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

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 83 Planet BrSpSi Āryabhaṭīya Khaṇḍa- Varāha- Later or khādyaka Sūrya- modern siddhānta Sūrya- (PSi.) siddhānta Moon 57,753,300 57,753,336 57,753,336 57,753,336 57,753,336 Sun 4,320,000 4,320,000 4,320,000 4,320,000 3,320,000 Mars 2,296,828.522 2,296,824 2,296,824 2,296,824 2,296,832 Jupiter 364,226.455 364,224 364,220 364,220 364,220 Saturn 146,567.298 146,564 146,564 146,564 146,568 Moon's apogee — 488,219 488,219 488,219 488,203 Venus 7,022,389.492 7,022,338 7,022,388 7,022,388 7,022,376 Mercury 17,936,998.984 17,937,020 17937,000 17,937,000 17,937.060 Moon's nodes 232,311.168 232,226 232,226 232,226 232,238 TABLE II Longitudes of the apogees of the orbits of Planets. Āryabhaṭīya Khaṇḍa- Varāha Modern Planets khādyaka Sūrya-sid- Sūrya-siddh- dhānta. ānta Sun 78° 80° 80° 77° 07′ Mercury 210° 220° 220° 220° 26′ Venus 90° 80° 80° 79° 49′ Mars 118° 110° 110° 130° 00′ Jupiter 180° 160° 160° 171° 16′ Saturn 236° 240° 240° 236° 37′ TABLE III Dimensions of the epicycles of Apsis Planets Ārybhatīya Khaṇḍa- Varāha. Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta Sun 13°-30′ 14° 14° 13½°-14° Moon 31°-30′ 31° 31° 31½°-32° Mercury 22½°-31½° 28° 28° 28°-30° Venus 9°-18° 14° 14° 11°-12° Mars 63°-81° 70° 70° 72°-75° Jupiter 31½-36½° 32° 32° 32°-33° Saturn 40½-58½ 60° 60° 48°-49°

84 BRĀHMASPHUṬA SIDDHĀNTA & KHAṆḌAKHĀDYAKA Table IV Dimensions of the Śīghra epicycles (i.e. conjunctions) Planet Āryabhaṭīya Khaṇḍa- Varāha Modern Khādyaka Sūrya- Sūrya- siddhānta siddhānta Saturn 36½°- 40° 40° 40° 39°- 40° Jupiter 67½°- 72° 72° 72° 70°- 72° Mars 229½°-239½° 234° 234° 232°-235° Venus 256½°-265½° 260° 260° 262°-262° Mercury 130½°-139½° 132° 132° 132°-133° Table V Longitudes of the nodes of the orbits of planets Planets Āryabhaṭīya Khaṇḍa- Varāha Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta siddhānta Saturn 40° 40° Not stated Have to be Jupiter 20° 20° in the calculated Mars 80° 80° Text from the Venus 60° 60° data of the Mercury 100° 100° text Table VI Orbital inclinations (geocentric) to the ecliptic Planets Āryabhaṭīya Khaṇḍa- Varāha Modern khādyaka Sūrya- Sūrya- siddhānta siddhānta Mars 90' 90' 10' 90' Mercury 120' 120' 135' 120' Jupiter 60' 60' 101' 60' Venus 120' 120' 101' 120' Saturn 120' 120' 135' 100' The Mahābhāskarīya of Bhāskara I (522 A. D.) contains a passage which corroborates the fact that Āryabhaṭa I was the author of both the audayika and the ardharātrika systems of Indian Astronomy. According to Pṛthūdakasvāmin, whose

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 85 commentary on the Brāhmasphuṭasiddhānta we have the privilege of presenting to the public, it is clear that in certain respects Bhāskara and others may be wrong but the Āryabhaṭa's authenticity cannot be questioned. Pṛthūdakasvāmin while commenting on the Brāhmasphuṭasiddhānta, XI. 26 writes: Such a mistake may have been made by Bhāskara and others; they have not understood his (Āryabhaṭa's) intention. The passage in the Mahābhāskarīya giving constants of the ardharātrika system runs as follows (we are giving the translation from Kripa Shankar Shukla's edition on the Mahābhāskarīya) : The astronomical processes which have been set forth above come under the sunrise day - reckoning (audayika system). In the midnight day-reckoning (ardharātrika system) too, all this is found to occur: the difference that exists is being stated (below). ¹ The next fourteen stanzas relate to the midnight day- reckoning of Āryabhaṭa I. (i) Civil days and omitted lunar days in a yuga and revolution numbers of Mercury and Jupiter are thus given : (To get the corresponding elements of the midnight day-reckoning) add 300 to the number of civil days (in a yuga) and subtract the same (number) from the number of omitted lunar days (in a yuga); and from the revolution numbers of (the śīghrocca of) Mercury and Jupiter subtract 20 and 4 respectively. ² Thus according to the midnight day-reckoning, we get civil days in a yuga = 1,577,917,800 omitted lunar days in a yuga = 25,082,280 revolution number of the śīghrocca of Mercury = 17,937,000 revolution number of Jupiter = 364,220

