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Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2: A : Madhyamādhikāra (Adhikāra on Mean Places)

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

Language
English

Contents

A : Madhyamādhikāra (Adhikāra on Mean Places)

Chapter I

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*History of astronomical Works and Computation of mean places of Planets etc.*

Foreward

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As mentioned in the INTRODUCTION,[*] the author proposes to discuss the History of the science of Astronomy from about 500 years before Śaka era to this day; and in the beginning, this first chapter of the Madhyamādhikāra (i.e. a section on mean places) under the GAṆITA SKANDHA (i.e. a Branch of Mathematics) will deal with the history of astronomical works and the question of computing the mean motions and places of planets.

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The knowledge of astronomy as developed during the Vedic and Vedāṅga Jyotiṣa periods and described in Part I, was considerable as compared with the pace of general progress during the period; but it would appear very meagre, when compared with the ability (developed in a later period) to predict the true positions of planets. It appears that some works might have been compiled during the interval between these periods. There may be some Saṃhitā works of that type; but they are either not available now or have not been seen by him. The period of Siddhāntic astronomy can somehow be linked with the ancient period. This point will be discussed later on; but we have no information as to how the knowledge of astronomy reached the highest stage of calculating the true motions and places of planets, how observations used to be taken and how the motions of planets were finally fixed after comparing the different observations. The oldest of astronomical (Siddhānta) works reveal a sudden rise in the standard of astronomical knowledge. Those who raised the standard of such knowledge through their works were naturally regarded as superhuman, and thus arose the popular belief that the available ancient works on mathematical astronomy are regarded as ‘apauruṣeya’ (i.e. not compiled by any mortal man), and it is clear that this belief has been formed later.

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Because these works were regarded as superhuman, they naturally did not include the description of subjects like observations. There seems to be another very strong reason for this omission. Looking to the conditions of those days when as a rule ‘shorter the works, the easier they were to commit to memory’, such works dealt with only the rules of calculating the motions and places of planets, and they appear to have avoided length by omitting the underlying theory.

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∗ Pulished in part I of the book

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The author proposes to deal with all works on astronomy in chronological order in this chapter on mean places. Their points of differences, if any, with matters pertaining to other Adhikāras from different works, as also some special points worth mentioning, have been treated in subsequent chapters ; otherwise, all information about the works has been given in this very chapter on mean places. Some works are considered divine while some authors have more than one book to their credit ; hence, the following account is arranged under the names of authors, and at places, under the titles of their works.

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The oldest known works on astronomy are the five Siddhāntas—the Sūrya Siddhānta and others. These are regarded as divine. They are again of two kinds. The Pañcasiddhāntikā of Varāhamihira mentions the Saura and other four Siddhāntas ; but at present they are not available. The Pañcasiddhāntikā simply provides clues to their elements. The author calls them the ‘ancient Siddhānta Pañcaka’ or ‘group of five ancient Siddhāntas’. There are five other Siddhāntas likewise entitled the Saura, which are at present available. He calls them ‘the modern group of five Siddhāntas’. These will presently be dealt with. First he takes up for consideration the group of five ancient Siddhāntas. These belong to the 5th century before Śaka era. Some of them may belong to an even earlier period.

The group of five ancient Siddhāntas

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The following are the Siddhāntas mentioned by Varāhamihira :—

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पौलिशरोमकवसिष्ठसौरपैतामहास्तु पंचसिद्धांताः ।।

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Pauliśa, Romaka, Vasiṣṭha, Saura and Paitāmaha are the five Siddhāntas.

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The elements described by the Pañcasiddhāntikā show that the Siddhāntas were different from the five modern Siddhāntas. Not only are these not available at present, but even the original work, entitled Pañcasiddhāntikā is not available and so, not much is known to any one on this side. Two copies of the work brought from Kashmir by Dr. Buhler are preserved in the Government collection of Manuscripts in the Deccan College (See Reg. No. 37 of 1874-75 and No. 338 of 1879-80) ; but these are very incorrect and incomplete and one is unable to find, at places, where one āryā (couplet) ends and the next one begins. The author has written out for himself a copy from the two versions, and calculations made therefrom have shown that the Sūrya and other Siddhāntas described in it, are different from the modern ones, in that they differ in the length of the year and the motions of planets. The astronomical works compiled during the last 800 years do not show that any one knew that there existed before a Sūrya-Siddhānta different from the modern one. He came to know of their existence in 1887 and no doubt reed be entertained about it, since this can be proved by calculations and by other evidence. The booklet of the Pañcasiddhāntikā is very incorrect and the major portion of it is unintelligible for want of a commentary ; however mary important points have been understood.. *The author would, therefore, describe briefly the five Siddhāntas in the order of the dates of their compilation as found by him.

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∗ *Dr. Thibaut published in 1889 A.D. the Pañcasiddhāntikā based on the Deccan College versions. It also gives a new commentary by Sudhākara Dvivedī. The author could not find time to read the whole of it uptil now. However all the important information which he could glean from the Pañcasiddhāntikā has already been given above.

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Varāhamihira, in the very first chapter of Pañcasiddhāntikā, observes as follows ;—

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पौलिशाति* विस्फुटसौ तस्यासन्नस्तु रोमक: ।।
स्पष्टतर: सावित्र: परिशेषौ दूरविभ्रष्टौ ।। ४ ।।

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“The Siddhānta made by Pauliśa is accurate ; near to it stands the Siddhānta by Romaka ; more accurate still is the Sāvitra (Saura). The two remaining ones are far from truth.”

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This shows that the Pauliśa Siddhānta was very clear at the time of the compilation of Pañcasiddhāntikā, which means that its calculations very much agreed with the actual observed positions. The Romaka was nearer to it in correctness. The Sūrya-Siddhānta was better than both and the remaining two (Vasiṣṭha and Pitāmaha) were far removed from correctness meaning that their calculations did not agree with the observed results. The Pitāmaha Siddhānta and Vasiṣṭha Siddhānta must have been the oldest of the five, and of the these two, in the author’s opinion, Pitāmaha Siddhānta must have been the more ancient. The reasons for this will be given later on. Let us now consider the Pitāmaha Siddhānta.

Pitāmaha Siddhānta

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The Subject matter

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The basic principles underlying the Pitāmaha Siddhānta have been given in the 12th chapter of the Pañcasiddhāntikā. That chapter contains only five āryās (couplets). Nowhere else in the Pañcasiddhāntikā is found anything about this Siddhānta. The first two of the five couplets run thus :—

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रविशशिनो: पंच युगं वर्षाणि पितामहोपदिष्टानि ।। अधिमासस्त्रिंशद्भिर्मासैरवमस्त्रिषष्ट्याह्नां ।।
दूनं शकेन्द्रकालं पंचभिरुद्ध्य शेषवर्षाणि ।। युगं माघसितार्थं कुर्याद्युगुणस्तदन्युदयात् २ ।।

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“According to the teaching of Pitāmaha, five years constitute a yuga of the sun and the moon. The adhimāsas occur after thirty months and an omitted lunar day (avam) once in sixty three days. Lessen the time of the Śaka King by two and divide the remainder by five ; with the remaining years find the ahargaṇa, counting from the first day of the light half of Māgha. The Ahargaṇa thus found begins with sunrise”.

