Surya Siddhanta

This article aims at providing a thorough (but not verse by verse) exposition of most important topics of and problems related to Surya Siddhanta and its comparison with ancient and modern astronomy, together with its use in astrology. Concrete proofs are provided in this article concerning many cardinal problems in the field of ancient astronomy.

In ancient India 18 original theoreticians were frequently mentioned in the field of theoretical astronomy known as SIDDHANTA , but Varah Mihir could get tangible proofs of only five, which he mentioned in his Panch-siddhantika ('Of Five Theories'), among which Surya Siddhanta was the only theory which was complete in itself and therefore highly regarded by Varah Mihir as the most accurate among all theories about heavens.

In India, the practical formulae of Surya Siddhanta are orally preserved by brahmachari brahmanas (lifelong celibates) and are regarded as sacrosanct by traditional scholars of this field, and are never published, because it is believed that the practical method of using this text ought to be given only to worthy persons, who does not make a living out of it (-Manusmriti). No commentator has ever published the practical methods of making actual planetary computations from Surya Siddhantic formulae. These ancient and modern commentators themselves did not know these formulae. The Makaranda-Tables (AD 1478) are Surya Siddhantic, made from some approximate formulae of Surya Siddhanta, which implies that Makarandacharya possessed at least the approximate formulae and publicised the tables made from those formulae, but did not made public those formulae. There is error in mean Mars of Makaranda because Makarandacharya forgot to undertake beej-samskara (see the sections 'The Mean Motions of the Planets' and 'Siddhantic Beej Samskaara') in Mars, while the Surya Siddhantic commentary of Aryabhata contained beej corrections for Mars too' and was therefore the only reliable commentary of Surya Siddhanta for at least over a millenium which had any practical utility for theoreticians and almanac-makers. This was the real reason behind immense prestige accorded to Aryabhata, but this Aryabhata was different from Aryabhata I or Aryabhata II, as Al Beruni rightly believed. Unfortunately, this excellent Surya Siddhantic commentary by Aryabhata was lost during or after 16th century, and now most of people wrongly believe that this Aryabhata was the same as the author of Aryabhatiya. All other ancient and modern commentaries of Surya Siddhanta are useless as far as their utility for actual computations is concerned, and all of them contain incomplete formulae of making true planets, but the author of Grahalaghava (Ganesh Daivajna,1519 AD) explicitly mentions that he used the beej corrections of Aryabhata for three outer planets,whose value tally with those of Surya Siddhantic tables of Makarandacharya and not with those of Aryabhatiya. This topic is discussed in the section 'Siddhantic Beej Samskaara'.

Author of Aryabhatiya was not a follower of Surya Siddhanta, and differs from the latter in all major practical aspects, e.g, number of revolutions per mahayuga, sizes of epicycles, sequence of corrections to be made in the mean planet to get true one, etc. Aryabhatiya was widely criticised by supporters of canonical astronomy in India, especially by Brahmagupta and his followers. It is noteworthy that the philosophical and astronomical or astrological statements made by epic and puranic authors of India conform to Surya Siddhantic views . Hence, Surya Siddhanta is the sole book which can be said to constitute the bedrock of India's traditional astronomical/astrological mathematics. Other traditional text are are either incomplete or inconherent.

Recently, many governmental and non-governmental organisations in India have started publishing annual almanacs based upon a freely distributed Surya Siddhantic software, whose developer refuses to publish the formulae, saying publicising it is prohibited in Surya Siddhanta itself (Vinay Jha,2006). But many secret formulae and concepts of Surya Siddhanta are graduadually being published by this software developer. It is still claimed by many astrologers of India that predictions based upon Surya Siddhanta are absolutely perfect, but professors of astrology in Indian universities neither refute nor accept such claims. Currently a high court in India (Patna High Court) is sitting over a public interest litigation which charges that Surya Siddhantic formulae are not known to university professors and therefore wrong knowledge and wrong degrees are being awarded to students of Surya Siddhanta !

In 2006, a book was published in Hindi in which all major constants of modern astronomy were mathematically deduced, by means of theorems, from Surya Siddhanta ! But the author stated that Surya Siddhantic planets occupy different positions in heaven than material planets (Vinay Jha,2006),which amounts to stating that there are at least two universes, one phenomenal universe of matter open to human senses, and the other noumenal universe of gods who are presiding deities of material planets of phenomenal world (martya-loka or the world of mortals). If the claims made in this Hindi book is accepted, it will mean that constants of modern astronomy were known to ancient Surya Siddhantic scholars with a high degree of precision ! Surya Siddhanta is said to describe this noumenal world, whose planetary deities control the destinies of creatures living in the material world. Any attempt to confuse Surya Siddhanta with the phenomenal world leads to misinterpretation of this mystic text (Surya Siddhanta is described as a secret text in its last stanza, whose knowledge is said to be equivalent to Brahma-jnana or omniscience,i.e., "brahma-samjnitam-rahasyam" !). Indian astrologers used to worship Surya Siddhanta till the end of 19th century (Sudhakar Dwivedi mentions this practice in the introduction to his commentary of Surya Siddhanta),when Western scholars and modernised scholars of India started criticising Surya Siddhanta as an outdated and inaccurate text and gradually its worth declined. None of these commentators of Surya Siddhanta possessed any knowledge of the formulae of Surya Siddhanta, yet they posed as experts of this text and wrote worthless commentaries, which fail to elucidate how to compute actual positions of Surya Siddhantic planets (proofs of this fact are provided in subsequent sections).

The problem with Surya Siddhanta is that its masters preserved its practical formulae and crucial concepts as closely guarded secrets, because the text itself ordered so. No attempt was ever made to refute the false commentaries of ignorants, because supporters of Surya Siddhanta believe that it is futile to argue in favour of a divine Non-Sensory Universe during a materialistic Kali Age. This article provides proofs showing that there were genuine scholars of Surya Siddhanta, e.g., Aryabhata the Elder (prior to the author of Aryabhata) in ancient India and Makarandacharya in mediaeval India. After Aryabhata, no one tried to write down any genuine commentary of Surya Siddhanta showing practical methods of computations. The lost Surya Siddhantic commentary by Aryabhata the Elder was the source of those practical manuals of almanac making in India for millenia which followed Vedic-Puranic tradition(see the section "Lost Surya Siddhantic Commentary of Aryabhata : New Lights").