1. निबन्धः कर्मणां प्रोक्तो योऽसावौदयिको विधिः । अर्धरात्रे त्वयं सर्वो यो विशेषः स कथ्यते ॥ —MBh. VII. 21
2. त्रिशती भूर्दिने क्षेप्या ह्यवमेभ्यो विशोध्यते । ज्ञ गुर्वोर्भगणेभ्योऽपि विंशतिश्च ततोऽब्धयः ॥ —MBh. VII. 22

86 BRAHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA (ii) Diameters of the Earth, the Sun and the Moon are thus given : (In the midnight day-reckoning) the diameter of the Earth is ( stated to be ) 1,600 yojanas; of the Sun 6,480 (yojanas) and of the Moon, 480 (yojanas)¹. (iii) Mean distances of the Sun and the Moon are as follows : The (mean) distance of the Sun is stated to be 689,358 (yojanas), and of the Moon 51,566 (yojanas).² (iv) Longitudes of the apogees of the planets are as follows: 160, 80, 240, 110, and 220 are in degrees the longitudes of the apogees of Jupiter, Venus, Saturn, Mars and Mercury respectively.³ (v) Manda and Śīghra epicycles of the planets are as follows : The Manda epicycles ( of the same planets ) are 32, 14, 60, 70, and 28 (degrees) respectively; and the Śīghra epicycles are 72, 260, 40, 234, and 132 ( degrees ) respectively. The Sun's apogee and epicycle are the same as those of Venus (i. e. 80° and 14° respectively). The Moon's epicycle in the midnight day-reckoning is stated to be 31 (degrees). ⁴ (vi) The positions of the so called manda and śīghra pātas of the planets are given below: ( The following directions for) the degrees of the (manda and śīghra) pātas of the planets as devised

1. अष्टिशतगुणा व्यासो योजनानां भुवो रवेः । खाष्टाब्ध्यक्ष्यङ्गानि शीतांशोः शून्यवस्वब्धयस्तथा ॥ —23
2. वस्विन्द्रिय गुणच्छिद्रवस्वङ्गानि विभावसोः । अङ्गाङ्गेष्वेक भूतानि चन्द्रकर्णः प्रकीर्तितः ॥ —24
3. अष्टिरष्टौ जिनारुद्रा विंशतिर्द्वयधिकाः क्रमात् । दशघ्ना गुरुशुकार्कि भौमज्ञांशाः स्वमन्दजाः ॥ —25
4. मन्दवृत्तानि द्वात्रिंशन्मनवः षष्टिरेव च । खाद्रयो वसुदस्त्राः स्युः शीघ्रवृत्तान्यथ क्रमात् ॥ —26 द्वयद्वयः खाङ्कनेत्राणि खाब्धयोऽब्ध्यग्निदस्रकाः । द्वयग्नीन्दवो रवेर्मन्दं शुक्रवद् वृत्तमेव च ॥ —27 एकत्रिंशत्क्षपाभर्तु र्धरात्रे विधीयते । —MBh. VII 26-28 (i)

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 87 (under the midnight day-reckoning) should be noted carefully by learned scholars : Add 180° to the longitudes of the mandoccas (apogees) apd śīghroccas of Mercury and Venus, and subtract 3 signs from the mandoccas (apogees) and śīghroccas of the remaining planets. Then are obtained the longitudes of the manda and śīghra pātas of the planets. (Also) add 2 degrees to the longitudes of the manda pātas and śīghroccas of Venus, Saturn and Jupiter, and 1½ degrees to those of Mars and Mercury. (It should be noted that) the śīghra pātas have been stated for all the planets excepting Mercury. (Mercury does not have a śīghra pāta).¹ That is to say, the longitudes of the manda pātas of Mars, Mercury, Jupiter, Venus, and Saturn are 21.5, 41.5, 72, 262 and 152 degrees respectively; and the longitudes of the śīghra pātas of Mars, Jupiter, Venus and Saturn are (śīghrocca--88.5°), (śīghrocca —88°), (śīghrocca+182°) and (śīghrocca--88°) respectively. (vii) A rule for finding the celestial latitude of a planet is as follows : (From the longitude of a planet severally) subtract the longitudes of its (manda and śīghra) pātas and there- from calculate (as usual) the corresponding celestial latitude of that planet. Add them or take their differ- ence according as they are of like or unlike directions. Then is obtained the true celestial latitude of that particular planet. The true celestial latitude of any other planet is also obtained in the same way. The remaining (astronomical) determinations are the same as stated before. This all is in brief the difference of the other tantra (embodying the midnight day-reckon-

