HISTORY OF MATHEMATICAL ASTRONOMY IN INDIA

I. VEDIC ASTRONOMY

II. BABYLONIAN ASTRONOMY IN INDIA

III. GRECO-BABYLONIAN ASTRONOMY IN INDIA

IV. COSMOLOGY

THE GREEK PERIOD

VI . THE ĀRYAPAKṢA

VIII. THE SAURAPAKSA

IX. ECLECTIC ASTRONOMERS

X. THE GAṆEŚAPAḲA

XI. ISLAMIC ASTRONOMY IN INDIA

XII. CONCLUSIONS

XIII. EPILOGUE

NOTES

Astronomy¹ shares with other scholarly disciplines in India the characteristic of being repetitive. Indian astronomers did not usually attempt innovations in theory; they wished to preserve their tradition as intact as possible. Most of their energies, therefore, were devoted to devising computational techniques. And they delighted both in simplifications or approximations and in needless complications; each type of change displayed the skill of the master. Much of the history of this science in India, then, must be simply an account of the means by which the traditions were preserved, and a recitation of the often bizarre modifications and elaborations of the basic formulas. In the present essay these basic formulas are presented primarily in section V, and will not be repeated thereafter.

That Indian astronomy was not completely static is due almost entirely to the repeated intrusion of new theories from the West, hive times have such intrusions occurred–in the fifth century B.C., from Mesopotamia via Iran; in the second and third centuries A.D., from Mesopotamia via Greece; in the fourth century A.D., directly from Greece: in the tenth to eighteenth centuries, from Iran; and in the nineteenth century, from England. But, although the character of Indian astronomy at each such intrusion was changed, to a greater or lesser extent, these changes were accompanied by the minimum possible alterations of the earlier traditions, none of which ever completely died. The coalescence of these several traditions necessitated transformations and adjustments of both the older, native or assimilated theories and the new, foreign ones that were being adapted. It also meant that some contradictions would always persist. Thus, internal consistency was not expected in any Indian astronomical system, and this tolerance of inconsistency is undoubtedly one important factor that prevented Indian astronomers from making any advances in theory; they were not motivated to examine the logical foundations of mathematical astronomy, but only to tinker with the computational superstructure. The lack of a tradition of observational astronomy also permitted them to iterate models and parameters that did not at all satisfactorily predict the phenomena.

Moreover, Indian astronomers had the misfortune to receive, in the first three transmissions from the West, theories that were either antiquated in their countries of origin or also deviant and second-rate. For instance, despite the enormous influence of Greek astronomy on Indian astronomy in the period contemporary with and immediately subsequent to Ptolemy, the Indians remained ignorant of most aspects of Ptolemaic astronomy until the seventeenth century. For the historian of Hellenistic astronomy, however, this fact and the conservatism of the Indian tradition are of enormous assistance, for much of what we know of Greek astronomy between Hipparchus and the fourth century A.D. is to be found in Sanskrit texts. Admittedly, because of the Indian tendency to modify intellectual imports and also the corrupt nature of the earliest surviving texts, it is often difficult to determine precisely the nature of the Greek texts upon which the Sanskrit are based; but one hopes that careful analyses and comparisons of the sources will gradually achieve a more accurate description of Hellenistic astronomy than is possible from a consideration of the surviving Greek tradition in isolation²

In the preceding paragraph reference was made to the corrupt tradition of the earliest surviving Sanskrit texts on astronomy. The cause of this corruption is usually that the texts had become unintelligible; and this unintelligibility is not unrelated to the style developed by Indian astronomers. The texts proper were composed in verse in order to facilitate memorization, with various conventions for rendering numbers into metrical syllables. The exigencies of the meter often necessitated the omission of important parts of mathematical formulas, or contributed to the imprecision of the technical terminology by forcing the poet to substitute one term for another. This did not matter too much as long as the texts were taught in a school that preserved their interpretation orally, or after a commentary had been published; but the teaching of the earliest texts ceased and no commentaries, if any were ever composed, have survived. Only the texts were copied, without comprehension or care, so that what we have are frequently fragmentary, and certainly incorrect, representatives of the original. Often with such texts one must guess at the problem that is to be solved, invent a solution that seems reasonable within the context of the rest of the work, and then see whether the manuscript readings can somehow be made to coincide with this solution. Success in such endeavors is not of necessity attainable.

Even the existence of a commentary, however, does not always provide the reader with an understanding of the text. The commentators themselves are almost never interested in explaining the derivation of rules; they are more concerned to repeat them in clearer language and to exemplify them. Sometimes they cannot even repeat them correctly. What the commentary does do in all cases, however, is to offer some guarantee of the status of the text in the time of the commentator. Frequently this is a very valuable aid.

In consideration of the character of Indian astronomy as outlined above, this essay is divided into sections, each of which is devoted to a particular tradition. The basic chronological divisions are as follows (the sections in which they are discussed are indicated in parentheses):

1. Vedic, ca. 1000 B.C.-400 B.C. (I).
2. Babylonian, ca. 400 B.C.-200 (II)
3. Greco-Babylonian.ca. 200-400(III)
4. Greek.ca.400–1600(V-X)
5. Islamic, ca. 1600–1800(Xl)

In the Greek period there originated the five paksas, or schools of astronomy, that characterized the science in medieval times and to which most of our extant texts belong. The paksas, the approximate dates of their inception, and the areas in which they were especially prevalent are

1. Brāhmapaksa, ca. 400, western and northwestern India (V).
2. Āryapaksa.ca. 500, southern India(VI)
3. Ārdharātrikapaksa ca. 500, Rājasthān, Kashmir, Nepal, and Assam (VII).
4. Saurapaksa. ca. 800. northern, northeastern, and southern India (VIII)
5. Gaṇeśapaksa, ca. 1500, western and northwestern India (X)

The basic features of each paksa will be delineated when the initial text is described; later texts within each paksa will largely repeat this material, and only their more significant innovations will require description. Furthermore, texts in one paksa frequently repeat rules or techniques from another. These repetitions and borrowings are too numerous to be noticed in this essay.

The corpus of texts to which reference is made is restricted because of the lack of editions, good or bad, of the majority of Sanskrit astronomical works. Obviously, the future explorations of the manuscripts will necessitate corrections of many details in this essay. I can only hope that my inadvertent omissions and errors will not require many more.

I. VEDIC ASTRONOMY

Although speculations abound concerning alleged astronomical data in the Vedas, in the Brāhmanas, and in other early Sanskrit texts³ there is no convincing evidence that any method of mathematically describing celestial phenomena was known to Indians before the latter half of the first millennium B.c. Rather, the Vedic texts refer almost exclusively, when they do refer unquestionably to astronomical phenomena, to the times for performing sacrifices. In this connection they mention various yugas (periods), samvatsaras (years), ayanas (half-years), rtus (seasons), māsas (months), adhimāsas (intercalary months), paksas (half-months), and specific nights (indicated by ordinal numerals in the feminine, modifying rātri [night] and not tithi), as well as naksatras (constellations).⁴ All of these elements survived into later periods, and profoundly affected the form into which Indian astronomers molded the foreign systems that they adapted to their own use. As characteristic features of Indian astronomy they must now be briefly described, even though they do not in themselves pertain to mathematical astronomy.

In the following discussion, no attempt is made to distinguish sources chronologically since, even where this is possible –as in the case of the priority of the Rgveda to the Brāhmanas–it is not always clear that the views expressed in the latter developed historically after the composition of the former. All texts that can reasonably be dated before ca. 500 B.C. are here considered to represent essentially a single body of more or less uniform material.

Yugas of two, three, four, five, and six years are mentioned at various places in this vast literature. But, although there was no standard yuga, one of the more frequent variants was the yuga of five years named, respectively: samvatsara, parivatsara (or idāvatsara), idāvatsara (or iduvatsara), iduvatsara (or idvatsara or anu vat sara), and vatsara (or udvatsara). This yuga of five years is presumably that which influenced the author of the Jyotiṣavedānga

The length of a samvatsara is normally taken to be 360 days, or twelve months of thirty days each. Such an “ideal” year is also known in Mesopotamia,⁵ Since months were in practice synodic, however, this “ideal” year could not really have been in use.

The year contains two ayanas: in the uttarāyana the sun travels north; in the dakṣiṇāyana, south. Presumably these directions were determined by some such means as observing the point on the horizon at which the sun rises on successive days. There is no compelling reason to believe that the solstices were accurately determined.

There are three rtus in the Rgveda: vasanta (spring), grīṣma (summer), and śarad (autumn). In most texts of the Vedic period, however, there are five: vasanta, grīsma, vārṣa (rains), śarad, and hemantaśiśira (winter). A final enumeration of rtus, which remains standard in India, is vasanta, grísma, vārsa, śarad, hemanta (winter), and śiśira(cold).

The māsa in this period is either the “ideal” month of thirty days, a twelfth of a 360-day year, or a synodic month of twenty-nine or thirty days, which may have been measured either from new moon (amāntamāsa), as in Mesopotamia, or, less probably, from full moon (pūrṇimāntamāsa). There are two systems of naming the months: descriptively and after the naksatras. Naturally, many synonyms can be substituted for the names listed in Table 1.1.

In some texts Phālguna is the first month of vasanta. An intercalary month, or adhimāsa, was from time to time added to the normal twelve months of the year, presumably in order to make the beginnings of the sun’s ayanas fall in the correct months. Despite the efforts of many scholars to prove the contrary, no systematic intercalation scheme can be attributed to this period.

Paksas are the periods of fourteen or fifteen days between new moon and full (pūrvapaksa: later called śuklapakṣa) and between full moon and new (aparapakṣa; later called kṛṣṇapakṣa).

Since the normal term for a nychthemeron in this period is rātri (night), the nychthemeron presumably began at sunset, as in Mesopotamia.

The naksatras are twenty-seven or twenty-eight stars or groups of stars with one of which the moon is supposed to conjoin each night. Which stars were given the names of the naksatras preserved from this period cannot be determined; we are on secure ground in making such identifications only with texts written in the fifth century of the Christian era and after. With each naksatra is associated a deity, after whom the naksatra is sometimes named. All lists of naksatras in this period begin with Krttikā; a composite list is provided in Table 1.2.

When only twenty-seven naksatras are listed Abhijit is omitted. We have no information that would allow us to identify the stars to which these names refer.

II. BABYLONIAN ASTRONOMY IN INDIA

Probably late in the fifth century b.c. . while the Achaemenids dominated northwestern India, there occurred a massive transmission of Mesopotamian literary material to India. This influence is discernible in the Gargasamhitā⁶ and its successors that deal with omens in essentially the same manner as did the Babylonians. It also can be discovered in the Jyotiṣavedān̢ga of Lagadha (or Śuci) and in a series of other Sanskrit texts that are analyzed below.⁷

The Ṛk recension of the Jyotisavedāhga is evidently earlier than the Yajur recension; the former may be as old as ca. 400 B.C., the latter as late as ca. A.D. 500. The problem that Lagadha set out to solve in the Rk recension is that of establishing an intercalation cycle: this problem had also engaged the attention of Mesopotamian, Greek, Egyptian, and Iranian astronomers in the fifth century B.C. Lagadha’s solution involves the use of parameters. mathematical models, time units, and instruments of Mesopotamian origin. There can be no doubt that his system is not indigenous to India.

The basic period relation that he hypothesizes is

where a tithi (a Mesopotamian time unit) is one-thirtieth of a synodic month and all units of time are treated as mean. The yuga of five years perhaps appealed to Lagadha because of its appearance in the Vedic texts, but its choice ultimately depended on astronomical rather than sentimental considerations. The Egyptian twenty-five-year intercalation cycle, introduced during Achaemenid rule, in the middle of the fourth century B.C. is based on the period relation

whereas five of Lagadha’s yugas give the relation

Since for both Lagadha and the Egyptians it holds that

it is apparent that the Egyptian period relation, in which

is more accurate than Lagadha’s, in which

It should be noted that a solar year of 365 civil days may have been adopted in Iran as early as the second half of the fifth century B.C.

In order to designate the positions of the sun and the moon on the ecliptic. Lagadha uses the list of twenty-seven naksatras to divide the ecliptic into twenty-seven equal arcs, each of which contains 13;20°, in the same way that Babylonian astronomers in the late fifth century b.c. had divided it into twelve equal arcs, each of which was named after a constellation. Since Lagadha places the beginning of the uttarāyana at the beginning of Sraviṣṭhā, the twenty-first naksatra in a list of twenty-seven or twenty-second in a list of twenty-eight, it appears that he retained the old naksatra list of the Vedic period, beginning with Krttikā. and distributed the solstices and equinoxes symmetrically. on the assumption that the vernal equinox occurs when the sun is in Kṛttikā. His adaptation of the earlier lists without any adjustment makes these data useless for determining his own date.

The progress of the sun is stated to be a naksatra in sidereal days, since in five years it traverses 135 naksatras, and

Lagadha elsewhere teaches, however, that the sun progresses through a naksatra in days; this parameter yields an impossible 1,845 days in a yuga, He correctly states that the moon travels through a naksatra in sidereal days, since it traverses 1,809 naksatras in a yuga and

In both cases only the mean motion of the luminary is considered.

In one final matter Lagadha demonstrates his dependence on Babylonian astronomy: his determination of the length of daylight. His timemeasuring instrument is an outflowing waterclock such as those attested in cuneiform texts from about 700 b.c. on. From approximately the same date the Babylonians used 3;2 as the ratio of the longest to the shortest day in the year; and this is precisely the parameter used by Lagadha, although it is applicable only in part of northwestern India. He fürther divided the day into thirty equal parts, called muhnūrtas. probably in analogy to the division of the synodic month into thirty equal tithis. Then Lagadha established a linear zigzag function, such as the Babylonians had used for several centuries before the fifth century b.c. to describe periodic and regular deviations from a mean, to determine the amount of water to be put into the waterclock so that on each day it will measure the length of daylight. Unfortunately, he does not give the mean value that would measure fifteen muhūrtas at the equinoxes, but says only that one must add a prastha of water each day that the sun is in the uttarāyana, and remove one each day that it is in the dakṣiṇāyana. Lagadha also gives a rule to determine the length of daylight in muhūrtas that is based on a linear zigzag function:

where .x is the time before or after the winter solstice expressed in days, d(x) is the length of daylight in muhūrtas on that day, 12 muhūrtas is the length of daylight at the winter solstice, 6 muhürtas is the difference between the longest and shortest days of the year, and 183 is the number of (sidereal) days in an ayana. The parameters, then, are those of Table II.1.

Precisely the same five-year yuga as is found in the Rk recension (II, 1) and the same linear zigzag function for determining the length of daylight (Table II.1) are found in a Pattāmaiuisiddhānta, the epoch of which is 11 January A.D. 80 and that is known to us through a summary in chapter 12 of Varāhamihira’s Paācasiddhāntikā. Varāhamihira⁸ understood the five-year yuga to contain 1,830 civil rather than sidereal days. This was a common mistake in the sixth century. The Paitāmaha also contains equivalents of Lagadha’s rules for determining the mean longitudes in naksatras of the sun and moon. It adds to the data of the Rk recension only the idea of the yoga, which is the period in which the sum of the motions of the sun and moon equals a naksatra, or 13 ;20°. Since in a yuga the sun makes five rotations and the moon sixty-seven, which gives a total of seventy-two, and each combined rotation includes an entire series of the twenty-seven yogas, there are seventy-two series in a yuga of 1,830 sidereal days. Therefore, the number of lapsed yogas at a given ahargana (sum of days) within the yuga is given by the formula

where c is the ahargana.

The Paitāmaha names the seventeenth yoga, Vyatipāta; therefore, it presumably already had a complete list of the twenty-seven, such as is reproduced in Table II.2.

Further, the method of expressing the rule and the longitude of the conjunction of 11 January A.D. 80 imply that the list of naksatras was already considered to start with Aśvinī. This new list of naksatras and the ecliptic longitudes of their beginnings is given in Table II.3.

As we have seen, Varāhamihira in the sixth century understood the Paitāmaha’s period relation to be

This leads to the disturbing conclusion that a solar year contains 366 civil days. A similar misunderstanding of Lagadha’s Rk recension is found in the Yajur recension of the Jyotisavedānga, which otherwise is merely a rearrangement of Lagadha’s work with a few additional verses; I would date the Yajur recension to between the third and the fifth centuries of the Christian era. This same misinterpretation is repeated in the verses of Garga cited by Somākara in his bhāsya on the Yajur recension, and also in the Jaina canonical astronomical works, which we know in a recension of the early sixth century.

Before concluding this section, it should be noted that two texts of the early Christian era— the Śārdūlakarnāvadāna, which is earlier than A.D. 148, and the second book of the Arthaśāstra of Kautilya, which was given its present form probably in the second century, preserve rules for telling time similar to those of Lagadha, employing the same Babylonian parameters, zigzag functions, and instrument. Another instrument of Mesopotamian origin appears in these texts, however — the gnomon, which the Indians normally divide into twelve digits, although one passage in the Śārdūla-karnāvadāna refers to a gnomon of sixteen digits. Normally the noon shadow at the summer solstice is assumed to be zero digits, which may indicate that Ujjayinī, which lies very close to the Tropic of Cancer, was already a center of astronomical studies; and the length of the noon shadow at the winter solstice is assumed to be twelve digits, with the monthly (or for each zodiacal sign) difference being two digits. The parameters are shown in Table II.4.

The ratio of the longest to the shortest day is still assumed to be 3:2, which is valid about 11° north of Ujjayinī. Such gross inconsistencies usually did not disturb Indian astronomers, but they often make it difficult to determine the meaning of Sanskrit texts.

III. GRECO-BABYLONIAN ASTRONOMY IN INDIA

In A.D. 149/150, in the realm of the Western Ksatrapa, Rudradāman I, and probably in Ujjayinī, one Yavaneśvara translated a long Greek astrological treatise into Sanskrit prose. We now possess a large portion of the versification of this made by Sphujidhvaja in 269/270 and entitled Yavtmajâtaka. Chapter 79 of the Yavanajötaka is devoted to mathematical astronomy. In part Sphujidhvaja relies on the tradition of the Jyotisavedǟga, and in part upon Greek adaptations of the Babylonian astronomy of the Seleucid period; he also mentions Vasistha (YJ 79, 3), who will be discussed below.

Sphujidhvaja’s lunisolar yuga, the epoch of which is sunrise of 23 March 144, is based on the relation (YJ 79. 3–10)

From this relation follow those of Table III.1.

Elsewhere in his work, however, Sphujidhvaja slates the relations given in Table III.2 (YJ 79, 5, 11–13, and 34).

Of the items in Table III.2, the first, fifth, sixth, and eighth were intended to be correct, the rest approximate; but none are related to the yuga. The year length in the eighth, if a slight correction is accepted, is that of Hipparchus and Ptolemy (see Table 111.14). Sphujidhvaja already employs the four kinds of time measurement that became standard in Indian astronomy: saura (solar), candra (lunar), nâksatra (sidereal), and sāvana (civil).

since each yuga is assumed to be precisely identical with every other yuga, in order to determine the mean longitudes of the planets (in ihis case, of the sun and moon), it will suffice to determine the number of days lapsed since the beginning of the yuga (the ahargana), and then to apply the “rule of three” (proportion). Indian astronomers traditionally find the ahargana from the lunar calendar date, and Sphujidhvaja is no exception. In the formulations of these rules I use the following symbols:

a: adhimāsa

c: civil day

훆: epact

m: synodic month

n: saura month

r: rotation

s: saura day

t: tithi

u: ūnarātra

y: solar year

Capital letters refer to units in a yuga, lowercase letters to lapsed units. Subscript letters refer to the current larger units within which the referent letters have elapsed.

Sphujidhvaja’s procedure (III.2) is given below (YJ79, 15–18):

Clearly, (b) and (c) are not correct, since n = m and 30n‵ = s ≠ t. Instead, we need (III.3)

Further, Sphujidhvaja, at least as his text is now preserved, never gives instructions for finding u, such as

He does, however, give a rule for finding u, which he would need had he given (III.3a) (YJ 79, 19):

where ϵ = 11;11 tithis. This parameter means that one solar year contains 371;11 tithis; elsewhere Sphujidhvaja has given the approximate value 372 tithis. After (III.5) he states correctly (YJ 79, 20) that

This formula obviously is superior to (III.2a—c).

Once one has found c. one can use (III.16). Sphujidhvaja, however, refrains from taking this simple step, and instead gives crude rules for determining the mean longitudes tif the luminaries. These rules assume that the sun’s mean progress is 30° in a synodic (instead of a saura!) month, and that the mean moon traverses 30° for every 2;30° traversed by the mean sun. Then Sphujidhavaja describes two linear zigzag functions for obtaining the daily velocities, the parameters of which are given in Table III.3 (YJ 79, 23–24).

Unfortunately, the data for the moon are incomplete in the unique manuscript; and for neither the sun nor the moon does Sphujidhvaja provide enough information so that these functions could actually be used.

The Yavanajātaka gives two other linear zigzag functions. The first, for the rising times of the zodiacal signs, is that of Babylonian System A, which also is known from Greek texts.⁹ The units are called “muhūrtas,” but in fact each equals ten time degrees, as shown below (YJ 1,68; 79,26).¹⁰

The second gives the length of daylight in muhürtas when the sun is in each zodiacal sign; it employs the Jyotisavedānga’s ratio 3;2 (Table II.4). This scheme is reproduced in Table III.5 (YJ 79, 31).

