Ancient and medieval writers observed a distinction between the Science
and the Art of Arithmetic. The classical treatises on the subject, those
of Euclid among the Greeks and Boethius among the Latins, are devoted to
the Science of Arithmetic, but it is obvious that coeval with practical
Astronomy the Art of Calculation must have existed and have made
considerable progress. If early treatises on this art existed at all
they must, almost of necessity, have been in Greek, which was the
language of science for the Romans as long as Latin civilisation
existed. But in their absence it is safe to say that no involved
operations were or could have been carried out by means of the
alphabetic notation of the Greeks and Romans. Specimen sums have indeed
been constructed by moderns which show its possibility, but it is absurd
to think that men of science, acquainted with Egyptian methods and in
possession of the abacus,[2*] were unable to devise methods for its use.
[Footnote 2*: For Egyptian use see Herodotus, ii. 36, Plato, _de
Legibus_, VII.]