Bhāskara II writes down the Pythagorean theorem — already 2,000 years old in India when he did

# Bhāskara II writes down the Pythagorean theorem — already 2,000 years old in India when he did

In 1150 CE, in verse 134 of the *Lilavati*, Bhāskara II — Indian mathematician, astronomer, head of the astronomical observatory at Ujjain — wrote down the Pythagorean theorem in a single Sanskrit sentence:

> The square-root of the sum of the squares of those legs is the > diagonal. > > — *Lilavati* §134, trans. H. T. Colebrooke (1817)

That's the theorem. Right triangle, two legs $a$ and $b$, hypotenuse $c$. The rule: $c = \sqrt{a^2 + b^2}$. Three other Indian mathematical traditions had stated essentially the same rule before Bhāskara — by the time he wrote it down, the result had been part of Indian geometry for approximately 2,000 years.

This article is not about who got there first. It's about what it means that two mathematical traditions, in two different parts of the world, both arrived independently at the same result and held it continuously for two millennia each.

The Sulbasutras: ~800 BCE

The earliest known Indian statement of the rule appears in the **Sulbasutras** — texts on the geometry of fire-altar construction attached to the Vedic ritual literature. The Baudhāyana Sulbasutra, generally dated to ~800 BCE, gives the rule explicitly:

> The rope stretched along the length of the diagonal of a rectangle > makes an area which the vertical and horizontal sides make together. > > — *Baudhāyana Sulbasutra* 1.12 (approximate paraphrase; the > Sulbasutras are not in the Experli source library yet)

That is the Pythagorean theorem, in coordinate-free geometric language: the square on the diagonal equals the sum of the squares on the two sides. The Sulbasutras then use it concretely to construct right angles for ritual altars — given two perpendicular ropes of length $a$ and $b$, a third rope of length $\sqrt{a^2 + b^2}$ completes the right triangle.

Three Sulbasutras have survived: Baudhāyana (~800 BCE), Apastamba (~600 BCE), and Katyayana (~400 BCE). All three state the rule. All three predate Pythagoras (~570-495 BCE) by roughly the same margin that Pythagoras predates Euclid.

Pythagoras: ~530 BCE

The Greek tradition attributes the theorem to Pythagoras through later sources — Proclus, writing around 450 CE, says Pythagoras proved it. We have no surviving Pythagorean text; the attribution is ~1,000 years downstream of the alleged proof.

The earliest *extant* Greek proof is Euclid's, *Elements* I.47, ~300 BCE — a geometric dissection-and-area proof that became the standard European school presentation for the next 2,200 years.

The independent question — did Pythagoras *learn* the theorem from an earlier tradition? — has no consensus answer. Pythagoras reportedly traveled to Egypt and possibly Babylon, and the Babylonian tablet **Plimpton 322** (~1800 BCE) lists what appear to be Pythagorean triples — $(3,4,5)$, $(5,12,13)$, $(8,15,17)$, and more — suggesting Babylonian mathematicians knew the relation in some form a thousand years before either the Sulbasutras or Pythagoras. The Plimpton tablet, however, does not state the underlying rule abstractly. It lists examples.

So the *stated rule* appears in three independent traditions:

1. The Sulbasutras (~800 BCE) — geometric formulation, used for altar construction. 2. The Greek tradition (~530 BCE → Euclid ~300 BCE) — proven as a theorem of plane geometry. 3. The Babylonian tradition (~1800 BCE, Plimpton 322) — examples without an articulated rule, but enough triples that the underlying relation was clearly known.

There is no extant evidence of communication between the three traditions before all three had the result. The standard scholarly read is that the theorem was found independently — and probably more than three times — by mathematicians attentive enough to the geometry of squares and right angles.

What Bhāskara added

So if the Indian tradition already had the rule by 800 BCE, what is Bhāskara doing restating it in 1150 CE?

The same thing every mathematical tradition does with its foundational results: teaching them.

