ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

BRAHMAGUPTA'S OWN RULE 275 is effected without forming an equation of the factum Why then was it done so ?¹ Datta and Singh think that the reference in the latter por- tion of this rule is to the method of the unknown author : “Kiṁ kṛtaṁ tadataḥ” ? The principle underlying Brahma- gupta's method is to reduce, like the Greek Diophantus (c.275 A.D.), the given indeterminate equation to a simple determi- nate one by assuming arbitrary values for all the unknowns ex- cept one. So undoubtedly it is inferior to the earlier method. We now take an illustrative example from Brahmagupta : On subtracting from the product of signs and degrees of the Sun, three and four times (respectively) those quantities, ninety is obtained. Determining the Sun within a year (one can pass as a proficient) mathema- tician. If we presume x to denote the signs and y the degrees of the Sun, then the equation would be : xy - 3x - 4y = 90 Pṛthūdaka Svāmī solves it in two ways : (i) Let us assume the arbitrary number to be 17. then x = 1/1 ( (90.1 + 3.4) / 17 + 4 ) = 10 y = 1/1 (17 + 3) = 20 (ii) Let us assume arbitrarily y = 20. On substituting this value of y in the above equation, we get 20x - 3x = 170 whence x = 10.

1. भावितके यद्घातो विनष्टवर्णेन तत्प्रमाणानि । कृतेष्टानि तदाहत वर्णैक्यं भवति रूपाणि ॥ वर्ण प्रमाणभावित घातो भर्वातेष्ट वर्ण संख्यैवन् । सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ —BrSpSi. XVIII. 62-63
2. भानो राश्यंशवधात् त्रिचतुर्गुणितान् विशोध्य राश्यंशान् । नवतिं दृष्ट्वा सूर्यं कुर्वन्नावत्सराद् गणकः ॥ —BrSpSi. XVIII. 61. — : o : — Reference H.T. Colebrooke : Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhas- cara, London, 1817. B. Datta and A.N. Singh : History of Hindu Mathematics Pt. I and II, 1962.

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CHAPTER X Arabic and Indian Divisions of the Zodiac I It has long been a debated question whether the Indian and Arabian divisions of the zodiac had a common origin. Sir William Jones thought that they had not; but Colebrooke holds a contrary view. The coincidence, in the two systems of division is so exact that he thinks, it could not be due to chance. Coleb- rooke has discussed this point in details in one of his Papers entit- led "On the Indian and Arabian divisions of the zodiac", Asiatic Researches Vol. ix.. p. 323-376, reproduced in the Miscellaneous Essays, Vol. II., p. 321-373, 1872.

1. Aśvinī, now the first nakṣatra, but anciently the last but one, probably obtained its present situation at the head of the asterisms, when the beginning of the zodiac was referred to the first degree of Meṣa (the Ram). As measuring a portion of the zodiac, it occupies the first 13°20' of Meṣa; and its beginning follows immediately after the principal star in the last nakṣatra Revatī, reckoned by some exactly, by others nearly, opposite to the very conspicuous one, which forms the fourteenth asterism. As a constellation, Aśvinī comprises three stars (Aries α, β, γ) figured as a horse-head; and the principal, which is also the northern one, is stated by all ancient authorities, in 10°N and 8°E. from the beginning of the Meṣa. According to Arabs, the first manzil or lunar mansion is entitled Sheraṭan (by Persians, Sheratain), and comprises two stars of the third magnitude on the head of Aries. in lat. 6°36' and 7°51' N and long. 26°13' and 27°7'. With the addition of a

