Mahāvīra writes the first explicit ellipse-area formula in surviving Sanskrit mathematics

# Mahāvīra writes the first explicit ellipse-area formula in surviving Sanskrit mathematics

In 850 CE, in chapter VII verse 21 of the *Ganita-sara-sangraha*, Mahāvīra gave a recipe for computing the area of an ellipse — something Indian gardeners, jewelers, and metal-workers periodically needed to do when dealing with elliptical ponds, oval lockets, or elongated cooking vessels.

The verse, in two prose sentences:

> The longer diameter, increased by half of the shorter diameter > and multiplied by two, gives the measure of the circumference of > the elliptical figure. One-fourth of the shorter diameter, > multiplied by the circumference, gives rise to the area thereof. > > — *Ganita-sara-sangraha* VII.21, trans. M. Rangacarya (1912)

Two formulas. Circumference first, then area.

In modern notation: let $L$ and $S$ be the longer and shorter diameters of the ellipse, with semi-axes $a = L/2$ and $b = S/2$.

- Mahāvīra's circumference: $C_M = 2(L + S/2) = 4a + 2b$ - Mahāvīra's area: $A_M = (S/4) \cdot C_M = 2ab + b^2$

The modern exact values: - Circumference: $C = 4a \cdot E(e)$ where $E(e)$ is the complete elliptic integral of the second kind (no closed form; Kepler's 1609 approximation $\pi(a+b)$ is good to ~1%) - Area: $A = \pi \cdot a \cdot b$

How accurate is Mahāvīra's area formula

Test it against modern exact values for different ellipse shapes:

| Shape | $a/b$ ratio | $A_M = 2ab + b^2$ | Modern $A = \pi a b$ | Error | |-------|-------------|---|---|---| | Circle | 1 | $3 b^2$ | $3.14 b^2$ | −4.5% (the $\pi = 3$ approx) | | Slight oval | 1.5 | $4 b^2$ | $4.71 b^2$ | −15% | | Moderate | 2 | $5 b^2$ | $6.28 b^2$ | −20% | | Eccentric | 3 | $7 b^2$ | $9.42 b^2$ | −26% | | Very eccentric | 5 | $11 b^2$ | $15.7 b^2$ | −30% |

The formula is *exact for the circle limit* (recovering the standard π = 3 Indian circle-area approximation) and *increasingly inaccurate as eccentricity grows*. For garden-pond and storage-vessel shapes (a/b ≤ 2), it's within 20%. For very elongated ellipses it understates the area by 30%+.

That's not Newton's exact $\pi a b$. But it's also not bad for a 9th-century closed-form mensuration rule applied to a curved figure with no Euclidean compass-and-straightedge solution.

What the Greek tradition had

A footnote on chronology. The Greek geometric tradition studied conic sections extensively from **Apollonius of Perga** (~200 BCE) onward. Apollonius's *Conics* — eight books, four extant in Greek and three in Arabic translation — proves dozens of properties of ellipses, hyperbolas, and parabolas. But none of the surviving Apollonian theorems give an explicit closed-form area-of-an- ellipse formula. Apollonius treats ellipses as projections of cones, not as figures-with-area.

**Pappus of Alexandria** (~340 CE) in his *Mathematical Collection* may have given the area formula via the cross-section theorem (slicing a cylinder obliquely produces an ellipse whose area equals the perpendicular cross-section), but Pappus's relevant volumes survive only in fragments. Modern reconstructions suggest Pappus had the result; the textual evidence is incomplete.

The Greek tradition genuinely did have the geometric *idea* of ellipse-area, expressed through cylinder-section theorems. What the surviving Greek tradition does NOT have is an *explicit computational formula* — "to compute, do this, then that, get the area." Mahāvīra's *Ganita-sara-sangraha* gives the computational formula.

This is the recurring contrast between Greek and Indian mathematical style. The Greek tradition emphasises **why** (proof, geometric construction, theorem). The Indian tradition emphasises **how** (algorithm, procedure, computation). Both are mathematics. The two emphasis-styles produce different surviving artefacts.

The Newton resolution

The European tradition's exact $A = \pi a b$ comes through **Newton's** *Method of Fluxions* (written 1671, published 1736), where integration of $y = b\sqrt{1 - x^2/a^2}$ from $-a$ to $a$ gives $\pi a b$ directly. Newton's integration is the modern calculus-based derivation; it makes the formula exact and easy to derive.

Mahāvīra's approximation is what you get when you don't yet have calculus. Mahāvīra's circumference formula $4a + 2b$ is similarly an approximation — exact for the circle case (giving $\pi = 3$), not exact for ellipses. The two formulas together — circumference and area — are consistent: both use the same π = 3 approximation and both degrade gracefully for eccentric shapes.

Why this matters

Three reasons.

**The formula is real and applicable.** Indian mensuration literature used Mahāvīra's rule continuously through the medieval period for practical area-of-pond and area-of-locket calculations. For typical near-circular ponds and storage vessels (the everyday applications), the 5-10% error was tolerable.

**It's the earliest written explicit ellipse-area formula in any Sanskrit source.** Brahmagupta (628 CE) gave ellipse circumference but not area. Bhāskara II (1150 CE) restates Mahāvīra. The inheritance chain runs Mahāvīra → later Indian commentators through the Kerala school.

**It shows the Indian computational-mathematics style at work** on a problem that the Greek geometric-proof style approached differently. Both reach correct results (Greek via cylinder sections, Indian via approximate computation); both styles have strengths the other lacks.

What this leaves us with

A 1,170-year-old Sanskrit verse, giving two formulas in two sentences — circumference and area of an ellipse — using the standard Indian π = 3 approximation. The formulas are correct in shape, exact for the circle limit, and within 5-20% for typical ellipse eccentricities. Modern exact calculus-based derivation (Newton, 1671) is 821 years later.

The everyday mensuration of an elliptical pond, with no calculus available, had a tool that worked well enough — and the Sanskrit textbook that taught it kept teaching it for 1,000 years.

---

Sources

- [The Ganita-sara-sangraha of Mahāvīrācārya, M. Rangacarya trans., 1912](https://archive.org/details/RangacaryaTheGanitaSaraSangrahaOfMahavira1912) — verse VII.21 cited above. - Apollonius of Perga. *Conics*, 4 books extant Greek + 3 books Arabic translation. — the Greek conic-section reference. - Newton, I. (1736). *Method of Fluxions*. — modern exact derivation. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 5-6. — Mahāvīra's mensuration tradition.

Related claims

- [Mahāvīra writes down nCr](/c/50f870a0-ac17-52a9-ad76-d91de7ac23f6) — same author, same treatise, same algorithmic-procedural style. - [Aryabhata's pi approximation](/c/0b862684-d325-5002-b054-169bd2253ef9) — the better π approximation that was available in 499 CE but did not become the standard mensurational π until much later.