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The fifth couplet describes the method of calculating the length of the day :—

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द्विघ्नं शशिरस (६१) भवतं † द्वादशहीनं दिवसमानं ।।

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† The word ‘hinam’ in this couplet is incorrect ; the proper word must be ‘yuktam’. The first half of the couplet is incorrect and has not, therefore, been given here, but it means nothing more than that has been given above within brackets.

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“Multiplying (the number of days elapsed after winter solstice or the number of days to go before the end of the ayana beginning with summer-solstice) by 2 divide by 61. Add 12 (muhūrtas) to the quotient, and the result will be the length of the day”.

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∗ The author copied out the couplets from the Pañcasiddhāntikā, exactly as they are written in his book. Dr. Thibaut has introduced new readings in the next and he has accepted at places only such of them which were considered suitable.

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This verse and the following, describing the method of calculating the nakṣatra, and stating that the nakṣatra should be counted from the Dhaniṣṭhā onward, show that the Pitāmaha Siddhānta has some similarity with the Vedāṅga Jyotiṣa system.

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Date of Compilation

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The method followed by Varāhamihira in explaining the Pitāmaha Siddhānta involves Śaka era but it has been used simply to calculate the ‘ahargaṇa’. The methods attributed to other Siddhāntas also direct one to calculate the ‘ahargaṇa’ from Śaka 427. Just as this does not imply that the Siddhāntas were compiled (by Varāhamihira himself), in Śaka 427, it cannot be taken to imply that the Pitāmaha Siddhānta was compiled after the beginning of Śaka era. It is evident from its similarity with the Vedāṅga Jyotiṣa system that it must have been more ancient than the beginning of Śaka era. There are, however, no means to fix up the date of its compilation.

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Āryabhaṭa I has, in the beginning of *Daśagītikā* written the following benedictory verse :—

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प्रणिपत्यैकमनेकं कं सत्यां देवतां परं ब्रह्म ।। आर्यभटस्त्रीणि गदति गणितं कालक्रियां गोलं ।।१।।

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“After saluting the Pitāmaha (kam) who is manifold and one at the same time, the god of truth and the Parabrahma, Āryabhaṭa describes the three subjects viz. mathematics, method of calculating time and the celestial spheree and in his last couplet he observes,

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आर्यभटीयं नाम्ना पूर्वं स्वायंभुवं सदा सदसत् ।।

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“The science known as the Āryabhaṭīya is the same as the science originaly expounded by Brahmā himself”.

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This shows that the Pitāmaha Siddhānta must have been compiled long before Āryabhaṭa lived (i.e. Śaka 421)

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Brahmagupta (Śaka 550) , in his Siddhānta, observes,

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ब्रह्मोक्तं ग्रहगणितं महता कालेन यत्खिलीभूतं ।।३भिर्धृतं स्फुटं तत् जिष्णुसुतब्रह्मगुप्तेन ।।२।।

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“The calculation of planets’ places as propounded by Brahmā has, become disjointed on account of (lapse of) long time ; Brahmagupta, the son of Jiṣṇu, is describing the same in clearer terms”.

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There are three Brahma Siddhāntas available at present. One is the Brahma Siddhānta by Brahmagupta, the second, the Brahma Siddhānta mentioned in Śakalya Saṃhitā and the third that included in Viṣṇudharmottara tarapurāṇa. In the Author’s view, to be explained later, the Viṣṇudharmottara and the Śakalyokta Brahma Siddhāntas are not older than the Brahmagupta’s time, but compiled much later. The basic principles, propounded by the Śakalya Brahma Siddhānta, even it be more ancient than Brahmagupta, are exactly the same as those propounded by the modern Sūrya-Siddhānta. It may as well be said to be in current use and not antiquated or disjointed ; and it will be shown further that the Viṣṇudharmottara Brahma Siddhānta is not similar to the Brahmagupta Siddhānta. This shows that the Brahma Siddhānta to which Brahmagupta has referred as containing mathematics explained by Brahmā, must have been different from the Brahma Siddhāntas of Śakalya Saṃhitā and Viṣṇudharmottarapurāṇa and it must be the same as the Pitāmaha Siddhānta of the Pañcasiddhāntikā. The Vedāṅga Jyotiṣa does not deal with the calculation of planets other than the sun and the moon; and the Pitāmaha Siddhānta of the Pañcasiddhāntikā also describes the calculations of the sun and the moon only. Varāhamihira has given the calculations of all planets in the case of only the Sūrya-Siddhānta out of the five Siddhāntas and mentions nothing about the planetary calculations contained in the Pitāmaha Siddhānta. It must have, however, contained the calculations of planets as remarked by Brahmagupta, and Varāhamihira appears to have omitted it, on finding that it did not agree with the actual observed places on account of lapse of time. A Pitāmaha Siddhānta, different from the one described in the Pañcasiddhāntikā, does not appear to have ever existed before Brahmagupta. The words “planetary calculations given by Brahmā” occurring in Brahmagupta’s work must certainly be referring to the Pitāmaha Siddhānta of the Pañcasiddhāntikā, and the remark, “a long time has elapsed” also refers to the same Pitāmaha Siddhānta. It follows, therefore, that it must have been compiled long before the Śaka era.

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Āryabhaṭa and Brahmagupta appear to have expressed veneration for the Pitāmaha Siddhānta only as a formality, because their siddhāntas and the Pitāmaha Siddhānta of the Pañcasiddhāntikā have nothing in common. It has already been pointed out in the course of the study of the Vedāṅga Jyotiṣa that Brahmagupta has openly found fault with the five-year Yuga system. Still, these arguments in no way affect the inference that there existed a Siddhānta known as the Pitāmaha Siddhānta before these two astronomers lived.

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The System

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The Pañcasiddhāntikā gives in the beginning two couplets relating to the Pitāmaha Siddhānta, the first of which contains the expression,

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अधिमासस्त्रिंशद्विमसैः

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meaning, “an intercalary month to be reckoned after 30 months”. It has been shown in the study of Vedāṅga Jyotiṣa that the adoption of an intercalary month after 30 months causes a grave error; but this very couplet has been cited by Bhaṭṭotpala in his commentary on Vṛhatsaṃhitā in connection with the verse “ekaikamabdeṣu” in Chapter 8. The reading there is “adhimāso dwyagnisamāisaiḥ” which means that an intercalary month is to be reckoned after 32 months. Again the same couplet is found in the commentary (Chapter I) by Mahādeo on Śrīpati’s Ratnamālā and it also gives “adhimāso dwyagnisamāiḥ” as the reading. It is strange that there is room for confusion due to doubtful readings at such an important place.

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If it be assumed that Utpala and Mahādeo changed the original reading viz. “triṃśatbhirmāsaiḥ”, why should they have substituted “dwyagnisamāiḥ” as the reading? The intercalary month occurs after more than 32½ months, and hence, they could have as well substituted some words meaning 32½ or 33. It appears, therefore, that “dwyagnisamāiḥ” must have been the original reading. According to the Vedāṅga Jyotiṣa one day (tithi) is suppressed after 62 days, while the above couplet mentions the suppression of one day as occuring after 63 days (tithis). This shows that the Vedāṅga Jyotiṣa and the Pitāmaha Siddhānta are not similar to each other completely, and this also lends an additional support to the view that “dwyagnisamaiḥ” must have been the original reading.

