Mathematics in Ancient India: Aryabhata, Varahamihira, Brahmagupta, etc
This is a collection of articles archived for the excellence of their content.
The name of this celebrated astronomer iswritten either Aryabhata or Aryabhatta, but generally with one t only. In an old manuscript of the Brahma Sphuta SiddMnta of Brahma gupta, copied in Samvat 1678, or a.d. 1621, the name occurs about thirty-three times,1 and is invariably written Aryabhata; and a double / cannot be introduced without violating the Arya metre. Bhatta Utpala, in his commentary on the Varaha Sanhita, cites a passage from Varaha Mihira as follows:- (See Picture-1)
Here the word has only one t, and would not scan with two. This scholiast almost always writes, when quoting Arya bhata. In a commentary by Somesvara on the Aryabhatiya Sutra, of which the manuscript in my possession was copied about three hundred years ago, the name is spelt Aryabhatiya with only one t. (See picture-2)
In a copy of the Mah& Aryasiddhanta, dated Saka 1676, a.d. 1598, is the following line:- (See Picture 3)
Bhatta Utpala and Somesvara sometimes call him Acharyabhata or Acharya Aryabhata; Brahmagupta, in his Siddhanta, chap. x. 62, Aryah, and in chap. xxi. 40, Acharyabhata. In his Khanda Khadya Karana, copied Samvat 1783, he is called Achftrya Aryabhata or Aryabhata. In a commentary on it by AmarSja, he is simply called Acharyabhata. Hence it ap pears tome clear that the proper spelling of this name is Aryabhata.
Achftrya Aryabhata or Aryabhata. In a commentary on it by Amaraja, he is simply called Acharyabhata. Hence it appears to me clear that the proper spelling of this name is Aryabhata. The works attributed to Aryabhata, and brought to light by European scholars are:
An Aryasiddhanta (Maha Arya Siddhanta), written, acccording to Bentley, year 4423 of the Kali Yuga, or a.d. 1322.
Another Aryasiddhanta, called Laghu, a smaller work, which Bentley supposed was spurious,2 and the date of which, as stated in the text, was interpreted tomean the year of the Kali Yuga 3623, or a.d. 522. Of both these works Mr. Bentley possessed imperfect copies. He assumed a compara tively modern work, attributed to Aryabhata, and written in a.d. 1322, as the genuine Aryasiddhanta, and, reasoning on this false premiss, has denounced as spurious the real and older work, and has, further, been led into the double error of condemning the genuine works of Varaha Mihira, Brahma gupta, Bhatta Utpala, and Bhaskaracharya, tions and references to the older work, as modern impostures and of admitting as genuine a modern treatise (the Jatakar nava) as the work of Varaha Mihira.
Colebrooke, not having the works of Aryabhata before him, suggested that the older work might be a fabrication, but, from citations and references to Aryabhata in the works of Brahmagupta and Bhatta Utpala, came to a singularly accurate conclusion as to the age of Aryabhata, whose works he thought were different from either treatise in the possession of Bentley. Wc shall, however," writes Colebrooke, take the fifth [century] of Christ as the latest period to which Aryabhatta can on the most moderate assumption, be transferred. In one place, indeed, Colebrooke correctly guesses that the Laghu Arya Siddhantika is either Aryashtasata or the Dasagitika.
The following passage in theMaha Aryasiddhanta explains itself:- (See Picture 4)
"That (knowledge) from the Siddhanta, propounded by Aryabhata, which was destroyed, in recensions, by long time, I have, inmy own language, thus specified."
In another copy, the verse commences differently, having Vrddha for iti; i.e. the first Aryabhata is called Vrddha or old, whilst himself is the modern Aryabhata.
Strange to say, the dato corresponding to a.d. 1322, mentioned by Bentley, is not to be found in my copies. But I believe he was here, for once, correct.
In the first volume of the Transactions of the Madras Literary Society, a paper was published by Mr. Whish, evidently founded on the works of Aryabhata senior. But, although Mr. Whish's paper is not available to me, I am positive he did not recognize his Aryabhatiya Sutra as the work of Aiyabhata senior.
Professor Lassen has some admirable remarks on Aryabhata.
