Part III — Chapter 9

Brahmagupta to the Kerala School

Medieval Indian mathematics — from zero and negative numbers to infinite series for $\pi$ and trigonometric functions, centuries before Newton and Leibniz

9.1 Aryabhata (476–550 CE)

Aryabhata was born in 476 CE, probably in the region of modern-day Patna (Bihar), and is the earliest Indian mathematician whose work survives in detail. His masterwork, the Aryabhatiya (499 CE), is a concise treatise of 121 verses covering arithmetic, algebra, plane and spherical trigonometry, and astronomy. Written when the author was just 23 years old, it became one of the most influential mathematical texts in Indian history.

The Aryabhatiya is divided into four sections: Gitikapada (astronomical constants), Ganitapada (mathematics), Kalakriyapada (time reckoning), and Golapada (the celestial sphere). The mathematical section is the most relevant for our purposes.

The sine table. Aryabhata constructed a table of sines (jya) for angles at intervals of 3.75° (i.e., for 3.75°, 7.5°, 11.25°, ..., 90°). He expressed the sines not as decimals but as differences between consecutive values, using a radius of $R = 3438$ (in minutes of arc — since there are $360 \times 60 = 21600$ minutes in a circle, and$R = 21600 / (2\pi) \approx 3438$).

His first few sine values (in this system) are:

$\sin(3.75°) \approx 225$

$\sin(7.5°) \approx 449$

$\sin(11.25°) \approx 671$

$\sin(15°) \approx 890$

$\sin(30°) \approx 1719$

$\sin(45°) \approx 2431$

$\sin(60°) \approx 2978$

$\sin(90°) = 3438$

In modern units, $\sin(30°) = 1719/3438 = 0.5000$, which is exact. Aryabhata's values are accurate to 3–4 decimal places, a remarkable achievement for the 5th century.

Place-value system. Aryabhata used a novel alphabetic notation for numbers (the Aryabhata cipher) in his astronomical tables. While this specific notation did not survive, the decimal place-value system that underlies his calculations is the ancestor of our modern numeral system. The concept of zero as a placeholder was implicit in Aryabhata's work and would be made fully explicit by Brahmagupta a century later.

Earth's rotation. Aryabhata made the extraordinary claim that the apparent rotation of the heavens is actually due to the rotation of the Earth on its axis. He stated: “Just as a man in a boat moving forward sees stationary objects as moving backward, so are the stationary stars seen by people at Lanka as moving exactly toward the West.” This heliocentric-adjacent insight was controversial in India and was rejected by some later commentators, but it demonstrates the boldness of Aryabhata's scientific thinking.

Aryabhata's Approximation of Pi

Aryabhata stated (verse 10 of the Ganitapada): “Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle whose diameter is 20,000.”

$$\pi \approx \frac{(100 + 4) \times 8 + 62000}{20000} = \frac{62832}{20000} = 3.1416$$

This is correct to four decimal places. Aryabhata noted that this value is approximate (asanna = “approaching”), suggesting he understood that $\pi$ is irrational, though he did not prove it.

9.2 Brahmagupta (598–668 CE)

Brahmagupta was born in 598 CE in Bhillamala (modern Bhinmal, Rajasthan) and served as the head of the astronomical observatory at Ujjain, one of the great centres of Indian mathematical astronomy. His Brahmasphutasiddhanta(“Correctly Established Doctrine of Brahma,” 628 CE) is one of the most important mathematical texts in history, containing the first systematic treatment of zero and negative numbers as fully-fledged mathematical objects.

Zero as a Number: Brahmagupta's Rules

In Chapter 18 of the Brahmasphutasiddhanta, Brahmagupta stated explicit rules for arithmetic with zero (shunya) and with positive (dhana = “fortune”) and negative (rina = “debt”) numbers:

Addition:

$a + 0 = a$

“The sum of a positive and a negative is their difference; if equal, zero.”

$a + (-a) = 0$

Subtraction:

$a - 0 = a, \quad 0 - a = -a$

“A negative subtracted from zero is positive.” $0 - (-a) = a$

Multiplication:

$a \times 0 = 0$

$(-a) \times (-b) = ab$ (negative times negative is positive)

$(-a) \times b = -(ab)$ (negative times positive is negative)

Division:

$0 \div a = 0$

“Zero divided by zero is zero.” (This is the one rule Brahmagupta got wrong.)

“A positive or negative divided by zero is a fraction with zero as denominator.”

Brahmagupta's treatment of $0 \div 0 = 0$ and his ambiguity about division by zero are noteworthy. The latter would be clarified by Bhaskara II five centuries later (who declared that $a/0$ is infinite, or khahara). The full resolution awaited the rigorous limit concept of 19th-century analysis.

