WEMSK24

Geometry II

I. To a great extent, the story of medieval geometry is the story

of Euclid's Elements. This was the standard text used in

instruction during the `university period'. On the place of

geometry in the curriculum: B. L. Ullman, "Geometry in the

Mediaeval Quadrivium," Studi di bibliografia e di storia in onore

di Tammaro de Marinis (Rome, 1964), 4.263-285, with plates.

1. Your first port of call, and perhaps your only, ought to be:

John E. Murdoch, "Euclid," Dictionary of Scientific Biography, ed.

Charles C. Gillespie, 15 vols. (NY: , 1970-78); 4 (1971), 437-59.

Somewhat clearer, but not so thorough, is Heath's history in his

translation of Euclid's Elements (see below).

2. The standard Edition of Euclid, with Latin translation and solid

notes: Johan L. Heiberg and H. Menge, Euclidis opera omnia, eds.,

8 vols. + supplement. Teubner Bibliothek (Leipzig: Teubner, 1883-

1916). Also available on their CD-ROM. The first 5 vols. are

devoted to the Elementa. New edition: Euclidis Elementa I, Libri I-

IV, cum appendicibus, ed. E. S. Stamatis (Leipzig: Teubner, 1969).

3. English translation: The Thirteen Books of Euclid's Elements,

tr. Sir Thomas L. Heath, 3 vol., 2d ed. (Cambridge: CUP, 1926).

Republished also by Dover. Read the excellent introduction for
a

history of the medieval Euclid. The translation is included in The

Great Books of the Western World.

4. Bibliography: Max Steck, Bibliographia Euclideana: Die

Geisteslinien der Tradition in den Editionen der Stoicheia des

Euklid (um 365-300): Handschriften, Inkunabeln, Fruehdrucke (16.

Jh.), textkritische Editionen des 17.-20. Jhs., Editionen der opera

minora ..., nach dem Tode des Verfassers herausgegeben von Menso

Folkerts (Hildesheim: Gerstenberg Verlag, 1981).

II. As with so many influences of classical authors, one can divide

the influence of Euclid into: 1. The Greek-Latin Phase; 2. The

Arabic-Latin Phase; 3. The Latin Phase.

1. The Greek-Latin Phase:

a. George D. Goldat, "The Early Medieval Traditions of Euclid's

Elements," Diss. Wisconsin, 1957. University Microfilms, 20,236.

b. If you are only casually interested, skip this section. The

first Latin mention of Euclid is by Cicero (De oratore III, 132);

the Romans, true to their reputation, seem not to have been

interested in the theoretical parts of the Elements, rather only

the practical ones, useful to the so-called agrimensores. Martianus

Capella, for example, turns the chapter on geometry into one on

geography. There seem to have been four attempts to translate

Euclid into Latin: 1. A fragment of Censorinus, reprinted in F.

Hultsch's edition, De die natali (Leipzig: Teubner, 1867), 60-63.

2. A palimpsest (with nutgall treatment, as so often) from the

Biblioteca Capitolare in Verona, 5th C. Cf, Euclidis latine facti

fragmenta Veronensia, ed. Marius Geymonat (Milan: Istituto

Editoriale Cisalpino, 1964). 3. Boethius made some sort of

translation, mentioned by Cassiodor (Institutiones II, 6, 3; Variae

I, 45, 4. cf. PL 63.1307-1364. See Menso Folkerts, "Boethius"

Geometrie II: ein mathematisches Lehrbuch des Mittelalters

(Wiesbaden: Steiner, 1970). Cf. also David Pingree, "Boethius'

Geometry and Astronomy," Boethius. His Life, Thought and Influence,

ed. Margaret Gibson (Oxford: Blackwell, 1981), 155-161. 4. A

fragment, found in Munich (MS 20752), Lueneburg and Paris

fragments. Cf. Menso Folkerts, "Anonyme lateinische

Euklidbearbeitungen aus dem 12. Jahrhundert," Denkschriften der

Oesterreichischen Akade,mie der Wissenschaften. Math.-Naturwiss.

Klasse (1970), 5-42.