278 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC third, also in the head of the Ram, the asterism is denominated Āshrait. The bright star of the second or third magnitude which is out of the figure of the Ram, according to Ulugh Beg, but on the nose according to Hipparchus, cited by this author from Ptolemy, is determined Nātih : It is placed in lat. 9°30'N and long. 1 0° 43', and is apparently the same with the principal star of the Indian asterism; for Muhammad of Tizin, in his table of declination and right ascension, expressly terms it the first star of the Sheratain. 2. Bharaṇī, the second asterism, comprises three stars (35, 39, 41 Aries) figured by the yoni or pudendum muliebre and the principal and southern star of this nakṣatra is placed in 12°N. On the Arabian system, the second manzil, entitled Butain is placed by Ulugh Beg in lat. 1°12' and 3°12', and this cannot possibly be reconciled with the Indian constellation. But Muhammad of Tizin assigns to the bright star of Butain a decli- nation of 23°N exceeding by nearly 2° the declination allotted by him to Nātih or his first star in Sheratain. This agrees with the difference between the principal stars of Aśvinī and Bharaṇī; and it may be inferred, that some among the Mohammadan astrono- mers have concurred with the Hindus in referring the second constellation to stars that form Musca. 3. Kṛttikā, now the third, formerly the first, nakṣatra consists of six stars figured as knife or razor, and the principal and southern star is placed in 4½ or 5°N and in 65 sixths of degrees (or 10°50′) from its own commencement (cf. the Sūryasid- dhānta), or 37° 28′ to 38° from the beginning of the Meṣa (the Siddhānta-Śiromaṇi or the Graha Lāghava) respectively. This longitude of the circle of declination corresponds nearly with that of the bright star in the Pleiades, which is 40° of longitude distant from the principal star of Revatī. The stars indicated by Ulugh Beg for Thurayyā, also corres- pond exactly with the Pleiades. 4. Rohiṇī, is the fourth nakṣatra, the Arabic name for the fourth mansion is Debarān (or with the article Aldebarān). It corresponds to the bright star called the Bull's eye, and which is unquestionably the same with the principal and eastern star of Rohiṇī, placed in 4½° or 5°S and 49½°E by the Hindu writers on Astronomy. This nakṣatra is

figured as a wheel cart, and comprises five stars, out of the seven which the Greeks named the Hyades. The Arabs, however, like the Hindus, reckon five stars only in the asterism. Sir William Jones supposes them to be in the head and neck of the Bull; they probably are α, ρ, γ, δ, ε Tauri, agreeably to Mons. Bally's conjecture. 5. Mṛgaśirā, the fifth nakṣatra, represented by an antelope head, contains three stars; the same which constitute the fifth lunar mansion Hakāh; for the distance of 10°S assigned to the northern star of this nakṣatra, will agree with no other but one of the three in the head of Orion. The difference of longitude (24° to 25½°) from Kṛttikā corresponds with sufficient exactness; and so does the longitude of its circle of declination (62° to 63°) from the end of Rewatī; since the true longitude of λ Orionis, from the principal star in Revatī (ζ piscium) is 63½°. 6. Ārdrā, the sixth nakṣatra, consists of a single bright star, described as a gem, and placed in 9°S (by some in 11°) and at the distance of 4⅓ to 4° in longitude from the last asterism. This indicates the star in the shoulder of Orion (a orionis). The sixth lunar mansion is named by the Arabs as Hanāh; and comprises two stars in the feet of the second twin, according to Ulugh Beg, though others make it to be a shoulder. Mohammad of Tizin allots five stars to this constellation; and the Kāmūs, among various meanings of Hanāh, says, that it is a name for five stars in the left arm of Orion; remarking also, that the lunar mansion is named Tahāyī, comprising three stars called Tahyāt. Obviously here the Indian and Arabian asterisms are irreconcileable. 7. Punarvasū (used in a dual number) is the seventh nakṣtra, and is represented by a house, or even a bow, and it includes four stars, among which the principal and eastern one is 30° or 32° from the fifth asterism; but has been placed by all authorities in 6°N. This agrees with (β Geminorum) one of the two stars in the heads of the Twins, which together constitute the seventh lunar mansion ziraā, according to Mohammad of Tus and Mohammad of Tizin and other Arabian authorities. The seventh lunar mansion of Arabs is named ziraā ul ased according to Jauhari and other cited by Hyde in his Commen-

250 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC tary on Ulugh Beg, and that the Kāmūs makes this term to be the name of eight stars in the form of a bow. 8. Puṣya, the eighth asterism, is described as an arrow, and consists of three stars, the chief of which being also the middle most, has no latitude, and is 12° to 13° distant from the seventh asterism, being placed by Hindu astronomers in 106° of longitude. This is evidently δ Cancri; and does not differ widely from the eighth lunar mansion Nethrah, which according to Ulugh Beg and others consists of two stars, including the nebula of Cancer. The Indian constellation comprises two other stars besides δ Cancri, which are perhaps γ and β of the same constellation. 9. Aśleṣā, the ninth asterism, contains five stars figured as a potter’s wheel, and of which the principal or eastern one is placed in 7°S, and according to different tables, 107°, 108° or 109° E. This appears to be intended for the bright star in the south- ern claw of Cancer (α Cancri), and cannot be reconciled with the lunar mansion Tarf or Tarfah, which comprises two stars near the lion’s (siṁha) eye, the northernmost being placed by Mohammad of Tīzīn in 24° of N. declination. 10. Maghā, the tenth asterism, contains like the last, five stars, but which are figured as a house. The principal of the Southern one has no latitude; and according to all authorities, has 129° longitude. This is evidently Regulus (α Leonis) : which is exactly 129½° distant from the last star in Revatī. The tenth lunar mansion of Arabians is Jebhah, which comprises three (some say, four) stars, nearly in the longitude of the lion’s heart. In this instance, therefore, the Indian and Arabian divisions of the zodiac coincide. This nakastra consists of α, γ, η and ν Leonis. 11. Pūrva-Phālgunī is the eleventh nakṣatra and is repre- sented by a couch or bedstead; it consists of two stars deter- mined by the place of the chief star (the northernmost, accord- ing to the Sūrya-Siddhānta) in 12°N and 144°E., or according to Brahmagupta, the Śiromaṇi and the Grahalāghava 147° or 148° E. They are probably δ and θ Leonis. The Arabian name for this lunar mansion is zubrah or Khertan.

