Linear Zigzag Function

Definition

A linear zigzag function is the main arithmetic scheme of Babylonian mathematical astronomy. It tracks a quantity — the Moon's latitude, day-length, the sunset-to-moonset interval, a planet's speed — that climbs by a fixed step d at each interval, from a minimum m up to a maximum M, then falls by the same step d back to m. The result is a symmetric triangular wave: its period is P = 2(M − m)/d, and its mean is μ = (M + m)/2. The zigzag is the signature scheme of System B, in contrast to the step-function of System A.

In Tradition

Hunger-Pingree, Neugebauer, and Rochberg all treat the linear zigzag as the core arithmetic building block that Babylonian schematic astronomy uses for anything cyclic. The earliest known instance is EAE Tablet 14 Table A — the equinoctial-night scheme, with m = 1,0; M = 3,0 UŠ; d = 12 — and the precursor tablet BM 17175+17284 confirms an Old Babylonian origin. The function predates Greek geometric modelling of celestial motion and underlies the Seleucid-period ACT ephemerides.

In Practice

A Babylonian astronomer sets up a linear zigzag function with five quantities: m, M, μ, d, and P. It is written as a single column in a procedure-text or ephemeris, and its successive values are worked out by adding d over and over up to M, then subtracting d down to m, then adding again. In the Seasonal Hours Report K 2077+, a daylight-length zigzag has M = 20 UŠ, m = 10 UŠ, μ = 15 UŠ, d = 0;50 UŠ, and P = 24. In Tablet 14 Table A, the equinoctial-night zigzag has m = 1,0 = 2 bēru and M = 3,0 = 6 bēru, with d = 12. The column-Φ lunar-latitude zigzag of System B has Δ = 0;19,16,51,6,40 and d = 0;2,45,55,33,20. Modern historians of mathematics treat the linear zigzag as a forerunner of the trigonometric and Fourier modelling of repeating phenomena — reached here by purely arithmetic means.

Historical Origin

The linear zigzag is attested in cuneiform from the Old Babylonian BM 17175+17284 daylight-nighttime tablet (early second millennium BCE) through EAE Tablet 14 Table A (assembled in the late second millennium BCE), MUL.APIN, the seventh-century BCE Seasonal Hours Report, and the Seleucid System B ACT ephemerides. Modern critical treatments: Neugebauer, *Astronomical Cuneiform Texts* (1955) and *A History of Ancient Mathematical Astronomy* (1975); Hunger-Pingree, *Astral Sciences in Mesopotamia* (1999); and various JNES studies by Britton in the 1990s and 2000s.

Further Reading

• Hermann Hunger & David Pingree, Astral Sciences in Mesopotamia
• Otto Neugebauer, Astronomical Cuneiform Texts
• Otto Neugebauer, A History of Ancient Mathematical Astronomy