Mahaviracarya's Ganita Sara Sangraha, Written Around 850 CE Under the Rashtrakuta King Amoghavarsha, Packs Nine Chapters and More Than 1,130 Verses of Rules and Worked Problems From Fractions to the Ellipse

Mahaviracarya’s Ganita Sara Sangraha, written around 850 CE, is a Jain mathematician’s textbook in nine chapters. Britannica, in an entry by the historian of mathematics Takao Hayashi, counts more than 1,130 verses of rules and worked examples. They run from number names and fractions through equations, series, areas and volumes to shadows.

It was composed in the southern Deccan in the reign of the Rashtrakuta king Amoghavarsha. Its opening chapter names twenty-four decimal places, gives rules for arithmetic with zero and negative numbers, and the later chapters pose problems about plantain heaps, a dropped purse and arrows in a quiver. Brahmagupta’s rules for zero came more than two centuries earlier, and MacTutor describes Mahaviracarya’s book as an update of Brahmagupta’s.

A textbook with a king named in its opening verses

The Ganita Sara Sangraha does not give its own date. After two verses of salutation to the Jain teacher Mahavira, the first chapter spends six verses praising a king and wishing his rule prosperity. The verses call him Amoghavarsha and Cakrikabhanjana, and the last blessing names Nripatunga. The text’s 1912 editor and translator, Rangacarya, identified him as the Rashtrakuta king Amoghavarsha Nripatunga. He wrote that epigraphical research puts the reign from 814 or 815 to 877 or 878, and that the book therefore belongs to the middle of the ninth century.

Inscriptions fit that span. Fleet, cataloguing the records of Amoghavarsha I in the Kannada-speaking districts in 1882, lists two at the Kanheri caves, one dated Saka 773 (851 to 852 CE, a figure he gives as a correction of 775) and one dated Saka 799 (877 to 878 CE). Britannica’s entry on the dynasty gives the same reign, about 814 to 878, and says the Rashtrakutas spoke Kannada. It adds that Amoghavarsha was the author of part of the Kavirajamarga, the earliest known Kannada poem. Britannica also credits the dynasty’s earlier king Krishna I with the Kailasa temple at Ellora.

Of the mathematician himself, Britannica says only that he was a Jain and wrote during Amoghavarsha’s reign. Sanatan Koley, in a 2025 paper in the Indian Journal of History of Science, calls him a Digambara Jain who was not an astronomer, unlike his predecessors, and whose work is confined to mathematics. MacTutor says he was possibly born at or near Mysore. David Eugene Smith, who wrote the introduction to Rangacarya’s edition, held that Mahaviracarya “appears to have lived” in the king’s court. That is an inference from the opening verses, and Rangacarya’s own phrase is that the author was “in some way attached” to the court. No patron is named by MacTutor.

Twenty-four places and the rules for zero

Chapter I lists the places of a decimal number from the units place upward. The sixth is the lakh and the eighth the crore, and the list continues to the twenty-fourth place, called mahakshobha. Each place, the preface of Rangacarya notes, is worth ten times the one before. The same chapter, in verses 49 to 52, sets the rules for zero and for positive and negative quantities.

A number multiplied by zero is zero, the verse says, and addition of zero leaves a number as it was. Two negatives or two positives multiplied or divided give a positive, and one of each gives a negative. A positive quantity subtracted becomes negative, and a negative one subtracted becomes positive. The square of a positive or a negative quantity is positive. The verse reasons that a negative is not a square and so has no square root, a point Britannica also records.

On division, verse 49 says a number divided by zero remains unchanged. A footnote by Rangacarya says plainly that this is wrong, and that Bhaskara later gave the quotient as infinity. MacTutor’s history of zero says the same, noting that Mahavira’s attempt to improve on Brahmagupta’s statements on dividing by zero led him into error. The wider story of the decimal system and the zero sign is in the article on India and zero, and Bhaskara’s work is in the article on the Lilavati.

Fractions, pulverizers and the problem verses

MacTutor says Mahaviracarya gave methods for splitting a fraction into unit fractions, with the example 2/17 = 1/12 + 1/51 + 1/68. The three fractions do add to 2/17, since over a common denominator of 204 they make 17, 4 and 3, which total 24. In chapter III he sets a related puzzle, asking the reader to find the denominators when five, six or seven different fractions with 1 on top add up to 1.

Chapter VI, on mixed problems, contains the kuttikara or pulverizer, a method for whole-number solutions of equations that MacTutor describes as based on the Euclidean algorithm. A typical example, verse 117 and a half, asks for the size of one of 63 equal heaps of plantains. Seven more fruits are added and the total is shared among 23 travellers with none left over. The answers appendix gives 5, and the arithmetic holds: 63 times 5 plus 7 is 322, which is 23 times 14.

