Gaṇita & Jyotiṣa

Every time you write “2026” you use an idea that took shape in India: nine digits and a zero, each worth more by its place. That idea came out of a tradition of gaṇita, “calculation”, which grew up beside jyotiṣa, the science of the lights of heaven. The Map of the Śāstras gave them one paragraph. This page walks the line from the Vedic altar to the Kerala school — and is honest about the one part of jyotiṣa that is not science.

Two sciences, one limb

Jyotiṣa began as one of the six Vedāṅgas (see Layers of a Veda): the knowledge needed to time a rite. Later writers divided it into three “branches”: gaṇita or siddhānta (mathematical astronomy), saṃhitā (omens and portents) and horā (horoscopy). Mathematics proper — arithmetic, algebra, geometry — was usually written up as chapters inside astronomical treatises, because astronomers needed it.

Tradition’s view The Sūrya Siddhānta presents itself as revealed by the Sun to the asura Maya at the end of the Kṛta age; the science is as old as the cosmos it measures. Historians see a datable story of borrowing and invention, set out below. Both can be held at once, but they answer different questions.

1. c. 800–200 BCEThe ŚulbasūtrasRules for laying out fire-altars with cords: the diagonal theorem, √2, squaring the circle approximately.
2. Last centuries BCEVedāṅga Jyotiṣa; PiṅgalaA five-year ritual calendar attributed to Lagadha; Piṅgala’s counting of metres.
3. 499 CEĀryabhaṭa’s Āryabhaṭīya121 verses of mathematics and astronomy, written in Kusumapura (Patna) when he was 23.
4. c. 550VarāhamihiraThe Pañcasiddhāntikā sums up five older astronomies, some showing Greek influence.
5. 628BrahmaguptaThe Brāhmasphuṭasiddhānta: rules for zero and negative numbers.
6. 1150Bhāskara IIThe Siddhāntaśiromaṇi, including the Līlāvatī and Bījagaṇita.
7. c. 1340–1550The Kerala schoolMādhava’s infinite series; Nīlakaṇṭha’s Tantrasaṅgraha (c. 1500); the Malayalam Yuktibhāṣā.

Altars and cords: the Śulbasūtras

Vedic fire-altars had to be built to exact shapes and areas — a falcon of bricks, a square that must be doubled without changing its form. The Śulbasūtras, “rules of the cord”, attached to the ritual manuals of Baudhāyana, Āpastamba, Kātyāyana and Mānava, give the geometry. Baudhāyana’s is the oldest:

That is, the square on the diagonal equals the two squares on the sides — what Europe calls Pythagoras’ theorem, stated here as a builder’s rule, without a proof. Baudhāyana also gives √2 as 1 + 1/3 + 1/(3·4) − 1/(3·4·34), correct to five decimal places. Scholars dispute the dates (c. 800–200 BCE is the usual range), and whether Mesopotamian builders knew the same rule earlier is a separate question; what is clear is that these are the oldest geometry texts in Sanskrit.

Counting metres: Piṅgala

Another Vedāṅga, metre, led to combinatorics. Piṅgala’s Chandaḥśāstra (perhaps the last centuries BCE) asks how many patterns of light and heavy syllables a line can have, and gives procedures for listing them and finding the position of any pattern — in effect a binary code. Later commentators drew the triangle of numbers Europe calls Pascal’s (explicitly by Halāyudha, 10th century), and prosodists counting metres by beats found the sequence 1, 2, 3, 5, 8, 13… (Virahāṅka, Gopāla and Hemacandra, before Fibonacci’s 1202 book). Poetry, in this tradition, taught people to count.

