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Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t19: Brahmagupta

Bhāratīya Jyotiḥśāstra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t19

Language
English

Brahmagupta

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Date

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Brahamagupta writes in his work, *Brāhmasphuṭasiddhānta*

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श्रीचापवंशतिलके श्रीव्याघ्रमुखे नृपे शकनृपाणां ।। पंचाशत्संयुक्तवर्षशतैः पंचभिरतीतैः ५५० ।। ७ ।। ब्राह्मः स्फुटसिद्धान्तः सज्जनगणितगोलवित्प्रीत्यै ।। त्रिंशद्वर्षेण कृतो जिष्णुसुतब्रह्मगुप्तेन ।। ८ ।।

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From this it seems that Brahmagupta compiled this work in Śaka 550 when King Vyāghramukha of Cāpa dynasty was ruling. His father’s name was Jiṣṇu. Brahmagupta was 30 years old when he wrote Brahma Siddhānta in Śaka 550, which shows that his birth year was Śaka 520.

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∗ Brahmagupta has criticised Varāhamihira for not stating that Rāhu, who envelops the moon while it enters the earth's shadow, was the main cause of the eclipse ; but this is really not a defect ; and in reality even Brahmagupta did not mean to blame him.

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Place

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Brahmagupta was the resident of Bhinamāla. This village is situated on the northern border of Gujarat in South Marwar, 40 miles to the northwest of Abu, between the mount Abu and the river Luni. It is now a small village. Formerly it was known as Bhilamāla or Śrīmāla. It was the birth place of poet Māgha. It was the capital of north Gujarat, when Hiuen-Thsang, the Chinese traveller visited this country in the 7th century A.D. Brahmagupta compiled his Siddhānta during the reign of Vyāghramukha of the Cāpa dynasty and he called himself Bhillamālakācārya*. Some descendant kings of the Cāvadā or Cāpotkaṭa dynasty ruled at Anhilwāda from 756 to 941 A.D. ; and even at present, they are rulers of petty states in north Gujarat. This Cāvadā dynasty must be identical with the Cāpa dynasty referred to by Brahmagupta. Hiuen-Thsang has mentioned Bhilamāla as the Capital of Gujarat about the time of Brahmagupta himself, and the traditional account that Brahmagupta was the resident of Bhilamāla, is still found in Gujarati works on astronomy. This shows that Brahmagupta may be the resident of Bhilamāla**.

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His Works

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The Brāhmasphuṭasiddhānta and the Karaṇa work, named Khaṇḍakhādyaka are his well known works, and in the latter he has adopted Śaka 587 as the epochal year and this shows that he compiled this work at the age of 67. He writes in the Brahma Siddhānta.

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गणितन फलं सिद्धं ब्राहमे ध्यानग्रहे यतोध्याये ।। ध्यानग्रहो द्विसप्तन्यार्याणां न लिखितोत्र मया ।। अध्या. २४.

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" Since the chapter named 'dhyānagraha' gives sure results of mathematics, I have not included the chapter, dhyānagraha, consisting of 72 verses in this (Siddhānta)."

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It therefore shows that he had written a chapter named 'dhyānagraha' consisting of 72 couplets and he had mentioned some results in it. He states that it has not been included in the Siddhānta work, and it is not at present available as an independent work. It is not, therefore, known if it contained any astrological deductions or some results like those of Saṃhitā works ; but the above couplet suggests that it was a very important chapter and meant to be taught secretly to students.

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Other Versions of Brahmasiddhānta

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A couplet is already given on page 4 which states that the science of astronomy is being described in a correct form in it, since, the science, as revealed by Brahmā, has become loose on account of lapse of time. There is a well-known Brahma Siddhānta by Śakalya, which is supposed to have been described by god Brahmā to Nārada ; but it has already been pointed out on page 50 that it was not compiled before Śaka 743, and that it does not resemble the Brahmagupta Siddhānta in respect of the numbers of revolutions and other elements given by Brahmagupta or in other respects. This shows that the two are not related at all. One of the 'Purāṇas' known as " Viṣṇudharmottara Purāṇa †" contains a Brahma Siddhānta. Bhoṭotpal has taken

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*Varuṇa, the commentator of Khaṇḍakhādyaka, has given him some such title as Bhillamālakācārya, and it is found at the end of some of the manuscripts. **Indian Antiquary, XVII, page 192, July 1888. †There is a copy of this in the Deccan College collection.