The conceptual boldness of Brahmagupta's work cannot be overstated. The Greeks had no concept of zero as a number. The Romans had no symbol for it. Even the Islamic mathematicians initially used zero only as a placeholder in positional notation, not as a number to be operated on. Brahmagupta was the first to treat zero and negative numbers as full participants in arithmetic — a conceptual revolution that underlies all of modern mathematics.

Brahmagupta also gave the important formula for the quadratic equation. For$ax^2 + bx = c$ (with $a, b, c > 0$):

$$x = \frac{\sqrt{4ac + b^2} - b}{2a}$$

He also accepted negative roots, which was revolutionary. For the equation $x^2 - 10x + 9 = 0$, he would recognise both $x = 1$ and $x = 9$ as valid solutions.

9.3 Brahmagupta's Formula

One of Brahmagupta's most celebrated results is his formula for the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle). This beautiful result generalises Heron's formula for the area of a triangle.

Brahmagupta's Formula for the Area of a Cyclic Quadrilateral

Let a cyclic quadrilateral have sides $a, b, c, d$ and semiperimeter$s = \frac{a+b+c+d}{2}$. Then its area is:

$$A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$

Note: When $d = 0$, the quadrilateral degenerates to a triangle, and Brahmagupta's formula reduces to Heron's formula:

$$A_{\triangle} = \sqrt{s(s-a)(s-b)(s-c)}$$

where now $s = \frac{a+b+c}{2}$.

Worked Example: Brahmagupta's Formula

Find the area of a cyclic quadrilateral with sides $a = 3, b = 4, c = 5, d = 6$.

Semiperimeter: $s = \frac{3+4+5+6}{2} = 9$

$$A = \sqrt{(9-3)(9-4)(9-5)(9-6)} = \sqrt{6 \cdot 5 \cdot 4 \cdot 3} = \sqrt{360} = 6\sqrt{10} \approx 18.97$$

Brahmagupta also discovered a remarkable result about the diagonals of a cyclic quadrilateral. If the diagonals have lengths $p$ and $q$, then:

$$p = \sqrt{\frac{(ab+cd)(ac+bd)}{ad+bc}}, \quad q = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}}$$

and the product of the diagonals satisfies Ptolemy's theorem:

$$pq = ac + bd$$

These results demonstrate Brahmagupta's deep understanding of cyclic geometry. His proofs used a combination of algebraic and geometric methods that were centuries ahead of European developments.

9.4 Pell's Equation

The equation $x^2 - Ny^2 = 1$ (where $N$ is a positive non-square integer) is one of the most important equations in number theory. It is called “Pell's equation” due to a misattribution by Euler, but it should properly be called Brahmagupta's equation or the Brahmagupta-Bhaskara equation, since Indian mathematicians studied it systematically a thousand years before Pell.

The Bhavana (Composition) Principle

Brahmagupta discovered that if $(x_1, y_1)$ and $(x_2, y_2)$ are solutions to $x^2 - Ny^2 = k_1$ and $x^2 - Ny^2 = k_2$respectively, then:

$$(x_1 x_2 + N y_1 y_2)^2 - N(x_1 y_2 + x_2 y_1)^2 = k_1 k_2$$

This “composition of solutions” (bhavana) is a multiplicative principle: it allows one to combine solutions of related equations to produce new solutions. In particular, if $(x_1, y_1)$ solves $x^2 - Ny^2 = 1$, then composing this solution with itself generates infinitely many solutions from a single “fundamental” solution.

The most powerful method for solving Pell's equation was the chakravala (“cyclic”) method, developed by Brahmagupta and perfected by Jayadeva (c. 950) and Bhaskara II (c. 1150). It is an iterative algorithm that generates progressively better rational approximations to $\sqrt{N}$ and eventually produces the fundamental solution.

Solving x² - 61y² = 1 by the Chakravala Method

This is one of the most famous examples in the history of number theory. The fundamental solution is enormous, demonstrating the power of the method.

Step 1: Initial approximation. Since $\sqrt{61} \approx 7.81$, start with $m = 8$: $8^2 - 61 \cdot 1^2 = 64 - 61 = 3$. So $(x, y, k) = (8, 1, 3)$.

Step 2: Chakravala iteration. At each step, choose $m$ such that $(a + bm)/k$ is an integer and $|m^2 - N|$ is minimised. Then update:

$$a' = \frac{am + Nb}{|k|}, \quad b' = \frac{a + bm}{|k|}, \quad k' = \frac{m^2 - N}{k}$$

Iteration sequence (abbreviated):

$(a, b, k) = (8, 1, 3)$

$m = 7: \quad (39, 5, -4)$

$m = 8: \quad (164, 21, 3)$ (Not done yet... but approaching.)