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The number of intercalary months in 8 years comes to be 3 at the rate of one intercalary month in 32 months. This gives 99 as the number of lunar months in 8 years and 2970 as the number of tithis (lunar days); and 47 1/6 lunar days will be suppressed during this number of tithis at the rate of one suppressed tithi per 63 lunar days. Hence, 8 years will be equal to 2922 6/7 sāvana days or one year will be equivalent to 365 days and 21 3/4 ghaṭis. This measure of a year is more accurate than that found in the Vedāṅga Jyotiṣa.

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The Pitāmaha Siddhānta existed before Āryabhaṭa, Varāhamihira, and Brahmagupta. As it had fallen into disuse in their times it is evident that it must have beer compiled long before them. It is similar to the Vedāṅga Jyotiṣa but differs from it much. Brahmagupta’s statement shows that it (the Pitāmaha Siddhānta) contained the calculations of Mars and other planets, which is not given in the Vedāṅga Jyotiṣa. It proves that a more accurate work known as the Pitāmaha Siddhānta was compiled sometime after the Vedāṅga Jyotiṣa and this is an important fact. If it were known how the places of Mars and other planets used to be calculated according to the Pitāmaha Siddhānta, it would have been found very useful in tracing the growth of Indian astornomy; but we have almost no hopes of obtaining the Pitāmaha Siddhānta now in its original form.

Vasiṣṭha Siddhānta

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The Date

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The Pañcasiddhāntikā contains 13 āryās (couplets) relating to Vasiṣṭha Siddhānta. The system described by it is somewhat different from that met with in other siddhāntic works. This fact and also the statement of Varāhamihira that “Vasiṣṭha is very inaccurate” leads one to infer that it must have been more ancient than the remaining three siddhārtas excepting the Pitāmaha Siddhānta.

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The System

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The thirteen couplets show that they mention nothing about planets other than the sun and the moon. The method of calculating tithis and nakṣatras is not similar to that of the present day. It mentions the rāśi (sign), aṃśa (degree) and Kalā (minutes) as the units and the subject of ‘shadow’ has been considered at length. Something has been told about the length of the day and the word ‘lagna’ (ascendant) has been used in a somewhat present day sense. The Vasiṣṭha Siddhānta, available at present, is in no way similar to the one existing before Varāhamihira’s time and did not exist in his time. This question will be discussed again later on.

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Different versions of Vasiṣṭha and Romaka Siddhāntas

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Two versions of the Vasiṣṭha and Romaka Siddhāntas were known at the time of Brahmagupta (Śaka 550). The case of Romaka Siddhānta will automatically be considered along with the evidence on the basis of which it is to be proved that there were two kinds of Vasiṣṭha Siddhānta. Let both of them, therefore, be considered together.

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Brahmagupta says at one place in his Siddhānta,

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पौलिशरोमकवासिष्ठसौरपैतामहेषु यत्प्रोक्तं ।। तन्नक्षत्रयनं नार्यभटोक्तं तदुक्तितरतः ।। ४६ ।।

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“I describe the ‘nakṣatrānayana’ i.e. the method of calculating nakṣatras, which has been given by other Siddhāntas viz. Pauliśa, Romaka, Vāsiṣṭha, Saura and Paitāmaha but not given by Āryabhaṭa”.

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He observes at another place,

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अयमेव कृतः सूर्येंदुपुलिशरोमकवसिष्ठयवनाद्यैः ।।

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“This very (beginning of a yuga) has been adopted by Sūrya, Indu, Puliśa Romaka, Vasiṣṭha and Yavana”.

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Brahmagupta has cited as authority of the Sūrya and other Siddhāntas because these were in his favour. The Brahmagupta Siddhānta as a whole, appears to have launched a vigorous attack against Āryabhaṭa and others. He is, as it were, showering volleys of vituperation. Even then he has not made any direct attack against the group of first (Sūryādi) five Siddhāntas except the Romaka, and he has explicitly found fault with the Pattern, ony once, as may be seen from the following couplet :—

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युगमन्वंतरकल्पाः कालपरिच्छेदकाः स्मृतायुक्ताः ।। यस्मान्न रोमके ते स्मृतिबाह्यो रोमकस्तस्मात् ।। १३ ।।

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“The smṛti works mention yuga, manvantara and kalpa as the broad units of time. The Romaka has violated the Smṛtis since it does not mention them”.

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Brahmagupta, at another place, observes,

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लाटात्सूर्यशशांकौ मध्याद्विद्वच्चचंद्रपातो च ।। कुजबुधशीघ्रबृहस्पतिसितशीघ्रशनैश्चरान् मघमान् ।। ४८ ।। युगयातवर्षभगणान् वासिष्ठान् विजयनांदिकृतपादान् ।। मंदोच्चपरिधिपातस्पष्टीकरणाद्यार्यभटात् ।। ४९ ।। श्रीषेणेन गृहीत्वा रत्नोच्चयरोमकः कृतः कंथा ।। एतान्येव गृहीत्वा वासिष्टो विष्णुचंद्रेण ।। ५० ।।

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*The gist*

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“Śrīṣeṇa has compiled a ‘kanthā’ (i.e. a patch work) entitled as Romaka, by borrowing elements from different Siddhāntas, e.g. (i) mean sun and moon, moon's apogee, moon's node, Mars, Mercury, Jupiter, Venus, and Saturn from Lāṭa's work (ii) elapsed years of yugas and bhagaṇa (sidereal revolutions) from Vasiṣṭha (iii) the pāda (quadrants) from Vijayanandi's works and (iv) mandocca, aphelia, paridhi (epicycle), nodes of planetary orbits and calculations of true places from Āryabhaṭiya. Viṣṇucandra has similarly, compiled the Vasiṣṭha Siddhānta by borrowing the same elements".

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It has been said above that Viṣṇucandra has borrowed the same elements from other works in compiling the Vasiṣṭha Siddhānta as Śrīṣeṇa had borrowed in compiling the Romaka Siddhānta, and it has also been observed that Śrīṣeṇa took the bhagaṇas and the elapsed years of yugas from the Vasiṣṭha This shows that Viṣṇucandra also adopted the same elements from the Vasiṣṭha Siddhānta and other information from other Siddhāntas and compiled another Vasiṣṭha Siddhānta. It, therefore, shows that there existed two kinds of Vasiṣṭha Siddhāntas and that this was known to Brahmagupta. One of them was the original Vasiṣṭha Siddhānta and the other was Viṣṇucandra's Vasiṣṭha Siddhānta, compiled by borrowing some elements from the first.

v2.c12.s2.t5.21

It has already been remarked before that Brahmagupta has abused Romaka Siddhānta as 'a violator of Smṛti', because it does not give the time units of yuga, manvantara and kalpa; but it has also been shown above that Brahmagupta himself says that Śrīṣeṇa picked up the figures for 'elapsed years of yuga', from the Vasiṣṭha Siddhānta while compiling the Romaka Siddhānta. Similarly, he again observes,

v2.c12.s2.t5.22

तद्युगबधो महायुगमुक्तं श्रीषेणविष्णुचंद्राद्यैः ।।
अ० ११ आर्या १५.
मेषादितः प्रवृत्ता नार्यभटस्य स्फुटा युगस्यादौ ।। श्रीषेणस्य कुजाद्याः
अ० २ आ० ४६.

v2.c12.s2.t5.23

"That Śrīṣeṇa, Viṣṇucandra and other authors have mentioned 'Mahāyugas as a multiplicity of yugas' ...