1. पातभागाश्च विज्ञेयाः पण्डितैः परिकल्पिताः ॥ —28 मन्द शीघ्रोच्चयोः क्षेप्यं चक्रार्धं बुधशुक्रयोः । राशित्रयं तु शेषाणां पात्यते पातसिद्धये ॥ —29 शुक्राऽर्किदेव पूज्यानां भागौ द्वावेव संयुतौ । मन्दपाताच्च शीघ्रोच्चात् सार्धांशस्तु कुजज्ञयोः ॥ —30 विबुधानां च सर्वेषां शीघ्रपाताः प्रकीर्तिताः । MBh. VII. 28 (ii) 31 (i)

88 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA ing of Āryabhaṭa I).¹ (viii) A rule for finding the longitude of the true mean planet according to midnight-day reckoning is as follows : Apply half the Śīghraphala and (then) half the mandaphala to the longitude of the planet's own mandocca (reversely). From the resulting longitude of the planet's mandocca cal- culate the (mandaphala and apply it to mean longitude of the planet : the resulting longitude of planet is stated to be) the true-mean longitude of the planet. This is stated to be another difference (of the midnight day-reckoning) ² (ix) Length of the circle of the sky and derivation of the lengths of the orbits of the planets are given as follows : Multiply the revolutions of the Moon (in a yuga) by 3, 240,000 and then discard the zero in the unit's place : (this is the length of the circle of the sky in terms of yojanas). (Severally) divide that by the revolutions of the planets (in a yuga): thus are obtained the lengths of the orbits of the respective planets in terms of yojanas.³ From these stanzas (from 20-35), it is evident that one yojana of the sunrise day-reckoning is one and a half times that of the midnight day-reckoning. Now from stanza 22, it appears that 300 is to be added to the number of civil days in a Mahāyuga. According to the Āryabhaṭīya. the number of civil days in this cycle is 1,577,917, 500, which increased by 300, becomes 1,577,917,800, the number of

1. शोधयित्वा क्रमात् पातान् विक्षेपार्शान् प्रसाधयेत् । —31 योगविश्लेषणोत्पत्तिरेकानेकस्वदिग्वशात् । विक्षेपः स स्फुटो ज्ञेयो ग्रहस्यैकस्य कीर्तितः ॥ —32 अन्यस्याप्येवमेव स्याच्छेषाः प्रागुक्त कल्पनाः । एतत्सर्वं समासेन तन्त्रान्तरमुदाहृतम् ॥ —MBh. VII. 31 (ii)–33
2. शीघ्रमन्दोच्चचापाधैसंस्कृता त् स्वीयमन्दतः । स्फुटमध्यग्रहः स धै विशेषः परिकीर्तितः ॥ —MBh. VII. 34
3. वेदाश्विरामगुणितान्य युताहतानि । चन्द्रस्य शून्यरहितान्यथ मण्डलानि ॥ र : स्वैर्हृतानि भगणैः क्रमश्रो ग्रहाणां । कक्ष्या भवन्ति खलु योजनमण्ड्रस्था ॥ —MBh. VII. 35

civil days in a Mahāyuga according to Brahmagupta as referred to in the Khaṇḍakhādyaka. Again, the same stanza tells us to subtract 20 and 4 respec- tively from the revolutions of Mercury and Jupiter, and we arrive at the figures 17,937,000 and 364, 220, which are the revolutions of Mercury and Jupiter in a Mahāyuga according to Brahma- gupta as given in the Khaṇḍakhādyaka. Again we can compare the figures for the diameters of the Earth, the Sun and the Moon given in the Mahābhāskarīya, in the present day Sūrya-siddhānta and the Āryabhaṭīya. Diameter Mahābhāskarīya Modern. Āryabhaṭīya of Sūrya-siddhānta Earth 1,600 yojanas 1,600 yojanas 1,050, yojanas Sun 6,480 6,500 4,410 Moon 480 480 315 Then in stanza 24, we are given the distances of the Sun and the Moon as 689, 358 and 51, 566 yojanas respectively. The same figures are worked out by Lalla according to the Āryabhaṭīya and quoted in the Śiṣyadhwṛddhida (IV.3.4) and they come to be 459,585 and 34,377. The stanza 25 states the longitudes of the aphelia of planets these figures tally with the corresponding figures given by Brah- magupta in the Khaṇḍakhādyaka : Longitude of aphelion of Jupiter 160°, of Venus 80°, of Saturn 240°, of Mars 110° and of Mercury 220°. Similarly the stanza 26 gives the peripheries of planets’ epicycle of apsis, which also is in concordance with the values given by Brahmagupta : Periphery of epicycle of apsis of Jupiter 32°, of Venus 14°, of Saturn 60°, of Mars 70° and of Merury 28°. In the stanza 27 of the Mahābhāskarīya, we have the dimensions of the epicycles of conjunction for planets; these figures are also the same as given by Brahmagupta : Epicycles of conjuction for Jupiter 72°, for Venus 260°, for Saturn 40°, for Mars 234° and for Mercury 132°. In the stanza 28, we have the Sun’s epicycle having a periphery of 14° and the Moon’s epicycle 31° ; the longitudes of

