaccurate of the five *siddhāntas* noticed. Its astronomical elements are the same as those of the *Vedāṅga Jyotiṣa*. Thus it teaches a quinquennial cycle containing 5 revolutions of the sun, 67 revolutions of the moon, 60 solar months, 62 synodic months, 2 intercalary months, 1,830 *sāvana* days and 1,860 *tithis*. The cycle begins with the conjunction of the sun and the moon at the first point of the *nakṣatra Dhaniṣṭhā*. The rule for the shortest and the longest day is the same as that given in the *Vedāṅga Jyotiṣa*.

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*The Vasiṣṭha-siddhānta*

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Although rated inaccurate in the case of its *tithis* compared to the *Pauliśa*, the *Romaka* and the *Sūrya-siddhānta*, the teachings of the *Vasiṣṭha-siddhānta* were more advanced than those of the *Paitāmaha*. In this system, the traditional *nakṣatras* were supplemented with a zodiac with its subdivisions of signs, degrees and minutes for the first time. Lagna, that is the ecliptic point on the eastern horizon, is mentioned and rough rules for finding it from the shadow and vice versa are given. We also find rules for determining day-lengths. Rough rules for calculating mean longitudes from the length of the midday shadow are given. This *siddhānta* reveals knowledge of the anomalistic month and gives its values as 248/9 and 3031/110. More importantly, the *Vasiṣṭha-siddhānta* also deals with the true motions of the five planets, Venus, Jupiter, Saturn, Mars and Mercury, and recognizes their motions as direct, stationary (*anuvakra*), retrograde (*vakra*) and again direct. The synodic periods in days of these planets are given as follows:

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| Venus | .. | 584 1/11 days |
| Jupiter | .. | 399 1/9 days |
| Saturn | .. | 378 1/11 days |
| Mars | .. | 780 2/45 days |
| Mercury | .. | 115 days 52 *nāḍikās* 45 *vināḍikās* |

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The *siddhānta* also gives the equivalence between synodic and sidereal revolutions for each planet, from which the sun's sidereal revolutions in a given number of days, or in other words the length of the solar year, can be easily calculated. For Jupiter, the text gives 36 sidereal revolutions as equivalent to 391 synodic revolutions. This means that the sun executes 427 sidereal revolutions during the same time as the Jupiter makes 391 synodic revolutions, that is in the course of 391 × 399 days (omitting 1/9 day). The length of the solar year, according to the *Vasiṣṭha-siddhānta*, works out to 365·36 days approximately, a considerable improvement upon the value adopted in the *Vedāṅga Jyotiṣa*. At another place,a the *siddhānta* specifically gives the sidereal year length as 365¼ days, as has been shown by T. S. Kuppanna Sastry. The *Vasiṣṭha-siddhānta* was recast by Viṣṇucandra whose name is mentioned by Brahmagupta and al-Bīrūnī.

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a *Psi.*, ii, 1.

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The Paulīśa-siddhānta

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Varāha states that ‘the *siddhānta* made by Paulīśa is accurate’. The elements of this *siddhānta* are presented in chapters 3 to 7 and partly in chapter 1. These elements include calculation of *ahargaṇa*, that is the number of civil days elapsed from a certain epoch up to a given date, the determination from the *ahargaṇa* of the mean places of the sun and the moon and thence their true places. The rules for finding the longitude of the moon are analogous to the *vākyam* method found later on in the Tamil astronomy. This *siddhānta* also contains miscellaneous rules for finding direction, place and time, sine tables and simple calculations of eclipses. From a statement that there are 43,831 civil days in 120 years, the length of the year works out to 365·2583 days.

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The *Paulīśa-siddhānta*, in giving the quantities to be added to or subtracted from the mean sun in order to obtain the true places, reveals knowledge of the anomaly (*kendra*) and the equations of the centre. According to Thibaut’s interpretation, the rules indicate that for anomalies 10°, 40°, 70°, 100°, 130° and 160° the corresponding equations should be 21′, 96′, 139′, 140′, 108′ and 50′. But how the equations are obtained is not mentioned; that is there is no hint as to any epicyclic or eccentric model upon which to calculate them.a

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The sine table given in chapter 4 may be due to Paulīśa or may be a common feature of all the three accurate *siddhāntas*. The radius adopted for calculating the table is 120′ and not 3438′ used in later astronomical texts. There is also evidence of the beginnings of spherical trigonometry in the *siddhānta*. The formula given for the variation of day-length takes the form: $R sin (ascensional difference) = R tan φ tan δ$ where φ = latitude and δ = the sun’s declination.

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Paulīśa’s calculations of lunar and solar eclipses are rough and less accurate than those given in the *Romaka-* and the *Sūrya-siddhānta*. The sidereal period of the revolution of the moon’s nodes is given as 6794·6854, the semi-diameter of the moon as 17′ and that of the sun as 16·8′. The moon’s greatest latitude is given as 270′ (4° 30′). The parallax in longitude is expressed in time, that is by the difference in time between the instant of the new moon and the instant of conjunction as it appears to the observer. The expression is given by $Parallax = (4 R sin (sun's hour angle))/R nāḍikās (1 nāḍikā = 1/60 day).$

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a In a private communication to the author, Sri T. S. Kuppanna Sastry has suggested that the method is possibly the same as that of the *Vākyakaraṇa*.

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Al-Bīrūnī, who derived his information on Hindu astronomy both from this *siddhānta* as summarized by Varāha as also from Brahmagupta’s *Brāhmasphuṭa-siddhānta*, expressed the view that its author Paulīśa was a Greek from the city of Saintra which he supposed to be Alexandria.a Besides Varāha’s summary of the *Paulīśa-siddhānta*, references to, and elements from, a *Paulīśa-siddhānta* were given by Bhaṭṭotpala and Pṛthūdakasvāmin. These elements differ from Varāha’s Paulīśa and agree more closely with those found in the *Saura* of the *Pañcasiddhāntikā* and in Āryabhaṭa’s *ārdharātrika* system. Possibly the original work known to Varāhamihira underwent one or several emendations with the passage of time.