v2.c12.s2.t5.24

"Śrīṣeṇa has not given true positions of Mars and others from the commencement of a yuga, as is done by Āryabhaṭa, but from the beginning of Meṣa".

v2.c12.s2.t5.25

Thus, according to the statement of Brahmagupta himself, the Romaka Siddhānta by Śrīṣeṇa did contain the yuga system and from this (it can be inferred that) there were two siddhāntas named Romaka, at the time of Brahmagupta— the one known as the original Romaka Siddhānta and the other, that 'compiled by Śrīṣeṇa'.

v2.c12.s2.t5.26

Most of the names of astronomers who lived before Brahmagupta and who have been mentioned in his Siddhānta, are found in the Pañcasiddhāntikā. However, the names of Śrīṣeṇa and Viṣṇucandra are not found in the Pañcasiddhāntikā which mentions only one Siddhānta each, named as Vasiṣṭha and Romaka. This goes to show that Śrīṣeṇa's Romaka Siddhānta and Viṣṇucandra's Vasiṣṭha did not exist before Śaka 427, and that only the original Romaka Siddhānta and the original Vasiṣṭha Siddhānta existed then and these are given in the Pañcasiddhāntikā only in summary form. According to Brahmagupta, Śrīṣeṇa and Viṣṇucandra borrowed the method of finding true places from Āryabhaṭa. This also shows that they compiled their respective Siddhāntas after Śaka 421 while the Pañcasiddhāntikā leads to the conclusion that they were compiled after the Śaka 427.

Romaka Siddhānta

v2.c12.s2.t6.1

It has been pointed out above that out of the two Romaka Siddhāntas described, only the original Romaka Siddhānta existed at the time of the Pañcasiddhāntikā. Let us now consider this Romaka Siddhānta.

v2.c12.s2.t6.2

A large portion of the Pañcasiddhāntikā has been devoted to Romaka Siddhānta. Three couplets of the first chapter beginning from the 8th describe the method of finding the ‘ahargaṇa’. The fifteenth couplet mentions the intercalary months and the suppressed tithi. All the 18 ‘āryās’ of the 8th chapter are devoted to Romaka Siddhānta. They describe the calculations of the sun and moon, their true places and the method of calculating solar and lunar eclipses.

v2.c12.s2.t6.3

The very first couplet giving the method of calculating the ‘ahargaṇa’ according to the Romaka Siddhānta runs thus :—

v2.c12.s2.t6.4

सप्ताक्षिवेद (४२७) संस्यं शककालमपास्य चैत्रशुक्लादौ ।। अर्धास्तमिते भानौ यवनपुरे भौमदिवसाद्यः ।। ८ ।।

v2.c12.s2.t6.5

“Deduct the Śaka year 427 from the number of that year for which the ahargaṇa is wanted at the beginning of the light half of Caitra, when the sun was halfset in Yavanapura, at the beginning of Tuesday”.

v2.c12.s2.t6.6

This shows that the first lunar day of Caitra was a Tuesday.

v2.c12.s2.t6.7

A Karaṇa work is required to give the positions of planets as at the beginning or epoch of that work, for finding the planetary positions. These positions are termed ‘kṣepaka’. The ‘kṣepakas’ as mentioned by the Pañcasiddhāntikā prove to be true for the mean Meṣādi (Sun’s entry into Aries), falling on the 14th Lunar day of the dark half of ‘amānta’ Caitra of Śaka 427 ; i.e. on Sunday, the 20th of March, 505 A.D. There is no doubt that some of them are true for the moon and others for the midnight of that date. This point will be more clearly explained in the study of the Sūrya-Siddhānta later on. The next Śukla ‘pratipadā’ (i.e. the first lunar day) after the 14th day of the dark half of Caitra, i.e. the first day of the light half of Vaiśākha, is seen to fall on Tuesday. Evidently this very ‘pratipadā’ was called the ‘Caitra-śukla-pratipadā’, by Varāhamihira (otherwise, by no other method the Caitra-śukla-pratipadā of Śaka 427, can be shown to fall on Tuesday), and it was no doubt correct that the ‘ahargaṇa’ is required to be calculated from that day. Varāhamihira appears to have adopted this epoch, because it is convenient to calculate positions from those on the 1st day of the light half. It is a well known fact of mathematical astronomy that the ‘ahargaṇa’ calculated with the help of any ‘Karaṇa work’ proves to be correct within a day, and it has to be verified with the help of the day of the week. It may now be questioned as to why the Vaiśākha śukla-pratipadā could have been called the Caitra-śukla-pratipadā by Varāha. I have absolutely no doubt that the śukla-pratipadā following the day for which the Kṣepakas are given is the first day of the Vaiśākha-śukla-halfmonth of Śaka 427 according to the ‘amānta’ system. The Caitra is defined as that ending lunar month which ends while the sun is in Meṣa (Aries). As the sun is found to be in Meṣa at the end of that amāvāsyā which fell on the day, next to the day for which the elements have been given irrespective of whether a mean or a true Meṣa was reckoned; and hence, according to the above definition the amānta lunar month which ended with the amāvāsyā must be termed as Caitra, and Vaiśākha commenced on Tuesday i.e. the next day. But a lunar month belonging to the pūrṇimānta system ends on the full moon day, and hence, the full moon day belonging to that light half which commenced just after the ‘kṣepaka’ day will coincide with the end of the lunar month. The calculations made on the basis of ‘kṣepakas’ given by Pañcasiddhāntikā show that the sun still occupied the Meṣa sign, and hence that lunar month naturally received Caitra as its name. The author does not find any other convincing theory to explain how that month could be termed Caitra. The pūrṇimānta system has remained in vogue in Northern India from a very ancient time. Even though the ‘pūrṇimānta’ system is still in vogue there, the lunar months are not named according to the system described above; but it appears that the system must have been in use in its pure form at the time of Varāhamihira.

v2.c12.s2.t6.8

The first couplet of the 8th Chapter gives the method of finding the sun’s place according to the Romaka Siddhānta :—

v2.c12.s2.t6.9

सेमकसूर्यो दृगणात् क्षतिथ (१५०) घ्नात् पंचकर्तु (६५) परिहीनात् ॥ सप्ताष्टकसप्तकुतेन्द्रियो ५४७८७ दृतान्मध्यमार्कः सः ॥

v2.c12.s2.t6.10

“Multiply the ahargaṇa by 150, deduct 65, and divide the remainder by 54787; the result is the mean longitude of the sun”.

v2.c12.s2.t6.11

The sun’s position is obtained by multiplying the ahargaṇa by 150, subtracting 65 from the product and by dividing the remainder by 54787. The number 65 is to be subtracted from this because of the Kṣepaka. The figure denoting the sun’s longitude is obtained in terms of bhagaṇas etc. (the bhagaṇa means a complete sidereal revolution of a planet through the Zodiac.) There is no doubt about this, although it is not explicit in the couplet. The sun makes 150 complete revolutions in 54787 days and hence, one revolution requires exactly 365 days 14 ghaṭis 48 palas. This is, therefore, the measure of a year according to Romaka. The modern Sūrya-Siddhānta gives 365d 15g 31p 31.4v. as the measure. Brahmagupta has blamed Romaka for not giving measures of yuga and other units, as is done by other Siddhāntas and the following discussion will show that it is true. In order to facilitate the comparison of Romaka with other Siddhāntas, below the figures are given indicating the number of revolutions and other measures relating to the moon in one Mahāyuga i.e., in 4320000 years, which are derived from the elements given by Romaka. The couplets from the Pañcasiddhāntikā on the basis of which these have been calculated are as follows :—

v2.c12.s2.t6.12

रोमकयुगमर्कंदोर्वर्षाकांशपंचबसुपक्ष: (२८५०) खेंद्रियदिशो (१०५०) धिमासा: खरकृतविषयाष्टय: (१६५४७) प्रलया: ।। १५ ।।

v2.c12.s2.t6.13

“Romaka’s yuga consisted of 2850 years. During this period, the number of intercalary months is 1050 and that of pralaya's or suppressed tithis is 16547".