90 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA the nodes of the planets to be the same as in the Āryabhaṭīya. All these are the same as given by Brahmagupta in the Khaṇḍa- khādyaka. In the stanza 33 we have rules for finding the geocentric longitudes of planets which may be taken to be the same as in the Khaṇḍakhādyaka¹; compare these values with those in the Sūryasiddhānta of Varāhamihira in the Pañcasiddhāntikā, XVII. 6, but slight different from the Āryabhaṭīya.² The last stanza of the Mahābhāskarīya (35) gives the dimen- sions in yojanas of the orbits of planets ; these are the same as in the modern Sūryasiddhānta. Thus we find a great semblance in the constants as given by the Mahābhāskarīya of Bhāskara I, of the Sūryasiddhānta as given by the Pañcasiddhāntikā and understood by Varāhamihira, and also the constants as given by Brahmagupta in his treatises, specially the Khaṇḍakhādyaka. It must not be forgotten that the same Āryabhaṭa I who is the celeberated author of the Āryabhaṭīya is also the author of another treatise very often referred to as the Tantra. I shall quote Prabodha Chandra Sengupta in connection with these similarities, and the great influence of Āryabhaṭa on Indian Astronomy. He writes in his Introduction to the Khaṇḍakhādyaka as follows : We have shown that there is much resemblance in the constants between the Sūryasiddhānta of Varāha and the Khaṇḍakhādyaka and for the matter of that with the Tantrāntara of Āryabhaṭa I. In my papers “Ārya- bhaṭa and Āryabhaṭa's Lost Work”, I have establish- ed the fact that the Sūryasiddhānta as it existed before the time of Varāha, was made more accurate by him by borrowing the constants from Āryabhaṭa's ārdharātrika system. That there was a Sūryasiddhānta १. शीघ्रफलाद्धं मध्ये मन्दफलाद्धं च मन्दशीघ्रफले । सकले मध्ये स्पष्टः शीघ्रं मध्योनकं केन्द्रम् ॥ —KK. II. 18 २. मन्दोच्चाच्छीघ्रोच्चादर्धमृणधनं ग्रहेषु मन्द्येषु । मन्दोच्चात्स्फुट मध्याच्छीघ्रोच्चाच्च स्फुटा ज्ञेयाः ॥ शीघ्रोच्चादर्धोनं कर्तव्यमृणं धनं स्वमन्दोच्चे । स्फुटमध्यौ तु भृगुबुधौ सिद्धान्मन्दात्स्फुटौ भवतः ॥ —Arya. III. 23-24

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 91 before the time of Varāha, is seen from Section 6 of the Table on page xii given before. This point is made clear from another consideration, viz., the star table in the modern Sūryasiddhānta, which unmistakably points to the conclusion that the longitudes of some stars, e.g., Spica etc., correspond to a time much ante- rior to that of Āryabhaṭa I. The great fame of Ārya- bhaṭa I induced Varāha, the first maker of a neo-Sūrya- siddhānta to use the elements of Āryabhaṭa's ardha- rātrika system to supplant the older materials in it. No wonder, therefore, that there is an opinion in favour of the hypothesis that Āryabhaṭa I was the author of the Sūryasiddhānta. If there were a shadow of truth in it, Varāha would have admitted it. Albe- runi indeed says that the Sūryasiddhānta was compos- ed by Lāṭa (Alberuni's India, translated by Sachau, Vol. I.p. 153). We now know that this Lāṭa or Lāṭa- deva was one of the first pupils of Āryabhaṭa I. He was the expounder of the Romaka and Paulīśa Siddhāntas as we learn from Varāhamihira's Pañcasiddhāntikā, (I.3). As Alberuni's statement is not corroborated by Varāha, we are not inclined to take it as correct. None of the earlier writers suggest that the Sūryasiddhānta was in any way modified or changed by Ārya- bhaṭa I. It has now been established beyond doubt that the same Āryabhaṭa was the author of the Aryabhaṭīya and another Tantra which is now lost. There is reason in support of hypothesis that this Tantra itself was the first work of Āryabhaṭa I and that the Aryabhaṭīya was the second work from the order in which Varāha mentions them in the Stanza quoted earlier. If this hypothesis be true, the stanza in the Aryabhaṭīya¹, which was translated by me as : “Now when sixty times sixty years and three quarter yugas also have elapsed, twenty increased by three years have elapsed since my birth.”

1. षष्ट्यब्दानां षष्टिर्यदा व्यतीतास्त्रयश्च युगपादाः । व्यधिका विंशतिरब्दास्तदेह मम जन्मनोऽतीताः ॥ —Ārya. III. 10.