$\vdots$

After several more iterations...

$(a, b, k) = (1766319049, 226153980, 1)$

Final answer:

$$x = 1766319049, \quad y = 226153980$$

Verification:

$$1766319049^2 - 61 \times 226153980^2 = 1$$

That the fundamental solution involves a 10-digit number for the modest value $N = 61$ shows why this equation is so difficult and why the chakravala method is so remarkable.

The chakravala method is essentially equivalent to the continued fraction algorithm for$\sqrt{N}$, which was not developed in Europe until the 18th century by Euler and Lagrange. The Indian achievement is all the more impressive given that it was carried out without any of the theoretical framework of continued fractions — it was a purely algorithmic discovery, guided by deep numerical intuition.

9.5 Bhaskara II (1114–1185)

Bhaskara II (also called Bhaskaracharya, “Bhaskara the Teacher”) was born in 1114 CE in Bijapur (Karnataka) and was the head of the astronomical observatory at Ujjain, the same position held by Brahmagupta five centuries earlier. He is widely regarded as the greatest Indian mathematician of the classical period.

His two main mathematical works are the Lilavati (named after his daughter, covering arithmetic and geometry) and the Bijaganita (algebra). Together with the Siddhanta Shiromani (astronomy) and the Vasanabhasya (commentary), they form a comprehensive mathematical encyclopedia.

The Lilavati is notable for its charm and accessibility. Problems are posed in the form of riddles and stories, often addressed to Lilavati herself: “Beautiful and dear Lilavati, tell me: if a bamboo 32 cubits high is broken by the wind...” Despite this playful style, the mathematics is rigorous and advanced.

Early differential calculus. Bhaskara II's most remarkable contribution, from a modern perspective, is his investigation of instantaneous rates of change. In the Siddhanta Shiromani, he studied the daily motion of planets and asked: at what moment is the rate of change of a trigonometric function zero?

Bhaskara II's Proto-Derivative of Sine

Bhaskara II observed that when the sine function reaches its maximum (at$\theta = 90°$), its rate of change is zero. He stated a result equivalent to:

$$\frac{d}{d\theta}(\sin \theta) = \cos \theta$$

His argument proceeded as follows: consider a small increment $\Delta\theta$ in the angle. The change in the sine is:

$$\Delta(\sin\theta) = \sin(\theta + \Delta\theta) - \sin\theta$$

Using the identity $\sin(\theta + \Delta\theta) - \sin\theta = 2\cos\left(\theta + \frac{\Delta\theta}{2}\right)\sin\left(\frac{\Delta\theta}{2}\right)$, and noting that for small $\Delta\theta$, $\sin(\Delta\theta/2) \approx \Delta\theta/2$, he obtained:

$$\frac{\Delta(\sin\theta)}{\Delta\theta} \approx \cos\theta$$

This is the derivative of sine, obtained through a limiting argument five centuries before Newton and Leibniz.

Bhaskara II also gave the result (in modern notation) that $\frac{d}{d\theta}(\cos\theta) = -\sin\theta$, and he used these results to find the maximum and minimum values of planetary equations. He did not, of course, have the formal concept of a limit or a general theory of differentiation — these were isolated results, obtained through geometric and trigonometric reasoning in the context of astronomy. But they demonstrate that the fundamental ideas of calculus were within reach of Indian mathematicians centuries before the European invention of calculus.

Bhaskara II also solved $x^2 - 61y^2 = 1$ using the chakravala method, obtaining the fundamental solution $x = 1766319049$, $y = 226153980$— a computational tour de force of medieval mathematics.

A Problem from the Lilavati

“The square root of half the number of bees in a swarm has flown to a jasmine bush; eight-ninths of the whole swarm has remained behind; one female bee flies about a male bee that is buzzing within a lotus. Tell me the number of bees.”

Let $n$ be the total number of bees. Then:

$$\sqrt{\frac{n}{2}} + \frac{8n}{9} + 2 = n$$

Simplifying: $\sqrt{\frac{n}{2}} = n - \frac{8n}{9} - 2 = \frac{n}{9} - 2$

Let $u = \sqrt{n/2}$, so $n = 2u^2$:

$$u = \frac{2u^2}{9} - 2 \quad \Longrightarrow \quad 2u^2 - 9u - 18 = 0$$

$$u = \frac{9 \pm \sqrt{81 + 144}}{4} = \frac{9 \pm 15}{4}$$

Taking the positive root: $u = 6$, so $n = 2 \times 36 = 72$ bees.

