permit really intricate
computations to be made. By the aid of this system they sketched out all
the great theories of astronomy at a very early age. In the course of a few
centuries, they carried that science to a point never reached by the
Egyptians.[93]

The chief difficulty in the way of a complete explanation of the Chaldaean
system of arithmetic lies in the interpretation of the symbols which served
it for ciphers, which is all the greater as it would seem that they had
several different ways of writing a single number. In some cases the
notation varied according to the purpose of the calculation. A
mathematician used one system for his own studies, and another for
documents which had to be read by the public. The doubts attending the
question are gradually being resolved, however, by the combined efforts of
Assyriologists and mathematicians. At the beginning of their civilization
the Chaldaeans did as other peoples have done when they have become
dissatisfied with that mere rough opposition of unity to plurality which is
enough for savage races, and have attempted to establish the series of
numbers and to define their properties. "They also began by counting on
their fingers, by _fives_ and _tens_, or in other words by units of _five_;
later on they adopted a notation by _sixes_ and _twelves_ as an improvement
upon the primitive system, in which the chief element, the _ten_, could be
divided neither into three nor four equal parts."[94] Two regular series
were thus formed, one in units of six, the other in units of five. Their
commonest terms were, of course, those that occur in both series. We know
from the Greek writers that the Chaldaeans counted time by _sosses_ of
sixty, by _ners_ of 600, and by _sars_ of 3,600, years, and these terms
were not reserved for time, they were employed for all kinds of quantities.
The _sosse_ could be looked at either as _five twelves_ or _six tens_. So,
too, with the _ner_ (600) which represents _six hundreds_, or a _sosse_ of
_tens_, or _ten sosses_ or _fifty twelves_. The _sar_ may be analysed in a
similar fashion.

A system of numeration was thus established which may be looked at from a
double point of view; in the first place from its _sexagesimal_ base, which
certainly adapts itself to various requirements with greater ease than any
other;[95] in the second from the extreme facility with which not only
addition, but all kinds of complex calculations may be made by its use.