v2.c12.s2.t6.14

शून्यैकैका (११०) म्यस्तन्नवशून्यरसा (६०९) त्वनाहिनसमूहात् ।। रूपत्रिवगुण (३०३१) भक्ताकेंद्रं शशिनोस्तगमवंत्याम् ।। ५ ।। व्यष्टक (२४) गुणिते द्वाद्रसर्तुमषडकपंचकान् (५६२६६) राहोः ।। भवरूपाग्यष्ट (१६३१११) हृते...।। ८ ।।

v2.c12.s2.t6.15

"(5) Multiply the ahargaṇa by 110, add 609, and divide by 3031; the quotient gives the position of the moon's kendra at Sunset at Avantī.

v2.c12.s2.t6.16

(8) Multiply the ahargaṇa by 24, add 56266 and divide by 163111; the result is the successive position (in revolutions, signs, etc.) of Rāhu (i.e. the moon's ascending node), reckoning backwards from the end of Pisces (i.e. the first point of Aries). (24 revolutions of Rāhu are supposed to take place in 163111 sāvanadays)".

v2.c12.s2.t6.17

The following figures denoting several astronomical measures are derived from these couplets and from the above couplet describing the calculation of the sun's position and that describing the calculation of ahargaṇa.

v2.c12.s2.t6.18

| Number giving | During one Mahāyuga 4320000 years | During a yuga of 2850 years |
| Revolutions of stars | 1582185600 | 1043803 |
| Revolutions of the Sun | 4320000 | 2850 |
| Sāvana days | 1577865600 | 1040959 |
| Revolutions of the Moon | 57751578 18/19 | 38100 |
| Revolutions of the Moon's apogee | 488258 13768/57589 | 322 228/3031 |
| Revolutions of Moon's ascending node | 232165 109085/163111 | 153 26889/163111 |
| Solar Months | 51840000 | 34200 |
| Intercalary Months | 1591578 18/19 | 1050 |
| Lunar Months | 53431578 18/19 | 35250 |
| Tithis | 1602947368 8/19 | 1057500 |
| Suppressed Tithis | 25081768 8/19 | 16547 |

v2.c12.s2.t6.19

The numbers showing the revolutions of the moon etc. in one Mahāyuga are not integral numbers. Hence, the sun and the moon, according to Romaka, will not, like other Siddhāntas, come together in the beginning of the Kaliyuga or a Mahāyuga. Similarly, the number of lunar months is also not an integral number and the Romaka’s yuga has been stated to consist of 2850 years. This shows that the Romaka Siddhānta has not followed the system of adopting 4320000 years as the measure for a Mahāyuga.

v2.c12.s2.t6.20

The couplet describing the method of calculating the moon’s place is very incorrect. The author could not calculate the number of the moon’s revolutions from it; these have been calculated by a different method.

v2.c12.s2.t6.21

The kṣepakas at the epoch of the Karaṇa work are found to be as follows :---

v2.c12.s2.t6.22

| Sun | . | . | . | 11ˢ | 29° | 34′ | 23″ | Moon’s Kendra | 2ˢ | 12° | 19′ | 57″ |
| Moon | . | . | . | 11 | 29 | 18 | 50 | Moon’s ascending Node | 7 | 25 | 49 | 3 |

v2.c12.s2.t6.23

The Kṣepakas are true for the moment of sunset at Ujjayinī on Sunday the 14th lunar tithi of Caitra Kṛṣṇa, Śaka 427 (i.e. 20th March 505 A.D.).

v2.c12.s2.t6.24

Hipparchus, the Greek astronomer, lived about 150 B.C. His figure for the length of the year exactly tallies with that of the Romaka (viz. 365d—14gh.—48pal.)

v2.c12.s2.t6.25

The work of Hipparchus is not available at present, but he had compiled tables for calculating the positions of the sun and the moon only and not for calculating the planets. Well known European astronomers say that the latter were compiled by Ptolemy on the basis of the principles of Hipparchus. They also admit* that the principles of Greek astronomy had already reached India long before Ptolemy’s time. The Romaka Siddhānta gives the calculations of only the sun and the moon, and its measure of the year is found in no other Siddhānta. It does not describe ‘the universally accepted yuga system’ and its name. Romaka appears to be western. All these things go to show that the original Romaka Siddhānta was compiled on the lines of the work of Hipparchus, and it must have been compiled after 150 B.C. and before Ptolemy’s time i.e. 150 A.D.

v2.c12.s2.t6.26

It has already been shown above that the Paitāmaha and Vasiṣṭha Siddhāntas were older than the Romaka. Similarly, it is thought that even the Sūrya Siddhānta and the Pulīśa Siddhānta of the Pañcasiddhāntikā are more ancient than the Romaka; for, it is obvious from Brahmagupta’s work, that the other four Siddhāntas were regarded with more veneration than the Romaka. He has nowhere blamed any of the four works. After the time of Brahmagupta, the Romaka Siddhānta appears to have gone quite out of use, both in its original form and the form in which it was recast by Śrīṣeṇa. Utpala has nowhere made references to the Romaka on any occasion in his discussion of planetary calculations in his commentary on the Bṛhatsaṃhitā, but he has cited the authority of the four Siddhāntas, viz., those by Pulīśa, Sūrya, Āryabhaṭai or by Brahmagupta. The original Romaka Siddhānta seems to have disappeared in its original form at the time of Utpala. The Romaka Siddhānta of the present day gives elements according to the modern Sūrya Siddhānta and not according to any other Siddhānta; and even this Rómaka Siddhānta is not much known to any one now-a-days. This shows that the veneration which the other Siddhāntas, out of the group of five, enjoy, is due to their being much more ancient than the Romaka.

v2.c12.s2.t6.27

∗ See Grant’s History of Physical Astronomy [Introuction page (iii)] and page 439); similarly, see English translation of Sūrya Siddhānta, by Burgess, page 330.