Surya Siddhantic 'Sun' is stated to be a deity, only 5.496 million kilometres from Earth. A deity cannot be seen by ordinary senses, argue the proponents of Surya Siddhanta. The only proof of Surya Siddhanta is its astrological predictions, which are held to be absolutely perfect. Since astrology based upon modern astrononomy has already proven to be a fake science, Surya Siddhantic astrology needs to be tested for its claims. Another proof of Surya Siddhanta is a series of secret theorems which show that major constants of modern astronomy can be accurately deduced from Surya Siddhantic notions and equations( see the section "Deduction of Modern Astronomical Constants from Surya Siddhanta"). Some of these Surya Siddhantic or Vedic theorems suggest solutions of many unresolved cardinal problems of modern astronomy, and therefore deserve serious attention.

The problem with western commentators is that Surya Siddhantic system has many similarities with that of Almagest, on account of which Surya Siddhanta is declared to be influenced by Almagest. But detailed investigation of Surya Siddhantic system does not support this view. Almagest also does not seem to be directly influenced by Surya Siddhanta. There seems to be a long history of distortions through translations, which may place original Surya Siddhanta in distant prehistory. Many secret ideas of Surya Siddhanta were exported to Greece, but could not be fitted into the general framework of Western astronomy and were subsequently forgotten. The concept of Philolaus that Earth and all heavenly bodies revolve round a central fire which could never be seen since there was a counter earth between the earth and this fire was related to Puranic and Surya Siddhantic tradition of a heavenly Meru which was away from the centre of Earth and was related to a terrestrial Mt Meru (see the section "Meru : Centre of All 14 Universes"). The idea of trepidation originated from Surya Siddhanta, in which 360° was multiplied with 30% to get 108°, which had four parts of 27° each, in both positive and negative phases. Some prehistoric translator multiplied this 27° with 30% again and deduced the maximum value of trepidation to be 8° only, which was current in Babylon and Europe till 15th century AD, when modern astronomy proved that material universe does not manifest any to and fro oscillation or trepidation of equinoxes at all. But Surya Siddhantic system cannot work without trepidation ( = ayanaamsha ), and Indian astrology will die if ayanaamsha is removed. If astrology has any merit at all, it must be based upon Surya Siddhantic system , because this system has a complete and coherent system as well as a complete parallel universe of gods who regulate the destinies of living creatures. Those commentators who confuse Surya Siddhantic planets to be same as material planets observed by astronomers declare this text to be imprecise, and such commentators do not even try to test the validity of Surya Siddhantic system for astrology based upon Parashara. There are many facets of Surya Siddhantic system which were highly esteemed by ancients but are now regarded to be false notions,e.g., the idea of trepidation or of Meru as Centre of Cosmos. Therefore, no serious attempt is made to understand the original logic behind such ideas which captured their imagination for millenia. Such exotic ideas are discussed in detail in subsequent sections, and they prove to be right provided we do not judge them out of context.

Surya Siddhanta gives the locations of several stars other than the lunar nakshatras, which are not the stars of material universe according to proponents of Surya Siddhanta.

Later Indian mathematicians and astronomers such as Aryabhata and Varah Mihir made references to this text, while later Arabic and Latin translations were very influential in the Middle East and Europe.

Dating the Surya Siddhanta : Different Views

The Surya Siddhanta is one of the earliest astronomical treatises of India, but the form in which it has come to us is generally believed to date after circa 400. It has rules laid down to determine the true motions of the luminaries, which do not exactly conform to the positions of material objects in the sky. It is not known who wrote the Surya Siddhanta or when it was first compiled. It is stated in Surya Siddhanta that it was given by the Sun-God to an Asura named Maya in 2163101 BCE. Such a date cannot be accepted by modern scholars,but we do not have definite alternatives. On the basis of concurrence of Surya Siddhantic tropical True Sun with that of modern astronomy, 782 AD is the date for such a concurrence in mean sun, and 908 AD is for true sun. Since Varah Mihir had stated around 550 AD that Surya Siddhanta was the most accurate and prominent of all theories, 800 or 900 AD cannot be the date of composition of original Surya Siddhanta. It is,therefore, proposed that 1° of error in Surya Siddhantic True Sun may place it around 400 AD. A deferrent epicycle revolving round an equant is the most important Surya Siddhantic device used for accurate description of true planets, and it is same as that in Almagest. Hence Surya Siddhanta's date had to be placed subsequent to that of Ptolemy's, so that Indian astronomy could be proved to be a result of Greek teaching. No detailed examination of mathematical systems was undertaken. Equations of Surya Siddhantic true planet were a secret, and that was why modern commentators could not realize that Surya Siddhantic sysytem was not only internally coherent but was vastly different from that of Almagest in crucial respects also. These peculiarities of Surya Siddhantic system could not have been innovations, because an internally coherent system cannot evolve as a result of haphazard patchworks of borrowings plus improvisations. Hence the causes behind superficial similarities between Surya Siddhanta and Almagest needed to be investigated more thoroughly. Important original facts had to be deliberately forgotten or distorted in order to prove posterity of Surya Siddhanta with respect to Almagest.Some of these misconceptions are examined here.

While declaring that scientific investigation into Surya Siddhantic positions of tropical Sun revealed that it was composed around 800 AD (±400 years if 1° error is assumed),it was not mentioned that other planets do not fit into this scheme. If sidereal Sun is chosen to be a criterion for dating, we get a date nearabout 2000 AD or 150000 BCE ! Other planets gives various dates, showing vast differences. There is no period in history when the descrepancy between Surya Siddhantic and scientific values of true positions of all planets fall within tolerable margins. Around 400 to 900 AD,all five planets show differences widely fluctuating bewteen ±12°,which is too much to enable any dating based on this method. Hence, this method is highly questionable. But it has been accepted because it neatly fits into the currently believed dates for Vedic and post-Vedic history. But to accept the result a doubtful method just because it supports a premise is tantamount to prove a premise on the basis of premise itself.

