The Babylonian World - Historia Antigua

— David Brown —
one zodiacal sign to the next, and various other phenomena, all for a given forthcoming
year. Although the attested examples date to the Hellenistic period, and the
computations they contain could have derived from the ephemerides (discussed below),
it is far likelier that they also were based on characteristic intervals and a record of
initial observations (Hunger and Pingree 1999: 167).
Concomitant with the non-mathematical astronomical texts of the last five 17 centuries
BC are the so-called astronomical cuneiform texts (ACTs). These represent the highpoint
of Babylonian astronomical endeavour. Some 300 texts are known, most of
which are termed (rather incongruously) ‘ephemerides’, some procedure texts which
outline how one might construct an ephemeris, and some unusual, but important,
often earlier, texts which describe other ways in which an entire set of astronomical
phenomena might be predicted using only one initial observation of position and
location and a series of parameters describing the mean motion of a body and the
variation about that mean. That variation was modelled using piece-wise linear
functions, which we thus term ‘arithmetic’, in contrast to the trigonometric functions
of Greek kinematic astronomy (described below). A good command of the basic
mathematical operations with large numbers is shown in these texts, though the
difficulties of non-terminating fractions often determine the particular parameters
chosen. There is little evidence of a consistent treatment of approximations, or of
systematic error. Calculations are peppered with redundant accuracy. The highly
accurate mean values were determined from the shorter characteristic intervals used
in the non-mathematical astronomical texts, and estimates as to the respective
observational errors in the latter. Mars’ accurate mean value for the interval after
which it returns to its initial longitude and exhibits the same phase is 284 years, for
example, derived from a combination of its shorter characteristic intervals of 79 and
47 years, and an estimate that the error in the former is a third of that of the latter,
and of opposite sign. Thus 3(79) + 47 = 284.
A prediction of where Mars next might rise and when, based on its characteristic
interval of 79 years and a record of that planet’s behaviour 79 years earlier, does not
require either that the luni-solar calendar be regulated, or that locations in the sky
be assigned relative longitudes. The ACTs and related texts did, however, for they
predicted, for example, the date and location of Mars’ rising on each occasion over
the following years, given one initial starting point. Successive risings do not occur
after whole numbers of years, nor at the same point in the ecliptic, the path on which
the planets travel. Using the parameter of 284 years and the fact that during that
interval Mars performs 133 phenomena of one type, it follows that the mean temporal
and spatial intervals between successive phenomena of that type can be calculated.
Variations about that mean can then be added according to some scheme depending
either on location in the ecliptic (described by a step function), or on which number
in the cycle of 133 has been reached (described by a zig-zag function). In order to
express the temporal variation in terms of dates, however, some fixed value giving the
number of months in a year was required, given that 12 lunations fall short of a year,
and periodically a thirteenth was needed. Furthermore, it needed to be determined
when best to intercalate these additional months. The ratio 235 months = 19 years
was known at least by 500 BC, and a particular scheme placed the intercalary months
either after the twelfth month or after the sixth in particular years of the 19-year
cycle. Later ephemerides used still more accurate relationships. 18 In order to be able
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