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in Bṛhatsaṃhitā a number of couplets from Brahmagupta Siddhānta and he has referred to all such places in such words as ‘Brahma Siddhānta’ or ‘so says Brahmagupta’. Nowhere has Utpala referred to these couplets as ‘from Brahma Siddhānta of Śakalya’ or ‘from Brahma Siddhānta included in Viṣṇudharmottara Purāṇa’. This shows that the two Siddhāntas, it existing then at all, were not much known; and according to Utpala, at any rate the Brahmagupta Siddhānta was the one compiled by Brahmagupta. Even Brahmagupta calls his Siddhānta everywhere Brahmasphuṭa Siddhānta or simply Brahmasiddhānta. The author has also called this Siddhānta ‘Brahma Siddhānta’ for convenience in this work.

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Elements

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The numbers of revolutions and other elements mentioned in the Brahma Siddhānta are given below :—

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| | | In a Kalpa which is a period of 4320000000 years | | |
| | | Measures like sāvana days, etc. | Revolutions in longitudes | Revolutions of | |
| | | | | apsides | nodes |
| Revol. of stars | . | 1582236450000 | | | |
| The Sun | . | . | 4320000000 | 480 | |
| Sāvana days | . | . | 1577916450000 | | |
| Moon | . | . | 57753300000 | | |
| Moon's apogee | . | 488105858 | | | |
| Moon's ascending node | . | 232311168 | | | |
| Mars | . | . | 2296828522 | 292 | 267 |
| Mercury | . | . | 17936998984 | 332 | 521 |
| Jupiter | . | . | 364226455 | 855 | 63 |

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| | | In a Kalpa which is a period of 4320000000 years | | |
| | | Measures like sāvana days, etc. | Revolutions in longitude | Revolutions of | |
| | | | | apsides | nodes |
| Venus | . | . | 7022389492 | 653 | 893 |
| Saturn | . | . | 146567298 | 41 | 584 |
| Solar months | . | . | 51840000000 | | |
| Intercalary months | . | 1593300000 | | | |
| Lunar months | . | . | 53433300000 | | |
| Tithis | . | . | 1602999000000 | | |
| Suppressed tithis | . | . | 2508255000 | | |

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| | d | gh | p | vip | pvp |
| Length of the year = | 365 | 15 | 30 | 22 | 30 |

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All the measures in this table are given for a Kalpa. No number can be found which can completely divide the numbers denoting the revolutions of all planets, and hence, according to this Brahma Siddhānta the mean planets cannot all come together at one place, at any moment other than at the beginning of Kalpa. All the planets come together by mean motions at the beginning of Kaliyuga according to the first Ārya Siddhānta or the two Sūrya Siddhāntas, but they are not so conjoined according to the Brahma Siddhānta. This Siddhānta does not, like the modern Sūrya-Siddhānta, assume any period spent over creation. According to this, the moment of commencement of the Kalpa coincides with that of the planets starting to move.

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Length of Year

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The first thing to be remembered is that the length of the year, viz. 365d 15g 30p 22vip 30pvp, as adopted by this Siddhānta, is less than that adopted by any Indian Siddhānta except the Pulīśa and Romaka of the Pañcasiddhāntikā. It has been clearly pointed out in the discussion of the ancient and modern groups of five Siddhāntas that the Pulīśa and Romaka Siddhāntas of Pañcasiddhāntikā were not in use in Brahmagupta's time. The first Ārya Siddhānta and original Sūrya Siddhānta were in use in his time. Of them, Brahmagupta's length of the year is shorter than that of the original Sūrya Siddhānta by 67½ vipalas and by 52½ vipalas than that of the first Ārya Siddhānta. This difference appears to be very small, but because of this (small) variation the moment of the mean sun's entry into Aries in Śaka 540, according to Brahmagupta, occurred 54gh 14½ pals earlier than that according to the first Ārya Siddhānta and 54gh 43⅞ pals earlier than that according to the original Sūrya Siddhānta. What then is the cause of this variation? The cause seems to be only one that he assumed the solar ingress into Aries to have occurred on the day on which the day and night were of equal length, i.e. on the equinoctial day, when the sun rises on the horizon exactly in the east. This moment of the solar ingress is nothing but the entry of the sun in the sāyana (tropical) sign of Aries. The moment of the sun's actual entry into the tropical sign of Aries on or about the day when Brahmagupta might have taken actual observation, coincided with the moment of the same phenomenon as calculated from Brahma Siddhānta. The moment of the true sun's entry into Aries in Śaka 509, as calculated from Brhamagupta Siddhānta comes to be 56gh—40p after mean sunrise at Ujjayinī on Tuesday, the 3rd lunar day of Caitra Śukla, the 18th March 587 A.D. ; and the time of the entry of the sun into sāyana Aries (i.e. sun's longitude being 0ˢ 0° 0') comes to be the same moment on the same day during that year.* Brahmagupta was born in Śaka 520. He might have begun the taking of observations from about Śaka 540. Hence, the calculation, according to Brahmagupta, for the year Śaka 540, shows that the apparent Aries ingress appears to have occured at 57gh—22p (after sunrise) on Saturday, the first lunar day of the dark half of Caitra and the tropical longitude of the sun comes to be 0° 0° 30', which shows that the sāyana Aries ingress took place about 30gh before the moment calculated from Brahma Siddhānta. But the sun's declination increases only by 12 minutes in 30 ghaṭis nearabout the equinoctial time. Hence, in Śaka 540, the sun must have been to the north of the equator by 12 minutes at the moment of the sun's entry into Aries according to Brahma Siddhānta, and if the sun had come to the first point of Aries (Meṣa Saṃkramaṇa) according to Brahmagupta, at sunrise, the sun's centre would have appeared 12 minutes to the north of the east point. But it is not that the equinox always takes place at the time of sunrise. Any one experienced in the taking of observations will easily admit that an error of 12 minutes arc is possible because errors of some minutes can occur in the determination of directions and also the fact that the instruments of observations used to be crude; and these considerations lead me to be convinced that he must have taken the sun's entry into the tropical sign of Aries as the moment of Meṣa Saṃkrānti. He observes in the Siddhānta,