The Yavanajātaka’s approximate rule for the relation of the time since sunrise, t, in hours, to the length of the shadow of a twelve-digit gnomon, s, in digits (the noon shadow is denoted s_n), is (YJ 79, 32)¹¹

where 12C is the length of daylight.

For the planets Sphujidhvaja repeats a Greek adaptation of Babylonian planetary theory of the Seleucid period.¹² First he establishes for each planet a yuga, or number of solar years, in which occur, approximately, a given number (II) of synodic periods (YJ 79, 35–36): these relations are given in Table III.6 (compare Tables III.10– 11).

He then gives, with some gaps in the manuscript, the intervals in degrees (and times) between the occurrences of successive Greek-letter phenomena (YJ 79. 40–47); these are repeated in Table III.7 (compare Table III.13), where the symbols have the following meanings.

Not only is this information incomplete; one is also given no epoch positions, so that the whole is of no practical value. Yet its ultimately Babylonian origin is clear.

One final bit of planetary theory introduced by Sphujidhvaja is the elongation between each planet and the sun necessary for the planet’s visibility (YJ 79, 50): these arcs are given in Table III.8.

Again the method is crude, but the values are similar to the elongations necessary for Г found in the Pauliśasiddhāta (Table III.17).

In general, the Yavanajātaka could not (as we presently have the text, at least) have made possible the computation of the longitudes of the planets necessary for the casting of horoscopes. But the occurrence of this material in such a context gives a strong indication of the motivation of the Indians who adapted Greco-Babylonian, and later purely Greek, planetary théories.

As Vasistha is cited by Sphujidhvaja. some version of a siddhānta ascribed to htm existed before a.d. 269/270. But what we know of —aside from the Vasisthasiddhānta of Viṣnucandra¹³ and a late Vrddhavasisthasiddhānta that will be discussed in section VIII —is a work similar to YJ 79. but preserved only in a summary by Varāhamihira. The epoch of this version of the Vasisthasiddhānta was 3 December 499, although the contents in general probably are derived from a second- or third-century adaptation of a text translated from Greek,

The solar year according to Vasistha contains 6,5;15 civil days —that is, it is a Julian year. Solar motion is represented by a table of the number of quarter-days that the sun remains in each zodiacal sign (PS 2, 1).

Two Babylonian period relations provide the basis of Vasistha’s lunar theory (PS 2, 2–6):

For each period of 3.031 days (see Table III.14) the moon’s longitude increases by 5,37;32° (this implies a mean daily motion of 13;10,34,52,46….°); for each anomalistic month it increases by 3;4,50° (this implies a mean daily motion of 13; 10,34,43,3,…°). On the epoch date the moon’s mean longitude was 69;7,1° and its anomaly 180°; for euch period of 3.031 days thereafter one adds the first increment noted above, for each anomalistic month within the current period of 3,031 days, the second increment: the true progress in longitude of the moon during the current anomalistic month is found from the summing of a linear zigzag function of lunar daily velocity based on the parameters of Table III.9 (see Tables III.3 and III.15).

From this it follows that the maximum lunar equation is 5;5°.

In its rules for finding the moon’s naksatra, the muhūrta (here defined as 1/30 of the time that the mean moon is within a naksatra), and the tithi, the Vasisthasiddhānta employs the relations (PS 2,7)

and, since a mean tithi occurs when the elongation of the mean moon from the mean sun is 12°, each 30° of mean elongation occurs in 5/2 tithis. Vasistha (PS 2, 8) repeats the linear zigzag function of Sphujidhvaja for finding the length of daylight (Table III.5) and Kautilya’s linear zigzag function for determining the length of the noon shadow (Table II.4) (PS 2, 9–10). What he introduces that is new is a crude formula, related to one of Sphujidhvaja’s (III.7), for finding the longitude of the ascendant, λ(H), in zodiacal signs(PS 2, 11–13).

This is based on the assumption that, at noon,

It seems likely that the first sixty verses of adhyāya 17 of the Pañcasiddhāntikā preserve Vasistha’s planetary theory. Like Sphujidhvaja’s it is closely related to Babylonian material of the Seleucid period, but probably was also transmitted to India through a Greek intermediary.

Like his lunar theory described above, Vasistha’s planetary theory involves computing lapsed synodic periods since epoch, and then finding the planet’s progress within the current synodic period. The epoch dates and longitudes all fall within the year 505, and appear to have been computed (in the case of Venus, wrongly) by Varāhamthira. The synodic periods in days (compare Table III.3) and increments in longitude, , are tabulated in Table III.10.

Only the of Venus is actually given in the text¹⁴ (the Babylonian value is 215;30°). The rest can be derived from the basic period relations of Vasistha given in Table III.11; they are all attested in cuneiform texts.¹⁵ These consist of statements that there are II occurrences of a phenomenon (synodic periods) in Z sidereal rotations of the planet; I have added a column of solar years.

Vasiṣṭha omits the period relation of Venus.

If the distances between consecutive occurrences of the same phenomenon were always Δ≈macr;, the longitudes of all the Π points on the ecliptic would be equidistant from each other, since

However, since more Π points occur in some arcs of the ecliptic than in others, the Babylonians divided the ecliptic into a number of sections of unequal length for each planet, as does Vasistha (see Table III.12).

Unfortunately. Varāhamihira has not understood this procedure, and his summary does not enable us to reconstruct this aspect of Vasistha’s system.

Once the date and longitude of the beginning of the current synodic period have been determined, it is necessary to determine the progress of the planet within the synodic period. This is accomplished by means of schemes similar to those of Sphujidhvaja (Table III.4); they are given in Table III.13.

This sixfold division of the ecliptic is of Babylonian origin,¹⁶ So is the second of two variant schemes for retrogression introduced into the middle of the one given above:

The retrograde arcs in this last scheme are identical with the Babylonian System R.¹⁷

This section of the text is corrupt and incomplete.

The scheme for Mercury has, in part, a direct parallel in cuneiform:¹⁸

The Romakasiddhānta or Roman Siddhānta probably was originally produced in the third or fourth century; of this original version¹⁹ we know that in it the discrepancy between the naksatras marking the solstices in the Jyotisavedānga and the zodiacal signs was explained by the theory of the precession of the equinoxes.²⁰ This implies the use of a tropical year; and, in the summary of Lātadeva’s version of the Romaka presented by Varāhamihira, the length of the solar year is Hipparchus 6,5;14. 48^d (PS 1, 9–10), as it also had been at one point in the Yavanajātaka (Table III.2). Combining this parameter with the nineteen-year Metonic cycle, the Romaka obtains a yuga of 2,850 years, in which there are 35,250, synodic months and 1,040,953 civil days (PS 1, 15). The epoch is 21 March 505, sunset at Yavanapura (Alexandria), which lies 7⅓3 nādīs (2;56 hours) west of Ujjayinī (PS 1. 8). Lātadeva corrects this by 0;43,52,… days; 0;45 days would correct it precisely from sunset to midnight.

Although the yuga contains 35,250 · 30 = 1,057,500 tithis and 1,040,953 civil days, in the rule for computing the ahargana Lātadeva uses the approximation also found in the Brāhmasphnutsid-dhānta (BSS 1, 42–43):

The only further computation of the Romaka about which Varāhamihira informs us is that of solar eclipses (PS 8). This involves first the determination of the longitudes of the sun, the moon, the lunar node, for which the Romaka uses the well-known rule of proportion

The ratios of R to C are given in Table III.14.

The equations of the center for the sun and moon are found by interpolation in tables, presumably of Greek origin, in which the interval between arguments is 15°. The solar apogee is placed at 75°; the maximum solar equation, 2;23,23° (Ptolemy’s is 2;23°), occurs at an anomaly of 90°, although the table is not computed by the “method of sines.” The maximum lunar equation, which also occurs at an anomaly of 90°, appears to be 4;46°: one would expect 4;56°.

The rules for computing parallax (see Figure 1) in the Romaka are essentially the same as those in the Pauliśa; presumably both have a Greek origin. The horizontal (maximum) parallax, π₀, is assumed to be 4 nādīs (0;4 days), which represents about 0;49° of elongation between the sun and the moon or 0;53° of lunar motion. The maximum mean lunar parallax, according to Ptolemy, is 0;53,34° and the maximum solar parallax is 0;2,51°. It is assumed that the longitudinal parallax, π_λ, is 0 on the meridian, and varies sinusoidally between horizon and meridian. Therefore, if the depression of the conjunction from the meridian, measured in nādīs, is designated Δ tⁿ and if the Radius, R, is taken to be 120 (this value of R is not necessarily that of the Romaka), the formula is

To find the latitudinal parallax, π_β, which also is assumed to have a maximum of 0;53° at the horizon, one must first determine the zenith distance of a point on the lunar orbit, U, of which the degree of longitude on the ecliptic coincides with the nonagesimal, V, of the ecliptic (see Figure 2). It is assumed that (III.18)

where ω is the distance of λ_V from the moon’s node. Correctly, the formula should be

where i is the inclination of the lunar orbit; but since β is small, the approximation

can be used it R = 120. Then it follows that

and then

Having found β_U, one forms (see Figure 2) (δ symbolizes declination) (III. 19)

and then (see Figure 3) (z symbolizes zenith distance):

This is true only if U lies on the meridian; the mistake is often repeated. The latitudinal parallax is found from

where is the daily lunar velocity, R = 120, and 1/15 of a day is 4 nādīs.

The lunar latitude, , in minutes is found from (III.21):

Again (see III. 18d),

There is a discrepancy between (III.18g) and (III.21e). The corrected lunar latitude, β_α , is determined by

The mean diameters, d̄, of the sun and moon are defined as 0;30° and 0;34°, respectively; their apparent diameters, d, vary as their velocities according to a Greek principle that was used in determining the distances of the planets:²¹

Then the half-duration of a solar eclipse, AB, is given by the formula (see Figure 4)

where r symbolizes the apparent radius. It is here assumed that the lunar orbit is parallel to the ecliptic. Finally, the magnitude of the eclipse at its middle, CD, is found from

Probably in the third or fourth century yet another Greek astronomical text, the Pauliśaiddhānta,²² was translated into Sanskrit; the name Paulisa may be a transliteration of the Greek . This work, like the Romaka, was revised by Lātadeva, and is now known to us only through the summary of Látadeva’s version included in the Pañcasiddhāntikā.

The year in the Pauliśa is sidereal, as (PS 1, 11–13):

This year length is attributed to the “Egyptians and Babylonians” by al-Baltāmī;²³ it also is expressed as a yuga in which (PS 3, 1)

The mean longitude of the sun can be determined from this yuga by means of (III.16). The true longitude is obtained by applying the equation of the center, found by interpolation in a table, presumably Greek in origin, in which the equations are given for mean solar longitudes at the beginnings of the zodiacal signs (PS 3, 2–3). The maximum equation is 1;12°; the text fails to give the necessary instruction to double these equations. The solar apogee is at 80°.

Pauliśa gives rules very similar to Vasistha’s (Table III.9) for finding the longitudinal increment of the moon during what has passed of the current anomalistic month. This is in the form of a linear zigzag function for lunar velocity, the parameters of which are recorded in Table III.15 (PS 3, 4–9).

This is based on (III.8), but is modified in a very strange and inexplicable fashion; and the maximum lunar equation in it is only 4^o. Presumably Pauliśa determined the mean lunar longitude at the beginning of the anomalistic month by means of a procedure similar to Vasistha’s.

The planetary theory incorrectly and incompletely expounded by Varāhamihira in Pañcasiddhāntikā 17, 64–80, probably is derived from the Pauliśa. His confusion concerning the epoch longitudes and the “ahargana” makes it impossible to utilize the system as he presents it. The structure is apparent, however, and is Greco-Babylonian.

One first operates with the mean synodic arc, , of each planet, expressed in the form of a ratio, a^o:b synodic periods; these mean synodic arcs, given in Table III.16, are almost identical with attested Babylonian values²⁴ (see Tables III.7, III.10, III.13).

Then the mean synodic arcs are divided into sections by the Greek-letter phenomena, and each section is characterized by a certain solar motion and a certain elongation of the planet from the sun. Such elongations for the occurrence of the Greek-letter phenomena are found in a Greek text ascribed to Rhetorius.²⁵ The elongations according to the Pauliśa and to “Rhetorius” are given in Table III.17.

Much of Pañcasiddhāntikā 3 is based on the Pauliśasiddhānta, although it is clear that some extraneous matter has been introduced. In any case, since all of it seems to belong to the Greco-Babylonian period, it ought to be described here. Some is similar to what we have already encountered, such as the definitions of naksatras and tithis (PS 3, 16; see III.10) and the scheme for daily solar velocity in which M = 1;1° and m = 0;57° (PS 3, 17; see Table III.3)²⁶. Others are new.

A karana is 1/60 of a synodic month, or half a tithi. Those near conjunction have special names; the other fifty-six form eight series of seven each, as in Table III.18(PS 3, 18–19).

There are two special configurations of the sun and the moon that are called pätas (PS 3, 20–22). In this earliest text defining them, they are vaidhrta, when the sun and the moon are equidistant from and on opposite sides of an equinox, so that

and vyatjpāta, when the sun and the moon are equidistant from and on opposite sides of a solstice, so that

Varāhamihira introduces into this simple scheme the idea of a trepidation of the solstices and equinoxes—an idea that he evidently derived from a Greek source, and then modified.²⁷

The sadaśtimukhas divide the ecliptic into four equal arcs of 86° each and one remaining arc of 16° (PS 3, 23–24). The beginnings of these arcs are at Libra 0°. Sagittarius 26°, Pisces 22°. Gemini 18°, and Virgo 14°, The origin and purpose of the sadaśītimukhas remain obscure.

The sankrānti is the time in nādīs, tⁿ, during which the sun passes a boundary between two zodiacal signs (PS 3, 26). It is found from

where is the diameter of the sun in minutes and its daily motion in minutes per day.

Previous Indian methods of determining the length of daylight depended on simple linear zigzag functions (Tables II.1 and III.5); the Pauliśa gives a series of three coefficients that, when multiplied by the noon equinoctial shadow, s₀, give the differences between the lengths of daylight in vinādīs for the moments when the sun’s longitude is 0°, 30°, 60°, and 90°; symmetry will provide the lengths at other solar longitudes (PS 3, 10– 11). These coefficients are given in Table III. 19.

The use of s₀ implies that ø is involved in the computation (see Figure 5), and the computation that lies behind the values of γ can be reconstructed as follows.

The day circle, the diurnal path of the sun parallel to the equator, is a fundamental concept of Indian spherical trigonometry, which operates, if it possibly can, with projections and analemmas.²⁸ The radius of the day circle, r, is found from (see Figure 6)

It is clear, from Figures 5, 6, and 7, that

where ω is half of the equation of daylight and Sin_r ω is what is later called the “earth sine,” e. From (III.32) it follows that

From this and (III.18 c) we obtain

And, since ω° = 10 ω vinādīs, the values of Γ should come from this formula

in fact, this gives results very close to the attested numbers.

Another, but less successful, attempt to solve a problem in spherical trigonometry occurs in this section of the Piñcasiddhāntikā (PS 3, 14). The deśāntara is the difference in geographical longitudes between two localities, B and C. If the direct distance in yojanas, d, between the two is known (a virtual impossibility in antiquity) and the terrestrial latitude of each, ø_B and ø_C, is also known, so that Δø = ø_B - ø_C, then the computation is the following (see Figure 8). It is assumed that the circumference of the earth is 3,200 yojanas (as in the Ārdharātrikapakṣa). so that

and

Even though Δ l is not an equatorial arc unless C lies on the equator, the text converts it into a time difference in nāḍīs by

A final section of Pañcasiddhāntikā 3, which probably is not connected with Pauliśa, concerns the computation of the moon’s latitude, (PS 3, 30–31). The retrograde motion of the lunar node from its given epoch position on 22 March 505 is stated to be 8° in 151 civil days, to which must be added a bīja (correction) of 0;1° for every revolution; in daily motion this is 0;3,10,44,14,18,…. When the longitude of the node has been found, and thence its elongation from the Moon, ω°, the text states that

We have seen the maximum lunar latitude of 4;40° previously in the Romaka (III.21 e), although the Romaka does not use (III.39).

The eclipse theory of the Pauliśasiddhānta is summarized in Pauliśasiddhānta 6 (lunar) and 7 (solar). For lunar eclipses, it is first necessary to compute the longitudes of the sun, the moon, and the center of the earth’s shadow at sunrise in the usual manner (PS 6, 1). Then the distance of the moon from the shadow’s center is converted into a time difference in nāḍīs, Δ tⁿ, on the assumption that the elongation (which is equal, of course, to the elongation between the moon and the sun in-

creased by 180°) increases at a rate of 12° per day or 0;12° per nāḍī. The longitudes of the several bodies and of the lunar node are then computed for the new time of opposition. A lunar eclipse will occur if the moon is less than 13;36° from its node, and a darkening (penumbra) if it lies between 13° and 15° from the node (PS 6, 2). The given eclipse boundary, 13;36°. implies that

that is, the value of (III.18g).

The Pauliśa’s rule for the duration of a lunar eclipse in degrees is similar to (III.24) (PS 6, 3):

where r_u = 0;38° is the radius of the earth’s shadow and = 0;17°. This is converted into time, Δ t, by

The derivation of the formula for the duration of totality of a lunar eclipse, τ, is far more complicated. That formula is (PS 6,5)

where Δλ is the elongation of the node from the center of the shadow, U (see Figure 9). Taking the example where the moon is totally eclipsed for a moment with its center at B₀, and assuming that a small spherical triangle is plane, it is clear that (III.44)

Further, combining (III.41) with this, we find that

The duration of totality in minutes, τ;, is equal to 2AB · 60; therefore, substituting in (III.45) the values of Sin i and Δλ₀ in (III.44e and f), we have

This is another way of expressing (III .43).

A characteristic of Indian eclipse computations, which originally may have been motivated by some omens in the Sin section of the Babylonian omen series Enūma Anu Enlil,²⁹ is the projection of the eclipse that permits one to determine the directions of points on the circumferences of the sun or the moon at different phases relative to an “east-west” line that is perpendicular to the great circle passing through the center of the eclipsed body and the north and south points on the local horizon. It is assumed that the angle between the equator and the “east-west” line, the “deflection,” λ, has two components. The first, or aksavalana, depends on ø the second, the ayanavalana, on Δ. Pauliśa, as preserved by Varāhamihira, gives a rule only for the aksavalana, λ₁, (PS 6, 8):

where t⁰ is the “depression” of the eclipsed body from the meridian-that is, the hour angle in time degrees. For, γ₁ one assumes that the ecliptic and equator coincide, so that at midheaven λ₁ = 0 and at the horizon λ₁ = ø.

Another eclipse projection is preserved by Varāhamihira (PS 6, 12– 15). In this, thirteen lines are drawn parallel to the lunar orbit, their limits being the parallels that pass through the lunar node and through the center of the shadow; they represent the eclipse limit of 13°, and can be used to measure the magnitude of the eclipse. In addition, the direction of the phases with respect to the lunar orbit can be determined by graphic means utilizing three concentric circles with radii r_u, and, r_u + and, by the use of diameters, reflecting lunar positions onto a quadrant of the innermost circle, which represents the moon. Again, the original motivation was probably to establish omen criteria.

One final section of this chapter on lunar eclipses is also related to omens of a type found in Enūma Ann Enlil.³⁰ This involves criteria for determining the direction of impact and the color of the eclipsed body. The latter in the Pauliśasiddhānta involves the altitude of the eclipsed body, its relation to the ascendant or descendant, and its magnitude; the colors are blood-red, reddish-brown, variegated, smoke-colored, and cloud-colored (PS 6, 9–10). The later texts that deal with the colors of lunar eclipses make them depend on the phase (see Table V. 19).

Pauliśa’s computation of solar eclipses is essentially identical with that of the Romaka, except that the radii of both the sun and the moon are assumed to be 0;17°; the inclination of the lunar orbit is 4°, as in (III.40). The eclipse limit for a solar eclipse is 8;36° (PS 7, 5). Longitudinal parallax is found by (III.17) (PS 7. 1). Latitudinal parallax is determined by a formula equivalent to (III.21); this is (PS 7.2)

where and

The rule for finding the duration of the eclipse in nā;ḍīs. tⁿ,is (PS 7, 6).

where Δλ₀ is the appropriate eclipse limit. We have already seen that the duration of totality in minutes, τ, is given by (III.43). To convert minutes of arc into nādīs, we divide by 0;12° of elongation per nādī, which produces

Pauliśa replaces 7/10 by 3/4 for indiscernible reasons .