ARABIC AND INDIAN DIVISIONS OF THE ZODIAC 281 It may be mentioned here that Brahmagupta and Bhāskara selected the southern for the principal star; while the Sūrya- Siddhānta took the northern. Hence the latitude stated by several Hindu authorities is the mean between both stars; and the difference of longitude, compared to the preceding and subse- quent asterisms, may be exactly reconciled upon this suppo- sition. 12. Uttara-Phalgunī, which is the twelfth nakṣatra, con- sists of two stars, and is figured as a bed or cot. These stars are ascertained by the place of one of them (the northernmost) 13°N. and 155° E. This indicates β Leonis; the same which singly constitutes the Arabian Lunar mansion Serfah, though Moham- mad of Tīzīn seems to hint that it consists of more than one star. 13. Hasta, the thirteenth nakṣatra, has the name and figure of a hand; and is suitably made to contain five stars. The principal one towards the west, next to the north-western star, is placed according to all authorities in 11° and 170° E. This can only belong to the constellation Corvus ; and accordingly five stars in that constellation (α, β, γ, δ and ε Corvi). The thirteenth lunar mansion of Arabs is Awwā, which is also described to contain five stars, situated under Virgo and so disposed as to resemble the letter Alif. They are placed by Ulugh Beg in the wing. Here obviously, there is nothing common between the Hindu and Arabian specification of the asterism. The agreement is only in the number of stars and in the longi- tude. 14. Citrā, the fourteenth nakṣatra, is figured as pearl. It is placed by the Sūrya-Siddhānta in 2° S and 180° E. and by Brahmagupta, the Śiromaṇi and Grahalāghava in 1¾ or 2°S and 183°E. This agrees with the Virgin's spike (α Virginis). The same star constitutes the fourteenth lunar mansion of the Arabs named from it Simāc ul aāzi. 15. Svāti, the fifteenth nakṣatra, is represented by a coral bead. The Sūrya-Siddhānta, Brahmagupta, the Śiromaṇi and Grahalāghava, all concur in placing it at 37°N. They differ one degree in longitude of its circle of declination, three of them

282 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC making it 199° and the other 198°. The Indian asterism totally disagrees with the lunar mansion Ghafr which is the fifteenth Arabian mansion. and which consists of three stars in the Virgin's (Kanyā) foot, according to Ulugh Beg. but in or near the balance (Tulā), according to others. 16. Viśākhā, the sixteenth nakṣatra, consists of four stars described as a festoon. All the authorities place the principal and northernmost star in 1°, 1°20′ or 1°30′ S and in 212°, 212°5′ or 213° E. The latitude seems to indicate the bright star in the Soutehrn Scale (α Librae), though the longitude disagrees (suggest- ing possibly a remote star κ Librae). Colebrooke suggests the four stars to be α ν ι Librae and γ Scorpii. The sixteenth lunar mansion according to Arabs is Zubanah or Zubāniyah according to Mohammad of Tizin, the bright star in the northern scale (β Librae). 17. Anurādhā, the seventeenth nakṣatra, consists of four stars and is described as a row of oblations in a right line. Its chief or middlemost star is placed in 3°, or 2° or 1°45′ S and in 224° or 224°5′E, thus placing it near the head of the Scorpion (Vṛścika) (δ Scorpionis) and the asterism comprises β, δ, π, and ρ Scorpionis. The seventeenth lunar mansion of Arabs is called Iklīl or Iklīlu'l- jebhah, which is said to contain 4, 3, or 6 stars lying in a straight line. Those assigned by Ulugh Beg for this mansion are β, δ, ν and π Scorpionis. Thus here the Indian and Arabian astronomers both concur exactly. 18. Jyeṣṭha, the eighteenth nakṣatra, comprises three stars figured as a ring. The principal and middlemost star is placed in 4°·3½° or 3° S and in 229°, 229°5′ or 230°E; this position indicates Antares or the Scorpion's heart (α Scorpionis), which is also the eighteenth lunar mansion, named Kalb or Kalbul'akrab. The three stars of Indian asterism may be α, σ and τ Scorpionis. 19. Mūla, the nineteenth nakṣatra, is represented by a lion's tail, and it contains eleven stars, of which the charac- teristic one, the easternmost, is placed in 9°, 8½° or 8° S and in 241° or 242° E. This probably (not exactly) indicates ν Scor- pionis. This agrees with the eighteenth lunar mansion of Arabs known as Shaulah, consisting of two stars near the Scorpion's