The most ancient date for Indian astronomy was 1400 BCE (Colebrooke), which was supposed to be the date of original observations cited in Vedanga Jyotisha. Rgvedic and Yujurvedic treatises of Vedanga Jyotisha mention that Uttarayana (winter solstice) coincided with the simultaneous entry of Sun and Moon into Dhanishtha in the beginning of bright half of lunar month of Magha. Beginning of Dhanishtha means sidereal 293°:20', which is equal to tropical 316° now-a-days(due to +23° ayanamsha). Around 1400 BCE, sidereal 293°:20' was equivalent to tropical 270° due to -23°:20' ayanamsha. Since, tropical 270° was tropical Uttarayana,Colebrooke concluded that it was the date when observations were originally noted dowmn which were later used in the treatises of Vedanga Jyotisha. To date, all scholars are using this conclusion of Colebrooke as a proven fact, and no one bothers to wonder that the beginning of bright half of lunar month of Magha,which is explicitly mentioned as a condition in original texts, was totally impossible around 1400 BCE from any system of reckoning. A comprehensive understanding of Indian astronomy was not the objective of these scholars. That is why no modern commentator even raised the question that original equations of Surya Siddhanta need to be re-discovered. They had to prove an Aryan Invasion Theory at any cost,and they proved it at the cost of facts.It was deliberately forgotten that beginning of bright half of lunar month of Magha should coincide with aforementioned event. Every almanac maker of India knows that the Kali Age began with Magha, when siddhantic sidereal Mean Sun stood at 360°. Now-a-days, an average of 60 years has shown that Magha starts when siddhantic sidereal Sun stands at 293°(Dhanishtha),which is believed to be Sidereal Uttarayana by Indians. Hence, in 1400 BCE, siddhantic sidereal Sun stood at 338° nearly, and tropical Sun was at 315° when Magha bright half began. But tropical Uttarayan needed tropical Sun at 270°. Colebrooke's computations showed an error of about 45° in Sun's longitude, which is not a small error. Colebrooke and his followers, therefore,decided that Indian system of reckoning should not be considered at all. But even with modern system of reckoning, beginning of Magha cannot be proven around 1400 BCE. At the meridians of New Delhi, winter solstice ( tropical Sun at 270°)occurred on 21 Dec,2000 AD at 19:06:51 PM, Julian date 2451900.067257, when Tithi (= Moon minus Sun, the result divided by 12) was equal to 26.08875 scientifically and 26.00450787 siddhantically. Same event of winter solstice took place on Jan 1,1400 BCE at 17:04 PM when Tithi was 20.11363426 scientifically and 19.7426117 siddhantically. Difference in scientific and siddhantic Tithis was 0.08424213 in 2000 AD and 0.37102256 in 1400 BCE. It is a marginal difference compared to the error of 45° in Sun's longitude at the stsrt of Magha found in Colebrooke's reckoning,which is equivalent to an error of nearly 45.7 tithis, or of 15.7 tithis if one lunar month is substracted ! The detailed equations of finding lunar month in distant eras is discussed in the section "Surya Siddhantic Time-Cycles and Age of Universe". These equations are not very intricate, but have been neglected by Indian and Western scholars alike for centuries, because they have lost interest in deciphering the methods of the ancient astronomers. But these equations are essential for traditional almanacs of India,because observance of many religious rites demands an accurate determination of tithis for all manvantaras and yugas ! A careful examination of these equations reveals an organic relationship between the Surya Siddhantic system of reckoning and the general body of Vedic-Puranic tradition, which is helpful in determining the date of composition of Surya Siddhanta as well. When we delve into intricate equations of Surya Siddhanta, we find that pre-historic astronomers of India had a highly complex and organic system of astronomy and mathematics which was forgotten during the post Christian era, especially during 1500-2000 AD. Commentators like G.Thibaut have described Paitamaha Siddhanta as an underdeveloped system, but there are tangible proofs to the contrary,shown in subsequent sections here,which these scholars neglected.

During 400-800 AD, Sury Siddhantic and scientic positions of planets show differences amounting upto ±12°. It is inconceivable that errors of 12° went unnoticed by all ancient astronomers ! The reason is different,which is deliberately neglected by all modern commentators. Surya Siddhanta explicitly states that its planets are various incarnations of God and are therefore deities. According to Surya Siddhanta, the whole material universe vanishes during the night of Brahma when Surya Siddhantic Sun(and other planets) continues moving, otherwise Brahma will not know that His night has elapsed. Hence,Surya Siddhantic planets were believed to be divine and therefore distinct from material planets. The main purpose of astronomy was predictive astrology in all ancient societies, hence actual position of material bodies in the sky was not a matter of primary concern to most of the ancient Indians. Modern commentators assume Surya Siddhantic planets to be same as material objects of the sky, and therefore deduce that Surya Siddhanta is an inaccurate text as far as physical astronomy is concerned. No detailed examination of the original scheme of Surya Siddhanta was deemed necesasary. But in the light of many new facets of this mysterious text unearthed in a Hindi book (Vinay Jha,2006), dating the Surya Siddhanta needs a reevaluation. If we fail to find out a definite date, we should state it explicitly and should not enforce a fictious date by deliberately distorting the facts.

Old and New Versions of Surya Siddhantas

In his famous treatise Panchsiddhaantika (Five Theories),Varah Mihira concluded that Surya Siddhanta was the most accurate and clear among all theories known to him. But in his version of Surya Siddhanta the revolutions of planets during a mahayuga of 4320000 years (known as Yuga-bhagan) does not tally with the extant version of Surya Siddhanta. This descrepancy induced European scholars to call the Panchsiddhaantic version of Surya Siddhanta as the old one, and the extant version as the modern one. This nomenclature was used by G.Thibaut,the commentator of Panchsiddhaantika. But this decision was made without any scientific investigation into the causes of this descrepancy.