This completes the analysis of all that can be definitely attributed to this phase of the development of Indian astronomy . It is characterized by the use of essentially Babylonian techniques for determining lunar and planetary positions, although they are modified, presumably by Greek intermediaries; there exist, in fact, Greek parallels in the texts ascribed to Rhelorius (Table III.17) and to Hetiodorus (Table V.9). Other elements are survivals from the Babylonian period (such as daylight and noon-shadow schemes), and yet others purely Indian developments (such as karanas and pātas). Two of the most interesting areas, however, reflect Greek astronomical traditions of which we otherwise are little informed. These are in the development of projections to solve problems in spherical trigonometry (employing the Sine function) and The rough methods of predicting eclipses found in the Romaka and Pauliśa, which also involve the Sine function and material related to the interpretation of Babylonian omen literature. Because they fit in with these general characteristics, although no authors are named, chapters 4, 5, and 14 of the Pañcasiddhāntikā can be regarded as largely derived from the texts produced in the Greco-Babylonian period. The possibility remains, however, that some of the more sophisticated methods in these chapters come from the later Greek phase–for instance, from the Sūryasiddhānta of Lātadeva.

The Sine table found in chapter 4 (PS 4, 6–15) is based on R = 120—that is, double the value used in Ptolemy’s Chord table. The intervals in the argument are 90°/24 = 3;45°, as became standard in Indian Sine tables; this may be related to the Hellenistic βαx03D1;μoi (15°)³¹ halved and halved again. This Sine table is reproduced in Table III.20; there are added, for the sake of comparisons, columns showing R sin ϑ and Sin ϑ from Ptolemy’s Chord table(Sin¹²⁰ϑ = Chrd⁶⁰2ϑ).

The relation of the diameter of a circle, d, to its circumference, c, is given as the crude (PS 4, 1; see [V.1])

The following formulas for deriving Sines (PS 4, 1–5) are presented (III.51):

The text gives also an inaccurately computed table of the differences between declinations at intervals of 7;30° (PS 4, 16–18): in it 훆 = 23;40ϵ. although elsewhere one finds the rounded and more normal Indian value e= 24° (PS 4, 23–25).

The cardinal directions are determined by describing a circle about a gnomon (g = 12), marking the points where the shadow enters and leaves this circle (it is assumed that there is no change in solar declination), and bisecting the line connecting those two points by means of intersecting arcs (PS 4, 19). The relations to be derived from observed noon shadows of this gnomon (PS 4, 20–23) (see Figures 5 and 10) are (III.52)

Where S_n is the noon shadow; and

Other rules are (see Figure 10) (III.53)

where α is the altitude of the sun, its coaltitude; and

The altitude of the sun when it is on the prime vertical, α_p can also be determined from (Figure 10) (PS 4,32–33)

And, since (see Figure 6)

it follows also that (PS 4, 35–36)

The right ascensions of ecliptic arcs, α are found by the formula (PS 4, 29–30)

This follows from Figure 6, where, for example,

which is equivalent to (111.57).

The oblique ascension, ρ, is found from (PS 4, 31)

The text tabulates the values of Sin δ, d, and α for λ = 30°, λ. = 60°, and ϵ. = 90° (R = 120 and 훆 = 24°) that are given in Table 111.21 (compare Table V.17 (PS 4, 23–25, 29–30).

We are also given (PS 4, 26) a rule for computing ω measured in vinādīs that is equivalent to (III.33):

and another, equivalent formula where ω is measured in nād+s (PS 4, 34):

A concept frequently used in Indian astronomy is that of the “earth sine,” e (see III.32). From Figure 10 it is clear that (PS 4, 27–38)

From the same figure it is obvious that the Sine of the rising amplitude of the sun, Sin η, which is BO, can be found from (PS 4, 39–40)

and thence that

Varāhamihira gives the following new rules involving gnomon shadows (PS 4,41–44):

where t′ = 6tⁿ + ω° and tⁿ is the nā;dīs elapsed since sunrise or to come until sunset. Thus, in Figure 6, for λ = 60°, when the sun is on the prime vertical, Sin 6tⁿ is H_eL_e and Sin ω⁰ is L_e O. Hence (III.66)

which is (III.65). Then, from Figure 11:

Varāhamihira also presents the inverse of this procedure, which allows one to compute tⁿ from s (PS 4, 45–47).

A more complicated analemma involves the triangle ∑′ BM′ in Figure 10 (PS 4, 52–54). In this the sahku (the Sine of altitude, Sin α) is ∑′ M′ (which equals ∑ M in Figure 11); it is found from

Then the śankutala, which is BM’. is found from

Furthermore, combining (III.55) and (III.63), we have

and then

where the koti is the perpendicular distance of the tip of the shadow from the east-west line running through the base of the gnomon (see Figure 12) and

Pañcasiddhāntikā 5 is devoted to the problem of lunar visibility, in the solution of which appears the first approximation to what became known as the ayanadrkkarma, which is the computation of the longitudinal difference, λλ*, between the ecliptic longitude and the polar longitude of the moon, a star-planet, or a star (see V.33). If (see Figure 13).

since the small spherical triangle may be considered to be plane. Further, under the same consideration,

Then, clearly,

and its right ascension is Δα*. The moon is visible, according to the text, if Δα* ≥ 2 nādīs= 12°.

The sickle of the lunar crescent,σ, is measured in units of which the moon’s diameter contains fifteen. Then the number of these units that are illuminated

along the diameter that, when extended, reaches the sun is made to vary with Δλ, so that(PS 5,4–7)

In adhyāya 14, Varāhamihira gives us our oldest information about the coordinates of the yogatā;rās of at least some of the naksatras (PS 14, 33–37). These are all of small enough latitudes that they can still plausibly Be considered to serve the function of “junction stars” with the moon and planets; some of the yogatārās in the later star catalogs do not meet this requirement. Table III.22 lists the Pañcasiddhāntikā’s coordinates along with some possible identifications with stars included in Ptolemy’s star catalog.³²

That Varāhamihira here intends to give ecliptic rather than polar coordinates is proved by the computation, in the following verse, of the distance between the moon at latitude and a yogatā;rā at its latitude (PS 14, 38). The chapter concludes with a statement of the rising time in oblique ascension for Ψ = 24° of the ecliptic arc between the sun and 90° necessary for the visibility of Agastya –that is, Canopus (α Arg.)(PS 14, 39–41).

IV. COSMOLOGY

Before discussing the Indian adaptations of Greek spherical astronomy, it is necessary to describe briefly the early cosmology of the Purānes, some of the basic concepts of which were taken over by the siddhāṇtakāras. The text source of the cosmological section of the Purā;rtas probably was written in the early centuries of the Christian era: some of the concepts it reflects go back to Vedic times, and some show an affinity with Iranian thories.

In the Purānas³³ the earth is a flat-bottomed, circular disk, in the center of which is a lofty mountain, Meru. Surrounding Meru is the circular continent Jambūdvīpa, which is in turn surrounded by a ring of water known as the Salt Ocean. There follow alternating rings of land and sea until there are seven continents and seven oceans. In the southern quarter of Jambūdvīpa lies India–Bhāratavarsa.

Above the earth’s surface and parallel to its base are a series of wheels the centers of which lie on the vertical axis of Meru, at the tip of which is located the North Polestar, Dhruva.³⁴ The wheels, bearing the celestial bodies, are rotated by Brahma by means of bonds made of wind. The order of the celestial bodies varies; the earliest seems to be sun, moon, naksatras, and Saptarsis (Ursa Major), Some Purānas place the graphas (planets) between the moon and the naksatras; in others, interpolated verses add Mercury, Venus, Mars, Jupiter, and Saturn (in that order) between the naksatras and the Saptarsis.

Meru provides the explanation for the alternation of day and night. The variation in the sun’s rising amplitude is ascribed to the existence of 180 paths for the sun on its wheel; it travels the next, successive path on each day of a half-year. It is not clear how this model could explain some other phenomena, such as the varying periods of visibility of the naksatras due to their varying declinations.

The Jaina canonical works,³⁵ which were codified in the early sixth century a.d., are based on this Purānic cosmology, but confuse it by hypothesizing that each celestial body has its double 180° from it on its wheel; the field of vision from a locality in Bhāratavarṣa is only a quadrant, so that, while in a nychthemeron the sun’s wheel rotates 180°, half of that nychthemeron is day and half is night. The order of the celestial wheels also is somewhat different; fixed stars, sun, moon, naksatras, Mercury, Venus, Jupiter, Mars, Saturn. Also, there are 184 paths of the sun, not just 180.

With the introduction of the Greek concept of a spherical earth surrounded by the spheres of the planets and that of the fixed stars, it became necessary to discard these older ideas; they are assiduously attacked by Varāhamihira, Brahmagupta, and others. But it was also found possible to preserve some of them, although in altered form.

The circumference of the disk of the earth became the equator; at its four quadrants are located Lankā at the prime meridian and, proceeding westward, Romakavisaya (the Roman territory), Siddhapura (the city of the perfected ones), and Yamakoti (the peak of Yama). At the terrestrial North Pole stands Meru; opposite it lies Vadavamukha (the Mare’s Mouth). Above Meru still shines Dhruva, and the axis passing through these two and extended to the opposite Pole has wrapped about it bonds of wind that cause it to rotate with the diurnal motion. The individual motions of the planets in their orbits, and the deviations from their mean longitudes–known as their manda and śīghra equations–are also caused by cords of winde; the deviations are due to the pulls on these cords by demons located at the mandoccas and śīghroccas. This reluctance to part with any segment of their tradition that could possibly be saved is one of the most noteworthy characteristics of Indian astronomers, and one that has provided us with information about obsolete systems that in many other cultures would surely have been forgotten.

THE GREEK PERIOD

V. The Brāhmapaksa

Probably in the late fourth or early fifth century, during the Gupta domination of western India, a non-Ptolemaic tradition of Greek astronomy, influenced by Aristotelianism, was transmitted to India. The occurrence of several planetary models and sets of parameters in Sanskrit texts indicates that more than one Greek work was translated; it also is clear that the Greek material was modified considerably, both to fit in with the established traditions of Indian astronomy (discussed in the preceding sections) and to conform to a concept of time that was first described in Sanskrit texts of the second century³⁶ This concept is one of a system of yugas, of which a Caturyuga or Mahāyuga of 4,320,000 years contains four smaller yugas in the ratio to each other of 4 ; 3 ; 2 ; 1 ; a Kaliyuga(in one of which we now are) contains 432,000 years (this is a Babylonian period); the Dvāparayuga, 864,000 years: the Tretāyuga, 1,296,000 years: and the Krlayuga, 1,728,000 years. A Kalpa contains 1,000 Mahāyugas; and often there are further multiples of the Kalpa forming larger yugas, although these larger yugas were not used for astronomical purposes.

The astronomers who adopted Greek planetary theory expressed the mean motions of the planets as integer numbers of rotations within a yuga, an idea that also lay behind the Hellenistic magnas annus.³⁷ The earlier Indian system, which seems to date from the early fifth century, assumes a true conjunction of the planets, their mandoccas, and their nodes at a sidereally fixed Aries 0° at the begining and end of a Kalpa (PS 3, 20); a later system simplifies the numbers by assuming a mean conjunction of only the seven planets at the beginning and end of a Mahāyuga. Both systems assume a mean conjunction or near conjunction of the seven planets at a siderally fixed Aries 0° at the beginning of the current Kaliyuga, which is either midnight of 17/18 February or 6 A.M. of 18 February-3101 Julian at Lanka and Ujjayini. Table V.1 shows the mean tropical longitudes of the planets at 6 A.M. of 18 February-3101, as computed by means of Ptolemy’s Almagest, and their distance from ζ Piscium, whose longitude according to Ptolemy was then 320;37°.

If one knew, from a Greek set of tables, the “correct” mean tropical (or sidereal) longitudes of the planets at some time in the fifth century and their period relations, one could derive for each by means of an indeterminate equation an integer number of rotations in a Kalpa such that the conjunction of -3101 and the “correct” mean longitudes in the fifth century would both be accounted for. This, I suggest, is what the Indians did.

The Brāhmapaksa, it is claimed, was revealed by Brahma (Pitāmaha); it flourished for some 1,500 years, primarily in western and northwestern India. The basic text of this school is the Paitāmahasid-dhānta,³⁸ written in the early fifth century; one element from it appeared in a Sassanian work of ca. 450,³⁹ and the work was familiar to Aryabhata I, as will be demonstrated below. It is recognized as the origin of the Brāhmapaksa by Kamalākara (STV 1. 62). Unfortunately, however, it is very imperfectly preserved in a vast compilation of the sixth or seventh century, entitled Viṣṇudharmottarapurāna.

In the Paitā;maha, the Kalpa contains 1,000 Mahāyugas, grouped into fourteen Manvantaras, each of which consists of seventy-one Mahāyugas or 306,720,000 years: alternating with these fourteen Manvantaras are fifteen Sandhis, each of which is equal to a Krtayuga or 1,728,000 years (Pait. 3, 4). The revolutions, R. of the celestial bodies in a Kalpa according to the Paitāmaha are recorded in Table V.2 (Pait. 3, 5), along with the corresponding yearly and daily mean motions. The latter are computed from the number of days, C, in a Kalpa–1,577,916,450.000–which implies a year length of 6.5;15.30,22,30 days (Pait. 3. 32), The Paitāmaha, pretending to be prophetic, gives no indication of the fraction of the Kalpa that has elapsed.

The śīghra motions of the planets are their proper motions on their śīghra epicycles, as counted from a point on a radius of that epicycle parallel to the radius of the deferent that connects the center to Aries 0°: they are. for the inferior planets, the sums of the mean solar motion and the planets’ anomalies. The śīghras of the inferior planets are used in Indian tables because of the Greek principle (a variant of which is found in the Romakasiddhānta) according to which the velocity of the planet is inversely proportional to its distance from the center of the earth.⁴⁰ The mean motions of the inferior planets are, of course, identical with the sun’s, and Venus’ anomalistic motion is less than the sun’s: therefore, the only way to preserve the principle and keep Venus in an orbit below the sun was to use the śīghra motion.

The application of this principle to the problem of the distance of the planets’ orbits led the author of the Paitāmaha to make the rather arbitrary assumption that each of the 21,600 minutes of the moon’s orbit contains 15 yoojanas. so that the circumference is 324,000 yojanas.⁴¹ Then it is assumed that each planet travels the same number of yojanas in a Kalpa; that number, O, which is also the circumference of heaven, is 324,000 yojanas · 57,753,300,000 = 18,712.069.200.000.000 yojanas (Pait. 3. 6–7). The circumferences of the planetary orbits are given in Table V.3: the circumference of the naksatras’ is 60 times that of the sun, or 259,889.850 yojanas.

The diameters of these orbits are found by using

the value found in (III.50) (Pait. 3. 6): this value of π is traditionally used in the computation of the radii of the planetary orbits in preference to more correct ones.

The numbers of rotations of the sun and the moon yield the units in a Kalpa given in Table V.4.

The ahargana, c, or number of lapsed days, is computed from the beginning of the Kalpa (Pait. 4, 1), although it is not stated how far back in the past that is (V.2):

where L is the lapsed Manvantaras and y_L the lapsed years of the current Manvantara

(it is incorrect to add together n and m_y and to assume that t_m = S_n):

To compute the remaining fractional part of the current intercalary month in saura days. — ϵ. the author posits (figures between vertical lines are integers: r represents remainders) (Pait. 4, 2) that (V.3)

And, to compute the remaining part of the current avama in lithis, — v (Pait. 4, 3), one proceeds as below (V.4):

Both —ϵ and — v could be used in ahargana computations, Since ϵ and r_u are used by Brahmagupta (V.52–54). Our version of the Paitāmahasiddhānta, however, pursues the subject no further.

With c, one is instructed to find the mean longitudes of the planets by means of (111.16) (Pait. 4, 6). These mean longitudes are to be corrected by the planets’ mean motions in half the equation of daylight of that day and in the time interval corresponding to the deśāntara. or longitudinal difference, of that locality. For computing the deśāntara it is assumed that the distance in yojanas between Lanka and the point where one’s local circle of terrestrial longitude crosses the equator has been found by comparing the computed and observed times of an eclipse (Pait. 3. 27): the circumference of the earth is stated to be 5.000 yojanas.

The normal Indian model of the planets is derived from a Greek attempt to account for the inequalities of planetary motion while retaining the Aristotelian principle of concentricity: the model seems to have originated in Greek philosophical circles in the second century.⁴² It is depicted in Figure 14. The center of the earth and of each planetary orbit is at O. The mean planet rotates on the concentric deferent. The mean position of the planet. . is surrounded by epicycles of given circumferences: one (the manda epicycle) about the mean sun and the mean moon, and two (the manda and the śīghra epicycles) about the mean star-planels (the mean longitudes of Venus and Mercury are equal to the mean longitude of the sun). On each of these epicycles is an ucca— M on the manda. .S on the śīghra-the longitudes of which, measured from the epicycle radius pointing toward Aries 0°. are equal to those of the planet’s manda and śīghra: for all planets, OM₀ is parallel to . and for the superior planets the deferent radius pointing toward ., is parallel to S. The points where the lines MO and SO, extended if necessary, cut the deferent are the positions to which each epicycle, if acting alone, would displace the mean planet at : their combined effects in the case of the star-planets are determined by various integrations. The sun’s true longitude can be found directly by the “method of declinations” (III.28):

The circumferences, c, of the epicycles according to the Paitāmaha are listed in Table V.5. along with the corresponding maximum equation, μ_max and σ_max for each. The circumferences are measured in units of which there are 360 in the deferent (PS 3, 10–11).

In the Paitāmahasiddhānta a correct geometric solution is given for both μ and σ (Pait. 4, 10). To use σ as an example, in Figure 14 we have (V.6)

For the superior planets the method of integration is(V.7)

The process is iterated until differences disappear.

For the inferior planets, one first finds σ and then, with the corrected λ′, one finds μ,

The preceding computations have utilized the Sine function. A rule for determining the second differences, ΔΔ. in a Sine table where R = 3,438 is given by the Paitāmaha (Pait. 3, 12):

where s₁ is the first Sine, that for an argument of 3;45°, and S_n is a given tabular Sine. It has been argued persuasively that 3,438, which is derived from the relation

was also the Radius in Hipparchus’ Chord table.⁴³ The rule (V.8) is not accurate, as can be seen from Table V.6, where I use the Δ’s of Āryahhata I, since the Paitāmaha gives no numbers.

The Paitāmaha repeats many of the problems connected with spherical trigonometry that we have previously considered. Here, as later, we shall refrain from specifying these repetitions, and shall indicate only those new formulas that have appeared to us most significant. Some have undoubtedly escaped our notice that should be mentioned, but brevity has been our general aim.

The main elaboration of the projections of the Greco-Baby Ionian period that we find in the Paitāmaha is the cheda, or divisor. D (Pait. 6, 1); this is illustrated in Figure 15, where D is ∑G; it is ∑′B in Figure 10. In Figure 10 the time degrees since sunrise are represented by ∑′ H, and the time degrees since 6 o’clock by ∑′ E = ∑′ H — ω. Then (see III.65a)

With D known, one can find the Ĕanku, or Sine of the sun’s altitude, Sin α, which is ∑′ M′ in Figure 10 (see III.65b);

Then one can find the shadow with (III.67), which involves also finding the Sine of the zenith distance, called the drgjyā in Sanskrit. We have already seen in (III.53b) that the Paitāmaha calls Sin = Cos α _n the anasta, and prescribes (Pait. 6, 2)

which is illustrated in Figure 16.

Furthermore, in Figure 15. S_nB is the noon cheda, D_n, such that (Pait. 6, 3)

From this it is clear that h_n and D_n are immediately computable if one knows . Then, observing the gnomon’s shadow at any given time, one can find h by the Pythagorean theorem; and

which is certainly not true. With D one can compute backward to find the time degrees since sunrise: and with those time degrees, the solar longitude, and the local oblique ascensions, one can find the longitude of the ascendant.

The latitude theory of the Paitāmaha is different for the moon, the superior planets, and the inferioir planets. Each planet has a given orbital inclination, i(Pait.3, 18). Then, for the moon, (see III.18b)

For the superior planets, however,

where H is the final śīghra hypotenuse. The nodes of the superior planets and the moon lie on their respective orbits, while those of the inferior planets lie on their śīghra epicycles, so that, for them, ω is the difference between the longitudes of the planet’s slghrocca and its node. Thus, the plane of the śīghra epicycle is tilted with respect to the plane of the ecliptic at the angle iso that (see Figure 17) when ω = 90° and H = R –r_∑(;r_∑ being the radius of the śīghra epicycle), the maximum latitude occurs. For the inferior planets, then, given their special definition of ω. the relation (V.17) is also valid. In Table V.17 are tabulated the values of í and of

The Paitāmaha’s fragmentary rules For computing eclipses (Poit. 5, 2–4; 9. 5–7) show some development beyond the computations of the Greco-Babylonian period, but the text is both corrupt and incomplete, The diameters of the sun and the moon, as of the other planets (see Table V.8), are given in yojanas (V.18):

The true radius of the planet’s orbit at any given time, measured in yojanas, k^y, is found from (Pait. 5.2)

where H is the manda hypotenuse for the sun and the moon, Since the apparent diameter in minutes, d^m varies inversely with , k^y it follows that

since From the data given in Table V.3

we know that the mean distances from the earth of the sun and the moon are, respectively, 684,869 and 51,229 yojanas. Therefore, we find that (V.21)

The formula for finding the distance from the earth of the tip of the earth’s shadow, k_u, is given by the Paitāmaha as (Pait. 5, 3)

where r_e and d_e are, respectively, the radius and the diameter of the earth, the circumference of which has been given as 5,000 yojanas. As can be seen from Figure 18, one should have

The distance of the end of the shadow cone from an observer on earth is approximately k_u - r_e; this presumably is what the Paitāmaha intended to find.