9.6 Madhava of Sangamagrama (c. 1340–1425)

More than two and a half centuries before Isaac Newton discovered the sine and cosine series and James Gregory the arctan series, the Indian astronomer and mathematician Madhava (c. 1340–c. 1425) gave expressions for $\sin x$, $\cos x$, and $\arctan x$ as infinite power series. Madhava's work may have been motivated by his studies in astronomy, since he concentrated mainly on trigonometric functions.

The mathematician-astronomers of medieval Kerala lived, worked, and taught in large family compounds called illams. Madhava, believed to have been the founder of the Kerala school of astronomy and mathematics, worked in the Bakulavihara illam in the town of Sangamagrama (near modern Irinjalakuda, a few miles north of Cochin). He was an Emprantiri Brahmin, then considered socially inferior to the dominant Namputiri (or Nambudri) Brahmin caste. This position does not appear to have curtailed his teaching activities; his most distinguished pupil was Paramesvara, a Namputiri Brahmin.

No mathematical works of Madhava have been found, though three of his short treatises on astronomy are extant. The most important of these describes how to accurately determine the position of the moon at any time of day. Other surviving mathematical works of the Kerala school attribute many very significant results to Madhava. Although his algebraic notation was almost primitive, Madhava's mathematical skill allowed him to carry out highly original and difficult research.

Madhava's discoveries are among the most remarkable in the history of mathematics. He discovered infinite series expansions for $\pi$, the sine, cosine, and arctangent functions — results that are conventionally attributed to Gregory, Leibniz, Newton, and Taylor, all of whom lived 250–300 years later. There appears to be no connection between the work of Madhava's school and that of Newton and other European mathematicians. In spite of this, the Keralese and European mathematicians shared some similar methods and results — both were fascinated with transformation of series, though they used very different methods.

Madhava-Leibniz Series for Pi

Madhava discovered that:

$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}$$

This is equivalent to $\arctan(1) = \pi/4$, expanded as a power series:

$$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$

evaluated at $x = 1$. In Europe, this series was discovered independently by James Gregory (1671) and Gottfried Leibniz (1674).

The series converges very slowly — after 100 terms, it gives $\pi$ to only about 2 decimal places. Madhava was well aware of this and developed convergence acceleration techniques. He added correction terms to the partial sums:

Madhava's Correction Terms for the Pi Series

After summing $n$ terms of the series, Madhava applied correction terms:

$$\frac{\pi}{4} \approx \sum_{k=0}^{n-1} \frac{(-1)^k}{2k+1} + \frac{(-1)^n}{2} \cdot \frac{1}{(2n)^2 + 1} \cdot \frac{4n}{1}$$

More precisely, his correction terms took the form:

$$\frac{(-1)^n \cdot n}{4n^2 + 1}$$

which dramatically accelerates convergence. With this correction and $n = 50$ terms, Madhava obtained $\pi$ correct to 11 decimal places:$\pi \approx 3.14159265359$.

Madhava's Power Series for Sine and Cosine

Madhava discovered the infinite series:

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}$$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$$

The Kerala mathematicians also discovered a series for $\sin^2\theta$:

$$\sin^2\theta = \theta^2 - \frac{\theta^4}{(2^2 - \frac{2}{3})} + \frac{\theta^6}{(2^2 - \frac{2}{3})(3^2 - \frac{2}{3})} - \frac{\theta^8}{(2^2 - \frac{2}{3})(3^2 - \frac{2}{3})(4^2 - \frac{2}{3})} + \cdots$$

where $\theta$ is measured in radians (Madhava used the equivalent measure in terms of arc length divided by radius). These are precisely the Taylor series for sine and cosine, attributed in Europe to Brook Taylor (1715) and Colin Maclaurin (1742) — nearly 300 years after Madhava.

In the surviving proofs, the range of $\theta$ for the arctangent series was$0 \le \theta \le \frac{\pi}{4}$, and for the sine and cosine series it was$0 \le \theta \le \frac{\pi}{2}$. Although the series for sine and cosine converge for all real values, the concept of periodicity of the trigonometric functions was discovered much later.

Verification: Sine Series for x = pi/6

We know $\sin(\pi/6) = 1/2$. Let $x = \pi/6 \approx 0.5236$:

Term 1: $x = 0.5236$

Term 2: $-\frac{x^3}{6} = -\frac{0.1434}{6} = -0.02392$

Term 3: $+\frac{x^5}{120} = +\frac{0.03956}{120} = +0.000330$

Term 4: $-\frac{x^7}{5040} = -\frac{0.01036}{5040} = -0.00000206$

Sum: $0.5236 - 0.02392 + 0.000330 - 0.00000206 \approx 0.50001$

Already after 4 terms, we have $\sin(\pi/6) \approx 0.5000$ to 4 decimal places.

