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Jyotihshastra: Astral and Mathematical Literature c4: Chapter II. Astronomy

Jyotiḥśāstra: Astral and Mathematical Literature c4

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English

Contents

Chapter II. Astronomy

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¹ A detailed survey of this field is given in D. Pingree, "History of Mathematical Astronomy in India," DSB, vol. 15, New York 1978, pp. 533—633 (henceforth cited as "History").

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Many Vedic sacrifices are to be performed at specific times determined by the position of the Sun relative to its northern (*uttarāyana*) or southern (*dakṣiṇāyana*) path, the synodic month and the night within it, or the position of the Moon with respect to the *nakṣatras*. The first attempt in India to describe the motions of the Sun and the Moon relative to the *ayanas*, the *nakṣatras*, and each other was the Jyotiṣavedāṅga composed by Lagadha. He clearly states his purpose (JV Ṛk 36; repeated at the end of Paitāmahasiddhānta 9, 8):

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For the Vedas have come forth for the sake of sacrifice, and sacrifices are established in the orderly succession of times. Therefore, he who knows this *jyautiṣa*, the science of determining times, knows the sacrifices.

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This purpose shifted somewhat later on when it was no longer as important to decide when Vedic sacrifices should take place as it was to fix the proper moments (*muhūrta*) for performing the *saṃskāras* and to compute the calendar (*pañcāṅga*) which indicated festivals, auspicious and inauspicious times (*tithis*, *yogas*, or *karaṇas*), entries of the Sun into zodiacal signs (*saṅkrāntis*), eclipses (*grahaṇas*), and the like. Another motive for the development of astronomy in India, at least after the second century A.D., was the need to compute with reasonable accuracy the positions of the planets for the purpose of casting horoscopes. And a final goal of the astronomers, though only to a slight degree realized, was the application of this science to geographical problems. Such practical aims, of course, must often have been secondary to individuals whose primary motivation was intellectual curiosity.

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The history of Indian astronomy can be divided into five main periods, depending in most cases on the foreign origin:

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1. Vedic (ca. 1000 B.C.—400 B.C.), in which some of the basic Indian calendaric terms were introduced,² but in which only a crude and rudimentary mathematical structure is evident. As there is no astronomical literature as such from this period it shall not be considered further in this book.

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² "History," pp. 534—536.

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2. Babylonian (ca. 400 B.C.—200 A.D.).

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3. Greco-Babylonian (ca. 200–400).
4. Greek (ca. 400–1600).
5. Islamic (ca. 1600–1800).

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Though the fundamental approach and many of the models and parameters of each period were determined by the foreign sources, the basic traditions of Indian astronomy imposed on these external systems its peculiar stamp, and transformed the science of Mesopotamia, Greece, or Iran into something unique to India.

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*Babylonian*³

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³ “History,” pp. 536–538. ⁴ The Yajus recension with variants from the Ṛk and the generally unrewarding commentary by Somākara Śeṣanāga was edited by A. WEBER, Über den Vedakalender Namens Jyotisham, Berlin 1862; the text of the Yajus recension along with the non-Yajus verses of the Ṛk recension were edited by G. THIBAUT, “Contributions to the Explanation of the Jyotisha-Vedāṅga,” JAS Bengal 46 (1877), 411–437; the Ṛk recension with a Marāṭhī translation was published by J. B. MOḌAKA of Thana in 1885; both recensions were edited and commented on by BĀRHASPATYA (LĀLA CHOṬELĀL) in the Hindustan Review for 1907 (reprinted Allahabad, 1960) and again, with Somākara’s Bhāṣya on the Yajus, by S. DVIVEDIN (Benares, 1908). The Yajus recension was published with his own Sanskrit commentary and an English version by R. SHAMASASTRY (Mysore, 1936). ⁵ The fundamental interpretation of the JV is D. PINGREE, “The Mesopotamian Origin of Early Indian Mathematical Astronomy,” JHA 4, 1973, 1–12. It has recently been most unconvincingly argued that the two Mesopotamian instruments for measuring time used in this period—the outflowing water-clock (ghaṭikā) and the gnomon (śaṅku)—were indigenous to India; see S. S. LISHK and S. D. SHARMA, “Season Determination through the Science of Sciatherics in Jaina School of Astronomy,” IJHS 12 (1977), 33–44, and “Length of the Day in Jaina Astronomy,” Centaurus 22 (1978), 165–176.

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The literature on astronomy in Sanskrit is headed by the Jyotiṣavedāṅga, which has come down to us in two recensions: that of the Ṛgveda in 36 verses, ascribed to Lagadha (or to Śuci following Lagadha), and that of the Yajurveda in 43 or 45 verses, of which 29 occur also in the Ṛgveda recension.⁴ The earlier of these two recensions is clearly the shorter, that of the Ṛgveda, as it alone preserved the original period relation of Lagadha’s intercalation cycle:

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5 solar years = 1830 sidereal days = 62 synodic months = 1860 tithis (JV, Ṛk 32 = Yajus 5; Ṛk 4 = Yajus 13; Ṛk 18 = Yajus 39; Ṛk 8 = Yajus 9; and Ṛk 12 = Yajus 27).⁵

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The Yajus has introduced two verses (28–29) that misinterpret this relation by taking the 366 sidereal days (= 365 civil days) in each year to be themselves civil days.

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The age of Lagadha’s work can be fixed from the following considerations. Its language is post-Vedic, and it imitates Piṅgala’s Chandaḥsūtra in using the final or first syllables of the names of the nakṣatras as their designations;⁶ its period relation is copied in the oldest Paitāmahasiddhānta, whose epoch is 11 January 80;⁷ and its astronomy reflects that of Mesopotamia in the Achaemenid period. It is likely, therefore, that it was composed not very many years before or after 400 B.C., when the Achaemenids controlled Gandhāra.

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⁶ WEBER, pp. 5–7.

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⁷ PS 12.

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Lagadha has retained the Vedic list of twenty-seven nakṣatras beginning with Kṛttikā, but treats them not as constellations, but as measurements of arcs on the ecliptic of 13;20° each beginning with the vernal equinox. This adaptation means, of course, that the actual position of the equinoctial and solsticial colures with respect to the fixed stars can not be used for dating the JV. He introduced into Indian astronomy the Mesopotamian concepts of the possibility of describing mathematically the periodicity of celestial motions; their use of linear zig-zag functions to describe periodic deviations from the mean (Ṛk 7 = Yajus 8 and Ṛk 22 = Yajus 40); and their artificial time-unit, a thirtieth of a mean synodic month (named tithi in Sanskrit). He gives rules for computing the mean longitudes of the Sun and the Moon in terms or nakṣatras, but mentions neither the planets nor the zodiacal signs. Other texts that reflect this level of astronomy are Kauṭilya’s Arthaśāstra (2, 20) and the Buddhist Śārdūlakarṇāvadāna (Divyāvadāna 33). The revised system of the Yajus recension is used in the Jaina Sūriyapannatti and Caṃḍapannatti and in the verses ascribed to Garga in Somākara’s Bhāṣya.

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*Greco-Babylonian*⁸

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⁸ “History,” pp. 538–554.