The *Lilavati* is a textbook. Named for Bhāskara's daughter (the name means "she who is playful" — the book is structured as a series of problems posed to her), it became the standard Indian school text in mathematics for the next 700 years. Its 277 verses cover arithmetic, algebra, geometry, mensuration, combinatorics, and number theory at the level of a strong undergraduate course. The Pythagorean rule appears at §134 not because Bhāskara discovered it but because every Indian student of geometry needed to know it.

The same chapter (Bhāskara's "Plane Figure" chapter) gives the inverse forms — given the hypotenuse and one leg, find the other, using the same $\sqrt{c^2 - a^2}$ structure. It works through explicit numerical examples: "where the upright is four and the side three, what is the hypotenuse?" Answer 5. "Where the hypotenuse is seventeen and the sum of side and upright is twenty-three, find the side and upright?" Answer 8 and 15.

Those are exactly the same right triangles — $(3, 4, 5)$ and $(8, 15, 17)$ — that appear on Plimpton 322 three thousand years earlier. Mathematicians find the same examples.

The unbroken transmission

What's actually remarkable about Bhāskara's verse is that he is the late-medieval point in an unbroken chain. The Pythagorean rule appears continuously in the Indian mathematical literature:

- Sulbasutras (Baudhāyana, ~800 BCE) - Apastamba Sulbasutra (~600 BCE) - Katyayana Sulbasutra (~400 BCE) - Aryabhata's *Aryabhatiya* (499 CE) — uses it implicitly in the sine-table construction - Brahmagupta's *Brāhmasphuṭasiddhānta* (628 CE) — used extensively in mensuration - Mahavira's *Ganita-sara-sangraha* (850 CE) — explicit statements in the geometry chapter - Sridhara's *Patiganita* (~900 CE) — restated - Bhāskara II's *Lilavati* (1150 CE) — the version above - Nilakantha Somayaji's *Tantrasangraha* (1500 CE) — used in the Kerala school's sine-series derivations

Twenty-three centuries. Nine known transmissions. The rule never left the Indian curriculum.

The Greek tradition, by contrast, has a major break: the Pythagorean theorem disappears from Western European mathematical literature for roughly the millennium between the closure of the last Roman schools (~500 CE) and the recovery of Euclid through Arabic translations in the 12th century. The Arabic tradition kept it, drawing on both Greek and Indian sources — Al-Khwarizmi's *Algebra* (~830 CE) uses Indian numerical methods alongside Greek geometric ones — but Latin Europe forgot it for ~700 years before re-importing it.

The honest comparison

So what does Bhāskara's Lilavati §134 actually show us?

Not that India had the Pythagorean theorem first. Both Indian and Babylonian traditions had it before any extant Greek statement, and the historical record is too thin to say which came first or whether "first" is even the right framing.

What it shows is that **the result was held continuously, in living mathematical practice, in India for at least 1,950 years before Bhāskara wrote it down again, and continued for another 850 years after him**. The Lilavati is not a discovery. It is a *waypoint* in a transmission chain so stable that the same theorem worked the same way for a hundred generations of Indian students.

Pythagoras, if he existed and proved what Proclus said he proved, was a contemporary of Apastamba. Two mathematicians on two sides of the same Indian Ocean, working out the same geometric relation at roughly the same century. That coincidence is the actual interest. Not priority. Convergence.

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Sources

- [Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhāscara, H. T. Colebrooke trans., 1817](https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-br) — Lilavati §134 cited above. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, chs. 2-3. — the Sulbasutras tradition and its continuation. - Datta, B. (1932). *The Science of the Sulba: A Study in Early Hindu Geometry*. Calcutta University Press. — the canonical study of the Sulbasutras as mathematical texts. - Joseph, G. G. (2011). *The Crest of the Peacock: Non-European Roots of Mathematics* (3rd ed.). Princeton University Press, chs. 5, 8. — comparative treatment of Indian, Greek, and Babylonian sources.

Related claims

- [Bhāskara II's chakravala for Pell's equation](/c/e9b99c06-981f-5f4d-8628-0357df5417f0) — same author, same century, the deepest single result of medieval Indian mathematics. - [Brahmagupta's cyclic-quadrilateral formula](/c/5c1b33ac-50a8-5b98-86d5-feb2ac2109f9) — the parallel geometric tradition Bhāskara inherited.