Madhava also obtained the series for the arctangent function and used it with various substitutions to compute $\pi$. For example, using:

$$\frac{\pi}{6} = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{1}{\sqrt{3}} - \frac{1}{3 \cdot 3\sqrt{3}} + \frac{1}{5 \cdot 9\sqrt{3}} - \cdots$$

which converges much faster than the $\arctan(1)$ series. With this and his correction terms, Madhava computed $\pi$ as:

$$\pi \approx 3.14159265358979324$$

correct to 11 decimal places — the most accurate value of $\pi$ computed anywhere in the world at that time.

9.7 The Kerala School Lineage: From Paramesvara to Sankara Variyar

Paramesvara (c. 1380–c. 1460)

Paramesvara, Madhava's most distinguished pupil, was from the village of Asvattagram (modern Alattur in Kerala). He lived and worked in the Vatasreni illam, continuing the illam tradition of mathematical scholarship. Paramesvara was one of the most dedicated astronomical observers in medieval history: he carried out systematic observations of eclipses over a 55-year period from 1393 to 1432, meticulously recording data that would be passed on to his successors.

His mathematical contributions include an important formula for the radius of a circle circumscribing a cyclic quadrilateral — a result sometimes called Paramesvara's formula. Given a cyclic quadrilateral with sides $a, b, c, d$ and diagonals $p, q$, the circumradius$R$ is:

$$R = \frac{abc}{4K}$$

where $K$ is the area of the triangle formed by three of the vertices. More generally, for the full quadrilateral with diagonals $p$ and $q$:

$$R = \frac{pq \cdot \sqrt{(ab+cd)(ac+bd)(ad+bc)}}{4 \cdot \text{Area of quadrilateral} \cdot (ab+cd)}$$

Damodara (c. 1410–c. 1510)

Damodara, the son of Paramesvara, was a crucial link in the Kerala school's chain of transmission. Though few of his own works survive, his importance lies in the fact that he served as the teacher of two of the school's most brilliant later figures: Nilakantha Somayaji and Jyesthadeva. Through Damodara, the entire body of mathematical knowledge developed by Madhava and refined by Paramesvara was transmitted to the next generation.

The Kerala School Lineage

c. 1340–1425Madhava (founder) — Bakulavihara illam, Sangamagrama

↓

c. 1380–1460Paramesvara (pupil of Madhava) — Vatasreni illam, Asvattagram

↓

c. 1410–1510Damodara (son of Paramesvara)

↓ ↓

c. 1444–1544Nilakantha Somayaji (pupil of Damodara)

c. 1500–1570Jyesthadeva (pupil of Damodara)

↓

c. 1500–1560Sankara Variyar (pupil of Nilakantha)

Nilakantha Somayaji (1444–1544)

Nilakantha Somayaji, a pupil of Damodara, was the most important successor of Madhava in the Kerala school. He observed eclipses himself — notably in 1467 and again in 1501 — and was a strong advocate of correcting traditional astronomical parameters on the basis of direct observation, even when this meant departing from revered authorities.

His masterwork, the Tantrasangraha (1501), is a comprehensive treatise on mathematical astronomy that incorporates and extends the discoveries of Madhava. The Tantrasangrahais notable for its systematic and rigorous approach: it develops the entire theory of planetary motion using the refined mathematical tools inherited from Madhava and Paramesvara.

Nilakantha improved on Madhava's series for $\pi$ by discovering a more rapidly convergent series:

Nilakantha's Series for Pi

$$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \frac{4}{8 \cdot 9 \cdot 10} + \cdots$$

In general form:

$$\pi = 3 + \sum_{k=1}^{\infty} \frac{(-1)^{k+1} \cdot 4}{2k(2k+1)(2k+2)}$$

This converges much faster than the Madhava-Leibniz series: the terms decrease as$1/k^3$ rather than $1/k$.

Nilakantha's Series: First Few Terms

$\pi \approx 3 + \frac{4}{24} = 3 + 0.1\overline{6} = 3.1\overline{6}$

$\pi \approx 3.1\overline{6} - \frac{4}{120} = 3.1\overline{6} - 0.0\overline{3} = 3.1\overline{3}$

$\pi \approx 3.1\overline{3} + \frac{4}{336} = 3.1\overline{3} + 0.01190\ldots \approx 3.14524$

$\pi \approx 3.14524 - \frac{4}{720} = 3.14524 - 0.00556 \approx 3.13968$

After just 10 terms, this series gives $\pi \approx 3.14159$, correct to 5 decimal places.