The Mahayuga-bhaganmaana (revolutions per 43200000 years) of five planets in old and modern versions of Surya Siddhanta are as follows (Actual SS implies actual Surya Siddhanta which is elaborated in the section "Siddhantic Beej Samskaara"):

                 Old SS     Modern SS           Differences
                                        Old SS    Actual SS   Aryabhatiya
      Mercury   17937000     17937060      +60     +12.58       -40
      Venus      7022388      7022376      -12     - 9.43       +12
      Mars       2296824      2296832      + 8     + 2.6        - 8
      Jupiter     364220       364220        0     - 6.29       + 4
      Saturn      146564       146568      + 4     + 9.43       - 4

There were three methods of making almanacs (panchangas) : Siddhanta, Tantra,and Karana. In the siddhaantic method, computations were made from the beginning of Creation. 499 AD was 1955883600 years after Creation according to Surya Siddhaanta, which was equal to 452.7508 Mahayugas. A difference of 60 revolutions during one mahayuga means a difference of 452.75 * 60 = 27165.05 revolutions since Creation ! Such a difference makes a mockery of everything. Hence Varah Mihir's figure cannot be related to Siddhaantic methodology,i.e.,reckoning from the beginning of Creation.

In the Tantric Method (which has nothing to do with Tantra of religion or philosophy), difference of 60 revolutions during 4320000 years implies a difference of 18° during 3600 years ,i.e., between the onset of Kali Age and AD 499 , when trepidation or ayanamsha was zero and therefore the author of Aryabhatiya chose this year for his reference point,which was near the time of composition of Aryabhatiya. Varah Mihira is believed to have written his texts nearlt five decades later. An error of 18° could not be accepted by anyone. Since the modern Surya Siddhanta is the basis of a large number of Indian almanacs,being not much removed from actual positions of material planets as well, it is the Old Surya Siddhaanta of Varah Mihir which must contain errors upto 18° in the position of planets(+18° for Mercury,-3.6° for Venus,+2.4° for Mars,zero error for Jupiter,and +1.2° for Saturn).

There is a third possibility of Karana Method, in which any neighbouring year is chosen as reference point and Yuga-bhagans are readjusted accordingly in order to facilitate the task of almanac-makers. If the Old Surya Siddhanta had a Karana reference point only a few centuries before Varah Mihira, the error in mean planets will vastly decrease. Hence, it may be argued that the Old Surya Siddhanta quoted by Varah Mihira was actually quoted from not a Sidhhanta but a Karana text. This last possibility is really significant,because it has some remote connections with actual Beej-samskaara for AD 499 as shown in the table above. Although Varah Mihira's figures show large errors in magnitude, the sign of differences are never reversed. Hence, we must conclude that Varah Mihira had access to some almanac making Karana text supposedly based upon Surya Siddhanta, which used wrong values of beej-samakaara (See the section 'Siddhantic Beej Samskaara' for further details on Beej-Samskaara, which are essential corrections in mean planets). The figures given by Varah Mihira are certainly related to some lost Surya Siddhantic Karana text, but the values are wrong, which will result in a very faulty almanac for any period. For instance, Mercury will show a beej-samskaara of +8°,while its actual beej-samskaara ought to be a little over 6° during the age of Varah Mihira. An error of 12° could not have been accepted by anyone. Had Varah Mihira possessed any idea of beej samskaara, he must have elucidated it by differentiating the Siddhantic bhagans(e.g.17937060 for Mercury) from beej samskaara (+12.6 or roughly +13 for Mercury). When he tried to find out the values of mahayuga-bhagans from some almanac making karana text, he failed to notice that Karana texts contain not the Siddhantic bhagans but beej-corrected bhagans(i.e.,corrections already made in the tables for mean planets).Varah Mihir was not an almanac maker, and therefore could not discern this nuance of traditional Indian astronomy, which has always remained a trade secret of almanac-makers. It is intereasting to note in this respect that all modern Surya Siddhantic almanac makers of India are using tables made in AD 1478 (known as Makaranda Tables), because these almanac makers do not know how to update the tables, and therefore refuse to accept the views of those who actually know the secret. Hence, it is not surprising that Varah Mihir did not know this closely guarded secret and therefore gave wrong values in his treatise. No good almanac for any period can be made from the values given by Varah Mihira. Hence, it is wrong to label his version as Old Surya Siddhanta, because he did not possess any version of Surya Siddhanta at all and merely tried to get something out of karana text which he failed to understand properly. Thibaut was totally in the wrong in proposing manes like Old and Modern versions of Surya Siddhanata. He did not try to understand the actual practices of traditional almanac making. The socalled modern Surya Sidhhanta is the only version ever available, and whatever Varah Mihir quoted was not from Surya Siddhanta but from a wrong interpretation of some almanac making text supposedly based upon Surya Siddhanta.(These sentences are being written by a person who is the maker of six governmental and non-governmental Surya Siddhantic almanacs in different states of India at present. Cf. Vinay Jha,2006).

The table of Mahayuga-bhagans shown above also shows the figures of Aryabhatiya, which apparently seem to be even more spurious than those of Varah Mihir,because the figures of Aryabhatiya have signs inverted in the case of all planets. The only plausible explanation behind such a blunder by Aryabhata is that he substracted beej corrections instead of adding them to mean planets. Even if this view is accepted, the values viven in Aryabhatiya will lead to a faulty almanac, containing errors of 6° in the case of Mercury if comparison with Surya Siddhanta are made. Aryabhatiya does not show good results even if comparison with modern scientific methods are made. Moreover,the sequence of corrections made in mean planet to get true planet is wrong in Aryabhatiya, which is discussed in the section 'Corrections in Mean Planet to Make True Planet'.In this connection, the section 'Lost Surya Siddhantic Commentary of Aryabhata : New Lights ' should be viewed as well.