He observes : "Of Aryabhatta's writings we have the following. He has written a short outline of his system, in ten strophes, which composition he therefore called Dasagitaka; it is still extant. A more extensive work is the Aryashtasata, which, as the title informs us, contains eight hundred dis tichs, but has not yet been rediscovered. The mean between these works is held by the Aryabhaitlya, which consists of four chapters in which the author treats of mathematics in one hundred and twenty-three strophes.3 In it he teaches the method of designating numbers by means of letters, which I shall mention again by and by. Besides, he has left a comentary on the Surya Siddhanta, which has been elucidated by a much later astronomer, and is, probably, the work called Tantra by Albiruni.
This may be the same which was com municated to the Arabs, with two other Siddhantas, during the reign ofthe Khalif Almansur, (which lasted from a.d. 754 till 775), by an Indian astronomer who had come to his court, but of which only the book properly so called, i.e. that of Brahma gupta, had been translated into Arabic, by order of that Khalif, by Muhammed bin IbrdhimAlfazdri, and had received the title of the great Sind-hind. (See Colebrooke's Misc. Ess. ii. p. 504 seqq.) From this juxtaposition it appears that sufficient materials are at hand for investigating the doctrines of this founder of mathematical and astronomical science in India.
Therefore it would be very desirable if a mathematician and astronomer, provided with a competent knowledge of Sanskrit, were to undertake to fill up this great gap in the knowledge which we have hitherto possessed of the history of both these sciences."
To my learned friend Dr. Fitzedward Hall we are in debted for the first and accurate statement tbat, "as reference is made, in the Arya Siddhanta, toVrddha Aryabhatta, there should seem to havo been two writers called Aryabhatta." This correct reference Dr. Hall was enabled tomake from having "possessed himself of two copies of the Arya Siddhanta, both imperfect and very incorrect.
"This treatise is in eighteen chapters; and I more than suspect it to be the same composition which Mr. Bentlcy also had seen in a mutilated form," [i.e. the Maha Arya Siddhanta].
In an Additional Note on Aryabhatta and his Writings,"by the Committee of Publication, appended to Dr. Hall's paper, the learned writer under the initials W. D. W. brings" to light the contents of Bhuta Vishnu's "Commentary on the Dasagitika of Aryabhatta," from a manuscript of the Berlin Library, a copy of which was supplied to him by Prof. Weber.
From the nature of tho contents given in Appendix A, it is clear to me that the treatise which is described as "a brief one, containing only about one hundred and fifty stanzas," consists not only of the Dasagiti Sutra, with a commentary by Bhuta Vishnu, but also of the Aryashtasata of Aryabhata, which was hitherto believed to be unrecovered. Tho learned writer correctly remarks that the treatise is undoubtedly the same as "Bentley's Laghu Arya Siddhanta, and also that the other Arya Siddhanta, judging it from the account given of it by Bentley, appears to be, in comparison with this, a quite ordinary astronomical treatise, representing the general Hindu sys tem with unimportant modifications." Yet he falls very nearly into the same error as Colebrooke, when he proceeds to remark:
"Yet it seems clear that Brahmagupta and others have treated them as works of the same author, and have founded upon their discordances a charge of inconsistency against Arya bhatta." The fact is, as we shall see, that Brahmagupta, Bhatta Utpala, and Bhaskara Acharya know and cite only the elder Aryabhata.
The next and last paper is on some fragments of Aryabhatta, by Dr. 11.Kern in the Jour. Roy. As. Soc. vol. xx. pp. 371 seqq. After briefly noticing the works known to former writers as tho works of Aryabhata, and after alluding to the conclusion Dr. Hall arrived at, that there were two authors of the same name, he adds: "If the same course were adopted in regard to all the works ascribed to Aryabhatta, or to an Arya bhatta, if the contents were compared with the numerous fragments scattered in different works, chiefly commentaries, one might indulge the hope that the question of the author ship ofAryabhatta would be settled in a satisfactory manner."
Dr. Kern proceeds "to contribute a small share towards solving the question," by giving extracts from the commen tary of Bhatta Utpala on the Varahi Sanhit& of Varaha Mihira. In an additional note, at the conclusion of the paper, the learned author states that he was enabled, "by the kindness of Prof. Weber, to ascertain that all the quotations of Utpala, with the exception of one half-stanza, occur in a manuscript of the Berlin Library." Some of the passages puzzle him, especially the date a.d. 475, about which he observes: "Un fortunately, it is not clear whether the date of Aryabhatta is meant, or that of Bhutavishnu."