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*The Romaka-siddhānta*

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The elements of the *Romaka-siddhānta* are summarized partly in chapter 1 where matters concerning *yuga*, *ahargaṇa*, intercalation of lunar months and such general topics are discussed and in chapter 8 dealing with eclipse calculations. The title suggests its foreign origin which is further supported by its astronomical elements and the year being tropical. It introduces a luni-solar cycle of 2,850 years containing 1,050 intercalary months and 16,547 omitted lunar days. The total number of synodic months in this period will be 2,850 × 12 + 1,050 or 35,250; the total number of civil days will be 35,250 × 30 — 16,547 or 1,040,953. The lengths of the year and the synodic month calculated from these elements compare favourably with those given by Hipparchus and Ptolemy as follows:

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| | *Romaka-siddhānta* | *Hipparchus-Ptolemy* |
| Year .. | 365 days 5 h 55′ 12″ | 365 days 5 h 55′ 15·8″ |
| Synodic month .. | 29 days 12 h 44′ 2·2″ | 29 days 12 h 44′ 2·5″ |

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The anomalistic month is given as 3031/110 days or 27·554 days. This makes the moon’s daily motion in anomaly as 13° 3′ 53″ 58ᴵᴵᴵ 55ᴵⱽ 51ⱽ 45ⱽᴵ in close agreement with Ptolemy’s 13° 3′ 53″ 56ᴵᴵᴵ 29ᴵⱽ 38ⱽ 38ⱽᴵ in sexagesimal units.

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It is to be noted that the total number of years and synodic months in the luni-solar cycle used in the *Romaka-siddhānta* are both divisible by 150. The result of this reduction is that 19 solar years comprise 235 synodic months. This is the well-known 19-year cycle of Meton, the Athenian astronomer (432 B.C.). The reason for multiplying Meton’s cycle by a factor of 150 was probably to arrive at a *yuga* containing a whole number of civil days in conformity with the established Hindu practice.

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The information given in the *Romaka-siddhānta* regarding the equations of centre for the sun and the moon is more detailed, but again without any indication as to how these corrections were obtained. In finding the anomaly, that is the difference between the longitude of the mean sun and that of the apogee, the longitude of the apogee is stated as 75° as against Ptolemy’s value of 65° 30′. The following equations of centre of the sun and the moon for their anomalies at intervals of 15° are given (Table 2.2).

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a Sachau, I., p. 153.

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TABLE 2.2

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*The equations of centre of the sun and the moon in the Romaka-siddhānta*

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| Anomaly | 15° | 30° | 45° | 60° | 75° | 90° |
| *Romaka* | | | | | | |
| Equation of centre | | | | | | |
| Sun .. | 34′ 42″ | 1° 8′ 37″ | 1° 38′ 39″ | 2° 2′ 49″ | 2° 17′ 5″ | 2° 23′ 23″ |
| Moon . | 1° 14′ | 2° 25′ | 3° 27′ | 4° 15′ | 4° 44′ | 4° 55′ |
| *Ptolemy* | | | | | | |
| Equation of centre | | | | | | |
| Sun . | | 1° 9′ | | 2° 1′ | | 2° 23′ |
| Moon | | | | | | 5° 1′ |

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For purposes of comparison, a few values given by Ptolemy are quoted. The equations for the sun agree very closely, but this is not so for those for the moon. The text gives latitudes of the moon, parallaxes in longitudes and latitudes and a few other details.

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At one time Śrīseṇa appeared to be the author of the original *Romaka-siddhānta*. Brahmagupta himself, in his *Brāhmasphuṭa-siddhānta*, refers to Śrīseṇa as the author of the work. From a more careful study of the passage, Thibaut showed that Śrīseṇa was at best a poor compiler who possibly worked on the original *Romaka-siddhānta* and incorporated into it planetary elements from other authorities, namely Lāṭa, Vijayanandin and Āryabhaṭa I. In fact, Brahmagupta also severely criticized Śrīseṇa’s effort and passed the remark that he had turned a heap of jewels, which the original *Romaka-siddhānta* was, into a patched rag. In this connection, the name of Lāṭa is important, for Varāha himself referred to Lāṭa as a commentator of the *Romaka-siddhānta* and Brahmagupta held him in respect.

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The Sūrya-siddhānta

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The *Sūrya-siddhānta*, also called the *Saura-siddhānta*, was according to Varāhamihira the most accurate of the five *siddhāntas*. The main features of this work summarized in the *Pañcasiddhāntikā* appear in chapters 9-11, 16 and 17; the *yuga* elements of the sun and the moon are given in chapter I. 14; chapters 13 to 15 discuss some of its general matters, e.g. cosmogony, geography, etc. In the history of Hindu astronomy, the *Sūrya-siddhānta* occupies a very important position in that it does not appear to be the work of a single individual, but underwent periodic revisions and served as a standard astronomical text in different periods of time. Thibaut called Varāha's summary the old version of the *siddhānta* to distinguish it from the modern version we now possess. P. C. Sengupta has shown that there was certainly an old *Sūrya-siddhānta* dated about A.D. 400, which was brought up to date by Varāha himself on the basis of Āryabhaṭa's *ārdharātrika* (midnight) system; and that the modern *Sūrya-siddhānta*, with additions of improved data from Brahmagupta and others, continued to develop till the twelfth century A.D. or even later.

