I though the following posting I once sent to Sherman Stein [with a courtesy
copy to the MAA mailing list] might be helpful:
====
Date: Mon, 16 Dec 1996 20:27:55 -0500
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
To: Sherman Stein <stein@math.ucdavis.edu>
Subject: Re: Roman arithmetic
Cc: math-history-list@maa.org
On 14/12/1996 -0500, Sherman Stein wrote:
> I have recently been asked how the Romans computed with Roman numerals.
> I don't know the answer. Is there a good reference? My guess is that
> they didn't.
"It is widely believed that calculation with Roman numerals is a primitive
process, so clumsy that operations beyond simple addition and subtraction
become almost impossible. [...] On the contrary, the Roman system
incorporates features permitting algorithms that make multiplication and
division even easier than in our place value decimal system, as will be
demonstrated here." [Kennedy, James G.: "Arithmetic with Roman numerals,
AMM, vol 88, pp 29-32, January 1981.]
> Instead they just used an abacus (the Britannica has a photo of
> one from Roman times.) In any case there ought to be extant evidence,
> since Arabic numerals weren't introduced before 1200.
"The intimate connection between Roman numerals and the abacus has often
been noted. It may be described by saying that the Roman numeral for a
number describes the state of an abacus when it represents that number,
the number of occurrences of each simple numeral corresponding to the
number of counters at play in the appropriate columns of an abacus when
it represents that number." [Detlefsen M., Erlandson D. K., Heston J. C.,
Young. C. M.: "Computation with Roman Numerals", AHES, vol 15, no 2,
pp 141-148, 1976.]
My favourite reference is Jose Augusto Sanchez Perez's "La aritmetica en
Roma, en India y en Arabia", Madrid, Granada, 263 pages, 1949.
Further references:
See those in Georges Ifrah's bibliography (pp. 499-502): "From One to
Zero: A Universal History of Numbers", NY: Viking Penguin Inc., XVI,
503 pp., 1985.
Friedlein Gottfried: "Die Zahlzeichen und das elementare Rechnen der
Griechen und Romer und des christlichen Abendlandes vom 7. bis 13.
Jahrhundert", Erlangen, 1869.
Taisbak Christian Marinus: "Roman Numerals and the Abacus", Classica et
Mediaevalia, Vol. 26, 1965. Journal of philology and history on Graeco-
Roman Antiquity and the Middle Ages. ISSN 0106-5815.
Giacardi Livia & Roero Clara S.: "L'origine della numerazione romana:
un'ipotesi di Giuseppe Nicasi sul modo di contare dei contadini di Morra;
con una nota di M. Raffaella Trabalza, Foligno: Edizioni dell'Arquata,
109 pages, 1987. Series: Centro studi "Cristiano Mancini" per la storia
del pensiero matematico; 1.
Julio Gonzalez Cabillon