Music and the Making of Modern Science

58 Chapter 4

It is properly debated whether irrational numbers are true numbers or fictions. For if we lack rational

numbers in geometrical figures, their place is taken by irrationals, which prove precisely those things

that rational numbers could not; certainly from the demonstrations they show us we are moved and

compelled to admit that they [irrational numbers] really exist from their effects, which we perceive

to be real, sure, and constant.

On the other hand, other things move us to a different assertion, namely that we are forced to

deny that irrational numbers are numbers. Namely, where we might try to subject them to numera-

tion and to make them proportional to rational numbers, we find that they flee perpetually, so that

none of them in itself can be precisely grasped: a fact that we perceive in the resolving of them, as

I will show below in its place. Moreover, it is not possible to call that a true number which is such

as to lack precision and which has no known proportion to true numbers. Just as an infinite number

is not a number, so an irrational number is not a true number and is hidden under a sort of cloud of

infinity. And thus the ratio of an irrational number to a rational number is no less uncertain than that

of an infinite to a finite.^15

Here, the “ cloud of infinity ” is the infinite sum of fractions needed to represent an

irrational quantity; such an “ actual infinite ” was rejected by Aristotle. Stifel ’ s distaste

for this infinitude finally outweighs his geometrical and musical arguments that irratio-

nal quantities can “ take the place ” of rational numbers in every effective respect. Thus,

even though his musical arguments had led him to affirm irrational numbers, his concern

to avoid the infinite ultimately moved him to demote them from the class of “ true

numbers. ”

Stifel ’ s arguments show the effect of musical considerations on mathematical concerns,

indicating the possibility of shifting and surprising alliances between the various parts of

the quadrivium: geometric irrationalities, excluded from arithmetic, could find a place in

music. Though Stifel himself finally gave precedence to an Aristotelian rejection of the

actual infinite, others would take these arguments in a different direction precisely by

placing new emphasis on the musical side.

Among these, the famous mathematician, physician, and polymath Girolamo Cardano

has special importance, even though his writings on music are less well known than the

Figure 4.2

Michael Stifel ’ s diagram showing the equal division of a whole tone, from Arithmetica integra (1544). He uses

to denote a square root and places 8, 72 , and 9 in proportion.