Date: Dec 16, 1996 8:39 AM Author: Durham, John Subject: Re: Roman arithmetic You are almost correct in speculating that the Romans calculated with an

abacus. This was probably true for most people in ordinary life. Keep in

mind that the abacus was more of a concept than an artifact -- anyone with

something that kept lines fairly straight and a few pebbles ("calcluli")

could (and probably did) make an abacus on demand. This is why we don't

seem to have many abaci left. Probably there are little pebbles all over

the Mediterranean (and maybe elsewhere) which were once used as stones in

an abacus, then discarded. There is evidence of certain kinds of tokens

with hand-signals for numbers on them; these might have been gaming pieces

or might have been specialized for fancy abacus-use, such as might have

been the case of a rich merchant.

Probably there were made ruled boards or cloths on which the stones could

be moved conveniently and quickly; the little hand-held calculators which

are preserved are probably miniature versions. The boards or cloths, of

course, were very common (and continued to be during Medieval times), and

(just as modern computers are used both for calculation and gaming) became

handy places to play games. Several of our most common games, including

chess, checkers, and backgammon, are all played on versions of these boards

(the common practice of printing modern gaming boards with one side for

chess/checkers and the other for backgammon is merely a reflection of the

fact that both of these boards were places for calculation).

As to the details of how calculation was performed, the numerals I, X, C,

and M were usually represented by pebbles on the lines, while V, L, and D

were represented by pebbles in the spaces between the lines (the lines were

probably called "digitus" -- modern "digit" -- and the spaces were termed

"articulus" -- which originally meant the "joint" between the fingers; the

terminology survived in Medieval works on arithmetic). The original

versions of Roman numerals exactly reflect the usage of the abacus.

Probably even the apparently late and fancy version which uses IV for 4, IX

for 9, etc., reflects a specialized placement of stones on or between the

lines.

Used in this way, the abacus and Roman numerals were very efficient for

ordinary commercial work, and could easily be adapted to handle fractions

(the surviving examples show how -- it's really very simple, just adding

lines on one side and setting a rule for how many stones could be on each

line). In fact, this kind of abacus remains the only calculator ever

devised (to my knowledge) which conveniently handles arithmetic in which

each symbol in a number can be in a different base.

The classical Roman word for what we call today by the term "abacus" was

"tabula", and the counters which moved on it were called "calculi".

Obviously the words calculus and table (in modern usage) somewhat reflect

the ancient association of these terms with mathematics.

For fancy calculations, such as any kind of multiplication or division

which could not be accomplished quickly by repeated addition (or

subtraction, as the case may be) ancient people probably went to a

specialist who used tables. We still have many of these tables,

particularly the ones preserved in Ptolemy's works and in the many Medieval

copies of the "Calculus" of Victorius of Aquitaine.

The term "abacus" in ancient times referred to a flat slab (presumably of

marble) which was used as a kind of pad for the many columns which were

such a characteristic part of Greco-Roman civic architecture. The word is

not to be connected with Semitic words for dust, etc., and did not come to

be used to mean what we today call an "abacus" until late Medieval times.

The origins of the abacus are lost in antiquity; it could easily predate

civilization itself. Denise Schmandt-Bessarat (sp?; I don't have my

references at hand) wrote some twenty or thirty years ago about the various

specialized counters or tokens used in ancient Near Eastern proto-literate

societies. These might well have been used in early versions of the

abacus. Particularly interesting about them is that numbers seem to have

been associated with what they counted or measured. Thus in the earliest

versions of the abacus one could add (using the right tokens) two goats to

three goats to get five goats, but not two goats to five bushels of wheat.

The pure abstraction of number seems to have been a later development,

perhaps around the end of the third millennium BC