23 hours, 56 minutes, 4.1 seconds: Aryabhata times the Earth's spin (499 CE)

# 23 hours, 56 minutes, 4.1 seconds: Aryabhata times the Earth's spin (499 CE)

A day is not 24 hours. The 24-hour day is the Sun's day — spin plus a bit of orbit. Relative to the fixed stars, the Earth turns once in 23 hours 56 minutes 4.091 seconds: the sidereal day, the true period of the planet's rotation.

The *Āryabhaṭīya* opens with a data verse — a table of the cosmos compressed into meter — and its first entries read:

> "In a yuga the revolutions of the Sun are 4,320,000, of the Moon > 57,753,336, of the Earth eastward 1,582,237,500…" > > — *Āryabhaṭīya* I (Gītikā) 1, trans. Clark (1930)

Do the division the verse invites. In one yuga the Earth turns 1,582,237,500 times while the Sun makes 4,320,000 circuits of the sky; each solar circuit costs one turn, so the yuga contains 1,582,237,500 − 4,320,000 = 1,577,917,500 solar days. The sidereal day is therefore 1,577,917,500 / 1,582,237,500 of 24 hours:

86,400 s × (1 − 4,320,000/1,582,237,500) = **86,164.10 seconds** = 23h 56m 4.10s.

The modern value, measured by radio interferometry against quasars: 86,164.09 seconds. Aryabhata's figure is off by about **one hundredth of a second** — a tenth of a part per million — written down in 499 CE by a twenty-three-year-old, in an era whose best clocks were dripping water.

Where a hundredth of a second comes from

Honesty first: no one in the fifth century could *time* a rotation to 0.01 s, and the article should say how the trick actually works. The precision lives in the ratio, not the stopwatch. Any single observation — a star crossing the meridian, an eclipse's midpoint — is good to minutes at best. But rotations *accumulate*: if you know from records that the sky has made N revolutions between two dated observations centuries apart, your error in the period is your observational error *divided by N*. A few minutes' uncertainty spread across a few hundred thousand rotations is microseconds per rotation. Long archives, not fine instruments, are the engine — the same archives that let [Varāhamihira catch the solstice drifting](/c/206df158-ef72-5be2-b140-02527823a098) a generation later. Aryabhata's integer — rotations per yuga — is the natural container for exactly this kind of accumulated knowledge, and [the same verse-format carries the Moon's period at similar precision](/c/51a4fbd2-3f5b-5928-af58-c37fc27253d3) in the sister tradition.

By that standard the achievement is shared: Babylonian lunar theory and Hipparchus's year-length work encode the same order of precision, wrung from the same trick of long baselines. Day-length precision was the common summit of ancient astronomy. What is *not* shared is the bookkeeping — and that is the second half of this claim.

"Of the Earth eastward"

Read the verse again. Aryabhata does not say the stars revolve westward 1,582,237,500 times. He says the *Earth* turns *eastward* that many times. The rotating Earth — [argued by analogy in his Gola chapter with the famous boat verse](/c/3017aee5-d50c-53cd-b581-fd25905916e8) — appears here not as a philosophical position but as a *parameter*: a column in the data table, with a direction attached. Among all ancient astronomical canons — Babylonian, Greek, Chinese, Indian — this verse is the only opening parameter-list that books the daily rotation to the planet's account.

His tradition couldn't quite stomach it. Brahmagupta attacked the rotation doctrine; later copyists and commentators quietly recast this very parameter as "revolutions of the asterisms westward," neutralizing the heresy while keeping the number (the arithmetic is indifferent to who moves). Clark's text preserves the original reading, which modern scholarship (Plofker 2009) accepts as Aryabhata's own, consistent with the Gola chapter. The number survived unanimously; the *eastward* had to be smuggled through the centuries.

One more honest note: the verse's number is exactly right about the *ratio* of its era. The Earth's rotation slows by roughly 2 milliseconds per day per century (tidal braking), so no fixed value is timeless; Aryabhata's figure and today's differ by an amount within the range that slow drift and his method's precision jointly allow. The 0.01 s agreement is genuine, not curve-fitted — but it is agreement between two snapshots of a slowing clock.

Legacy

The sidereal day is modern astronomy's working clock — every telescope mount that tracks the stars turns at the sidereal rate, and the figure it uses is the one Aryabhata's division yields, to the first decimal of a second. His parameter set became the computational backbone of the āryapakṣa school, recomputed and commented for a thousand years; the encoding technique that packs ten-digit integers into two syllabic feet is [its own claim in this corpus](/c/b24b8d92-0600-5542-a179-8014ef059d75).

And there is the quiet historical irony: the tradition that rejected his spinning Earth kept his spin *rate*. Fourteen centuries later, Foucault's pendulum settled the doctrine, and the number everyone then needed — how fast does it spin? — had been sitting in the opening verse of a Sanskrit textbook all along, correct to a hundredth of a second, with the direction marked: eastward.

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Sources

- [*The Aryabhatiya of Aryabhata*, W. E. Clark trans., 1930](https://archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930) — Gītikā 1 cited (p. 9); Clark's notes on the variant readings. - Plofker, *Mathematics in India*, 2009 — the rotation reading and the āryapakṣa parameter tradition (secondary synthesis). - IERS Conventions — modern mean sidereal day 86,164.0905 s (referenced for the comparison only).

Related claims

- [Aryabhata: the Earth rotates — the boat argument (499 CE)](/c/3017aee5-d50c-53cd-b581-fd25905916e8) - [The Sūrya-Siddhānta's lunar orbit, to one second](/c/51a4fbd2-3f5b-5928-af58-c37fc27253d3) - [Aryabhata's alphabetic number code (499 CE)](/c/b24b8d92-0600-5542-a179-8014ef059d75)