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Coming to Varāha's version, the *siddhānta* taught that 180,000 years contained 66,389 intercalary months and 1,045,095 omitted lunar days. This gives 65,746,575 days in the *yuga* thus conceived. For the sake of comparison with the elements given in the modern *Sūrya-siddhānta*, the period of 180,000 years can be converted to a *mahāyuga* of 4,320,000 years by multiplying the former by 24 and the total number of days in such a period worked out. The mean motions of planets are given in chapter 16. The integral numbers of revolutions performed by the planets in a *mahāyuga* can be easily calculated, and they are set down in Table 2.3. The values

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TABLE 2.3 *Planetary revolutions in a Mahāyuga*

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| Planet | *Sūrya-siddhānta* (Summarized by Varāha) | Āryabhaṭa I (*Ārdharātrika* system reproduced in Brahmagupta's *Khaṇḍakhādyaka* and in Bhāskara I's *Mahābhāskarīya*) | Modern *Sūrya-siddhānta* |
| Sun .. | 4,320,000 | 4,320,000 | 4,320,000 |
| Moon | 57,753,336 | 57,753,336 | 57,753,336 |
| Mars | 2,296,824 | 2,296,824 | 2,296,832 |
| Jupiter .. | 364,220 | 364,220 | 364,220 |
| Saturn .. | 146,564 | 146,564 | 146,568 |
| Mercury | 17,937,000 | 17,937,000 | 17,937,060 |
| Venus .. | 7,022,388 | 7,022,388 | 7,022,376 |
| Moon's apogee | 488,219 | 488,219 | 488,203 |
| Moon's node .. | 232,226 | 232,226 | 232,238 |
| Number of civil days in a *mahāyuga* .. | 1,577,917,800 | 1,577,917,800 | 1,577,917,828 |

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given in the modern *Sūrya-siddhānta* and in the *ārdharātrika* system of Āryabhaṭa are also shown in the table for the sake of comparison.

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It will be seen that the planetary revolution numbers in a *mahāyuga*, as given in the *ārdharātrika* system of Āryabhaṭa I and in the old *Sūrya-siddhānta* summarized by Varāha, are the same, but these elements and other astronomical constants differ in several cases from those of the modern *Sūrya-siddhānta*. This identity of astronomical elements in the old *Sūrya-siddhānta* with those of Āryabhaṭa's *ārdharātrika* system has given rise to the view that the former might have been recast by Varāha after the teachings of his older contemporary Āryabhaṭa I.

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Mahāyuga

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The concept of *mahāyuga* is the central feature of the Indian astronomical *siddhāntas*. It developed from the idea of *yuga* or cycle which is very old indeed and is met with in the *Brāhmaṇa* literature. It is a period at the beginning of which all planetary bodies are in conjunction, during which they all perform integral numbers of revolutions, and at the end of which, therefore, they are again in conjunction. In selecting the length of such a cycle, care was taken that the apsides and the nodes also have whole numbers of revolutions. The *mahāyuga* of 4,320,000 years is such a period. Biot demonstrated that the length of the year as 365 days 6 h 12′ 35·56″ is such that the least number of years containing a whole number of civil days is 1,080,000. The *mahāyuga* also called the *caturyuga* is just 4 times this number. The number 108 is again 4 times 27, the number of *nakṣatras*.

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According to Āryabhaṭa I, a *mahāyuga* (simply called a *yuga*) is divided into four equal parts or *yugapādas*, each consisting of 1,080,000 years. In the modern *Sūrya-siddhānta*, a *mahāyuga* is subdivided into four mundane ages, e.g. the *Kṛta* or Golden Age, the *Tretā* or Silver Age, the *Dvāpara* or the Brazen Age and the *Kali* or the Iron Age.a The durations of these ages are in the descending order of 4, 3, 2 and 1 and have the following solar years:

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| The *Kṛta* | .. | 1,728,000 | solar years |
| The *Tretā* | .. | 1,296,000 | ,, |
| The *Dvāpara* | .. | 864,000 | ,, |
| The *Kali* | .. | 432,000 | ,, |

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These divisions are in keeping with the traditions of the *Smṛti* and the *Purāṇas*. Their use in an astronomical *siddhānta* is met with for the first time in the works of Brahmagupta, from which it might have been incorporated in the modern *Sūrya-siddhānta* by its unknown redactor. A special importance attaches to the *Kali* or the current Iron Age through which we are at present passing. At the commencement of this epoch, the planets are assumed to be in conjunction for the last time at the initial point of the Hindu sphere. As all astronomical data and calculations given in the *Sūrya-siddhānta* and works following more or less the same system depend on this assumption, it is important to know its date. This epoch started at midnight at the meridian of Ujjayinī between February 17 and 18, 3102 B.C.

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Revolutions of the apogees of the sun and the planets are not recognized in Varāha’s version of the *Sūrya-siddhānta* although it is a characteristic feature of the modern *Sūrya-siddhānta* and other later astronomical texts. The *ahargaṇa* method of finding the mean longitudes of the sun, the moon and the planets is the same as that found in the modern version.

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a Fleet, pp. 479-96.

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The old *Sūrya-siddhānta*, like the modern, makes use of an epicyclic model for calculating the true planetary positions. This and other geometric models as used by the Hindus will be explained in what follows. For such calculations, the important elements are the longitudes of the apogees and the dimensions of epicycles for the *manda* (equation of centre) and the *śīghra* (equation of conjunction) corrections. Table 2.4 gives these elements in the old and the modern *Sūrya-siddhānta*.

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TABLE 2.4

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*Longitudes of apogees and dimensions of manda and śīghra epicycles in Varāha's Sū. Si. and the modern Sū. Si.*

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| | Longitude of apogee | Longitude of apogee | *Manda* epicycle | *Manda* epicycle | *Manda* epicycle | *Śīghra* epicycle | *Śīghra* epicycle | *Śīghra* epicycle |
| | *Sū. Si.* (Varāha) | Modern *Sū. Si.* | *Sū. Si.* (Varāha) | Modern *Sū. Si.* (odd) | Modern *Sū. Si.* (even) | *Sū. Si* (Varāha) | Modern *Sū. Si.* (odd) | Modern *Sū. Si.* (even) |
| Sun | 80° | 77° 14′ | 14° | 13° 40′ | 14° | – | – | – |
| Moon .. | | | 31° | 31° 40′ | 32° | – | – | – |
| Mars .. | 110° | 130° | 70° | 72° | 75° | 234° | 232° | 235° |
| Jupiter .. | 160° | 171° 16′ | 32° | 32° | 33° | 72° | 72° | 70° |
| Saturn .. | 240° | 236° 37′ | 60° | 48° | 49° | 40° | 40° | 39° |
| Mercury .. | 220° | 220° 36′ | 28° | 28° | 30° | 132° | 132° | 133° |
| Venus .. | 80° | 79° 49′ | 14° | 11° | 12° | 260° | 260° | 262° |