Nine digits and a dot

Decimal place value with a zero is India’s best-known gift, and its history is still being written. Some milestones on which historians agree:

• The Jain cosmology Lokavibhāga (dated by its own colophon to 458 CE) writes large numbers in place-value order using number-words.
• Āryabhaṭa (499) assumes place value throughout, and invents a syllabic code for numbers.
• Brahmagupta, in the Brāhmasphuṭasiddhānta of 628, gives the first known rules for calculating with zero (śūnya) and with negative quantities, which he calls “debts” beside positive “fortunes”: a debt minus zero is a debt; zero times anything is zero. He stumbles only on dividing by zero.
• The oldest dated zeros carved in stone are in Southeast Asia, in Śaka-era dates: the Sambor inscription in Cambodia (605 Śaka, 683 CE) and the Old Malay Kedukan Bukit inscription from Palembang in Sumatra (604 Śaka) — the world of Śrīvijaya, next door to the Malay peninsula. In India, the Gwalior temple inscription of 876 CE writes “270” with a round zero.

The Bakhshali manuscript, a birch-bark arithmetic dug up near Peshawar in 1881, uses a dot for zero. In 2017 the Bodleian Library announced radiocarbon results placing three of its leaves roughly in the 3rd–4th, 7th–8th and 9th–10th centuries, and hailed “the oldest zero”. Kim Plofker and four colleagues replied that a manuscript is copied at one time, so the latest leaf dates the writing — which fits earlier estimates from script and content. The debate is open; treat headlines about “the earliest zero” with care. From India the system reached Baghdad (al-Khwārizmī, c. 825) and then Europe, where it is still mis-named “Arabic numerals”.

Āryabhaṭa: a verse that hides π

Āryabhaṭa’s Āryabhaṭīya is short and dense: a chapter of parameters, one of mathematics, one of time-reckoning and one on the sphere. He tells us he wrote it when 3,600 years of the Kali age had passed and he was twenty-three — that is, in 499 CE. Here is his value of the circle:

Work it through, one step at a time:

The same book gives a table of 24 jyā (sine) differences at intervals of 3¾°, the ancestor of every later Indian trigonometry; the kuṭṭaka or “pulveriser” for solving equations like ax − by = c in whole numbers; and a startling astronomical claim — that the stars stand still and the earth turns:

(“Laṅkā” here is a reference point on the equator, not Sri Lanka.) Most later astronomers, Brahmagupta among them, rejected the rotating earth, and some commentators reinterpreted the verse. Āryabhaṭa’s system was still earth-centred: the planets go round the earth.

From Varāhamihira to Bhāskara

Varāhamihira of Ujjain (6th century) wrote the Pañcasiddhāntikā, a digest of five astronomical systems current in his day. Two of them, the Romaka and Pauliśa, carry clear marks of Greek astronomy, and Indian astronomy of this period is best understood as a creative reworking of ideas that travelled along the trade routes. Varāhamihira also wrote on omens (Bṛhatsaṃhitā) and horoscopy (Bṛhajjātaka) — one person, three branches.

Bhāskara II (born 1114) completed his Siddhāntaśiromaṇi in 1150. Its arithmetic, the Līlāvatī, poses problems in verse to a “fawn-eyed girl”, and was taught for centuries; its algebra, the Bījagaṇita, perfects the cakravāla (“cyclic”) method, building on Jayadeva and Brahmagupta, for equations like 61x² + 1 = y², whose smallest solution has x = 226,153,980. Fermat set the same equation as a challenge in 1657. Later legend The story that Līlāvatī was Bhāskara’s daughter, whose wedding hour was lost to a pearl falling into a water-clock, is first told centuries later and is not in the book.

The Kerala school

Between about 1350 and 1550, in a cluster of Brahmin households in central Kerala, many near the Nīlā (Bhāratappuḻa) river, a line of teachers and students did something remarkable. Mādhava of Saṅgamagrāma (c. 1340–1425) found infinite series for the sine, cosine and arctangent, and so for π. None of his own writings on them survive; we know them from his successors, who quote verses like this:

In modern notation: circumference = 4D(1 − 1/3 + 1/5 − 1/7 + …), the series Europe credits to Gregory and Leibniz in the 1670s. (“Ocean” means four, for the four oceans; “arrow” means five, for Kāma’s five arrows — numbers in verse were coded as words.) Because the series converges slowly, the Kerala mathematicians also worked out correction terms for where you stop, and computed π to eleven decimal places.