The Paitāmaha ascertains the diameter of the earth’s shadow at the moon’s distance in minutes thus (Pait. 5,4):

This formulation is incomplete, since we have, from (V.20) and the relation , that

The remainder of the computation of a lunar eclipse in the Paitāmaha is lost. For the computation of a solar eclipse, all that remains is a corrupt version of rules for obtaining parallax. The longitudinal parallax seems to be found from the right ascension of the difference between the longitudes of the moon and the point where the ecliptic crosses the meridian; one then should employ something like (III .17). To find the latitudinal parallax the Paitāmaha’s rule is (Pait. 9, 6)

where Cos α _n, the solar drgjyā for noon, also called the anasta (see III .53b), is Sin (Ø ± δ). The text might have been expected to use the lunar drgjyā for noon, which is rather than the relative velocity, . (see III.20). The Paitāmaha also should have used the zenith distance of the nonagesimal rather than that of a body on the meridian.

In addition, the Paitāmaha contains a section on the visibility of the moon, the planets, and the stars

(Pait, 8, 2–3) that represents an advance over the Greco-Babylonian computation (III.73–75). In this computation, following a convention of Hipparehus,⁴⁴ one operates with polar rather than ecliptic coordinates; polar latitude is the angular distance of the body from the ecliptic measured along the declination circle passing through the planet, and polar longitude is the longitude of the point where that declination circle cuts the ecliptic.

The first step is to find the longitudinal difference, λλ*, between λ and λ* ;this procedure is called the ayanadrkkarma, and is similar to the computation of the ayanadrkkarma, and is similiar to the computation of the ayanavalana (see V.94). In Figure 19, since, when λ = 0°. ∠ PSP' = ϵ, and when λ = 90°, ∠ PSP ' = 0° therefore

For solving triangle λ Sλ*, which is assumed to be small and virtually plane –β = Sλ is assumed to be very small-the present text of the Paitāmaha has (Pait. 8.2)

This is a corrupt version of what is found in the Brāhmasphutasiddhānta (BSS 6. 3):

In fact, Brahmagupta has found λ M rather than λλ*. More accurate would be

Moreover, one could easily find λλ* with (V.31)

Furthermore, one can then find β* = Sλ* from

The corruption in the Paitāmaha may be related to the formulation (see VI. 4)

which follows from (III.55) and (V.29).

When a body with latitude β* crosses the local horizon, one must find the ecliptic arc between λ* and the horizon. This computation is called the aksadrkkarma, and is related to the computation of the aksavalana (see III.93). In Figure 19, the required ecliptic arc is Dλ*; to find it, the present text of the Paitāmaha prescribes (Pait. 8, 2)

Again, this is a corrupt version of Brahmagupta’s rule (BSS 6,4)

Brahmagupta has found NM rather than Dλ*; and a more accurate formula would be

Moreover, one again could have determined Dλ* accurately. Since ∠ DSλ = Ø - Δ (λ + 90°), it follows that (V.37)

The time at which it is computed that D is on the horizon is also the time that the body at S is on the horizon.

The body will be visible if the sun at that time is sufficiently far below the horizon. Indian astronomers convert this distance below the horizon into time; for each planet and for the fixed stars there is an equatorial arc measured in kālāmśas or time degrees, k (Pait. 3, 9), that is the minimum difference in the right ascensions, Δα, of λ* of the body and at which the body can be seen. Time until or since first visibility or first invisibility can be computed by permutations of (Pait. 8, 3)

The planets’ kālāmśas are related to their diameters in yojanas (Pait. 3, 8), which must be considered to be their apparent diameters at the moon’s distance, where 15 yojanas = 1′, These are tabulated in Table V.8.

The Paitāmaha gives another rule for determining the times of the first and last visibilities and first

and last stations of the planets, in which the effects of terrestrial and celestial latitude are ignored. This method is based on the śīghra anomalies, κ₀ necessary for the occurrences of these phenomena; it is, thus, closely related to a Greek text computed from Ptolemaic parameters and erroneously attributed to Heliodoras.⁴⁵ In Table V.9 are tabulated κ₀ for the first visibilities and first stations; the others are symmetrical (Pait. 3, 31). They are, of course, computed from the sizes of the śīghra epicycles.⁴⁶

For a superior planet, the time until phase, t, is found from (Pait. 4, 15)

for an inferior planet, from

To find the actual distance in degrees between the sun and the moon, the Paitāmaha establishes a very complicated procedure (Pait. 9, 1–2) that later authors such as Brahmagupta (BSS 7, 15– 16) converted into physical models. For both the sun and the moon one finds Sin α, the śankutala, and Sin η from (III.68–70). Then, for each, the distance of the base of its Sin 훂 from the east-west line, the bāhu (see Figure 20), is found from (see III.71)

Then one finds the drgjyā, cos α, of each from (see III.67)

By the Pythagorean theorem one finds the koṭis, AO and BO, of the moon and the sun; their difference is AB = M′ L′ = ML. The text of the Paitāmaha becomes corrupt at this point, but the rest of the procedure can be supplied from Brahmagupta (BSS 7.7–10). If NL′ = SS′ = Sin α, then the difference of the Sin α’s of the sun and the moon is NL,and

Further, the difference of the bāhus is S′L′ = SN; therefore

MS is the chord of the arc of the great circle that passes through the sun and the moon.

The illuminated portion of the moon’s diameter, σ, is, according to the Paitāmaha (Pait. 9.3), found by(V.45)

in the first and fourth quadrants, and

in the other two quadrants, where Δλ is the difference between the longitudes of the sun and the moon. Clearly, what is required is the equivalent of (III,76): this is given by Brahmagupta (BSS 7, 11–12) (V.46):

The Paitāmaha’s text again seems to be corrupt.

To find the sūtra, s, which is the radius of the circle the circumference of which lies along the inner edge of the moon’s sickle, the Paitāmaha (Pait. 9, 4), as preserved, gives what appears to be a fragment of Brahmagupta’s rule (BSS 7, 14):

This at least meets the requirement that when σ = 0, and there is no “sickle”; and when , S = ∞ and the inner edge of the “sickle” is a straight line passing through the center of the moon and perpendicular to σ.

We have noted previously that the Paitāmaha uses polar coordinates to determine the times of first and last visibility of the planets. In Table V.10 are listed the polar coordinates of the yogatārās (“junction stars”) of the naksatras according to the Paitāmaha (Pait. 3, 30), the ecliptic coordinates computed with adaptations of (V.30) and (V.32), and tentative identifications with stars listed by Ptolemy.⁴⁷ The extreme latitudes of some of these stars suggest that they are not members of the Vedic naksatras; nor is it conceivable that any astronomer without an armillary sphere could determine the polar coordinates of the stars, and there is no evidence that the Indians had armillary spheres. Therefore, it is likely that the author of the Paitāmaha somehow adapted a Greek star catalog to his purposes: however, either his source or he himself created such numerous errors that definite identifications are impossible. The non-Indian origin of these coordinates is also deducible from the discrepancies between Table V.10 and Table III.22.

The Paitāmaha redefines the pātas of the sun and the moon given in (III.28–29); they are now said to be the instants when the two luminaries have equal declinations (Pait. 5, 9). Vaidhrta occurs when they are on opposite sides of an equinox, vyatipāta when they are on opposite sides of a solstice.

The second representative of the Brāhmapaksa. the Brāhmasphutasiddhānta, was composed by Brahmagupta at Bhillamāla in southern Rājasthān in 628. The first ten adhyāyas. the Daśādhyāyī, are a summary of the Paitāmaha; adhyāyas 13–17 contain additions. At some points Brahmagupta displays familiarity with the Mahābhāskarīya of Bhāskara I or its source. Adhyāya 25, the Dhyānagrahopadeśa, is a karana (that is, it uses a contemporary epoch); adhyāya 21, a description of the celestial spheres. In this essay only these adhyāyas will be discussed, and in some detail; they are fundamental for all later Indian astronomy.

Brahmagupta’s division of the Kalpa is identical with the Paitāmaha’s (BSS 1, 7–12), although he adds that the interval between the beginning of the Kalpa and the beginning of the current Kaliyuga (sunrise at Lanlikā on 18 February -3101) is 1,972,944,000 years or 4,567 Kaliyugas (BSS 1, 26–28). His values for the rotations of the celestial bodies (BSS 1, 14–22) also are identical with the Paitāmaha’s (see Table V.2), as are, necessarily, the time relations in Table V.4. His computation of the ahargana is very similar to (V.2), but for (V.2 b–e) hesubstitutes (BSS 1, 29–30) (V.48)

n=12Y+M_y

In adhyāya 13 he adds another series of rules for computing the ahargana (BSS 13, 11–19). From (V.3b) he derives

and, from (V.4a)(V.50),

He then proceeds to develop several more such rules for finding the ahargana, of which the most important is (V.51)

therefore

substituting T for U + C. Moreover,

Thence,

This last quantity, , is called the “accurate remainder of the intercalary months”; we will denote it r′_a. Then

Brahmagupta gives rules for finding the accumulated epact, ϵ —that is, the time between one year’s last mean conjunction of the sun and the moon (Caitraśsuklapratipad) and the beginning of the next mean solar year (Meṣasȧrikrānti) (BSS l, 39–40). These rules also are not unrelated to (V.3). In the following formulas, a subscript zero indicates the unit in a year (V.52):

where r = 훆 in tithis, From (V.52 f) we have

To compute the civil days, c_m’, between Caitraśuklapratipad and Meṣasȧrikrānti from the tithis, t_m’, in the same interval (BSS l, 42–43), one must first compute the corresponding avamas, u_m then (V.53)

There are two components of u_m the fractional avama accumulated at Meṣasȧrikrānti – which, from (V.52e) is , which Brahmagupta converts into civil days by multiplication by - and any avamas resulting from the difference between the true interval, t_m, and the mean interval, ϵ. Using the approximation in (III.15), Brahmagupta’s final formulation is

He also gives a rule for finding the civil days, c_nm’, between mean Caitraśsuklapratipad and mean Meṣasȧkrnānti (BSS l, 57–58). This interval in tithis is ϵ the avamas, u_n, again are both those in ϵ and the fractional avama accumulated at Meṣasȧrikrānti. Brahmagupta’s formula, then, is

The basic formula for finding the mean longitude of a planet in the Brāhmasphutasiddhānta is that given in (III.16). Brahmagupta, however, adds several variants designed to shorten the computations. In the first, one proceeds as follows. The mean longitude at the last mean Meṣasȧrikrānti is found from (BSS l.31)

Then the ahargaṇa since mean Meṣasaıkrānti is multiplied by the mean daily motion of each planet except the moon; these are given in Table V.II (BSS 1, 44–49).

For the moon Brahmagupta gives the special equation (BSS 1.45)

where t_c represents the integer tithis elapsed since mean Caitraśuklapratipad, and the mean sun and the mean moon in the fractional avama —that is, in the time between the end of the mean tithi before mean Meṣasaıkrānti and the next crossing of the local six-o’clock circle by the mean sun.

In the second method Brahmagupta first gives the kṣepas of the planets for the beginning of Kaliyuga, sunrise at Laṅkā on 18 February -3101 (BSS 1, 51–56). The kṣepas are the remainders, ρ, from the application of (V.55). wherein = I tabulate in Table V .12 the interger rotations, ǀrǀ; the remainders, ρ: and the corresponding mean longitudes, λ (compare Table V.1).

Given these ksepas, one could proceed to apply the first method, with y being the years lapsed since the beginning of the present Kaliyuga.

The published text of the Brāhmasphutasiddhānta at this point inserts bījas or corrections to the planets’ mean longitudes, (BSS 1, 59–60); the verses describing these bījas, however, seem to be interpolated. (See Table V.22.)

In adhyāya 13 Brahmagupta adds some more rules for finding mean and true longitudes (BSS 13, 20–39). First, for the sun and the moon, where r_u · converts the fraction of a civil day between the beginning of the mean tithi and the beginning of the next mean sunrise day, r_u = c · U — C · ǀuǀ, into a fraction of an intercalary month, and where is the fraction of an intercalary month that has accumulated at the beginning of that sunrise day, it follows that

Then (V.58)

The calendar days, c_y, since mean Caitrasuklapratipad are found from (V .59)

where m_y is the lapsed synodic months, u the corresponding

avamas, and t_m the lapsed tithis of the current month until the beginning of the current tithi, which precedes the current mean sunrise day . Therefore,

and the mean progress of the moon since mean Caitraśuklapratipad is

The lapsed synodic months from mean Caitraśuklapratipad until the beginning of the current mean sunrise day are found by (V.60)

Since n = m — a. it follows that

and the mean progress of the sun is

Brahmagupta’s remaining rules for finding mean longitudes are simply playful, and need not be explicated here. The mean longitudes are corrected for the deśāntara by a rule similar to (III.37), although the circumference of the earth is taken to be 5,000 yojanas, as in the Paitāmaha (BSS 1. 33–38):

where d is the shortest distance between two localities, ΔØ the difference of their terrestrial latitudes, and Δ their longitudinal difference in yojanas. Clearly one of the two localities must lie on the prime meridian; for the formula to be correct, it should be Lanka itself. This criticism was also made by Bhāskara I (MB 2,5– 6).

The basic planetary model in the Brāhmasphuiasiddhānta is identical with that in the Paitāmah (Figure 14). The manda epicycles of the sun, the moon, and Venus and the śīghra epicycle of Venus are, however, pulsating (BSS 2, 20–21. 34–35). For Venus the maximum manda epicycle, c_m, and minimum śīghra.epicycle, c_m, occur at anomalies of 0° and 180°, the minimum manda epicycle and maximum śīghra epicycle at anomalies of 90° and 270°. The corrected epicycle, c′ will, then, be (BSS 2, 13)

where c₀ is the mean epicycle. The pulsation of the epicycles of the luminaries depends on the position of the body relative to the local meridian and six-o’clock circles and on the amount of the anomaly, These pulsating epicycles are so arranged that it appears they may be intended to account for horizon refraction, although that is in fact independent of the anomaly; and the magnitude of the effect of Brahmagupta’s device, at its maximum effectiveness on the horizon, is only a tenth of that of refraction.

The longitudes of the mandoccas and nodes of the planets in 628, according to Brahmagupta’s parameters, are given in Table V. 13.

Brahmagupta’s values of the circumferences of the epicycles are closely related to, but not identical with, the Paitāmaha’s (Table V.5); they are given in Table V.14 (BSS 2. 21–22, 34–39), where the Paitāmaha’s values of c are in italics.

The differences in the maximum equations between the Paitāmaha and Brahmagupta are due to computing with different values of R. Brahmagupta uses the normal Indian “method of sines” to find the manda equation(BSS 2. 17).⁴⁸

Brahmagupta’s method of integrating the two equations for all the planets save Mars (BSS 2, 36) is similar to that of the Paitāmaha for the superior planets (V.7); the difference lies in the fact that the corrected anomalies are found by algebraically applying all of the equation rather than just half. For Mars, Brahmagupta (BSS 2, 37–40) begins the process by modifying κ_μ by an amount that depends on κ_σ;the maximum correction is 6;40° when κ_σ is 45°, 135°, 225°, or 315°. Such a computation for Mars had already been stated, although obscurely, by the Paitāmaha (Puit. 4, 8). The integration rule for Mars uses the halving of the equations that Brahmagupta avoids for the other planets,

Brahmagupta gives rules for computing the increase or decrease to the equation, x, resulting from the pulsation of the manda epicycles of the sun and the moon as the angle of depression from the meridian, n, varies (BSS 2, 27–28). For the sun this is

The proof of this is as follows. To express μ in minutes, we can form (since μ ∠ 3;45° for the sun) (V.64)

From Table V.13 we know that the change in the size of the manda epicycle of the sun is ·

and, from (a) and (b),

which is rounded off to (V.63).

For the moon the change in the size of the manda epicycle is · Therefore (v.65)

But, since the effect of the pulsating epicycle for the moon is most noticeable when μ > 3;45°. we use the second sine difference, 213, in

in which μ is expressed in minutes. Then

Brahmagupta’s formula is

One component of the equation of time is that due to the variability of the daily solar velocity; it is simply μ_s. The influence of this upon any planet’s argument, x_κ, which is called the bhujāntara, is given by the formula (BSS 2, 29)

where ν_p is the planet’s daily velocity and all terms of the equation are in minutes.

The correction,x_μ to the mean daily motion, , due to the manda equation described by Brahmagupta is an approximation to the cosine rule already expounded in the Sūtryasiddhānta of Fātadeva (PS 9. 11–14). The expression in the Brāhmasphutasiddhānta is (BSS 2,41–44)

where Δ_κ is the daily increment in the manda anomaly, and

The correction, x_σ due to the śīghra equation, on the other hand, is inexplicable. It is

where ( is the difference between the mean daily motion of the planet’s ślghra and the mean daily motion of the sun) and The presence of σ and of in the formula, however, seems to be incorrect.

Brahmagupta also lists the śīghra anomalies necessary for the occurrences of the Greek-letter phenomena (BSS 2,48–53). This is identical with the Paitāmaha’s (Table V.9), except that the first stations of Mars and Mercury occur when κ_σ equals 163° and 145°. respectively.

In adhyāya 14 Brahmagupta briefly explains a planetary model with an eccentric circle producing the manda equation and an epicycle producing the śīghra equation (BSS 14. 10–18), This model, which is surely Hellenistic in origin, was also known to Ārvabhata (A Kālakriyā 17– 19).

Brahmagupta also refers to the portions (bhogas) of the naksatras (BSS 14, 47–52), I bhoga being equal to 790′ (the mean daily motion of the moon). Certain naksatras (Bharanī. Ārdrā. Āślesā. Svāti, Jyesthā, and Śalabhisaj) have half a bhoga or 395′: others (Punarvasu, Uttaraphālgunī, Viśākhā, Uttarāsādhā, and Pūrvabhādrapadā) have one and a half bhogas or 1185′. The rest of the twenty-seven naksatras have one bhoga apiece. Since the sum of these does not equal 21,600′, the remainder —270′ — is made the bhoga of Abhijit.

In this chapter he also gives, following Bhāskara I (MB 7, 17–19). a remarkable rule for finding .where r is the radius of the epicycle, without first finding Sin κ it is, in fact, a general formula for finding Sin ϑ. The rule is (BSS 14, 23 – 24)⁴⁹

This is equivalent to

In this formula R = 1 since, if ϑ = 90°,

The computation of a few values of sin ϑ with this formula in Table V.15 and their comparison with the entries in a modern table demonstrate its general approximative value.

The reverse of (V.69) is also given by Brahmagupta in the form (BSS 14,25–26)

So that

The actual Sine table in the Brāhmasphutasiddhānta uses the odd value R = 3,270. The table itself, which is very accurately computed, is reproduced in Table V.16 (BSS 2, 2–5).

Brahmagupta also gives a table of Versines (BSS 2, 6–9), as is usual.

Brahmagupta presents many elaborations of the Paitāmaha’s formulas in spherical trigonometry, but he also adds some new ones. He analyzes the gnomon shadow (see Figure 21) as follows (BSS 3,4):

Then the bhuja. bg, which was called the koti in (III.71) (see Figures 10 and 12), is found from

The other side, here called the koti, again is found by the Pythagorean theorem.

For finding the right ascension, α. of an arc, λ Brahmagupta presents a rule (BSS 3, 14–17) previously given by Āryabhata (A Gola 25):

which is evident from Figure 6.

As well as further exploiting the relation of the cheda triangle to the gnomon triangle (V.10–15), Brahmagupta introduces the istāntyā, I, which is Σ′ G′ in Figure 15 (BSS 3, 29):

where t is the time degrees elapsed since or remaining before six o’clock. The istāntyā. therefore, is the Sine of the time degrees that have lapsed since sunrise or remain until sunset. Thus (BSS 3, 30–32)

and I can be universally substituted for D. And. as the Paitāmaha found the noon antyā, so Brahmagupta finds the noon istāntyā. I_n, which is S′ A in Figure 15 (BSS 3.34):

Finally, along these same lines, he adds (BSS 3, 37)

where n is the hour angle of the sun from the meridian — Σ′ 0S′ in Figure 15. Brahmagupta explores the many ramifications of this relationship (BSS 3,38–46).

But the most extraordinary rule that Brahmagupta gives is the following, for finding the solar altitude (BSS 3,54–56):

This is true only when the sun is in the northern hemisphere and the angle between the shadow and the east-west line is 45°. Clearly there are easier ways to find Sin α for such a moment.

From Figure 22 it can be seen that

under the conditions stated above. Further, if gσ is the shadow of the sun (at E in Figure 23) on the equator at an hour angle of 45°. then

Now, from Figure 23 it is clear that (V.84)

To find SJ (using V .83) :

Then, using (V.82) and (V.84 d):

To find JF=IL:

Therefore,

But

With (V.84 f) and (V.811):

Thererfore,

and IL + SJ = Sin α has the form of (V.81).

