Fiqh & astronomy · from the dissertation

Why tamkīn? — A thousand years of precision corrections

The standard formula of prayer-time calculation computes an idealised world: a point-like earth, a point-like sun, no atmosphere, constant parameters. What an observer actually sees in the sky differs measurably. tamkīn is the classical framework that translates geometric into apparent solar altitudes — taken for granted in Islamic astronomy since the beginning, yet ignored by many modern calendars. Chapter 4.2 of the dissertation systematises it; here are its building blocks and its history.

The five corrections

The tamkīn framework remedies the simplifications of the standard formula — precisely the five factors the IZA prayer-times working group identified as indispensable:

1. Solar parallax — the formula computes for an observer at the earth’s centre; at the surface the sun stands minimally lower (up to ~9 arcseconds — small, but properly accounted for).
2. Horizon dip and elevation — an elevated observer sees a dipped horizon; the apparent horizon is decisive. The magnitude is real: from the Palandöken summit above the Erzurum plateau, the dip is 1.16°.
3. Extent of the solar disc — the fiqh sunset is only when the disc’s upper limb has entirely disappeared, not its centre. Reliable calculation uses the largest possible solar radius (16′16″ at perihelion).
4. Atmospheric refraction — the air apparently lifts the sun; on the horizon 34′ on average, in reality up to about 1°. Extreme mirages (the Novaya Zemlya effect with a doubled solar disc) remain theologically irrelevant: what counts is the disappearance of the original disc.
5. Extent of cities and daily variation of parameters — times are issued for whole cities, not for a point: the latest city edge counts for a time’s beginning, the earliest for its end; declination and the equation of time also vary through the day (classically absorbed by a flat ±2 minutes).

al-Bīrūnī’s mountain: how the dip measured the earth

That the horizon dip was understood from the very beginning is proven spectacularly by al-Bīrūnī (d. 1061) — he inverted the problem and determined the earth’s radius from the dip. In his astrolabe treatise he describes the method:

For the determination of the earth’s circumference there is another method — conceived in thought, yet correct by proof. Carrying it out is difficult due to the smallness of the instruments. Climb a mountain, observe the sunset and measure the depression of the horizon. Then determine the height of the mountain, multiply it by the sine of the depression’s complement and divide by the versed sine of the depression …

al-Bīrūnī, astrolabe treatise — in modern notation: r = h·cos α / (1 − cos α).

In al-Qānūn al-Masʿūdī he reports the execution: a mountain in India above a plain “whose flatness was replacing the smoothness of the ocean’s surface” — measured dip 34′, mountain height 652.05 cubits (triangulated from two standpoints), the sine value in Babylonian sexagesimal to four places. Result: an earth radius of 6409.6 km — only 0.6% off the true local value. The dissertation stays honest: the hit was partly luck (his own sine value and the ignored refraction of the dipped line of sight would have produced ~13% deviation). What matters is the principle: the dip correction was standard craft in the 11th century — sunrise and sunset were classically calculated for the most elevated parts of a city.

Refraction with a system: Ibn Yūnus and Taqī al-Dīn

Refraction, too, was treated systematically early on. Ibn Yūnus (d. 1009) is credited with a seasonal scheme: 47′ at the equinoxes, rising to 62′ at the summer and falling to 32′ at the winter solstice. Taqī al-Dīn al-Rāṣid (d. 1585), founder of the Istanbul observatory, developed a scheme with different refraction for sunrise and sunset — just as modern measurements show — and explicitly applied the corrections to the durations of Fajr and evening redness as well. His verdict on the day-length correction of up to ~3°: “fa-lā jarama — truly no small amount!”

And on the solar semidiameter the tradition delivers a gem: Ibn al-Shāṭir (d. 1375) gives 16′16″ — practically identical with the modern maximum of 16′16.4″. Even the daily variation of the input quantities was classically absorbed: “add or subtract two minutes wherever necessary” — the historical root of the ±2-minute span.

From manuscript to app

For the new IZA calendar, the prayer-times working group did not only verify the necessity of tamkīn theoretically but confirmed it empirically in numerous sighting expeditions; Acaroğlu’s algorithm automates the corrections completely. This app implements them transparently as switches: elevation correction (dip), “full solar disc” (upper limb instead of centre), city-bounds span (latest/earliest city edge) and the adjustable tamkīn span (±2 minutes by default) — each one with a thousand-year pedigree.

This text is a factual introduction, not a fatwa. Your local scholars and community remain authoritative.

More from the knowledge section

The polar problem — where local sighting reaches its limit · Thuluth al-layl — the third-of-the-night method and its justification · Where do 18° and 17° come from? — Twelve centuries of the angle debate · Are the prayers even obligatory then? — The schools’ dispute · Which calendar gets it right? Germany’s prayer-time calendars compared