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The difference in the values of the above elements clearly points to the modifications introduced into this standard text since the days of Varāhamihira. In the old *Sūrya-siddhānta*, the dimensions of the epicycles were assumed to remain constant over odd and even quadrants; in the modern version, presumably following Āryabhaṭa I and others, epicycles have been made to vary between odd and even quadrants. The formula for determining the *manda* correction, i.e. the equation of centre, is given bya

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$sin μ = (sin α . 2π e)/(2π R) = (sin α . ⊙_μ)/360$

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where μ = the equation of centre; e = eccentricity or radius of the *manda* epicycle; R = radius of the deferent circle (the Hindu radius); ⊙_μ = circumference of the *manda* epicycle. This formula is the same as that given in the modern *Sūrya-siddhānta*.

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Rules for finding the *śīghra* correction for planets are given in chapter 17, 4–6. If γ be the argument, that is the angle between the mean planet and the sun, r the radius of the *śīghra* epicycle, and R the radius (*trijyā*) of the deferent circle, the correction σ is given by $R sin σ = (R . r sin γ)/(√((r sin γ)² + (R+r cos γ)²)).$

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a *P.Si.*, ix. 7–8.

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These rules, in their essential features, are in agreement with those found in the modern *Sūrya-siddhānta*. P. C. Sengupta attempted to show that Varāha, in preparing his version of the old *Sūrya-siddhānta*, borrowed the epicyclic method of calculating the equations of centre and argument from Āryabhaṭa I.a

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The old *Sūrya-siddhānta* gives rules for calculations of the solar and lunar eclipses which have been carried forward in the modern version with further details.

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The old *Sūrya-siddhānta*, even after its modernization after the works of Āryabhaṭa I and Brahmagupta, continued to be in use in different parts of the country. In the beginning of the ninth century, the Nepalese astronomer Sumati wrote a *karaṇa*, whose elements were based on the old *Sūrya-siddhānta*. Śatānanda’s calendrical work *Bhāsvatī* (eleventh century A.D.) also utilized this work to a great extent.b

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*The Modern Sūrya-siddhānta*

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It would have perhaps been proper to deal with the modern *Sūrya-siddhānta* later on after noticing the works of Āryabhaṭa I, Bhāskara I, Brahmagupta, Mañjulācārya and others inasmuch as the form in which we now have it developed out of the efforts of these distinguished astronomers. But, after having started with the old version and made frequent references to some of the elements in which it differed from the modern, a few remarks about the modern version here would appear more preferable. To emphasize its high antiquity, the text opens with the statement that the knowledge of astronomy presented therein was revealed to an *asura* called Maya by the sun god himself. The mention of Maya at one time led to much speculation. Observing that the name of the Egyptian king, Ptolemaios, appeared as *Turamaya* in Indian inscriptions, Weber conjectured that *asura* Maya might be the altered name of the celebrated Greek astronomer Ptolemy himself. Such conjectures are no longer held seriously even in the background of unmistakable influence of Greek astronomy (possibly pre-Ptolemaic) on Hindu astronomy of the *siddhāntic* period. According to al-Bīrūnī, the *Sūrya-siddhānta* was composed by Lāṭa, a contemporary of Āryabhaṭa I. Possibly Lāṭa composed a commentary on it.

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b Shukla, Introduction to the *Sūrya-siddhānta*.

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The work is divided into 14 chapters, e.g. (i) Mean motions of the planets (*madhyamādhikāra*), (ii) True places of the planets' (*spaṣṭādhikāra*), (iii) Determination of direction, place and time (*tripraśnādhikāra*), (iv) Lunar eclipse (*candragrahaṇādhikāra*), (v) Solar eclipse (*sūryagrahaṇādhikāra*), parallax, (vi) Projection of eclipses (*chedyakādhikāra*), (vii) Planetary conjunctions (*grahayutyadhikāra*), (viii) Conjunction of planets with junction-stars of *nakṣatras* (*nakṣatragrahayutyadhikāra*), (ix) Heliacal risings and settings of planets (*udayāstādhikāra*), (x) Moon’s risings and settings and the elevation of the moon’s horns (*śṛṅgonnatyadhikāra*), (xi) Phenomena called *pāta* (*pātādhikāra*), (xii) Cosmogony and geography (*bhūgolādhyāya*), (xiii) Astronomical instruments (*jyotiṣopaniṣadadhyāya*) and (xiv) Reckoning of time (*mānādhyāya*). The text is highly condensed and written in a manner which gives the impression that its main purpose was not to teach and explain the principles of astronomy, but to provide a set of rules for memorization. In this the *Sūrya-siddhānta* was no exception as most Hindu astronomical texts were really compendiums of astronomical rules, the task of exposition being left to the teachers and the commentators. As to brevity and condensation, the work covered the subject in 500 verses whereas the *Brāhmasphuṭa-siddhānta* needed 1,008 verses, Śrīpati’s *Siddhānta-śekhara* 890 verses and Bhāskara II’s *Siddhānta-śiromaṇi* 962 verses. Even then the text is far less cryptic and condensed than the *Āryabhaṭīya* which attempted to present its system in 120 verses, including the 33 stanzas on mathematics and the 10 *gīti* stanzas.

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a Sengupta, Introduction to Burgess' translation of the *Sūrya-siddhānta*.