Brahmagupta devotes most of the remainder of this adhyāya to problems involving Sin_h η and Sin η. The only rule that need be repeated here is the following: To find the length of the string, e′ w′, connecting the points on the circumference of a circle the center of which is the base of the gnomon and the radius of which is r′, at which the sun’s shadow falls at sunrise (e′) and at sunset (w′) (it is assumed that there is no change in the solar declination during daylight), he prescribes (BSS 3, 64 (V.85):

In adhyāya 15 the principal new relations that Brahmagupta investigates are those involving the time interval between the moments of the sun’s being on the prime vertical and its being on the meridian. In Figure 10 the Sine of the corresponding time degrees is DS₀, and the Sine of the time degrees corresponding to the time interval between the moment of sunrise and that when the sun is on the prime vertical is OD ± Sin ω. Brahmagupta’s first formula is (BSS 15. 19–20)

Then, if we multiply all sides of right triangle OAS_n by we have (V .87)

Further,

and therefore

At one point Brahmagupta makes the absurd mistake of substituting R² - DS² for OD¹. so that he slates wrongly (BSS 15,24–25):

But later, using the relation

he obtains the correct (BSS 15, 35-38)

Then, considering the triangle BOC in Figure 10, he observes that

Combining this with (III.63) and (III.70), he finds (BSS 15,21–23):

Finally, Brahmagupta gives values for the right ascension, α, of the first three zodiacal signs (BSS 15, 32–33): in Table V.I7 I add Ptolemy’s values, the minutes corresponding to Brahmagupta s divided by 6, and the vinādīs≈ according to the Pañcasiddkāntikā (Table III.21).

Adhyāya 4 of the Brāhmasphutusittdhānta is devoted to lunar eclipses, in which five significant phases are distinguished: first contact, beginning of totality, mideclipse, end of totality, and last contact (BSS 4, 1–3). Brahmagupta’s rule for finding the latitude of the moon is identical with the Paitāmaha’s (Table V.7 and formula V.16) (BSS 4, 5). But for finding the apparent diameters of the sun, the moon, and the earth’s shadow, he develops a procedure different from the Paiitāmaha’s as described in (V. 18–21). If the daily motion of a planet be denoted ν he prescribes (BSS 4,6) (V.89):

From this the mean values given in Table V. 18 are derived.

Brahmagupta’s rules for finding the duration of a lunar eclipse, Δ t_e, are equivalent to the Pauliśa’s (III.41–42); the duration of totality, Δ t^τ, is found from (BSS 4, 8)

where the dividend is AU in Figure 24. To find the changes in longitude during the half-durations of the esclipse and of totality, Brahmagupta forms (BSS 4, 9)

To find the magnitude of an eclipse for a given time, t_n, we must first find the elongation of the moon from the shadow center at that time (BSS 4, 11–12) (V.92):

Then, in Figure 24, the eclipse magnitude, B′ C, is (BSS 4, 12)

These rules, with appropriate substitutions, are also considered valid approximations to the durations and magnitudes of solar eclipses, although in adhyāya 5 the changes in lunar latitude are taken into consideration (BSS 5, 13–15).

Brahmagupta gives a complete set of rules for computing the two elements of deflection (BSS 4, 16– 18) like those previously given by Lātadeva in his Sūryasiddhānta (PS 1 1). The aksavalana. λ₁ is found from (see III .47)

The ayanavalana, λ₂, which is the angle between the ecliptic and the equator, is found from

The projection is illustrated in Figure 25.

Brahmagupta repeats Āryabhata’s colors (A Gola 46) for the different phases of a lunar eclipse; they are noted in Table V. 19 (BSS 4, 19).

In computing parallax, Brahmagupta makes the

same error as does the Paitāmahu — that is, he regards the zenith distance of the nonagesimal, ZV, to be the same as the zenith distance of a body on the meridian (see V.26). Thus he states (BSS 5, 3)

If the ecliptic passed through Z (see Figure 1), then it would be true that

where ZS = VS is the elongation between the sun and the zenith and π₀ = 4 nādīs. When the ecliptic does not pass through Z, it is assumed that the ratio (where α[V] is the altitude of the nonagesimal) can be used to transform the longitudinal parallax to the ecliptic. Then (BSS 5.4)

since . Moreover, for latitudinal parallax, one must find the approximate zenith distance of the moon,ZM. from (BSS 5, 9)

From (V.95) one finds the solar latitudinal parallax (BSS 5, 10–11)

and the lunar latitudinal parallax

and the combined parallax (BSS 5, 22–24)

In all three formulas one would expect the true rather than the mean daily velocity to be used.

By computing the parallaxes for different phases of the eclipse and iterating the procedure until all differences vanish, the true durations and times between phases can be determined, as can the true magnitudes at different times during the eclipse. The projections are similar to those in a lunar eclipse. If the magnitude of a solar eclipse is less than 1/12 of the sun’s apparent diameter, or if the magnitude of a lunar eclipse is less than 1 /16 of the moon’s apparent diameter, Brahmagupta states that the eclipse is not visible (BSS 5, 20).

Adhyāya 16 is devoted to practical rules Tor constructing projections. Brahmagupta does, however, add the method for finding the longitudinal difference (the desamara) from the difference between the calculated and the observed times of eclipses (BSS 16, 27–28): he repeats (his from the Paitāmaha (Petit. III , 27). He also gives the rules for “predicting” eclipses from the parvan, which is a period of six months, or 177 civil days. To the mean longitudes at one parvan of the sun, moon, moon’s manda, and moon’s node are added the cālanas, or increments in mean motion for 177 days, listed in Table V. 20 (BSS 16, 29–33).

In adhyāya 6, Brahmagupta deals with the first and last visibilities of the planets. In the course of this he gives the correct formulations of the rules for performing the ayanadrkkarma (V.29) and the aksadrkkarma (V.35), which are corruptly preserved in the Paitāmahu. He also repeats the Paitāmaha’s kālāmśas for visibility (Table V.8) (BSS 6, 6) and its rule for finding the time until first or last visibility (V.38) (BSS 6, 7). Brahmagupta adds only a correction in the kālāmśsas of visibility for Mercury that depends on its śīghra equation, one for Jupiter that depends on its manda equation, and one for Venus that takes into account the fact that its apparent diameter is smaller at superior than it is at inferior conjunction (BSS 6, 10–11).

Adhyāya 7 of the Brāhmasphmasphutasiddhānta is devoted to the determination of the illuminated portion of the moon. One performs the drkkarma as in adhyāya 6 in order to find then (BSS 7, 5)

With one can find the radius of the day circle passing through the moon by means of (III.31), and thereby the time in ghalikās that the moon will be above the horizon. To find the arc on the great circle that passes through the sun and the moon, Brahmagupta employs the construction given by the Paitāmaha (V.41–44) (BSS 7. 7–10). His rules for finding the illuminated portion have already been given (V.46–47). Brahmagupta concludes with instructions on how to transform these calculations into a projection in scale in order to astound one’s patrons (BSS 7. 15–17). Adhyāya 17 gives further rules for this projection.

In adhyāya 8 Brahmagupta discusses the moon’s shadow, , which is approximately

First one computes the ghatikās that the moon is above the horizon, as indicated in adhyāya 7; then one converts the ecliptic arc between the longitude of the ascendant and the polar longitude of the moon, into ghatikās by means ofthe local oblique ascensions. If those ghatikās equal then is on the meridian and one can find Sin from (III.53 b) (with ). Otherwise, with the time in ghatikās since moonrise or until moonset, one can find the altitude of the moon from (V.10–12). The shadow, however, is assumed to be cast by the upper rim of the moon, so that one is to form (BSS 8,6–7)

with the radius of the moon expressed in minutes of arc. And this new altitude is affected by parallax to form

Thence

One can perform a similar operation to find the shadow of the sun.

In adhyāya 9 Brahmagupta tackles the problems of the conjunctions of planets with other planets and with fixed stars. The first step toward any solution is to find the planetary latitudes. He derives the values for the orbital inclinations from the Paitāmaka (Table V.7) (BSS 9. 1), but “corrects” the Paitāmaha formula for finding β (V, 17) while retaining the separate models for superior and inferior planets. The distance of the mean longitude of an inferior planet from its node on its śīghra epicycle, ω, is corrected by its first manda equation: the distance of the mean longitude of a superior planet from its node on its orbit is corrected by its first manda equation and its first śīghra equation (BSS 9, 8–11).

The apparent diameters of the planets depend, according to Brahmagupta, on two factors; their kālāmśas of visibility and their śīghra hypotenuses, H. The Paitāmaha determines their mean diameters in yojanas by a process of halving in which the order of the planets depends on the kālāmśas of visibility (Table V.8). Brahmagupta’s rule to find the mean diameter of a planet in minutes, d is (BSS 9,2)

where k represents the kālāmsas of visibility. The resulting mean diameters in minutes are presented in Table V.21

Brahmagupta’s formula (BSS 9, 3–4) for finding the true apparent diameter, d, is (V. 108)

But, with (V.107). this is equivalent to

This can be transformed into the following two formulas:

The factor , which occurs when R - H or H - R equals r, the radius of the śīghra epicycle (one must substitute 1 for Sin Σ), is much too large: one needs something on the order of

With the longitudinal difference between the two planets. Δλ and the difference of their velocities, Δν, one can find a first approximation to the time until conjunction. Δ t (BSS 9, 5):

For each planet the cālana, c which is the difference between the longitude ofthe conjunction and the longitude of the planet at time Δ t. is found from (BSS 9, 6)

And the longitude of the planet at conjunction, λ_c. is given by (BSS 9, 7)

But this is the computation of a conjunction only if both planets (or the planet and the fixed star) are on the ecliptic, for to Brahmagupta conjunction means that both bodies lie on the same great circle passing through the poles; in other words, their polar longitudes, λ*. must be the same. As was done in adhyāyas 7 and 8, one finds Δ* of each body at its rising and its setting, and the ghatikās that it is above the horizon during the night, g. By comparing these elements for the two bodies, one can discover whether or not a conjunction will occur during the night. If one will, one must find the velocities of the planets (measured in minutes per ghatikā), ν, during the time g from (BSS 9, 15)

where Δλ*; is the difference between the planet’s λ* at rising, λ*_u and that at setting, λ*., Then the time between the rising of the bodies and the conjunction, λ t, will be approximately (BSS 9, 16–18) (V.113)

where Δλ* is the difference of the λ*’s of the two bodies; and the time between the conjunction and their setting is

The cālana for the polar longitude, c*, will be, for each planet:

Then a first approximation to λ _c will be

However, since the latitudes of the planets will have changed in the interval λ t_u one must recompute the polar longitudes for that time and find the true time and longitude by a process of iteration. By comparing the final values of β* for the two bodies and their respective radii, one can determine whether an obscuration occurs. A fürther pair of verses asks that the time of the conjunction be modified by the longitudinal parallax of the planet, that the new β* ’s be computed forthat moment, and that they be modified by the latitudinal parallax of the planet (BSS 9, 20–21). The planetary parallax, π_p is found from the proportion

Finally, at the end of the adhyāya Brahmagupta adds a mistaken rule for checking the computation of the conjunction and correcting the time if an error is found. First he prescribes finding, from the day circles of the two planets at conjunction, the respective ghatikās that have risen, ft and s ; we have already computed the ghatikās that each is above the horizon, g_p and g_q Then he assumes that when there is a conjunction (BSS 9, 22),

That this is not true is clear from Figure 26; it is equivalent to saying But Brahmagupta proceeds to attempt to determine the time when it will be true (BSS 9, 23–24). To accomplish this he finds two time differences. Δ t₁ and Δ t₂, (V. 118):

where n′ represents the ghatikās that have risen for each planet in a time, Δ t_c, that is guessed at as the time between the previously computed time of

conjunction, t_c, and the time when (V. 117) will hold. From this he forms a new estimate of that interval, Δ t_c′, using

Finally, he asserts that (V. 117) will hold at the time t_c′, found from

The reasoning behind this procedure is not apparent. It is checked by computing new values of n_p and n_q for the time t′ _c: if (V.117) still does not hold, one must iterate (V.118) until it does.

Adhyāya 10 of the Berāhmasphutasiddhānta is devoted to the computation of conjunctions of planets with fixed stars. Brahmagupta gives polar longitudes and polar latitudes of the yogatārās of the naksatras that are in most cases identical with those of the Paitāmaha (Table V.10); his changes are given in Table V.22 (BSS 10, 1–9,35–37.40). In only one case does Brahmagupta seem to be thinking of a star other than the Paitamaha’s; he substitutes 훅 Sagittarii for γ Sagittarii as yogatārā of Purvāsādhā. Both are third-magnitude stars. Otherwise, his corrections to β are usually in the wrong direction; only for the yogatārās of krttikā, Svāti, Anurādhā, and Mūla (slightly) are his values better than those he corrects. It is tempting to believe that for the yogatārās of Rohini. Citrā, Viśākhā, and Jyesthā he is merely using the Paitāmaha’s β’s (usually poorly computed) as his own β’s. In any case, he does not demonstrate any greater aptitude for observational astronomy than did other Indian astronomers.

Much of the rest of adhyāya 10. then, is parallel to what we have already discussed in connection with the preceding adhyāyas; it involves finding λ★ and β of the planet, δ* of both planet and fixed star, and then, as in adhyāya 9, transforming all differences into time differences —that is, into equatorial arcs. Here, however, Brahmagupta uses a somewhat different procedure (BSS 10, 16–23).

This procedure is illustrated in Figure 27 (compare Figure 18). Using (III. 31–33), one can find the equatorial arcs ω(= N_eE ≈ D_eE) and ω* (= λ* E); then the sum or difference of ω and ω*, depending on the relative directions of Δ and Δ*. will be approximately D_eλ_e*,, which is the right ascensional arc of the longitudinal difference of the rising point (or setting point) of the ecliptic and the polar longitude of the planet. Thence the longitude of the ascendant (or descendant) when the planet, S. crosses the horizon is easily found. Forafixed star, which has no stated w one can simply perform the aksadrkkarma. substituting β* for β in (V.35). With these longitudinal differences and a knowledge of the longitude of the current ascendant (or descendant), one can predict the time until the planet and the star rise above (or set below) the horizon.

The rules for constructing a model illustrating the conjunction follow those in adhyāya 7 (BSS 10, 24); the rules for the “shadow” follow those in adhyāya 8 (BSS 10, 25): and the rules for the time of the conjunction follow those in adhyāya 9 (BSS 10, 25–26). Further, comparison of the right ascensions of a planet and the sun allows one to predict the time at which the Greek-letter phenomena will occur in a manner different from the procedure in adhyāya 6 (BSS 10, 33–34). Finally, in the Paitāmaha the ghatikās of visibility for the yogatārās were given as 2, and the corresponding kālārnsas are 12° (Pait. 8, 4); Brahmagupta prescribes 2 1/6 ghatikās (13 time degrees) for Sirius and 2 1/3 ghatikās (14 time degrees) for the yogatārās (BSS 10, 32.35–38).

In adhyāya 18 Brahmagupta applies the “pulverizer”

–a method for solving indeterminate equations–to problems involving planetary longitudes and the ahargana. In adhyāya 21 he includes the Paitāmaha’s rules for determining the circumferences of the orbits of the planets (Table V.3) (BSS 21, 11– 14) in yojanas, and in this connection he repeats the Paitamaha’s crude value of π, (BSS 21, 15): this statement by Brahmagupta of the value of π root 10 (compare BSS 21, 16–23) formed the basis of Sripati’s Sine table (Table V.33), in which Brahmagupta gives new values for the diameters in yojanas of the sun, the moon, and the earth, reported in Table V.23 (compare Table VII.7) (BSS 21, 32).

His formula here for finding the diameter of the earth’s shadow, d_u, at the moon’s distance is (BSS 21, 33)

This follows directly from Figure 17. The most interesting adhyāya is the last, the Dhyānagrahādhyāya, which is an abbreviated karana the epoch of which is 21 March 628. In this adhyāya Brahmagupta introduces a new Sine table in which R = 150; this is reproduced in Table V.24 (BSS 25,16).

The yearly mean motions of the planets, listed in Table V.25, differ slightly from those of adhyāya I of the Brāhmasphutasiddhānta (Table V,2) (BSS 25,26–31).

Brahmagupta also gives the epoch longitudes, and then the crude mean daily motions, which are constants, c, multiplied by the mean daily motion of the sun. The values of c are indicated in Table V.26 (BSS 25,33–36).

The longitudes of the mandoccas, given in Table V.27, also differ from those in the earlier part of the work (Table V.13) (BSS 25, 37).

The manda equations are derived by the “method of Sines,” with the coefficients listed in Table V.28 (BSS 25, 38–39); also given there are the maximum

equations, which differ from those in Table V.14.

The śīghra equations are tabulated at intervals of 13;20° (a naksatra); the maximum equations are listed in Table V.29 (compare Table V.14) (BSS 25.42–57).

Finally. Brahmagupta computes the half-equation of daylight in vinādīs, ω with the formula

where the values of γ/2 are very nearly those in the Pauliśa (Table III.19); they are listed in Table V.30 (BSS 25, 61).

In the late seventh or early eighth century a work based largely on the Brāhmasphutasiddhānta was written; it apparently was entitled the Mahāsiddhanta. This was the basis of the Zīj al-Sindhind al-kabīr composed by al-Fazārī in the late eighth century, and thereby became the foundation of the Sindhind tradition in Islamic astronomy, which had a particularly profound impact on western European science through its translation into Latin by Adelard of Bath in 1126 and the translation of associated texts by others in the twelfth century; it also influenced Byzantine science.⁵⁰ In most particulars it agreed with Brāhmasphutasiddhānia 1–5. Al-Fazārī disagrees or has other sources in the following particulars:

1. He gives the rotations of Saturn in a Kalpa as 146,569,284 instead of 146,567,298; this implies a mean daily motion of 0;2,0,22,57,36,16,….°.
2. He knows not only the value of R (3,270) used in the Brākmasphutasiddhānta , but also Ārya-bhata’s R = 3,438 and the Khandakhādyaka’s R = 150; this last value, as we have seen, is also found in Brāhmasphutasiddhānta 25.
3. His models (eccenter and epicycle) and parameters for computing the equations of the planets are adapted from the Sasanian Zīj al-Shāh, which was influenced by the Ārdharātrikapaksa.
4. His eclipse limits are derived from Varāhamihira’s Pañcasiddhāntikā.
5. Al-Fazari derived his method for computing planetary latitudes from adhyāya 9 of the Brāhma-sphutasiddhānta, although the parameters come from the Uttarakhandakhädyaka.

The Siddhāntaśekhara of Srīpati is based largely on the Brāhmasphuiasiddhānia of Brahmagupla; Śrīpati is known to have written other works in 1039 and 1056. His description of the Kalpa and its subdivisions is identical with Brahmagupta’s (SSS 1, 16–19), although he goes fürther in saying that a nychthemeron of the Creator contains two Kalpas: a year, 360 such nychthemera; and the life of the Creator, 100 such years (SSS 1, 20). He repeats Brahmagupta’s interval from the beginning of the current Kalpa to the beginning of the current Kaliyttga (SSS 1, 23–25). He also iterates the rotations of the planets according to the Brähmapaksa (Table V.2), with the exception that he has Saturn’s manda rotate fifty-four rather than fourty-one times in a Kalpa (SSS 1, 27–31): this means that about A.D. 1000 its longitude was 267° instead of Brahmagupta’s 261°.

Śrīpati gives a rule for finding the mean daily motions in rotations, which are simply reductions of R/IC (SSS 2, 33), and also gives much cruder ratios for daily motions in zodiacal signs and so on, as recorded in Table V.31 (SSS 2, 45–49) (compare Table V.11).

Śrīpati lists the mean longitudes of the planets at the beginning of Kaliyuga precisely as in column 4 of Table V.12 (SSS 2, 51–54). He gives rounded

values for the yojanas in the orbits of the planets listed in Table V.3 (SSS 2, 59–65), and he records (SSS 2,91– 92) the bījas that are inserted into some manuscripts of the Brāhmusphutasiddhānta. The period of increase and decrease in these bijas is 12.000 years: there were exactly 164,412 such periods between the beginnings of the Kalpa and of the current Kaliyuga. Table V.32 lists the yearly bimacr;jas according to the interpolation in Brahmagupta. the bimacr;ja amplitudes, and the bimacr;ja in A.D. 628.

Śrīpati’s rule for the desāntara is equivalent to (V.61) (SSS 2, 94), although he correctly criticizes it (SSS 2, 104), as had Bhāskara I. He lists the following localities as lying on the prime meridian: Lanka. Kumāri, Kāncī, Pānāta, Sitādri, Sadāsya, Vatsagulma, Māhismati, Ujjayinī, Pattasiva. Gar-garāta, Rohita, Sthānvīsvara, Sītagiri, and Sumeru (SSS 2,95–96).

Śrīpati’s table of Sines uses the unusual value. R = 3,415, that results from the formula in adhyāya 21 of the Brâhmusphuttisiddiíānta: it is reproduced in Table V. 33(SSS 3, 3–6).⁵¹

Śrīpati’s circumferences of the manda and Śīghra epicycles (SSS 3, 19–38) are identical with Brahmagupta’s (Table V.14) except that he makes the minimum of Venus’ Śīghra epicycle 263° and its maximum 268°.