sting. The Hindu asterism probably includes all the stars in the Scorpion’s tail (ε, μ, ζ, η, θ, ι, κ, λ, υ and ν Scorpionis). 20. Pūrva-Āṣāḍha, the twentieth nakṣatra, is figured as an elephant’s tooth or as a couch, and it consists of two stars, of which the most southern one is placed in 5½°, 5⅓° or 5° S and 254° or 255° E. This corresponds well with δ Sagittarii, and which also corresponds with the twentieth lunar mansion of Arabs called Nāaim. The Arabian mansion consists of four, or according to some eight, stars. The Indian nakṣatra corres- ponds to δ and ε Sagittarii. 21. Uttara-Āṣāḍha, the twenty-first nakṣatra, is represented by a couch or by an elephant’s tooth. The principal or the most northerly star is placed in 5° S and 260° or 261° E, agreeing with a star in the body of Sagittarius (τ Sagittarii), and the other star is perhaps the one marked ζ. The Arabian lunar mansion corresponding to it is Baldah, consisting of six stars, two, of which are placed by Mohammad of Tīzīn in declination 21° and 16°. One of these must be a star in the head of Sagit- tarius. Some authors, on the contrary, describe the lunar mansion as destitute of stars. Here the Arabs and Hindus do not show reconciliation. 22. Abhijit, the twenty-second asterism, consists of three stars figuring as a triangle or as a nut of floating Trapa (in modern Indian astronomy, it does not occupy an equal portion of the ecliptic with other nakṣatras). Its brightest star is very remote from the zodiac, being in 60° or 62° N. The longitude of its circle of declination is 265°, 266° 40′ or 268° according to different authorities. The corresponding lunar mansion of Arabs is Zābih, consisting of two stars (according to some, four) in the horns of Capricorn. This totally disagrees with Indian asterism. 23. Śravaṇa, the twenty-third nakṣatra, is represented by three footsteps, and contains three stars of which the middlemost is placed in 30° N (all authorities agree), and longitude 280° (Sūrya-Siddhānta) or 278° (Brahmagupta and Śiromaṇi), or 275° (Grahalāghava). The assigned latitude indicates the bright star in the Eagle, whence the three may be inferred to be α, β

284 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC and γ Aquilae. According to Arabs, the twenty-third lunar mansion is Balā, which consists of two stars in the left hand of Aquarius. Here again Arabian and Hindu divisions are at variance. 24. Dhaniṣṭhā, the twenty-fourth nakṣatra, is represented by a drum or tabor. It comprises four stars, the westernmost of which is placed in 36° N and according to Brahmagupta, Śiromaṇi and the Sūrya-Siddhānta in 290° E (Grahalāghava gives 286°). This longitude of the circle of declination and the distance of the star on it from the ecliptic indicate the Dolphin : and the four stars are α, β, γ and δ Dolphini. The correspond- ing lunar mansion of Arabs is Sāud, which comprises two stars in Aquarius (β and ζ Aquarii). Here again the two divisions disagree completely. 24. Śatabhiṣak, the twenty fifth nakṣatra, is a cluster of 100 stars figured by a circle. The principal or the brightest has no latitude; or only a third, or at utmost half, a degree of south latitude; and longitude 320°. This best corresponds with λ Aquarii. According to Arabs, the twenty-fifth lunar mansion is known as Akhbiyah which consists of three stars only, placed in the wrist of the right hand of Aquarius. However, it appears from Ulugh Beg's tables, as well as from Mohammad of Tīzīn's, that four stars are assigned to this mansion. The Indian and Arabian systems of division differ considerably but less widely according to some. 26. Pūrva-Bhādrapada, the twenty-sixth nakṣatra, consists of two stars represented by a couch or bed, or else by a double headed figure, one of which is placed in 24° N and 325° or 326° E. The only conspicuous star nearly in that position is the bright star in Pegasus (α Pegasi) and the other may be the nearest considerable star in the same constellation (ζ Pegasi). The twenty-sixth Arabian lunar mansion is Mukaddim, consisting of two brightest stars in Pegasus (α and β). Here the Indian and Arabian divisions show concurrence. 27. Uttar-Bhādrapada, the twenty-seventh nakṣatra, consists of two stars, figured as a twin or a person with double face, or else as a couch. The position of the most northerly of the two

