Aryabhata gives Earth a diameter — 1,050 yojanas, depending on which yojana you mean

# Aryabhata gives Earth a diameter — 1,050 yojanas, depending on which yojana you mean

In 499 CE, in verse I.5 of the *Aryabhatiya*, Aryabhata wrote:

> A yojana consists of 8,000 times a nṛ [the height of a man]. The > diameter of the Earth is 1,050 yojanas. The diameter of the Sun is > 4,410 yojanas. The diameter of the Moon is 315 yojanas. > > — *Aryabhatiya Dasagitika* I.5, trans. W. E. Clark (1930)

A 23-year-old Indian astronomer, working without telescopes, declared the Earth was a sphere of diameter 1,050 yojanas. He was near-exactly right — *if* you accept the standard mid-period conversion of one yojana to roughly 7.5 miles.

That conversion is where this gets interesting.

The arithmetic, in three ways

| Yojana convention | 1,050 yojanas in miles | Modern Earth diameter | Error | |-------------------|------------------------|------------------------|-------| | 8 mi/yojana (high reading) | 8,400 mi | 7,917 mi | +6.1% | | **7.5 mi/yojana (mid reading)** | **7,875 mi** | **7,917 mi** | **−0.5%** | | 5 mi/yojana (low reading) | 5,250 mi | 7,917 mi | −34% |

The mid-reading match is the one popularised in modern accounts of the *Aryabhatiya*: a 6th-century Indian astronomer giving Earth's diameter to within half a percent.

The low-reading puts Aryabhata in the same accuracy range as Eratosthenes (~240 BCE), who measured the Earth's circumference at ~250,000 stadia — also off by something like 15-30% depending on which stadion length you use. The high-reading puts him a little worse than the modern truth. Only the mid-reading gives the "spectacularly accurate" result.

So which yojana is right?

Yojana conventions through Sanskrit history

The *Arthashastra* (~300 BCE, attributed to Kautilya) defines a yojana as 4 *krośa* of 1,000 *dhanus* each, with a *dhanus* ("bow length") of about 6 feet. That gives ~4.5 miles per yojana.

The *Manusmriti* (~200 CE) uses a similar but not identical definition.

The *Surya-Siddhanta* (~5th c. CE, an astronomical text contemporary with Aryabhata) appears to use a yojana of roughly 5 miles, per Burgess's 1860 reconstruction working backwards from the implied Earth-Sun distances.

The *Aryabhatiya* itself defines a yojana as "8,000 times a *nṛ*" where *nṛ* is "the height of a man." If a man is 5 feet 6 inches, that's 8,000 × 5.5 ft = 44,000 ft = **8.33 miles per yojana**. If a man is 6 feet, it's about 9.1 miles. The Aryabhatiya's *own* definition points to the high end.

The mid-period 7.5 mile/yojana figure — the one that makes Aryabhata's Earth-diameter spectacular — is something of a compromise consensus, drawing on the average of various Sanskrit astronomical texts of the 5th-12th century.

The honest reading is: **the *Aryabhatiya* gives a self-consistent numerical model of the Earth-Sun-Moon system**, and *if* you map its internal "yojana" to ~7.5 modern miles, the absolute distances come out remarkably close to modern measurements. The mapping itself is what's underdetermined.

What is yojana-independent

The verse also gives the **diameters of the planets as fractions of the Moon's diameter**:

| Planet | Aryabhata's ratio (to Moon) | Modern ratio | |--------|------------------------------|--------------| | Venus | 1/5 = 0.20 | 3.49 | | Jupiter | 1/10 = 0.10 | 40.0 | | Mercury | 1/15 ≈ 0.067 | 1.40 | | Saturn | 1/20 = 0.05 | 33.4 | | Mars | 1/25 = 0.04 | 1.94 |

These ratios are *much* worse than modern values — Aryabhata substantially underestimates the gas giants. The reason is visible-disk geometry: the planets were known only by their unresolved-point appearance in the sky, and the inverse-square law between disk size and distance wasn't yet quantified. Aryabhata likely backed planet diameters out of *brightness* assumptions that don't hold.

So the Earth-Moon-Sun set is the strong part of the verse; the planet diameters are the weak part. Both are real, both are in the same verse, and both predate any equivalent in the contemporary European literature.

Earth and Moon — the most defensible reading

Independent of the yojana ambiguity, the **ratio** of Earth's to Moon's diameter from Aryabhata's numbers:

$$\frac{D_\text{Moon}}{D_\text{Earth}} = \frac{315}{1050} = 0.30$$

Modern value: 0.273. Aryabhata is about 10% high.

That ratio doesn't depend on what yojana means. Whatever yojana Aryabhata had in mind, *his Moon is 30% the size of his Earth*. The modern Moon-Earth ratio is 27%. A 1,000+-year-old pre-telescopic measurement, accurate to within 10% on a *ratio* — that's the genuinely robust claim from this verse.

Brahmagupta's correction

A footnote that matters: 129 years after the *Aryabhatiya*, in 628 CE, Brahmagupta in the *Brāhmasphuṭasiddhānta* explicitly "corrected" Aryabhata's Earth-diameter to 1,581 yojanas.

With the same yojana convention, Brahmagupta's number gives $1{,}581 \times 7.5 = 11{,}857$ miles — about 50% too large. By modern standards, Aryabhata's 7,875 was much closer to the true value than Brahmagupta's correction.

That isn't a unique pattern in the Aryabhata-vs-Brahmagupta debate. The same century-after attack hit Aryabhata on [the eclipse mechanism](/c/2953b2c9-59e9-5333-8207-8b227117c4e0) and [Earth's rotation](/c/3017aee5-d50c-53cd-b581-fd25905916e8) — and in both cases, Aryabhata was the more accurate of the two. The mainstream Indian astronomical tradition followed Brahmagupta; Aryabhata's better numbers were preserved as a minority reading until the Kerala school (~1500 CE) revived them.

What this leaves us with

One Sanskrit verse, two sentences, a numerical Earth-diameter that sits within 1% of the modern measurement *if* you accept the most-cited mid-period yojana conversion. Sun, Moon, and planet diameters in the same verse — the Moon match is solid, the planet matches are weak, the Sun match depends entirely on the yojana convention.

The honest summary: **Aryabhata had a numerical model. The model's Moon/Earth diameter ratio matches modern values to within 10% unconditionally. The model's absolute Earth diameter matches modern values to within 1% under the standard yojana reading.** Either way, the *Aryabhatiya* is the earliest known Sanskrit text giving specific Earth, Moon, and Sun dimensions numerically — pre- telescopic, by a 23-year-old, in metric verse.

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Sources

- [Aryabhatiya, W. E. Clark trans., 1930](https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930) — verse I.5 cited above. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 4. — the mid-period yojana convention used in the hook. - Burgess, E. (trans., 1860). *Translation of the Surya-Siddhanta*. American Oriental Society. — Burgess's reconstruction of the shorter Surya-Siddhanta yojana.

Related claims

- [Aryabhata on Earth's rotation](/c/3017aee5-d50c-53cd-b581-fd25905916e8) — the same chapter (Dasagitika), the same Earth, the rotation vs. shadow-illusion argument. - [Aryabhata's eclipse geometry](/c/2953b2c9-59e9-5333-8207-8b227117c4e0) — same Earth, same Brahmagupta-corrected-Aryabhata-incorrectly pattern.