Astronomy
One second per month: the Sūrya-Siddhānta's lunar orbit
Published July 5, 2026
# One second per month: the Sūrya-Siddhānta's lunar orbit
The Moon takes 27.321662 days to circle the Earth once against the stars — the sidereal month, one of the most precisely known numbers in astronomy, today measured by laser ranging against retroreflectors left on the lunar surface.
The Sūrya-Siddhānta, the standard astronomical treatise of classical India, encodes the same quantity as a pair of whole numbers. In an Age (mahāyuga) of 4,320,000 years, it states:
> "Of the moon, fifty-seven million, seven hundred and fifty-three > thousand, three hundred and thirty-six" > > — *Sūrya-Siddhānta* i.30, trans. Burgess (1860)
— that many sidereal revolutions, in an Age that the same chapter (i.37) fixes at 1,577,917,828 civil days. One division:
1,577,917,828 ÷ 57,753,336 = **27.321674 days.**
Modern value: 27.321662. The difference is 0.000012 of a day — about **1.1 seconds** per month, half a part per million. You can check every step of that with a pocket calculator, which is rather the point: the precision isn't an interpretation, it's an integer ratio sitting in a Sanskrit verse.
Why whole numbers?
The Siddhānta's method is worth pausing on, because it solves a problem every ancient astronomy faced: how do you write down a period that isn't a whole number of days, in a culture without decimal fractions? The Babylonians used sexagesimal places. The Greeks inherited those. The Indian siddhāntas chose a third way: make the *cosmic cycle* huge enough that every period fits into it a whole number of times. Within the Age, the Moon makes exactly 57,753,336 circuits, the Sun exactly 4,320,000, and every planetary period becomes a ratio of integers — exact by construction, arbitrarily precise, and immune to copying errors in a way decimal strings are not (corrupt one digit of a written fraction and you get a plausible wrong number; corrupt one digit of 57,753,336 and the astronomy visibly breaks).
It is the same instinct that makes a modern programmer store money in integer cents. The mahāyuga is, among other things, a common denominator — chosen so large that the sky divides into it evenly.
The honest comparison
Precision lunar theory is the *shared* summit of ancient astronomy, and the claim is explicit about it. Babylonian astronomers of the Seleucid era (System B) fixed the mean synodic month at 29;31,50,8,20 days sexagesimal — accurate to well under a second — by the 2nd century BCE at latest, and Hipparchus adopted their value outright. Whether the Sūrya-Siddhānta's lunar parameter descends from that lineage (Greek astronomy demonstrably reached India in the early centuries CE), was tuned against Indian eclipse records, or both, is not settled; the extant Siddhānta's parameter set is conventionally dated c. 400–500 CE with later redaction, the same caveat the corpus's other Sūrya-Siddhānta claims carry.
What is distinctively Indian is the encoding and the longevity. The integer-ratio idiom is the siddhānta tradition's own, and this particular ratio stayed in continuous computational use — for calendars, eclipse prediction, and horoscopy — for well over a thousand years. Pañcāṅga calendar-makers were still computing with Sūrya-Siddhānta parameters when the telescope arrived in India.
One more honest note: not every Siddhānta parameter is this good. Its planetary apogees drift, its obliquity is serviceable, its [earth-circumference rule is rough](/c/678c17d7-b096-5f80-9cbb-349c451f7534)-and-ready. The lunar and solar mean motions are the crown jewels — which makes sense, because the Moon is what eclipse prediction lives or dies on, and eclipse prediction was the tradition's public test. The numbers are best exactly where the accountability was.
Legacy
The Moon's period reached this precision three times in antiquity — cuneiform tablets, Greek geometry, Sanskrit verse — by three different notational technologies, with transmission threads between them that scholars still untangle. The Siddhānta's version is the one that stayed longest in daily service: every traditional Indian calendar printed today still runs on sidereal lunar arithmetic descended from these verses, precession-corrected but structurally intact.
And the verse pairs naturally with its physical picture: the same tradition that fixed the Moon's period to a second also taught that the Moon is [a dark sphere lit by the Sun](/c/1648b8d5-ada6-5bf8-9428-ee8ad4424f68), circling [a globe hanging in the ether](/c/678c17d7-b096-5f80-9cbb-349c451f7534). Right numbers, right geometry, fifteen centuries ago — and the number checks out to one second on a calculator you own.
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Sources
- [Burgess, *Translation of the Sûrya-Siddhânta*, 1860](https://archive.org/details/SuryaSiddhantaTranslation) — i.29-30 (lunar revolutions), i.34-37 (civil days) cited; Burgess's commentary tables for the derived month. - Plofker, *Mathematics in India*, 2009 — siddhānta parameter systems (secondary synthesis, for context). - Neugebauer, *A History of Ancient Mathematical Astronomy*, 1975 — Babylonian System B lunar theory (referenced for the comparison only).
Related claims
- [The Sūrya-Siddhānta's sidereal year (~5th c. CE)](/c/87117c14-fd33-516c-9691-ffc92c334315) - [Aryabhata: moonlight is borrowed light (499 CE)](/c/1648b8d5-ada6-5bf8-9428-ee8ad4424f68) - [The Sūrya-Siddhānta: a globe with no "up" (~5th c. CE)](/c/678c17d7-b096-5f80-9cbb-349c451f7534)
References
- [1]Surya-Siddhanta I.29-34 (~5th c. CE) gives the revolutions of each celestial body in one mahayuga (4,320,000 years). The sidereal year derives by arithmetic: (asterism revolutions − sun revolutions) / sun revolutions = (1,582,237,828 − 4,320,000) / 4,320,000 ≈ 365.2587 civil days per year. Modern sidereal year: 365.25636 days. The text is 3.5 minutes / ~7 parts per million long, pre-telescopic. Ptolemy's Almagest (~150 CE) gives 365.2467 — off by ~14 minutes. Source: Translation of the Surya-Siddhanta (T1)
- [2]Āryabhaṭīya IV (Gola) 5, 499 CE: half of the Earth, the planets, and the asterisms is dark — shadowed by the body itself — and the half turned toward the Sun is light. Applied to the Moon, this is the reflected-sunlight account of moonlight and of lunar phases. The insight has earlier independent precedents (Anaxagoras, c. 450 BCE, in Greece); Aryabhata's formulation embeds it in a quantitative astronomy curriculum used continuously in India for over a millennium. Source: The Aryabhatiya of Aryabhata (T1)
- [3]Sūrya-Siddhānta XII.53 (c. 400–500 CE core text, Burgess 1860 translation) states that the Earth is a globe in space with no absolute up or down: every observer takes their own place to be uppermost. Verses 51–52 apply it concretely — dwellers at opposite points of the globe each suppose the other underneath. Greek astronomy established terrestrial sphericity earlier (Aristotle, c. 350 BCE); the Siddhānta's plain statement of the relativity of "up" is among the clearest in any ancient text. Source: Translation of the Surya-Siddhanta (T1)
- [4]Sūrya-Siddhānta i.30 fixes the Moon's sidereal revolutions per Age at 57,753,336; i.37 fixes the Age's civil days at 1,577,917,828. The implied sidereal month, 27.321674 days, differs from the modern 27.321662 by about 1.1 seconds — roughly 0.5 parts per million. Babylonian-Greek lunar theory reached comparable precision by other routes; the Siddhānta's whole-number encoding is the Indian tradition's own and stayed in computational use for over a millennium. Source: Translation of the Surya-Siddhanta (T1)