v2.c12.s2.t6.28

One more important proof about the Romaka being more modern than the five Siddhāntas is given below :

v2.c12.s2.t6.29

*Measures of the year according to different works on astronomy.*

v2.c12.s2.t6.30

| | Days | Ghaṭī | Pala | Vipaḷa | Prativipaḷa |
| Vedāṅga Jyotiṣa | 366 | .. | .. | .. | .. |
| **Pañcasiddhāntikā group :** | | | | | |
| 1 Pitāmaha Siddhānta | 365 | 21 | 25 | .. | .. |
| 2 Vasiṣṭha Siddhānta | .. | .. | .. | .. | .. |
| 3 Pulisa Siddhānta | 365 | 15 | 30 | .. | .. |
| 4 Sūrya Siddhānta | 365 | 15 | 31 | 30 | .. |
| 5 Romaka Siddhānta | 365 | 14 | 48 | .. | .. |
| First Ārya Siddhānta | 365 | 15 | 31 | 15 | .. |
| **Brahmagupta Siddhānta (Modern group)** | 365 | 15 | 30 | 22 | 30 |
| Sūrya, Vasiṣṭha, Śākalya, Romaka and Soma Siddhāntas | 365 | 15 | 31 | 31 | 24 |
| Second Ārya Siddhānta | 365 | 15 | 31 | 17 | 6 |
| Rājamṛgāṅka | 365 | 15 | 31 | 17 | 17½ |
| Karaṇa Kutūhala etc. | | | | | |

v2.c12.s2.t6.31

Out of these figures indicating the measure of the year, none except that of Romaka are found to be smaller than 365d—15gh—30p, and none, excepting that of the Vedāṅga Jyotiṣa and the Pitāmaha Siddhānta, greater than 365d—15gh—32p. In other words, leaving aside the case of Vedāṅga Jyotiṣa and Pitāmaha Siddhānta, none of the rest, except the Romaka, differ from one another by more than 2 palas. Had Romaka been older than the Pulisa Siddhānta and the Saura Siddhānta of the Pañcasiddhāntikā, all of them would have taken the same year measure, as the Romaka or one with a slight variation, and they could not have strayed very far from the Romaka. This proves beyond doubt that the Pulisa and Saura Siddhāntas were older than the Romaka. It seems beyond doubt that all the Siddhāntas of the Pañcasiddhāntikā belonged to a pre Śaka period.

v2.c12.s2.t6.32

It is Dr. Thibaut's opinion that the Romaka and the Pulisa Siddhāntas of the Pañcasiddhāntikā are “not more modern than 400 A.D.” He means* to say that these two were compiled about the year 400 A.D. and the other Siddhāntas of the Pañcasiddhāntikā group were also compiled about the same year. But the above discussion will show that his view is erroneous.

v2.c12.s2.t6.33

The figures showing the number of revolutions and other elements, as given in the Romaka Siddhānta, which is available at present, are given later on. A comparison of these figures with the foregoing ones will show that there is absolutely no similarity between them. This shows that the modern Romaka Siddhānta did not exist before Śaka 427.

v2.c12.s2.t6.34

The question whether the modern Romaka Siddhānta is the same as that compiled by Śrīṣeṇa and whether the modern Vasiṣṭha Siddhānta was compiled by Viṣṇucandra, will be discussed later on.

v2.c12.s2.t6.35

∗ See Introduction page LX of Pañcasiddhāntikā by Dr. Thibaut

Puliśa Siddhānta

v2.c12.s2.t7.1

A large part of the Pañcasiddhāntikā is devoted to the Puliśa Siddhānta. It is stated in the 10th couplet of the first Chapter that Romaka’s ‘ahargaṇa’ is very nearly equal to that of the Puliśa Siddhānta. Then follows the calculation of the place of the luminaries etc. and of the eclipses of the sun and the moon.

v2.c12.s2.t7.2

It has not at all been stated what the motions and places of Mars and other planets are according to the Puliśa Siddhānta; but the last couplet states that “the planets have thus been described according to the Puliśa Siddhānta” and which shows that about 16 ‘āryās’ in the end mention something from Puliśa Siddhānta about their direct and retrograde motions, and the rise and set of planets.

v2.c12.s2.t7.3

The elements pertaining to the Puliśa Siddhānta are found to be as follows :

v2.c12.s2.t7.4

खार्क (१२०) घ्नेग्निहुताशन (३३) मपास्य रूपाग्निवसुहुताशकृतैः (४३८३१) ।। हृत्वा क्रमादिनेशो मध्यः... ।। १४ ।। अष्टगुणे दिनराशौ रूपेंद्रियशीतरश्मिभि (१५१) भंक्ते ।। लब्धा राहोरंशा भगणसमाश्च क्षिपेलिप्ताः ।। ४१ ।। वृश्चिकभागा राहोः षड्विंशतिरेकलिप्तिकागुप्ताः ।। ४२ ।।

v2.c12.s2.t7.5

“(14) Multiply the ahargaṇa by 120, deduct 33, and divide by 43831 the result is the mean longitude of the sun in due order. (41) Multiply the ahargaṇa by 8 and divide the product by 151; the quotient indicates the degrees of Rāhu (i.e. the moon’s ascending node) to which as many minutes have to be added, as there are complete revolutions. (42) This is a stanzas stating certain correction to be applied to the place of the moon’s node as found according to the above rule. Apparently 25 minutes have to be deducted from that place. We do not know what is meant by “Vṛścika-bhāgā Rāhoḥ”.

v2.c12.s2.t7.6

These couplets are found in the chapter following the one consisting of the first 25 āryās, and they form part of the passages attributed to the Puliśa Siddhānta. The elements derived from these āryās are as follows :—

v2.c12.s2.t7.7

| The measure of the year | . | 365d 15gh 30pal |
| Number of Sāvana days in one Mahāyuga | . | 1577916000 |
| No. of revolutions of Rāhu | . | 232227 65703915/67946855 |
| Period of Rāhu’s one revolution | . | 6794d 41gh 18pal |

v2.c12.s2.t7.8

This gives for the measure of the year a figure different from that of other Siddhāntas. Similarly, the period of the revolution of Rāhu (the moon’s ascending node) is also somewhat different.

v2.c12.s2.t7.9

The Pañcasiddhāntikā mentions other things from the Puliśa Siddhānta which include the question of true places of the sun and the moon. It describes the method of finding ‘carakhaṇḍas’ (groups of ascensional differences) from the ‘palabhā’ (the noon shadow on equinoctial day), and calculating therefrom the length of the day; terrestrial longitudes too have been considered.

v2.c12.s2.t7.10

The method of calculating the tithi and nakṣatra is similar to one of the present day. It includes the explanation of the ‘karaṇas’ and ‘Mahāpātas’ (or the parallels of the declination of the sun and the moon). The question of eclipses is also considered and the method is almost similar to that of other modern Siddhāntas. The study of the direct and retrograde motions of planets is similar to that described in the work entitled Khaṇḍakhādya. The study ‘cara’ (ascensional difference) quotes the following couplet.

v2.c12.s2.t7.11

यवनाच्चरजा नाड्यः सप्ता (७) वंत्यास्विभाग (१/३) संयुक्ताः ॥ वाराणस्यां त्रिकृतिः (६) साधनम्यत्र वक्ष्यामि ॥

v2.c12.s2.t7.12

“The Yavana mentions 7gh 20pal. as the ‘cara’ for Ujjayinī and 9 ghaṭis for Vārāṇasī. The method of calculating it, is described elsewhere”.

