Length of Daylight (Babylonian arithmetical scheme)

Definition

The length of daylight is the time from sunrise to sunset, varying through the year with the season. Babylonian mathematical astronomy treated this quantity by purely arithmetical schemes — never by spherical trigonometry — and developed two complementary models: System A is a strictly linear scheme in which the daylight length changes by a constant difference per zodiacal sign; System B is a refined zigzag in which the difference doubles in the middle of the increasing and decreasing branches. Both are built on arithmetic progressions of the rising times of the zodiacal signs and require no concept of geographical latitude.

In Tradition

Neugebauer treats the length-of-daylight scheme as one of the key applications of the Babylonian linear (arithmetical) methods — the calendrically motivated scheme that underwrote new-crescent-visibility prediction: month-start required knowing sunset, and sunset required the daylight-length-from-sunrise scheme. He emphasises that the next month's daylight length is obtained by adding and subtracting appropriate zodiacal-sign rising times. Hunger-Pingree concur, treating it as a System-A / System-B paired ACT construct.

In Practice

For the reader of a Babylonian lunar procedure text or ephemeris, the length-of-daylight scheme appears as a column whose values vary by season: in System A, by a constant difference per zodiacal sign of the Sun; in System B, by a zigzag with two slopes (the difference doubles in the middle of each branch). The scheme is anchored to the rising-times-of-zodiacal-signs sequence — once the rising times are known, daylight length over any consecutive arc of the ecliptic can be summed. The Seasonal Hours Report K 2077+ supplies a worked zigzag with M = 20 UŠ, m = 10 UŠ, μ = 15 UŠ, d = 0;50 UŠ, P = 24; EAE Tablet 14 Table A supplies the equinoctial nighttime scheme with m = 1,0 UŠ and M = 3,0 UŠ (d = 12). The Babylonian scribes applied it only at their own latitude. Second-century-BCE Alexandrian Greeks adapted it by a linear variation of the extremal daylight length — the move that generated the seven "climates" of equal longest day dominating ancient and medieval geographical lore.

Historical Origin

Attested across the Late-Babylonian ACT lunar Procedure Text + Ephemeris corpus from Babylon and Uruk (c. 250-50 BCE), with antecedents in EAE Tablet 14 Table A (~late second millennium BCE assembly), MUL.APIN, the seventh-century BCE Seasonal Hours Report K 2077+, and the Old Babylonian BM 17175+17284 daylight-nighttime tablet. Modern critical treatments: Otto Neugebauer, *Astronomy and History: Selected Essays* (Springer 1983), essay [5] §5 pp. 113-114 and essay [7] p. 162; Hunger-Pingree, *Astral Sciences in Mesopotamia* (Brill 1999), Ch. II §C4.

Further Reading

• Otto Neugebauer, Astronomy and History: Selected Essays
• Hermann Hunger & David Pingree, Astral Sciences in Mesopotamia