- MEDIEVAL GEOMETRY 329
early books of Euclid, often drastically edited. The tradition of attributing these
works to Boethius continued even in the twelfth century, when translations from
Arabic began to appear, as one can see in the booklets of Folkerts (1970, 1971), the
second of which compares an anonymous Latin version ascribed to Boethius with
the translation (from Arabic) of Athelhard of Bath. In a series of papers (1999,
2000) Zaitsev has argued that the pseudo-Boethius was enlarging the commen-
taries on Euclid by including material from surveying and geometric astronomy.
As a result,
[I] ç the writing process geometric concepts were systematically
translated into the language of surveying, and the resulting melange
of surveying and geometry was used as the basis for discussing
the theological-cosmological significance of the discipline. [Zait-
sev, 2000, p. 222]
Thus, in the West also, mathematics became once again mixed with philosophy,
but this time with the philosophy of Christianity.^9 Zaitsev also notes (2000, p.
223) that the idea of multiple layers of meaning was dear to the authors of medieval
texts; but in contrast to biblical commentaries, which were strictly separated from
the texts used as a source, commentaries and sources were routinely intermixed in
the geometric work.
Mathematics sank to a rather low level in Europe after 500 CE, recovering
only slightly if at all in the Carolingian Renaissance of the ninth century. One
of the better-informed scholars of the tenth century was Gerbert of Aurillac (ca.
940-1002), who reigned as Pope Sylvester II during the last three years of his life.
Even though Gerbert was one of the leading scholars of his day, who advocated the
use of Hindu-Arabic numerals, one of his letters to a certain Adalbold of Liege is
occupied with a discussion of the rule for finding the area of a triangle given its base
and altitude! The general level of geometry, however, was not so bad as the corre-
spondence between Gerbert and Adalbold seems to imply. In fact, Gerbert wrote,
but did not finish, Geometria, a practical manual of surveying based on what was
probably in Boethius' textbook. This work, which can be read online,^10 consists
of 89 brief chapters devoted to triangles, circles, spheres, and regular polygons. It
gives the names of standard units of length and finds the areas of such simple figures
as a trapezoid (Chapter XVLIII) and a semicircle (Chapter LXXIX, where the rule
is given to multiply the square of the diameter by 11 and divide the product by 28).
A specimen that may be typical of the level of geometric knowledge used in civil en-
gineering, architecture, surveying, and geometric astronomy in the twelfth century,
just as translations from Arabic works began to circulate in Europe, is provided by
Hugh of St. Victor's Practical Geometry (Homann, 1991), in which one can find a
description of the construction and use of an astrolabe (a fundamental tool used by
(^9) Compare with the quotations from the pseudo-Boethius and Gerbert in Chapter 3.
(^10) http:Wpld.chadvyck.com, a commercial site that provides the Patrologia Latino, of Jacques-
Paul Migne (1800-1875). Search for the title "geometria."