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The *mahāyuga* system and the theories of planetary motions have already been referred to in our discussion of the old *Sūrya-siddhānta*. Regarding planetary motions, the modern version, interestingly enough, contains an account which appears to have been developed before the epicyclic theory found general acceptance. According to this, the planets are attached, by cords of air, to invisible beings called forms of time which are stationed at the *uccas* (*mandocca* and *śīghrocca*) and the *pātas* (nodes). The planets are impelled in their motions by the provector wind (*pravāhavāta*) uniformly, but the invisible gods at the *uccas* pull or push them in such a way that their motions appear variable.

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Special mention should be made of the treatment of the precession of equinoxes, which is missing from its earlier version. The precessional rate and the rules of correcting the longitudes by taking account of it are given in chapter 3, verses 9–12. From the fact that these rules do not appear in their proper context either in chapter I or II where methods of finding mean and true planetary motions are discussed, but are surreptitiously introduced where these are least expected, Burgess concluded that these were interpolations found necessary as a result of afterthought.a As we shall see later, two theories regarding precessional motion were propounded by different schools of Hindu astronomers. The first and the more ancient view is the theory of libration or oscillation of the equinoxes about a fixed point. In this theory the equinoxes, like the swing of a pendulum, at first move eastward, reach the maximum amplitude and then move westward. The number of oscillations executed in a *mahāyuga* is given as 600 making the time period 7,200 years. The maximum eastward or westward deviations from the fixed point are set down differently by different authorities, accounting for differences in the annual precessional rates. The *Sūrya-siddhānta* has adopted this libration theory and given the maximum eastward or westward deviation as 27° whence the annual precessional rate works out to 54″ as against the modern value of 50·25″. A similar libration theory is met with in the *Soma-*, *Śākalya-* and *Laghu Vāśiṣṭha-siddhānta*. The other theory is that of a complete revolution of the equinoxes through the circle of asterisms, propounded for the first time in India by Mañjulācārya (A.D. 932) and clearly explained by Bhāskara II. Mañjula states that the equinoxes revolve 199,669 times in a *kalpa* (that is 1,000 *mahāyugas*), whence the period of one revolution works out to 21,635·8073 years and the annual precessional rate 59·9″.

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a *See* Burgess’ notes to his translation of *Sū-Si.*, iii. 9–12.

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The *Sūrya-siddhānta* in its modern form enjoyed great popularity for several centuries. This is attested by a large number of commentaries written by able astronomers and by the richness of the manuscript materials available in the libraries of India and abroad.a Some of the commentators include Bhaṭṭotpala (A.D. 966), Caṇḍeśvara and Mallikārjuna Sūri (both twelfth century A.D.), Parameśvara (A.D. 1432), Yallaya (A.D. 1472), Bhūdhara (A.D. 1572), Nṛsiṃha Daivajña (A.D. 1586), Raṅganātha (A.D. 1603), Kamalākara (A.D. 1618), Viśvanātha (A.D. 1628) and Dādābhāi (A.D. 1719). Besides these, there exist several anonymous commentaries as well as astronomical tracts based on the rules of this great *siddhānta*.

Leading Astronomers of the Period

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Āryabhaṭa I (b. A.D. 476)

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The importance of Āryabhaṭa lies in the fact that he probably was in the vanguard of the new astronomical movement which resulted in the recasting of this new branch of knowledge about the fifth century A.D. Piecemeal efforts might have started earlier as is evident from Varāhamihira's account of the five *siddhāntas*, and before and about the time when Āryabhaṭa flourished there were certainly astronomers of repute who were variously engaged in reforming astronomy, but little is known about their contributions as their works have not survived. As matters stand, the *Āryabhaṭīya* is the earliest preserved astronomical text bearing the name of an individual of the scientific period of Indian astronomy. Moreover, his place at the head of this movement is assured by the great influence his works and teachings wielded among astronomers of subsequent ages, and by the existence of a long line of followers who propagated his views through excellent commentaries.

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From his *Āryabhaṭīya*,b we learn that he was 23 years of age in A.D. 499 when he wrote his famous work, which gives A.D. 476 as the year of his birth. From another statement of hisc that he sets forth in his work the science which is held in high esteem at Kusumapura (a place near modern Patna), scholars have thought for a long time that Āryabhaṭa was either born in Kusumapura or lived and taught in that great city of ancient India. Such a view now appears untenable in the light of recent studies on the works of Bhāskara I and his commentators and also of the medieval commentators of Āryabhaṭa. In these works, Āryabhaṭa is frequently referred to as an *āśmaka*, that is one belonging to the Aśmaka country which is the name of a country in the south, possibly Kerala. His work is sometimes designated as *Āśmaka-sphuṭa-tantra*. These findings coupled with the fact that commentaries of, and works based on, *Āryabhaṭīya* have come largely from South India, from Kerala in particular, certainly constitute a strong argument in favour of Kerala being the main place of his life and activity.

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a Sen (5), pp. 217–19.

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b *Ā. Kā̄la.*, 10.

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c *Ā. Gaṇita.*, 1.

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Āryabhaṭa's mathematical-astronomical masterpiece, the *Āryabhaṭīya*, is composed in four sections, e.g. *Gītikāpāda*, *Gaṇitapāda*, *Kālakriyāpāda* and *Golapāda*. Of these the *Gaṇitapāda* deals exclusively with mathematics and the *Daśagītikā* deals, among others, with an alphabetical system of expressing numbers, which will be considered under mathematics in a separate chapter. The astronomical elements and parameters are given in the *Daśagītikā* in his novel alphabetical system. The two remaining sections, the *Kālakriyā* and the *Gola*, deal with astronomical principles and methods of computation in a highly condensed form. *Kālakriyā* means the reckoning of time, and as such we find in this section a discussion of such topics as the division of time and the circle, definitions of solar year, lunar month, civil day, sidereal day, intercalary months, omitted lunar days, planetary orders and movements, the eccentric-epicyclic models, use of these models for the calculations of true planetary positions, calculation of true distances of planets from the earth and related matters. Thus this section deals more or less with such topics as we find discussed under chapters called *madhyamādhikāra* and *spaṣṭādhikāra* of later astronomical texts. Āryabhaṭa does not give methods for the *ahargaṇa* for calculating the mean longitudes of planets, the knowledge of which he possibly took for granted. His treatment of the eccentric-epicyclic model is important, for it appears possible that Varāhamihira might have adopted it in recasting the older *Sūrya-siddhānta* in the form given in his *Pañcasiddhāntikā*. In this model, Āryabhaṭa used the hypotenuse in calculating the equation of the centre as also for the conjunction correction. This was followed up by astronomers of his school. But Brahmagupta and others used the hypotenuse for the conjunction correction and the radius as an approximation for calculating the equation of the centre.