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Planetary astronomy was introduced into India from Greece in conjunction with astrology as it was essential to possess some means of determining planetary positions in order to be able to cast horoscopes. The form of planetary astronomy originally transmitted represented Greek adaptations of the Babylonian astronomy of the Seleucid period; this was supplemented by other elements of early Greek astronomy, especially elements associated with Hipparchus.⁹ Some of the calendaric and other timekeeping conventions of the JV were also adopted by the practitioners of this type of astronomy in India.

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⁹ D. PINGREE, “The Recovery of Early Greek Astronomy from India,” JHA 7 (1976), 109–123, esp. 111–115.

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Two Sanskrit texts have preserved for us what remains of Greco-Babylonian astronomy in India. Sphujidhvaja (269/70) describes some aspects of it in the last chapter (79) of the Yavanajātaka,¹⁰ a text based in large part on the translation of a Greek text from Alexandria made by Yavaneśvara in 149/150; Sphujidhvaja (YJ 79, 3) refers to a *siddhānta* by Vasiṣṭha. Varāhamihira in the sixth century summarized in his Pañcasiddhāntikā¹¹ a Vasiṣṭhasiddhānta whose epoch is 3 December 499 (PS 2; and, probably, 17, 1–60), a Romakasiddhānta (*siddhānta* of the Romans) of the third or fourth century (PS 1, 9–10 and 15; and 8), and a Pauliśasiddhānta (*siddhānta* of Paulus ?) of the same period (PS 1, 11–13; 3; and 17, 64–80). The last two *siddhāntas* were known to Varāhamihira in recensions produced by Lāṭadeva; his epoch for them was 21 March 505. The original version of the Romakasiddhānta was probably the one used by Bhāskara (629) in his Bhāṣya on A (pp. 183, 186 and 202).

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¹⁰ D. PINGREE, The Yavanajātaka of Sphujidhvaja, 2 vols., Cambridge, Mass. 1978.

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¹¹ O. NEUGEBAUER and D. PINGREE, The Pañcasiddhāntikā of Varāhamihira, 2 vols., Copenhagen 1970–71. See also K. S. SHUKLA, "The Pañcasiddhāntikā of Varāhamihira (I)," Gaṇita 24 (1973), 59–73, and IJHS 9 (1974), 62–76, and T. S. KUPPANA SASTRI, "Some Misinterpretations and Omissions by Thibaut and Sudhakara Dvivedin in the Pañcasiddhāntikā of Varāhamihira," VIJ 11 (1973), 107 to 118.

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The Babylonian component of the astronomy expounded in these works is especially noticeable in the planetary theory which is based on the well known period relations for the planets, and the divisions of the synodic periods at the Greek-letter phenomena (YJ 79, 35–36, and PS 17), and in the elaborate lunar theory of the Vasiṣṭhasiddhānta (PS 2, 2–6) and the Pauliśasiddhānta (PS 3, 4–9). It would not have been possible to compute planetary positions with only the information in the texts as extant; their original forms were presumably more complete. The Hipparchan elements include the year-length 6,5;14,48 civil days (probably YJ 79, 34), which is combined with the Metonic cycle in the Romakasiddhānta (PS 1, 9–10 and 15); values close to his maximum solar and one of his maximum lunar equations, also in the Romakasiddhānta (PS 8); and the use of analemmata to solve problems in spherical trigonometry in the Pauliśasiddhānta (PS 3). Also from Hipparchus came the concept of the precession and trepidation of the equinoxes,¹² and, most likely, the polar coordinate system for fixing the positions of the fixed stars and the value 3438 for the Radius in tables of Sines.¹³ This value of R first appears in India in the early fifth century (Pait. 3, 12); Varāhamihira uses R = 120 (PS 4, 6–15).

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¹² D. PINGREE, "Precession and Trepidation in Indian Astronomy before A.D. 1200," JHA 3 (1972), 27–35.

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¹³ G. J. TOOMER, "The Chord Table of Hipparchus and the Early History of Greek Trigonometry," Centaurus 18 (1973), 6–28.

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The other main topics dealt with in these texts, in so far as we know them, were the computations of lunar (PS 6: Pauliśasiddhānta) and solar (PS 7: Pauliśasiddhānta; and 8: Romakasiddhānta) eclipses. Both seem to reflect Hellenistic astronomy as it was influenced by Mesopotamia, though no extant Greek texts can be pointed to as sources. It is likely that the formulas for solving problems in spherical trigonometry and the analemmata that lie behind them in PS 4 also were originally transmitted to India from the Roman Empire.

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Greek¹⁴

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¹⁴ “History,” pp. 555–625.

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By the early fifth century when the Brāhmapakṣa began there had been introduced into India a more sophisticated form of Greek astronomy characterized by planetary models comprising the combination of an eccentric deferent with an epicycle (but without Ptolemy’s equant or special model for Mercury), or a Peripatetic model with two independent epicycles whose centers are carried by a concentric deferent,¹⁵ or one in which each inequality is accounted for by a concentric circle with an equant.¹⁶ The inequalities in the motions of the Sun and the Moon are computed by means of an eccentric, a concentric deferent with an epicycle (the preferred model), or a concentric circle with an equant; in the earlier part of the Greek period Ptolemy’s second lunar inequality was not known, though it was introduced into India from Islamic sources in the tenth century.

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¹⁵ D. PINGREE, “On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle,” JHA 2 (1971), 80–85.

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¹⁶ D. PINGREE, “Concentric with Equant,” AIHS 24 (1974), 26–29.

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As these geometric models of planetary motion based on the idea of the circularity of their orbits were introduced into India, it became necessary to modify the traditional Indian cosmology as expressed in the Purāṇas and other texts. This was done by transforming the disc of Jambūdvīpa into a sphere and Meru into the terrestrial North Pole; along the equator at 90° distance from each other lie Laṅkā on the prime meridian, Romakaviṣaya, Siddhapura, and Yamakoṭi; opposite Meru at the South Pole is Vaḍavāmukha. The axis of the universe passes through Meru and Vaḍavāmukha and the poles of the spheres of the planets and of the nakṣatras; the spheres of the planets rotate from West to East at a uniform velocity, driven by bonds of wind, while they and the sphere of the nakṣatras rotate diurnally from East to West. The deviations of the planets from their mean motions are caused by demons stationed at their manda (accounting for the equation of the center) and śīghra (accounting for the equation of the anomaly) uccas pulling them towards themselves by cords of wind. The relative distance of each planet’s sphere from the center of the system—that is, the center of the earth—can be computed by comparing their sidereal periods since they move equal spatial distances (but unequal angular distances) in equal times. Absolute values for planetary distances depend on the assumption of an arbitrary number of yojanas in the orbit of one of them (that of the Moon was normally chosen). In order to retain the Greek order of the planets in which the Sun is further distant from the earth than Venus, the Indians operated with the motion of Venus’ (and then, by analogy, Mercury’s) śīghra; in Ptolemaic terms the śīghra for an inferior planet is the sum of its (i.e., the Sun’s) mean motion and its anomalistic motion.

