Brahmagupta

How to use this course

Brahmagupta wrote two books that survive, both in verse and both about astronomy. Two chapters of the first, on calculation and on algebra, contain some of the most important mathematics of the first millennium: the first full set of rules for computing with zero and negative numbers, a general rule for quadratic equations, a composition law that turns one solution of \(Nx^2+k=y^2\) into many, and exact formulas for the area and diagonals of a quadrilateral inscribed in a circle. His life is known only from a few lines in his own works and from later astronomers. The course keeps these two kinds of evidence apart.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. Verse references such as 12.21 or 18.44 give chapter and verse of the Brāhmasphuṭasiddhānta (abbreviated BSS) as they are usually cited in the modern literature (for example by Kim Plofker); editions differ by a verse or two. Brahmagupta stated rules in words, never in symbols. The formulas in this course are modern transcriptions.

Timeline

All dates CE. Only 598, 628 and 665 rest on Brahmagupta’s own words.

Bhillamāla and the sources

Near the end of the Brāhmasphuṭasiddhānta, in chapter 24, Brahmagupta gives the facts that date him. He wrote the book, he says, when 550 years of the Śaka kings had elapsed, during the reign of King Vyāghramukha of the Cāpa (Cāvaḍa) dynasty, and he was thirty years old. Śaka 550 is 628 CE, so he was born in 598. He also names his father, Jiṣṇugupta, and later astronomers sometimes call him Jiṣṇusuta, “son of Jiṣṇu”.

Bhillamāla

The Cāpa kings ruled from Bhillamāla, modern Bhinmal in southern Rajasthan, then a capital of the Gurjara country. His ninth-century commentator Pṛthūdakasvāmin calls him Bhillamālakācārya, “the teacher from Bhillamāla”, and this is the main evidence that he lived and worked there. The epithet does not prove that he was born there, though most accounts assume it.

Ujjain

Many modern accounts, MacTutor among them, say that Brahmagupta headed the astronomical observatory at Ujjain, the city of Varāhamihira and the reference meridian of Indian astronomy. His own works do not say so, and no early source does either. Ujjain mattered to him as the prime meridian for all his calculations. The claim of an institutional post should be treated as later tradition.

Religion and outlook

The book’s title means “the corrected treatise of Brahmā”: Brahmagupta presented his work as a restoration of an older revealed astronomy, the Paitāmahasiddhānta of the creator god. This framing let him claim authority while changing parameters, and it explains the polemical tone toward rivals who, in his view, had strayed from tradition.

The schools of astronomy

By 600 Indian astronomy was a set of competing systems. Each gave a long time cycle, the integer numbers of revolutions of the planets in it, and rules for corrections by epicycles. Varāhamihira’s Pañcasiddhāntikā in the sixth century summarized five older systems. Āryabhaṭa’s Āryabhaṭīya of 499 founded the school later called the Āryapakṣa, with a day counted from sunrise, and his lost second work founded the Ārdharātrika school, counting from midnight.

Brahmagupta’s book founded or renewed the Brāhmapakṣa, the school of Brahmā. It kept the traditional cosmic cycle of a kalpa of 4,320,000,000 years, and the traditional division of the great age into unequal quarters, which Āryabhaṭa had abandoned. Its parameters dominated astronomy in western and northern India for centuries, and Bhāskara II in the twelfth century still worked within it.

Why parameters mattered

A system’s revolution numbers determine the computed positions of the planets, and hence the dates of festivals, the timing of eclipses and the casting of horoscopes. Small differences in revolution numbers grow into visible errors over centuries. Disputes between schools were therefore practical, and they were also religious: a system’s claim to revealed origin was part of its authority.