2. The Arabic-Latin Phase. The second phase begins with the

translations of the Elements from Arabic. On the process, see:
De

Lacy O'Leary, How Greek Science Passed to the Arabs (London:

Routledge and Kegan Paul, 1979; repr. from 1949). The major text

is that of Adelard of Bath.

a. Marshall Clagett, "The Medieval Latin Translations from the

Arabic of the Elements of Euclid, with special emphasis on the

versions of Adelard of Bath," Isis 44 (1953), 16-42. First to

distinguish the three versions.

b. The first Latin translation of Euclid's "Elements" commonly

ascribed to Adelard of Bath: books I-VIII and X.36-XV.2, ed. Hubert

L. L. Busard. Studies and texts,/ Pontifical Institute of Mediaeval

Studies 64 (Toronto: Pontifical Institute of Mediaeval Studies,

1983).

c. A very nice vocabulary is included in: Marshall Claggett,

Archimedes in the Middle Ages. Volume 1. The Arabo-Latin Tradition

(Madison, WI: University of Wisconsin Press, 1964): "A Selective

Index of Latin Geometrical Terms," 691-708.

d. A useful book on Adelard, with a solid chapter on his

translation of the Elements: Louise Cochrane, Adelard of Bath: the

First English Scientist (London: British Museum Press, 1994).

e. A collection of articles on Adelard, two on his translation of

the Elements: Adelard of Bath: an English Scientist and Arabist of

the Early Twelfth Century, ed. Charles Burnett (London: Warburg

Institute, University of London, 1987).

3. Latin Phase. There are numerous attempts at bettering Adelard,

though his text remains the standard. I append a few of these

here. Unless you are deep into it, you might skip this section,

also:

a. The Latin translation of the Arabic version of Euclid's Elements

commonly ascribed to Gerard of Cremona, introduction, edition and

critical apparatus by Hubert L. L. Busard (Leiden: Brill, 1984).

b. Robert of Chester's (?) redaction of Euclid's Elements, the

so-called Adelard II version, ed. Hubert L.L. Busard, Menso

Folkerts, 2 vols. Science Networks Historical Studies, 8-9 (Basel:

Birkhaeuser, 1992).

c. Thomas Bradwardine, Geometria speculativa: Latin text and

English translation with an introduction and a commentary, George

Molland. Boethius 18 (Stuttgart: Steiner, 1989).

d. The mediaeval Latin translation of Euclid's Elements made

directly from the Greek, ed. Hubert L. L. Busard. Boethius 15

(Stuttgart: Steiner, 1987). Transl. by Henricus Aristippus, Eugene

the Emir and an anonymous translator.

e. Paul M. J. E. Tummers, "The Commentary of albert on Euclid's

Elements of Geometry," Albertus Magnus and the Sciences:

Commemorative Essays, 1980, ed. James A. Weisheipl. Pontifical

Institute of Mediaeval Studies, Studies and Texts 49 (Toronto,

1980), 479-99.

II. As mentioned, the Romans had little interest in theoretical

geometry, but there was a large amount of interest in surveying,

architecture, and the like. The story of Gefjon, who plowed the

borders of her land, reveals perhaps the influence of the Roman

agrimensores, since that was their common method of outlining the

fields and the town. We know from the layout of such Viking forts

as Fyrkat and Trelleborg, with their use of the Roman foot as a

standard and the fact that the ratio of the outer perimeter to the

inner perimeter follows the Golden Mean, that Roman influence was

felt all over.

a. Oswald A. W. Wentworth, The Roman Land Surveyors: An

Introduction to the Agrimensores (Newton Abbot: David and Charles,

1971). An excellent survey.

b. The texts are gathered in: Friedrich Blume, Karl Lachmann, A.

Rudorff, Die Schriften der roemischen Feldmesser (Gormatici

veteres), 2 vols. (Berlin, 1848-52; repr. 1962). TOC in Wentworth.

c. Excellent survey: Paul Tannery, "La geometrie au xie siecle,"

Memoires scientifiques, ed. J. L. Heiberg et al., 5 (Paris, 1922),

79-102; and: "Histoire des sciences: Geometrie," Memoires

scientifiques 10 (Paris, 1930), 37-59.

There were many other endeavors which required a knowledge of

practical geometry, such as architecture, page layout, etc.

a. Lon R. Shelby, "The Geometrical Knowledge of the Medieval Master

Masons," Speculum 47 (1972), 409-410.

b. Stephen K. Victor, ed. & tr. Practical Geometry in the High

Middle Ages: "Artis cuiuslibet consummatio" and the "Pratike de

geometrie," Memoirs of the APS 134 (Philadelphia, 1979).

c. Villard de Honnecourt showed how to construct buildings

geometrically, even calling his tools "geometrical." For an easy

look at his work: The Sketchbook of Villard de Honnecourt, ed.

Theodore Bowie (Bloomington: Indiana University Press, 1959). See

also: Carl F. Barnes, Villard de Honnecourt -- The Artist and his

Drawings: A Critical Bibliography (Boston: G. K. Hall, 1982).