Many crucial aspects of Surya Siddhanta have not been thoroughly researched as yet by anyone. For instance, Surya Siddhantic equatorial diameter of Earth is stated to be 800 yojanas, one yojana equalling 7.97274625 kilometres. On this basis, the distance from Girivraja (prehistoric capital of Magadha) to Mathura comes out to be 98.51 yojanas. In Mahabharata, it is stated that Mathura was 99 yojanas from Girivraja  ! This story of Jarasandha of Girivraja is part of the basic story of Mahabharata and cannot be brushed aside as an interpolation. Moreover, only the Surya Siddhantic value of one yojana equalling 7.97274625 kilometres satisfies the statement in Mahabharata. During 500-600 AD, Earth's circumference was stated to be 3200 to 3300 yojanas by Aryabhata and Varah Mihira, 6600 yojana by Aryabhata II later, 4867 by Bhaskaracharya in 11th century and nearly 4000 by Ganesh of Grahalaghava in AD 1519. The implication is that the author of Mahabharata knew how to compute the diameter and other aspects of Earth, and possessed an accurate value of pi ! There are,however, some sections in Mahabharata which do not form part of the main story and give such values of yojana which are incompatible with Surya Siddhanta. Such portions must be interpolations,because they are not related to the main story. Narada Purana gives a detailed explanation of mant topics of astronomy,which are in harmony with Surya Siddhantic framework. Intricate equation of determining lunar month od distant age confirms the statement of Bhagvat Gita that Margashirsha was the foremost month. Etymologically, it means "at the head of Sun's Path". Another popular ancient name of the month of Margashirsha is Agra-hayana, which etymologically means "at the start of Year". Surya Siddhantic month-equation shows that Agrahayana was the first month of Creation. If etymological meanings confirm actual meanings, they must belong to a very early period when language was itself taking shape and names of months were being worked out. The month-formula which make these computations possible are related to the so-called Metonic Cycle, which was borrowed by Greece from Babylon. But the relations of Metonic Cycle with prehistoric India when month-names were being discovered makes it clear that Metonic Cycle was known to Indian astronomers in very remote prehistoric ages. These formulae are discussed in the section " Surya Siddhantic Time-Cycles and Age of Universe".

Lost Surya Siddhantic Commentary of Aryabhata : New Lights

Aryabhata the Elder is reputed for having written the first known commentary of Surya Siddhanta, which had been mentioned by many ancient authorities but is no longer available. The author of Aryabhatiya (of Kusumpur) is wrongly believed by many to be the same as this Surya Siddhantic commentator. But the author of Aryabhatiya does not accept the system of Surya Siddhanta in many crucial fields. For instance, mean motions of planets in Aryabhatiya differ widely from the socalled Old as well as Modern versions of Surya Siddhanta and the values as well as sighs of seed corrections ( beej samskaara ) in mean motions also differ widely. Epicycles also show differences, and the sequence of four corrections made in mean planet to get true one in different from that of Surya Siddhanta ; all other theories of ancient India follow the sequence of Surya Siddhanta. Aryabhata's concept of moving Earth also keeps him apart. Hence the original commentator of Surya Siddhanta was a different Aryabhata. Varaha Mihir did not possess either a good version of Surya Siddhanta or a good commentary on it. But Aryabhata was a highly respecterd astronomer and renowned for his commentary upon Surya Siddhanta in ancient India. The Mahasiddhanta of Aryabhata II adheres to the crucial concepts of Surya Siddhanta but differs in some significant details. Hence the commentatator of Surya Siddhanta cannot be identified either with Aryabhata I or with Aryabhata II, but cannot be placed in a lated period because he was known to ancients. Hence, the commentator of Surya Siddhanta must have existed long before Aryabhata I ( the author of Aryabhatiya). Grahalaghava of Ganesh Daivajna was written around AD 1519, and in it Ganesh says that his mean Mars, Jupiter and Saturn are based upon those of Aryabhata. Actual computations show that the mean positions of Mars, Jupiter ans Saturn according to Grahalaghava for different periods broadly conform to those of Surya Siddhanta of Makarandacharya (except in the case of Mars, in which Makarandacharya forgot beej corrections but Aryabhata the Elder did not err). Mean motions of Aryabhata I or Aryabhata II do not conform to those of Grahalaghava. Hence, the author of Grahalaghava was certainly referring to the Surya Siddhantic Aryabhata, whose commentary was extant around AD 1519. Grahalaghava is among the most popular treatise from which traditional almanacs are still being made in a large part of India, and the statement of its author carries much weight. G. Thibaut expressed surprise that the Aryabhata enjoyed great fame which cannot be satisfactorily accounted for (G.Thibaut and Sudhakar Dwivedi,Panchasiddhantika, 1888). But he was confusing the famed commentator of Surya Siddhanta with the author of Aryabhatuya who was renounced by many renowned astronomers, notably Brahmagupta. The commentator of Surya Siddhanta must be placed in an unknown era, well before 500 AD. Al Beruni was right in assuming that the author of Aryabhatiya was a different person than the Aryabhata the Elder (who wrote a commentary of Surya Siddhanta). Al Beruni was merely expressing what he heard, but modern scholars are willing to forget the greatest and most renowned astronomer of India, mainly because be belonged to the mainstream of Indian tradition, unlike the author of Aryabhatiya. Proofs of Aryabhata the Elder are provided in the section " Siddhantic Beej Samskaara".

Deduction of Modern Astronomical Constants from Surya Siddhanta

Kamlakara Bhatt(author of Siddhant-tattva-viveka,as yet untranslated),an ardent supporter of Surya Siddhanta and an opponent of Bhaskara II,had strongly advocated in 16th century that Surya Siddhantic planets are to be distinguished from the matererial planets. In the beginning of 20th century,terms like Drik-paksha and Saur-paksha came into vogue in India, to distinguish planets and phenomena of Sensory World from that of Surya Siddhanta. Drik-paksha meant the world perceived by means of sense organs, and therefore it denoted the foeld of modern astronomy, while Saurpaksha denoted the gods of Next World bearing same name as the material planets but being non-material. Ketaki system of almanac used these concepts in actual practice. But the Surya Siddhantic viewpoint of Drikpaksha was never elaborated by anyone.Unfortunately, after the disappearance of the Surya Siddhantic commentary of Aryabhata the Elder, even the Saurpakshiya mathematics became obscure, and all the commentators kept on repeating hackneyed phrases whose practical significance was clear to none. Ranganath,Kamlakar Bhat,Sudhakar Dwivedi, Kapileshwar Shashtri, etc wrote voluminous commentories on Surya Siddhanta, elucidating everything except the practical ways of using the formulas and the Merucentric geometrics. Let us examine some orally transmitted occult theorems of Surya Siddhantic school which show that Drikpaksha can be deduced from Saurpaksha mathematically, without the aid of any observatory(Vinay Jha,2006).