Nilakantha also made a remarkable contribution to astronomy. In the Tantrasangrahaand his commentary on the Aryabhatiya, he proposed a partially heliocentric model in which Mercury, Venus, Mars, Jupiter, and Saturn orbit the Sun, while the Sun orbits the Earth. This model is mathematically equivalent to Tycho Brahe's system (1588), proposed 87 years later, and captures the correct relative motions of the planets.

Jyesthadeva (c. 1500–c. 1570)

Jyesthadeva, also a pupil of Damodara, is the author whose works give us all the surviving proofs of the Kerala school's mathematical results. His Yuktibhasa (c. 1530) — written in Malayalamrather than the scholarly Sanskrit used by most other Kerala mathematicians — is a remarkable text that provides detailed, step-by-step proofs of the results of Madhava and Nilakantha.

The Yuktibhasa is the first known text in any mathematical tradition to present proofs of infinite series results, including the derivation of the power series for sine, cosine, and arctangent from first principles. It is, in effect, a textbook of mathematical analysis written a century before the European invention of calculus. Its use of Malayalam (rather than Sanskrit) suggests it may have been intended for a wider audience, making the sophisticated mathematics of the Kerala school accessible beyond the traditional scholarly elite.

Sankara Variyar (c. 1500–1560)

Sankara Variyar, a pupil of Nilakantha, continued the school's tradition of commentary and exposition. He composed the Yuktidipika (a commentary on the Tantrasangraha) and began the Kriyakramakari (a commentary on Bhaskara II's Lilavati), which was completed after his death by his student Mahisamangalam Narayana. These commentaries are invaluable because they preserve and elaborate on results that might otherwise have been lost, providing alternative proofs and additional worked examples.

Continued Fractions: From Bhaskara's Kuttaka to the Kerala School

One of the most striking connections between Bhaskara II's Lilavati and the Kerala school lies in the theory of continued fractions. Although the continued fraction expansion of $\pi$ is not mentioned explicitly in any extant works of the Kerala school, their successive approximants — the partial sums and correction terms applied to the Madhava–Leibniz series — yield rational approximations to$\pi$ that coincide with the convergents of its continued fraction expansion. This indicates that the Kerala mathematicians must have known the continued fraction, at least implicitly, even if they did not formulate it as such.

In fact, continued fractions appear in much earlier Indian works. The Lilavati of Bhaskara II (c. 1150) presented the kuttaka (“pulveriser”) method for solving first-order indeterminate (Diophantine) equations of the form:

The Kuttaka Method — Solving Linear Diophantine Equations via Continued Fractions

Given integers $a$, $b$, $c$ with$\gcd(a, b) \mid c$, find integers $x$, $y$ such that:

$$ax - by = c$$

The method proceeds by applying the Euclidean algorithm to $a$ and $b$, which is equivalent to computing the continued fraction expansion of $a/b$:

$$\frac{a}{b} = q_0 + \cfrac{1}{q_1 + \cfrac{1}{q_2 + \cfrac{1}{q_3 + \cdots}}}$$

where $q_0, q_1, q_2, \ldots$ are the successive quotients. The convergents$p_k/q_k$ of this continued fraction satisfy the key identity:

$$p_k q_{k-1} - p_{k-1} q_k = (-1)^{k+1}$$

The kuttaka uses this identity to construct a solution to $ax - by = c$ by working backwards through the chain of quotients, “pulverising” the problem at each stage until a trivially soluble equation is reached.

Kuttaka Example from the Lilavati

Solve $137x - 60y = 1$.

Step 1. Apply the Euclidean algorithm to 137 and 60:

$$137 = 2 \cdot 60 + 17, \quad 60 = 3 \cdot 17 + 9, \quad 17 = 1 \cdot 9 + 8, \quad 9 = 1 \cdot 8 + 1$$

The quotients are $q_0 = 2,\; q_1 = 3,\; q_2 = 1,\; q_3 = 1,\; q_4 = 8$, giving the continued fraction:

$$\frac{137}{60} = 2 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{8}}}}$$

Step 2. Work backwards (the “pulverising”). Starting from the last remainder equation $9 - 1 \cdot 8 = 1$ and substituting back:

$1 = 9 - 1 \cdot 8$

$\;\;= 9 - 1 \cdot (17 - 9) = 2 \cdot 9 - 17$

$\;\;= 2(60 - 3 \cdot 17) - 17 = 2 \cdot 60 - 7 \cdot 17$

$\;\;= 2 \cdot 60 - 7(137 - 2 \cdot 60) = 16 \cdot 60 - 7 \cdot 137$

Therefore $137 \cdot 7 - 60 \cdot 16 = -1$, giving $137 \cdot (-7) - 60 \cdot (-16) = 1$. Adjusting: $x = 60 - 7 = 53$, $y = 137 - 16 = 121$. Check: $137 \times 53 - 60 \times 121 = 7261 - 7260 = 1$. ✓

The kuttaka method originated much earlier than Bhaskara II — it traces back to Aryabhata (c. 499), who gave the first systematic description in the Aryabhatiya, and was refined by Brahmagupta (628). But it was Bhaskara II's Lilavati that presented the method in its most polished and pedagogically accessible form, complete with numerous worked examples. The kuttaka was essential in Indian astronomy for solving problems involving planetary conjunctions and calendar computations — finding, for instance, the number of days elapsed in a yuga (cosmic cycle) given residues modulo various orbital periods.

Sankara Variyar's Kriyakramakari (c. 1540), as a detailed commentary on the Lilavati, naturally engaged deeply with the kuttaka method. What makes the Kriyakramakari especially important is that Variyar did not merely repeat Bhaskara's exposition but enriched it with the mathematical insights of the Kerala school. He provided alternative proofs, additional examples, and extended explanations that drew on the two centuries of mathematical development since Madhava. In particular, the Kriyakramakaricontains one of the earliest known derivations of the Madhava–Leibniz series for$\pi/4$, connecting the classical kuttaka tradition to the Kerala school's revolutionary work on infinite series.

The Kriyakramakari as a Bridge Text

The Kriyakramakari occupies a unique position in Indian mathematical literature. As a commentary on Bhaskara II's 12th-century Lilavati, written by a 16th-century member of the Kerala school, it serves as a bridge between two great mathematical traditions:

• Bhaskara's classical tradition: the kuttaka (continued fraction) method for Diophantine equations, combinatorics, arithmetic of fractions, and quadratic equations.
• The Kerala school's innovations: infinite series for $\pi$ and trigonometric functions, convergence acceleration, and proto-calculus techniques.

By embedding the Kerala school's results within a commentary on the most widely studied mathematical text in India, Variyar ensured that these results reached the broadest possible audience of Sanskrit-literate scholars.

The implicit presence of continued fractions in Kerala school work goes beyond the kuttaka. Madhava's correction terms for the slow-converging series$\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \cdots$ yield successive rational approximants to $\pi$ that match the convergents of its continued fraction expansion:

Kerala Approximants and the Continued Fraction for Pi

The continued fraction expansion of $\pi$ begins:

$$\pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \cdots}}}}$$

whose convergents are:

$$3,\quad \frac{22}{7},\quad \frac{333}{106},\quad \frac{355}{113},\quad \frac{103993}{33102},\quad \ldots$$

Madhava's correction terms, when applied to partial sums of his series, produce rational approximations that coincide with or closely track these convergents — most strikingly the approximation $355/113$, which is accurate to six decimal places. The fact that these approximants arise naturally from the Kerala school's series manipulations, without any explicit theory of continued fractions, is a powerful indication that the underlying structure was understood implicitly.

Surviving Texts of the Kerala School

• Tantrasangraha (Nilakantha, 1501) — comprehensive treatise on mathematical astronomy; the school's crowning systematic work. Written in Sanskrit.
• Yuktidipika (Sankara Variyar) — Sanskrit commentary on the Tantrasangraha, providing proofs and explanations of its results.
• Yuktibhasa (Jyesthadeva, c. 1530) — the only text in Malayalam, containing all surviving proofs of the infinite series. All other major works of the school are in Sanskrit.
• Kriyakramakari (started by Sankara Variyar, completed by his student Mahisamangalam Narayana) — Sanskrit commentary on Bhaskara II's Lilavati with extensions and additional results.
• Karanapaddhati (Putumana Somayaji) — traditionally dated to c. 1700, though four recent translators have argued that Somayaji was a junior contemporary of Nilakantha and composed the work between 1532 and 1566. It preserves computational methods and series expansions from the school's heyday.

9.8 Priority Questions — Did Kerala Influence Europe?

The discovery that Indian mathematicians had obtained infinite series for $\pi$, sine, and cosine 250 years before their European counterparts naturally raises the question: did the Kerala results reach Europe and influence Newton, Leibniz, Gregory, or their predecessors?

