Indian mathematics

• 1. Introduction

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  Indian mathematics carries a secret hiding in plain sight: every time you write a number, you are using a system that was first recorded in India. The decimal place-value notation now universal across the globe traces its earliest surviving evidence to the Indian subcontinent and southeast Asia from the middle of the first millennium CE. A copper plate from Gujarat mentions the date 595 CE written in decimal place-value notation. Stone inscriptions from Indonesia and Cambodia record the years 683 CE in the same system, in regions where Indian cultural influence ran deep.

  But the story of Indian mathematics stretches far beyond numerals. From at least 1200 BCE through the end of the 18th century, scholars on the Indian subcontinent developed ideas about zero, negative numbers, algebra, trigonometry, and what we now recognize as early calculus. Many of these ideas traveled to the Middle East, then to Europe, quietly reshaping the foundations of modern science. Others never left the region at all, and only came to the attention of Western scholars centuries after they were first worked out.

  The questions worth sitting with: Who were the people doing this work? How did they preserve and transmit it across generations before writing was widespread? And what did mathematicians on the southern tip of India discover about infinite series that Europeans would not reach for two more centuries?

• 2. Before Writing: The Oral Architecture

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  Mathematicians of ancient and early medieval India were almost all Sanskrit pandits, trained in language and literature and possessing a shared foundation in grammar, exegesis, and logic. The word pandit literally means "learned man," and these scholars operated within a culture that treated the accurate transmission of knowledge as a sacred obligation.

  All mathematical works were transmitted orally until approximately 500 BCE. What this required was not simple memorization but a rigorous, redundant system of recitation designed to catch errors the way a checksum catches corrupted data. Memorization of the sacred Vedas, for instance, included up to eleven forms of recitation of the same text, which were then compared against each other as a form of proof-reading.

  One recitation method, described in the source as "mesh recitation," required that every two adjacent words be spoken first in their original order, then reversed, then repeated in original order again, before moving on. Another, "flag recitation," paired the first two words with the last two, then worked inward. The most complex form, "dense recitation," wove words together in a pattern of forward, backward, and extended sequences that could occupy a scholar for years. The most ancient Indian religious text, the Rigveda, dating to around 1500 BCE, survives as a single text without variant readings, a testament to how effective these methods proved.

  Mathematical texts rode the same oral infrastructure. The form they took was the sutra, a Sanskrit word meaning "thread." A sutra was intentionally compressed to the point of near-incomprehensibility without a teacher's guidance, devoid of ambiguity while also shedding every non-essential word. The Baudhayana Sulba Sutra from around 700 BCE demonstrates this in a construction recipe that describes an altar's brick layout in just two lines, relying on the practitioner's training to supply the unstated gender of adjectives and the implied direction of brick orientation.

• 3. Ritual Geometry and the Pythagorean Theorem

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  The Sulba Sutras, composed between roughly 700 and 400 BCE, were practical manuals for building sacrificial fire altars. Their title translates as "Aphorisms of the Chords," and chords, ropes, and pegs were the actual tools priests used on the ground. The mathematical problems the sutras address grew almost entirely from one religious requirement: altars of different shapes had to occupy the same area.

  Each altar needed five layers of burnt brick, with 200 bricks per layer, and no two adjacent layers could share the same arrangement. The geometry this demanded was not decorative; it was load-bearing for the ritual itself.

  In solving these problems, the authors of the Sulba Sutras stated what we now call the Pythagorean theorem. Mathematician Hayashi has identified this as "the earliest extant verbal expression of the Pythagorean Theorem in the world," while noting it had already been known to the Old Babylonians. Baudhayana, who composed the best-known Sulba Sutra around the 8th century BCE, stated it for squares: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." He also gave a general statement for rectangles and listed specific Pythagorean triples.

  Baudhayana also calculated the square root of two to a value accurate to five decimal places, the same accuracy as a Mesopotamian tablet from the Old Babylonian period between 1900 and 1600 BCE. Mathematician S. G. Dani has observed that while Babylonian tablets containing sophisticated Pythagorean material predate the Sulbasutras by several centuries, the Sulba Sutras were primarily architectural manuals, and their mathematical knowledge likely did not represent the full depth of understanding at the time. The question of what Indian mathematicians knew independently, and what was borrowed, remains open.

• 4. The Jain Contribution: Zero, Infinity, and Independence

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  Between about 400 BCE and 200 CE, Jain mathematicians made contributions that would prove historically decisive, and they made them by doing something their predecessors had not done: they freed mathematics from its ritual context.

  Jain texts on mathematical topics were mostly composed after the 6th century BCE. Their authors were captivated by the problem of very large numbers and, eventually, by infinity itself. They classified numbers into three categories: enumerable, innumerable, and infinite. And they were not content to treat infinity as a single concept. Jain texts define five distinct types of infinity: the infinite in one direction, in two directions, in area, everywhere, and perpetually.

  Jain mathematicians also appear to have been the first to use the Sanskrit word shunya, literally meaning void, to refer to zero. This single word's journey is itself a minor marvel of cultural transmission. Shunya was calqued into Arabic as sifr, then borrowed into Medieval Latin as zephirum, traveled through one or more Romance languages including French and Italian, and arrived in English as the word zero.