92 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA should now be translated thus : "In this Mahāyuga when sixty times sixty years and three quarter yugas also had passed, twenty increased by three years had elapsed since my birth." Now Bhāskara I the author of the Mahābhāskarīya and the Lagubhāskarīya, wrote a commentary on the Āryabhaṭīya. The author commenting on this stanza observes that : "Or this was addressed by Āryabhaṭa when expound- ing the science to Pāṇḍuraṅgasvāmin, Lāṭadeva Niḥśaṅku and other pupils."¹ This direct pupil of Āryabhaṭa I also says that this stanza does not show that the Āryabhaṭīya was composed when Ārya- bhaṭa I was only 23 years old, but refers to the time when he probably began his career as a teacher of Astronomy. Senagupta out of his discussion concludes that we are not justified in accepting that the Āryabhaṭīya was composed when Āryabhaṭa was only 23 years of age. This treatise as it exists in the present form must have been the composition of a mature age; it is a treatise highly finished in form; the date mentioned in this great work refers to a date when its author became a reputed guru or teacher. Alberuni and Brahmagupta Dr. E.C. Sachau in his translation of Alberuni's India (vol. II, p. 304) speaks of Brahmagupta in the following words : Brahmagupta holds a remarkable place in the history of Eastern civilization. It was he who taught the Arabs astronomy before they became acquainted with Ptolemy; for the famous Sindhind of Arabian litera- ture, frequently mentioned but not yet brought to light, is a translation of his Brahmasiddhānta; and the only other book on Indian astronomy, called Atarkand, which they knew, was a translation of his Khaṇḍa- khādyaka. Brahmagupta, the celebrated author of the Brāhmasphuṭa- siddhānta, has another great work as we have said before to his credit which goes by the name Khaṇḍakhādyaka. This has

1. एतदेवाचार्य्यार्य्यभटस्य शास्त्रव्याख्यान समये वा पाण्डुरङ्गस्वामिलाटदेवनिःशङ्कुप्रभृतिभ्यः प्रोवाच ।

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 93 already been said that perhaps to meet the popular demand, Brahmagupta in this treatise took upon himself the task of sim- plifying Āryabhaṭa's ardharātrika system or the system of midnight day reckoning. Alberuni, the author of the Indika has made several references or quotations from the Khaṇḍakhādyaka proper and also its supplement, known as Uttara-Khaṇḍa- khādyaka. (a) There is a reference to the accepted circumference of the Earth, as given in the Khaṇḍakhādyaka (Sachau's Alberuni, Vol. I, p. 312) Multiply the difference in longitude (from Ujjayinī) by the (mean) daily motion of a planet (in minutes) and divide by 4,800; apply the quotient taken as minutes negatively in places east of the meridian line of Ujja- yinī and positively in places lying west.¹ (b) The rules for finding the ahargaṇa as given in the Khaṇḍakhādyaka in I. 3-5 (Sachau's Alberuni, Vol. II, 46-47), to which Dr. Schram adds a valuable anno- tation, the constants being taken from the later Pauliśa Tantra as known to Bhaṭṭotpala. This Pauliśa astronomy is derived from Āryabhaṭa I's ardharātrika system.² (c) A quotation from the Uttara Khaṇḍakhādyaka (Sachau's Alberuni, Vol. II, pp. 84-86) which Sengupta has given in his translation, Chapter X, pp. 148-152. (d) A quotation also probably from the Uttara Khaṇḍakhā- dyaka (Sachau's Alberuni, Vol. II, p. 87). These stanzas are found in the Brāhmasphuṭasiddhānta, XIV, 47-52, also quoted by Bhaṭṭotpala as occurring in the Brahma Siddhānta in his commentary on the Bṛhat-Saṁhitā, IV, 7. The manuscripts which Sengupta used did not show them as occurring in the Uttara Khaṇḍakhādyaka. These relate to the dimensions of the nakṣatras as seen, as distinguished from the same as calculated.

1. उज्जयिनी-याम्योत्तर-रेखायाः प्रागृणं धनं पश्चात् । देशान्तर भुक्तिवधात् ख खाष्टवेदैः कलाद्याप्तम् ॥

• KK. I, 15.

2. • KK, Pt. B. Misra's edition, p. 145.

94 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA (e) Two quotations from the Uttara Khaṇḍakhādyaka rel- ating to the celestial co-ordinates of Canopus and Sirius (Sachau's Alberuni, Vol. II, p. 91). Present manus- cripts do not show these stanzas, which are probably the same as stanzas 35-36 and 40 of Chapter X of the Brāhmasphuṭasiddhānta. (f) Two quotations from the Khaṇḍakhādyaka proper as alleged by Alberuni (Sachau's Alberuni, Vol.II, p. 116). According to Āmarāja, the first is a couple of stanzas of which the author is Bhaṭṭotpala and not Brahmagupa. The second quotation cannot be traced. These relate to finding the possibility of an eclipse whether of the Sun or of the Moon. (g) Two quotations from the Khaṇḍakhādyaka proper as asserted by Alberuni (Sachau's Alberuni, Vol. II. p. 119). These relate to finding the Lords of the year and of the month. According to Āmarāja the rules in question were given by Bhaṭṭotpala and not by Brah- magupta. Pṛthūdaka in his commentary on the first chapter at its concluding portion says ; “In this work the Khaṇḍakhādyaka, the teacher (Brahmagupta) has not given the rules for finding the Lords of the year and the month¹.”