Adopting Dr, Kern's suggestion, I proceed to state briefly the result of an examination of the various works attributed to Aryabhata. In a diligent and expensive search for old and rare Sanskrit, Prakrit, Arabic, and Persian manuscripts, noise lessly conducted for many years past, I have succeeded in procuring the following works whose authorship is attributed to Aryabhata.
A. Three copies of what has been called the Vrddha or Laghu Arya Siddhanta, but, correctly speaking, Aryabhatiya Sfitra, consisting of the Dasagiti Sutra, or Dasagitika, and the AryashtaRata, or one hundred and eight couplets. These copies contain the following number of Aryas:-
There are two introductory stanzas at the commencement of copy a, evidently an after-addition, and not in the Arya metre. This is a copy of a manuscript written in Saka 1760, or ad. 1838.
Copy b, Tho date of the original manuscript is omitted.
Copy c is an old manuscript, without date; but, from the character of the letters, and from the worn-out paper, it appears to be more than three hundred years old. It has, in addition, aBhaslrya, or commentary, by Somesvara, containing about 64,000 letters. Unfortunately, portions are illegible or destroyed. The scholiast gives no information about himself, but adds that his commentary is founded upon one by Bhas kara. At the conclusion are the words ill somesvara-virachite dchdryabhatiyam bhdshyam samdptam iti.
B. Of the Maha Arya Siddhanta I have a copy from Gujerat, containing eighteen chapters, copied in Saka 1676, or a d. 1754.
Two others are fresh copies of one or two originals, one of which has the date Saka 1762, or a.d. 1840. The Aryas are from 600 to 612. There is no commentary.
I hope soon to bo in possession of fresh copies of both these Siddhantas, with commentaries.
C. The Brahma Sphuta Siddhanta, or Brahma Siddhanta, of Brahmagupta, from Gujerat; transcribed in Saka 1544, or a.d. 1622. Of this I have sent a copy tomy learned friend, Professor Whitney.
D. The Khanda Khadya Karana, of Brahmagupta, with a Bhashya, by Aina Sarman, son of Pandita Mahadeva, of Anandapura; copied in Samvat 1783, or a.d. 1726. (Anandapura is the modern Wadanagara, in Katyawar). This manuscript was sent tome by a learned Pandit, of Benares, acquaintance I was glad tomake at Delhi last year.
E. The Varaha Sanhita, with the commentary of Bhattotpala. Another copy of the text only.
F. The Brhaj Jataka of Varaha Mihira; two copies, one lithographed in Bombay.
G. The Laghu Jataka of Varaha Mihira.
The Vasishtha Siddhanta. Samvat 1810, or a.d. 1733. The Vyasa Siddhanta.
Tho Brahma Siddhanta.
The Romaka Siddhanta. Copied Samvat 1727,ora.d. 1670.
The Siirya Siddhanta, with the Vasanabhashya.
The Sarvabhauma Siddhanta.
The Tattva Viveka Siddhanta (imperfect)
A commentary on the Siddhanta Siromani by Vachaspati.
The Sundara Siddhanta.
I am glad to announce that, in the Aryabhatiya, or Arya bhatiya Sutra, we have got all the works of the elder Aryabhata, at least all those which were known to Brahmagupta, Bhatta Utpala, and Bhaskara Acharya.
My copies of the Aryabhatiya, or Laghu Arya Siddhanta, are evidently identical with the one which Whish possessed, and with the Berlin manuscript; the latter containing about twenty-seven verses or Aryas more, in the shape of a commentary on the Dasagitika, by Bhutavishnu, which I have not.
Lassen's Aryabhatiya is, undoubtedly, the same work. Brahmagupta having cited and controverted a work of Arya bhata, as Aryashtasata, Colebrooke understood and published "that Aryabhat^a was the author of the Aryashtasata (eight hundred couplets)." That Aryashtasata means eight hun dred couplets is also assumed in the passage I have quoted from Lassen. None of the learned scholars who have written so ably on Aryabhata have impugned the correctness of the translation. Professor Whitney goes so far as to state that Dr. Hall has farther made it at least a probable supposition that the treatise in question (i.e., the Maha Arya Siddhanta) is, in conformity with Colebrooke's earlier conjecture, to be identified with that so often credited to Aryabhatta by the name of Aryatshtasata.