A thousand verses at thirty

The Brāhmasphuṭasiddhānta has 24 chapters, about 1008 verses in the āryā metre; some manuscripts add a 25th chapter of versified tables. The first ten chapters form a complete astronomical treatise: mean and true longitudes of the planets, the three problems of time, place and direction, lunar and solar eclipses, risings and settings, the Moon’s crescent, the Moon’s shadow, and conjunctions of planets with each other and with stars. The rest is a series of supplements:

• Chapter 11, an examination of earlier treatises, largely a critique of Āryabhaṭa, Śrīṣeṇa, Viṣṇucandra and others.
• Chapter 12, Gaṇita, calculation: arithmetic operations, fractions, the rule of three, series, plane figures including the cyclic quadrilateral, volumes, and shadows.
• Chapters 13–17, further astronomical problems and refinements.
• Chapter 18, Kuṭṭaka, the pulverizer, which is in fact a chapter of algebra: the pulverizer itself, rules for positive, negative and zero quantities, unknowns, equations, and the square-nature equation.
• Chapters 19–24, on the gnomon, metres and prosody of numbers, the celestial sphere, instruments, measures, and a concluding chapter.

Style

Like the Āryabhaṭīya, the book states rules without proofs. Unlike it, it is long and often discursive, and it includes problems for the student. The algebra chapter ends with a famous boast attached to the equation \(92x^2+1=y^2\): whoever can solve it within a year is truly a mathematician. Lesson 07 shows how he did it.

Zero, fortunes and debts

For the general reader. Zero as a placeholder in written numbers was in use in India before Brahmagupta. What he did was new: he treated zero, and negative quantities, as numbers that can be added, subtracted, multiplied and divided, and he wrote down the rules. He called positive numbers “fortunes” (dhana) and negative ones “debts” (ṛṇa). Most of his rules are exactly ours. One is not.

The rules

Write \(a,b>0\) for fortunes and \(-a,-b\) for debts. Brahmagupta defines zero (śūnya) as the result of subtracting a number from itself, \(a-a=0\), and then states:

Rules 1–4 and 6 are those of the ring \(\mathbb Z\) and the field \(\mathbb Q\). Rule 5 is the exception. Brahmagupta leaves \(a/0\) as a formal expression, which is defensible, but sets \(0/0=0\), which is not. Later Indian authors tried other answers: Bhāskara II in the twelfth century treated \(a/0\) as an infinite quantity unchanged by adding anything to it.

Proof. For any \(z\), \(0\cdot z=(0+0)z=0z+0z\), so \(0z=0\ne1\). If \(a/0=w\) means \(w\cdot0=a\), then \(a=0\). Setting \(0/0=0\) is consistent with \(w\cdot0=0\), but so is every other \(w\), so the definition picks one value arbitrarily and fails the cancellation rule \((x\cdot0)/0=x\).

Equations and the quadratic rule

For the general reader. Brahmagupta’s algebra uses words for unknowns: the first is yāvat-tāvat, “as much as so much”, and further unknowns are named after colours, kālaka (black), nīlaka (blue) and so on. With these he states rules to solve linear equations in one unknown and quadratic equations, as well as systems in several unknowns. His rule for the quadratic is the earliest clear general statement of the formula we learn at school.

Linear equations (BSS 18.43)

For an equation with the unknown and a constant on each side, \(bx+c=dx+e\), Brahmagupta instructs: subtract the coefficient of the unknown on one side from that on the other, and the constant on the other side from this one; divide the difference of constants by the difference of coefficients. In symbols \(x=(e-c)/(b-d)\).

The quadratic

Derivation. Multiply \(ax^2+bx=c\) by \(4a\): \(4a^2x^2+4abx=4ac\). Add \(b^2\): \((2ax+b)^2=4ac+b^2\). Take the root and solve for \(x\). This is completing the square, and the verse transcribes these steps.

The next verse (18.45) gives a second form, obtained by first dividing by \(a\) and halving the middle coefficient: \(x=\big(\sqrt{ac+(b/2)^2}-b/2\big)/a\). Both are standard in later Indian texts. The constant sits on the right, so with Brahmagupta’s rules for negatives the rule covers every case in which the discriminant \(4ac+b^2\) is a fortune.

