Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t24: Āryabhaṭa II (About Śaka 875)
Bhāratīya Jyotiḥśāstra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t24
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Āryabhaṭa II (About Śaka 875)
v2.c12.s2.t24.1
His Work
v2.c12.s2.t24.2
There is another Ārya Siddhānta in addition to the siddhānta of Āryabhaṭa described before. There is a copy of the work, kept in the Deccan College Collection, but it is entitled as Laghu-Ārya Siddhānta. But the author himself calls it neither ‘bṛhat’ (extensive) nor ‘laghu’ (short). In the very first verse he observes.
v2.c12.s2.t24.3
∗ Colebrooke's Miscellaneous Essays pp. 442, 450, 470.
v2.c12.s2.t24.4
विविधखगामपटीकुट्टकबीजादिष्टशास्त्रेण ।।
आर्यभटेन क्रियते सिद्धांतो रुचिर आर्याभि ।। १ ।।
v2.c12.s2.t24.5
“This beautiful Siddhānta has been compiled in the ‘Ārya’ metre by Āryabhaṭa who has studied various sciences on planetary motions, elementary mathematics, problems in arithmetic, and algebra”.
v2.c12.s2.t24.6
In this, he calls his work a Siddhānta. This author is more modern than the earlier Āryabhaṭa and I have called him as Āryabhaṭa II and his work, Second Ārya Siddhānta, because it is convenient to do so.
v2.c12.s2.t24.7
His Date
v2.c12.s2.t24.8
He has not mentioned his date. He has given in his Siddhānta the mean places mentioned by another Siddhānta known as Parāśara Siddhānta. He describes both these siddhāntas,
v2.c12.s2.t24.9
एतत्सिद्धांतद्वयमीष्यांतै कलौ युगे जातं ।। २ ।।
v2.c12.s2.t24.10
meaning “compiled when a small part of Kaliyuga had elapsed”. In this verse he intends to show to the world that the two Siddhāntas were compiled very soon after the Kaliyuga had started. But I am quite sure that he lived after Brahmagupta, because, even though he maintains that his Siddhānta was compiled very soon after the beginning of Kaliyuga, he includes himself among the authors of “pauruṣa” (human) works. There is no other proof to show that the length of year or other measures adopted by him were in use before Brahmagupta, and all criticisms levelled by the latter against Āryabhaṭa apply to the First Ārya Siddhānta and not at all to this. No subject dealt with in this Siddhānta has been referred to by Brahmagupta. Had this Siddhānta existed in his time, Brahmagupta would rot have failed to criticise it in some respect or the other. The Pañcasiddhāntikā does not appear to have mentioned the equinoctial motion. It is not found in the works of Āryabhaṭa I, Brahmagupta or Lalla. But it is given in this Ārya Siddhānta and the authors appear to have attempted to remove the blemishes for which Brahmagupta has criticised Āryabhaṭa I. His work describes the Yuga-system, and the Kalpa begins on Sunday and Brahmagupta has criticised Āryabhaṭa I that his work recommends the planetary calculation, from the beginning of the Yuga when only the ‘mean planets’ are said to come together and not the ‘true planets’ (see 46th couplet, Chap. 2). But according to this Āryabhaṭa’s work, the true positions of all planets are given to be together at the beginning of Creation. All these facts have convinced the author that he lived after Brahmagupta i.e. after Śaka 587. This is the farthest limit of his date. As regards the nearest limit, Bhāskarācārya has quoted him (in his work). In the 65th verse of the chapter on true places, in Siddhānta Śiromaṇi, he observes “Āryabhaṭa and others have mentioned the rising of the ‘dṛkkāṇa’ i.e. the third part of a sign or 10°, to ensure accuracy”. Āryabhaṭa I has mentioned the duration of ascendants in terms of arcs of 30° and not of 10°; but Āryabhaṭa II has mentioned the durations for the ascension of ‘dṛkkāṇas’ that is arcs of 10°; in couplets 38 to 40 in the 4th chapter, and such a mention of the duration of ‘dṛkkāṇa’ is not at present found in any work except in that of Āryabhaṭa II. This shows that Āryabhaṭa referred to by Bhāskarācārya in the above line is not the first Āryabhaṭa, but the second. From this it is clear that Āryabhaṭa II must have lived before Śaka 1072. He has described the method of calculating ayanāṃśa; but the equinoctial motion as calculated from it is not found always to be constant, but to increase or decrease much (more discussion about this will be made in the study of the motion of equinoxes). But there is no harm in saying that the equinoctial motion is always constant; the variation in it is exceedingly small. The modern Sūrya Siddhānta gives a constant motion for it; but the date of its compilation is not definitely known. The work, Rājamṛgāṅka (Śaka 964) has adopted a constant motion for the equinox for all times. No definite proof of the measures adopted (by) earlier (writers) is at present available. From this, Āryabhaṭa II appears to have lived before the equinoctial motion was correctly known. The Bhaṭotpala's commentary (Śaka 888) quotes from several works, but not from the second Ārya Siddhānta. It shows that even if Āryabhaṭa II lived before Bhaṭotpala, he must have preceded him only by a few years.
v2.c12.s2.t24.11
The time when the ayanāṃśas, obtained from the Second Ārya Siddhānta, would be equal to the sun's tropical longitude at the true vernal equinox, comes to be about Śaka 900. If he had lived before this year, the date must have been only a few years earlier.
v2.c12.s2.t24.12
From all these considerations, he seems to have lived about the Śaka 875. It has already been shown before (page 33) that the date of his Siddhānta and that of Parāśara, as found by Bentley, are incorrect.