ARABIC AND INDIAN DIVISIONS OF THE ZODIAC 285 is in 26° or 27°N and 337° E. which probably indicates the bright star in the head of Andromeda, and the other star to be the one in the extremity of the wing of Pegasus (γ Pegasi). This exactly agrees with the twenty-seventh lunar mansion of Arabs named as Muakkher. Ulugh Beg assigns those stars to it. 28. Revatī, the twenty-eighth nakṣatra, comprises thirty- two stars figured as a tabor. The principal star is the southern- most one, it has no latitude, and two of them assert no longitude, but some make it ten minutes short of the origin of the ecliptic, viz. 359° 50'. This clearly marks the star on the ecliptic in the string of the Fishes (ζ Piscium). The ascertainment of this star is important in regard to the adjustment of the Hindu sphere. The Arabic name for this mansion is Risha, signifying a cord. But the constellation as described by Jauhari and cited by Golius, consists of a multitude of stars in the shape of a fish and termed Betnu'lhūt; in the navel of which is the lunar mansion. Moham- mad of Tīzīn alse makes this lunar mansion to be the same with Betnu'lhūt, which appears, however, to be the bright star in the girdle of Andromeda (β Andromedae) though others describe it as the northern fish, extending, however, to the horns of Ram. The lunar mansion and the Indian asterism, therefore, are not reconcileable in this last instance. I leave it to the readers to draw an inference as to the concurrence of the divisions of zodiac in Indian and Arabian systems. I would personally agree with Sir William Jones that the agreements are by chance. Arabs derived the idea of dividing zodiac in 27 or 28 mansions from Indians, or may have got it from Greeks, and then they proceeded in their own way for details. I do not agree with those scholars who sometimes state that the Hindus took the hint of dividing the ecliptic from Greeks. The Atharvaveda devotes a number of Suktas or hymns on Nakṣatras, and I have shown elsewhere that inspired by these hymns, Gārgya was the first Ṛṣi who detailed out the nakṣatras. This happened much before Greeks developed even their first notions of astronomy. While the concept of 27 nakṣatras is Vedic and most ancient and of purely Indian origin the concept of 12 Rāśis (signs) or twelve constellations is proba- bly inspired from Greeks. [The names Kanyā, (virgo), Tulā

286 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC (Libra),Vṛścika (Scorpio), Dhanu, (Sagittarius), Makara, (Capri- corn), Kumbha (Aquarius), Mīna (pisces), Meṣa (Aries), Vṛṣa (Taurus), Mithuna (Gemini), Karka (Cancer), and Siṁha (Leo) were not used for Rāśis or signs in the Vedic times]. I shall con- clude this description with a passage from Colebrooke : The result of comparison shows, I hope satis- factorily, that the Indian asterisms, which mark the divisions of the ecliptic, generally consist of nearly the same stars, which constitute the lunar mansions of the Arabians : but in a few instances, they essentially differ. The Hindus have likewise adopted the divi- sion of the ecliptic and zodiac into twelve signs or constellations, agreeing in figure and designation with those of the Greeks; and differing merely in the place of the constellations, which are carried on the Indian sphere a few degrees further west than on the Grecian. That the Hindus took the hint of this mode of dividing the ecliptic from the Greeks, is not perha- ps altogether improbable; but if such be the origin of it they have not implicitly received the arrangement suggested to them, but have reconciled and adapted it to their own ancient distribution of the ecliptic into twenty-seven parts. In like manner, they may have either received or given the hint of an armillary sphere as an instrument for astronomical observation ; but certainly they have not copied the instrument which was described by Ptolemy, for the construction differs considerably. Names, Shapes, and Number of the Stars of the Nakṣatras The Muhūrta-cintāmaṇi provides a list of shapes associated to the nakṣatras (MuC. II. 59-60), In this list we are giving the number of stars as indicated by Varāhamihira, Brahmagupta and Lalla. The identification given here is as indicated by E. Burgess, in his Translation of the Sūrya-Siddhānta 1935, p. 378. (Calcutta). This table has been reproduced here from the Mahābhāskarīya of Bhāskara I, edited by K.S. Shukla.