The pulverizer and residues

For the general reader. Āryabhaṭa gave a method, the kuṭṭaka or pulverizer, for finding whole numbers that leave given remainders on division by given numbers. Brahmagupta named his algebra chapter after it, restated it more clearly, and showed how astronomers could use it to run their calculations backwards: from an observed planet position to the number of days since the beginning of the cosmic cycle.

Method. For two moduli, \(N=m_1x+r_1=m_2y+r_2\) gives \(m_1x-m_2y=r_2-r_1\). Run the Euclidean algorithm on \(m_1,m_2\), keep the quotients, and work back up to a solution, as in the Āryabhaṭa course. Having \(N\equiv N_{12}\pmod{\operatorname{lcm}(m_1,m_2)}\), combine it with the third congruence in the same way, and so on. Brahmagupta’s instructions follow this scheme and are clearer than Āryabhaṭa’s on the back-substitution.

Astronomy backwards

Suppose a planet makes \(R\) revolutions in \(D\) days of the cycle. After \(t\) days its mean position, in revolutions, is \(Rt/D\). An observer knows only the fractional part, measured as the residue \(Rt \bmod D\). Recovering \(t\) from the residue is a linear congruence \(Rt\equiv s\pmod D\). With several planets, one gets a system of such congruences, and the pulverizer gives the elapsed days. Chapter 18 poses problems of exactly this kind.

Square-nature and composition

For the general reader. Which whole numbers \(x,y\) satisfy \(y^2=92x^2+1\)? Small trials give nothing; the smallest answer has \(y=1151\). Brahmagupta found a way to multiply solutions of related equations together, which lets one climb from an easy near-solution to a true one in two steps. Europeans call the equation Pell’s equation, after a misattribution by Leonhard Euler. In India it was varga-prakṛti, “square-nature”, and Brahmagupta’s law of composition, bhāvanā, was its key.

Brahmagupta writes the equation as \(Nx^2+k=y^2\), with \(x\) the “lesser root” and \(y\) the “greater root”. Below we use the modern form \(X^2-NY^2=k\) (so \(X\) is his greater root) and call \((X,Y;k)\) a triple for \(N\).

Proof. In \(\mathbb Z[\sqrt N]\), the norm \(\mathcal N(X+Y\sqrt N)=X^2-NY^2\) is multiplicative, because \(\mathcal N(\alpha)=\alpha\bar\alpha\). Multiply \((X_1+Y_1\sqrt N)(X_2+Y_2\sqrt N)=(X_1X_2+NY_1Y_2)+(X_1Y_2+X_2Y_1)\sqrt N\). Brahmagupta stated the identity as a rule, without this language; expanding both sides verifies it directly.

Two consequences

• Many solutions from one. Composing a solution \((X,Y;1)\) with itself gives \((X^2+NY^2,\,2XY;\,1)\), and repetition gives infinitely many.
• Reducing \(k\). Composing \((X,Y;k)\) with itself gives \((X^2+NY^2,\,2XY;\,k^2)\). Dividing both roots by \(k\) gives a rational solution with \(k=1\). When \(k=\pm2\) or \(\pm4\), Brahmagupta shows how to reach integers.

The famous example

For \(N=92\): \(10^2-92\cdot1^2=8\), so \((10,1;8)\) is a triple. Composing it with itself gives \((100+92,\ 20;\ 64)=(192,20;64)\). Dividing the roots by 8: \((24,\tfrac52;1)\), a rational solution. Composing this with itself: \(\big(24^2+92\cdot\tfrac{25}{4},\ 2\cdot24\cdot\tfrac52;\ 1\big)=(576+575,\ 120;\ 1)=(1151,120;1)\). Check: \(1151^2=1{,}324{,}801=92\cdot14{,}400+1\).

What Brahmagupta lacked was a procedure guaranteed to find a starting triple with \(k\in\{\pm1,\pm2,\pm4\}\) for every \(N\). That came with the cakravāla (“cyclic”) method of Jayadeva and Bhāskara II, who used it to solve \(x^2-61y^2=1\) with \(x=1{,}766{,}319{,}049\), \(y=226{,}153{,}980\).