v2.c12.s2.t24.13
Description of his Work
v2.c12.s2.t24.14
His work consists of 18 chapters, which contain 625 couplets. The first 13 chapters deal with all the subjects usually discussed in different chapters in karaṇa works. The 14th chapter deals with the celestial sphere and problems concerning it. The 15th chapter consisting of 120 couplets is devoted to Pāṭīgaṇita (i.e. arithmetic and mensuration); and it contains almost all the questions dealt with in Bhāskarācārya's Līlāvatī. The 16th chapter is devoted to bhuvanakośa (Universe) i.e. the description of the three worlds. The 17th Chapter gives a theory of the mean motions of Planets, and the 18th chapter deals with algebra, and particulary the ‘kuṭṭaka’ problems in it. It gives some special information not given by Brahmagupta.
v2.c12.s2.t24.15
Numerical Code
v2.c12.s2.t24.16
He has adopted the usual conventional code to denote numbers in ‘Pāṭīgaṇita’ (i.e. arithmetic and mensuration) only; otherwise he has used, everywhere else, the letters of the alphabet to denote numbers. These letter values are different from those used by Āryabhaṭa I. They are :—
v2.c12.s2.t24.17
| Consonants | Numbers denoted | Consonants | Numbers denoted |
| Ka, ṭa, pa, ya | 1 | ca, ṭa, ṣa | 6 |
| kha, ṭha, pha, ra | 2 | cha, ṭha, sa | 7 |
| ga, ḍa, ba, la | 3 | ja, ḍa, ha | 8 |
| gha, ḍha, bha, ba | 4 | jha, ḍha | 9 |
| ṅa, ṇa, ma, śa | 5 | ña, ṇa | 0 |
v2.c12.s2.t24.18
While denoting numbers by letters, Āryabhaṭa I has not abandoned the general rule that’, “digits are written from right to left” but this Āryabhaṭa has differed from it, and has adopted the system of writing the digits from left to right. Example : According to this system the word ‘ghaḍapha’ denotes 432.
v2.c12.s2.t24.19
It has been pointed out in the account of Āryabhaṭa I how confusion is caused by adopting his system of code letters. The same remark applies to this Āryabhaṭa also.
v2.c12.s2.t24.20
Below are given the numbers of revolutions and other elements etc. in one Kalpa as given by his Siddhānta as well as by that of Parāśara.
v2.c12.s2.t24.21
| | Second Ārya Siddhānta | Parāśara Siddhānta |
| Years spent over creation | 3024000 In one Kalpa | 0 In one Kalpa |
| Revol. of stars . . . . . | 1582237542000 | 1582237570000 |
| Revol. of Sun . . . . . | 4320000000 | 4320000000 |
| Sāvana days . . . . . . | 1577917542000 | 1577917570000 |
| Moon’s revolutions . . . | 57753334000 | 57753334515 |
| Moon’s apogee : revolutions . | 488108674 | 488104634 |
| Moon’s node : revolutions . | 232313354 | 232313235 |
| Mars . . . . . . | 2296831000 | 2296833037 |
| Mercury . . . . . . | 17937054671 | 17937055474 |
| Jupiter . . . . . . | 364221682 | 364219955 |
| Venus . . . . . . | 7022371432 | 7022372148 |
| Saturn . . . . . . | 146569000 | 146571813 |
| Solar months . . . . . | 51840000000 | 51840000000 |
| Intercalary months . . . | 1593334000 | 1593334515 |
| Lunar months . . . . . | 53433334000 | 53433334515 |
| Tithis . . . . . . | 1603000020000 | 1603000035450 |
| Suppressed days . . . . | 25082478000 | 25082465450 |
v2.c12.s2.t24.22
| Planets | Revolutions of apsides in a Kalpa | Revolutions of nodes in a Kalpa |
| | Second Ārya Siddhānta | Parāśara Siddhānta | Second Ārya Siddhānta | Parāśara Siddhānta |
| Sun . . . | 461 | 480 | | |
| Mars . . . | 299 | 327 | 298 | 245 |
| Mercury . . | 339 | 356 | 524 | 648 |
| Jupiter . . | 830 | 982 | 96 | 190 |
| Venus . . | 654 | 526 | 947 | 893 |
| Saturn . . | 76 | 54 | 620 | 630 |
v2.c12.s2.t24.23
Length of year, according to Ārya Siddhānta.—365ᵈ—15ᵍʰ—31ᵖᵃˡ—17ᵛⁱᵖ—6ᵈᵛᵖ Length of year, according to Parāśara Siddhānta—365—15—31—18—30.
v2.c12.s2.t24.24
According to Ārya Siddhānta it is assumed that some years have been spent over creation. No such assumption has been made in Parāśara Siddhānta. According to both the Siddhāntas the planets appear to come together not in the beginning of Kaliyuga, but at the beginning of Creation. The length of the year according to both is nearly equal to that of the rectified Brahma Siddhānta. This Āryabhaṭa has mentioned the number of revolutions of the “saptarṣi” stars (Great Bear), on the assumption that they have some motion. But as a matter of fact the Saptarṣis have, practically, no motion.
Cites (8)
- Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
- Sūryasiddhānta — Maya, Sanskrit
- Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
- Āryabhaṭīya — Āryabhaṭa, Sanskrit
- Pañcasiddhāntikā — Varāhamihira, Sanskrit
- Āryasiddhānta — Āryabhaṭa, Sanskrit
- Siddhāntaśiromaṇi — Bhāskara II, Sanskrit
- Rājamṛgāṅka — Bhoja, Sanskrit