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In order to compute the mean longitudes of the planets the Indians adapted the Greek idea of a “great year” which begins and ends with a mean planetary conjunction to the already existing theory of the Kalpa.¹⁷ The choice of the subdivisions of the Kalpa (see Table 1) and of the number of rotations of each of the planets within it (see Table 2) are two of the main features that distinguish one pakṣa or school of astronomy from another, the others being the geometrical models for planetary motion and their parameters (see Table 3). It was demonstrated by R. Billard that the mean longitudes computed by the two systems of Āryabhaṭa converge in accuracy on about 500, and this led him to suggest that Āryabhaṭa founded Indian astronomy in the form which we have called "Greek" on the basis of his own extraordinarily precise observations.¹⁸ However, the same phenomenon is explained in a way that fits better into the known facts of the history of Indian astronomy: that "Greek" astronomy existed in India before Āryabhaṭa; that the geometrical models solve problems faced by Greeks rather than by Indians, and must have been accompanied by means of computing mean longitudes when they were transmitted to India; and that all of the Indian evidence is against the theory of their being able to make accurate observations. This alternate explanation assumes that two procedures were used.¹⁹ One, utilized by the Brāhmapakṣa, consists of an expansion of Babylonian-type period relations to numbers of rotations of the planets in a Kalpa by assuming a fixed time between the beginning of the Kalpa and the beginning of the current Kaliyuga on 18 February - 3101, and a close proximity to a mean conjunction of the planets on the latter date. This problem results in a series of indeterminate equations that can be solved by the kuṭṭaka. Āryabhaṭa could have arrived at his surprising accuracy by using a reasonably accurate set of Greek astronomical tables to compute the mean longitudes of the planets on 21 March 499, and by assuming a true mean conjunction of the planets 3600 years earlier, on 18 February - 3101.

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¹⁷ D. PINGREE, "Astronomy and Astrology in India and Iran," Isis 54 (1963), 229-246.

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¹⁸ R. BILLARD, L'astronomie indienne, Paris 1971.

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¹⁹ D. PINGREE in JHA 7 (1976), 116-121.

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The five main pakṣas that developed in India are the Brāhma,²⁰ the Ārya,²¹ the Ārdharātrika,²² the Saura,²³ and the Gaṇeśa;²⁴ the Brāhma and the Saura both were revised in the medieval period. There are, as well, some texts which belong to no pakṣa,²⁵ and some derived from Islam.²⁶ These find their expression in various literary forms, which were classified under three main headings from the seventh century on. Siddhāntas are comprehensive treatises deducing mean motions from the beginning of the Kalpa or the current Kaliyuga; karaṇas are more concise expositions of astronomy in which the mean longitudes for a time close to the date of composition are given, and the mean longitudes at later times are computed therefrom; and koṣṭhakas are sets of astronomical tables, for determining planetary positions or for solving other problems in astronomy. There are as well many specialized works on aspects of astronomy, of which the most important group is on observational instruments (yantra).

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²⁰ "History," pp. 555-589.

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²¹ "History," pp. 590-602.

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²² "History," pp. 602-608.

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²³ "History," pp. 608-618.

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²⁴ "History," pp. 624-625.

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²⁵ "History," pp. 618-624.

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²⁶ "History," pp. 625-629, and D. PINGREE, "Islamic Astronomy in Sanskrit," JHAS 2 (1978), 315-330.

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Brāhma:

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| Kalpa | = 4,320,000,000 years |
| Kalpa | = 14 Manvantaras |
| | (Manvantara = 71 Mahāyugas = 306,720,000 years) |
| | + 15 Sandhis (Sandhi = Kṛtayuga = 1,728,000 years) |
| Kalpa | = 1000 Mahāyugas or Caturyugas |
| | (Mahāyuga = 4,320,000 years) |
| Mahāyuga = | Kṛtayuga (1,728,000 = 432,000 · 4 years) |
| | + Tretāyuga (1,296,000 = 432,000 · 3 years) |
| | + Dvāparayuga (864,000 = 432,000 · 2 years) |
| | + Kaliyuga (432,000 years) |
| Beginning of Kalpa to beginning of current Kaliyuga: | |
| | 1,972,944,000 = 432,000 · 4567 years |

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Ārya = Ārdharātrika:

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| Kalpa | = 4,354,560,000 years |
| Kalpa | = 14 Manvantaras |
| | (Manvantara = 72 Mahāyugas = 311,040,000 years) |
| Kalpa | = 1008 Mahāyugas (Mahāyuga = 4,320,000 years) |
| Mahāyuga = | Kṛtayuga (1,080,000 years) |
| | + Tretāyuga (1,080,000 years) |
| | + Dvāparayuga (1,080,000 years) |
| | + Kaliyuga (1,080,000 years) |

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Saura: same as Brāhma, but with an initial period of non-motion at the beginning of the Kalpa amounting to 17,064,000 years.

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Rotations of the planets in a Mahāyuga

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| | Brāhma | Ārya | Ārdharā-trika | Saura | Adjusted Saura |
| Saturn | 146,567.298 | 146,564 | 146,564 | 146,568 | 146,580 |
| Jupiter | 364,226.455 | 364,224 | 364,220 | 364,220 | 364,212 |
| Mars | 2,296,828.522 | 2,296,824 | 2,296,824 | 2,296,832 | 2,296,832 |
| Venus’ śīghra | 7,022,389.492 | 7,022,388 | 7,022,388 | 7,022,376 | 7,022,364 |
| Mercury’s śīghra | 17,936,998.984 | 17,937,020 | 17,937,000 | 17,937,060 | 17,937,076 |
| Moon | 57,753,300.000 | 57,753,336 | 57,753,336 | 57,753,336 | 57,753,336 |
| Lunar node | —232,311.168 | —232,226 | —232,226 | —232,238 | —232,246 |

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¹ The revised Brāhmapakṣa and the Gaṇeśapakṣa do not express their mean motions in terms of integer numbers of rotations in a Kalpa or a Mahāyuga.

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Table 3

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A. Mandoccas and pātas[1]

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| | Brāhma | Ārya | Ārdharātrika | Saura | Gaṇeśa |
| Saturn | | | | | |
| ucca | 260;55° | 236° | 240° | 236;37° | 240° (8 · 30) |
| pāta | 103;12° | 100° | | 100;24° | |
| Jupiter | | | | | |
| ucca | 172;32° | 180° | 160° | 171;18° | 180° (6 · 30) |
| pāta | 82;1° | 80° | | 79;41° | |
| Mars | | | | | |
| ucca | 128;24° | 118° | 110° | 130;2° | 120° (4 · 30) |
| pāta | 31;54° | 40° | | 40;4° | |
| Sun | | | | | |
| ucca | 77;55° | 78° | 80° | 77;17° | 78° |
| Venus | | | | | |
| ucca | 81;15° | 90° | 80° | 79;50° | 90° (3 · 30) |
| pāta | 59;47° | 60° | | 59;43° | |
| Mercury | | | | | |
| ucca | 224;54° | 210° | 220° | 220;27° | 210° (7 · 30) |
| pāta | 21;11° | 20° | | 20;43° | |

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¹ These rotate slowly in the Brāhma and Saura pakṣas; the parameters given are for 628 and 850 respectively.

B. Maximum Equations

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¹ Deviations from these basic parameters occur in some texts of each *pakṣa*.