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Theorem of Drikpakshiya Sidereal Year : Saurpakshiya eccentricity of Sun's elliptic orbit round the centre of Cosmos(meru) is exactly equal to 1/60,although saurpakshiya equation of centre requires an equant,which will be elaborated in the section 'The True Places of Surya Siddhantic Planets'. Let us denote 1/60 by ε. Then, Ys' = [1/(π^2 * ε^2)] + [(1/2)(1+ε^2)] = (3600 / π^2) + 0.5 + (1/7200) = 365.256400001304866086855 days This is the limiting value of scientific sidereal year. The Vedic (i.e.,Surya Siddhantic) theorem of scientific Tropical Year Yt (=365.24219878125) will be demonstrated later,let us first get the value of mean sidereal year with the help of following equation : Ys = (Ys'+1) / (1+ (1/Yt)) = 366.256400001304866086855 / (1+ (1/365.24219878125)) = 365.2563612258166724169 days

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Now we can get the Period of Precession PP : PP = Yt / ( Ys - Yt ) = 25789.4883232765701616 years This mean value needs two complex correction which are too intricate to be shown here. Let us deduce the value of scientific Tropical Year first.We will not explain all the intermediate terms here, which can be easily recognised by students of modern astronomy. Let sidereal lunar month Mss = 27. 321660641391789747802454274321 days, which will be proven later. Then, synodic month Ms will be : Ms = Ys / [ (Ys / Mss) -1 ] = 29.53058780664716371374 days. Metonic Year Ym is equal to : Ym = 235 Ms / 19 = 365.246743924320182775185653635 days. PPM1 = 1 / [(Ys / Ym)-1] = 37978.09022183997109169737 years. PPS1 = 1/[(1/PP ) - (1/PPM1 )] = 80356.674413324332490977057144470 years. PPS2 = 1/[Ys * (1/Yt ) - (1/Ym )] = 80356.674413324332490977057250561 years. A1 = PPS1 / PPM1 = 2.1158692799964388041303958720096. A2 = PPS2 / PPM1 = 2.1158692799964388041303958748028. PPS1 + A1 = 80358.790282604328929781187540342 PPS = 80358.790282604328929781187540342 PPS1 + A1 = 80358.790282604328929781187540342 There are three equations for obtaining scientific Tropical Year : Yt.1 = Ym / [ 1 + { 1 / ( PPS1 + A1 )}]

    = 365.24219878124999999999999999999638527125

Yt.2 = Ym / PPS

    = 365.24219878124999999999999999999638595267

Yt.3 = Ym / [ 1 + { 1 / ( PPS2 + A2 )}]

    = 365.2421987812499999999999999999999999972349

Drikpakshiya Tropical Year is the most precise constant known to modern astronomy,whose empirical value is 365.24219878125 ± 0.00000000058 days.

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Now let us see the Vedic theorem of Lunar month. M1 = 365.256400001304866086855 / (42/π) = 27.321114831446531255657 K1 = M1 / ( Mss - M1 ) = 50056.095658915529 K2 = 42000(Ys-Yt) = 594.8226718002415 Now raise (Ys/360) to the power (1/K2): Z1 = (Ys/360)^(1/K2) = 1.014601^(1/594.82267) = 1.000024369635568 degrees. K3 = 1-[(180/π)* {(Sin(Z1+1)-Sin(Z1)}]

  	   = 1-[57.296*{(Sin(2.000024369635568)-Sin(1.000024369635568)}]
          = 0.0003553741530559558546620855628939

K4 = K3 * 1000000 = 355.3741530559558546620855628939 K5 = 1+(1/K1) Now we get the value of Drikpakshiya synodical or lunar month : Ms = [(K4 / K5)-1}/12 = 29.53058780664716371373841555 days. Sidereal lunar month will be : Mss = Ys / [(Ys/Ms)+1] = 27.321660641391789747802454274321

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Now we show some more intricate Vedic theorems. First of all, let us see the Lunar Binomial Theorem: A1 = 12/(K4-1) = 1 / 29.5311794213296538 A2 = Ys / 365.256400001304866086855 A = A1 * A2 * (42/π) = 0.45270842758190827172 Ax2 + x - Ys = 0 Roots of this binomial are : M1 = [-1 + Sqr(1-(4A*Ys)] / 2A = -29.5305886713712313156 days. M2 = [-1 - Sqr(1-(4A*Ys)] / 2A = +27.3216613815891770963 days. M2 - Mss = 0.063953054266910187950698752 seconds. This apparent 'error' is equivalent to the error of 104.643228673117 years in 4.1748 billion years ( = 14 manavantara of 71 mahayugas each, each Drikpakshiya mahayuga being of 4.2 million years).This is the value of Drikpakshiya correction in Kalpa-Mandochcha, for which Bhaskaracharya deduced the value 93 in Siddhantashiromani and stated Kalpa-Mandochcha to be equal to 480 (= Saurpakshiya Kalpa Mandochcha 387 + 93 Drikpakshiya correction).

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Surya Siddhanta states Saurpakshiya period of precession to be of 24000 years exactly, while modern value is near the Drikpakshiya value of PP deduced above ( = 25789.4883233 years). Let us see its logic. 1/K' = (1/24000) - (1/25789.4883233) = 1/ 345879.71975438125 Mt = Mss - (Mss/K') = 27.32158164959469683453 days. This constant Mt is the modern value of tropical sidereal lunar month !

Surya Siddhantic Theory of the Rotation of Material Universe

Now we show the Vedic theorem of Rotation of the Material Universe. Surya Siddantic Kalpa is equal to 4.32 billion years. The Creator (Brahma) took 47400 divine yuears to create the Creation, which is equal to 47400 * 360 human years. Hence the total Age of Creation = 4.32 billion - (47400 * 360) = 4302936000 years. 4302936000 / 24000 = 179289 is the extra years due to Saurpakshiya precession. Hence total number of Saurpakshiya tropical years in one creation is equal to 4302936000 + 179289 = 4303115289 years. Divide this number with (Saurvarsh / Chandravarsh) = (Saurpakshiya Sidereal Year / Twelve Saurpakshiya synodical months) = 365.258756481481481 / (12*29.53058794607) = 1.0307356481481. The result is 4174800101.976788423. In it, 4174800000 is the duration of Drikpakshiya Creation ( = 4200000*71*14), and 101.976788423 is the exact value of Drikpakshiya correction in Kalpa-Mandochcha, for which we had got a crude value 104.643228673117 above, and Bhaskaracharya had got 93. A quantity of 101.976788423 years in 4.1748 billion years is equal to 0.107065 hours in 500 years. Nirmal Chandr Lahiri was the secretary of Panchanga Reform Committee of Government of India. He analysed the differencebetween Drikpakshiya and Saurpakshiya tithi (elongation of moon), and found a difference of 0.11 hours in 500 years,which he assumed to be due to error in Surya Siddhantic values(NC Lahiri,1968,p.90). But Surya Siddhantic values do not belong to this physical Universe. This apparent error of 0.107065 hours in 500 years is a result of extra 102 rotations of the Drikpakshiya solar orbit during one Creation : Saurpakshiya value is 387 while Drikpakshiya value is 489 (Bhaskaracharya-II gave 480 only in Siddhantashiromani). This Drikpakshiya rotation of solar ellipse is in addition to the normal Drikpakshiya rotation per 136000 years which is the cause behind anomalistic year.

