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

RATIONAL INSCRIBED QUADRILATERALS 267 circumscribed circles will be expressible in integers. Such quadrilaterals we shall call as Brahmagupta Quadrilaterals. The solution of this formidable problem has been given by Brahmagupta as follows : The upright and bases of two right-angled triangles being reciprocally multiplied by the diagonals of the other will give the sides of a quadrilateral of unequal sides : ( of these ) the greatest is the base, the least is the face, and the other two sides are the two flanks.¹ Taking Brahmagupta's integral solution, the sides of the two right triangles of reference are given by : (1) m²-n², 2 mn, m²+n²; (ii) p²-q², 2 pq, p²+q²; where m, n, p, q are integers. Then the sides of the Brah- magupta Quadrilateral are. (m²-n²)(p²+q²), (p²-q²)(m²+n²), 2mn(p²+q²), 2pq(m²+n²) (Arrangement A) Pṛthūdaka Svāmī has- illustrated the rational inscribed quadrilateral by taking an example of the right angle triangles. (i) (3,4,5) (m²-n²=3, m²+n²=5, whence m=2, n=1) (ii) (5,12,13)(p²-q²=5, p²+q²=13, whence p=3, q=2) Fig. 22 Substituting these values in the above equations, we get the sides of the quardilateral as ( 39, 25, 52 and 60).²

1. जात्यद्वय कोटिभुजाः परकर्णगुणा भुजाश्चतुर्विषमे । अधिको भूर्मुखंहीनो बाहुद्वितयं भुजावन्यौ ॥ —BrSpSi. XII. 38 2 The diagonals of this quadrilateral are given by Bhāskara II as 56 (=3.12+4.5) and 63 (=4.12+3.5). (Cont. on page 268)

270 BRAHMAGUPTA AS AN ALGEBRAIST Put in other words, this means that one has to solve the following equations : (i) 5x—25 = y² (ii) 10x—100 = y² (iii) 83x—7635 = y² Pṛthūdaka Svāmī, the commentator on the Brāhmasphuṭa- siddhānta proceeds to solve these equations as follows : (1.1) Suppose y = 10; then x = 125. Or put y = 5; then x = 10. (2.1) Suppose y = 10; then x = 20. (3.1) Assume y = 1; then x = 92. He then remarks that by virtue of the multiplicity of suppositions there will be an infinitude of solutions in every case, But no method has been given either by Brahmagupta or his commentator to obtain the general solution. Double Equations of the First Degree Perhaps we have the earliest reference of the simultaneous indeterminate quadratic equations of the type x ± a = u² x ± b = v² in the Bhakaśālī Manuscript (Folio 59, recto). Brahmagupta gives the solution of such simultaneous inde- terminate quadratic equations of a general case as follows : The difference of the two numbers by the addition or subtration of which another number becomes a sq- uare, is divided by an optional number and then incre- ased or decreased by it. The square of half the result diminished or increased by the greater or smaller (of the given number) is the number (required).¹ Expressed in the language of algebra, shall have : = ½ { ½ ( (a - b)/m ± m ) }² ∓ a

1. याभ्यां कृतिरधिको नस्तदन्तरं हृत युतो न मिष्टेन । तद्दल कृतिरधिकोऽधिकयो रविको न यो राशिः ॥ —BrSpSi. XVIII. 74

DOUBLE FIRST DEGREE EQUATIONS 271 or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b where m is an arbitrary number. Datta and Singh has given the rationale of this method as follows : u² = x ± a; v² = x ± b, From them, we have u² - v² = ± a ∓ b Therefore u - v = m and u + v = (± a ∓ b) / m , where m is arbitrary. Hence u = ½ ( (± a ∓ b) / m + m ) = ± ½ ( (a - b) / m ± m ) Sincc it is obviously immaterial whether u is taken as posi- tive or negative, we have u = ½ ( (a - b) / m ± m ) Similarly v = ½ ( (a - b) / m ∓ m ) Therefore x = { ½ ( (a - b) / m ± m ) }² ∓ a, or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b, where m is an arbitrary number. Now we shall take up another particular case, for which Brahmagupta has given a rule : The sum of the two numbers tne addition and sub- traction of which make another number (severally) a square, is divided by an optional number and then diminished by that optional number. The square of half the remainder increased by the subtractive number is the number (required)¹. In the algebraic notations, we shall express it as follows :

1. यैरूनो यैश्च युतो रूपैर्वर्गस्तद्द्वयमिष्ट हृतम् । इष्टोनं तद्दल कृतिरूनाऽभ्यधिका भवति राशिः ॥ —BrSpSi. XVIII. 71

274 BRAHMAGUPTA AS AN ALGEBRAIST y = 1/a ((ad + bc)/m + b) if b > c and m > (ad + bc)/m. If these conditions be reversed then x and y will have their values interchanged. Datta and Singh have given the following rationale of these solutions : axy = bx + cy + d, or a²xy − abx − acy = ad, or (ax − c) (ay − b) = ad + bc. Suppose ax − c = m, a rational number; then ay − b = (ad + bc)/m. Therefore x = 1/a (m + c) y = 1/a ((ad + bc)/m + b) Or, we may put ay − b = m; in that case, we shall have ax − c = (ad + bc)/m; whence x = 1/a ((ad + bc)/m + c), y = 1/a (m + b). Brahmagupta’s own rule. Whilst the rule given above is ascribed to an unknown author, Brahmagupta’s own rule for the solution of a quadratic indeterminate equation involving a factum is as follows : With the exception of an optional unknown, assume arbitrary values for the rest of the unknowns, the product of which forms the factum. The sum of the product of these (assumed values) and the (respective) coefficients of the unknowns will be absolute quantities. The continued products of the assumed values and of the coefficient of the factum will be the coefficient of the optionally (left out) unknown. Thus the solution

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