I. Manda

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| | Brāhma | Ārya | Ārdharātrika | Saura | Gaṇeśa |
| Saturn | 4;46,47° | 9;32° | 9;36,55, ...° | 7;40° | 9;18° |
| Jupiter | 5;15,35° | 5;43° | 5;6° | 5;6° | 5;42° |
| Mars | 11;12,41° | 13;7° | 11;13° | 11;32° | 13;0° |
| Sun | 2;10,30° | 2;9° | 2;14° | 2;10,32° | 2;10,45° |
| Venus | 1;45,3° | 1;26° | 2;14° | 1;45° | 1;30° |
| Mercury | 6;33,33° | 3;35° | 4;28° | 4;28° | 3;36° |
| Moon | 5;2,7° | 5;1° | 4;56° | 5;2,48° | 5;1,40° |

II. Śīghra

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| | Brāhma | Ārya | Ārdharātrika | Saura | Gaṇeśa |
| Saturn | 5;34,46° | 5;44° | 6;20° | 6;22° | 5;42° |
| Jupiter | 10;53,19° | 10;53° | 11;30° | 11;31° | 10;48° |
| Mars | 42;37,39° | 44;53° | 40;30° | 40;16° | 40;0° |
| Venus | 46;22,54° | 53;37° | 46;15° | 46;24° | 46;6° |
| Mercury | 21;31,30° | 21;57° | 21;30° | 21;31° | 21;12° |

C. Maximum Latitudes

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| | Brāhma | Ārya = Ārdharātrika = Saura |
| Saturn | 2;26° | 2;15° |
| Jupiter | 1;34° | 1;15° |
| Mars | 5;38° | 4;26° |
| Venus | 8° | 7;37° |
| Mercury | 4° | 1;36° |
| Moon | 4;30° | 4;30° |

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*Siddhāntas*

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The earliest *siddhānta* that has come down to us is a Paitāmahasiddhānta²⁷ of the early fifth century. This is preserved because of its incorporation into the Viṣṇudharmottarapurāṇa²⁸ (2, 166–174), though it also (or at least the astronomical section, chapters 168–174) has been copied as an independent treatise.²⁹ Its position as the source of the Brāhmapakṣa is made evident by Brahmagupta’s dependence on it in composing the first ten chapters (Daśādhyāyī) of the BSS, and was explicitly recognized by Kamalākara (Siddhāntatattvaviveka 1, 62). Its chronological position is fixed by the fact that Sassanian astronomers in about 450 knew one of its characteristic parameters;³⁰ this makes it more than likely that Pitāmaha is the Svayambhū referred to by Āryabhaṭa (A Gola 50), whose Kalpa-system is a modification of the Brāhmapakṣa’s (he even accepts the Brāhma’s rotation of the *uccas* and *pātas* [A Daśagītikā 7 and Gola 2] despite the fact that in his system that is unnecessary), whose table of Sines (A Daśagītikā 10 and Gaṇitapāda 12) is derived from the Pait. (3, 12), and who copies various other formulae from the same source.

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²⁷ CESS A 4.

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²⁸ Bombay, 1912.

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²⁹ Ed. by V. P. DVIVEDI in Jyautiṣasiddhāntasaṅgraha, BSS 39, 2, Benares, 1912; trans. by D. PINGREE, “The Paitāmahasiddhānta of the Viṣṇudharmottarapurāṇa,” Brahmavidyā 31–32 (1967–68), 472–510.

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³⁰ D. PINGREE, “The Persian ‘Observation’ of the Solar Apogee in *ca.* A.D. 450,” JNES 24 (1965), 334–336.

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The Pait. is cast in the form of a lecture by Brahmā to Bhṛgu. Its first two chapters deal with astrology. Chapter three discusses time-measurements, the parameters of the planetary system, Sines and Versines, the declination of the Sun and latitudes of the planets, gnomon-problems, right and oblique ascensions, a star catalogue, the Greek-letter phenomena of the planets, and the calendar; this confused congeries is also indicative of the age of the Pait. as later *siddhāntas* are arranged far more systematically. Chapter four covers the computation of the *ahargaṇa* (number of days lapsed since epoch), the astrological lords of the years, days, and *parvans*, the computation of the mean and true longitudes of the planets and of their latitudes, and the times of the occurrences of the planets’ first and second stations. Chapter five deals with the earth’s shadow (for lunar eclipses), *saṅkrāntis*, *tithis*, *karaṇas*, *yogas*, and the *vaidhṛta* and *vyatīpāta* *pātas*. Chapter six covers some problems in spherical trigonometry and their solution by means of analemmata. Chapter seven is devoted to the computation of the ascendant and of the time lapsed since sunrise. Chapter eight presents the method of computing the times and longitudes of the first and last visibilities of the planets and fixed stars. And chapter nine gives rules for computing the illumination of the Moon and longitudinal and latitudinal parallax (for solar eclipses). Several of these rules are incomplete or wrong as presented in the two editions, and only the preliminaries for computing lunar and solar eclipses are given. One suspects that the original Pait. was a more competent work than what we now possess of it.

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Āryabhaṭa³¹ was 23 years old in 3600 Kali = A.D. 499 (A Kālakriyā 10) and wrote his Āryabhaṭīya³² at Kusumapura (A Gaṇitapāda 1), which is identified by Bhāskara in his commentary on this verse with Pāṭaliputra (p. 45). The same Bhāskara, in a series of examples relating to the kuṭṭaka, refers to the Aśmakasya gaṇita (p. 138), or Aśmakīya (pp. 140, 146, 147, and 148), or Aśmaka as an authority on the subject, and he applies to the Āryabhaṭīya the epithet Aśmaka (e.g., in MB 1, 3). Scholars, beginning at least with Nīlakaṇṭha (on A Gaṇitapāda 1), have contended on this basis that Āryabhaṭa was born in the Aśmaka country between the Godāvarī and the Narmadā; but the epithet may only reflect Bhāskara’s own origin and training in Aśmaka.

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³¹ CESS A 1, 50b–53b; A 2, 15b; A 3, 16b; and A 4.

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³² Edited with the Bhaṭadīpakā of Parameśvara by H. KERN, Leiden 1875, reprinted (with a Hindi translation by UDAYA NARAYANA SINGH), Madhurapur, Etawah, 1906, and Osnabrück 1973; with the Bhāṣya of Nīlakaṇṭha (omitting the Daśagītikā) by K. SAMBASIVA SASTRI (2 vols., TSS 101 and 110) and S. K. PILLAI (TSS 185), Trivandrum 1930–1957; with Sanskrit and Hindi commentaries by B. MISHRA, Patna [1966]; with an English translation by K. S. SHUKLA and K. V. SARMA, New Delhi 1976; with the commentaries of Bhāskara (to Golapāda 6) and Someśvara (Golapāda 6–50) by K. S. SHUKLA, New Delhi 1976; and with the commentary of Sūryadeva Yajvan by K. V. SARMA, New Delhi 1976. The Gaṇitapāda was published with a German translation and commentary by K. ELFERING, Die Mathematik des Āryabhaṭa I, München 1975. There are English translations of the whole work by P. C. SENGUPTA, JDL/UC 16 (1927), art. 6, and by W. E. CLARK, Chicago 1930.