v2.c12.s2.t7.13

The word mentions, as in Vedāṅga Jyotiṣa, dinamānavṛddhi, or the increase in the length of a day at the end of Udagayana as compared with that at the end of the Dakṣiṇāyana. According to the sayana almanac, the minimum and the maximum length of the day at Ujjayinī are 26gh. 26pal. and 33gh. 34 pal. respectively; that is, the difference between the two amounts to 7gh. 8p. According to the Grahalāghava these values are 26gh. 21p. and 33gh. 39p respectively and the difference is equal to 7gh. 18p. These have been calculated by assuming 5-8 as the palabhā for Ujjayinī. Pandit Bāpūdeva śāstri gives in his almanac, the maximum length of the day at Vārāṇasī to be 33gh. 56p. and the minimum length as 26gh. 4HP. The difference between the two comes to be 7gh. 52p. This has been calculated on the basis of 5-40 as the palabhā at Vārāṇasī. Assuming this very value for Vārāṇasī, the Grahalāghava calculation shows 8gh. 4pal. as the difference between the two lengths. Assuming 6-15 as the palabhā, one gets a difference of about 9 ghaṭis on the basis of Puliśa’s ‘Carakhaṇḍa’ as given by the Pañcasiddhāntikā.

v2.c12.s2.t7.14

The third couplet in the Pañcasiddhāntikā shows that Lāṭadeva has written a commentary on the Paulīśa Siddhānta.

v2.c12.s2.t7.15

The author has neither seen nor heard of any Puliśa Siddhānta of any kind being available at present. Utpala has incidentally quoted about 35 couplets from the Puliśa Siddhānta as authority in his commentary on the Bṛhatsaṃhitā. They contain the number of revolutions and other important elements given by the Puliśa Siddhānta. Those couplets as collected in one place are, therefore, quoted below :—

v2.c12.s2.t7.16

अष्टाचत्वारिंशत्पादविहीनाः क्रमात् कृतादीनां ॥ अंशास्ते शतगुणिता ग्रह्तुल्ययुगं तदेकत्वं ॥

v2.c12.s2.t7.17

(The meaning is not quite clear)

v2.c12.s2.t7.18

“The number of years* in a Kṛta yuga is obtained by multiplying 48 by 100 and that in the successive Yugas is obtained after multiplying 100 the “number 48 diminished by its quarter (i.e. 12) successively”.

v2.c12.s2.t7.19

This is a unit (ekatvam) to start with in the case of each kind of ‘planetery’ yugas.

v2.c12.s2.t7.20

[*Footnote by the translator :— Although the text quotes ‘aṃśaḥ’ as the word, the manuscript, which I could see by the kind permission of the local Scindia Institute, gives the word “abdaḥ”, and the explanation therein shows that the Mahāyuga (which was rendered as the ‘divine yuga’ by the later astronomers) consisted of (4800+3600+2400+1200=12,000 years). According to the prevailing system, the year was of 4 kinds and consisted of 4 kinds of months, each of 30 days of the corresponding denomination].

v2.c12.s2.t7.21

सावनमकृतं १५५२००००० चांद्रं सूर्येंदसंगमान् दिनीकृत्य (१६०३००००८०) ।। सौरं भूदिनराशि: १५७७९१७८०० शशिभगणदिनानि १७३२६०००८० नाक्षत्रं ।।

v2.c12.s2.t7.22

“The natural yuga is termed ‘sāvana’ (civil). That which is reckoned after calculating number of days in the joint revolutions of the sun and the moon as Cāndra (Lunar); the Saurā or solar (yuga) is “the heap” of solar days; and the number of days in the revolution of the moon constitutes a Nakṣatra (sidereal) yuga”.

v2.c12.s2.t7.23

परिवर्तैरयुतगुणद्वित्रिकृतै (४३२००००) भास्करो युगं भुंक्ते ।। रसदहनहुतवहानलशरमन्यद्रीशवचंद्र: (५७७५३३३६) ।।

v2.c12.s2.t7.24

“The sun completes 4320000 revolutions in one yuga, while the moon makes 57753336 revolutions (in the same period)”.

v2.c12.s2.t7.25

अधिमासका: षडग्नित्रिकदहनछिद्रशररूपा: (१५९३३३६) ।। भगणांतरशेषं यत् समागमास्ते द्वयोर्गृहयो: ।। तिथिलोपा: खवसुद्विकदसाष्टशून्यशरपक्ष: (२५०८२२८०) ।।

v2.c12.s2.t7.26

“The intercalary months (adhimāsas) in a yuga amount to 1593336. The number of luni-solar conjunctions is equal to the difference between the bhagaṇas of the sun and the moon. The number of suppressed tithis is 25082280”.

v2.c12.s2.t7.27

दसार्थबाणतिथयो लक्षहता: (१५५२०००००) सावनेन ते दिवसा: ।। विषया (?) ष्टौ ऋचतुष्कं विश्व…षोडश चांद्रमानेन ।। वसुसप्तरूपनवमुनिनगतिय: शतगुणाश्च (१५७७९१७८००) सौरेण ।। आक्षैण खाष्टक्षत्रयरसदसगुणनिल (?) शशांका: (१७३२६०००८०) ।।

v2.c12.s2.t7.28

“The measure of the Sāvana yuga is 1555200000 days, that of Lunar yuga is 1603000080, of Solar yuga is 1577917800 and of Sidereal yuga is 1732600080.”

v2.c12.s2.t7.29

षद्‌ प्राणास्तु विनाडी, तत्षष्ट्या नाडिका, दिनं षष्ट्या ।। एतासां तंत्रिंशन्मासस्तद्वैशभिरब्द: ।।

v2.c12.s2.t7.30

“Six ‘praṇas’ make a ‘vināḍī’, sixty vināḍīs make a nāḍikā (or ghaṭikā), sixty nāḍikās make a day; thirty such days make a month and twelve months make a year.”

v2.c12.s2.t7.31

षष्ट्यातु तत्पराणां विकला, तत्षष्टिरपि कला, तासां ।। षष्ट्यांशस्ते त्रिंशद्राशिस्ते द्वादश भचक्रं ।।

v2.c12.s2.t7.32

“Sixty ‘tatpars’ make a ‘Vikalā’, sixty ‘vikalās’ make a ‘kalā’ (minute), sixty ‘kalās’ make an ‘aṃśa’ (degree), thirty degrees make a rāśi (sign) and twelve rāśis make a ‘Bhacakra’ (Zodiacal circuit).”

v2.c12.s2.t7.33

चांद्रै: सावनवियुतै: प्रचय (४७८०००८०) स्तैरपचयोर्कदिनै: (२५०८२२८०) युगवत्सरै: प्रयच्छति यदिमानचतुष्टयं किमेकेन ।। यदवाप्तं ते दिवसा विज्ञेया: साबनादीनां ।। “We get the number of suppressed days with the help of the number of solar days in the yuga and the difference (47800080) between the measures of the lunar and sāvana yugas. When all the four results can be obtained by means of the above figures, where is the need of remembering the results individually ? Whatever results we thus obtain, indicate the number of days in the Sāvana and other kinds of Yugas.”

