Literary Evidence for Roman Arithmetic with Fractions

Roman arithmetic is a perennially troubling subject for both classicists and mathematicians. Scholars universally comment on the difficulty posed by Roman alphabetical notation both in expressing simple figures and in doing written calculations. For example, Lloyd Motz and Jefferson Weaver with exasperation ask, "How ... can anyone do any arithmetic with DCCCLXXXVIII, the Roman equivalent of 888?," and Florian Cajori concludes that the Romans must have resorted to the abacus in order to multiply a number like 723 (DCCXXIII) by 364 (CCCLXIV).' Yet our sources, literary and inscriptional, indicate that the Romans were capable of highly sophisticated calculations, and, of course, it is well recognized that they had great facility with the abacus and with finger reckoning. The arithmetic problems that appear in Latin literature have been treated in depth by only one author, Gottfried Friedlein (1866 and 1869). This is surprising because Latin literature provides a rich supply of material that deals with arithmetical problems and calculations. This paper will examine these materials in an attempt to determine, first, what the Romans actually did with their number system, and second, how the problems of arithmetic in classical literature were solved.