The mathematics was always here.
We only forgot whose name was on it.

A complete generative grammar for Sanskrit, written in roughly 4,000 algorithmic rules. Not a description — a compiler. The Aṣṭādhyāyī uses meta-rules, recursion, and substitution semantics that the modern computer scientist recognises immediately.

Pāṇini's grammar is the most remarkable production of human intelligence … the construction is so extraordinary that no language to this day has produced anything to equal it. — Leonard Bloomfield (1929)

The structure is identical to Backus–Naur Form. We re-discovered it in 1959 to write ALGOL. Pāṇini did it in 500 BCE — to compile a language already spoken.

Piṅgala encoded Sanskrit metres as laghu (light · 0) and guru (heavy · 1) and built a complete combinatorial calculus on these two symbols. Binary numerals. Two thousand years before Leibniz.

He gives, in passing, the 2ⁿ rule for counting metres of length n, the algorithm we now call the meru-prastāra — Pascal's triangle, drawn by Pingala twelve centuries before Pascal — and a bit-shift trick for converting metre to integer that is identical to modern binary encoding.

The earliest physical evidence of the symbol 0 used as a placeholder and as a number — a small dot, written confidently among other digits — pushed back the history of zero by five hundred years when radiocarbon dated in 2017 by the Bodleian. The dot evolves into the modern circle by Brahmagupta's time.

Without zero there is no place value. Without place value there is no algebra. Without algebra there is no calculus. Every line of mathematics descended from this dot.

Four chapters. 121 verses. Inside them: a sine table at 3.75° intervals; the Earth as a rotating sphere; an estimate of π "approximate" at 3.1416 (correct to four places); the kuṭṭaka algorithm — what we now call the extended Euclidean algorithm, used today in RSA cryptography; and an explicit statement that solar and lunar eclipses are caused by shadows, not by Rāhu.

āsannaḥ — approximate. Āryabhaṭa knew π was irrational. He marked it as such, fifteen hundred years before Lambert proved it.

The first book in human history to lay down arithmetic rules for negative numbers — for what Brahmagupta called ṛṇa (debt) — including the rule that the product of two debts is a fortune. The first book to define zero in operational terms. The first solution of the general quadratic ax² + bx = c, including the negative root.

"A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is zero." — written when Europe was still arguing whether nothing could even be a number.

Bhāskara I gives a rational-function approximation for sine — sin x ≈ 16x(π−x) / [5π² − 4x(π−x)] — accurate to within 1.9 × 10⁻³ across the entire half-period. It is the first non-trivial closed-form approximation to a transcendental function in any culture.

He also systematised the decimal place-value notation in writing, and was the first to mark zero with a small circle ° — the ancestor of the digit we still write today.

The first textbook of mathematics in the world — a self-contained school book covering arithmetic, fractions, geometry, mensuration, the śūnyatā (zero) operations, permutations, and the formula nCr = n!/(r!(n−r)!). Mahāvīra also provides the formula for the area of an ellipse, an explicit statement of nothing-times-anything-is-nothing, and a clean treatment of zero division (which he correctly leaves undefined, unlike Brahmagupta).

Bhāskara II writes, in plain Sanskrit verse, the differential calculus — five hundred years before Newton and Leibniz. He gives the differential of sin x. He gives the mean-value theorem. He gives Pell's equation in full and solves a case (x² − 61y² = 1) that Fermat would later pose to Europe as a challenge problem six centuries later, and that Europe would not solve until Lagrange in 1767.

…in this lemma, by what amounts to use of the differential calculus, the relation d(sin x) = cos x · dx is given… — David Bressoud, A Radical Approach to Real Analysis

Two and a half centuries before Newton, Mādhava writes infinite series for π, sin x, cos x, and arctan x. The series for π that the West would later call the Madhava–Leibniz series appears in the Yuktibhāṣā (Jyeṣṭhadeva, c. 1530) — a complete proof, in Malayalam prose, of π/4 = 1 − ⅓ + ⅕ − ⅐ + …, including the convergence acceleration term.

The Yuktibhāṣā contains the foundations of analysis — Taylor series, term-by-term integration, the geometric proof of the arctan series — written in the early 1500s. Jesuits at Cochin had access to these manuscripts. The chain of transmission to Europe is contested but the priority is not.

Geometry as ritual altar-construction. Baudhāyana states the so-called "Pythagorean" theorem — "the diagonal of a rectangle produces the sum of the squares of the sides" — five centuries before Pythagoras. He also gives √2 ≈ 1 + 1/3 + 1/(3·4) − 1/(3·4·34) — accurate to five decimal places, the first known irrational approximation in mathematics.

Kaṇāda — whose name literally means "the atom-eater" — proposed that all matter is composed of indivisible aṇu (atoms), that atoms cluster into dvyaṇuka (di-atomic) and tryaṇuka (tri-atomic) molecules, and that these combinations explain the diversity of substances. He proposes this by argument from divisibility — the same logical structure Democritus would use a century later in Greece.

184 chapters, 1,120 illnesses, 700 medicinal plants, 121 surgical instruments. The first description of cataract surgery, plastic surgery (the Indian rhinoplasty, still the textbook technique today), caesarean section, and the use of leeches, anaesthetic wines, and aseptic procedure — at least 2,400 years before modern surgery rediscovered any of it.