A century after the immovable center began to move, another seeming impossibility began
to seem necessary: a new concept of number that encompassed both integers and irrational
quantities. The transformation of the ancient concept of number underlies modern math-
ematics, and hence also much of modern science. Though a number of social, economic,
cryptographic, and even legal perspectives have shed light on this mysterious shift, music
(both theoretical and practical) helps illuminate the hesitation about the nascent concept
of irrational number in the work of three close contemporaries: Michael Stifel, Girolamo
Cardano, and Nicola Vicentino. All three worked at the frontier between mathematics and
music around 1550; all three discussed the possibilities of “ irrational numbers, ” more or
less hesitantly. In the end, their different mathematical conclusions strongly reflected their
different approaches to music. Only music connected Heaven and Earth, theory and experi-
ence, mathematics and feeling.
Sixteenth-century mathematicians worked in the shadow of the ancient concepts of
number and magnitude, but struggled with these ancient distinctions. For instance, though
Robert Recorde ’ s The Whetstone of Witte (1557) notes that “ Euclide , Boetius , and other
good writers ” acknowledge only “ whole numbers, ” Recorde also includes “ nombres
irrationale, ” approximated as closely as desired by infinite series of fractions. Thus, as
Katharine Neal notes, Recorde broadened his “ number concept while simultaneously using
labels that signaled his awareness of the unacceptability, by traditional standards, of the
new numbers. ” Recorde observed that his number terms draw on algebra, then known as
the “ cossic art, ” whose solutions include both rational and irrational quantities. This art
has many practical aspects, as Cardano and other Italian mathematicians had noted;
Recorde dedicated his book to the “ venturers ” of the Muscovy Company, offering practical
examples of military formations, bricklaying, and geography and promising a further book
on navigation.^1
Both practical and theoretical considerations moved Fran ç ois Vi è te to make crucial
symbolic innovations that linked these different number concepts more closely. In his
Canon mathematicus (1579), Vi è te advocated the use of decimal fractions to replace the
sexagesimal calculations traditionally used for astronomy; such decimals could express
barré
(Barré)
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