This is another sticky wicket. We are still lacking a decent study

on the way numbers were handled in the Middle Ages. One important

thing to remember is that modern arithmetic is unlike medieval

arithmetic. We have Arabic numbers, they had for the most part

Roman. For most of the Middle Ages, arithmetic was just counting.

The ease with which we multiply and divide was unknown to them, and

much was done by iteration. We have regular numbers and square

numbers; they had triangular, pentagonal, stellate, etc. (see

Hogben under Geometry). They believed in Wisdom 11.21: omnia
in

mensura et numero et pondere disposuisti `thou hast ordered all

things in measure, and number, and weight', so numbers had

significance as well as being numbers. Much of the arithmetic
was

for the purpose of calendar calculation. BTW, remember the

etymology of calculus. They did use counting boxes and the like.

1. A good start on elementary arithmetic is: Gottfried Friedlein,

Die Zahlzeichen und das elementare Rechnen der Griechen und Roemern

und des christlichen Abendlandes vom 7. bis 13. Jh. (Wiesbaden:

Saendig, 1968; repr. of 1869 ed.).

2. Computus. The medieval computus was for the most part devoted

to the calculation of the church year, time, cycles and the like;

see the section on the Calendar.

a. Bedae Opera de temporibus, ed. Charles W. Jones (Cambridge:

Mediaeval Academy of America, 1943). This would be the best place

to start. There are English translations, e.g. Bede, the Reckoning

of Time, tr. with intro., notes and commentary, by Faith Wallis.

Translated Texts for Historians 29 (Liverpool: Liverpool UPress,

1999).

b. Just added for our Old English friends: Heinrich Henel, Studien

zum altenglischen Computus. Beitraege zur englischen Philologie,

Heft 26 (Leipzig: Tauchnitz, 1934). Mostly devoted to Byrhtferth.

3. As we moved away from counting and into `real' arithmetic, a

number of texts on `algorismus' arose. A good survey is offered

by:

a. The Earliest arithmetics in English, ed. with introduction by

Robert Steele. EETS, Extra Series, 118 (London, Published for the

Early English Text Society: Oxford University Press, 1922.

Reprinted, (New York: Kraus, 1973). Contains: The crafte of

nombrynge, a translation and amplification of one of the glosses on

the De algorismo of Alexander de Villa Dei (Egerton ms. 2622)- The

art of nombryng, a translation of John of Holywood's De arte

numerandi (Ashmole ms. 396, fol. 48) - Accomptynge by counters,

reprinted from the 1543 edition of Robert Record's Arithmetic,

printed by R. Wolfe.- Appendix I. A treatise on the numeration of

algorism (From a 14th century ms.) II. Carmen de algorismo, by

Alexander de Villa Dei (B.M. ms., 8 C IV., with additions from

12 E. 1 & Eg. 2622)

4. Numbers were, as we have said, symbolic, and they were used in

the construction of everything:

a. Michael J. Batts, "Numerical Structure in Medieval Literature,"

in Formal Aspects of Medieval German Poetry: A Symposium, ed.

Stanley N. Werbow (Austin, TX: UTexasP, 1969), 93-121. A fair

survey, with bibliography. Easily supplemented by MLA.

b. Robert E. McNally, S. J., "Der irische Liber de numeris. Eine

Quellenanalyse des pseudo-isidorischen Liber de numeris" (Diss.

Munich, 1957). An excellent source study. An outline is found in

the Medtextl archives.

c. For symbolic numbers in general, a good read: Karl Menninger,

Number Words and Number Symbols. A Cultural History of Numbers, tr.

Paul Broneer (Cambridge: MITPress, 1969). Translation of his

revised ed., Zahlwort & Ziffer (Goettingen: Vandenhoeck & Ruprecht,

1958.

d. For a great list of medieval examples: Hieronymus Lauretus,

Silva allegoriarum totius sacrae scripturae (Barcelona, 1570),

photomechanical reprint of the 10th ed. (Cologne, 1681) (Munich:

Fink, 1971).

e. Heinz Meyer & Rudolph Suntrup, Lexikon der mittelalterliche

Zahlenbedeutung. Muenstersche Mittelalter-Schriften (Munich: Fink,

1987). A good survey with bibliographies.

5. It is good to look at times at the literatures on the edge, for

they often have collected better than the center:

a. Alfraedhi islenzk, ed. Kr. Kaalund. Samfund til udgivelse af

gammal nordisk Litteratur (Copenhagen: S. L. Mo/ller, 1917-18). A

good survey of just about everything symbolic. The Cisiojanus is

illustrated. We ought to have a bibliography on the hand and
all

the things we can do with it. An outline of the Alfraedhi is found

in the medtextl archives.