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The last section *Gola* means the sphere. Here the methods of representing planetary motions in a celestial sphere are explained and such terms as prime vertical, meridian, horizon, hour circle, equator, ecliptic, etc., are defined. The sun, the nodes of the moon and planets and the earth's shadow move along the ecliptic which is inclined to the equator at an angle of 24°. The rotation of the earth about its axis is given as an explanation for the apparent westward motion of the stationary asterisms, through the well-known simile of a man in a moving boat, to whom a fixed object appears to move in the opposite direction. Āryabhaṭa was the first among the Indian astronomers to have mentioned the rotation of the earth to explain the apparent daily motions of the fixed stars, although he was in this opposed and even bitterly criticized by later astronomers. His followers, however, explained the verse in the usual sense. The diurnal rotation of the earth was known to the Greeks several centuries earlier, for in the fourth century b.c. Heraclides of Pontus and his contemporary Ecphantus had clearly held this view. Other topics discussed in the *Gola* include parallax, the eclipses and related matters. Thus in the *Gola* are compressed a set of rules which usually find their treatment in chapters called *tripraśnādhikāra*, *candragrahaṇādhikāra* and *sūryagrahaṇādhikāra* of later *siddhāntas*.

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The system of astronomy taught in the *Āryabhaṭīya* is sometimes called the *audayika* inasmuch as the day beginning is reckoned from sunrise at Laṅkā. From Varāhamihira, Brahmagupta and others we learn that Āryabhaṭa was the originator of another, the *ārdharātrika* system, in which the day is reckoned from midnight at Laṅkā. Varāha's statement on this is as follows: ‘Āryabhaṭa maintains that the beginning of the day is to be reckoned from midnight at Laṅkā; and the same teacher again says that the day begins from sunrise at Laṅkā.’ Brahmagupta, in his *Brāhmasphuṭa-siddhānta*, also mentions Āryabhaṭa as the author of the sunrise and the midnight systems and himself followed the latter, along with its astronomical elements, in writing his *Khaṇḍakhādyaka*. Thus, although a great critic of Āryabhaṭa, Brahmagupta himself preserved for posterity his rival's *ārdharātrika* system. Al-Bīrūnī mentions that Āryabhaṭa also wrote a *tantra*, but this work has not come down to us. It is possible that the same *Āryabhaṭīya* was meant as it also came to be known as a *tantra* (Bhāskara I).

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Of the direct pupils of Āryabhaṭa, the names of Pāṇḍurāṅgasvāmī, Lāṭadeva, Prabhākara and Niḥśaṅku are mentioned by Bhāskara I. Lāṭadeva, the expounder of the old *Romaka*- and *Sūrya-siddhānta*, earned the appellation ‘*sarvasiddhānta-guru*’ and was held in high esteem by Varāha. But the man who appears to have contributed most to the propagation of Āryabhaṭa's astronomy, both through commentaries and independent works based on his master's principles and parameters, was Bhāskara I, a contemporary of Brahmagupta and himself an accomplished astronomer. In the eighth century Lalla wrote a work based on the *Āryabhaṭīya*. Some of the medieval commentators include Bhūtaviṣṇu, Nīlakaṇṭha Somasutvan, Parameśvara, Sūryadeva and Yallaya.

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*Varāhamihira* (c. a.d. 505)

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The author of the *Pañcasiddhāntikā* we have discussed above appears more as a compiler and a historian of astronomy than as an astronomer of originality. Nevertheless such a compiler could be an astronomer of no mean order. He was, however, more well known as an astrologer and wrote compendious astrological treatises such as *Bṛhatsaṃhitā*, *Bṛhajjātaka*, *Laghujātaka* and *Yogayātrā*. According to Bhaṭṭotpala, Varāha had originally belonged to Magadha and later on lived and worked in Ujjayinī. He flourished in the middle of the sixth century A.D. His astrology and *horāśāstra* (horoscopes) were of Greek origin, and it is therefore not surprising that his works abound in Greek technical terms.

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Bhāskara I (c. A.D. 600)

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The greatest exponent of Āryabhaṭa’s system of astronomy, Bhāskara I was a contemporary of Brahmagupta. T. S. Kuppanna Sastri placed him between A.D. 550 and 628. Shukla showed that he lived beyond this limit, his commentary on the *Āryabhaṭīya* being written in A.D. 629. Stray references to his works appear to indicate his association with Surāṣṭra (Western India) and Aśmaka (South India, possibly Kerala). It is possible that he was a native of either of these two regions and migrated to the other.

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Bhāskara I wrote three works, e.g. the *Mahābhāskarīya*, the *Laghubhāskarīya* and a *bhāṣya* on the *Āryabhaṭīya*. The *Mahābhāskarīya* is an elaborate exposition of the three astronomical chapters of the *Āryabhaṭīya*, arranged in eight chapters devoted to the following topics: (1) Mean longitude of planets and indeterminate analysis (*kuṭṭaka*); (2) Longitude correction; (3) Time, place and direction, spherical trigonometry, latitudes and longitudes of junction-stars; (4) True longitudes of planets; (5) Solar and lunar eclipses; (6) Rising, setting and conjunction of planets; (7) Astronomical constants; and (8) *tithi* and miscellaneous examples. The work is occasionally characterized by innovations and the author’s own methods. For example, to find the mean longitudes of planets, the *śīghra* of venus and mercury and the perigee and node of the moon, Bhāskara I gives a new method called the *pratyabda-śodhana*. It is true Āryabhaṭa gave rules for indeterminate analysis (*kuṭṭaka*), but it was the unquestionable merit of Bhāskara I to fully explain the method and its application to astronomy. In his formulą for the equation of the centre, he uses *manda karṇa* or hypotenuse to obtain the correction more accurately, whereas other Indian astronomers remained generally satisfied with the approximate value due to the use of the radius only.