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∗ The sāyana calculations have been made with Keropant's Planetary Tables. As these Tables are not quite accurate, the phenomenon will perhaps be found to occur a year earlier or later. Similarly, the secular equation has not been applied to the sun's place in the above calculation, it would come to about two minutes, which may cause a variation of a year or two.

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यदि भिन्नाः सिद्धान्ता भास्करसंक्रान्त्योपि भेदसमाः ।।
स स्पष्टः पूर्वस्यां विषुवत्यर्कोदयो यस्य ।। ४ ।।

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अ. २४.

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" If Siddhāntas are different, so must the moments of the sun's entry into signs; but when the sun is on the equator, it is actually seen rising exactly in the east."

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The purport of the verse is that the moments of the sun's crossing as seen in the sky will not appear* to be occuring at different moments. This refers to the sun at the time of sunrise on the equinoctial day. It is clearly the place of the tropical sun and Brahmagupta has clearly recorded the place by actual observation. Brahmagupta did not know that the equinox has motion and even if it were known before his time, he undoubtedly did not take it into consideration. Hence, he does not differentiate between the sāyana place of the sun and that found by calculation from his works (i.e. nirayaṇa). He attempted to see that the place calculated by his work will exactly tally with that of the tropical sun; but this was a correct step only so far as his own time was concerned. The reason for this is that though equinox occurred 54 ghaṭis before the calculated time, but yet he could not ignore the traditional belief that the mean sun was at the first point of Aries at the commencement of the Kaliyuga (i.e. at sunrise, on Friday, according to his view). He, therefore, distributed the error of 54 ghaṭis over the period of 3730 years (the period between the commencement of Kaliyuga and the date of compilation of the Brahma Siddhānta ) and he so effected the adjustment that his work should give the moment of equinox to tally with the entry of the sun into the tropical sign of Aries, that is, the moment when the sun actually appeared to rise exactly in the east. This adjustment diminished the length of the year only by a few vipalas. If he had not to encounter the difficulty of distributing the error over the period from the Kaliyuga to his own date, and if he had just considered how much earlier than a particular date the equinox had begun to occur in his time, he would have done one of the two things—accepting, for the length of a year, the measure of the tropical solar year, viz. 365d 14gh 32p, or to retain the traditional length of the year and to assume some motion for the equinox. He compiled the work Khaṇḍakhādyaka 37 years after the Siddhānta and he adopted in it the length of the year given by the original Sūrya Siddhānta. This shows that he must have inclined to adopt some motion for the equinox after retaining the length of the year once adopted or even after being convinced that he must adopt, the actual measure of the tropical year, as the length of the year; he was not bold enough to discard the traditional length of the year and to alter the one already adopted by him while compiling the Siddhānta. Bhāskarācārya has remarked “how is it that proficient astronomers like Brahmagupta have not mentioned the equinox* ?”. It shows that Brahmagupta’s original works make no mention of the equinoctial motion.

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∗ It is because of this kind of disagreement that he remarks " Brahma Siddhānta is the only real siddhānta, while others are more compilations " and often criticises other siddhāntas. The Saṃkrānti according to others, occurs later by one day than according to his work.

