# Āryabhaṭīya

**Type:** Work  
**Domain:** Indian Mathematics  
**Composed:** 499 CE  
**Author:** Āryabhaṭa I (b. 476 CE, Kusumapura / Pāṭaliputra)  
**Language:** Sanskrit (āryā metre)  
**Codex URL:** /mathematics/indian-mathematics/aryabhatiya/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Curious Level

The Āryabhaṭīya is a Sanskrit scientific treatise written in 499 CE by the mathematician and astronomer Āryabhaṭa I, when he was approximately twenty-three years old. In 121 terse verses he set down results that would shape mathematics and astronomy across Asia and eventually Europe.

Key contributions include:

- **π ≈ 3.1416** — accurate to four decimal places, explicitly called *āsanna* ("approximate") by Āryabhaṭa himself
- A **sine table** with 24 values — the ancestor of trigonometric tables used in Islamic and European astronomy
- A **place-value decimal system** encoded in Sanskrit syllables
- The claim that **stars appear to move because the Earth rotates** on its axis — a proto-heliocentric insight
- The **kuṭṭaka** ("pulveriser") — the world's first general algorithm for solving equations of the form *ax ≡ b (mod c)*

The text was translated into Arabic around 820 CE and influenced Al-Khwārizmī. The word "sine" in European mathematics traces directly to Āryabhaṭa's Sanskrit term *jyā* via Arabic *jiba* and Latin *sinus*.

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## Exploring Level

### Structure

The Āryabhaṭīya has four sections (*pāda*):

| Section | Verses | Subject |
|---|---|---|
| Gītikāpāda | 13 | Astronomical constants, large number names |
| Gaṇitapāda | 33 | Arithmetic, geometry, series, linear algebra |
| Kālakriyāpāda | 25 | Time reckoning, calendar system |
| Golādhyāya | 50 | Spherical astronomy, celestial mechanics |

### The π Approximation

Gaṇitapāda verse 10 states: "Add 4 to 100, multiply by 8, add 62,000 — this is approximately the circumference of a circle whose diameter is 20,000."

(4 + 100) × 8 + 62000 = 62832 → π ≈ 62832 / 20000 = **3.14160**

The word *āsanna* (approximate) suggests Āryabhaṭa understood π to be irrational. This value is accurate to four decimal places and surpassed contemporary Greek estimates.

### The Sine Table

Āryabhaṭa tabulated *ardha-jyā* (half-chord) values at 3.75° intervals for the first quadrant (24 entries). He encoded the values in a compact Sanskrit syllabic notation and provided a **second-difference interpolation scheme** — a precursor to finite difference methods. The etymology of "sine": Sanskrit *jyā* → Arabic transliteration *jiba* (meaningless in Arabic) → misread as *jaib* (bay/bosom) → Latin *sinus* (bay).

### The Kuṭṭaka Algorithm

For the linear congruence *ax ≡ b (mod c)*, the kuṭṭaka applies repeated division analogous to the Euclidean algorithm, then works back through the quotients. This is the earliest known systematic algorithmic solution to linear Diophantine equations — predating similar European results by over a millennium.

### Astronomical Accuracy

| Quantity | Āryabhaṭa's value | Modern value |
|---|---|---|
| Sidereal day | 23h 56m 4.1s | 23h 56m 4.091s |
| Sidereal year | 365d 6h 12m 30s | 365d 6h 9m 10s |
| Earth's circumference | 39,968 km | 40,075 km |

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## Deep Dive

### Critical Edition and Transmission

The standard critical edition is Shukla and Sarma (INSA, 1976), which collates manuscript variants and provides mathematical commentary. The oldest surviving manuscripts are from Kerala, where the text was intensively studied and commented upon (Bhāskara I, 629 CE; Nīlakaṇṭha Somayājī, c. 1500 CE). The Nīlakaṇṭha commentary in particular contains significant mathematical extensions.

### Dating

The text is self-dated. Golādhyāya 32–33 states that 3,600 years of the Kali Yuga have elapsed — placing composition at 499 CE by the standard Kali Yuga epoch of 3102 BCE. This absolute date is unusually secure for ancient Indian mathematics.

### Kuṭṭaka — Formal Statement

Given integers *a*, *b*, *c* with gcd(*a*, *c*) | *b*, find integer *x* such that *ax ≡ b (mod c)*.

Āryabhaṭa's procedure: apply the Euclidean algorithm to (*a*, *c*) recording quotients *q₁, q₂, …, qₙ*, then construct the solution by back-substitution through the quotient chain. This is precisely the extended Euclidean algorithm, demonstrated algorithmically in verse rather than symbolic notation.

### The Rotation of the Earth

Golādhyāya 9–10: "The sphere of stars does not move. The apparent westward motion of the stars is caused by the eastward rotation of the Earth on its axis." This is a kinematic claim (relative motion), not a full heliocentric model. However, combined with Āryabhaṭa's epicyclic planetary models, some historians (notably B.L. van der Waerden) have argued for a partially heliocentric conception. The claim was controversial even among Indian astronomers — Brahmagupta (628 CE) explicitly rejected it.

### Transmission to Islamic Mathematics

The Āryabhaṭīya reached the Baghdad translation movement through the *Zīj al-Sindhind* (c. 771 CE), a Sanskrit astronomical work brought to the Abbasid court. Al-Khwārizmī's *Kitāb al-mukhtaṣar* (c. 820 CE) — from which "algebra" and "algorithm" derive — incorporates Indian place-value arithmetic whose Indian pedigree passes through or parallel to Āryabhaṭa's tradition.

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## Sources

### Tier 1 — Primary Scholarly Editions
- Shukla, K.S. and Sarma, K.V. (1976). *Āryabhaṭīya of Āryabhaṭa*. Indian National Science Academy, New Delhi. [Critical edition, Sanskrit text, English translation, mathematical notes.]
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press, pp. 111–127.

### Tier 2 — Reliable Reference
- Clark, W.E. (1930). *The Āryabhaṭīya of Āryabhaṭa*. University of Chicago Press.
- O'Connor, J.J. and Robertson, E.F. (2006). Āryabhaṭa the Elder. MacTutor History of Mathematics Archive. https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_I/

### Tier 3 — Accessible Scholarly
- Joseph, G.G. (2011). *The Crest of the Peacock: Non-European Roots of Mathematics*. 3rd ed. Princeton University Press.

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## Relationships

- **Authored by:** Āryabhaṭa I
- **Part of tradition:** Indian Mathematical Tradition
- **Introduces:** Kuṭṭaka algorithm; Sine table (jyā)
- **Commented on by:** Bhāskara I (629 CE); Nīlakaṇṭha Somayājī (c. 1500 CE)
- **Influenced:** Brāhmasphuṭasiddhānta (Brahmagupta, 628 CE)
- **Transmitted to:** Al-Khwārizmī, Baghdad (c. 820 CE)

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*