measures, and the practical mathematical treatises of Fibonacci (Leonardo of Pisa)
culminated in the Fibonacci series for determining the convergence of a series of ratios.
Geometry. Geometry was studied in Boethius’s translation of Euclid and, from the
12th century, in the Practica geometriae of Hugh of Saint-Victor. In the 10th century,
Gerbert of Aurillac (the future Pope Sylvester II) incorporated practical surveyors’
manuals into the study of geometry. Practical geometry also included treatises on the use
of measuring instruments and overlapped with astronomy in the treatise on the use of the
astrolabe (known from the 10th century) and the equatory of planets. Jordannes and
Fibonacci also wrote geometry texts. The geometry of construction and sculpture found
its most interesting expression in the Sketchbook of the 13th-century designer Villard de
Honnecourt.
Astronomy. Astronomy followed the system of planetary relationships worked out in
Plato’s Timaeus, which was known in Latin translation, and the Almagest of Ptolemy.
Martianus Capella’s Book 8 also provided a good discussion of astronomy. Astronomical
calculations were necessary for figuring the dates of Easter; Bede’s table continued to be
used, and others were developed. Gerbert of Aurillac wrote a treatise on the astrolabe. In
the 1130s, Bernard Silvestris incorporated astronomical material into his Cosmographia,
as did Thierry of Chartres in his Heptateuchon, and later in the century Alain de Lille
discussed astronomy at length in the Anticlaudianus. William of Moerbecke translated
Aristotle from the Arabic versions, and practical school texts were written in the 13th
century by John of Sacrobosco.
Music. Boethius’s De institutione musica remained the chief textbook throughout the
Middle Ages. His discussion, like that of Martianus Capella, was based largely on
Pythagorean number symbolism and stressed harmonics. Gregorian chant developed fully
in the Frankish kingdom in the 8th century and was written in neumes, which appeared at
varying heights and in different shapes above the words, but with no staff lines or clear
indication of exact pitch or duration. The alphabet letters taken from Paul the Deacon’s
Ut queant laxis hymn to John the Baptist by Guido d’Arezzo in the 11th century gave the
tones ut, re, mi, fa, sol, la, from which the modern scale developed. These could be
written on a staff to indicate regular intervals and duration, and by the 13th century
neumes had become obsolete.
Jeanne E.Krochalis
[See also: ALAIN DE LILLE; BAUDRI OF BOURGUEIL; BERNARD
SILVESTRIS; BOETHIUS, INFLUENCE OF; ERIUGENA, JOHANNES SCOTTUS;
GERBERT OF AURILLAC; HUGH OF SAINT-VICTOR; IMAGE DU MONDE;
JOHANNES DE GARLANDIA; MARTIANUS CAPELLA; MUSIC THEORY;
MUSICAL NOTATION (NEUMATIC); MUSICAL NOTATION, 12TH-15TH
CENTURIES; PSEUDO-TURPIN; REMIGIUS OF AUXERRE; SCHOOLS
(CATHEDRAL); THEODULF OF ORLÉANS; VILLARD DE HONNECOURT]
Jehan le Teinturier d’Arras. Le mariage des sept arts par Jehan le Teinturier d’Arras, ed. Arthur
Långfors. Paris: Champion, 1923.
Eriugena, Johannes Scottus. Annotationes in Marcianum, ed. Cora Lutz. Cambridge: Mediaeval
Academy of America, 1939.
Martianus Capella. De nuptiis Philologiae et Mercurii, ed. A. Dick. rev. ed. J.Preaux. Stuttgart:
Teubner, 1969.
——. Martianus Capella and the Seven Liberal Arts, trans. W.H.Stahl, R.Johnson, and E.L.Burge.
New York: Columbia University Press, 1971.
The Encyclopedia 1033