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*Sāyana*

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Whether the almanac should follow the sāyana or the nirayaṇa system is at present a matter of controversy. A point favourable to the followers of the sāyana system is noticeable in the above discussion, and it is that it was Brahmagupta’s original view that the sun’s entry into a tropical sign was the actual ‘samkramaṇa’ and it was his desire to alter the length of the year, which he did accordingly. Had he carried on observations throughout his life and compared the results, it was not impossible for a scientist like him to hit upon the correct measure of the tropical year. It may be that he might have come to know it and still was not bold enough to discard the traditional one. The reason for his measure of a year which is shorter than that of others, has not been explained by the author because he was a follower of the sāyana system. Even the staunchest follower of the nirayaṇa system will have to admit its corrections.

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*Correcting the Planetary Elements and Observations*

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The numbers showing the revolutions of planets etc., as given by the Brahma Siddhānta are somewhat different from those of the other Siddhāntas. However, the comparison of mean places of planets in Śaka 421 obtained from the Brahma Siddhānta with those of modern European works (page 63) shows that there is not much difference between them. It shows that Brahmagupta adjusted the numbers of planetary revolutions so that the calculated planetary positions for his time would agree with observations. The result of the comparison of apsides and nodes made on page 68 above also shows his independent research in that direction. Hence, the length of the year, the numbers showing revolutions of planets, nodes and apsides, point to the fact that Brahmagupta was an independent research worker who used to take observations for himself; and this is the main important factor in astronomy. His works reveal at several places the spirit of independence and self-resp expected of such a personality. He says in the Chapter on true places,

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ब्रह्मोक्तमध्यरविशशितदुच्चतत्परिधिभिः स्फुटीकरणं कृत्त्वेवं स्पष्टतिथिदूरभ्रष्टान्यतत्रोक्तैः ।। ३१ ।।

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“The true places obtained from the element like epicycles, aphelia and the mean sun and the mean moon, as mentioned by Brahmasphuṭasiddhānta give the correct tithi; that obtained from other tantras is far from accurate.”

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*See Golabandhādhikāra—commentary on couplets 17-19.

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He maintains here that the tithi calculated from other tantras is far from accurate and that the tithi calculated on the basis of the sun and the moon according to Brahma Siddhānta is correct.

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आर्यभटस्याज्ञानान्मध्यमंदेच्चशीघ्रपरिधीनां ।। न स्पष्टा भौमाद्याः स्पष्टा ब्रह्मोक्तमध्यचैः ।। ३३ ।।

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अ. २.

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“The true places of Mars and other planets calculated from the ignorant Āryabhaṭa’s work giving mean positions of planets, aphelia and epicycles prove to be wrong, they are found correct when calculated from the mean positions and other elements given by Brahma Siddhānta.”

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He maintains that the positions of Mars and other planets are correct when calculated from the aphelia, epicycles and mean planets according to Brahma Siddhānta but are incorrect according to Āryabhaṭa. There are many such instances showing his pride. This pride has, in some cases, run to such excess, that one cannot help feeling that it is little short of arrogance. He has appended in his Siddhānta an independent chapter, named, ‘dūṣaṇādhyāya’ (No. 11) consisting of 63 couplets. Some of the charges levelled against Āryabhaṭa in it show sheer obstinacy on his part.

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Subject matter of Brahma Siddhānta

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He has incorporated in the first ten chapters of his Siddhānta, the chapters which are usually found occurring in almost all the Siddhāntas and been enumerated in the ‘Introduction of this book’. But he has treated many more subjects in the next 14 chapters and they are very important. One of them is the chapter of criticism. One deals with arithmetic and another with algebra. Yet another describes instruments. Most of the remaining chapters are devoted to the theory underlying the subjects dealt with in the first half. The 12th chapter is devoted to arithmetic, mensuration etc. the 56 couplets of which, include almost all the questions dealt with in Bhāskarācārya’s ‘Līlāvatī’. The 18th chapter mainly treats of algebra, and contains 102 couplets, but nowhere does it actually mention the word algebra. The chapter is entitled ‘Kuṭṭaka’. It contains a number of subjects found in Bhāskarācārya’s ‘Bījagaṇita’ (algebra). It contains a chapter headed ‘Kuṭṭaka’ and it is given, mainly for being used in the calculation of mean places of planets etc. in astronomy. The Brahma Siddhānta has 24 chapters and contains 1008 Couplets.

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Commentary

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The Deccan college collection has a copy of the commentary by PRTHŪDAKA on the first ten chapters. Colebrooke has recorded that he had obtained the complete commentary. Colebrooke translated the portions dealing with arithmetic and algebra from the Brahma Siddhānta into English in 1817.

