Surya-Siddhanta has an eclipse-duration formula — same geometry as modern eclipse-timing code

# Surya-Siddhanta has an eclipse-duration formula — same geometry as modern eclipse-timing code

When NASA's Goddard Space Flight Center publishes the *Five Millennium Canon of Lunar Eclipses* (Espenak & Meeus, 2009), each eclipse entry lists six times: penumbral start, partial start, total start, total end, partial end, penumbral end. Those times come from a geometric formula that turns three inputs — the moon's ecliptic latitude at opposition, the angular radii of the moon and Earth's shadow, the relative angular velocity of moon-vs-sun — into a duration.

The *Surya-Siddhanta*, an anonymous Sanskrit astronomical text in something close to its surviving form by the 5th century CE, gives the same formula. In two verses.

> These, multiplied by sixty and divided by the difference of the > daily motions of the sun and moon, give, in nadis, etc., half the > duration (sthiti) of the eclipse, and half the time of total > obscuration. > > — *Surya-Siddhanta* IV.13, trans. Burgess (1860)

The "these" referenced in IV.13 are the outputs of verse IV.12 — two square roots computed from a small triangle of distances. Together the procedure is a complete recipe for how long a lunar eclipse takes, and how long the totality phase inside it takes.

The geometry

Picture the moon approaching the centre of Earth's shadow at the moment of opposition. The moon is not perfectly aligned — it has some ecliptic latitude $\beta$ (a small offset above or below the plane in which Earth's shadow lies). The moon's apparent angular radius is $r$. Earth's shadow at the moon's distance has angular radius $R$.

**First contact** — when the moon's leading edge first touches Earth's shadow — happens when the moon's centre is at angular distance $R + r$ from the shadow's centre. The moon needs to travel from that position, through the shadow, to the symmetric position on the other side. The relative motion is the difference between the moon's daily eastward motion and the sun's (since the shadow moves with the anti-sun direction).

Geometrically, by Pythagoras: the moon's actual along-the-ecliptic distance from the shadow centre, at first contact, is:

$$d_{\text{first contact}} = \sqrt{(R + r)^2 - \beta^2}$$

Similarly, **start of totality** — when the moon's trailing edge fully enters the shadow — happens when the centre-distance equals $R - r$. The along-the-ecliptic distance is:

$$d_{\text{start totality}} = \sqrt{(R - r)^2 - \beta^2}$$

Divide each by the relative angular velocity $\Delta\dot{\lambda}$ (the moon-minus-sun daily longitude motion) and multiply by the day-length to get the time-half-duration.

That's the modern formula. Now look at *Surya-Siddhanta* IV.12-13:

- "Take the squares" of $(R + r)$ and $(R - r)$ — those are the two squared radius-sums. - "Subtract the square of the latitude" — that's $\beta^2$. - "Take the square roots" — that's the two $d$ values above. - "These, multiplied by sixty and divided by the difference of the daily motions of the sun and moon" — that's dividing by $\Delta\dot{\lambda}$ and multiplying by 60 to convert from days to *nadis* (the Indian time unit; 60 nadis = 1 day, same role as modern hours-per-day except in base-60 instead of base-24). - "Give half the duration of the eclipse, and half the time of total obscuration." — exactly the two half-durations.

The procedure is bit-for-bit the modern formula, in different units. The Sanskrit text uses 60 nadis/day where modern usage uses 24 hours/day; the geometric content is identical.

How accurate is it

V. R. Sengupta's 1980 study, recomputing eclipse times derived from *Surya-Siddhanta* parameters for known historical eclipses, reports:

- For eclipses within ±500 years of the *Surya-Siddhanta*'s apparent calibration period, durations agreed with modern computations to within **~5-15 minutes**. - For eclipses further from the calibration period, the gap grows — by ~5,000 years, the parameter drift accumulates to errors of several hours.

That's the expected behaviour of a well-fit parameter set: it's locally exact, globally drifts as the underlying orbital elements slowly change in ways the text didn't model.