1. अथाऽत्र खण्डखाद्यके वर्षाधिपमासाधिपा नयनमाचार्येण नाभिहितम् । —: ० :— Reference P.C. Sengupta : The Khaṇḍakhādyaka, 1934. K.S. Shukla : Mahābhāskarīya, 1960. K.S. Shukla : Sūrya-Siddhānta, 1957.

Chapter IV Brahmagupta’s Originality in the Khaṇḍakhādyaka Sengupta in his Indroduction to the Commentary of the Khaṇḍakhādyaka has discussed this point. We shall reproduce here some of the points mentioned by him. Brahmagupta’s Khaṇḍakhādyaka (i) Brahmagupta does not accept the system of Āryabhaṭa but has simplified it in the Khaṇḍakhādyaka proper ; and here he has given the system which he thinks to be correct. Uttar Khaṇḍa Khādyaka (ii) In the Uttara-Khaṇḍakhādyaka, he has further correc- ted some of his results, given earlier in the Khaṇḍakhādyaka proper. In the proper Khaṇḍakhādyaka, Brahmagupta assignes to the longitude of the Sun's apogee the value 80º, whereas in the Uttara text he corrects it to 77º (UKK. 4) : As the process of finding the apparent places of planets as given by Āryabhaṭa does not make them agree with observation, I shall, therefore, speak of this process. Of the Sun the apogee is at two signs and seventeen degrees (2 signs 17º=60 plus 17 degrees=77º).¹ Compare this with the value given in the Khaṇḍakhādyaka proper (I.13)² : The longitude of the Sun's apogee is 80º [KK.I,13] (The Sun's apogee is 80º or two signs plus 20 degrees) inocco means

1. न स्फुटमार्य्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये । भानुमतो मन्दोच्चं राशिद्वयमंशकाश्च सप्तदश ॥ UKK. IX 4
2. भागाशीतिरितिनोच्चं शशिनः पादोनकृत शरकृतोनाद् । भगणादि द्विस्त्रिर्दैर्वसुशुनव यम नव गुणैः सकलम् ॥ KK. I, 13

96 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA mandocca of the Sun). The value given in the Pañcasiddhān- tikā, IX.7-8) is also the same. Let us compare it with the present value. According to the astronomical constants as given in the Conn. des Temps., the longitude of the Sun's apogee in 499 A.D. (i.e. 1,400 years before 1900 A.D.) was —77° 19′19.44″ according to Conn.des Temps’ equation. —76° 40′37.22″ according to Newcomb's equation. The mean of these two values is very nearly 77⁰ as given by Brahmagupta in the Uttara text. Thus the value given by Brahmagupta is more correct than the value given by Āryabhaṭa. The Āryabhaṭīya gives the value 78⁰ which is less correct. Brahmagupta more correct than Āryabhaṭa (iii) Brahmagupta detected that Āryabhaṭa had made the Moon's apogee quicker and nodes slower, than they really are. In both the cases, Brahmagupta made rather an over-correction. We shall give the extract from Uttara-Khaṇḍakhādyaka in this connection : Multiply the ahargaṇa by 110, increase the product by 511 and divide by 30, 31; subtract the result taken as revolutions, etc., from the mean Moon; the final result is the Moon's apogee.¹ Evidently Brahmagupta assumes that the anomalistic month = 3031/110 days. This convergent to the anomalistic month was known to the author of the Vasiṣṭha Siddhānta as summarised in the Pañcasiddhāntikā² (II-2-6). According to Brahmagupta, the length of the anomalistic month = 1582236450000—4320000000 days. (BrSpSi. I.15,16,18, ───────────────────────── and 20) 577533000000—488105858 = 27.55454641 days which is for 1900 A. D. = 27.5545502 days according to Radau. = 27.554602 according to the Āryabhaṭīya.

1. द्युगणात् ख रुद्र गुणिताद् भवशरयुक्ताच्छशित्रिखाग्नि हृतात् । भगणादि फलं शोध्यं घसचन्द्राच्छशाङ्कोच्चम् ॥ (UKK. IX. 5)

UTTARAKHAṆḌA KHĀDYAKA 97 Here also Brahmagupta is more accurate. Again, the length of the sidereal period of the Moon's apogee = 1577918450000 / 488105858 days = 3232.732048 days. Āryabhaṭa's value of the same is 3231.987844 days, and the modern value is 3232.3754 days. Hence Brahmagupta's result is by 0.3566 of a day out, while Āryabhaṭa's is by 0.3876 of a day in. Further in the Uttara-Khaṇḍakhādyaka(IX.10) we have : Deduct 354½ from the ahargaṇa, divide the remainder by 6792 ; subtract the quotient that is obtained in revolutions etc. from the circle : the result is the longitude of the ascending node.¹ (IX.10) Here Brahmagupta gives the approximate period of the sidereal revolution of the Moon's node to be=6792 days. This according to his Brāhmasphuṭasiddhānta = 1577916450000 / 232311168 days = 6792.25396 days, which according to Lockyer would be 6793.39108 days and according to the Khaṇḍakhā