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Interpolation and Yogas

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Brahmagupta has mentioned the number of verses at the end of every chapter. He seems to have taken this precaution, because he knew from experience that changes are often made later in the original. In spite of this precaution there appears to be a discrepancy in respect of a few couplets in the work. Three couplets are actually found in the book without any commentary but they are not found at all in the commentary by Pṛthūdaka. One of the three couplets in the chapter on true places is particularly noteworthy since it deals with Viṣkambha and other yogas. It describes the method of calculating yogas, but it is not found in the annotated edition. One is, therefore, inclined to believe that Viṣkambha and other 26 yogas which form a part of the present almanac, did not exist in Brahmagupta's time, that is to say Vyātīpāta, Vaidhṛti and other yogas did not exist in his time. They are not given even in the Pañcasiddhāntikā. This point will be discussed elaborately in the study of almanacs.

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Khaṇḍakhādya

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It is now proposed to review briefly his work ‘Khaṇḍakhādya’. The name ‘Khaṇḍakhādya’ is strange and the object of giving such a name is not known. This has two parts Pūrva (first) and Uttara (second). The first part consists of 9 chapters which contain 194 couplets. The second par consists of 5 chapters comprising of 71 couplets. Brahmagupta observes in part one, at the very outset;

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वक्ष्यामि खण्डखाद्यकमाचार्यार्यभटतुल्यफलं ।। १ ।।

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प्रायेणार्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः ।।

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उद्वाहजातादिषु तत्समफललघुतरवितरतः ।। २ ।।

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“(1) I compile the work, Khaṇḍakhādya, which gives results equivalent to those given by the great scholar, Āryabhaṭa (2) Since it is impossible to carry on every day affairs with Āryabhaṭa’s work, this work is being compiled so as to give easily equally accurate results relating to matters like birth, marriage and the like.”

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In these verses he states that he is compiling a Karaṇa work, the calculations from which give equally correct results, or in other words which would give places of planets similar to those obtainable from the Āryabhaṭa’s work which is impossible to use in day-to-day life. The Khaṇḍakhādya has adopted the length of the year (365ᵈ—15ᵍ—31ᵖ—30ᵘᵖ) given by the original Sūrya Siddhānta and not that of the Ārya-Siddhānta ; and hence, he had to assume the beginning of the yuga at midnight and not at sunrise as assumed by him in his own Siddhānta, or as in the Ārya Siddhānta. The epoch in Khaṇḍakhādya is Śaka 587, and the apparent first lunar day of the light half of Vaiśākha (according to the ‘amānta’ system) falls on Sunday in that year. The epochal positions in it are true for the midnight of Saturday, the new moon day of amānta Caitra i.e. for the midnight preceding Sunday; and the ahargaṇa is to be calculated from that moment. The mean Aries Ingress according to the original Sūrya Siddhānta falls at 12ˢ—9ᵖ* on the same Saturday. The epochal positions given are as follows :—

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| | ˢ | ° | ′ | ″ | | ˢ | ° | ′ | ″ |
| Sun | 0 | 0 | 32 | 22 | Mercury | 9 | 0 | 44 | 49 |
| Moon | 0 | 9 | 9 | 43 | Jupiter | 6 | 4 | 25 | 16 |
| Moon's apogee | 10 | 8 | 28 | 9 | Venus | 10 | 0 | 19 | 14 |
| Rāhu | 0 | 18 | 47 | 23 | Saturn | 9 | 6 | 41 | 16 |
| Mars | 3 | 10 | 13 | 6 | | | | | |

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If the places of planets, true for the midnight of Saturday, the Caitra Amāvasyā of Śaka 587, be calculated on the basis of the numbers of revolutions and other elements given before on page 23—from the original Sūrya Siddhānta, all of them except the moon's apogee and node, are found exactly to agree with the above positions. They do not agree with those calculated from the Ārya Siddhānta. It appears from this that the Khaṇḍakhādya-Karaṇa agrees with the original Sūrya Siddhānta with respect to all items like the length of the year, the initial moment for computing ahargaṇa and almost all the epochal positions. The revolutions of the moon's nodes are not met with in the original Sūrya Siddhānta. The place of moon's apogee does not agree with that of the original Sūrya Siddhānta, but it does not also agree with those of the Ārya Siddhānta or the Brahma Siddhānta. The moon's node too does not agree with either of the last two works. As the length of the year and the initial moment of the year adopted for Khaṇḍakhādya were different from those adopted in the Brahma Siddhānta, it is clear that it was no use adopting for Khaṇḍakhādya the moon's apogee and node adopted in the Brahma Siddhānta. It is true that the Khaṇḍakhādya does not agree with the Āryabhaṭa Siddhānta completely ; still, as some of the elements in the Āryabhaṭīya were equal to those in the original Sūrya Siddhānta, the mean positions of planets calculated from Khaṇḍakhādya for Śaka 587, almost resembled those from the Ārya Siddhānta.

