Bhaskara II Completed the Siddhanta Shiromani in 1150, and Its Algebra Solved the Equation Fermat Set as a Challenge to Europe in 1657

Bhaskara II completed the Siddhanta Shiromani in 1150 CE, and by his own count he was 36 years old. Its algebra section, the Bijaganita, works through the equation 61x² + 1 = y² and gives the smallest answer in whole numbers: x = 226,153,980 and y = 1,766,319,049. Pierre de Fermat set the same equation to the mathematicians of Europe and England as a challenge in 1657, according to MacTutor, the history-of-mathematics archive at the University of St Andrews.

The same work holds the Lilavati, a 13-chapter text of arithmetic and geometry written as problems in verse, and a two-part astronomy covering eclipses, planetary positions and instruments. David Pingree, in the Dictionary of Scientific Biography, lists 35 commentaries on the Lilavati, the first of them Narayana’s of 1356. The Kerala mathematicians who found infinite series for sine and pi around 1400 are the subject of a separate piece. This one follows Bhaskara.

What he says about himself

Henry Thomas Colebrooke, who translated the Lilavati and the Bijaganita into English in 1817, wrote that the date of Bhaskara’s work was “ascertained with unusual precision”. Bhaskara says in the book that he finished it in Shaka year 1072, aged 36, and that he was born in Shaka 1036, Colebrooke reports. That gives 1150 for the book and, in most modern sources, 1114 for his birth. Pingree’s entry is headed 1115. A shorter manual for working out astronomical positions, the Karanakutuhala, is dated 1183 in both Pingree’s entry and the Biographical Encyclopedia of Astronomers.

His father was Maheshvara, of the Shandilya lineage, who wrote a popular work on astrology, Pingree notes. The closing verses of the Siddhanta Shiromani name his city as Vijjadavida. Pingree reads it as Bijapur, now in Karnataka, and adds the caveat “if this identification is correct”. Both MacTutor and the encyclopedia give Vijayapura, the same city, as his birthplace. At Ujjain, then the leading mathematical centre in India, where Varahamihira and Brahmagupta had worked, MacTutor says he became head of the astronomical observatory. MacTutor and the encyclopedia also agree on Ujjain and 1185 for his death.

The family kept to the same work. Pingree names Bhaskara’s son Lakshmidhara, appointed chief of the pandits by King Jaitrapala, and his grandson Changadeva, chief astrologer to King Singhana.

The Lilavati: thirteen chapters of problems in verse

Chapter by chapter, MacTutor lists definitions, arithmetical terms, interest, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon (an upright stick used to read time and direction from its shadow), the kuttaka, and combinations. Known as the pulveriser, the kuttaka is the method for equations whose answers must be whole numbers, which Aryabhata’s mathematics chapter had set out in 499.

Verse problems carry the book. In one, a king marching to seize an enemy’s elephants covers two yojanas, an old measure of distance, on the first day and must reach a city 80 yojanas away in a week. Bhaskara shows, MacTutor reports, that he must add 22/7 yojanas to each day’s march. In another, a swarm of bees has divided so that the square root of half the swarm and eight-ninths of the whole have gone to a jasmine shrub, while one male and one female remain. Colebrooke’s translation gives the answer as 72 bees.

One of the two multiplication methods in the Lilavati puts off carrying figures until the final addition. MacTutor follows Georges Ifrah’s account of it, using 325 × 243: Bhaskara writes the multiplicand once for each digit of the multiplier, multiplies digit by digit without carrying, and adds the pieces to reach 78,975.

For the circle, the translation gives two rules. Multiply the diameter by 3,927 and divide by 1,250 for the near circumference, or multiply by 22 and divide by 7 for a rougher figure that suits everyday use. Worked out, the first ratio is 3.1416 to four decimal places. Colebrooke notes that Bhaskara adds it “apparently from some other authority”. Counting closes the book: how many numbers of a given length have digits adding to a given sum, MacTutor says.

The Bijaganita: negative numbers, two roots and zero

Twelve chapters make up the Bijaganita, per MacTutor and Pingree. They cover positive and negative numbers, zero, the unknown, surds (square roots that do not come out even), the kuttaka, quadratic equations and equations with several unknowns. Negative numbers get a dot above them. In Colebrooke’s translation the rule is that the square of a positive or a negative quantity is positive, that a positive quantity has two roots, one positive and one negative, and that “There is no square-root of a negative quantity”.

Bhaskara also accepts equations with two valid answers. In his problem of the apes, a pack where the square of one-eighth of the animals plays in a forest and twelve sit on a hill, the quadratic gives 16 or 48, and he calls both admissible, MacTutor says. In MacTutor’s judgment, he understood number systems and the solving of equations at a level Europe did not reach for several centuries. Colebrooke adds a smaller point: the first of the two proofs of the Pythagorean theorem in the Bijaganita is, he says, the one that Wallis gave in his treatise on angular sections.

The equation Fermat set Europe in 1657

Pell’s equation asks for whole numbers x and y with px² + 1 = y² for a given p. Smallest answers can be very large. Bhaskara worked it for p = 8, 11, 32, 61 and 67. For 67 the pair is x = 5,967 and y = 48,842. For 61 it is the nine-digit x quoted at the top, and the pair checks out: 61 × 226,153,980² + 1 equals 1,766,319,049².

MacTutor credits Bhaskara with the chakravala, or cyclic method, in 1150. It starts from a pair that almost fits and improves it step by step until it reaches a case that Brahmagupta’s method of 628 finishes. Colebrooke judged in 1817 that the Hindu way of deriving many answers from one found by trial came as near to a general solution as anyone did until Lagrange.