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The Āryabhaṭīya, which is the basic text of the Āryapakṣa, is not, strictly speaking, a siddhānta since it combines, in extreme conciseness, the fundamentals of both astronomy and mathematics. It is divided into four pādas. The Daśagītikā gives the planetary parameters, the divisions of the Kalpa, and a table of the first order differences of the Sines; at one point Āryabhaṭa refers to the diurnal rotation as a rotation of the earth (A Daśagītikā 3; see also Golapāda 9–10) but elsewhere (A Golapāda 16) he mentions the mathematically equivalent theory of the rotation of the fixed stars (bhagola). The theory of the diurnal rotation of the earth was fiercely attacked by many later Indian astronomers, beginning with Varāhamihira (PS 13, 6–7). The Gaṇitapāda, on mathematics, will be briefly discussed in chapter III. The Kālakriyāpāda defines time units, describes two geometrical models of planetary motion (eccentric with epicycle and concentric with two epicycles), and gives rules for computing the true longitudes of the planets. Finally, the Golapāda expounds Āryabhaṭa’s cosmology and geography, explains the apparent motions of the heavenly bodies, and gives solutions for problems in spherical trigonometry and rules for computing eclipses. The Āryapakṣa which evolved from the A was for most of its history popular mainly in South India, though the earliest extant commentaries were composed in Gujarāt—the Āryabhaṭīyabhāṣya of Bhāskara³³ at Valabhī in 629 and that of Someśvara in the eleventh or twelfth century; equally attesting to its popularity in Western India in the eighth and early ninth centuries is its use by Arabic astronomers of the period, who call Āryabhaṭa Arjabhar.³⁴ Al-Bīrūnī, however, while he possessed a copy of the A in the Panjāb in the 1020’s, was confused about its identity.³⁵ The later Indian commentators on the A, almost all of its manuscripts, and most of the texts following the Āryapakṣa³⁶ come from South India. The Sanskrit commentaries in addition to those named above were composed by Sūryadeva Yajvan, who was born in 1191, at Gaṅgāpura in Cola country; by Parameśvara³⁷ in about 1450 at Aśvatthagrāma in Kerala; by Yallaya at Skandasomeśvara in Āndhrapradeśa in 1480; by Nīlakaṇṭha³⁸ in Kerala shortly after 1501; by Raghunātha Rāja at Ahobila in Āndhrapradeśa in 1597; by Bhūtiviṣṇu,³⁹ probably near Kāñcī in Tamilnadu; and by Ghaṭīgopa⁴⁰ in Kerala in about 1800. There are also commentaries in Telugu and Malayālam and a translation into Marāṭhī.

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³³ CESS A 4.

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³⁴ D. PINGREE, “The Greek Influence on Early Islamic Mathematical Astronomy,” JAOS 93 (1973), 32—43, esp. 37—38.

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³⁵ D. PINGREE, “Al-Bīrūnī’s Knowledge of Sanskrit Astronomical Texts,” The Scholar and the Saint, New York 1975, pp. 67—81, esp. 68—69.

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³⁶ K. V. SARMA, “Tradition of Āryabhaṭīya in Kerala: Revision of Planetary Parameters,” IJHS 12 (1977), 194—199.

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³⁷ CESS A 4.

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³⁸ CESS A 3, 175b—177b, and A 4.

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³⁹ CESS A 4.

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⁴⁰ CESS A 2, 147a—147b, and A 4.

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Āryabhaṭa wrote a second work that formed the basis of the Ardharātrikapakṣa, apparently entitled Āryabhaṭasiddhānta. This is now lost, though a number of astronomers have mentioned its views, and some commentators—especially on the Sūryasiddhānta—have quoted its verses.⁴¹ Especially prevalent in these quotations are verses describing Āryabhaṭa’s simple and crude observational instruments (e.g., by Mallikārjuna Sūri⁴² and by Tammayajvan⁴³) and his clepsydras (by Tammayajvan and by Rāmakṛṣṇa). The first treatise to follow the Ārdharātrikapakṣa of the Āryabhaṭasiddhānta was a revised version of the Sūryasiddhānta. This was composed by a pupil of Āryabhaṭa named Lāṭadeva⁴⁴ according to the information available to Al-Bīrūnī; and there is no good reason to doubt this attribution. As summarized by Varāhamihira in the PS it is a karaṇa rather than a siddhānta; its epoch is midnight 20/21 March 505. It is nuclear whether this form is that of Lāṭadeva’s original, or was imposed on it by Varāhamihira.

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⁴¹ See especially K. S. SHUKLA, “Āryabhaṭa I’s Astronomy with Midnight Day-reckoning,” Gaṇita 18 (1967), 83—105, and “Glimpses from the Āryabhaṭasiddhānta,” IJHS 12 (1977), 181—186.

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⁴² CESS A 4.

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⁴³ CESS A 3, 85a—86a.

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⁴⁴ PS ed. NEUGEBAUER—PINGREE, vol. 1, pp. 12—15; A with the commentary of Bhāskara ed. SHUKLA, pp. lxi—lxiv.

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The earliest extant *siddhāntas* to adhere to something approaching the classical form belong to the Āryapakṣa. They were composed by that Bhāskara who commented on the A at Valabhī in 629; of these the Mahābhāskariya⁴⁵ was written before the Āryabhaṭīyabhāṣya which quotes from it under the title Karmanibandha (pp. 33, 40, 131, 179, 203, 204, 205, and 225–226), while the Laghubhāskariya⁴⁶ summarizes the MB, and probably postdates the Āryabhaṭīyabhāṣya (and the BSS).

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⁴⁵ Edited with the commentary, Karmadīpikā, of Parameśvara by B. D. APATE, ASS 126, Poona 1945; with the Bhāṣya of Govindasvāmin and the super-commentary, Siddhāntadīpikā, of Parameśvara by T. S. KUPPANNA SASTRI, Madras GOS 130, Madras 1957; and with an English translation and commentary by K. S. SHUKLA, Lucknow 1960.

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⁴⁶ Edited with the commentary, Pārameśvara, of Parameśvara by B. D. APATE, ASS 128, Poona 1946; with the Vivaraṇa of Śaṅkaranārāyaṇa by S. VENKATA-SUBRAMONIA IYER and S. KOCHUKANJU ASARI, TSS 162, Trivandrum 1949; and with an English translation and commentary by K. S. SHUKLA, Lucknow 1963.

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An arrangement less rigid than that of the classical *siddhānta* is evident in the contents of the MB’s eight chapters:

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1. *ahargaṇa*; mean longitudes; planetary *kuṭṭaka*.
2. *deśāntara* correction.
3. spherical trigonometry; catalogue of stars.
4. planetary equations (including the model involving a concentric with equant); *tithis*, *karaṇas*, *nakṣatras*, *pātas*, and *yogas*.
5. solar and lunar eclipses.
6. first visibility and illumination of the Moon; first and last visibilities of the planets; planetary conjunctions.
7. parameters of the Āryapakṣa and of the Ārdharātrikapakṣa.
8. examples.

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The arrangement of the eight chapters of the LB, which is probably influenced by that of the Daśādhyāyī of the BSS, represents the first stage of the fully developed form:

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1. *ahargaṇa*; mean longitudes; *deśāntara* correction (*madhyama*).
2. planetary equations; *nakṣatras*, *tithis*, *karaṇas*, and *pātas* (*spaṣṭa*).
3. spherical trigonometry (*tripraśna*).
4. lunar eclipses (*candragrahaṇa*).
5. solar eclipses (*sūryagrahaṇa*).
6. first visibility and illumination of the Moon (*śṛṅgonnati*).
7. first and last visibilities and conjunctions of the planets (*udayāsta* and *grahayuti*).
8. catalogue of stars; conjunctions of planets with stars (*bhagrahayuti*).