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Brahmagupta, observes in the very beginning in the latter portions of the Khaṇḍakhādya, that he would describe the method of finding apparent places of planets because those calculated from Āryabhaṭa's work did not agree with observation. On this, Varuṇa, the commentator, remarks, “Brahmagupta has declared that he was going to compile a work as good as that of Āryabhaṭa, and he did the same in the first half of the work. In the second half he has mentioned an equation from his own Siddhānta to ensure results agreeing with observation. Now only those things which have not been mentioned therein, should be accepted from Āryabhaṭa's Karaṇa work”. This remark and other chapters in the second half show that he has made only such changes while compiling the Khaṇḍakhādya as would give accurate results comparable by observation. He has borrowed the following important items from Āryabhaṭa's work ; the length of the year, mean motions of planets, their epochal positions and the moment of the beginning of yuga. The above remark of Varuṇa and other things show that Āryabhaṭa's work, referred here, is not his Siddhānta now available, but this Karaṇa work.

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∗ It has been mentioned in the account of Varāhamihira that an epoch wherein the moment of mean Aries Ingress nearly coincided with the moment of the new moon was selected as a convenient moment by the Pañcasiddhāntikā also. The two resemble each other in other respects also.

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It is really strange that instead of compiling a Karaṇa work equal in merit to his own Siddhānta, he porposed to compile a work and for the most part did compile one equal in merit to that of a staunch rival like Āryabhaṭa on whom he had showered a shower of criticism. There are two reasons for this, one of them must be the fect that Āryabhaṭa’s work might have been so popular that he was unable to ignore it and the second reason was that in Śaka 587 when he compiled the Khaṇḍakhādya, the moment of Aries Ingress according to his own Siddhānta occurred 55gh—36g pals before that of the original Sūrya Siddhānta and 54gh—55g pals before that of Āryabhaṭīya. And because of this much difference, the two works would show different intercalary months. The difference in intercalary month and the occurrence of the saṃkrānti a day earlier, are things easily noticeable even by an ignorant person; and these created an unfavourable public opinion with regard to the introduction of the mesure of his own Siddhānta. These two reasons, it appears, could not make him bold enough to compile a ‘karaṇa’ which would be equal in merit to his own Siddhānta, when Brahmagupta could not introdus his own year-measure, because his saṃkrānti differed by about less than a day, it is worth considering how difficult is would be to bring into use either Keropant’s almanac, whose saṃkrānti occurs about 4 days ealier than that of the old works or the sāyna almanac, whose saṃkrānti takes place about 22 days earlier

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Commenmentaries on Khaṇḍakhādya

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Varuṇa and Bhaṭotpala have written commenmentaries on Khaṇḍakhādya. Pṛthūdaka too is likely to have written one, but it is nowhere available. One more incomplete commentary has been found which does not mention the name of the commentator; but he appears to be one from Kashmir as can be seen from the Śaka year 1564 adopted for solving examples, and from the differencess refer to Kashmir. The Deccan college collection has got a copy of the work entitled Pañcāṅga Kautuka (No. 537 of 1875-76 A.D.) which gives tables and methods of calculating figures for almanacs very easily. It has adopted Śaka 1580 as the epoch and the whole calculation has been made with the help of Khaṇḍakhādya. It does not mention that it is compiled in Kashmir; but it was found in Kashmir and it was made use of the popular local era prevalent in Kashmir. It clearly shows that the author belonged to Kashmir. It also shows without doubt that the Karaṇa, Khaṇḍakhādya, was in use in Kasmir till Śaka 1580; and from the three above mentioned commentaries on Khaṇḍakhādya and from the fact that the copies of the Pañcāṅga Kautuka’ be still in use in that province. Bhāskarācārya has refered to Khaṇḍakhādya. Albiruni, (Śaka 950) had obtained the Khaṇḍakhādya and quoted from-it.

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Spread of Brahma Siddhānta

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The fact that Brahmagupta complied Khaṇḍakhādya as a work different from his own Siddhānta shows that he was not sure that he would get any follower for the Siddhānta; and it is natural, as can be seen from following well known remark of Kālidāsa.

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आपरितोषाद्विदुषां न साधु मन्य...विज्ञानं ॥

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“No performance should be regarded as excellent, until it satisfies the learned”.