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The *Laghubhāskarīya*, as the name implies, is an abridged version of the author’s fuller work noticed above. Intended for the beginner, its treatment is more systematic. His *Bhāṣya* is an elaborate and at the same time a learned commentary on the *Āryabhaṭīya* and abounds in quotations from *Vyākaraṇa*, *Vedānta*, *Mīmāṃsā*, *Arthaśāstra*, *Manusmṛti* and similar texts. The great popularity of Bhāskara’s works, particularly in the south, is attested by several extant commentaries, of which mention should be made of those by Govindasvāmin (c. 800–850), Śaṅkaranārāyaṇa (c. 825–875), Udayadivākara (c. 1073) and Parameśvara (c. 1360–1460).

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Brahmagupta (c. A.D. 598)

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In his *Brāhmasphuṭa-siddhānta*, Brahmagupta states that he was the son of Jiṣṇugupta, lived in the reign of King Vyāghramukha, and composed this astronomical work in śaka 550 (A.D. 628) at the age of 30. According to al-Bīrūnī, Brahmagupta was from Bhillamala, a town between Multan and Ahilwara. Bühler identified Bhillamala with modern Bhīnmāl or Śrīmāl near the northern frontier of Gujarat.

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The *Brāhmasphuṭa-siddhānta* is a voluminous work comprising 1,008 verses (or 1,022 verses if the last chapter on generalities is included) and divided into 24 chapters. The main astronomical topics are dealt with in the first 10 chapters in the following order: mean planetary motions, true planetary motions, problems of time, space and distance, lunar and solar eclipses, risings and settings of planets, the moon's cusps and shadows, conjunctions of planets. Some of these topics are further discussed in chapters 13-17, 19-21. Chapter 22 is devoted to astronomical instruments. Chapters 12 and 18 deal with mathematics and reveal Brahmagupta's excellence and originality as a mathematician.

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The chapter 11, entitled *Tantra-parīkṣādhyāya* (examination of other astronomical systems), is full of historical importance. Fully conscious of his own abilities, Brahmagupta criticized the views of, and rules and methods given by, other astronomers, notably Āryabhaṭa I, Śrīṣeṇa, Viṣṇucandra, Lāṭa and Pradyumna. He was also very critical of foreign astronomical concepts and elements which by then had found their way into Indian astronomical works. He rejected the five-year *yuga* system of the *Vedāṅga Jyotiṣa* and criticized the absurd views of the Jainas who believed in the two suns and moons and two sets of *nakṣatras*. His undue bitterness towards Āryabhaṭa did not fail to attract the notice of al-Bīrūnī who, in spite of his regard for Brahmagupta, freely acknowledged the merit of Āryabhaṭa and his school. Brahmagupta attacked Āryabhaṭa for dividing the *yuga* into four equal parts, for upholding the rotatory motion of the earth, for believing in the eclipses being caused by the shadows of the moon and the earth and not in accordance with the traditional *Rāhu-Ketu* theory. All this is understandable if we remember that he followed in the main the *Brahma-siddhānta* and wrote his work at the age of 30 when he was certainly an enthusiastic supporter of the orthodox view.

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From this position he certainly travelled a long way when he composed at the age of 67 his *Khaṇḍakhādyaka*, based principally on Āryabhaṭa's *ārdharātrika* system. Compared to his earlier work it is a small tract in eight chapters, to which a small appendix (*Uttara Khaṇḍakhādyaka*) is added. Here the mathematical chapters are omitted and purely astronomical topics are dealt with in the following order: *tithis* and *nakṣatras*; mean and true places; time, space and direction; lunar and solar eclipses; rising and setting of planets; the moon's cusps; conjunction of planets. Although he reconciled himself, by and large, to Āryabhaṭa's system, he nevertheless followed his own rules in a number of cases, which he considered to provide better and more accurate results. He gave methods for finding the instantaneous daily motion of planets affected by *manda* and *śīghra* inequalities, correct equations for parallax in longitude and latitude, correct equations for *dṛkkarma* and better expressions for *valana*. In all this a better mathematician was obviously at work.

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Brahmagupta's works played an important part in the introduction among the Arabs of a new mathematics-based astronomy. Both *Brāhmasphuṭa-siddhānta* and *Khaṇḍakhādyaka* were translated into Arabic, with the assistance of Hindu pandits, by Muḥammad ibn Ibrāhīm al-Fazārī (d. 796 or 800) and Yaʻqūb ibn Ṭāriq (d. 796) under the Arabic titles of *Sindhind* and *Arkand* respectively. These early Arabic translations were defective and corrupt, but none the less exerted a great influence among the Arab scholars. Indian commentators of Brahmagupta include Pṛthūdakasvāmī, Āmarāja, Bhaṭṭotpala, Lalla, Someśvara, Śrīdatta and Varuṇa. Pṛthūdakasvāmī commented on both the works, whereas the rest concentrated on his *Khaṇḍakhādyaka* only, which incidentally proves its greater popularity.