Europe’s record starts later. MacTutor says that the methods of Brahmagupta, Bhaskara II and Narayana, who wrote a commentary on the Bijaganita in the 14th century, were completely unknown to European mathematicians of the 17th century. Fermat issued his challenge in 1657, and one of its problems was 61x² + 1 = y², the case Bhaskara had solved. Brouncker found a method, and Wallis published the exchange of letters in 1658. Euler named the equation after Pell, it is generally believed because he confused Pell with Brouncker, MacTutor says. Lagrange published a rigorous proof in 1771. Colebrooke also saw that the general solution of first-degree whole-number equations, first given in Europe by Bachet de Meziriac in 1624, is “precisely the method of the Hindu algebraists”.

Eclipses, spheres and an early rate of change

Pingree describes the Siddhanta Shiromani in two parts. Part one, the Ganitadhyaya, has 12 chapters on mean and true positions of the planets, eclipses, planetary latitudes, risings and settings, the new crescent moon and conjunctions. Part two, the Goladhyaya, has 13 chapters on the sphere, including an armillary sphere (a model of nested rings for the celestial circles), spherical trigonometry, instruments and the seasons. Bhaskara also wrote his own commentary, the Vasanabhasya. MacTutor adds that he seems more interested in trigonometry for its own sake than his predecessors, who saw it only as a calculating tool, and quotes his formula for the sine of a sum of two angles.

A claim of early calculus attaches to this book, and it has a named history. Pingree’s bibliography lists an 1858 paper by Bapudeva Sastri titled “Bhaskara’s Knowledge of the Differential Calculus”, followed in 1860 by a note from W. Spottiswoode on the “supposed discovery” of that principle by an Indian astronomer. A 2023 paper in the Indian Journal of History of Science, by Venketeswara Pai R. and M. S. Sriram, states in its abstract that derivatives were used to find the rates of motion of planets in Indian astronomy from the Laghumanasa (about 932 CE) onward. In the Vasanabhasya, it says, Bhaskara explains why instantaneous rates of motion (tatkalikagati) are needed, which involves the derivative of the sine function.

That is a rate rule inside astronomy, used to find a planet’s speed at a given instant. MacTutor reports that Madhukar Mallayya reads the zero in Bhaskara’s rule (a × 0) / 0 = a as a modern infinitesimal, a quantity smaller than any number, and adds that this may be seeing ideas beyond what Bhaskara intended. Nothing in MacTutor’s own account of his results mentions calculus. Infinite series came later, and they are told in the Kerala piece linked above.

The daughter, the pearl and what the text says

A story attached to the Lilavati has Bhaskara cast the horoscope of his daughter Lilavati and find the right hour for her wedding. He set a cup with a small hole in a vessel of water, timed to sink at that hour. She leaned over it, a pearl from her dress fell in and blocked the hole, and the hour passed. To console her, he wrote her a book of mathematics. MacTutor retells it from G. G. Joseph’s book and says the Persian translator Faizi told it in 1587.

This is a tradition, and the scholars treat it that way. MacTutor says it is “hard to see that there is any evidence for it being true”, that it is not certain Lilavati was his daughter, and that one theory makes her his wife. Pingree describes the book as addressed to a lady, “(his daughter or wife?)”. Within the text, Bhaskara speaks to a woman. In Colebrooke’s translation one example calls her “Beautiful and dear” and gives her eyes “like a fawn’s”. A footnote there says the name is “Seemingly the name of a female to whom instruction is addressed”, and adds that some commentaries take it, in line with its etymology, to mean “Charming”.

Division by zero, missing proofs and a forged seventh book

Bhaskara gave a rule for dividing by zero. In the Bijaganita he wrote that a quantity divided by zero “becomes a fraction the denominator of which is zero”, and called it an infinite quantity. MacTutor says this does not hold, because if n/0 were infinite then zero times infinity would equal every number n, and notes that the Indian mathematicians could not bring themselves to admit that division by zero is impossible. Brahmagupta’s earlier rules for zero are covered in the piece on zero.

MacTutor also argues that Bhaskara had no proof of the cyclic method, since the proof is long and difficult and would be well beyond 12th-century mathematics. Six works are accepted as his. A seventh, the Bijopanaya, claims to be from 1151, and Pingree concludes from the work of Kuppanna Sastri that it was forged in south India in the early 1810s.

A school in 1207, three Persian versions and two satellites

Found near Chalisgaon in Maharashtra, the Patna inscription is dated to a lunar eclipse in Shaka 1128, which the Epigraphia Indica equates with 9 August 1207. It records that Changadeva founded a college for the study of the Siddhanta Shiromani. Two Nikumbha brothers, Soideva and Hemadideva, feudatories of the Yadava king Singhana, endowed it. MacTutor adds that a medieval temple inscription praises him as “like the crest on a peacock”.

Faizi’s Persian Lilavati of 1587 was made at the order of the Emperor Akbar, Colebrooke records, and Pingree knows of two more Persian versions, from about 1663 and 1678. Shah Jahan had the Bijaganita translated in 1634 and 1635. Into English, J. Taylor’s version was printed at Bombay in 1816 and Colebrooke’s followed the next year.

In Pune the Bhaskaracharya Pratishthana was founded in 1976 by the mathematician Shreeram Abhyankar. It has been a recognised centre for the Regional Mathematics Olympiad since 1992, and the institute says that students it trained have won more than 50 medals at the International Mathematical Olympiad.

India’s first satellite, named for an earlier astronomer, has its own article. ISRO’s page for Bhaskara-I, which carried television cameras and a microwave radiometer, gives its launch date as 7 June 1979. Gunter’s Space Page lists a second satellite launched on 20 November 1981 and says the pair are named after the two Bhaskaracharyas.