WEMSK24: Geometry II


                            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,

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,

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).