But Aryashtasata, I venture to affirm, means a treatise of one hundred and eight couplets. Ashtadhikam satam Ash tasatam. Tho Arya from Brahmagupta, referring to this, Aryashtasata is as follows; it is in the Tantra D&shanadhy &ya, Chapter xi. Ary& 8 :- (See Picture 5)
"In the Aryashtasata, the Patas (nodes) revolve; in the Dasagitika, they are described as stationary."
These two statements, which led Brahmagupta to censure Aryabhata for inconsistency of doctrine, are to be found in my copies of tho Aryabhatiya Sfttra. They are as follows :- (See Picture 6)
In the twenty-fourth chapter (Sandhyadhyaya), Arya 10, of his Brahma Siddhanta, Brahmagupta tells us that- (See Picture 7)
Bhata Brahmacharya, the son of Jishnu, mathematician and astronomer, composed the Brahma Sphuta Siddhanta, in 1008 Aryas." The word Aryasbtasahasrena means 1008, and not 8000; and my copy of Brahmagupta's Siddhanta consists of the former number of couplets.
Analogous examples may be produced from the Smrtis, where numbers are given. But, to set aside all doubt of the correctness of my translation of the word Aiyashtasata, I produce nearly all the passages in the Aryabhatiya Sutra which have been controverted by Brahmagupta:- (See Pictures 8, 9, 10 and 11)
These extracts are given as in the original, without any attempt at correction. Colebrooke quotes and translates the following passage as from Aryabhata, cited by Prthudaka :- (See picture 12)
"The sphere of the stars is stationary; and the earth, making a revolution, produces the daily rising and setting of stars and planets."
With regard to this passage, Dr. Hall remarks that he has not sought it out in his manuscript of the Maha Arya Sid dhanta. Nor would he find it there. The fact is, the Laghu Arya Siddhanta ismetrical; and tho passage quoted is not so.
This is observed by Dr. Hall; but he went in the wrong track in stating that "this extract might go to prove that Arya, besides his works in verse, wrote others in prose."
I do not find the passage, literally, in either of the Siddhantas; but I have no doubt that it is only a paraphrase of the following line from Aryabhata in Prathudaka's own words; Aryabhatiya, Golapada, Arya ix.:- (See Picture 13)
" As a person in a vessel, while moving forwards, sees an immoveable object moving backwards, in the samo manner do the stars, however immoveable,seem to move (daily). At Lanka (i.e. at a situation of no geographical latitude) they go straight to the west (i.e. in a line that cuts the horizon at right angles, or, what is the same, parallel to the prime vertical at Lanka)."
I have no doubt that, in the following passage, which has been a theme of fruitful discussion, the first line only is from Arya bhata (it is the latter half of the fourth Arya of the Kfilapada chapter, in my MS.); the second line is, in all likelihood, an addition by Bhatta Utpala : (See picture 14)
" The revolutions of Jupiter, multiplied by the number of the signs (twelve), are the years of Jupiter, called Asvaj'uja, etc.; his revolutions are equal to the number of the Jinas, a couple, the Vedas, the seasons, the fires (i.e. 364,224)."
Aryabhata having an alphabetical notation of his own, it was surprising to find him make use of our arithmetical notation, which the second line quoted above implies. But now the enigma is solved: the second line is not of Aryabhata.
The Dasagiti Sutra, as the namo purports, is composed of ten Aryas; the three additional ones, in my copies, relating to:- 1. invocation, 2. the alphabetical notation, and 3. the fruit or advantage of knowing the Dasagiti Siitra. The Aryashta sata consists of three chapters, viz., 1. Ganita; 2. Kalakriya; and 3. Gola. As the Aryabhatiya consists of the Dasagiti Sutra and Aryashtasata, the treatise consists of four chapters, called padas, of which the Dasagiti Sutra is the first, and the remaining three as above. This arrangement was not clearly perceived by the learned writer of the additional note to Dr. Hall's paper.
Any one studying the Commentary of Munisvara, alias Visvarftpa, styled Marichi, on the Siddhanta Siromani of Bhaskaracharya, and also his Sarvabhauma Siddhanta, cannot fail to remark that he clearly notices two Aryabhatas. He calls, however, the author of the so-called Maha Arya Sid dhanta, Laghu Aryabhata, or Aryebhata junior; and, in quoting from the senior, calls him simply Aryabhata. Sometimes Aryabhata junior appears only as plain Aryabhata; in all likelihood, from the fault of copyists.