Why the combined version of the rule succeeds is worked out in Rangacarya’s notes. When one number must satisfy two such conditions at once, he shows, the next solution comes by adding the least common multiple of the two divisors to the first. That derivation is in his footnote, not in Mahaviracarya’s verse, which gives only the procedure.

Verses 236 and 237 pose three merchants who find a purse. The first says that with the purse he would have twice as much as the other two together, the second says three times, the third five times. MacTutor gives the smallest answer in whole numbers: the purse holds 15, and the merchants hold 1, 3 and 5. Checking: 15 plus 1 is 16, twice 3 plus 5. Fifteen plus 3 is 18, three times 1 plus 5. Fifteen plus 5 is 20, five times 1 plus 3.

Verse 288 gives a rule for counting the arrows bundled in a quiver. Add 3 to the number of arrows in the outer ring, square the sum, add 3 and divide by 12. The next verse asks for the case of 18 outer arrows, which gives 37. That is 1 in the middle plus rings of 6, 12 and 18. The rule rests on the fact that six equal circles fit around a seventh, as Rangacarya’s note explains, and Koley, who reads it as a correct formula, cites that note.

Circles, quadrilaterals and an ellipse

Areas fill chapter VII. For a circle, verse 19 takes the circumference as three times the diameter and the area as three times the square of half the diameter. Both are approximate, a footnote by Rangacarya says, because they take pi as 3. For the more exact work, Britannica adds, Mahaviracarya used the square root of 10, the traditional Jain value. Smith noted that Brahmagupta and Mahaviracarya both used 3 and the square root of 10, and that neither seems to have known the work of Archimedes or Heron.

Verse 50 gives the formula that takes the square root of the product of four quantities, each being the half-sum of the sides minus one side. Koley points out that it is Brahmagupta’s formula for the area of a four-sided figure, and that setting one side to zero gives the area of a triangle. On whether the text says it holds only when the four corners lie on a circle, the scholars differ. According to Smith, neither Brahmagupta nor Mahaviracarya observes this. In Koley’s reading it is implied, and the examples in verses 57 to 59 are all of that kind.

Koley singles out the ellipse. He writes that apart from this treatise, no Indian mathematical work contains a formula for the measure of an ellipse. Verse 21 says to add half the shorter diameter to the longer, double the sum to get the circumference, and multiply that by a quarter of the shorter diameter for the area. The worked example, with diameters 12 and 36, gives 84 and 252 in the answers appendix. Verse 63 offers a second, “exact” pair of rules. Both rules are incorrect by Koley’s judgement, since the true area is pi times the product of the two semi-axes. He still calls the attempt appreciable, because it introduced the study of conic sections in India, a study that began in Greece long before.

MacTutor says the text also gives rules for a conch-like plane figure, formed from two unequal semicircles joined along their diameters. It quotes Hayashi that such formulas have so far been found only in the works of Mahavira and Narayana. Permutations and combinations are in the book too, Britannica says, and it calls them traditional Jain subjects. The rule for the volume of a sphere in the eighth chapter is almost correct, Koley writes.

A book that stayed in use in the south

Rangacarya’s preface describes the manuscripts he used. They were five in all, one in Grantha script with a Sanskrit commentary on the first five chapters, two palm-leaf copies in Kannada script, one from the Government Oriental Library at Mysore, and one from a Jain monastery at Mudbidri in South Canara. The Mysore copy carried a short Kannada commentary by one Vallabha. The work seems to have been widely known and appreciated in southern India, Rangacarya writes, and a Telugu verse translation by Pavuluri Mallana was made in the eleventh century. Koley gives the same translation date.

MacTutor quotes Filliozat as saying that the book, though as condensed as other versified texts, “has a significant advantage over earlier texts” from a teaching point of view. In 1912 Smith wrote that the edition revealed a writer almost unknown to European scholars. Rangacarya says it is not easy to tell whether Bhaskara knew the book, since neither Bhaskara nor any known commentator of his seems to quote it or name its author, and this piece makes no claim of influence.

Where the sources differ

Smith, Koley, Britannica and MacTutor’s biography all date the book to about 850. MacTutor’s history of zero, a separate page, says 830. The life dates also vary: MacTutor gives about 800 to 870, Koley about 800 to 875, and Britannica says only that he was active about 850.

The claim that this is the earliest Indian text devoted wholly to mathematics rests on MacTutor, which says so of the book, and on Koley, who repeats it. Koley’s own introduction says that Sridhara’s Trisatika, from about 750 CE, treats only mathematics. The sources opened here do not reconcile the two statements, so this article does not use the superlative. Rangacarya signed his preface in Madras in February 1912.