Śrīpati repeats the cālanas of the parvans from the Brāhmusphunisiddhānta (Table V.20),although he gives 19;42,26° as the cālana of the moon’s manda (SSS 7, 3). He adds the cālanas of the paksas, which are the mean longitudinal differences between two successive syzygies separated by fifteen civil days. These are recorded in Table V.34 (SSS 7,4).

One component of the equation of time had been given previously by Brahmagupta (V.66) and is repeated by Śrīpati (SSS 3, 46). But Śrīpati also recognizes the second component, the udayāntara, which is due to the changing declination of the sun; (see Figure 28). This component, e, he gives correctly as (SSS 11,1)

having found α from

where r_min = Cos ϵ and r =Cos 훅. so that (V. 122) is equivalent to (V.76). He may have derived (V.121) from an Islamic source.

Śrīpati also introduces a rule to compute the evection of the moon, here called the (sphuta) caraphala; a similar rule had been given by Muñjāla in the tenth century (IX.1). Śīghra’s rule is (SSS 11, 2–4)

where ϑ is expressed in minutes and λ_A is the longitude of the moon’s apogee. The maximum correction

then, without the last term, is 2;40° when =R and = R—that is, the moon is at quadrature and its apsidal line points toward the sun (see Figure 29). The value for the maximum is almost that of Ptolemy (2;39°), and Śrīpati’s rule must be dependent on a knowledge of the Ptolemaic lunar model.

In the text the last term is expressed only as VersǀH– R (or VersǀR– H when R > H). This term will be 0/0 = 1 at the time of the maximum correction if the arc of which one is to take the Versine is and if H = R when the anomaly is 90° or 270°. The astronomical significance of this term is not apparent.

Finally, Śrīpati repeats the orbital inclinations of the planets given in Table V.7, except that the value of i for Jupiter is 1;26° (SSS 11,8). which gives aβ_max of 1;46°.

The Rājamrgānka is a small karana attributed to the Paramāra rājā of Dhārā, Bhojarāja (ca. 1005–1056): its epoch is 21 February 1042. Unfortunately, there exists in print of this important work only a redaction, apparently by one Rāma, of the first two adhikāras, which deal with the determination of the ahargana and the longitudes of the planets.⁵²

Bhojarāja changed the planetary mean motion parameters of the Brāhmapaksa; most later karanas and tables of this school follow his lead. These changes depend on the yearly bījas given in Table V.32. The new yearly and daily mean motions as they appear in the tables are given in Table V.35; it is clear that they are not precise. The mean daily motions are the mean yearly motions divided by the year length — 6,5; 15,31,17,17 civil days. The difference between this year length and that of the Paitāmaha — 6,5; 15,30,22,30 — is 0;0,0,54,47 days, or approximately the sun’s annual bīja.

Bhojarāja gives rules for finding the ahargana since his epoch (RM 1, 3–6), the interval between the beginning of the Kalpa and his epoch(RM 1, 8), the mean longitudes of the planets at the epoch (RM 1, 11–13), and the mean daily motions of the planets in minutes and seconds (RM 1, 14–15). From these elements it is simple to find the mean longitudes for any given day within a reasonable time after the epoch.

Bhojarāja lists the following localities as lying on the prime meridian: Lankā, Kāncī, Vatsagulma, Avanti, Gargarātaka, Sthāneśvara, and Sumeru (RM 1, 21). This list has obvious affinities to Śrīpati’s. Bhojarāja is the first known follower of the Brāhmapaksa to admit precession; he fixes its rate as 0;1° per year, and the year of coincidence as Saka 444 = a.d. 522 (RM 1, 24).⁵³ He also gives another method of finding the second component of the equation of time (see V.121–122). His rule is equivalent to that in the Siddhāntaśiramitni (V.126); e is measured in nādīs (RM 1,25):

The published text omits division by 90R. Clearly (V.124) produces the proper curve, for the maximum occurs when λ is 45°, 135°, 225° or 315°, and e = 0 when λ is 0°, 90°, 180°, or 270°.

Bhojarāja’s formula for the correction to k due to the first component of the equation of time is the same as (V.6) (RM 1, 26). When the two corrections due to the two components of the equation of time are combined with that due to the half-equation of daylight, one has the trying ω^r which corrects the mean longitude for mean sunrise into the mean longitude for true sunrise (RM 1, 32–33). The half-equation of daylight in vinādīs, ω^r is computed by Brahmagupta’s formula (V.120), and the values of γ/2 given in Table V.36 are nearly those in Table V.30 (RM 1,27).

Bhojarāja, at the beginning of the second adhikāra, lists the longitudes of the mandoccas and nodes of the planets in 1042; these are given in Table V.37 (RM 2. 1 and 6–7). Those that differ from Brahmagupta’s as listed in Table V.13 are in italics.

The equations and latitudes of the planets were tabulated, but these tables are not now available. Bhojarāja also lists the śīghra arguments required for the occurrences of the Greek-letter phenomena, as did Brahmagupta, except that the first visibility of Saturn occurs (as in Lalla) at 20°, the first station of Venus at 165°, the first visibility of Mercury in the west at 51°, and the last visibility of Mercury in the east at 309°(RM 2, 33–36).

Daśabala, the only Indian Buddhist known to have written on mathematical astronomy after the author of the śārdūlakarṇāvadāna, composed in Madhyapradeśa two works following the Brähmapaksa. The first, the Cintāmanisāranikā, has as epoch 1055 and apparently uses the lunar and solar elements of Table V.28. His second work, a karana entitled Karaṇakamahimānanda, the epoch of which is 1058, seems to have followed the Rājamrgāhka; unfortunately little is yet known of it.

Another karana with tables belonging to the Brāhmapaksa is the Grahajñāna composed by Āśādhara in Gujarat;⁵⁴ its epoch is Sunday, 20 March 1132. The tables of the mean motions of the planets for one to nine years, one to nine periods of ten years, one to nine periods of one hundred years, and one to nine periods of a thousand years are based on yearly mean motions differing minutely from those in Table V.35; they are listed in Table V.38 along with Āśādhara’s būjas.

Āśādhara’s year length is identical with Bhojaraja’s. He gives tables of the true longitudes of the planets of the type I have called “true linear”; in them Δλ = 13;20° (compare Table V.29), N = 0 to 26, and k= 14 days.⁵⁵

The only other material of interest to us in the Grahajñāna is Āśādhara’s list of the periods, in days, of retrogression, visibility, and invisibility of each planet. They are given in Table V.39 (compare Table Vl.19) (GJ 18–20).

An expanded version of Āśādhara ’s Grahajñāna is the Ganitacūdāmani composed by Harihara in Saurāstra about 1580.⁵⁶

The last sidhānta of the Brāhmapaksa that we can consider is the Sidākāntaśiromani, written by Bhāskara II in what is now Mysore State in 1150. This work basically follows the Brāhmasphutasidhānta of Brahmagupta, although with some reference to later works both within and outside the Brāhmapaksa and with some additions, of which I will mention those that seem most significant. In the first part of the Sidhāntaśiroamina, the Ganitādhyāya, adhikāra 1 repeats all the parameters of the Paitāmaha relating to mean motions. Instead of Table V.12, however, Bhāskara gives the mean longitudes of the planets at the beginning of the Kaliyuga in the form of the seconds by which each longitude is less than a circle; thus, in Table V.40 (SSB 1,1,3,19–20):

Bhāskara’s statements of the mean daily motions are crude, but are accompanied in his commentary by more accurate values; compare Brahmagupta’s (Table V. 11) and Śrīpati’s (Table V.31). Both sets

of values are given in Table V.41 (SSB 1, 1, 5, 15, 17–20).

For the moon Bhāskara gives an alternate form of Brahmagupta’s rule (V.56) (SSB 1,1,5.,16).

Bhāskara gives yet another set of crude values for the mean daily motions, presented here in Table V.42 (SSB 1, 1, 5, 22-23).

The prime meridian is said by Bhāskara to pass over Laṅkā, Ujjayinī, Kuruksetra, and Meru (SSB 1, 1,7,2).

Bhāskara’s table of Sines is that of the Paitāmaha as given in Table V.6, except that Sin 60° = 2,977–a more correct value than 2,978 (SSB 1, 2, 3–6). He also gives a Sine table in which R = 120 (as in Table lll.13) and the interval is 10°: this is recorded in Table V.43 (SSB 1,2,23).⁵⁷

Such a Sine table, together with Bhāskara’s advanced knowledge of trigonometry,⁵⁸ hints at the influence of Islamic astronomy, although this is yet to be substantiated.

BhĀskara’s circumferences of the planets’ epicycles are identical with the Paitamaha’s (Table V.5), save that Saturn’s manda epicycle is 50° and Mars’s Śīghra epicycle is 243;40° (SSB 1, 2, 22–23): the latter value is found in the BrĀhmasphutasidhānta (Table V.14). He correctly states that the equation of time, E. is found by (SSB 1, 2, 61–63; compare 2. 4, 19–20)

In a variant to Bhojarāja’s rule in (V.124), he gives the following formula for finding e, the component of E due to the obliquity, in nādīs (SSB 1, 2, 65):

Bhāskara repeats the theory of the naksatrabhogas expounded by Brahmagupta, except that he uses the more accurate mean daily motion of the moon, 790;35′, as one bhoga, so that 1 1/2 bhogas are 1185;52°, 1/2 bhoga is 395;17′, and Abhijits bhoga is 254;18′ (SSB 1,2,71–75).

The rest of the Ganitādhyāya also is mostly derived from Brahmagupta and from Śrīpati, with some modifications. Of these I note the following:

1. Bhāskara gives the radii of the orbits of the sun and the moon as 689,377 and 51,566 yojanas, respectively. He uses π = 3927/1250 = 3.1416 instead of the value, used by the Paitāmaha and Śrīpati. He does, however, keep Śrīpati’s values for the diameters of the sun, moon, and earth given in Table V.26(SSB 1,5,3–6).

2. Bhāskara modifies the traditional kālāmśas of visibility of the planets given in the Paitāmaha’ (Table V.8) so that they are those in Table V.44 (SSB 1,8,6).

3. Bhāskara gives values for the mean apparent diameters of the planets in minutes rounded off from Brahmagupta’s (Table V.21); these rounded values are listed in Table V.45 (SSB 1, 10, 1).

4. He minutely corrects β* of Krttikā to+ 4;30°, of Rohinī to -4;30°, of Viśākhā to -1;20°, of Anurādhā to - 1;45°, and of Satabhisaj to -0;20°; otherwise he follows Brahmagupta (Tables V.10 and V.22) (SSB 1, 11,1–7).

The Golādhyāya of the Sidhāntaśiromani is a more theoretical treatise than the Ganitādhyāya; in it Bhāskara explains the models on which astronomical computations are based. It contains also several parameters not in the Ganitādhyāya:

1. The circumference of the earth is 4,967 yojanas and its diameter 1,581 1/24 yojanas; here, then, π = 4967/1581 ; = 3.1413… (SSB 2, 3, 52).
2. Bhāskara refers erroneously to the theory of trepidation given by the Sāryasidhānta as one of a precession with 30,000 rotations in a Kalpa (only one rotation in 144,000 years or 1° in 400 years!). He also attributes to Muñjāla a theory of precession with 199,699 rotations in a Kalpa, which gives a motion of nearly 1° in sixty years; this theory is not, however, found in Munjala’s extant works (SSB 2,6, 17–18).⁵⁹
3. Bhāskara gives a rule for finding Sin (ϑ + 1°) if R 3,438(SSB Jyotpatti 16)⁶⁰:

which follows from his rule for finding Sin (α + β) (SSB Jyotpatti 21):

Bhāskara also wrote a karana, the Karanakutūhala, the epoch of which is Thursday, 24 February 1183; a set of astronomical tables based on this karana is also extant.⁶¹ In the tables the mean daily motions of the planets are virtually identical with those in Table V.35, and in the text approximations are given (KK 1, 7–12). Both are listed in Table V.46.

Bhāskara states that the prime meridian passes over the city of the Raksasas (Laṅkā), Devakanyā,

Kāntī, Sitaparvata, Paryalī, Vatsagulma, Ujjayinī, Gargarāta, Kuruksetra, and Meru (KK 1, 14). He gives as longitudes of the mandoccas of the planets those listed in Table V.47 (KK 2, 1–2).

The position of Saturn’s mandocca indicates that Bhāskara follows Brahmagupta (Table V.13) rather than Bhojarāja (Table V,37).

The Karanakutūhaki’s Sine table is the second one of the Sidhāntaśiromani (Table V.43), in which R = 120 (KK 2, 6). The equations of the center are found by multiplying the Sine of the anomaly by given constants; the equations also are tabulated in the tables. Table V.48 lists the text’s formulas and resulting maximum equations, which are also the tables’ maximum equations (KK 2, 9–10).

The text gives the radii of the śīghra epicycles, called parākhyas, in parts of the radius of the deferent (R = 120), and the correct geometrical solution for σ the tables give the values of σ for every

degree and the values of the śīghra hypotenuse, H_σ. Table V.49 lists the radii of the epicycles from the text and the maximum equations and corresponding anomalies from the tables (KK 2, 2).

The rate of precession adopted by Bhāskara in the Karanakutūhala is 1° in sixty years (KK 2, 17). In other respects the karana offers little that is unusual; but one may note the following:

1. Bhāskara gives a table of lunar latitudes divided by four at intervals of 15° in which β_max = 4;40°

(KK 4, 6) as in the Romaka (III.21e; compare 111.39).

2. He gives a table of longitudinal parallaxes for every 15° of elongation between the nonagesimal and the sun. The maximum is 4 nādīs at an elongation of 90°(KK 5,4).

3. He gives rules for computing the times of the heliacal risings of the planets. These are based on the date of a heliacal rising in the epoch year and the mean synodic periods of the planets (KK 6).

The next text belonging to the Brāhmapaksa is the Laghukhecarasidhi of Śrīdhara;⁶² its epoch is 20 March 1227. This work is accompanied by tables, from which the following elements appear.

The mean daily motions of the planets are given for one, ten, and one hundred years of 365 days each (Egyptian years), and for one, ten, and one hundred days. Comparison of Tables V.50 and V.35 will show that they belong to the new Brāhmapaksa.

The longitudes of the mandoccas for 1227 in Table V.51 are close to Bhojarāja’s for 1042

(Table V.37); the longitudes of the nodes are closer to Brahrmagupta’s for 628 (Table V.13).

Śrīdhara’s manda equations and śīghra equations are rather eclectic (see Table V.52).

The manda equation of Saturn is greater than it normally is in the Brāhmapaksa, that of Mercury is less, and none of the others is precisely the same; Saturn’s śīghra equation is less than, and Mercury’s is equal to, that of the Ārdharātrikapaksa (Table VII.6).

The Śīghrasidhi of Laksmīdhara,⁶³ the epoch of which is 1278, is the earliest set of tables yet known for determining the tithis, naksatras. and yogas. It is a double set based on the parameters of both the Brahmapaksa and the Āryapaksa, This is also the arrangement in the Pañcāngavidyādharī written by Vidyādhara at Rajkot, Saurāstra;⁶⁴ its epoch is 1 April 1643.

The Mahādevī of Mahādeva.⁶⁵ the epoch of which is 28 March 1316, is a set of tables of the true longitudes of the planets according to the “true linear” type used previously by Āśādahra . In the Mahādevī, Δλ = 6°, so that N = 0 to 59; and k = 14. The mean motion tables are based on the mean yearly motions given in Table V.35; other parameters characteristic of the Brahmapaksā are listed in Table V.53.

Related sets of tables were written by Dinakara at Bārejya in Gujarat; his epoch is 31 March 1578. The Candrārkī⁶⁶ is concerned with solar and lunar positions, and the Khetasidhi⁶⁷ with those of the planets. Dinakara also wrote a Tithisāranī.⁶⁸ which contains tables of the tithis, naksatras, and yogas; its epoch is 31 March 1583. A fürther set of tithi, naksatra, and yoga tables based on the parameters of the Brāmapaksa is the Tithikalpadruma of Kalyāna:⁶⁹ its epoch is 31 March 1605. This Kalyāna may also be the author of a set of planetary tables, the Khecaradīpikā⁷⁰ based on the Mahādevī; its epoch is 31 March 1649.

The Jagadbhūsana⁷¹ written by Haridatta II in Mewar, Rājasthān, is another set of astronomical tables based on the parameters of the Brāhmapaksa; its epoch is 31 March 1638. The tables of true longitudes of the planets in this work, as in the Anonymous of 1704,⁷² are of the “cyclic” arrangement, derived from the Babylonian “Goal-year” periods.⁷³

The existence of several anonymous sets of tables belonging to the Brāhmapaksa and datable to the sixteenth through nineteenth centuries attests to the vitality of this tradition in western and northwestern India.⁷⁴ The last full-scale work to which we can at present refer, however, is the Karanavaisnava of Sankara.⁷⁵ the epoch of which is 9 April 1766 (Gregorian).

Sankara’s yearly mean motions are generally those found in Table V.35, except that the lunar node’s is — 19;21,33,38°. His list of cities over which the prime meridian passes includes Lankā, Devasutāpuri, Kāntī, Karnātasitācala, Parjali, Vatsagulma, Ujjayinī, Gargarāta, Ātavāśrama, Rohītaka, Kuruksetra. Himālaya, and Sumeru.

Śankara’s longitudes of the mandoccas and nodes are those of Table V.51 rounded to the nearest degree (the longitude of Mars’s mandocca is 129°). He uses two Sine tables: in one R = 24, and in the other R = 700 and the entries are for increments of 5°; the second table is reproduced in Table V.54.

The first Sine table, in which R = 24, is used for finding the manda equation by the method illustrated in Table V.55 (compare Table V.48).

The Karanavaisnava lists the diameters of the śīghra epicycles of the planets (R = 24) as the Karanakutūhala did their radii (Table V.49;

R = 120); it also gives tables of equations. These are listed in Table V.56.

Śankara’s values for the kālāmśas of visibility and the maximum latitudes of the planets, given in Table V.57. differ somewhat from the standard Brāhmapaksa values (compare Tables V.7 and V.8).

Thus, for a millennium and a half this school of astronomy survived in India with only one major change—that which occurred in the eleventh century, when the planetary parameters were modified and when some elements apparently derived from Ptolemaic Islamic were introduced.

VI . THE ĀRYAPAKṢA

The Āryapaksa was founded by Āryabhata I with his Āryabhatīya, in which he mentions the year 499, when he was twenty-three. He states that the science of astronomy is based on a revelation from Brahma (A Gola 50), and it appears that the Paitāmahasidhāta of the Visnudharmottarapurāna was one of his sources. Although Āryabhata seems to have lived in Pātaliputra (modern Patna in Bihar), and the Āryabhatīya was still followed by some in northwestern India in the tenth century and its elements were known even later (for instance, by Ganesa). since the seventh century it has been associated primarily with southern India, where it still is influential.

The Āryabhatīya is organized in a fashion different from that of all other Indian sidhāntas. We will here describe the contents of the Daśagītikā, the Kālakriyāpāda, and the Golapāda, indicating similarities with and differences from the early texts of the Brāhmapaksa.

The Āryabhatīya assumes a Kalpa containing 1,008 Mahāyugas or 4,354,560,000 years; these 1,008 Mahāyugas are divided into fourteen Manus, each of which comprises seventy-two Mahāyugas (A Daśagītikā 3, Kālakriyāpāda 8). The Mahāyugas are divided into four equal Yugas, each of which contains 1,080,000 years. Since a mean conjunction of the planets at Aries 0° occurs at the beginning of each of these Yugas of 1,080,000 years, Āryabhata is forced to have each planet rotate an integer number of times within each of these Yugas rather than within the much larger Kalpa. Theoretically, this allows the users of the Kalpa to employ better parameters for mean motions than does Āryabhata; that they do not, indicates that they had no means to arrive at better parameters. But Brahmagupta severely criticizes Āryabhata for breaking with tradition, and this criticism caused followers of the Āryapaksa and Ārdharātrikapaksa in the eighth century to abandon Āryabhata’s equal Yugas.

The fundamental parameters of the mean motions of the planets in the Āryapaksa are given in Table VI.1; in the Daśagītikā they are expressed as rotations in a Mahāyuga, R (A Daśagītikā 1–2).

The solar parameter implies a year-length of 6,5;15,31,15 civil days (in excess of the Brāhmapaksa’s parameter by 0;0,0,52,30 days) and 1,577,917,500 civil days in a Mahāyuga. Since Āryabhata has the current Kaliyuga begin at the same moment as does the Brāhmapaksa—on 18 February-3101 at sunrise at Lanka-he places the beginnings of the mean solar years in the fifth and sixth centuries of the Christian era about one day later than does the Brāhmapaksa. He has, however, so chosen his lunar parameter that mean conjunctions of the sun and the moon will occur on the same day in both paksas, for the mean synodic month in the Brāhmapaksa is 29;31,50,5,43…. days (see Table V.4) and that in the Āryapaksa is 29;31,50,5,40…. days. His mean motions of the planets, however, seem unrelated to those of the Brāhmapaksa (see Table V.2), and apparently are computed from the assumed mean conjunction of the beginning of the Kaliyuga and the mean longitudes of the planets as found from a Greek table for exactly 3,600 years later-that is, for noon of 21 March 499.