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Both of these *siddhāntas* were, like most Āryapakṣa works, popular only in South India; it is there that most of their manuscripts were copied, and their commentaries written. These commentaries include that composed by Govindasvāmin⁴⁷ (ca. 800–850) in Kerala on the MB, that written by his pupil, Śaṅkaranārāyaṇa, at Kollapurī, Kerala, in 869 on the LB, that composed by Udayadivākara⁴⁸ in 1073 on the LB, and those composed by Parameśvara (ca. 1380–1460) in 1408 on the LB and after 1443 on the MB; Parameśvara also wrote after 1432 a super-commentary on the Mahābhāskarīyabhāṣya of Govindasvāmin.

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⁴⁷ CESS A 2, 143b–144a, and A 3, 35b. ⁴⁸ CESS A 1, 56b–57a. ⁴⁹ CESS A 4. ⁵⁰ Edited with his own Sanskrit ṭīkā by S. DVIVEDIN, Benares 1902, and with his own Sanskrit and Hindi ṭīkās and excerpts froın those of Pṛthūdakasvāmin and DVIVEDIN by R. SARMAN, 4 vols., New Delhi 1966. A new critical edition with the commentary of Pṛthūdakasvāmin, an English translation, and a commentary is being prepared by D. PINGREE. ⁵¹ Based on it was a Mahāsiddhānta which was translated into Arabic as Al-Zīj al-Sindhind al-kabīr in 771 or 773 by Muḥammad ibn Ibrāhīm al-Fazārī; see D. PINGREE, “The fragments of the Works of al-Fazārī,” JNES 29 (1970), 103–123. This tradition was also followed by al-Fazārī’s contemporary, Ya‘qūb ibn Ṭāriq; see D. PINGREE, “The Fragments of the Ya‘qūb ibn Ṭāriq,” JNES 27 (1968), 97–125. See also D. PINGREE, The Thousands of Abū Ma‘shar, London 1968. ⁵² Concerning the relation of the Sindhind tradition—particularly in the form of the Latin version of al-Khwārizmī’s Al-Zīj al-Sindhind and its commentaries—to the BSS and other Sanskrit works see D. PINGREE, “The Indian and Pseudo-Indian Passages in Greek and Latin Astronomical and Astrological Texts,” Viator 7 (1976), 141–195, esp. 151–169.

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Bhāskara’s contemporary rival as an astronomer was Brahmagupta,⁴⁹ who was born in 598 and completed his Brāhmasphuṭasiddhānta⁵⁰ in 628 at Bhillamāla in Rājasthāna during the reign of the Cāpa (Cāvoṭaka) monarch Vyāghramukha. The BSS, which belongs to the Brāhmapakṣa, was enormously influential on later Indian astronomy as well as on Islamic⁵¹ and Western European.⁵² It consists of twenty-four chapters, to which a twenty-fifth—a karaṇa—is appended in many manuscripts. These begin with a Daśādhyāyī (chapters 1–10), which summarizes and expands the teachings of the Pait. The relation to (and possible influence on) the LB will be clear from Table 4. This is followed by a chapter (11) in which Brahmagupta criticizes his predecessors and praises the Brāhmapakṣa (tantraparīkṣā) and another (12) on mathematics (gaṇita). Subject to the tantraparīkṣā’s animadversions are Āryabhaṭa, the Jainas, Pradyumna, Lāṭadeva, Varāhamihira, Vijayanandin, Viṣṇucandra’s Vasiṣṭhasiddhānta, Śrīṣeṇa’s Romakasiddhānta, and Siṃha. Additional information concerning those of these authors whose works are lost can be found in the PS, in the Āryabhaṭīyabhāṣya of Bhāskara, and in several later commentaries (e.g., those of Bhaṭṭotpala on the Bṛhatsaṃhitā and of Pṛthūdakasvāmin on the BSS). The next five chapters of the BSS contain additions and corrections to the Daśādhyāyī: chapter thirteen to chapter one, fourteen to two, fifteen to three, sixteen to four and five, and seventeen to seven. Brahmagupta followed this practice of providing supplementary chapters (uttara) in his Kh also. The last seven chapters of the BSS are more heterogeneous: eighteen is on the kuṭṭaka, nineteen on gnomon problems (śaṅku), twenty on prosody (chandas), twenty-one on cosmology (gola), twenty-two on instruments (yantra), twenty-three on measurements (māna), and twenty-four on the contents of the BSS and on its author (saṃjñā). The BSS was particularly popular in North and West India; the scholar who brought its derivative, the Mahāsiddhānta, to Baghdād in 771 or 773 came from Sind, and the two commentaries known to have been written on it were composed in the North. Balabhadra⁵³ wrote his at Kānyakubja, probably under Yaśovarman (ca. 725–750); it is now lost except for fragments preserved by Pṛthūdakasvāmin, Bhaṭṭotpala, and al-Bīrūnī. Pṛthūdakasvāmin⁵⁴ wrote his monumental Vāsanābhāṣya on the BSS at Sthāneśvara (Sthāṇvīśvara) in Kurukṣetra in the 860's; the commentary on chapters sixteen to twenty, on most of twenty-two, and on twenty-three and twenty-four is lost.

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⁵³ CESS A 4.

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⁵⁴ CESS A 4.

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Earlier another astronomer had apparently lived at Sthāneśvara, where, in the eighth century, someone wrote a new Pauliśasiddhānta.⁵⁵ This astronomer, while accepting the planetary parameters of the Ārdharātrikapakṣa and the main division of the Kalpa according to the Āryapakṣa, compromised with smṛti (and the Brāhmapakṣa) to the extent of accepting the traditional division of a Mahāyuga. In order to achieve a mean conjunction at the beginning of the current Kaliyuga (which he placed 3,888,000 years after the beginning of the current Mahāyuga), he delayed the beginning of celestial motion 648,000 years so that there would remain 3,240,000 years (three of Āryabhaṭa's yugas) before the beginning of the current Kaliyuga. The Pauliśasiddhānta is not extant in its entirety, but many fragments remain in the works of Pṛthūdakasvāmin, Bhaṭṭotpala, al-Bīrūnī, and Āmarāja.⁵⁶

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⁵⁵ CESS A 4.

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⁵⁶ D. PINGREE, "The Later Pauliśasiddhānta," Centaurus 14 (1969), 172–241.

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Probably also in the eighth century, though perhaps in the early ninth, Lalla composed his Śiṣyadhīvṛddhidatantra⁵⁷ following the Āryapakṣa, but with bījas for periods of 250 years subsequent to 498 (the numerators of some of these bījas are borrowed from Haridatta, who wrote in 683). Lalla may have lived in Daśapura in Mālava, which he mentions (SDV 2, 9, 10). His approximate date is indicated by his initial bīja-period, which ends in 748; by his sharing with the Pauliśasiddhānta and the Sūryasiddhānta the Brāhmapakṣa's divisions of a Mahāyuga while retaining Āryabhaṭa's parameters; and by the new organization of the siddhānta that he introduced and that was imitated by Vaṭeśvara, who wrote his VS in 904, and, in part, by Śrīpati⁵⁸ when he wrote his SŚŚ toward the middle of the eleventh century. This new organization of the siddhānta retained essentially the contents of the BSS's Daśādhyāyī in a first section on computation (grahaṇita), while gathering theoretical and cosmological material in a second section (gola). The ŚDV was commented on by Bhāskara⁵⁹ (b. 1114) and by Mallikārjuna Sūri⁶⁰—probably he who commented on the Sūryasiddhānta in 1178 in Sanskrit, and previously in Telugu. Thus Lalla’s work, as is to be expected of one belonging to the Ārya-pakṣa, was popular in the South; but the presence of manuscripts in Western India and the references to Lalla in, for instance, Āmarāja’s commentary on the Kh, show that it was also studied in Gujarāt and Rājasthāna.