Ganesa, in his commentary on Bhaskaracharya's Lilavati, had only to deal with Aryabhata junior; and, accordingly, he speaks of onljr one Aryabhata.
The Sundara Siddhanta of Jnanaraja also distinguishes a Laghu from a Vrddha Aryabhata. Besides the passages referred to by Brahmagupta, all those cited by Bhatta Utpala and Bhaskara Acharya are found in my copy of the Aryabhatiya. There cannot now, therefore, possibly be any doubt of our having a correct copy of the works of Aryabhata senior ; and I shall take tho earliest opportunity of publishing the original text, with the commentary of Somesvara, for the gratification of learned orientalists.
I shall now proceed to an examination of Aryabhata's age. Aiyabhata was born in a.d. 47G. This date is given, by him self, in the Aryashtasata, as follows :- (See Picture 15)
Dr. Kern quotes a corrupt version of it from the Berlin manuscript. The correct translation is : " When three of the four Age-quarters and 60 x GO [ = 3600] years are past, then are past upwards of twenty-three years from my birth."
This gives, for the birth of Aryabhata, the year a.d. 476; as the fourth Age-quarter, or Kali Yuga, commenced 3101 years be fore Christ.
Strange to say, the commentator Somesvara understands the verse to mean that 3623 years had elapsed of the Kali Yuga at the birth of Aryabhata. The commentator whom Mr. Whish consulted fell, perhaps, into the same error; as Mr. Whish does not appear to have given the exact date, but refers Aryabhata simply to the beginning of the sixth century of the Christian era. Aryabhata calls himself a native of Kusumapura, or Pataliputra. Beyond a doubtful allusion to the Brahma Siddhanta, ho never mentions any previous author, and rather prides himself on his originality. His work is written with great attention to conciseness. His system of notation is quite original. The account given of it by Lassen is not altogether correct. In the invention of a new system of notation, the Indian Aryabhata may be compared to the Grecian Archimedes. The ratio of the diameter to the circumference was given, by Archimedes, in his book De dimensione circuit, as seven to twenty-two; while that of Aryabhata is as one to 3.1416. Aryabhata has the following notice of the Buddhist system of measuring time:- (See Picture 16)
This arrangement is different from that of Hemacharya and other Jaina authorities.
As we have already seen, cites Aryabhata by name in the passage given by Bhatta Utpala from Varaha's Pancha Siddhantika Karana, quoted in his commentary on the Varahi Sanhita.2 Tho Pancha Siddhantika Karana I have as yet failed in recovering. Colebrooke assigned to him the close of the fifth century of the Christian era, from a calculation of the position of the colures affirmed as actual in his time by Varaha Mihira. Since then no further solid data have been discovered to fix the era of this celebrated astronomer. I havo failed to find out the date of his birth, but am glad to have discovered that of his death in a commentary on the Khanda Khadya of Brahmagupta, by Amaraja:- (See Picture 17)
"Varaha Mihira Acharya went to heaven in the 509th [year] of the Saka Kala, i.e. a.d. 587."
There is no reason to doubt the correctness of this statement.
The following passage, quoted by Bhatta Utpala, in his commentary on the Varahi Sanhita, from a Tantra by Varaha Mihira, shows that he flourished after Saka Kala 427, I.e. after A.D. 505:- (See Picture 18)
Tho astronomers of Ujjayini place Varaha Mihira in the beginning of tho fifth century. It is not unlikely that they have mistaken the Saka Kala for that of the Vikrama Samvat, or fallen into the same error as Albiruni.
Albiruni states that 526 years had passed up to his own date, a.d. 1031, from the date of the composition of the Pancha Siddhantika of Varaha Mihira;2 which event must, therefore, be placed in a.d. 505. But in this he is evidently in error. Varaha Mihira adopted the epoch of the Roinaka Siddhanta, for finding the number of civil or natural days (Ahargana) and Albiruni or his informants have assumed this to be the date of Varaha's Pancha Siddhantika, which adopts the epoch of the Romaka Siddhanta.
Varaha Mihira, as we have seen, on the authority of Amaraja, died in a.d. 587, i.e. a century and eleven years after the birth of Aryabhata. The so-called Khanda Kataka of Albiruni is evidently Brahmagupta's Khanda Khadya; and Arcand is a corruption of Karana. From various considerations, the middle of the sixth century has been allotted to Varaha by Colebrooke, and by the learned American translator and commentator of the Silrya Siddhanta. The passage I have quoted from Bhattotpala confirms these conjectures.