The time periods in Āryabhata’s Mahāyuga are enumerated in Table VI.2 (compare Table V.4). Aryabhata at times refers to the sidereal days as “rotations of the earth” (A Daśagītikā 1, Gola 9), and at times as “rotations of the naksatras” (A Gola 5). Like other Indian astronomers, he was aware of the mathematical equivalence of the two concepts; unlike others (such as Varāhamihira and Brahmagupta) he did not specifically reject the rotation of the earth on physical grounds.⁷⁶

Āryabhata’s rules for computing the orbits of the

planets in yojanas are based on the same principles as are those in the Paitāmaha; but he uses a longer yojana, ten of which equal a minute of arc in the orbit of the moon (A Daśagītikā 4). Therefore, the yojanas in the orbits of the planets in the Āryapaksa are as given in Table VI.3 (compare Table V.3).

The Daśagīkā gives the diameters of the planetary disks in yojanas as in Table VI.4 (A Daśagītikā 5); I add the diameters of the moon and the star-planets in minutes, computed on the assumption that these yojanas represent their apparent diameters at the moon’s mean distance.

The order of the planets is identical with that in the Paitāmaha (V.22 and Table V.8), although the diameters, when multiplied by 3/2, are not the same. Āryabhata states that the diameter of the earth is 1,050 yojanas, but its circumference 3,375 yojanas (A Daśagītikā 9); this yields 3 3/14 as the value of π, although elsewhere he uses and π = 21600/6876 = 3 27/191.

The orbital inclinations according to Āryabhata, as listed in Table VI.5 (A Daśagīacute;tikā 6), differ from those in the Brāhmapaksa (Table V.7); neither set is very accurate, Āryabhata’s represent values rounded to multiples of 0;30°.

Āryabhata accepts the rotation of the mandoccas and nodes demanded by the Kalpa system of the Brāhmapaksa (A Dasagātikā 7; Gola 2), but he assumes that their longitudes since the beginning of the present Kaliyuga can be regarded as fixed. These longitudes are given in Table VI.6 (compare Table V.13).

In the Āryabhatīya the manda epicycles and śīghra epicycles arc pulsating, except for the manda epicycles of the sun and the moon. For the superior planets the maximum manda epicycles occur when κ_μ = 90° or κ_μ = 270° and the maximum śīghra epicycles when κ_σ = 0° or κ_σ = 180°; for the inferior planets the maximum manda epicycles and śīghra epicycles occur when κ_μ or κ_σ = 0° or 180°. The effect of the pulsation is minimal, its purpose unfathomable. The dimensions are given in Table VI.7 along with the maximum equations from Haridatta (compare Table V.5) (A Daśagītikā 8–9).

In the Dasagātikā. finally, Āryabhata lists the differences in the Paitāmaha’s Sine table (Table V.6) (A Daśagītikā 10); and in the Ganitapāda he iterates

the Paitāma’s rule for determining the second differences in the Sine table (V.8) (A Ganitapāda 12).

In the Kālakriyā, Āryabhata defines the vyatipāta as in (III.29) [A Kālakriyā 3). and mentions the twelve-year Jovian cycle also referred to in the Paitāmaha (A Kālakriyā 4). More important, he discusses the equivalence of planetary models with double epicycles to those with eccenter and epicycle (A Kālakriyā 17-21); both models are of Greek origin. In integrating the effects of the two equations he prescribes essentially the procedure of the Paitāmaha (V.7)(A Kālakriyā 22-24).

In the Golapāda. Āryabhata deals with various astronomical problems. In the course of this diffuse chapter he gives the kālāmsas of visibility of the planets as the Paitāmaha had (Table V.8) (A Gola 4). He also repeats from the Paitāmaha several formulas of spherical trigonometry (A Gola 18-32); but his formula for right ascension is that later repeated by Brahmagupta (V.76) (A Gola 25)

Āryabhata’s rules for computing parallax, however, differ from those of the Paitāmaha (V.26) and of the Brāhmasphutasidhaānta (V.95-101). He mentions only the combined parallax, which he

considers to be 0 when the sun is at the nonagesimal at the zenith. His computation (A Gola 33-34) is illustrated in Figure 30, where triangle S_nNZ is assumed to be a plane right triangle, with the right angle at N. Then one can find Sin (S_nZ) = Cos α_n from(III.53b)and(VI.1):

Āryabhata then assumes that the elongation of the sun from the nonagesimal, SN, can be found from a plane right triangle:

This is, of course, grossly wrong.

Āryabhata does not proceed to tell one how to obtain the parallax; he only states that the horizontal parallax equals the radius of the earth. Indeed, since the radius of the earth is 525 yojanas and each minute in the orbit of the moon equals ten yojanas. it follows that

which is essentially correct for lunar parallax.

For finding the distance of the tip of the earth’s shadow and the diameter of the shadow at the moon’s distance, Āryabhata (A Gola 39–40) gives the correct (V.23) and (V.25), where the Paitāmaha, as the text is preserved, gives the incorrect (V.22) and (V.24). Āryabhata’s rules for finding the durations of eclipses, the durations of totality, and the eclipse magnitudes are the familiar rules in which it is assumed that the lunar orbit is parallel to the ecliptic (A Gola 41–44). Moreover, his formulas for the aksadrkkarma and the ayanadrkkarma (A Gola 35-36) are those that presumably were originally in the Paitāmaha (V.35, V.29), although the latter is replaced by the approximation (compare V.33)

Further, Āryabhata gives the formula for the aksavalana and refers obliquely to that for the ayanavalana, which are repeated by Brahmagupla (V.93-94) (A Gola 45); and he gives the eclipse colors that are found in the Brāhmasphutasidhānta (Table V.19) (A Gola 46). But he slates that a solar eclipse of 1/8 or less than the sun’s disk is invisible, whereas Brahmagupta puts the limit at 1/12 (A Gola 47).

The ablest commentator on the Āryabhatīya, and one of India’s most competent mathematicians, was Bhāskara I. whose Bhāsya on the Āryabhatīya was written at Valabhī in Saurāstra in 629, although he is usually associated with Aśmaka in central India. Apparently before composing the Bhāsya, he wrote a summary of the astronomical system of the Āryapaksa entitled Mahābhāskarāya, in which he especially displays his ingenuity in mathematics. The text is in the form of rules based on the parameters of the Āryapaksa. which parameters are stated by Bhāskara in adhyāya 7.

In adhyāya I, Bhāskara gives rules for finding the ahargana from Āryabhata’s beginning of the current Kaliyuga, and for finding the mean longitudes of the planets. The ahargana rules are identical with, or variations of, Brahmagupta’s(V.48) (MB 1,4-7). He also provides formulas to find the mean longitudes of the planets without the ahargana that are similar to those in Brākmasphufasidhānta 13 (MB 1. 13–19). Also, similar to (V.52) but based on the parameters of the Āryapaksa, he states (MB 1.22–28) (VI.5):

The remainder of adhyāya 1 gives rules for finding the mean longitudes of the planets from the grahatanu, which is the number of saura days between the beginning of Kaliyuga and the beginning of the current year, and from the dhruvaka, which is the ahargana in saura days (MB 1, 29–39), as well as applications of the kuttaka to problems in mean motions similar to Brahmagupta’s in Bràhmasphutastddhānta 18 (MB 1,41–52).

Adhyāya 2 is a brief chapter on the correction for the longitudinal difference. The prime meridian is stated to pass over Lanka, Kharanagarā, Sitorugeha, Pānāta. Misitapurī. Taparnī. Sitavara mountain, Vātsyagulma, Vananagarī. Avantī, Sthānesa, and Meru (MB 2, 1-2). For the des-āntara Bhāskara gives Brahmagupta’s rule (V.61), but with the circumference of the earth taken to be 3,298 17/25 yojanas (MB 2, 3–4): if the diameter is 1,050 yojanas, as Āryabhata states, and π = 3 27/191, the circumference would be 3,298 82/191. Bhāskara correctly criticizes this rule, however, on the grounds that the distance between two cities cannot be accurately known and that a parallel of latitude is not a great circle. To correct this, he proposes using a water clock to measure the time difference between computed moonrise for the prime meridian at one’s own latitude and the observed moonrise; from this time difference one can find the true longitudinal difference, and thence the true distance in yojanas of one’s locality from the prime meridian (MB 2, 5)

In Mahābhāskarīya 3, Bhāskara gives many of the usual rules relating to direction, place, and time. Using (III.35) and (III.55) and the value Sin ϵ = 1.397 (R = 3.438. ϵ = 24°). Bhāskara establishes as values for y in time minutes at solar longitudes of 30°, 60°. and 90° those in Table VI.8 (comparelll. 19)(MB3,8).

Bhāskara also gives the values for α1 α2 and α3 found in the Brāhmasphuutsidhānta (Table V.17) (MB 3, 10), and Āryabhata’s rule for Sin a (V.76) (MB 3, 9). The remaining rules in this adhyāya are repetitions of or elaborations on those occurring in other works. Bhāskara adds descriptions of some observational instruments and a catalog of the ecliptic longitudes and latitudes of the yogatārās; the laller are reproduced in Table VI.9 (MB 3, 62-71).

M ahābhāskarīya 4 presents the rules for computing the true longitudes of the planets. Both the epicyclic and the eccentric models are considered, and both are used to solve the concentric with equant model by iteration (MB 4. 9- 12. 19-20)⁷⁷

Adhyāya 5 is concerned with eclipses. Bhāskara

It is difficult to conceive of how Bhāskara arrived at these longitudes and latitudes.

repeats the diameters of the sun, the moon, and the earth in yojanas given by Āryabhata (Table VI.4) (MB 5, 4), but gives as approximate distances in yojanas of the sun and the moon (based on Table VI.3) the figures in Table VI.10 (MB .5, 2).

For Finding the true diameters he gives the rules in (VI.6) (compare V.89) (MB 5,6-7):

The diameter of the shadow should depend on the difference of the lunar and solar velocities rather than on the lunar velocity alone. From these relationships the mean values in Table VI.11 are (compare Table V.18)

Bhāskara finds the moon’s latitude by the rule (MB 5, 14MVI.7)

From (III. 18 b) we know that

if R = 3,438. The maximum lunar latitude, then, is 4;30°, as in Table VI.5. Bhāskara’s computation of parallax is a completed form of Āyabhata’s (Vl.l-VI.3) (MB 5, 16-32), and his rules for finding the durations and magnitudes of eclipses are elaborations of those in the Āryubhatya (MB 5, 33-41). For deflection, however, he gives the approximative formulas (compare V.93-94) (MB 5, 42-45)

Compare Āyabhata’s ayanadrkkarma (VI.4) with (VI.9). Bhāskara completes his discussion of solar eclipses with elaborate rules for constructing projections of eclipses (MB 5, 47–67). He erroneously prescribes the application of parallax to lunar eclipses as well as to solar (MB 5,68–70).

In Mahābhāskrīya 6, Bhāskara discusses the heliacal risings and settings of the planets and their conjunctions. In this he repeats Āryabhata’s rules for the aksadrkkarma (V.35) (MB 6, 1–2) and the ayanadrkkarma (VI.4) (MB 6, 2–3). For the illuminated portion of the moon Bhāskara’s formula, when the moon is in the second and third quadrants, is equivalent to Brahmagupta’s (V.46b); but when it is in the first and fourth he substitutes for (V.46a) the following (MB 6. 5-7):

Bhāskara devotes many verses to the problems of the elevation of the moon’s horns, its projection, and the time of moonrise(MB 6, 9–42). He repeats Āryabhata’s kālāmśas of visibility of the planets (Table V.8), but adds that those for Venus when retrograde (at inferior conjunction) are only 4°–4;30° (MB 6, 44–45). He computes conjuncttions with ecliptic rather than polar coordinates (MB 6,48–55).

In adhyāya 7, besides listing the parameters of the Āryapaksa (MB 7, 1–16). Bhāskara gives the formula for finding Sin ϑ that is also given by Brahmagupta (V.69) (MB 7,17–19). At the end of this adhyāya he describes the characteristics of the Ārdharātrikapaksa(MB 7, 21–35).

The Laghubhāskarīya repeats in abbreviated (and sometimes corrected) form the contents of the Mahābhāskarīya, but in it Bhāskara follows the order of the Daśādhyāyī. Here one need only note that he has changed some of the ecliptic coordinates of the yogatārās listed in Table VI. 9; these changes are displayed in Table VI.12 (LB 8,1–9)

A small karana, the Grahacāranibandhcina, based on the parameters of the Āryapaksa. was written by Haridatta, traditionally in 683. It is the fundamental text of the so-called Parahita system of astronomy, prevalent for many centuries in southern

ern India. Haridatta expresses the mean motions of the planets in the period relations of Table VI. 13 (compare Table VI. 1) (GCN 1,21–29). These expressions take the form of those in Lātadeva’s Sūryasiddhānta (Table VII.2).

The central portion of Haridatta’s work is versified tables of the sines of the planetary equations for arguments increasing by 3;45° (GCN 2, 1–15 and 3, 35–36); he calls these tables vakyas. whence the name of the Vākya system of astronomy. The maximum equations are those recorded in Table VI.7. He also gives the śīghra anomalies (nonsymmetricall) necessary for the occurrence of first and last stations, as in Table VI.14 (compare Table V.9)(GCN 3, 12–16).

Haridatta gives mean longitudes for the ahargana of 210,389; (= 365;15.31,15. 576) from the beginning of Kaliyuga (this epoch is 23 February–2525) (GCN 1, 12–18) and for the ahargana of 210.389 6 = 1,262.334 (this epoch is 20 March 355) (GCN 3,47-49).

Based on, and repeating much of, Haridatta’s work is an anonymous Grahacāranibandhanasahgraha, the epoch of which is represented by the ahargana 1,472,723 = 210,389 7 (25 March 931) (GCNS 4). This text preserves bīja corrections to the mean yearly motions since 522 that are ascribed to Haridatta himself by Sundararāja⁷⁸ and may be from the former’s lost Mahāmārganibandhancr, these bījas and the resulting mean yearly motions are given in Table VI. 15 (compare Tables VI. 16 sad VI. 18) (GCNS 17-18).

Immediately following these bījas the Grahacāranibandhanasangraha gives another set, also to be applied to the years since 522; they are given in Table VI.16(GCNS 19-22).

One of the earliest Arabic astronomical texts, written at Sind in 742, was the Zīj al-Harqan⁷⁹ which evidently utilized parameters of the Āryapaksa; its existence indicates the survival of this paksa in that region of India into the eighth century. although the Arabs primarily encountered texts belonging to the Brāhmapaksa or the Ardharātrika-paksa. Perhaps this Zīj al-Harqan the source of the knowledge of Āryabhata’s rotations of the planets in a Mahāyuga (Table VI.I) demonstrated by al-Ahwāzī, who flourished after about 830;⁸⁰ al-AhwāzT changes the rotations of Mars from 2,296,824 to 2,296,828. a figure found also in Vateśvara (Table VI.24). There also are Ārya—paksa parameters in the works of al-Fazān, who likely derived them from the Zīj al-Harqan

Probably also in the eighth century, although perhaps in the early ninth, Lalla wrote the Sās-yadhīvrddhidatantra, a work structured like Srīpati’s Siddhāntaśekhara and surely antedating it. He may have lived in Mālava, since he mentions

Daśapura (SVD 2, 9, 10). Lalla’s parameters are those of the Āryapaksa; but he describes the tour yugas in a Mahāyuga in the traditional fashion of the Brāhmapaksa rather than equal, as Āryahhata has them (SDV 1, 1, 14). This, of course, destroys Āryabhata’s structure and denies the occurrence of mean conjunctions of the planets at the beginnings and ends of the Mahāyugas. His contemporaries, the authors of the later Pauliśasiddhānta and of lhe Sūryasāddhānta, had done essentially the same thing, but had achieved harmony with the Brāhmapaksa’s Katpa system by delaying the inception of the planets’ motions for the requisite number of years to produce a mean conjunction at the beginning of the current Kaliyuga.

Lalla’s crude approximations to the mean daily motions of the planets are given in Table VI. 17 (SDV 1,1,39–46).

Lalla names only Lankā. Ujjayinī, and Himālaya as lying on the prime meridian (SDV 1 1. 55); he uses Āryabhata’s value for the diameter of the earth (1.050 yojanas) but gives its circumference as 3,300 yojanas, which is based on (SDV 1, 1, 56-57). But elsewhere he repeats Āryabhata’s aberrant value for the circumference of the earth (3.375 yojanas) and gives the diameter as 1,074 yojanas; here π = 3 51/358 (SDV 2, 8, 2). His bījas. listed in Table VI.18, are applied to the years following 498; although his period is 250 rather than 235 years, the numerators of the bāja fractions frequently are identical with those in the Grahacāranibandhanasahgraha (Tables VI.15 and (VI. 16) (SDV.1, 1, 59–60).

Lalla’s normal Sine table is that of the Paitāmaha

and Āryahhata (Table V.6) (SDV 1. 2. 1–8), although he also reproduces that in which R = 150 (Table V.24) (SDV 1,13. 2). He alsoretains the same longitudes of the apogees as Āryabhata’s (Table VI. 6) (SDV 1, 2, 9 and 28). His circumferences of the manda epicycles and śīghra epicycles are Āryabhata’s for κ = 0° or 180° (Table VI.7) (SDV 1. 2. 28–29). Among other rules Lalla repeats that of Bhaskara I for computing the concentric with equant model for the sun and the moon

by means of an epicycle with varying radius (SDV 1, 2, 44). But he lists Brahmagupta’s sīghra arguments for the occurrences of the Greek-letter phenomena, except that the first visibility of Saturn is said to occur at 20°, as it is also by Bhojarāja (SDV 1, 2, 47-50). Lalla records the retrograde periods, and the periods of visibility and invisibility of the planets in days, as in Table VI.19; his figures disagree with Āśādhara’s (Table V.39) (SDV 1,2.52-53).

In a list of values for γ/2 Lalla gives slightly different numbers from those of Bhāskara I (Table VI. 8); these are noted in Table VL.20 (SDV 1, 13.9)

Most of the rest of Lalla’s work is similar to Āryabhata’s. to Bhāskara’s. or to Brahmagupta’s. He does state, however, that the diameter of the moon’s disk is 320 yojanas rather than the 315 in Table V1.4 (SDV 1. 4, 6). Lalla also is the first to give the cālanas for parvans of 277 and 15 civil days according to the Āryapaksa. These are enumerated in Table VI.21 (SDV 1, 6, 1 I); they of course differ slightly from the Brāhmapaksa cālanas (Tables V.20 and V.34).

He repeats the standard kālāmśas of visibility of the planets (Table V.8), but adds that when Venus is retrograde, its kālāmśas are 8°, and when Mercury is retrograde, its kalarnsas are 12°(SDV I, 7, 5).

In his catalog of coordinates of the yogatārās, Lalla generally repeals those (ecliptic) in the Muitāhhāskarīya (Table VI.9) (SDV 1, II, 1-9); his changes are recorded in Table V1.22, along with polar coordinates of Brahmagupta (Table V. 10 and V.22) that influenced him. This mixture of ecliptic and polar longitudes can he explained only by Lalla’s incompetence or, possibly, by the uncertainty of the identifications of the yogatārās among Indian astronomers.

Al-Bīūinī refers several times to a Karanasāra composed by Vateśvara, the son of Mahadatta; the epoch of this work was 899. The same Vatesvara wrote a Valeśvarasiddhānta at Ānandapura in 904, of which a large part, although not all, survives. It shows a strong influence of the Āryapaksa and, save for some late tithi, naksatra, and yoga tables, is the last major representative of that paksa in northern, northwestern, or western India.

Unfortunately, the published text is both inaccurate and incomplete. Therefore, one cannot yet report definitively on its contents. At present it will suffice to say that Vatesvara, like Lalla and the authors of the later Pauliśddhānta and of the Sāryasiddhānta, has made certain changes in an attempt to accommodate some of the system of the

Brā;hmapaksa. In particular, he has the apogees and nodes of the planets rotate integer numbers of times in 72,000 Kalpas, as in Table VI.23, where R is the number of rotations in 72,000 Kalpas and Ρ is the longitudes at the beginning of the current Kaliyuga (compare Table VI.6) (VS 1. 1, 15–19 and 1,4,56–60).

Vateśvara also gives “new” parameters for the mean motions of the planets, and changes the number of civil days in a Mahayuga to 1,577,917,560 (see Table VI.2), so that a year is 6,5;15,31,18 days. In Table VI.24 his values for the planets’ rotations and the corresponding mean yearly and mean daily motions are recorded (VS 1, 1, 11 – 14). In fact, except in the cases of Mars and Mercury’s śīghra, where Vaṭeśvara’s numbers are less by 4, the rotations of the planets in a Mahāyuga are identical with those in the Saurapakṣa (Table VIII.1); but the Saurapakṣa has a slightly longer year (6,5;15,31,31,24 days). Vaṭeśmacrvara’s figure for the rotations of Mars is identical with al-Ahwāzī’s.

Vaṭeśvara’s table of Sines uses R = 3437;44, but is computed (to a sixtieth of a part) for ninety-six intervals in a quadrant (a fourth of the standard 3;45°); each interval, then, is 0:56,15° (VS 2, 1, 2–48). The table in the published text is very corrupt, but has been reconstructed.⁸¹ As presented in the edition, his dimensions of the manda epicycle and śīghra epicycle are unusual and perhaps corrupt; they are reproduced in Table VI.25. (VS 2, 1, 49–50).