In the same book NC Lahiri gives data of Surya Siddhantic beej corrections applied to lunar anomaly in comparison to modern scientific values, which shows that beej correction needed in lunar anomaly in order to get siddhantic tithi from scientific tithi increases at a rate of one revolution in 42000 years(NC Lahiri,1968,p.90). Difference between modern scientific tropical Sun and siddhantic Sun also show 360° change during 42000 years. Sun and moon do not move in same orbits. Hence we must conclude that the physical Universe itself is revolving at the rate of one revolution per 42000 years round some point very near to Earth's centre,which suggests that the centre of Universe is not far from Earth's centre. Before dealing with this centre (Meru or Mt Kenya in Africa),let us first elucidate the 42000 year cycle of the Sun. Siddhantic sidereal year (365.258756481481)and Drikpakshiya tropical year(365.24219878125) differ at the rate of one revolution or one year in 22059.75174 years. But in reality both divurge from each other at the rate of one revolution in 42000 years. For instance,Kaliyuga commenced at Ujjain midnight 17-18 Feb,3102 BCE,when siddhantic nirayan(=sidereal in Indian system) Mean Sun was at zero longitude. 5106 years later siddhantic zero Sun was to be found on 16 Apr,2005 at 5:03:15 AM (Ujjain). If mean Sun differs by 44.2106 days in 5106 years(taking into account 13 days of Gregorian reform), it should differ by one year in 42182.8 years. Due to non-linearity of elliptical paths,we get here 42182.8, the exact figure is an integer 42000. It raises a question : if mathematically siddhantic year and scientific year should show a difference of one revolution in 22059 years, why do they differ by one revolution in 42000 years in reality ? Where does 19941.24826 years come from ? We have here compared sidereal siddhantic year with tropical scientific year, hence this extra difference of 19941 years must be related to precession. Siddhantic period of precession is 24000 years and scientific period is 25789.4883233 years. Both form cycles of 100000 ± 12000 years with respect to 19941 in harmonic series. Thus, we are now getting close to constants of Milankowitz,just by means of analysing Surya Siddhantic constants ! The Vedic value for Drikpakshiya period of precession is 25789.4883233 years. But it needs slight modifications due to long term cycles.We had got 101.976788423 years per Drikpakshiya Creation for the exact value of Drikpakshiya correction in Kalpa-Mandochcha, which is equal to 101.976788423/4174800000 per year. Add this correction to 1/257898.488, we get 1/25773.252377 in the harmonic series. Again add 1/(4200000*71) to it and get 1/25771.025002. (4200000*71) is the duration of one Drikpakshiya Manvantara. A period of 25771.025 years is equal to 50.28903584 seconds of arc per sidereal year, or 50.288160017 seconds per Julian Year. Presently, 50.28796195 seconds per Julian Year is the accepted value.

Cosmogony and Geography

Surya Siddhantic system is neither heliocentric nor geocentric. It clearly states in Bhoogoladhyaya that Mt Meru resides at the centre (equator) of globe in the region of Zamboodweep. In Africa, Mt Kenya is situated upon equator in a region where many modern place names are reminiscent of Surya Siddhanta : Meru town near Mt Kenya, another Mt Meru slightly southwards, a place named kinyan-giri which means Mt Kinyan or Mt Kenya in sanskrit, river Zamboonadi > *zamboodi > *zambedi > *zambezi, Mu-zambique, Zambia, Zimb-abwe, Gabon (< *Zamboon), Congo (< *Gongo < *zambo),etc. Homo genus of mankind is known to have evolved in that region around 4 million years ago. Indian Puranic ttreadition also mention that modern races of mankind evolved near Meru in 3891194 BCE when the present Mahayuga commenced. Surya Siddhantic formulae of making true planets from mean ones require the use of distance from Earth's centre to a point in space 28.913 kilometres above the top of Mt Meru (Mt Kenya), which was believed to be centre of all universes by puranic authors.

Surya Siddhantic universe is much smaller in comparison to material universe, and Sun's distance from Earth is only 861.7 times of Earth's equatorial radius. Material Sun's distance is 23455 times of Earth's equatorial radius ! Ptolemy used a figure 1210, which is not much removed from Surya Siddhantic figure. Ptolemic system is well known, but Surya Siddhantic system is rather obscure, known to a few initiated brahmanas only. Due to lack of knowledge of orally transmitted and unpublished portions of original Surya Siddhanta, European commentators believe that Surya Siddhantic system was influenced by Ptolemy's Almagest. But those who know the secrets of Surya Siddhanta say that its framework is too complex and organically self-contrained to have been influenced by any other system. For instance, Surya Siddhantic daily motions of all planets are exactly equal to a constant, but this rule is not followed in Almagest. Surya Siddhantic system is based upon a cosmic centre at Meru, which is absent in Almagest. Surya Siddhantic solar epicycle is equal to 14 yojanas per degree, which is equal to 5040 yojanas for 360 degrees. Its diameter is 1604.3 yojanas, which is 4.3 yojanas more than Earth's equatorial diameter. 4.3 yojanas equals 5.199 kilometres ( height of Mt Meru or Mt Kenya)plus 28.913 kilometres. Solar epicycle equals to 14 yojanas, which gets reduced to 13:40 at perigee of this elliptical epicycle, which when divided by 2π gives 2:10:31 degrees, which is the maximum value of equation of centre (mandaphala = difference between mean and true Sun) for Sun. Surya Siddhantic theory, therefore relates yojana to degrees in an intrinsic manner, which makes it clear that it was not borrowed from Almagest. Earth's diameter is an integer 1600 yojana. Moon's diameter is also an integer 436 yojanas. These rations are perfectly scientific. Such integral values seem to be mysterious when they are confirmed with modern science. This value of yojana was not only prehistoric, manifest in the story of Jarasandha's 99 yojanas cited above, but was also intrinsically related to many native concepts of Surya Siddhanta, discussed in other sections of this article.