Nīlakaṇṭha Somayājī (1444–1544), in his Tantrasaṅgraha (c. 1500), revised the planetary model so that Mercury, Venus, Mars, Jupiter and Saturn circle the sun while the sun circles the earth — a scheme like Tycho Brahe’s, a century earlier. And Jyeṣṭhadeva’s Yuktibhāṣā (c. 1530) is the rare thing: a textbook of upapatti, demonstrations, written not in Sanskrit but in Malayalam, explaining why the series work.

Did this knowledge reach Europe? Here historians must be careful, and three positions are worth knowing:

Tamil skies: the vākya almanacs

The Tamil tradition prized calculation as highly as the Sanskrit one. Tiruvaḷḷuvar puts it in a single couplet:

In South India astronomy was carried less by new treatises than by working almanac-makers. The vākya system — “sentences” — encodes the moon’s and planets’ positions in short memorised phrases, so that a calculator needs only addition and a good memory. It was developed in Kerala and Tamil Nadu (the Vākyakaraṇa is usually dated to the 13th century) and still underlies the traditional Tamil vākya pañcāṅkam. In the 1760s the French astronomer Guillaume Le Gentil, stranded in Pondicherry by war, recorded how Tamil astronomers computed eclipses by these methods, and found them practical and impressively accurate. Today many Tamil almanacs, in India and Malaysia alike, follow either this vākya reckoning or the modern-calculation tirukkaṇita one — which is why temples sometimes keep a festival a day apart.

The calendar link

All this mathematics lands, every morning, on your temple’s wall. The pañcāṅga’s five limbs are astronomical quantities: a tithi ends each time the moon gains another 12° on the sun; a nakṣatra is the moon’s place among 27 segments of 13°20′. Computing them for a given town needs exactly the true-longitude methods of the siddhāntas. How the almanac works, and why festival dates move, is explained on The Panchanga; the great cycles of yugas and kalpas are on Time & Cosmology.

Horoscopy, honestly

Horā, the casting of birth charts, is a different kind of thing. Its core — twelve signs, planetary “houses”, the ascendant — came into India from the Hellenistic world: the Yavanajātaka (“Horoscopy of the Greeks”) is a Sanskrit version of a Greek-language work, and horā itself is the Greek hōra, “hour”. Indian astrologers then built a rich system of their own, joined to the older Indian nakṣatras.

Questions

Check your knowledge:

Key terms on this page

Sources & further reading

• Kim Plofker, Mathematics in India (Princeton University Press, 2009) — the standard survey, and this page’s main guide.
• David Pingree, Jyotiḥśāstra: Astral and Mathematical Literature (Otto Harrassowitz, 1981).
• K. S. Shukla and K. V. Sarma (ed. and trans.), Āryabhaṭīya of Āryabhaṭa (Indian National Science Academy, 1976).
• S. N. Sen and A. K. Bag, The Śulbasūtras (Indian National Science Academy, 1983).
• K. Ramasubramanian and M. S. Sriram, Tantrasaṅgraha of Nīlakaṇṭha Somayājī (Springer / Hindustan Book Agency, 2011).
• K. V. Sarma, K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, Gaṇita-Yukti-Bhāṣā of Jyeṣṭhadeva (Springer / Hindustan Book Agency, 2008).
• George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. (Princeton University Press, 2011).
• Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk, “The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating”, History of Science in South Asia 5.1 (2017).
• Shawn Carlson, “A double-blind test of astrology”, Nature 318 (1985); J. V. Narlikar and others, “A statistical test of astrology”, Current Science 96 (2009).