  The Anuyogadwara Sutra, dated roughly between 200 BCE and 100 CE, contains the earliest known description of factorials in Indian mathematics. The Sthanangas Sutra, from around the same period, and the Shatkhandagama from around the 2nd century CE rounded out the major Jain mathematical texts. Jain mathematician Bhadrabahu, who died in 298 BCE, authored astronomical works; Yativrisham Acharya, around 176 BCE, wrote a mathematical text called Tiloyapannati. Mahavira Acharya, who lived between roughly 800 and 870 CE and was patronized by the Rashtrakuta king Amoghavarsha, solved cubic, quartic, and even some quintic equations, worked out formulas for ellipses, and stated that the square root of a negative number did not exist.

• 5. The Classical Golden Age

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  Aryabhata was born in 476 CE and produced a treatise called the Aryabhatiya. In 332 verses, he covered quadratic equations, trigonometry, and the value of pi correct to four decimal places. He defined sine as the modern relationship between half an angle and half a chord, established the first known tables of sine, cosine, and versine values at intervals of 3.75 degrees from 0 to 90 degrees, and calculated astronomical constants for solar and lunar eclipses. He also stated a formula for the sum of cubes that would later be recognized as an important step toward integral calculus.

  Brahmagupta's major astronomical work, completed in 628 CE, devoted two entire chapters to arithmetic and algebra. Chapter 18 contained 103 Sanskrit verses and is considered the first systematic treatment of arithmetic operations involving zero and negative numbers. In that chapter, Brahmagupta also gave the first explicit solution of the quadratic equation in written form. He challenged his readers with a problem involving Pell's equation and added, in his commentary, that "a person solving this problem within a year is a mathematician."

  Bhaskara II, who lived from 1114 to 1185, wrote multiple treatises including the Siddhanta Shiromani, Lilavati, and Bijaganita. He recognized that a positive number has two square roots, stated an early form of Rolle's theorem, and computed pi correct to five decimal places. He also calculated the solar year to nine decimal places. In his astronomical work, Bhaskara gave a result that can be interpreted as the discovery that cosine is the derivative of sine, though he did not develop a formal theory of derivatives.

  Variahamihira, working between 505 and 587, produced the Pancha Siddhanta, a compilation of five earlier astronomical canons from Mesopotamian, Greek, Egyptian, Roman, and Indian sources, representing the cosmopolitan intellectual environment in which these mathematicians operated.

• 6. The Kerala School and the Edge of Calculus

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  Madhava of Sangamagrama founded what became known as the Kerala school of astronomy and mathematics in the southern Indian state of Kerala. The school flourished between the 14th and 16th centuries, and its most important results were series expansions for trigonometric functions: sine, cosine, and arc tangent.

  These expansions, now known as Taylor-Maclaurin series, were produced by the Kerala school two centuries before Isaac Newton and Gottfried Leibniz developed calculus in Europe. The proofs for the series for sine, cosine, and inverse tangent appeared about a century after the original results, in a work called the Yuktibhasa, written in Malayalam around 1500-1610 by Jyesthadeva. The Kerala school also used the series expansion of arc tangent to obtain the Leibniz formula for pi and derived a rational approximation of 104348/33215 for pi, accurate to nine decimal places.

  Nilakantha Somayaji, who lived from 1444 to 1544, composed the Tantra Samgraha and elaborated on Madhava's contributions. Parameshvara, who worked roughly between 1370 and 1460, discovered a version of the mean value theorem in his commentary on Bhaskara II's Lilavati.

  The Kerala school did not, however, invent calculus. They did not formulate a theory of differentiation and integration, did not develop the fundamental theorem of calculus, and did not have knowledge of exponential or logarithmic functions. Crucially, there is no evidence that their results were transmitted outside Kerala before the 19th century. An Englishman named C. M. Whish first wrote up the school's work for a Western audience in 1835. His findings were largely ignored for over a century until C. Rajagopal and his associates revisited them. David Bressoud has stated that "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."

• 7. Transmission and the Question of Credit

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  The decimal number system reached Europe through the Islamic world. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. This is the earliest clear external acknowledgment of the system. From there, the notation passed through Arabic scholarship and eventually into European mathematics.

  The transmission of Indian trigonometry followed a similar path. Later Arabic and Latin translations of texts touching on Aryabhata's work were influential in Europe and the Middle East. The contributions of Brahmagupta and Bhaskara II were, according to the source, later transmitted to the Middle East and Europe.

  But the Kerala school's calculus-adjacent work appears to have stayed put. Scholars have suggested that Jesuit missionaries or traders using the Kerala trade route to Europe around 1500 might have carried some of this knowledge westward, since Kerala was in continuous contact with China, Arabia, and from around 1500 onward, with Europe. The chronology is plausible. But no documentary evidence of such transmission has been found.

  The historian of mathematics Florian Cajori wrote that he and others suspected Diophantus got his first understanding of algebra from India, while also acknowledging that portions of Indian mathematics were of Greek origin. G. G. Joseph, writing on what he called ethnomathematics, argued that contributions from India and China have often been perceived either as borrowings from Greek sources or as minor contributions to the mainstream, a framing he considered inadequate given more recent research. Whether the predecessors of Newton and Leibniz, including Fermat and Roberval, may have encountered Islamic or Indian mathematical ideas through sources now unknown is a question being actively investigated, particularly in manuscript collections in Spain and the Maghreb, at the CNRS.