Cyclic quadrilaterals

For the general reader. The area of a triangle can be computed from its three sides alone. Brahmagupta gave the analogous rule for a quadrilateral whose four corners lie on a circle, and formulas for its diagonals in terms of the sides. He also showed how to build triangles and quadrilaterals whose sides, diagonals and areas are all whole numbers. These are among the most elegant results in the geometry of the first millennium.

Brahmagupta states this as the “exact” area, alongside a rough area (the product of half-sums of opposite sides). He does not say in the verse that the quadrilateral must be cyclic. Commentators and modern historians disagree about whether he assumed it; the formula is correct only in the cyclic case. Setting \(d=0\) gives Heron’s formula for a triangle.

Proof sketch. Let the diagonal \(p\) split the quadrilateral \(ABCD\) (with \(AB=a\), \(BC=b\), \(CD=c\), \(DA=d\)) into triangles with angle \(B\) between \(a,b\) and angle \(D\) between \(c,d\). Since the quadrilateral is cyclic, \(B+D=\pi\), so \(\sin D=\sin B\) and \(\cos D=-\cos B\). The law of cosines gives \(a^2+b^2-2ab\cos B=c^2+d^2+2cd\cos B\), so \(2\cos B\,(ab+cd)=a^2+b^2-c^2-d^2\). The area is \(K=\tfrac12(ab+cd)\sin B\). Then \(16K^2=4(ab+cd)^2-(a^2+b^2-c^2-d^2)^2\), which factors as a difference of squares into \((2(ab+cd)-a^2-b^2+c^2+d^2)(2(ab+cd)+a^2+b^2-c^2-d^2)=((c+d)^2-(a-b)^2)((a+b)^2-(c-d)^2)\), and each factor splits again into the four terms \(16(s-a)(s-b)(s-c)(s-d)\).

Derivation. From the proof above, \(\cos B=\frac{a^2+b^2-c^2-d^2}{2(ab+cd)}\), and \(p^2=a^2+b^2-2ab\cos B\). Simplifying, \(p^2(ab+cd)=(a^2+b^2)cd+(c^2+d^2)ab=(ac+bd)(ad+bc)\). The formula for \(q\) follows by symmetry, and the product is Ptolemy’s theorem.

Proof. Let the perpendicular from \(M\) to \(BC\) meet \(BC\) at \(E\) and \(AD\) at \(F\). The angles \(\angle FAM=\angle DAC\) and \(\angle DBC=\angle MBC\) are inscribed angles on the same arc \(CD\), so they are equal. In the right triangle \(BMC\), \(\angle MBC=90^\circ-\angle BCM=\angle CME\), and \(\angle CME=\angle AMF\) (vertical angles). So \(\angle FAM=\angle AMF\), and triangle \(AFM\) is isosceles: \(FA=FM\). The same argument with \(D\) gives \(FD=FM\). Hence \(FA=FD\).

Rational figures

Brahmagupta also shows how to build triangles with rational sides and area by gluing two right triangles along a common leg, and cyclic quadrilaterals with rational sides and diagonals from two right triangles. For example, the right triangles 5–12–13 and 9–12–15 glued along the side 12 give the triangle with sides 13, 14, 15 and area 84.

Sines and interpolation

For the general reader. A table gives values at fixed steps; astronomers need values in between. The simplest method draws a straight line between neighbouring entries. Brahmagupta improved on it: he let the step size itself change smoothly across the interval, using the two neighbouring differences. The result is a second-order interpolation rule, a special case of a formula that Isaac Newton and James Stirling would write down a thousand years later.