This calculation, it proceeds to tell us further, is that of the Romaka Siddhanta; and, judging by similar calculations and other Siddhantas, it appears to me highly probable that the Romaka Siddhanta was composed in Saka 427, or a.d. 505. Varaha Mihira founded bis Pancha Siddhantika Karana on the Romaka Siddhanta and four others. It is clear, therefore, that Varaha could not have lived before a.d. 505. In all likelihood, his birth was twenty or thirty years subsequent to this date; which would make Varaha Mihira about fifty or sixty years old at the time of his death in a.d. 587. In an essay on Kalidasa, I have attempted to show that the nine gems of Indian celebrity flourished at the Court of Ilarsha-Vikramaditya, at Ujjayini, in the sixth century of tho Christian era. As we have now established the date of Varaha Mihira beyond a doubt, my hypothesis that Matrigupta is identical with the celebrated Kalidasa gains further support.
The authors of the Paulisa, Romaka, Vasishtha, Saura, and Faitamaha Siddhantas being noticed by Varaha Mihira, it follows that they flourished before a.d. 505. Brahmagupta affirms that Srishena, the author of tho Romaka Siddhanta, bases his calculations on those of Lnt a, Vasishtha, Vijayanandin, and Aryabhata, and that Vishnuchandra, following the same guides, wrote the Vasishtha Siddhanta. As wo have the date of a.d. 508 for the Itomaka Siddhanta, it is evident that Lata, Vijayanandin, and Vasishtha flourished before that time. I presume that the Romaka Siddhanta was composed in accord ance with the work of some Roman or Greek author; just as the Paulisa Siddhanta was composed from the work of Paulus Alcxandrinus. Is not Latacharya also a foreign author?
Bhatta Utpala notices also a strange author, of the name of Sphujidhvaja or Asphujidhvaja, as a Yavancsvara, who com posed a new Sastra before the Saka era. His works appear, from the following passage,1 to have been consulted by Bhatta Utpala, who remarks that Varaha Mihira consulted the works of other Yavana authors not available to himself. I believe the word Sphujidhvaja is a corruption of the Greek name Speusippus. Diogenes Laertius mentions two authors of this name, one of whom was a physician called Hcrophileus Alcxandrinus, and may, possibly, be the astronomer whose works were translated and studied in India.
Varaha Mihira's knowledge of Greek technical astronomical terms and doctrines has been fully treated of by Weber and others. The verse in which he gives the Greek terms for the Sanskrit names of the signs of the Zodiac, has hitherto been presented to us, except by Mr. Whish, in a corrupt form;2 as the following will show that the last puzzling word is the veritable Greek IxQves or Pisces : (See Picture 19)
I have failed, as yet, to discover the Paulisa and Romaka Siddhantas. I have two old copies of a Romaka Siddhanta ; but it is, evidently, a modern production, taken from some Arabian author, and having nothing in common with the citations and references to the older Romaka Siddhanta preserved in the works of Varahamihira, Brahmagupta, and Bhattotpala. (See Picture 20)
has given his own date in the Brahmasphuta Siddhanta. I gave this extract in my Essay on Kalidasa, three or four years ago; but it has escaped the attention of one of our ablest and most accurate scholars. (See Picture 21)
"In the reign of Vyaghramukha, of the Sri Chapa dynasty, five hundred and fifty years after the Saka King (i.e. Sailivahana) or a.d.628, having passed, Brahmagupta, the son of Jishnu, at the age of thirty, composed the Brahmagupta Siddhanta, for the gratification of mathematicians and astronomers" (chap. 24. Aryas 7-8).