98 UTTARAKHAṆḌA KHĀDYAKA one-sixteenth.¹ This stanza says that in 499 A.D., Mars's aphelion point had a longitude of 127⁰ ; of Jupiter the longitude of the aphelion was 170⁰. ( KK.II. 6²) According to Newcomb's rule, the longitude of the aphelion point of Mars in 499 A. D. works out to have been = 128⁰28'12''. According to the Conn. des. Temps' rule, the same was 128⁰27'51''. Hence Brahmagupta's determination of Mars's aphelion is correct within 1⁰30' and is therefore, quite satisfactory. According to the Khaṇḍakhādyaka proper it was 110⁰, and according to the Āryabhaṭīya 118⁰. Of planets, beginning with Mars, the degrees of longitude of the apogees are respectively 11, 22, 16, 8 and 24, each multiplied by 10. (KK .II. 6) Thus the longitudes of apogees of Mars = 110 (3 signs 20⁰); of Mercury = 220⁰ (7 signs 10⁰) ; of Jupiter = 160⁰ (5 signs 10⁰); of Venus = 80⁰ (2 signs 20⁰) and of Saturn = 240⁰ (8 signs). Compare these values with those given in the Pañcasiddhāntikā XVII. 2 (the Sūrya-siddhānta). Again according to this stanza Jupiter's aphelion had a longitude of 170⁰ in 499 A.D. According to Conn. des. Temps' rule the same was 170⁰25'. Thus here too, Brahmagupta is very accurate. According to the Khaṇḍakhādyaka proper, Jupiter's aphelion had a longitude of 160⁰ (KK. II.6) and according to the Āryabhaṭīya, the value was 180⁰. Brahmagupta First to Use Second Differences All these illustrations reproduced here very well establish the point that the great Indian astronomers from Āryabhaṭa I to Brahmagupta were aware of the methods of separating the two distinct planetary inequalities, viz., that of the apsis and of conjunction in the cases of the five ‘star’ planets (PSi. Intro- duction Lii). In the Khaṇḍakhādyaka, Brahmagupta having given the “sines” and the equations of the Sun and the Moon

1. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः । सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।। पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ।। —UKK. IX. 11
2. मन्दांशा दशगुणिता रुद्रा द्वियमाश्व षोडशाष्टजिनाः । —KK. II. 6

BRAHMAGUPTA FIRST TO USE SECOND DIFFERENCES 99 at the interval of 15° of arc of the mean anomaly, in the Uttara Khaṇḍakhādyaka teaches, for the first time in the history of mathematics, the improved rules for interpolation by using the second difference. This very important feature I am reproducing here from the translation by Senagupta of the verse ¹ (UKK. 8) : Multiply the residual arc left after division by 900' (i.e by 15°), by half the difference of the tabular difference passed over and that to be passed over and divide by 900' (i. e. 15°) : by the result increase or decrease, as the case may be, half the sum of the same two tabular differences ; the result which, whether less or greater than the tabular difference to be passed, is the true tabular difference to be passed over. (UKK. 8) The rule given here applies to the case of all functions hitherto considered in the Khaṇḍakhādyaka, which are tabulated at the difference of 15° of arc of the argument. They are : (i) the tabular differences of the Sun's equation, (ii) the tabular differences of the Moon's equation. (iii) the tabular differences of the 'sines'. Sengupta has illustrated the rule by an example belonging to the table of sines. Illustration—To find the 'sine' of 57°. Brahmagupta's table of sines in the Khaṇḍakhādyaka is as follows : Thirty increased severally by nine, six and one; twenty- four, fifteen and five, are the tabular differences of sines at intervals of half-a-sign. For any arc, the 'sine' is the sum of the parts passed over, increased by the proportional part of the tabular difference to be passed over.² (*KK.*I.30 ; also III.6)

1. गतभोग्य खण्डकान्तरदलविकलवधात् शतैर्नवभिराप्त्या । तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥ —UKK. IX. 8
2. त्रिंशत् सन्नवरसेन्दुजिन (तिथि) विषयागृहार्द्धं चापानां । भर्द्ध्ज्या खण्डानि ज्या भुक्तैक्यं सभोग्यफलम् ॥ —KK. I. 30; also III. 6

100 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA This can be shown in the tabular form thus :

Now 57° = 3420 minutes = 900' × 3 + 720'. Thus three of the tabular differences are considered as passed over ; the last one being 31 and the one to be passed over is 24. The true tabular difference by the rule, for arc 57°, = (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 Hence the 'sine' of 57° = 39 + 36 + 31 + (720 / 900) [ (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 ] = 125.76 As worked out from the logarithm tables the same comes out to be 125.80. Again 'sine' of 57° from Brahmagupta's formula = 106 + (720 / 900) × 24 + ((31 - 24) / 2) × (720 / 900) - [(720 / 900)] × (31 - 24) / 2 = 106 + (720 / 900) × 24 + (720 / 900) { (720 / 900) - 1 } × (24 - 31) / 2 This in fact is the modern form the interpolation equation up to the term containing the second difference. Brahmagupta thus takes a decidedly improved step here and is undoubtedly the first man in the history of mathematics who has done this. One should also remember that in the case where the function is not tabula- ted at a constant interval, Brahmagupta's rule is remarkable.