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⁵⁷ Edited by S. DVIVEDIN, Benares 1886.

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⁵⁸ Śrīpati also modelled his Jyotiṣaratnamālā on Lalla's Jyotiṣaratnakośa.

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⁵⁹ CESS A 4.

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⁶⁰ CESS A 4.

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The Saurapakṣa was initiated in about 800 by the unknown author of the “modern” Sūryasiddhānta,⁶¹ who has modified many of the planetary parameters of the Ārdharātrikapakṣa (and thereby of Lāṭadeva’s Sūryasiddhānta) while retaining others along with the midnight epoch, and who has taken over from the Brāhmapakṣa the traditional divisions of both the Kalpa and the Mahāyuga, but delayed the beginning of the motions of the planets by 17,064,000 years so that the rotations of each within the remainder of the Kalpa are divisible by four. Other elements in the SS indicate its author’s knowledge of Brahmagupta’s BSS. The popularity of the work was enormous in all parts of India, but especially in the South and Northeast. Virtually every commentator, however, has rearranged the text, adding and subtracting verses ad libitum; the manuscript traditions of these many recensions, especially the earlier ones, will have to be fully investigated before a clear idea of the exact contents of the original can emerge. The most important of the extant Sanskrit commentators are: Mallikārjuna Sūri,⁶² who wrote a commentary in Telugu before composing his Sanskrit Sūryasiddhāntaṭīkā in 1178; Caṇḍeśvara,⁶³ a Maithila Brāhmaṇa whose Sūryasiddhāntabhāṣya was written in 1185; Madanapāla,⁶⁴ a Mahārājādhirāja of the Ṭāka family, who wrote his Vāsanārṇava between 1375 and 1400; Parameśvara,⁶⁵ who composed his Sūryasiddhāntavivaraṇa in Kerala in 1432; Yallaya, who wrote his Kalpavallī in Āndhrapradeśa in 1472; Rāmakṛṣṇa Ārādhya, whose Subodhinī was also composed in 1472; Bhūdhara,⁶⁶ who wrote his Sūryasiddhāntavivaraṇa at Kāmpilya in 1572; Tamma Yajvan,⁶⁷ who composed his Kāmadogdhrī at Paragīpurī in 1599; Raṅganātha, whose Gūḍhārthaprakāśaka was completed in Kāśī in 1603; Nṛsiṃha,⁶⁸ who wrote his immense Saurabhāṣya in the same city in 1611; Viśvanātha, also of Kāśī, who composed his Gahanārthaprakāśa in about 1628; Kamalākara,⁶⁹ another astronomer of Kāśī, whose Sauravāsanā on SS 1—10 was written after 1658; and Dādābhāī,⁷⁰ a Cittapāvana Brāhmaṇa, who composed his Kiraṇāvalī in 1719. The dates of three other commentators, all of whom lived in South India, are unknown; they are Kāmabhaṭṭa⁷¹ (Sūryasiddhāntaṭīkā), Cola Vipaścit⁷² (Gaṇakopakāriṇī), and Bhūtiviṣṇu⁷³ (Gurukaṭākṣa).

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⁶¹ Edited with the commentary of Raṅganātha by F. HALL and BĀPŪ DEVA ŚĀSTRIN, BI 25, Calcutta 1859, repr. Amsterdam [1974]; edited with the same commentary by JĪVĀNANDA VIDYĀSĀGARA, 2nd ed., Calcutta 1891; edited with a Hindī translation by UDAYA NARAIN SINGH, Biddoopur, Meerut, 1903; edited with his own commentary, Sudhāvarṣiṇī, by SUDHĀKARA DVIVEDIN, BI 173, Calcutta 1911, 2nd ed. Calcutta 1925; edited with his own bhāṣya, Tattvāmṛta, by KAPILEŚVARA ŚĀSTRIN, KSS 144, Banārasa 1936; edited with his own Malayālam commentary by PULIYOOR PURUSHOTTAMAN NAMPUTIRI, Trivandrum 1950; edited with the commentary of Raṅganātha and a Hindī translation by B. P. MISRA, Bombay 1956; and edited with the commentary of Parameśvara by K. S. SHUKLA, Lucknow 1957. The madhyama was edited with his own ṭīkā, Sudhādhavalā, by UMĀDATTA JOŚI, Hādiyāvāda [ND]. The Sūryasiddhānta was translated into English by E. BURGESS with the assistance of W. D. WHITNEY, JAOS 6, 2 (1860), 141--498, reprinted New Haven 1860 and Calcutta 1935, and by BĀPŪ DEVA ŚĀSTRIN, BI 32, Calcutta 1861, reprinted Amsterdam [1974].

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⁶² CESS A 4.

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⁶³ CESS A 3, 40b—41a.

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⁶⁴ CESS A 4.

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⁶⁵ CESS A 4.

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⁶⁶ CESS A 4.

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⁶⁷ CESS A 3, 85a—86a.

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⁶⁸ CESS A 3, 204a—205a.

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⁶⁹ CESS A 2, 23a.

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⁷⁰ CESS A 3, 97a—97b.

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⁷¹ CESS A 2, 31b.

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⁷² CESS A 3, 52b—53a.

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⁷³ CESS A 4; a fragment was edited by K. M. K. Sarma, “The Sūryasiddhāntavyākhyā of Bhūtiviṣṇu,” Brahmavidyā 5 (1941), 77—83.

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A siddhānta in which the influence of both the Āryapakṣa and the Saurapakṣa as well as that of Brahmagupta’s BSS are apparent is the Vaṭeśvarasiddhānta⁷⁴ composed by Vaṭeśvara at Ānandapura in Gujarāt in 904; the arrangement of the chapters follows that of Lalla’s ŚDV. One of the more interesting sections is that (VS 1, 1, 10) in which Vaṭeśvara criticizes Brahmagupta as Brahmagupta had criticized his predecessors, and especially Āryabhaṭa (BSS 11).

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⁷⁴ Adhikāras 1—3 are edited in an unsatisfactory manner by R. S. Sharma and M. Mishra, New Delhi 1962; see also T. S. Kuppanna Shastri, “The System of the Vaṭeśvara Siddhānta,” IJHS 4 (1969), 135—143, and K. S. Shukla, “Hindu Astronomer Vaṭeśvara and His Works,” Gaṇita 23 (1972), 65—74.

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In about 950—1000 a second Āryabhaṭa⁷⁵ wrote a Mahāsiddhānta⁷⁶ in large part based on the Brāhmapakṣa, but beginning the planets’ rotations 3,024,000 years after the beginning of the Kalpa in the fashion of the Saurapakṣa and substituting a new set of planetary parameters. He also reports another set of new parameters which he found in a treatise ascribed to Parāśara (MS 1, 2). And, while following Lalla in dividing his work into a grahagaṇita and a gola section, he follows Brahmagupta’s BSS in including chapters on arithmetic and geometry (2, 2 = 15) and on the kuṭṭaka (2, 5 = 18) in his siddhānta.

