Period Relation

Definition

A period relation in Babylonian mathematical astronomy is a numerical equivalence between a whole number of intervals of one kind and a whole number of intervals of another — for example 235 lunar months = 19 solar years (the Metonic relation underlying luni-solar intercalation), or 391 synodic phenomena of Jupiter = 36 revolutions of the ecliptic. Period relations are the empirical-arithmetic foundation on which both the lunar and the planetary ephemerides of Systems A and B are built.

In Tradition

Neugebauer calls period relations "the very backbone of Babylonian mathematical astronomy." He treats long-interval period relations and lunar eclipses as the principal empirical input of ancient astronomy; everything else, in his account, is mathematical theory. Period-relation values are deliberately chosen as round numbers convenient for arithmetic, so they reflect a compromise between observation and computational convenience rather than direct observational records.

In Practice

A linear zigzag function or step function is parametrised so that its number period — the number of steps after which it repeats — reproduces an empirically known period relation. The fundamental Babylonian planetary relation states that, over a whole number of years, the number of sidereal rotations a planet performs plus the number of synodic returns it makes equals the number of years: for Mars, 42 sidereal rotations + 37 synodic returns = 79 years. The same identity underlies the Greek epicyclic model. The Goal-Year-Text prediction method exploits these relations to project from a known past synodic date to a target year. Neugebauer derives the relation 810 years = 10019 months from a worked solar-velocity column and the simpler 225 years = 2783 months from sister columns, both yielding a year-length close to 12;22,8 months.

Historical Origin

Attested across Late-Babylonian procedure texts and ephemerides from Babylon and Uruk (Seleucid period, c. 250-50 BCE); the Mercury relation 848 years = 2673 risings, the 19-year Metonic luni-solar relation, and planet-by-planet relations for all five visible planets are recovered in modern critical editions. Modern treatments: Neugebauer, *The Exact Sciences in Antiquity* (1957) Ch. V §§44, 48-49, 57, 70; Neugebauer, *Astronomy and History: Selected Essays* (1983) essays [5] p. 101 and [25] p. 312; Hunger-Pingree, *Astral Sciences in Mesopotamia* (1999) §C5.

Further Reading

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