Brahmagupta

Dates: 598–c. 670 CE
Period: Roman (30 BCE–800 CE)

Brahmagupta was an Indian astronomer who worked at Bhillamala, now Bhinmal in Rajasthan, in the 7th century. At the age of 30, in 628, he finished the Brahmasphutasiddhanta, a treatise of 24 chapters on the motions of the planets, eclipses and the calculations they require. His mathematics fills 2 of those chapters.¹

His greatest attainment sits in chapter 18. There he gave zero and the negative quantities a set of working rules, calling positive numbers fortunes and negative numbers debts: “A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt.” With those rules a calculation could keep track of absence and of what is owed as well as of what is held.²

Zevism reads the work under Theuth, the God who in Plato's account invented number, calculation, geometry and astronomy. Brahmagupta served the same order under his own tradition's name: the title of his book means “Correctly Established Doctrine of Brahma.”³

LIFE AND CONTEXT

Brahmagupta was born in 598, the son of Jiṣṇugupta. He was known as Bhillamālācārya, “the teacher from Bhillamala,” and when he wrote his great book the city's king was Vyāghramukha, of the Cāpa dynasty of the Gurjaras.⁴ MacTutor makes him head of the observatory at Ujjain, “the foremost mathematical centre of ancient India at this time.”⁵

He didn't start from nothing. Most of his planetary parameters came from the older Paitāmahasiddhānta, and he gave a whole chapter, the 11th, to an examination of earlier astronomical treatises.⁴ He attacked Jain cosmology and rejected Aryabhata's claim that the Earth is a spinning sphere.⁶ In his own system the Earth stood still.⁵

The book covered the whole working astronomy of its day: the mean and true longitudes of the planets, lunar and solar eclipses, risings and settings, the Moon's crescent and the conjunctions of the planets with one another and with the fixed stars.⁵ Its mathematics fell into 2 fields. Pāṭīgaṇita, “mathematics of procedures,” covered arithmetic and measurement; bījagaṇita, “mathematics of seeds,” covered equations.³

He kept working for nearly 4 decades. The Khandakhadyaka, a handbook of 8 chapters, takes as its epoch Sunday, 15 March 665, when he was 67.⁴ In it he adopted Aryabhata's practice of starting each day at midnight, a concession to the rival he'd attacked at 30.⁶ His figure for the year moved as well, from 365 days 6 hours 5 minutes 19 seconds in the Brahmasphutasiddhanta to 365 days 6 hours 12 minutes 36 seconds in the Khandakhadyaka.⁵

He died some time after 665; MacTutor puts his death about 670. The books went on without him. Pṛthūdakasvāmin wrote a commentary in 864, and al-Biruni read both works in Sanskrit in the 11th century and cited them at length.⁴ In 1817 Henry Thomas Colebrooke published chapters 12 and 18 in English in London, in the same volume as the Sanskrit algebra of Bhāskara.⁷

ATTAINMENTS

• He defined zero as the result of subtracting a number from itself and gave rules for arithmetic among negative numbers. The product or quotient of 2 debts, his rule says, “is one fortune.”⁸
• He treated zero as a number to compute with: a number plus or minus zero stays unchanged, and “the product of zero multiplied by zero is zero.” His rule that “zero divided by zero is zero” fails, since every number multiplied by zero gives zero, and the record keeps the error beside the rules that hold.⁹
• In chapter 12 he gave the area of a quadrilateral as the square root of (s−a)(s−b)(s−c)(s−d), with s half the perimeter. The rule is exact when the 4 corners lie on a circle. For a rectangle with sides 5, 12, 5 and 12, s is 17 and the area comes out at 60.¹⁰
• In 628 he set out the method Indian mathematicians called composition, which builds new solutions of equations of the form Nx² + 1 = y² from a known one.¹¹ With it he solved the case N = 92, whose least solution is x = 120, y = 1151.¹²
• He stated the sum of the squares of the first n natural numbers as 1/6 n(n+1)(2n+1), and the sum of their cubes as the square of n(n+1)/2.⁵
• He measured the year twice and revised his own result, by 7 minutes and 17 seconds, between 628 and 665.⁵
• His astronomy crossed into Arabic. Britannica dates the Arabic translation of the Brahmasphutasiddhanta at Baghdad to about 771 and credits it with a major impact on Islamic mathematics and astronomy.⁶

KEY STORIES

The Challenge of 92

Chapter 18 sets problems as well as rules. One of them asks for a square that, multiplied by 92 and increased by 1, gives another square. Brahmagupta attached a dare to it: “One who can solve it within a year [is truly a] mathematician.”¹³