BRAHMAGUPTA AND SINE RULE 101 Brahmagupta First to Introduce Sine Rule in Indian Plane Trigonometry In this connection, we shall reproduce the following verse from the Khaṇḍakhādyaka : Multiply the ‘sine’ of the (Śīghra) anomaly by the ‘sine’ of the maximum Śīghra equation and divide by the ‘sine’ of the corresponding Śīghra equation, the result is the ‘Śīghra hypotenuse’ when the (Śīghra) anomaly is half a circle, this śīghra hypotenuse is equal to the radius diminished by the ‘sine’ of the maximum equation ; when the anomaly is equal to the whole circle, the same is equal to the radius increased by the same ‘sine’ of the maximum equation.¹ Let S, E and P be the positions of the Sun, the Earth and the planet, say Mars, respecti- vely. Complete the paralle- logram SEMP ; with M as centre and MP as radius, describe a circle. This circle is the epicycle of conjunction of Mars. Produce EM to cut this circle at K. The ∠PMK = ∠S′SP (the point S′ is on [Fig. 4] ES produced ), the angle gained by the Earth over Mars since the preceding conjunction. The ∠PMK is called the śīghra anomaly or anomaly of conjunc- tion. We take EM=360, and MP=234. The ∠PEM, which is equal to ∠PES, the annual parallax of Mars, is called the śīghra equation. The ∠MPE is equal to the ∠SEP, the elongation. The ∠PMK is given, and PM and ME are also given. Hence in the triangle MPE, we have tan ½ (P−E) = (EM−MP)/(EM+MP) tan ½ PMK = 126/594 tan ½ PMK ∴ L tan ½(P−E) = log [126/594] + L tan ½ PMK We have also ½ (P+E) = ½ ∠PMK

1. केन्द्रज्याऽन्त्यफलज्या गुणिता फलजीवया हृताकर्णः । त्रिज्यास्य फलज्योना चक्रार्द्धे संयुता चक्रे ॥ KK. VI. 1.

102 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA Now log [ 126 / 594 ] = 1.3265841. The values of the ∠PMK and the ∠PEM and Brahma- gupta's values as given in the verse¹ are presented below in a tabular form : | ∠PMK = | 28° | 60° | 90° | 121° | 135° | 148° | 164° | 173° | | ∠PEM = | 10°58′ | 23°1′ | 33°1′ | 39°56′ | 40°23′ | 37°31′ | 25°32′ | 12°35 | | Brahmagupta's ∠PEM = | 11° | 23° | 33° | 40° | 40°30′ | 37°30′ | 25°30′ | 12°30′ | It will be seen that Brahmagupta gives the values of the equation within 1/8th of a degree. It seems inexplicable why such discrepancies should remain in Brahmagupta's calculations. It is probable that he wanted to state his equations to the nearest half a degree. Now we shall take up the Śīghra equations of Mars, and then revert to the Sine Rule. We have in the Khaṇḍakhā- dyaka : Mars, by the degrees of Śīghra anomaly (i.e. anomaly of conjunction) of 28 getting at the corresponding equation of 11° rises (heliacally) in the east ; by the next 32° gets 12° more of the equation ; by the next 30°, 10° more ; by the next 31°, 7°, more ; by next 14°, half a degree ; these are positive ; by the next 13°, nega- tive 3° ; by the next 16° ; negative 12° after this he is retrograde ; by the next 9°, negative 13° ; by the next 7°, negative 12½°. After this the parts of the equations occur in the reverse order¹. On the basis of this we have the following table of the Śīghra equations for Mars : | Degrees of anomaly | Equation of | Phenomena | | of conjunction | conjunction | | | 0° | 0° | Motion direct. | | 28° | +11° | Rises in the east. |

1. भौमोऽष्ट्यग्नौ रुद्रान् भुक्त्वा पूर्व्वौदितोऽदरैर्कान् । ख_णैर्दशरूपगुणैः सप्तांशा मनुभिरर्द्धांशान् ॥ धनम् 'मग्नि शशाङ्कैस्त्रिनष्ट्या भास्करात्नतो वक्री । नवभिश्च योदशनगैर्द्वादशसार्द्धान् विलोमोऽतः ॥ KK.II.8-9