His age has been quoted from his work by Colebrooke and others : (See Picture 22)
(See Picture 23)
has stated, in his Siddhanta Siromani, the date of the com position of his work as follows :-
This date has been singularly confirmed by an inscription which I discovered, some years ago, in the neighbourhood of the railway station of Chalisgam, about seventy miles beyond Nassik. This inscription gives also the names of several of Bhaskaracharya's descendants, who taught his works in a college endowed in the neighbourhood of Chalisgam, at the foot of the hills which contain the Peetulkhora caves. In the reign of Sri Vyaghramukha,
In this inscription we have the names of the following kings of theYadu dynasty: Bhillaraa, Jaitrapala, and Singhana. Under this dynasty is the following genealogy of subordinate chiefs of the solar race: Krshnaraja, his son Indraraja, his son Govana, and his son Sonhadeva. This last makes a grant of certain privileges, in the shape of perquisites, or first-gifts, to a college established by Changadeva, (within six miles of the railway station at Chalisgam, and at the foot of the Pitalkhora caves), now entirely deserted and in ruins, in the year 1128 Saka-kala, i.e., a.d. 1206, on the occasion of a lunar eclipse.
Changadeva was the son of Lakshmidhara, who was patro nized by Jaitrapala, as the chief of his Pandits. Lakshmi dbara's father was the celebrated Bhaskarachfirya. His father was Ka vis vara Millies varacharya; and his father was Manoratha; his father, Prabhakara; his father, Govinda Sarvajna; his father, Bhaskara Bhatta, towhom Bhoja gave the title of Vidyapati. His father was Trivikrama. This learned family, we are further told, belonged to the gotra of Sandilya.
The names of Bhillama, Jaitrapala, and Singhana occur in two copper-plate grants, Nos. 9 and 10 of Mr. Wathen's series (Journal of the Royal Asiatic Soc. v. pp. 178, 183).
In these we have also the names of the successors of grants Singhana, e>.,Krshna, Jaitrapala (Singhana's son), his younger brother Mahadcva, who succeeded him, and Ramachandra, the son of Krshna, the dates of whose grants are Saka 1212 and 1194. The donor, Sonhadeva, in our inscription, dated Saka 1128, is a contemporary and subordinate of Singhana, whose third lineal descendant was Ramachandra, in whose reign the grants of Saka 1212 and 1194 are dated.
These dates, therefore, are perfectly compatible with, and confirmatory of, the accuracy of that of the inscription.
Bhaskaracharya, the author of the Siddhanta gives the date of his birth as follows: (See Picture 24)
"In the year 1036 of the Saka king, I was born; And at the age of 36,1 composed the Siddhanta Siromani."
This date is quite in accordance with that of the inscription. The following lines, in praise of Bhaskaracharya's accomplishments, are sometimes found to be MSS of the Lilavati : (See Picture 25)
An important fact to be noticed is, the bestowal of tho title of Vidyapati on Bhaskara Bhatta, an ancestor of Bhaskaracharya, by Bhoja.
This Bhoja is, undoubtedly, the monarch of Dhara, whom it is the fashion to speak of as distinguished for his patronage of learning.
In the Raja Mrganka Karana, attributed to Bhoja, a copy of which was brought to me, by my Pandits, from Jessulmere, whither I sent them in search of Jaina MSS. two years ago, its author recommends 964 to be subtracted from tho Saka era to find the Ahargana, i.e. "the sum of days," in civil reckoning.
Bhoja, therefore, it follows, flourished in 964+78 = 1042 a.d. At the end of the work he is called Rana Ranga Malla, a title also to be found in the Pataujala Yoga Sutra Vrtti. My pandits have succeeded in getting a very imperfect copy of the Tilaka Manjari of Dhanapala, who is said by Merutunga, the author of the Prabandha Chintamani, to be tho author of that work, and to have been a favourite of Bhoja.
Merutunga adds that fromVikrama Samvat 1076, or a.d. 1019, Bhiiiinraja reigned. Ho was succeeded by Karna in 1028. At the time the Chfilukya Bhima reigned in Gujerat, Bhoja says Mcrutuhga ruled in Maiava. According to this Jaina hierarch, Munja was imprisoned by Tailapa, the Chalukya, who commenced his reign in Saka 895, or a.d. 973, according to the Copper plate grants of Mr. Wathen. Dhanapala was a Brahmana, but, evidently, a Jaina by faith. He composed the Jina Stavana, or Gathas in Magadhi, in praise of Jina, of which a complete copy was brought me from Jessulmere. The Jainas are loud in the praises of Dhanapala.
Mr. Bentley speculated that Bhaskar&charya flourished after Akbar the Great. Our inscription furnishes a new proof that the unfavourable opinion of Hindu veracity which led to this conclusion was utterly unfounded.
Mathematics in Ancient India: Aryabhata, Varahamihira, Brahmagupta, etc