The trick began with a near miss. Multiply 1² by 92 and add 8, and the result is 10². Brahmagupta's composition, the bhāvanā, combined that triple with itself and produced 92 × 20² + 64 = 192². Dividing through by 64 gave a solution in fractions, x = 5/2 and y = 24. One more composition turned it into whole numbers: 92 × 120² + 1 = 1151².¹⁴ He didn't stop at 92. He handled 83 the same way, starting from 83 × 1² − 2 = 9² and composing his way to x = 9 and y = 82.¹¹

Europe took up the equation 1,029 years later. In 1657 Pierre de Fermat, writing to Frénicle de Bessy, challenged mathematicians with the same equation for N = 61. Its least solution is x = 226,153,980 and y = 1,766,319,049.¹⁵ Bhāskara II had already solved that case in India in 1150.¹⁴

The Dragon and the Eclipse

Brahmagupta fought his predecessors in print. He set himself against Aryabhata's spinning Earth, and the 11th chapter of his book puts the older treatises on trial.¹⁶

About 1030 a reader from another world turned the same scrutiny on him. Al-Biruni, a Muslim scholar writing his great book on India, read Brahmagupta's works in Sanskrit. He accused Brahmagupta, in the summary of his translator Edward Sachau, “of injustice and rudeness to his predecessor, Aryabhata.”¹⁷

The heavier charge concerned eclipses. Brahmagupta taught 2 theories side by side: the popular one, in which the dragon Rāhu devours the luminous body, and the scientific one. Sachau sums up al-Biruni's verdict: Brahmagupta “certainly committed the sin against conscience from undue concessions to the priests of the nation, and from fear of a fate like that which befell Socrates when he came into collision with the persuasions of the majority of his countrymen.”¹⁸

The Book That Went to Baghdad

In 771 or 773 an embassy from Sind reached the court of the caliph al-Manṣūr in Baghdad. Among its members was an Indian astronomer whose name hasn't survived. The caliph asked Muḥammad al-Fazārī to work with him on an Arabic translation of a Sanskrit astronomical text.¹⁹

Edward Sachau wrote that the scholars from Sind brought 2 books of Brahmagupta, the Brahmasiddhanta and the Khandakhadyaka, known in Arabic as the Sindhind and the Arkand, and that al-Fazārī, perhaps with Yaʿqūb ibn Ṭāriq, translated them with the help of the pandits.²⁰ Pingree later identified the text as a Mahāsiddhānta of the same school, whose closest cousin was the Brahmasphutasiddhanta. Al-Fazārī built his own Zīj al-Sindhind al-kabīr from it.²¹

So 143 years after its completion, the astronomy of Bhillamala was being worked in Arabic by the caliph's astronomers. Zevism honours such translators as keepers of the source: without the unnamed Indian and al-Fazārī, the chain from Brahmagupta to Baghdad would have broken at the border of a language.

THE ZEVIST READING

Plato's Phaedrus tells of Theuth, the Egyptian God of Naucratis: “He it was who invented numbers and arithmetic and geometry and astronomy, also draughts and dice, and, most important of all, letters.”²² Brahmagupta's book holds 4 of those arts in a single binding. Astronomy fills most of it, arithmetic and geometry stand in chapter 12, and equations in chapter 18. Zevism reads him as a servant of Theuth's domain.

He was a Hindu astronomer, and he named his work for Brahma. Zevism is the chain of all spirituality from its inception to today, and it reads the astronomer of Bhillamala as a link in that chain, serving the divine order under the name his own tradition gave it.

Zero and debt are the dark country of number: nothing in hand, and less than nothing. Zevism teaches that unexplored darkness isn't evil. It's the region one enters carrying light, and once explored it becomes Light, knowledge that others can use. Brahmagupta walked into that region with rules, and debts became quantities a person could add, multiply and divide. Where his light gave out, at zero divided by zero, the book records the failure plainly, and a later reader can see exactly where the work must continue.

Ma'at, the order of the cosmos held by Zeus as its Sovereign, is also the order of a true account. Egyptian religion named her “the personification of truth, justice, and the cosmic order” and set her against isfet.²³ A reckoning that drops the sign of a debt is a false record, a small form of Izfet. The rules of fortunes and debts keep the record true.

Zevism also holds that knowledge changes in transmission while the source keeps priority. The embassy from Sind, al-Fazārī's tables, al-Biruni's Sanskrit reading and Colebrooke's English all lead back to one treatise finished at Bhillamala in 628, and the Temple returns to that treatise as the source.