Trigonometry of Surya Siddhanta

                             Trigonometry

The Surya Siddhanta contains the roots of modern trigonometry. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant when discussing the shadow cast by a gnomon in verses 21–22 of Chapter 3:

Of [the sun's meridian zenith distance] find the jya ("base sine") and kojya (cosine or "perpendicular sine"). If then the jya and radius be multiplied respectively by the measure of the gnomon in digits, and divided by the kojya, the results are the shadow and hypotenuse at mid-day.

In modern notation, this gives the shadow of the gnomon at mid-day as

$s={\frac {g\sin \theta }{\cos \theta }}=g\tan \theta$

and the hypotenuse of the gnomon at mid-day as

$h={\frac {gr}{\cos \theta }}=gr{\frac {1}{\cos \theta }}=gr\sec \theta$

where $\ g$ is the measure of the gnomon, $\ r$ is the radius of the gnomon, $\ s$ is the shadow of the gnomon, and $\ h$ is the hypotenuse of the gnomon.

Practical Method of Casting a Surya Siddhantic Horoscope

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Methods for accurately calculating the shadow cast by a gnomon are discussed in both Chapters 3 and 13.

The astronomical time cycles contained in the text were remarkably accurate at the time. The Hindu cosmological time cycles, copied from an earlier work, are described in verses 11–23 of Chapter 1:

11. That which begins with respirations (prana) is called real.... Six respirations make a vinadi, sixty of these a nadi;

12. And sixty nadis make a sidereal day and night. Of thirty of these sidereal days is composed a month; a civil (savana) month consists of as many sunrises;

13. A lunar month, of as many lunar days (tithi); a solar (saura) month is determined by the entrance of the sun into a sign of the zodiac; twelve months make a year. This is called a day of the gods.

14. The day and night of the gods and of the demons are mutually opposed to one another. Six times sixty of them are a year of the gods, and likewise of the demons.

15. Twelve thousand of these divine years are denominated a caturyuga; of ten thousand times four hundred and thirty-two solar years

16. Is composed that caturyuga, with its dawn and twilight. The difference of the krtayuga and the other yugas, as measured by the difference in the number of the feet of Virtue in each, is as follows:

17. The tenth part of a caturyuga, multiplied successively by four, three, two, and one, gives the length of the krta and the other yugas: the sixth part of each belongs to its dawn and twilight.

18. One and seventy caturyugas make a manu; at its end is a twilight which has the number of years of a krtayuga, and which is a deluge.

19. In a kalpa are reckoned fourteen manus with their respective twilights; at the commencement of the kalpa is a fifteenth dawn, having the length of a krtayuga.

20. The kalpa, thus composed of a thousand caturyugas, and which brings about the destruction of all that exists, is a day of Brahma; his night is of the same length.

21. His extreme age is a hundred, according to this valuation of a day and a night. The half of his life is past; of the remainder, this is the first kalpa.

22. And of this kalpa, six manus are past, with their respective twilights; and of the Manu son of Vivasvant, twenty-seven caturyugas are past;

23. Of the present, the twenty-eighth, caturyuga, this krtayuga is past....

When computed, this astronomical time cycle would give the following results:

The average length of the Surya Siddhantic tropical year as 365.24353736662808641975308642 days is 115.65377667 seconds longer than the modern value of 365.24219878125 days. This estimate remained the most accurate approximation for the length of the tropical year anywhere in the world for at least another six centuries, until Muslim mathematician gave a better approximation, though it still remains more accurate than the value given by the Julian calendar, which gives the average length of the tropical year as 365.25 days.

The length of the Surya Siddhantic sidereal year is 365.258756481481481 days, which is longer than the modern value (365.2563612258) by 3 minutes 27 seconds. The Surya Siddhantic sidereal year is therefore declared to be inaccurate. But Surya Siddhantic sidereal year is not the year of this physical world, it is the period of revolution of the Sun God residing in Heaven (Next Universe) round Meru, according to the Surya Siddhantic or Vedic-Puranic system of thought. Surya Siddhantic sidereal year is related to the sidereal year of physical astronomy by means of a very intricate complex of equations, which are discussed in the section

Calendrical uses of Surya Siddhanta

The Indian solar and lunisolar calendars are widely used, with their local variations, in different parts of India. They are important in predicting the dates for the celebration of various festivals, performance of various rites as well as on all astronomical matters. The modern Indian solar and lunisolar calendars are based on close approximations to the true times of the Sun’s entrance into the various rasis.

Conservative "panchang" (almanac) makers still use the tables based upon Surya Siddhanta to compile and compute their panchangs. The panchang is an annual publication published in all regions and languages in India containing all calendrical information on religious, cultural and astronomical events. It exerts great influence on the religious and social life of the people in India and is found in most Hindu households.

References

• Ebenezer Burgess. "Translation of the Surya-Siddhanta, a text-book of Hindu Astronomy", Journal of the American Oriental Society 6 (1860): 141–498.
• G.Thibaut and Sudhakar Dwivedi."Panchasiddhantika",Chowkhambha,Varanasi, India,1888,reprint 1997
• Victor J. Katz. A History of Mathematics: An Introduction, 1998.
• Dwight William Johnson. Exegesis of Hindu Cosmological Time Cycles, 2003.
• Alaska Mark. Surya Siddhanta, Chapter I with Commentary and Illustrations, 2005.
• Vinay Jha,Deduction of Modern Astronomy from Surya Siddhanta,Sky Research Centre,364 Lancer's Road,Timarpur,Delhi-54,India; AD 2006. Tel: (+91)9431219630,(+91)9931360854,(+91)06272222730.
• N C Lahiri. "Advance Ephemeris", Astro-Research Bureau,17 Brindaban Mullick 1st Lane,Kolkata-9,India,1968,pp.115.

See also

• Indian science and technology
• Indian mathematics
• Hindu astronomy