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mathematics, according to Aristotle and everyone following him in the
medieval and early- modern university, comprised arithmetic and geom-
etry, the fi rst concerned with numbers, or “discrete quantity,” the second
with spatial extension, or “continuous quantity.” Pure mathematics was
therefore said to be about “general magnitude,” quantity in itself rather
than quantities of something. But this perspective also recognized that
mathematics was not restricted to the “pure” variety, and that many areas
of mathematics were indeed concerned with quantities of specifi c sorts
of things. Astronomy had been recognized as a mathematical discipline
since antiquity, because it applied geometrical reasoning to the motions of
the heavens. Music, on the other hand, was a mathematical science to the
extent that consonances could be understood in terms of number ratios,
typically as applied to different string lengths of equal tension (3:2 for the
interval of the fi fth, for example; or 2:1 for an octave). As prototypes of
mathematical sciences that used, in the fi rst case, geometry, and in the
second case, arithmetic, therefore, astronomy and music became estab-
lished in the liberal arts education of classical antiquity as two of the four
mathematical arts: geometry, arithmetic, astronomy, and music. However,
these four by no means exhausted the list of mathematical subjects. Oth-
ers, especially optics (using ray diagrams to represent the relationships
between seen objects, vision, and images) and mechanics (meaning statics
and referring to simple machines like the lever),^2 were also incorporated
under these ideal headings; by the seventeenth and eighteenth centuries
the list had become very long indeed, encompassing any subject of study
that made central use of “pure” mathematical reasoning regarding mea-
surable or quantifi able subject matter.
In the early Middle Ages in the Latin west, the four prototypical math-
ematical subjects became known, thanks to the late- fi fth- century writer
Boëthius, as the quadrivium, and together with the trivium (grammar,
rhetoric, and logic) constituted the “seven liberal arts.”^3 By the time of
the Renaissance, the quadrivium, seldom much emphasized in the arts
curriculum of the medieval university, began to be reemphasized in some
quarters as part of the humanist revival of classical culture. The most im-
portant name here is that of Johannes Müller, known as Regiomontanus,
who printed a number of texts in the second half of the fi fteenth cen-
tury that praised the project of restoring the mathematical project of the
ancients. Regiomontanus placed especial (although by no means exclu-
sive) stress on the mathematical science of astronomy, and a work that
he wrote with Georg Peurbach, the Epitome of the Almagest (published
posthumously in 1496) served as Copernicus’s chief access to Ptolemy’s