Brahmasphutasiddhanta

01At Bhillamala

Brahmagupta, son of Jishnu, tells us he wrote in the reign of King Vyaghramukha of the Chapa line, 550 years after the Shaka era began, when he was thirty years old. That is 628 CE, so he was born in 598. Bhillamala, today’s Bhinmal in Jalore district of Rajasthan, was then a seat of learning in western Bharat, and Bhaskaracharya five centuries later would call him ganaka-chakra-chudamani, the crest-jewel of the circle of mathematicians. No date in his life is disputed; he gives them himself.

02Brahma’s doctrine made clear

The title is the tradition’s own account of the book. Brahmagupta says the science of the planets spoken by Bhagwan Brahma had worn thin over a great length of time, and he sets it out again, sphuta, clear and corrected. His base was the Paitamaha Siddhanta, the teaching of the Grandsire, which survives in a short form in the Vishnudharmottara Purana. He kept its frame, the kalpa of 4,32,00,00,000 years beginning with the creation, and he brought its numbers into line with what could be seen in the sky. Where observation and an older author disagreed, he trusted observation.

03Twenty-four chapters

The first ten chapters form a complete siddhanta: mean and true places of the planets, the three questions of direction, place and time, lunar and solar eclipses, risings and settings, the moon’s crescent and its shadow, conjunctions of planets with each other and with stars. Chapter 11 examines earlier tantras. Chapter 12 is arithmetic and geometry. Chapters 13 to 17 come back to the first ten with harder problems. Chapter 18 is algebra, 19 the gnomon, 20 metres, 21 the sphere, 22 instruments, 23 measures of time, and 24 a closing summary with the author’s own details.

04Rules for zero

In the Kuttakadhyaya Brahmagupta calls positive numbers dhana, wealth, and negative numbers rina, debt, and zero kha or shunya, the empty. Then he gives the rules in six verses. Two debts added make a debt; a debt and zero make a debt; zero and zero make zero. A debt times a debt is wealth; a debt times wealth is a debt; zero times anything is zero. The square of a debt or of wealth is wealth. Division by zero he leaves as a quantity with zero as its divisor. One rule did not last: he says zero divided by zero is zero. Everything else is what children learn today.

05Bhavana and the 92 problem

The same chapter treats varga-prakriti, the square-nature equation Nx² + 1 = y². Brahmagupta’s bhavana, composition, takes two solutions and builds a third from them; written in modern form it rests on the identity (a² − Nb²)(c² − Nd²) = (ac + Nbd)² − N(ad + bc)². With it one solution breeds endless others. He sets a challenge: whoever can make 92x² + 1 a square within a year is a mathematician. The smallest answer is x = 120, y = 1,151. Bhaskaracharya’s chakravala later solved every such equation; Europe took the question up with Fermat in 1657.

06The cyclic quadrilateral

Chapter 12 opens by saying that whoever knows the twenty operations and the eight practical procedures, down to shadows, is a ganaka. Its geometry holds Brahmagupta’s best-known rule. For a four-sided figure inscribed in a circle, take half the perimeter, subtract each side in turn, multiply the four remainders and take the square root: that is the exact area. He gives the lengths of the diagonals from the sides, the radius of the circumscribing circle, and right triangles with rational sides. Heron’s rule for the triangle drops out of his as the case where one side is zero.

07Astronomy and the critique

The astronomical chapters work out eclipses to the minute, the height of the moon’s horns, and the shadow of a gnomon on any day, and chapter 22 describes instruments of observation. In chapter 11 Brahmagupta tests earlier works, Aryabhata’s above all, and is often sharp: he rejects the turning earth and Aryabhata’s four equal yuga quarters. Al-Biruni quotes him saying that heavy things fall to the earth because it is the nature of the earth to draw and hold them. His later handbook, the Khandakhadyaka of 665, adds a rule of second-order interpolation for reading between the lines of a sine table.

08The road to Baghdad

In the early 770s an Indian astronomer came with an embassy from Sindh to the court of the Caliph al-Mansur in Baghdad, bringing a siddhanta in the Brahmagupta tradition. Muhammad al-Fazari and Yaqub ibn Tariq turned it into Arabic as the Zij al-Sindhind, and al-Khwarizmi built his own tables on it. The Khandakhadyaka became the Arabic al-Arkand. Al-Biruni, who learnt Sanskrit in the eleventh century, cites Brahmagupta again and again. Through these books Indian numerals and the zero reached the Arab world, and from there Europe.