A description of the remainder of the text must await a better edition.

A popular karana belonging to the Āryapakṣa as corrected by Lalla’s bījas is the Karaprakāśa of Brahmadeva, probably written at Madurai in

southern India; its epoch is 11 March 1092. The mean daily motions of the planets are given by Brahmadeva in the form represented in the second column of Table VI.26 (kp 1,4–12). The Kanuiaprukāśa gives the rate of precession (or trepidation) as 1° in sixty years, and the year of coincidence as 522 (KP 2, 9).

Brahmadeva’s Sine table is based on R = 120 (compare Tables 111. 13 and V.43). The entries are recorded in Table VI.27 (KP 2, 1).

Keeping Āryabhaṭa’s longitudes of the mandoccas (Table VI.6) (KP 3, 1), Brahmadeva states the maximum manda equations and Sines of the maximum śīghra equations, as in Table VI .28 (KP 2, 5 and 3, 1 and 4).

Brahmadeva gives both the śīghra anomalies necessary for the occurrence of the Greek-letter phenomena (KP 3, 8 and 10) and the periods of retrogression, visibility, and invisibility (KP 3, 9 and 1 I); these are all presented in Tables VI.29 and VI.30. The synodic periods, which are not given by Brahmadeva, indicate how approximative his periods are.

In southern India, probably at the beginning of the second millennium after Christ and professedly based on the Parahita system of Haridatta, the Vākya system of predicting planetary longitudes was developed. This is based on period relations, which are integer numbers of days, and tables of positions for every day within each period. The lunar vākyas, ascribed to Vararuci, give lunar longitudes for each of the 248 days in nine anomalistic months;⁸² this is one of the period relations of Vasiṣṭha (III.8). From the ahargana since epoch all periods of 12,372 days or 449 anomalistic months are to be eliminated: since the beginning of the current period of 12,372 days all periods of 3,031 days or 110 anomalistic months are to be eliminated. This last period relation is also found in Vasiṣṭha (III .9), and 12,372 days is simply four periods of 3,031 days plus one period of 248 days. Various epoch dates have been given in different presentations of the vākyas; they range from 1184 to 1756. None, of course, can be shown to be even the approximate date of the inception of the Vākya system. For each period there is an increment in lunar longitude: 4,57;48,10° in 12,372 days; 5,37; 31,1° in 3,031 days;and 27;44,6°in 248 days.

One of the epoch dates utilized in the tradition is 22 May 1282; this also is the epoch date of the lunar vākyas in the anonymous Vākyakarana, which gives vākyas for all the planets. It was composed in about 1300, probably near Kāñcī.

In this text the year length is 365 + 1/4 + 5/576 days (VK 1. 2-3), which is the Āryapaksa’s 6,5;15,31,15 days. The true longitude of the sun in degrees is determined approximately as equal to the days elapsed since true Mesasahkrānti; this approximation is corrected by the entries in a vākya table in which the corrections for each ten days up to 370 days are recorded (VK 1, 4-5). The longitude of the moon is found as in the Candravākyas of Vararuci (VK 1, 9-11). It is assumed that the ascending node makes one rotation in 6,792 days (VK. 1, 17-19).

The longitudes of the planets are computed by means of cycles from an epoch position; one cycle is an approximate synodic period. In each cycle, at stated intervals in days, the planet’s true position and a corrective factor are recorded. The numbers of cycles of the planets, the days in each cycle, the

total numbers of days, and the increments in longitude are given in Table VI.31 (VK Appendix III).

Superimposed on these synodic periods are a number of larger cycles for each planet, which are tabulated in Table VI.32, where the cycles are given in days and the longitudinal increments in degrees (Sundararāja on VK 2, 18). These larger cycles, of course, represent mean rather than true motions: the italicized numbers in the last column are the mean synodic periods.

The remainder of the Vākyakarana is devoted to the problems related to time, position, and direction, to eclipses, and to the first and last visibilities of the planets. The rate of trepidation accepted by the author is 0;1° in 120/121 years over an arc 24° on either side of sidereal fixed Aries 0° (VK 3, 1). His table of sines is based on R = 43; it is given in Table VI.33 (VK 3,2–4). Otherwise, the text is unexceptional. The vākyas continued to be popular in Kerala, as is clear from various works of such authors as Mādhava (ca.

1340-ca. 1425),⁸³ Parameśvara (ca. 1380-ca. 1460), and Sundararāja (ca. 1475).

With Parameśvara’s institution in 1431 of the drgganita system, which is really only the Saurapaksa, the process already visible in Vateśvara’s work was accelerated and the Āryapaksa was replaced by the Saurapaksa. It lingered only in southern India in the Parahita system and in the Vākya system, although its solar and lunar parameters were used for computing tithis, yogas, and naksatras in western India. There is only one known set of tables belonging to this school,⁸⁴ although this situation is undoubtedly due to the paucity of southern Indian tables that have hitherto been examined.

VII. THE ĀRDHARĀTRIKAPAKSA

The Ārdharātrikapaksa, like the Āryapaksa, was founded by Āryabhata I in about 500. It is characterized not only by its own parameters but also by the fact that its epoch is midnight rather than dawn,

so that the present Kaliyuga begins at midnight of 17/18 February — 3101. The work of Āryabhata I from which this paksa is derived is lost except for testimonia and fragments; but one of his pupils, Lātadeva, is evidently the author of a revision of the original Sūryasiddhānta that makes it conform to the Ārdharātrikapaksa; there is a summary of Lātadeva’s Sūryasiddhānta in Varāhamihira’s Pañcasiddhāntikā. Lātadeva’s epoch is midnight 20/21 March 505; the epoch of the original Sūryasiddhānta was noon, and its parameters probably were not those of the Ārdharātrikapaksa.⁸⁵

The relation of the Ārdharātrikapaksa to the Āryapaksa is close. The former shares the latter’s division of the Mahāyuga into four equal Yugas. The difference in epoch dates of the present Kaliyuga, however, which comes to 0;15 days, amounts to an increment of 0;0,0,15 days per year when distributed over the 3,600 years between that epoch and 499. The Ārdharātrikapaksa’s year length, therefore, is 6,5;15,31,30 days instead of the Āryapaksa’s 6,5;15,31,15; but the rotations of the planets in a Mahāyuga remain essentially the same (see Table VI. 1). They are listed along with the mean yearly and mean daily motions in Table VII. 1.

Lātadeva represents the planets’ period relations as indicated in Table VII.2 (PS 9. 1–5 and 16, 1–9). Further, Lātadeva adds the yearly bījas given in Table VII.3 (PS 16, 10–11).

From the relations given above, it follows that the time periods in the Ārdharātrikapaksa are identical with those in the Āryapaksa (Table VI.2) except for those listed in Table VII.4.

The longitudes of the nodes in the Ārdharātrikapaksa coincide with those of the Āryapaksa (Table VI.6), although the longitudes of the mandoccas differ slightly. The latter are given in Table VII.5 (PS 16, 13).

The circumferences of the manda epicycles and of the śīghra epicycles according to Lātadeva are recorded in Table VII.6 (PS 9, 7–8 and 16, 12 and 14); the corresponding maximum equations are added (the śīghra equations and anomalies are those of the Khandakhādyaka).

Lātadeva’s model for the planets is that with two epicycles (PS 16, 15–22); his values for the kālāmsas of visibility are identical with those in the Paitāmaha (Table V.8)(/PS 16, 23).

In Pañcasiddhāntikā 9–10. with which 11 seems to belong, Varāhamihira summarizes the eclipse computations found in Lātadeva’s Sāryasiddhānta, First, in order to determine the true longitudes and true velocities, Lātadeva proposes the following two rules (PS 9, 9):

where Δμ is the daily increment or decrease to the equation and Δκ is the daily increment in the argument, all measured in minutes. For the sun , mean daily motion; for the moon , where is the mean daily motion of the moon’s manda. The second rule is (V.67), the cosine rule for finding the daily increment or decrease to the mean daily motion (PS 9, 13).

Further, Lātadeva assumes, with other Indian astronomers, that (PS 9, 14)

where H is the true distance of the planet in units of R; as presented in the Pañcasiddhāntikā. R = 120. He measures the actual distance in units of which there are eighteen in the radius of the earth, r_e. He also states that the circumference of the earth is 3,200 yojanas, so that its radius is about 509-1/3 yojanas (PS 9, 10): these parameters are slightly smaller than those of the Āryapaksa.

The distances of the sun and moon are computed by the following formulas (VII.3) (PS 9, 15–16):

Thence the mean values, when ρ = R = 120, are (VII.4)

From (VII.4 b) it follows that

where π₀ is the horizontal parallax. The value of 16041, should be derived, if (VII.2) is correct. from

where and are the rotations of the moon and the sun in a yuga. But then, with the Ārdharātrikapaksa’s parameters, we would have . One must suspect that the original Sūryasiddhānta derived in some other fashion.

To find the true diameters, d, of the sun and the moon, in the units in which and are measured, Lātadeva uses the relation

where is given as 517,080 and as 38,640. Using the mean values of k, one finds (VII.8)

Lātadeva’s rule for determining the diameter of the earth’s shadow at the distance of the moon in units of the radius, R, is related to (V.30), which was used by Āryabhata I: it is(PS 10, 1–2)

where the term in parentheses is the numerical expression of (V. 123).

In computing parallax, Lātadeva follows essentially the incomplete rules of Āryabhata I. He finds Sin SN from (VI.1) and (VI.2), and then (PS 9, 19–23)

And, with Sin ZN from (VI.1 b), he forms (PS 9, 24–25)

Otherwise, his rules for determining the durations and magnitudes of eclipses are unexceptional. The projection with aksavalana and ayanavalana described in the Pañcasiddhāntikā (PS 11, 1–5) is identical with that in the Brāhmasphutasiddhānta (V.93–94).

A text belonging to the Ārdharātrikapaksa, known as the Zīk i Arkand (”Tables of the Ahargana“), was available in Sasanian Iran in the middle of the sixth century and formed the basis of much of the Zīj al-Shāh of Anūshirwān, and of that of Yazdijird III in the 630’s.⁸⁶

The next Sanskrit text available to us is the Mahābhāskarīya, in which (MB 7, 22–35) Bhāskara I repeats the Ārdharātrikapaksa’s parameters. One variation is that the latitudes of the planets are regarded as a combination of the tilting of both the manda epicycle and the śīghra epicycle. Further, the following measures in yojanas are given, as in Table VII.7.

We have seen in Lātadeva that 1,600 yojanas is half of the circumference of the earth, not its diameter. And in general these diameters are closer to those in Table V.23 than to Latadeva’s. The given vālues of the mean distances of the sun and the moon correspond to 861–279/400 and 64-183/400 earth radii; these numbers do not agree with Latadeva’s either (VII.4). The correctly computed, although rounded, distances of the sun and moon according to the later Pauliśa (Table VII.11) are 689,378 yojanas and 51,566 yojanas: the former is 861-289/400 earth radii.

The principal text of the Ārdharātrikapaksa, however, is the Khandakhādyaka of Brahmagupta;⁸⁷ its epoch is 23 March 665. Like his Brāhmasphutasiddhānta, this work consists of an initial summary of an early text (in this case the Ārdharātrika text of Āryabhata I) followed, in the uttara, by corrections and additions; many of these corrections and additions in the Khandakhādyaka are derived from the Brāhmasphutasiddhānta.

Brahmagupta gives the mean daily motions of the planets (Table VII.1) in the form of Table VII.8 (Kh 1, 1, 8-13 and 1, 2, 1-5); compare Table VII.2, which contains some of the same relations, The mean daily motion of the lunar node is based on 232,218 rotations in a Mahāyuga instead of –232,226. In the uttara section of the Khandakhādyaka, Brahmagupta uses the relation (III.9) to obtain the longitude of the moon’s manda (Kh 2, 1, 2), and substitutes for the lunar node the approximation (Kh 2, 1, 3)

This parameter is close to that of the Brāhmapaksa

(Table V.2). Both parameters also appear in the Vākyakarana.

In the first section Brahmagupta repeats the longitudes of the mandoccas recorded in Table VII.5 (Kh 1, 1, 13 and 1, 2, 6); in the uttara he changes Jupiter’s to 180°, Mars’s to 117°, and the sun’s to 77° (Kh 2, 1, 1 and 2, 2, 1). Brahmagupta tabulates the sun’s and the moon’s manda equations at intervals of 15°; the manda equations of the starplanets are multiples of the sun’s. Their sighra equations are given for unequal intervals. The maximum sighra equations have been recorded in Table VII.6; the maximum manda equations are given in Table VII.9 (Kh 1, 1, 16–17 and 1, 2, 6-7), and half the synodic periods in Table VII.10 (Kh 1,2,8-17).

The longitude of the moon is fürther corrected by 1/27 of the sun’s manda equation, so that the maximum correction is 5;1° (Kh 1, 1, 18).

In the uttara Brahmagupta prescribes the following modifications (Kh 2, 2, 1). The manda equation of Saturn is decreased by 1/5, so that the maximum equation becomes 7;39,25,…°. The śīghrocca of Venus is diminished by 74′, so that

This in effect means that at the beginning of the Mahāyuga the center of Venus’ śīghra epicycle was the true rather than the mean sun. Also, the śīghra equation of Mercury is increased by 1/16, so that the maximum equation becomes 22;50,37,…° Further, the manda equation of the sun is decreased by 1/42. so that the maximum becomes 2;11°(Kh 2, 1, 5); and the manda equation of the moon is increased by 1/52, so that the maximum becomes 5;1,41,…°. These corrections generally bring the equations close to the values in Brāhmasphutasiddhānta 25 (Tables V.28 and V.29).

In the first part of the Khandakhādyaka, the computation of the longitudinal difference is based on the circumference of the parallel of latitude passing through Ujjayinī being 4,800 yojanas (Kh 1, 1, 15), In the uttara the earth’s circumference is given as 5,000 yojanas, as in the Paitāmaha; and the circumference of the parallel of latitude, c_ø, is found by (Kh 2, 1, 6)

Brahmagupta’s Sine table in the Khandakhādyaka is identical with that in Brähmasphutasiddhānta 25 (Table V.24) (Kh 1, 3, 6), as are his values of γ/2 (Table V.30) (Kh 1, 3, 1). The computation of eclipses in the Khandakhadyaka also is essentially identical with that in the Brāhmasphutasiddhānta (Kh 1, 4–5); in the uttara the cālanas for 177 days are also given (Table V.20) (Kh 2, 4, 21–22).

The first part of the Khandakhādyaka repeats the familiar kālāmśas of visibility of the planets (Table V.8) (Kh 1, 6, I), the Āryapaksa’s longitudes of their nodes (Table VI.6) (Kh 1, 8, 1), and its values of the orbital inclinations (Table VI.5)(Kh 1,8, 1): in the uttara Mercury’s orbital inclination is given as 2;30° instead of 2° (Kh 2, 5, 1). Further, the kālāmśas of visibility of Venus and Mercury are stated to be 10° and 14° at superior conjunction, and 8° and 12° at inferior conjunction (Kh 2. 5, 3–4).

In the Khandakhādyaka, Brahmagupta ascribes to the yogatārās of the naksatras the same polar longitudes and polar latitudes that he does in the Brāhmasphutasiddhānta (Table V.10 with Table V.22) (Kh 1, 9, 4–12). In the uttara he repeats the theory of the naksatrabhogas that he has expounded in Brāhmasphutastddhānta 14 (Kh 2, 1, 6–9). Both of his works, then, although fundamental for their respective paksas. represent the unassimilated conflation of material of diverse origins.

In the early eighth century an Arabic Zīj al-Arkand, written in Sind, was dependent on the Khandakliādyaka;⁸⁸ its epoch is 735.

At about the same time, in Sthāneśvara, the later Pauliśasiddhānta was composed; it is largely based on Ārdharātrika parameters, although the author accepts the traditional division of the Mahāyuga into four unequal yugas that is used in the Brāhmapaksa. Therefore he dates the first mean conjunction of the Mahāyuga 648,000 years after the beginning of the Krtayuga. which is 1,080,000 · 3 years before the beginning of the current Kaliyuga. Therefore, like the roughly contemporary author of the Sūryasiddhānía, he must have introduced a period of nonmotion of the planets at the beginning of the Kalpa, The later Pauliśa gives precisely the parameters of Table VII.1, and by and large those of Table VII.6, although Saturn’s śīghra epicycle is made 39;30° rather than 40°, and Mars’s 233° rather than 234°. Following Bhāskara I’s summary of the Ārdharātrikapaksa, the diameter of the earth is given as 1,600 yojanas (Table VII.7), and thence, with π = 3-177/1,250. its circumference as 5,026-14/25 yojanas. The prime meridian passes over Ujjayinī, Rohītaka, Kuruksetra, the Yamunā (Jumna) River, the Himālayas, and Meru. Because of the above value of π, which is Āryabhata’s. the later Pauliśa makes R = 3,437-967/1,309.

Basing the computation on the principles previously enunciated, the later Pauliśa gives the circumferences and radii of the heavenly orbits. measured in yojanas, as in Table VII.11.

The later Pauliśa’s mean diameters of the starplanets in minutes and of the luminaries in yojanas and minutes are displayed in Table VII, 12.

The diameters of the luminaries in yojanas are those found in Table VII.7; the order of the star-planets and their diameters are those of their kālāmsas of visibility in Table V.8.

Al-Bīrunī, in the late 1020’s, was acquainted with both the Khandakhādyaka and the later Pauliśasiddhānta, and makes extensive quotations from them. This fact attests to the great popularity of the Ārdharātrikapaksa in northwestern India in the eleventh century.

A somewhat later Ārdharātrika karana that is more difficult to localize is the Bhāsvatī of Śatānanda; its epoch is 1099. Śatānanda claims that his Bhāsvati is based on the Sūryasiddhānta summarized by Varähamihira.

The Ārdharātrikapaksa continued to flourish, especially on the fringes of India —in Kashmir, Nepal, and Assam. So far very little tabular material has come to light,⁸⁹ but this is undoubtedly due to the paucity of manuscripts from the areas mentioned above that have yet been examined.

VIII. THE SAURAPAKSA

As we have seen, one of the earliest texts of the Ārdharātrikapaksa was a Sūryasiddhānta—that of Lātadeva. Another Sūryusiddhānata closely allied to the Ārdharātrikapaksa was composed sometime before Vijayananda of Benares (966), and apparently before Vateśvara of Ānandapura (905) and Govindasvāmin of Kerala (ca. 800/850);⁹⁰ it seems reasonable to date this text to the late eighth or early ninth century, and to surmise that it was composed in southern India, By the twelfth century commentaries were being written on it in Mysore, by Mallikārjuna (fl. 1179[?]), and in Mithilā, by Candeśvara (1183). In the fifteenth century it was recognized as a strong rival to the Āryapaksa in southern India, and as the authentic astronomical

text in northern and eastern India. The text with which modern scholars are most familiar is the version by Ranganātha of Benares (1602).⁹¹

The Sūryasiddhānta follows completely the divisions of the Kalpa and Mahāyuga enunciated in the Brāhmapaksa (SS 1, 14-21). The rotations of the planets in a Mahāyuga and their corresponding mean yearly and mean daily motions are displayed in Table VIII.1; the number of civil days in a Mahāyuga is given as 1,577,917,828 (SS 1, 29-33,and 37).

Thus, although the equal-Yuga system of Āryabhata I has been abandoned by the author of the Sūryasiddhānta, as by Lalla, and by the author of the later Pauliśa, the principle remains that his values of R are divisible by 4. In order to produce a mean conjunction at the beginning of the current Kaliyuga, the author of the Sūryasiddhänta hypothesizes a period of creation equal to 17,064,000 years at the beginning of the Kalpa (SS 1, 24). The Saurapaksa, like the Ārdharātrikapaksa, uses midnight epoch.

The year length implied by the number of civil days is 6,5;15,31,31,24 days. It is this year length that generates differences in the mean longitudes of the planets computed according to the Ārya, Ārdharātrika, and Saura paksas. The relations of these pakṣas to each other are displayed in Table VIII.2.

The numbers of rotations imply orbits of almost the same sizes in yojanas as those in the later

Pauliśa (Table VII.11); they are listed in Table VIII.3 (SS 12,80-90).

The changed number of days also alters the following parameters in a Mahāyuga, listed in Table VIII.4 (SS 1, 34-40).

Unlike the Ārya and Ārdharātrika pakṣas, the Saura does not just give the current longitudes of the planets’ mandoccas and nodes, but records their rotations in a Kalpa, as the Brāhma had. Those are listed in Table VIII.5 (SS 1, 41–44); their approximate longitudes in 850 also are given.

The longitudes of the mandocca of the sun (approximately) and of the nodes are from the Āryapaksa (Table VI.6), those of the inferior planets’ mandoccas from the Ārdharātrikapaksa (Table VII.5): compare Table IX. 1.

Like the Khandakhādyaka, the Sūryasiddhānta gives the diameter of the earth as 1,600 yojanas (SS 1, 59); but, following the Paitāmaha, it uses to find the circumference, which then is 5,059 + yojanas. The circumference, c_φ, of the parallel of latitude of any locality is found by (SS 1, 60)

With this rule, also found in the Khandakhādyaka