v2.c12.s2.t7.34

वेदाशिवसुरसांतरलोचनदस्रं (२२६६८२४) खनिसूनुः ॥ अंबरगगनवियन्मुनिगुणविवरगेंदुभिः (१७४३७०००) शशिसुतस्य ॥ आकाशलोचनेक्षणसमुद्रषटकानलै (३६४२२०) जीवः ॥ अष्टवसुहुतवहानल (?) यमखनगै (७०२२३८८) भर्ग्वस्यापि ॥ कृतरसशर्तुमनुभिः (१४६५६४) सौरो; बुधभागर्वौ दिवाकरवत् ॥

v2.c12.s2.t7.35

“These verses give the number of revolutions which each planet makes in a yuga. Mars makes 2296824 ; Mercury’s epicycle 17937000 ; Jupiter; 364220; Venus’s epicycle, 7022388 ; Saturn 146564 ; Mercury and Venus make revolutions equal to those of the Sun. “Atha kakṣāmānāni”:—Now are given the lengths of the orbits of planets :—

v2.c12.s2.t7.36

अथकक्षामानानि-आकाशशून्यतिथिगुणदहनसमुद्रुधार्कशुक्राणां (४३३१५००) इंदोः सहस्रगुणितैः समुद्रनेत्राग्निभिश्च (३२४०००) स्यात् ॥ भूसूनुर्मुनिरामिछद्रतुसमुद्रशिशुसृभिः (८१४६९३७) ॥ रुद्रयमाग्रिचतुष्क्योमशांकै (१०४३२११) बुधोच्चस्य ॥ जीवस्य वेदष्टकस्वरविषयनगामिश्रीतकिरणायैः (५१३७५७६४) ॥ शुक्रोच्चस्य यमानलष्टकसमुद्रतुंरसद्रैः (२६६४६३२) ॥ भगणोर्कजस्य नवशिखिमुनींदुनगष्टकमुनिसूर्यैः (१२७६७१७३६) रिवखवियन्नवसुनवविषयेक्षण (२५६८४००१२) योजनर्भकक्षयाः ॥

v2.c12.s2.t7.37

“The lengths of planetary orbits in yojanas (i.e. 8 miles) are as follows :— Mercury, Venus and the Sun, 4331500 ; Moon, 324000 ; Mars, 8146937 ; Mercury’s aphelion, 1043211 ; Jupiter 51375764 ; Aphelion of Venus 2664632 ; Saturn 127671739 ; Zodiac 259890012”.

v2.c12.s2.t7.38

इष्टग्रहकक्षाम्यो यलब्धं चंद्रकक्षया भक्त्वा ॥ ता मध्यमा ग्रहाणां सौरादीनां कलाश्चंद्राः ॥ पंचदशाहतयोजनसंख्या तत्सगुणार्धविष्कंभः ॥ मोजनकर्णार्धस्यायोजनकर्णविधिना वा ॥

v2.c12.s2.t7.39

“Whatever is obtained after dividing the planet’s orbital length by the moon’s orbital length, is to be known as the planet’s mean distance in lunar-minutes of arc. When this number is multiplied by 15, it gives the semi-diameter of the planet’s orbit, which represents the planet’s geocentric distance. This can otherwise be calculated from the geocentric radius of the orbit of the heavens (i.e. that of the celestial sphere).”

v2.c12.s2.t7.40

वसुमुनिगुणांतराष्टकष्टकै (६८६३७८) दिननाथशुकसौम्यानां ॥ द्वादशलषद्केंद्रियशशांकभूतै (५१५६६) रजनिकर्तुः ॥

v2.c12.s2.t7.41

दस्त्राब्धिषट्करसनलोचनचंद्रै (१२६६४२) खनिसूनोः ॥ रूपाग्निशन्यषट्काष्टिसंमितः (१६६२३१) स्यादुच्चस्य ॥ अष्टकवसुरसमुनिशशांकवसुभिसु (८१७६६८८) जीवस्य ॥ वसुवसुशून्याष्टद्विकवेदै (४२८०८८) रपि भार्गवोच्चस्य ॥ एकार्णवार्थनवशशिदहनखदस्रै (२०३१९५४१) रविसुतस्य ॥ त्रिवसुरसद्विरसानलशशिवेदै (४१३६२६८३) राक्षपरिधकर्णार्ध ॥

v2.c12.s2.t7.42

"The semidiameters of planetary orbits in yojanas are given below :— 689378 of the Sun, Venus and Mercury, 51566 of the Moon ; 1296642 of Mars, 166231 of Mercury’s aphelion ; 8176688 of Jupiter, 428088 of Venus’s aphelion, 20319541 of Saturn and 41362683 of the Zodiacal belt”.

v2.c12.s2.t7.43

वृत्ता चक्रवदनिस्तमस्पारे विनिर्मिता धात्रा ॥ पंचमहाभूतमेया तन्मध्ये मेरुरमराणां तस्योपरि ध्रुवः खे न द्वंद्वं पवनरश्मिमिश्चक्रं ॥ पवनाक्षिप्तं भानामुदयास्तमयं परिभ्रमति । सर्वैजयिन उदकस्था दक्षिणदिक्स्थो जयी शुक्रः ॥

v2.c12.s2.t7.44

"Beyond darkness has been created by God Brahmā, this earth, round like wheel, was made up of five main elements. In its centre stands the mountain Meru, the abode of gods, and the pole occupies a place in the sky just above it. The wheel of stars, being propelled by wind, and creating rises and sets, revolves, as if pulled by ‘reins’ of wind. (The words ‘na dwandwan’ are not clear). All planets when occupying the north give success and only Venus, when in the south gives success.”

v2.c12.s2.t7.45

Although the Pañcasiddhāntikā does not explicitly state that the Pulisa Siddhānta in it postulated the Yugādi system, it appears from the couplets, which mention the intercalary months and suppressed tithis, that it did not postulate the Yuga system. Moreover Brahmagupta has blamed only Romaka on that account. This tends to show that the Pulisa Siddhānta of the Pañcasiddhāntikā did probably contain the description of the yuga-system. The Pulisa’s statement as quoted by Utpala includes it. The ‘sāvana māna’ (i.e. civil measure) alluded to in the statement is termed as ‘solar’ in other works, and the solar measures in the former are termed sāvana by the latter. The measures of ‘bhagaṇas’ etc., as quoted by Utpala from the Pulisa Siddhānta, taking the meanings of the words ‘sāvana’ etc. as given by other works are given below :—

v2.c12.s2.t7.46

| Revolutions of stars | . | 1582237800 |
| Revolutions of the sun | . | 4320000 |
| Sāvana days | . | 1577917800 |
| Revolutions of the moon | . | 57753336 |
| Revolutions of Moon’s apogee (from Beruṇi) | . | 488219 |
| Revolutions of Moon’s asc. node (from Beruṇi) | . | 232226 |
| Revolutions of Mars | . | 2296824 |
| Revolutions of Mercury’s epicycle | . | 17937000* |
| Revolutions of Jupiter | . | 364220 |

v2.c12.s2.t7.47

*This is the number of the planets’ conjunctions with the Sun.

v2.c12.s2.t7.48

Revolutions of Venus’s epicycle . . . . . . 7022388 Revolutions of Saturn . . . . . . . . 146564 Civil Months . . . . . . . . . . 51840000 Intercalary months . . . . . . . . . 1593336 Lunar months . . . . . . . . . . 53433336 Tithis . . . . . 1603000080 Suppressed days . . . . . . . . . . 25082280 The Length of a year . . . . . . . . 365ᵈ-15ᵍʰ-31ᵖ-30ᵛᵖ

v2.c12.s2.t7.49

This shows that the length of the year as given by the Pulisa Siddhānta of the Pañcasiddhāntikā and the Pulisa Siddhānta of Utpala are different. This means that the Pulisa Siddhānta of the Pañcasiddhāntikā is quite different from the Pulisa Siddhānta of Utpala. One more surprising fact is that Utpala himself has given the sentence as an ‘extract’ from the original Pulisa Siddhānta :-

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