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*Vaṭeśvara* (b. A.D. 880), *Mañjulācārya* (c. A.D. 932), *Āryabhaṭa II* (c. A.D. 950), *Śrīpati* (c. A.D. 999), *Śatānanda* (c. *eleventh century*)

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Several astronomers flourished between the time of Brahmagupta and Bhāskara II (twelfth century), who made varying contributions to the subject. Of them mention may be made of Vaṭeśvara, Mañjulācārya (also called Muñjāla), Āryabhaṭa II, Śrīpati and Śatānanda. Vaṭeśvara's importance lies mainly in his criticism of Brahmagupta. A follower of Āryabhaṭa's system, he attacked Brahmagupta in the same manner as the latter had attacked his master about 250 years before. The chapter 10 of section 1 of his *Vaṭeśvara-siddhānta* is devoted exclusively to criticizing and refuting the views of Brahmagupta. The book is written in three principal sections, e.g. mean motions, true motions and problems of time, space and distance, each divided into a number of chapters. From his own statement we learn that he was the son of Mahādatta Bhaṭṭa, a native of Ānandapura in the Panjab, and was born in śaka 802. Author of *Laghumānasa* in six chapters, Mañjulācārya introduced into Indian astronomy the corrections due to the precession of the equinoxes. This precession is in the sense of retrograde motion and not of oscillation, as we have already noticed. The author of one *Ārya-siddhānta* or *Mahā-siddhānta* calls himself Āryabhaṭa who flourished about the middle of the tenth century. The work is composed in 18 chapters and devoted to familiar astronomical topics. It has a special chapter on *kuṭṭaka*, i.e. integral solutions of indeterminate equations of the first degree. Āryabhaṭa II was a compiler and did not possess the merit of his illustrious predecessors. He did not also follow Āryabhaṭa I, but adhered instead to the orthodox views. He mentioned the precession of the equinoxes. Śrīpati, son of Nāgadeva, is well known for his *Dhīkoṭī*, a *karaṇa* work based on the *Āryabhaṭīya*, a fuller astronomical work *Siddhānta-śekhara* divided into 20 chapters and his *Gaṇitatilaka*, a work on mathematics. Śrīpati considered, among other things, the moon's second inequality. Śatānanda flourished in the eleventh century in Puri on the coast of the Bay of Bengal. His *Bhāsvatī* was composed more or less in imitation of the *Sūrya-siddhānta* and attained a measure of popularity among the almanac-makers. *Bhāsvatī* is a *karaṇa* work.

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*Bhāskara II (b.* A.D. 1114)

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Bhāskarācārya or Bhāskara II, to distinguish him from his namesake of the sixth/seventh century, was born in A.D. 1114 in Vijapur in the province of Karnata on the Western Ghats. His *Siddhānta-śiromaṇi* written in A.D. 1150 when he was 36 readily surpassed similar efforts by his predecessors in the field of astronomy and mathematics in lucidity of exposition, depth of treatment and occasional flashes of originality. The work is arranged in four parts, of which the first two, the *Līlāvatī* and the *Bījagaṇita*, deal with arithmetic and algebra respectively. Astronomy proper is treated in the two remaining sections, e.g. the *Gaṇitādhyāya* and the *Golādhyāya*. In the former section, the mean and the true motions of planets, the three problems relating to time, direction and place, the eclipses, risings, settings and the conjunctions of planets, etc., are treated in the same sequence as met with in the *Sūrya-siddhānta*. The *Golādhyāya*, or the chapter on the sphere where the same problems reappear, is more important from the viewpoint of theoretical astronomy. The epicyclic-eccentric theories to account for planetary motions are fully developed. The astronomical instruments described in the *Yantrādhyāya* are more numerous and perfect than those given in the *Sūrya-siddhānta* and other works.

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Sufficiently clear and detailed as his versified rules were, Bhāskara II wrote his own commentary *Vāsanābhāṣya* to further explain and illustrate his own rules wherever he thought necessary. In A.D. 1183, he wrote another work on planetary motions under the title *Karaṇakutūhala*. His works, both in their entirety and in parts, are available in a large number of manuscripts. Several commentaries were also produced later on by such medieval astronomers as Gaṇeśa, Lakṣmīdāsa, Munīśvara, Nṛsiṁha, Raṅganātha, Viśvanātha and others, which kept alive Bhāskara II's popularity for several centuries throughout the length and breadth of India.

Secondary Works and Commentaries

c6.s3.t3.1

The Sanskrit astronomical literature of the medieval period is extremely rich in secondary works and commentaries. The majority of this class of literature, it is true, was produced after Bhāskara II particularly from the fourteenth or fifteenth century, but it is highly unrealistic to draw any such line of demarcation. It is not quite true that the astronomical literature produced after Bhāskara II was barren of originality. As we have seen, some of the original authors before the twelfth century appeared as excellent commentators. Bhāskara I himself spent much of his time in elucidating his master Āryabhaṭa's works. In the ninth century A.D. we have Brahmagupta's scholiast Pṛthūdakasvāmī (c. 864) who wrote two important commentaries, *Brāhmasiddhānta-vāsanābhāṣya* and *Khaṇḍakhādyaka-vivaraṇa*. In the same century Govindasvāmī (*c.* 800–850) and his pupil Śaṅkaranārāyaṇa, of the Āryabhaṭa school in the south, elucidated the works of Bhāskara I. The former, in addition to his *Mahābhāskarīya-bhāṣya*, wrote a few other astrological tracts, whereas the latter commented upon the *Laghubhāskarīya*. At the beginning of the thirteenth century Āmarāja wrote a commentary on the *Khaṇḍakhādyaka*.

c6.s3.t3.2

In the fourteenth century, Mahendra Sūrī (*c.* a.d. 1320), a disciple of Madana Sūrī and native of Vṛgupura, flourished in the court of Emperor Firoz Shah Tughlak as one of his principal court astronomers. He wrote a tract entitled *Yantrarāja* or *Yantrarājāgama* which dealt with astronomical instruments based mainly on Persian sources, in the course of the following five chapters, *Gaṇita*, *Ghaṭanā*, *Yantraracanā*, *Yantraśodhana* and *Yantravicāraṇa*. His disciple Malayendu Sūrī prepared on it a gloss called *Yantrarājaṭīkā*.

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