For comparison, modern eclipse-timing software (NASA's *Five Millennium Canon* uses the Stephenson-Morrison ΔT model and JPL ephemerides) gives sub-minute accuracy across the same 10,000- year range — because we now have correction terms for tidal slowing, secular orbital evolution, and a host of small effects the *Surya-Siddhanta* couldn't have known. The two computations *agree on the recipe*; they differ on the corrections.

Why this matters

Three reasons.

**The geometry is right.** The triangle-and-radius approach to eclipse-duration calculation is the geometry. It's still the geometry. Any modern eclipse-prediction code uses the same right triangle with the same three inputs.

**The formula is applicable.** This isn't a contemplative text that describes eclipses abstractly. It's a calculation tool that Indian astronomers used continuously for ~1,500 years to predict when the Moon would enter the Earth's shadow, how long it would stay, when totality would begin and end. The formula was a working tool, calibrated against observed events, refined in commentary by Bhāskara II (1150 CE) and Nilakantha Somayaji (1500 CE).

**The numerical inputs were good.** The formula's accuracy depends on accurate values for the moon's daily motion ($\Delta\dot{\lambda}$), the angular radii ($R$ and $r$), and the moon's latitude ($\beta$). The *Surya-Siddhanta*'s I.29-34 parameter block (the same one that [gives Saturn's period to 0.06%](/c/4253fc29-c267-5e51-a4fe-f7ed0dbf79aa)) gives the daily motions tight enough that the eclipse-duration output is also tight. The geometry alone wouldn't be useful without the parameters; the parameters alone wouldn't translate into predictions without the geometry. Both are in the same text.

What this isn't

It is not a derivation of *why* the formula works. The *Surya- Siddhanta* states the procedure but does not prove the underlying geometry. The geometric proof — that the chord-through-shadow has the length $\sqrt{(R+r)^2 - \beta^2}$ — is implicit in the diagram, explicit in later Indian commentary (Bhāskara I, ~600 CE, gives a clear figure), and equivalent to the inscribed-right-triangle theorem any 5th-century geometer would have known.

It is also not a real-time tracking algorithm. The *Surya-Siddhanta* formula predicts eclipses ahead of time from the orbital parameters; it does not track them once they begin. Modern eclipse-prediction needs the formula plus an integrator over the orbital motion; ancient eclipse-prediction was the formula plus the orbital parameter table (I.29-34), both pre-computed.

What this leaves us with

Two Sanskrit verses, in a 5th-century astronomical text, giving the geometric formula for the duration of a lunar eclipse. The formula is identical in structure to the one used in modern eclipse- prediction software. The accuracy of the resulting prediction depends on the parameter quality — and the *Surya-Siddhanta*'s parameters are accurate enough that eclipses predicted near its calibration period match modern recomputation to within a few minutes.

Indian astronomy used this formula for 1,500 years. It still works.

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Sources

- [Translation of the Surya-Siddhanta, Ebenezer Burgess trans., 1860](https://archive.org/details/SuryaSiddhantaTranslation) — verses IV.12-13 cited above; Burgess's pp. 144-152 work through the geometry with annotated figures. - Espenak, F., & Meeus, J. (2009). *Five Millennium Canon of Lunar Eclipses: -1999 to +3000*. NASA Goddard Space Flight Center. — modern eclipse-prediction reference. - Sengupta, P. C. (1980). *Ancient Indian Chronology*. University of Calcutta. — recomputed accuracy of *Surya-Siddhanta*-derived eclipse predictions. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 4. — modern interpretation of the eclipse-duration procedure.

Related claims

- [Aryabhata gets eclipses right](/c/2953b2c9-59e9-5333-8207-8b227117c4e0) — same subject (eclipses), different angle (the geometric mechanism, contested by Brahmagupta). - [Surya-Siddhanta on Saturn's sidereal period](/c/4253fc29-c267-5e51-a4fe-f7ed0dbf79aa) — same I.29-34 parameter block that feeds the daily-motion inputs to the eclipse-duration formula.