The History of Mathematics: A Brief Course

48 3. MATHEMATICAL CULTURES II

last convulsive effort of Greek geometry which was now nearly

dead and was never effectually revived.

Greek mathematics held on longer in the Byzantine Empire than in Western

Europe. Although Theon of Alexandria had found it necessary to water down

the more difficult parts of Greek geometry for the sake of his weak students, the

degeneration in Latin works was even greater. The philosopher Boethius (480-

524) wrote Latin translations of many classical Greek works of mathematics and

philosophy. His works on mathematics were translations based on Nicomachus and

Euclid. Boethius' translation of Euclid has been lost. However, it is believed to be

the basis of many other medieval manuscripts, some of which use his name. These

are referred to as "Boethius" or pseudo-Boethius. The works of Boethius fit into

the classical quadrivium of arithmetic, geometry, music, and astronomy.

Politically and militarily, the fifth century was full of disasters in Italy, and some

of the best minds of the time turned from public affairs to theological questions.

For many of these thinkers mathematics came to be valued only to the extent that

it could inspire religious feelings. The pseudo-Boethius gives a good example of

this point of view. He writes:^3

The utility of geometry is threefold: for work, for health, and for
the soul. For work, as in the case of a mechanic or architect; for
health, as in the case of the physician; for the soul, as in the case
of the philosopher. If we pursue this art with a calm mind and
diligence, it is clear in advance that it will illuminate our senses
with great clarity and, more than that, will show what it means
to subordinate the heavens to the soul, to make accessible all the
supernal mechanism that cannot be investigated by reason in any
other way and through the sublimity of the mind beholding it, also
to integrate and recognize the Creator of the world, who veiled so
many deep secrets.
In the tenth century, Gerbert of Aurillac (940-1003), who became Pope Sylvester
II in 999, wrote a treatise on geometry based on Boethius. His reasons for studying
geometry were similar:

Indeed the utility of this discipline to all lovers of wisdom is the

greatest possible. For it leads to vigorous exercises of the soul, and

the most subtle demands on the intuition, and to many certain

inquiries by true reasoning, in which wonderful and unexpected

and joyful things are revealed to many along with the wonderful

vigor of nature, and to contemplating, admiring, and praising the

power and ineffable wisdom of the Creator who apportioned all

things according to number and measure and weight; it is replete

with subtle speculations.

These uses of geometry were expressed in the last Canto of Dante's Divine

Comedy, which describes the poet's vision of heaven:

(^3) This quotation and the next can be read online at http://pld.chadwyck.com, a commercial
website. This passage is from Vol. 63Ϊ the next is from Vol. 139. Both can be reached by searching
under "geometria" as title.