Two sine tables

In the BSS (chapter 2) Brahmagupta gives 24 sines at steps of \(3^\circ45'\), like Āryabhaṭa, but with radius \(R=3270\). In the Khaṇḍakhādyaka he uses a coarse table with \(R=150\) and steps of \(15^\circ\) (\(900'\)). Its values and differences are:

The verse phrases the correction as added or subtracted according to whether the half-sum is less or greater than the coming difference. For a sine past its steepest point, \(D_{r+1}<D_r\), so the correction is subtracted. Stirling’s central-difference formula is

with \(\Delta f_{r-1}=D_r\), \(\Delta f_r=D_{r+1}\), \(\Delta^2f_{r-1}=D_{r+1}-D_r\). Brahmagupta’s rule is exactly its truncation after the second difference. It reproduces \(f_r\) at \(\theta=0\), \(f_{r+1}\) at \(\theta=1\) and \(f_{r-1}\) at \(\theta=-1\): it is the parabola through three consecutive table points.

The astronomy of the Brāhmapakṣa

For the general reader. Brahmagupta was an astronomer first. His books compute the positions of the Sun, Moon and planets, predict eclipses, and explain the Moon’s phases. He held that the Earth is a sphere at rest at the centre of the cosmos, rejected Āryabhaṭa’s rotating Earth, and explained why things fall in terms that al-Bīrūnī later quoted with admiration.

The year

The BSS gives a sidereal year of 365 days 6 hours 5 minutes 19 seconds; the Khaṇḍakhādyaka, following the Ārdharātrika system, gives 365 days 6 hours 12 minutes 36 seconds. The modern sidereal year is 365 days 6 hours 9 minutes 10 seconds.

The Moon (BSS chapter 7)

Brahmagupta argues that the Moon is nearer than the Sun, and that the illuminated part seen from the Earth depends on the angular separation, the elongation, between them. In modern terms, if the Sun is far away and the elongation is \(E\), the illuminated fraction of the disc is

which is the versine of \(E\) divided by the diameter. Indian rules express this through the versine (utkramajyā), \(R(1-\cos E)\).

A static Earth, and why things fall

Brahmagupta rejected the rotation of the Earth that Āryabhaṭa had proposed. According to al-Bīrūnī’s India, he explained falling bodies by the nature of the Earth: it is the nature of the Earth to attract and hold things, as it is the nature of water to flow. Modern claims that he formulated a theory of gravitation read too much into this; it is a statement about natural places, close in spirit to Aristotle.

The handbook of 665 and the critic

In 665, thirty-seven years after the BSS, Brahmagupta wrote the Khaṇḍakhādyaka, a karaṇa or practical handbook for calendar makers. Its title is the name of a sweet, a hint at its intended ease of use. The first part reproduces, in convenient form, the astronomy of Āryabhaṭa’s midnight system. The second part, often called the Uttarakhaṇḍakhādyaka, corrects that system with Brahmagupta’s own refinements, including the interpolation rule of Lesson 09. The handbook was widely used and commented on, notably by Pṛthūdakasvāmin and Bhaṭṭotpala.

The critic of Āryabhaṭa

Chapter 11 of the BSS and passages throughout the book attack Āryabhaṭa: his equal division of the great age, his rotating Earth, his eclipse rules and many parameters. The tone is sharp. Yet the Khaṇḍakhādyaka is built on Āryabhaṭa’s midnight system. Historians explain the contrast in several ways: the handbook was meant for practitioners who already used that system, Brahmagupta’s polemic targeted the claim to authority more than the arithmetic, and the rival schools borrowed from each other while disputing in public.

Death

Nothing is known about Brahmagupta’s life after 665. The death date of about 668 found in many reference works has no documented basis.

Baghdad, Bhāskara II and Fermat

Baghdad

In the reign of the caliph al-Manṣūr (754–775), an Indian astronomer arrived at the Abbasid court with a Sanskrit astronomical text, which Muḥammad ibn Ibrāhīm al-Fazārī rendered into Arabic. The result, the Zīj al-Sindhind, transmitted parameters and methods largely of the Brāhmapakṣa, and Arabic astronomers also knew a work derived from the Khaṇḍakhādyaka under the name Arkand. Muḥammad ibn Mūsā al-Khwārizmī’s astronomical tables drew on this material. Around 1030 al-Bīrūnī, who read Sanskrit, discussed Brahmagupta repeatedly in his India, both praising and criticizing him.