NAMES, SHAPES AND NUMBER OF THE STARS 287

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CHAPTER XI Brahmagupta’s Astronomy : Its Highlights Beginning or Starting Point Very often in Indian astronomy, we come across a term ahargaṇa (literally meaning collection of days), which means the number of mean civil days elapsed at mean Sunrise at Laṅkā on a given lunar day (tithi), since the beginning of Kaliyuga. It is the beginning of Kaliyuga, which is taken as the starting point for the reckoning of ahargaṇa. This happened on Friday, Febru- ary 18, B.C. 3102, at mean sunrise at Laṅkā, when the Sun, Moon, and the planets are supposed to have been in conjunction at the first point of the nakṣatra Aśvinī (which is a fixed point situated near the star ζ-Piscium). According to Āryabhaṭa and Bhāskara I, the duration of Kaliyuga is 1,080,000 solar years. Four times this (4,320,000) is the duration in solar years of a bigger unit called Mahāyuga or even yuga. Laṅkā in Indian astronomy is a hypothetical place where the meridian of Ujjain (latitude 23° 11′ N, longitude 75° 52′ E from Greenwich) intersects the equator. It is one of the four hypothetical cities on the equator called Laṅkā, Romaka, Siddhapur and Yamakoṭi (or Yavakoṭi). The Sūrya-siddhānta describes Laṅkā as a great city (mahāpurī) situated on an island to the south of Bhāratavarṣa.¹ The present Ceylon is not the

1. समन्तान्मेरुमध्यात्तु तुल्यभागेषु तोयधेः । द्वीपेषु दिक्षु पूर्वादिन्नगर्यो देवनिर्मिताः ।। भूवृत्त पादे पूर्वस्यां यवकोटीति विश्रु ता । भद्राश्व वर्षे नगरी स्वर्णप्राकारतोरणा । याम्यायां भारतेवर्षे लङ्का तद्वन्महापुरी ।। (Cont. on page 290)

290 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT astronomical Laṅkā, as it is about six degrees to the north of equator. The astronomical Laṅkā is mentioned by Brahmagupta in the beginning of his very first Chapter¹. According to Brahmagupta all the four yugas of a Catur- yuga or mahāyuga are not of the equal duration : Kaliyuga is of 432,000 years. Dvāpara of 864,000 years, Tretā of 1,296,000 years and Kṛtayuga of 1,728,000 years; total of the four is 4,320,000 years. Āryabhaṭa regards all yugas of equal duration, 1,080,000 years². The Saka era, which is usually used in Indian astronomy for the reckoning of years commenced 3179 years after the beginning of Kaliyuga. The number of lunar months in a yuga does not coincide with the number of solar months. Thus we have the conception of the Intercalary months : the number of intercalary months in a yuga denotes the excess of the number of lunar months in a yuga over the number of solar months in a yuga. Thus in a yuga we have Lunar months 53,433,336 Solar months 51,840,000 Intercalary months 1,593,336 Lunar days 1,603,000,080 Civil days 1,577,917,500 Omitted lunar days 25,082,580 The number of omitted lunar days in a yuga is equal to the number of lunar days in a yuga minus the number of civil days in a yuga. (Cont. from page 289) पश्चिमेकेतुमालाख्ये रोमकाख्या प्रकीर्तिता । उदक्सिद्धपुरी नाम कुरुवर्षे प्रतिष्ठिता ॥ --Sūrya. XII. 36-39

1. चैत्रसितादेरुदयाद्भानोर्दिनमासवर्षयुगकल्पाः । सृष्ट्यादौ लंकायां समं प्रवृत्ता दिनेऽर्कस्य ॥ —BrSpSi. I. 4
2. युगदशभागो गुणितः कृतं चतुर्भिस्त्रिभिर्गुणस्त्रेता । द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥ युगपादानायैभटश्चत्वारि समानि कृतयुगादीनि । यदभिहितवान् न तेषां स्मृत्युक्तसमानमेकमपि ॥ —BrSpSi. I. 8-9

UNITS OF TIME 291 Units of time For the measurements of durations, it is necessary to have units of time. Brahmagupta gives the following units :¹ 6 prāṇas or Asus=1 Ṛkṣa-vināḍikā or nakṣatra-vighaṭikā or one pala (24 seconds) 60 palas =1 ghaṭikā (24 minutes) 60 ghaṭikās =1 divasa or dina (day) (24 hours) 30 dinas =1 māsa (month) 12 māsas =1 varṣa or year Similar to the divisions of time, we have the divisions of an arc :² Vikalā (or viliptā or viliptikā) =second of arc. 60 vikalās =1 kalā (minute of arc) 60 kalās =1 aṁśa (degree of arc) 30 aṁśas =1 rāśi 12 rāśis =1 bhagaṇa (complete circle, 360°) Unlike Āryabhaṭa and others, who take kali, dvāpara, tretā, and kṛta of equal number of years, Brahmagupta regards kali consisting of 432,000 years, dvāpara twice of it, consisting of 864,000 years, tretā thrice of Kali consisting of 1,296,000 years and kṛta four times of kali conisting of thus 1,728,000 years, all the four to be making a yuga of 4,320,000 years.³ Further in the beginning of kṛta there is a sandhyā of 1728000/12 years (=144,000 years), and at the end of Kṛta, there is a sandhyāṁśa of 144,000 years; similarly in the beginning of tretā, we have a

