Linear Methods

LIN-ee-ar METH-ohdz

babylonian: linear methods (modern scholarly term; Neugebauer's coinage)

Definition

Linear methods (Neugebauer's coinage; synonymous with "arithmetical methods" in the same author's usage) is the umbrella scholarly designation for the computational procedures of Babylonian mathematical astronomy: representing periodically varying quantities — solar velocity, lunar latitude, synodic arc, day-length — by linear zigzag functions and step functions built on first-order difference sequences, rather than by geometric or trigonometric models. The classification contrasts the Babylonian arithmetical-computational tradition with the geometrical tradition that culminates in Ptolemy's *Almagest*.

In Tradition

Neugebauer introduces the term as a methodological-class label in *The Exact Sciences in Antiquity* Ch. VI §66 — "I call this second class of procedure the arithmetical methods or the linear methods" — and treats "linear" as marking the contrast with geometrical methods, not merely that the sequences are first-order arithmetic progressions. Hunger-Pingree and Rochberg accept the classification.

In Practice

A reader who classifies an ancient astronomical procedure as "linear" is asserting two related claims: that the procedure operates on difference sequences (constant in System A, linearly varying in System B) rather than on a circular-motion model, and that it descends from the Babylonian arithmetical tradition rather than the Greek geometrical one. Neugebauer's diagnostic use of the label is decisive: when he finds linear-methods constants surviving in Greek astrological papyri, in Varahamihira's *Pañcasiddhāntikā*, in Indian and Islamic astronomy, and in the bulk of the astrological literature, he reads the survival as evidence of direct Babylonian transmission. A late-antique Greek horoscope of A.D. 137 describing computation "by greatest and smallest velocity" matches a linear zigzag function of solar velocity — Neugebauer suggests it may preserve the ancient name. The linear methods coexisted with spherical trigonometry throughout antiquity and the Middle Ages, used simultaneously and independently of the geometrical methods.

Historical Origin

The label is Neugebauer's, introduced in *The Exact Sciences in Antiquity* (1957/1969), Ch. VI §66 p. 158, and *Astronomy and History: Selected Essays* (1983), essay [2] p. 5 and essay [5] §4 pp. 111-112; the underlying computational practice is attested across the Late Babylonian ACT corpus (Babylon and Uruk, c. 250-50 BCE). Modern critical treatments also: Hunger-Pingree, *Astral Sciences in Mesopotamia* (1999), Ch. II §C.

Definition

In Tradition

In Practice

Historical Origin

Further Reading