Bhāskara II

The greatest later Indian astronomer of the Brāhmapakṣa, Bhāskara II, called Brahmagupta gaṇakacakracūḍāmaṇi, the crest-jewel of the circle of calculators. His Bījagaṇita of 1150 completed the theory of the square-nature equation with the cakravāla method and took a new position on division by zero.

Europe

In 1657 Pierre de Fermat challenged European mathematicians to solve \(x^2-Ny^2=1\) in integers, choosing \(N=61\) as a test; William Brouncker and John Wallis found a method. Euler, following a misreading of Wallis, named the equation after John Pell. Lagrange proved in 1768 that a non-trivial solution exists for every non-square \(N\), using continued fractions. Europeans learned of Brahmagupta’s algebra from Colebrooke’s translation of 1817. In modern algebraic number theory his composition is multiplication of norms, and its generalization to binary quadratic forms is Gauss’s composition of forms.

Assessment

• Arithmetic: the first systematic rules for zero and negative numbers as quantities.
• Algebra: a general quadratic rule, symbolic names for several unknowns, and the composition law that made the square-nature equation tractable.
• Geometry: exact area and diagonals of cyclic quadrilaterals, rational triangles and quadrilaterals.
• Numerical analysis: second-order interpolation.
• Limits: \(0/0=0\); an area rule stated without its cyclic hypothesis; no proofs in the text.

Connections

People linked to Brahmagupta across this collection. A name without a link has no page here yet. See Brahmagupta in the network of all 337 lives →

Influenced by

• Varahamihira

Influenced

• al-Biruni
• Muhammad ibn Musa al-Khwarizmi
• Pierre de Fermat
• Bhaskara II
• Lalla
• Muhammad ibn Ibrahim al-Fazari
• Pṛthūdakasvāmin

Colleagues and collaborators

• Bhaskara I

Rivals

• Aryabhata

Also mentioned in this life

• André Weil
• Aristotle
• Euclid
• Leonhard Euler
• Carl Friedrich Gauss
• Joseph-Louis Lagrange
• Adrien-Marie Legendre
• Isaac Newton
• Claudius Ptolemy
• John Wallis

Fields

• astronomy 50 lives
• mathematics 15 lives
• algebra 29 lives
• number theory 45 lives
• geometry 32 lives

On CoursesHub

• Brahmagupta to Kerala School · History of Mathematics
• Astrophysics for Beginners

Sources and further reading

Texts and translations

1. H. T. Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London, 1817 (BSS chapters 12 and 18).
2. R. S. Sharma (ed.), Brāhma-Sphuṭa Siddhānta, 4 vols., Indian Institute of Astronomical and Sanskrit Research, New Delhi, 1966.
3. B. Chatterjee, The Khaṇḍakhādyaka of Brahmagupta, with the Commentary of Bhaṭṭotpala, 2 vols., New Delhi, 1970.

Biography and history

1. Brahmagupta, MacTutor History of Mathematics.
2. Brahmagupta and Brāhmasphuṭasiddhānta, Wikipedia.
3. Brahmagupta, Biographical Encyclopedia of Astronomers, Springer, 2007.
4. K. Plofker, Mathematics in India, Princeton University Press, 2009, ch. 5.
5. D. Pingree, Brahmagupta, Dictionary of Scientific Biography, vol. 2, 1970.

Mathematics

1. A. K. Dutta, The bhāvanā in mathematics, Bhāvanā 1 (2017).
2. Pell’s equation, MacTutor history topic.
3. Pell’s equation (history section) and Brahmagupta’s interpolation formula, Wikipedia.
4. N. Handa and P. Taneja, Methods of interpolation in Indian astronomy, Indian Journal of History of Science 49 (2014) 251–259.
5. Brahmagupta theorem and Brahmagupta’s formula, Wikipedia.
6. A. Weil, Number Theory: An Approach through History from Hammurapi to Legendre, Birkhäuser, 1984 (on the square-nature equation).