1. प्राणैर्विनाडिकाऽर्क्षी षड्भिर्घटिका षष्ट्या । घटिका षष्ट्या दिवसो दिवसानां त्रिंशता मासः ॥ —BrSpSi. I. 6
2. मासा द्वादशवर्षं विकलालिप्तांशराशिभगणांतः । क्षेत्र विभागस्तुल्यः कालेन विनाडिकाद्ये न ॥ —BrSpSi. I. 6 वर्षं द्वादश मासास्त्रिंशद्दिवसो भवेत्स मासस्तु । षष्टि र्नाड्यो दिवसष्षष्टिश्च विनाडिका नाडी ॥ —Ārya. III. 1 गुर्वक्षराणि षष्टिर्विनाडिकार्क्षी षडेव वा प्राणाः । एवं काल विभागः क्षेत्र विभागस्तथा भगणात् ॥ —Ārya, III. 2
3. खचतुष्टयरदवेदा रवि वर्षाणां चतुर्यु गं भवति । सन्ध्या सन्ध्यांशैः सह चत्वारि पृथक् कृतादीनि ॥ युगदशभागो गुणितः कृतं चतुर्भिस्रिभिर्गुणस्त्रे ता । द्विगुणो द्वापरमेकेन संगुणः कलियुगं भवति ॥ —BrSpSi. I. 7. 8

292 BRAHMAGUPTA'S ASTRONOMY ITS HIGHLIGHT sandhyā of 1.296.000/12; i. e. 108.000 years and at the close of tretā a sandhyāṁśa of 108,000 years. Again, in the beginning of dvāpara we have a sandhyā of 864.000/12, i. e. 72.000 years and at the close of dvāpara a sandhyāṁśa of 72000 years; and similarly at the beginning a sandhyā and at the close a sandhyāṁśa of 432.000/ 1, i. e. of 36,000 years in the case of kali. In this respect Brahmagupta appears to follow Manu. the first author or giver of law. He regards further the following divisions of time :¹ 71 yugas =1 manu 14 manus =1 kalpa Again, in the beginning, at the middle and at the close of each manu, there are sandhis, each equal to the measure of kṛta. Thus, taken as a whole 1 kalpa=71×14 yugas+15 sandhyā-sandhyāṁśa =994 yugas+15×duration of kṛta =994 yugas+15×(4×432,000) years =994 yugas + 6 yugas =1000 yugas = 1 Brahma-dina (Brahmā's day) Thus Brahmā's day is regarded as 1 kalpa or one thousand caturyugis or 1000 yugas or the same as 1000 mahāyugas). Āryabhaṭa regards a manu to consist of 72 yugas and therefore a kalpa according to him would be of 14×72 yugas, or 1008 yugas.² Since in the foreign Siddhāntas like Romaka, there is no reference to yuga, manu and kalpa. Brahmagupta regards these systems to be unauthoritative.³ We have said that our starting point was the beginning of Kaliyuga, Friday February 18, B.C. 3102, at mean rise at Laṅkā, when the Sun, Moon and the planets are supposed to have been in conjunction at the first point of the Nakṣatra Aśviṇī. This type of conjunction would again happen after a period of kalpa.

1. मनुरेकसप्ततियुगः कल्पो मनवश्चतुर्दश मनूनाम् । आद्यन्तरान्त सन्धिषु कृतकालोऽस्माद्युग सहस्रम् ॥ —BrSpSi. I. 10
2. दिव्यं वर्ष सहस्रम् ग्रहसामान्यं युगं द्विषट्क गुणम् । अष्टोत्तरसहस्रं ब्राह्मो दिवसो ग्रहयुगानाम् ॥ —Ārya. III. 8
3. युगमन्वन्तरकल्पाः कालपरिच्छेदकाः स्मृतावुक्ताः । यस्मान्नरोमके ते स्मृतिबाह्यो रोमकस्तस्मात् ॥ —BrSpSi. I- 13