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He compiled the Khaṇḍakhādya at the age of 67. His Siddhānta might not have come into actual use till then and he must have compiled Khaṇḍakhādya after being disappointed to see that he had no followers even when so many years of his age had elapsed. Very few of the great researchers are fortunate enough to see their researches bear fruit in their own lifetime. It is not very creditable to him that instead of leaving the future to decide for itself, he disregarded his own thesis; but is it possible that the best work of such a master mind would fail to satisfy scholars? The astronomer Bhāskarācārya who was as great as he (Brahmagupta) has in fact recognized his work. Also two more Karaṇa works compiled before Bhāskarācārya are found to have followed the Brahma Siddhanta. All of them have mentioned one additionnl correction to the plantary places calculated from Brohma Siddhānta. This correction is first noticed in the “Rāja Mṛgāṅka Karaṇa’ compiled in Śaka 964. But the author came across an example in which the Brahma Siddhānta was found to have been used in its original form without any correction.

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Original Form

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The Uttarapurāṇa, a Jain work by Guṇabhadra found in the Poona college collection (No. 289 of 1883-84), states the date of its compilation in the followiag lines :—

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शकनृपकालाम्यंतरविंशत्यधिकाष्टशत ८२० मिताब्दान्ते ।।
मंगलमहार्यकारणी पिंगलनामनि समस्तजनसुखदे ।। ३५ ।।
श्रीपंचम्यां बुधार्त्युजि दिवसवरे मंत्रिवारे सुधाशौ
पूर्वायां सिंहलग्ने धनुषि धरणिजे वृश्चिकाकौ तुलागौ ।।
सूये शुक्रे कुलीरे गविच सुरगुरू

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“(This work was completed) on Thursday the 5th lunar day, in the auspicious year of Pingala saṃvatsara, numbering 820 of the Śaka era when the planets were occupying the signs mentioned herein.”

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The planetary positions dercribed in the verses were as follows :

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| Sun | —Occupying ‘Kulīra’ (Cancer) | Jupiter | in Gau (Taurus) |
| Moon | — ,, Pūrvā (Bhādrapadā) | Venus | ,, Kulīra (Cancer) |
| Mars | — ,, Dhanuḥ (Sagittarius) | Saturn (Arki) | ,, Scorpio |
| Mercury | — ,, Ārdrā nakṣatra | Rāhu (Agu) | ,, Libra |

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Śaka year 819 (elapsed) was known ar Pingala. The elapsed year 819 is the same as the current year 820. This at first gives rise to the doubt whether 819 should be taken for calculation or 820. Similarly, the verse does not mention any month or a half-month, but simply the tithi. The day is no doubt given as “mantri” vāra. It is conjectured that it may either be a Thursday or even a Friday. But the verse mentions the positions of all planets, and the best way to determine the vāra is to find out that day for which these planetary positions would be found to be simultaneonsly true. After attempting the calculations for a number of days in the two years, Śaka 819 and 820, the author has found Amānta Āṣādha Kṛṣṇa 5, Śaka 818 elapsed i.e. Thursday the 23rd June, 897 A.D. as the day on which the given positions of planets appear to be true for the period between the sunrise and about 24 ghatis after it.

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The verse gives Leo as the rising sign, whose duration on that day was the period between 4—9 ghaṭis after sunrise. This planetary condition is impossible for any day other than this in these two years. The moon's position mentioned in the verse, is not found to be true for any other day being earlier or later by one day. The object of mentioning this here is that these planetary positions agree only if the length of the year given by Brahmagupta in his Brahma Siddhānta be accepted and by that of no other Siddhānta. The sun, according to the Sūrya Siddhānta, appears to belong to Gemini on Thursday, the 5th lunar day of Āṣāḍha (Kṛṣṇa), and goes in to Cancer at about 5 ghaṭis after sunrise, on Friday. By no other Siddhāntas does it appear to be occupying Cancer on Thursday. The saṃkrānti, according to Brahma Siddhānta, appears to occur 61gh 31pal earlier than the saṃkrānti of the modern Sūrya Siddhānta in that Śaka year. Similarly, even Mars appears to be occupying the sign of Capricorn on Thursday according to the modern Sūrya Siddhānta and that of Sagittarius according to the Brahma Siddhānta. In short, the planetary positions are seen to agree quite well according to Brahma Siddhānta and even the consideration of the matter from several points of view leaves no doubt* about it. It proves beyond doubt that the Brahma Siddhānta was in use in its original form in Śaka 819. This old work was compiled in the Deccan when the King Akālavarṣa of Rāṣṭrakūṭa dynasty was ruling the Deccan. From this it appears that the Brahma Siddhānta was in use in its original form in the Deccan in Śaka 819. The corrections in it have been introduced by some one else later on.