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⁷⁵ CESS A 1, 53b—54a; A 2, 15b—16a; and A 4.

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⁷⁶ Edited with his own Sanskrit commentary by Sudhākara Dvivedin, BSS 36, Benares 1910; the grahagaṇita was edited with an English translation by S. R. Sarma, 2 vols., Marburg 1966.

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Śrīpati, who wrote various works at Rohiṇīkhaṇḍa (about 150 miles south of Ujjayinī) between 1039 and 1056, followed the Brāhmapakṣa when he composed the Siddhāntaśekhara,⁷⁷ though he adds the bījas to the planetary mean motions that characterize the later history of this school. He also adds to Brahmagupta’s astronomy rules for computing the second component (udayāntara) of the equation of time (SŚŚ 3, 46) and the evection of the Moon (sphuṭacaraphala) (SŚŚ 11, 2–4); the latter rule, which has ultimately an Islamic Ptolemaic origin, had previously been given by Muñjāla. Śrīpati arranged the twenty chapters of the SŚŚ in much the same order as Lalla’s ŚDV, though without the latter’s division into two sections and with the addition of two chapters (13 and 14) on arithmetic and geometry (vyakta-gaṇita) and on algebra (avyaktagaṇita) respectively. The commentary, Gaṇitabhūṣaṇa, composed by Makkibhaṭṭa⁷⁸ in 1377, is preserved only up to 4, 75.

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⁷⁷ Edited with the commentary on 1–4 of Makkibhaṭṭa and with his own commentary on 5–20 by BABUĀJĪ MIŚRA, 2 vols., Calcutta 1932–1947.

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⁷⁸ CESS A 4.

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But the most notable siddhānta produced in India is the Siddhāntaśiromaṇi⁷⁹ written by Bhāskara⁸⁰ at Vijjaḍaviḍa (?) probably in his native Sahyādri north of Devagiri in 1150. Modelled in form on Lalla’s ŚDV (though with the addition of a chapter on the trigonometrical functions, jyotpatti and of another describing the seasons) the SŚB is the foremost text of the Brāhmapakṣa after the BSS. Bhāskara changes a few parameters minutely, repeats some of the improvements introduced by earlier astronomy such as the equation of time, and presents for the first time the formula for finding Sin (α + β),⁸¹ but the most impressive quality of the SŚB is its comprehensiveness. Commentaries on the SŚB were composed by, among others, Bhāskara himself (Mitākṣarā); by Lakṣmīdāsa in 1501 (Gaṇitatattvacintāmaṇi); by Gaṇeśa⁸² at Nandipura in Gujarāt in the early seventeenth century (Śiromaṇiprakāśa); by Nṛsiṃha⁸³ at Kāśī in 1621 (Vāsanāvārttika) and by Munīśvara⁸⁴ at Kāśī before 1638 (Marici).

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⁷⁹ The gaṇitādhyāya with Bhāskara’s own Mitākṣarā was edited by L. WILKINSON, Calcutta 1842, reprinted Calcutta 1855; the golādhyāya with the Mitākṣarā by L. WILKINSON, Calcutta 1842, reprinted Calcutta 1856; edited with the Mitākṣarā by BĀPŪDEVA ŚĀSTRIN, Benares 1866, revised ed. by CANDRADEVA, Benares 1891, revised ed. by GAṆAPATIDEVA ŚĀSTRIN, KSS 72, Banarasa 1929; the golādhyāya with the Mitākṣarā by JĪVĀNANDA VIDYĀSĀGARA, 2nd ed. Calcutta 1899; the gaṇitādhyāya with the Mitākṣarā edited by JĪVĀNANDA VIDYĀSĀGARA, Calcutta 1881, revised ed. by AŚUBODHA VIDYĀBHŪṢAṆA and NITYABODHA VIDYĀRATNA, Kalikātā 1915; the golādhyāya with the Mitākṣarā and a Bengālī translation edited by RASIKAMOHANA CHAṬṬOPĀDHYĀYA, Calcutta 1887; the golādhyāya with the Mitākṣarā and a Bengālī translation published in Aruṇodaya 1 (1890), pt. 6; the golādhyāya published in Bṛhaspati 1 (1896/97); edited with the Mitākṣarā and a Hindī translation by UDAI NARAIN SINGH, Bombay 1905; adhyāyas 1–8 of the golādhyāya with Marāṭhī translation and commentary edited by V. P. KHĀNĀPURKAR, Mumbaī 1911; the golādhyāya with the Mitākṣarā and a Hindī commentary edited by GIRIJĀPRASĀDA DVIVEDIN, Lucknow 1911, reprinted Ahmadabad 1936; the gaṇitādhyāya with Marāṭhī translation and commentary edited by V. P. KHĀNĀPURKAR, Puṇem 1913; adhyāya 1 of the gaṇitādhyāya with the Mitākṣarā and the commentaries of Nṛsiṃha and Munīśvara edited by MURALĪDHARA JHĀ, Benares 1917; the golādhyāya with the Mitākṣarā and a Bengālī translation edited by RĀDHĀVALLABHA, Calcutta 1921; the gaṇitādhyāya with the Mitākṣarā and the commentary of Gaṇeśa edited by DATTĀTREYA APAṬE, ASS 110, 2 vols., Poona 1939–41; the golādhyāya with the Mitākṣarā and the commentary of Munīśvara edited by DATTĀTREYA APAṬE, ASS 122, 2 vols., Poona 1943–52; adhyāyas 1–2 of the gaṇitādhyāya with the Mitākṣarā edited by MURALĪDHARA THAKKURA, KSS 149, Banārasa 1950; and the gaṇitādhyāya with the Mitākṣarā and the commentary of Munīśvara (the latter not on adhyāya 1) edited by KEDĀRADATTA JOŚĪ, 3 vols., Vārāṇasī 1961–64. The gaṇitādhyāya was translated into Latin by E. ROER, JAS Bengal 13 (1844), 53–66; the golādhyāya was translated into English by L. WILKINSON, revised by BĀPŪDEVA ŚĀSTRIN, BI 32, Calcutta 1861.

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⁸⁰ CESS A 4.

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⁸¹ R. C. GUPTA, “Addition and Subtraction Theorems for the Sine and Cosine in Medieval India,” IJHS 9 (1974), 164—177.

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⁸² CESS A 2, 106b—107a.

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⁸³ CESS A 3, 205a—206a.

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⁸⁴ CESS A 4.

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For approximately 350 years after the composition of the SŚB no important siddhānta was written; the learning of astronomers was rather concentrated in commentaries, their ingenuity in koṣṭhakas. It was, however, probably during this period that a group of minor siddhāntas belonging to the Saurapakṣa, but attributed to deities or sages, were written. These are the Somasiddhānta,⁸⁵ on which a commentary, Gūḍhārthadīpikā, was composed by Nṛsiṃha⁸⁶ at Koṇḍavīḍu in Āndhrapradeśa in about 1400; a Vasiṣṭhasiddhānta⁸⁷ which claims to be based on that of Viṣṇucandra; a Vṛddhavasiṣṭhasiddhānta,⁸⁸ which is influenced by the Brāhmapakṣa in its planetary equations; and the Brahmasiddhānta⁸⁹ alleged to be part of a Śākalyasaṃhitā.

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