Step Function

STEP FUNK-shun

babylonian: step function (modern term; no single Akkadian designation)

Definition

A step function is one of the two arithmetical devices of Babylonian mathematical astronomy, complementary to the linear zigzag function. It holds a quantity constant within each of several zones of the ecliptic and changes abruptly from one constant value to the next at zone boundaries. The step-function form is the signature parameter scheme of System A: in the lunar theory it governs the solar velocity; in the planetary theory it governs the synodic arc.

In Tradition

Neugebauer, Hunger-Pingree, and Rochberg concur in pairing the step function with the linear zigzag function as the two arithmetical machines of the Babylonian ephemerides. Neugebauer notes that the System-A inventor deliberately chose the step form because it makes the corrections for variable solar velocity much simpler than the second-order difference sequences System B requires.

In Practice

Babylonian astronomers parametrise a step function by listing zone boundaries on the ecliptic, the constant value within each zone, and the ratio of values at the discontinuity. The canonical System A solar-velocity step-function in column B of the lunar ephemerides has w₁ = 30°/synodic month on one arc and w₂ = 28;7,30°/synodic month on the complementary arc, with jumping points at Virgo 13°/Pisces 27° in version B₁ or Pisces 13°/Virgo 27° in B₂. The Mercury ACT texts use three-zone step-functions for the Greek-letter phenomena; System A planetary treatments define two zones for Saturn and six for Mars. The output appears in procedure texts and ephemerides as a sequence of zone-keyed constants.

Historical Origin

Attested across the Seleucid-period ACT lunar and planetary procedure texts from Babylon and Uruk (c. 250-50 BCE). Modern critical treatments: Neugebauer, *The Exact Sciences in Antiquity* (1957) Ch. V §§49-50 and Ch. VI §70; Neugebauer, *Astronomy and History: Selected Essays* (1983) essay [6] p. 131; Hunger-Pingree, *Astral Sciences in Mesopotamia* (1999) §C4.2 pp. 240, 246, 253.

Further Reading

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