UNITS OF TIME 293 We shall have the same type of conjunction of grahocca, mandocca śīghrocca and pāta after a complete cycle of kalpa as we had in the beginning of creation. This is a natural observable cycle which is recognised in Indian astronomy and in no other foreign system; and hence only the Indian system recognises the time measure of kalpa. Ucca or apex is of two kinds : mandocca (apex of the slowest motion) and śīghrocca (apex of the fastest motion). The mandocca is that point of a planet's orbit which is at the remotest distance and where the motion of the planet is slowest. In the case of the Sun and the Moon, it is the apogee; and in the case of other planets, it is the apogee or aphelion, the geocentric longi- tude of the apogee being equal to the helio-centric longitude of the aphelion. The śīghrocca of a superior planet (Mars, Jupiter and Saturn) is defined as the mean Sun; that of an inferior planet Mercury or Venus) śīghrocca is an imaginary body which is supposed to move in such a way that its direction from the earth is always approximately the same as that of the actual planet from the Sun. The bhagaṇas or the revolution numbers of a planet have been given by Āryabhaṭa and Brahmagupta¹ both; they mean the number of revolutions that a planet performs in a certain period, say a kalpa of 4,320,000,000 years. The bhagaṇas of planets are given as follows :

1. कल्पेऽर्कबुधसितानां भगणाः शून्यानि सप्त रदवेदाः । प्राग्ब्रजता कुजगुरुशनि शीघ्रोच्चानां स्वकक्षासु ॥ पञ्चाम्बराणि गुण-गुण पञ्चमुनि स्वरैर्मिताः शशिनः । भौमस्य द्वियमशराष्टपञ्चवसूरसनवद्वियमाः ॥ कृतवसुखवाष्टनन्द-नवषट्‌त्रि नवागेन्द्वो ज्ञशीघ्रस्य । जीवस्य शरेष्वद्रि षट्पञ्चद्वि कृतरसरामाः ॥ सितशीघ्रस्य यमलगोवेदनवाष्टाग्निपञ्चयमखनगाः । अष्टनवपञ्च मुनि रसशररस मनवोऽर्कपुत्रस्य ॥ खाष्टाब्धयो वसुशरवसुपञ्चखचन्द्रक्सुवसुसमुद्राः । द्विनवयमा द्वित्रिगुणाः शरेषु वसवस्त्रि पञ्चरसाः ॥ शशिवेदा मन्दानामर्कादीनां विलोमपातानाम् । वसूरसरुद्रेन्दुगुणा द्वित्रियमाः सप्तरसपञ्चाः ॥ शशियमाशरा गुणरसास्त्रिनन्दवसवः समुद्रवसु विषयाः । चन्द्रादीनां पश्चात् व्रजतोऽश्विन्यादि भगणस्य ॥ BrSpSi. I. 15-21

294 BRAHMAGUPTA'S ASTRONOMY ITS : HIGHLIGHTS | Planet or a body | Bhagaṇas | | Ravi or Sun | 4,320,000,000 | | Budha or Mercury | 4,320,000,000 | | Śukra or Venus | 4,320,000,000 | | Candra or Moon | 57,753,300,000 | | Kuja or Bhauma or Mars | 2,296,828,522 | | Budha-śīghrocca | 17,936,998,984 | | Bṛhaspati or Jupiter | 364,226,455 | | Śukra-śīghrocca | 7,022,389,492 | | Śani or Saturn | 146,567,298 | | Arka or Ravi-mandocca | 480 | | Candra mandocca | 488,105,858 | | Kuja or Bhauma mandocca | 292 | | Budha-mandocca | 332 | | Bṛhaspati or Jīva-mandocca | 855 | | Śukra-mandocca | 653 | | Śani-mandocca | 41 | | Candra-pāta | 232,311,168 | | Kuja or Bhauma-pāta | 267 | | Budha-pāta | 521 | | Bṛhaspati or Guru-pāta | 63 | | Śukra-pāta | 893 | | Śani-pāta | 584 | By pāta is meant the ascending node of a planet's orbit (on the ecliptic). In a kalpa, the number of bha-bhramas (sidereal days) or also known as bha-parivartas is 51,040,000,000. If we subtract out from this number the bhagaṇa of the Sun, we get what is known as ku-dinas or Savana days or the solar or sacrificial days. (51,040,000,000—4,320,000,000=46,720,000,000 Sāvana days or kudinas). In a kalpa, the number of Ravi-bhagaṇas also correspond to the number of solar years (Saura-varṣas), i.e., 4,320,000,000; this number multiplied by 12 gives the number (i.e. 51,840,000,000) of solar months. The difference between the candra-bhagaṇas and the Ravi- bhagaṇas in a kalpa gives the number of lunar months (Cāndra-