Ad Quadratum Construction and Study of the Regular Polyhedra

157
The Millennium Sphere and the Liberal Arts Curriculum:
The early Pythagoreans were reputedly the first to link arithmetic, geometry, music and
astronomy together, and to organize their teaching into a curriculum 61 , on which the
earliest statement came down to us through Archytas, a contemporary of Plato (4 th C.
BC). That four-fold part of the curriculum to be known later as the quadrivium became
canonical in the schools of Greece and, through Rome and Boethius (470-525 AD), it reemerged
in the West with the Carolingian Renaissance. Scotus among others, was one of
its proponents. In a document of the ninth Century, the Musica Enchiriadis and its
commentary by an anonymous author, the Scholia Enchiriadis, we find a good example
of how early medieval school men thought about these matters. In a passage of the
Scholia, we find the following “dialog” in catechismal form between a teacher and his
student: 62
Pupil: How was harmony born of arithmetic as from a
mother? And what is harmony, and what is music?
Teacher: We regard harmony to be a mixed symphonia of
different sounds, altogether dependent on the theory of
numbers, like all the other mathematical disciplines, it is
only through numbers that we understand it.
Pupil: Which are the mathematical disciplines?
Teacher: Arithmetic, geometry, music, and astronomy.
Pupil: What is mathematics?
Teacher: A doctrinal science.
Pupil: Why doctrinal?
Teacher: Because it deals with abstract quantities.
Pupil: What are abstract quantities?
Teacher: Abstract quantities are those embraced only by
the intellect because they lack material, i.e., physical
admixture. And further: multitudes, magnitudes, their
opposites, forms, similarities, relations and many other
things … change when connected with physical substance.
These quantities are each directly treated in arithmetic, in
music, in geometry and in astronomy. It is thus because
these four disciplines are not skills of human invention but
important researches in holy works; and they support in the
most wondrous way acute minds in the understanding of
Creation.
This four-fold mathematical curriculum, together with the trivium of grammar, rhetoric,
and dialectic, flourished in the medieval universities and, with it, formed the curriculum
of the seven liberal arts, providing the foundations for higher studies in philosophy and
theology. The ad quadratum diagram can therefore be conceived of as an accurate
though stylized compendium of the quadrivium since it embodies some of the
61 David Wagner, Ed.: The Seven Liberal Arts in the Middle Ages. Indiana U.P. Bloomington, 1983.
62 Quoted in Nils Wallis’ article in David Wagner, Op. cit.