Evidence for possible transmission:

• Jesuit missionaries were active in Kerala from the mid-16th century onward. The Jesuits were deeply interested in astronomy and mathematics (they needed accurate calendars and astronomical tables for their missions). They maintained close ties with European scientific institutions.
• Jesuit astronomers in India are known to have collected and studied local astronomical and mathematical texts. Some of these texts may have contained the results of the Kerala school.
• The timing is suggestive: the Kerala school flourished from c. 1350–1600, and the European discovery of the same results occurred from c. 1600–1715, with a plausible transmission window via the Jesuits.
• Specific results (the arctangent series, the sine and cosine series) appear in European mathematics without a clear chain of prior European development leading to them.

Evidence against transmission:

• No documentary evidence has been found of a specific text or communication transmitting Kerala results to Europe. Despite extensive searches of Jesuit archives, no “smoking gun” has been discovered.
• The European development of calculus arose from a different intellectual context — the problems of tangent lines, areas under curves, and the motion of bodies — that has an independent and well-documented European genealogy (Cavalieri, Fermat, Barrow, Wallis, Newton, Leibniz).
• Newton and Leibniz developed not merely the same specific results but a general theory — the fundamental theorem of calculus, general rules for differentiation and integration — that goes far beyond the Kerala results.
• Parallel independent discovery is common in mathematics. The infinite series for$\pi/4$ is a natural consequence of studying the geometry of the circle, and it is not surprising that two mathematical traditions would discover it independently.

The Current Scholarly Consensus

Most historians of mathematics hold that direct transmission from Kerala to Europe has not been demonstrated, although it cannot be ruled out. The question remains open. Regardless of the answer, the achievements of the Kerala school stand on their own merits as one of the most remarkable episodes in the history of mathematics. Madhava's priority in discovering the infinite series for$\pi$, sine, and cosine is now widely recognised in the scholarly literature.

9.9 Legacy — Indian Contributions to World Mathematics

The contributions of Indian mathematicians from Aryabhata to the Kerala school form one of the great chapters in the history of mathematics. Their achievements include some of the most fundamental concepts in all of mathematics:

Summary of Indian Mathematical Contributions

• Decimal place-value system: The most important notational innovation in the history of mathematics, without which modern computation would be impossible
• Zero as a number: Brahmagupta's treatment of zero and negative numbers as objects of arithmetic
• Trigonometry: Aryabhata's sine tables and the systematic development of trigonometric functions
• Quadratic formula: Brahmagupta's explicit formula, including negative roots

• Pell's equation: The chakravala method, centuries ahead of European work
• Proto-calculus: Bhaskara II's derivatives of trigonometric functions
• Infinite series: Madhava's power series for $\pi$, sine, cosine, and arctangent
• Convergence acceleration:Madhava's correction terms, a precursor of numerical analysis

Timeline of Medieval Indian Mathematics

476–550 — Aryabhata: Aryabhatiya, sine table, place-value system, $\pi \approx 3.1416$
598–668 — Brahmagupta: Zero and negatives, cyclic quadrilateral formula, Pell's equation
c. 800–900 — Mahavira: Combinatorics, permutations and combinations
c. 950 — Jayadeva: Perfects the chakravala method
1114–1185 — Bhaskara II: Lilavati, Bijaganita, proto-derivatives
c. 1340–1425 — Madhava: Infinite series for $\pi$, sine, cosine; convergence acceleration
1444–1544 — Nilakantha: Improved $\pi$ series, partial heliocentrism
c. 1500–1575 — Jyeshthadeva: Yuktibhasa — proofs of infinite series results

The Indian mathematical tradition was distinctive in several ways. First, it was algebraic rather than geometric in orientation: where the Greeks built their mathematics on geometric constructions, Indian mathematicians worked primarily with numerical and algebraic methods. Second, it was computational: Indian mathematicians were virtuoso calculators who valued numerical results and efficient algorithms. Third, it was closely tied to astronomy: most of the major mathematical advances were motivated by astronomical problems, particularly the computation of planetary positions and eclipses.

The transmission of Indian mathematics to the Islamic world (particularly the decimal system and trigonometry) and thence to Europe was one of the most consequential cultural transfers in history. Without the Indian decimal system, the algebraic revolution of the Islamic Golden Age and the computational advances of the European Renaissance would have been impossible. And the infinite series of the Kerala school — whether or not they influenced Europe directly — demonstrate that the fundamental ideas of analysis were a universal human achievement, not the exclusive property of any single civilization.

The story of Indian mathematics is a powerful reminder that the history of mathematics is not a simple linear narrative from Greece to Europe, but a rich, polyphonic story in which multiple civilizations made essential and original contributions to the mathematical heritage of humanity.

Video Lectures: Mathematics in India

A comprehensive series of 40 lectures on the history of Indian mathematics from the NPTEL program, covering the full sweep from Vedic mathematics through the Kerala school to the modern era.