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Corrections

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Varuṇa's commentary on the Brahma Siddhānta appeared about Śaka 962. It does not refer to any corrections. The work Rājamṛgāṅkakaraṇa was compiled in Śaka 964 and it mentions the correction. It appears to have been first introduced then. The corrections include that for even the sun. This correction has changed his length of the year from the original Brahma Siddhānta viz. from 365d—15gh—30pal—22vip—30bvp to 365d—15gh—31pal—17vks that is, it is greater than that of the first Āryabhaṭa by about 2 vipalas. The worry of the Brahmapakṣa compiled after this date are found to be in conformed with the corrected Brahma Siddhānta. The Rājamṛgāṅka compiled in Śaka 964 is the first of such Karaṇa works. The second one is the Karaṇa work named ‘Karaṇa Kamalamārtaṇḍa’ compiled in Śaka 980. The next one, compiled after this in Śaka 1105 is the Karaṇakutūhala of Bhāskarācārya. The Mahādevī Sāraṇī, a work on planetary calculations, compiled in Śaka 1238 and the two works Kheṭakasiddhi and Candrārkī of the astronomer Dinakara and compiled in Śaka 1500, conform to the corrected Brahma Siddhānta. Of them, the Karaṇakutūhala is still in use in some places. The author of Grahalāghava has borrowed some positions of planets as in accordance with the Brahmapakṣa, and these have been taken from the Karaṇakutūhala. The Brahma Siddhānta might have remained in use in its original form up to Śaka 1000 at the most. It may have gone out of everyday use after Bhāskarācārya. Not only that, but because Bhāskarācārya's Siddhānta Śiromaṇi could serve the purpose as efficiently as the Brahma Siddhānta, it appears that Brahma Siddhānta itself might have gone out of use gradually. The quotations from he Brahma Siddhānta are rarely found in works compiled after Bhāskarā- cārya. The work Brahma Siddhānta is at present not found anywhere in Mahārāṣṭra, and the same may be true of other provinces as well.

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∗ The original verse as given in this book is very incorrect. This verse and the one corrected by me along with its explanation, may be seen on page 429—30 of Prof. Bhandarkar's Report on the search for Sanskrit manuscripts for the year 1883—84.

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Condition of Astronomy

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There is no harm in saying, on the whole, that all the branches of the system which go to make the science of astronomy in our country, appear to have been completely established in the time of Brahmagupta. The necessary variations in the positions of planets were made from time to time later on. It can safely be said that no special reform or research was afterwards made in the system except that of the equinoctial motion. It has already been pointed out above that Brahmagupta was an independent thinker as far as the revolutions of planets, the aphelia and nodes were concerned. The elements concerning the calculations of the true places of planets appear to be his own. Even in ‘tripraśnādhikāra’ (Adhikāra on three problems) he appears to show a greater skill than earlier writers. He has described the instruments of observation and it is my opinion that the “turīyayantra” (the Quadrant instrument) was his invention. The subject of Algebra is not found in any of the earlier works, which shows that he may probably be its originator. Sūryaḍāsa, the son of Jñyānarāja, the author of the work, Siddhānta-Sundara, wrote commentary in Śaka 1460 on Bhāskarācāryas Algebra. He regards Āryabhaṭa as the oldest writer on Algebra. The work of Āryabhaṭa I may be said to contain no discussion on Algebra; but that of Āryabhaṭa II does contain it. But it will be shown later that this work is more modern than that of Brahmagupta. Hence the information available at present leads one to conclude that Brahmagupta was the first writer on Algebra. He has not recorded in his work any boastful remarks in the chapter on Bījagaṇita that it was he who discovered the subject anew. From this it can be conjectured that the subject might have been known even before his time. But books are not available. Brahmagupta, was on the whole, a very ingenious research worker. Even a scholarlike Bhāskarācārya has praised him thus “May the work of jiṣṇuja (i.e. the son of Jiṣṇu, Brahmagupta) who is the supreme mathematician, succeed”. Similarly, at another place, he remarks, “when after a long lapse of time, a great deal of discrepancy will be caused, men of genius possessing the ability of Brahmagupta, will come to birth, and studying the planetary motions evolved by Brahmagupta, will compile work on these sciencec”. It is quite proper that he has been acclaimed as “a great discoverer of (correct) positions and motions of planets and a very intelligent author of scientific works.”

Cites (17)