09Commentaries and editions

Prithudaka Svami, called Chaturveda, wrote the Vasanabhashya on it in the ninth century, and a manuscript of his commentary is in the British Library. Bhaskaracharya built the Siddhanta Shiromani on Brahmagupta’s constants. H. T. Colebrooke translated chapters 12 and 18 into English in 1817, which first showed European readers how far Indian algebra had gone. Sudhakara Dvivedi printed the full Sanskrit at Benares in 1902, and Ram Swarup Sharma published it with a Hindi commentary in four volumes from New Delhi in 1966.

What it contains

1. 1
   Chapters 1–2: Madhyama and Sphuta

   The kalpa and its revolutions, the mean longitudes of the planets on any day, and their correction to true positions. These two chapters hold the constants on which the rest depends.

2. 2
   Chapter 3: Triprashna

   The three questions of direction, place and time, answered from the shadow of the gnomon. It covers finding the cardinal points, the latitude of a place and the hour of the day.

3. 3
   Chapters 4–5: Eclipses

   Lunar and solar eclipses: when they fall, how long they last, how much of the disc is covered. The solar chapter adds the correction for parallax.

4. 4
   Chapters 6–10: Risings, crescent, conjunctions

   Heliacal risings and settings of the planets, the elevation of the moon’s horns and the moon’s shadow. Then conjunctions of planets with one another and with the nakshatra stars.

5. 5
   Chapter 11: Tantrapariksha

   A critique of earlier astronomers, Aryabhata, Shrishena, Vishnuchandra and others. Brahmagupta lists where he thinks their numbers and doctrines fail.

6. 6
   Chapter 12: Ganita

   Sixty-six verses of arithmetic and geometry: the twenty operations, proportion, series, plane figures, the cyclic quadrilateral, volumes, excavations, saw-work and shadows. Colebrooke translated it in 1817.

7. 7
   Chapters 13–17: Further questions

   Uttara chapters that return to mean motion, true motion, the three questions, eclipses and the crescent with harder problems and their solutions. They test whether a student has really mastered the first ten chapters.

8. 8
   Chapter 18: Kuttaka

   About a hundred verses of algebra: the pulveriser, rules for positive, negative and zero, the unknown quantity, quadratic equations, equations with several unknowns, and varga-prakriti with the bhavana. It ends with problems set as challenges for a year’s work.

9. 9
   Chapters 19–21: Gnomon, metres, sphere

   Chapter 19 works with the shadow of the gnomon, chapter 20 counts the arrangements of syllables in Sanskrit metres, and chapter 21 explains the celestial sphere and its circles.

10. 10
    Chapters 22–24: Instruments, measures, summary

    Chapter 22 describes instruments of observation, chapter 23 the measures of time, solar, lunar, sidereal and civil. Chapter 24 closes the work and gives Brahmagupta’s name, father, king, city and age.

Famous verses

स्थूलफलं त्रिचतुर्भुजबाहुप्रतिबाहुयोगदलघातः। भुजयोगार्धचतुष्टयभुजोनघातात् पदं सूक्ष्मम्॥

The product of the halves of the sums of opposite sides of a triangle or quadrilateral is the rough area. Take half the sum of the sides four times, subtract each side, multiply: the square root is the exact area.Brahmasphutasiddhanta 12.21

धनयोर्धनमृणमृणयोर्धनर्णयोरन्तरं समैक्यं खम्। ऋणमैक्यं च धनमृणधनशून्ययोः शून्ययोः शून्यम्॥

The sum of two positives is positive, of two negatives negative; of a positive and a negative, their difference, and if they are equal, zero. The sum of a negative and zero is negative, of a positive and zero positive, of two zeros zero.Brahmasphutasiddhanta 18.30

ऋणमृणधनयोर्घातो धनमृणयोर्धनवधो धनं भवति। शून्यर्णयोः खधनयोः खशून्ययोर्वा वधः शून्यम्॥

The product of a negative and a positive is negative; of two negatives, positive; of two positives, positive. The product of zero and a negative, of zero and a positive, or of two zeros, is zero.Brahmasphutasiddhanta 18.33

राशिकलाशेषकृतिं द्विनवतिगुणितां त्र्यशीतिगुणितां वा। सैकां ज्ञदिने वर्गं कुर्वन्नावत्सराद्गणकः॥

The square of the remainder of minutes, multiplied by ninety-two or by eighty-three and with one added, made into a square on a Wednesday: whoever does this within a year is a mathematician.Brahmasphutasiddhanta 18.75