Music and the Making of Modern Science

Euler: The Mathematics of Musical Sadness 145

Euler went on to conduct many other inquiries into amicable and perfect numbers,

among a vast variety of topics related to the abundance of prime numbers of different

kinds, including his profound relation between the harmonic series and the prime numbers

( box 9.2 ). To be sure, Euler does not make any explicit connection between the harmonic

series and harmony, but he knew well that this series had its origins in music, for the

Pythagoreans already defined the harmonic ratio as a way of mediating between arithmetic

and geometric ratios (see box 1.1). 36 In making his arguments for this deep and surprising

result, Euler used the tools of analysis — that is, differential and integral calculus — as

well as those of traditional arithmetic, not only to find individual results such as this

one but to open a whole new field of mathematics. Andr é Weil observed that “ one may

well regard these observations as marking the birth of analytic number theory, ” as it came

to be called. 37

The influence of Euler ’ s musical work can also be seen in a very different arena of

his activity, his famous solution of the problem of the K ö nigsberg bridges. In that city

(now called Kaliningrad in Russia), the island Kneiphof in the river Pregel joins various

parts of the city via seven bridges ( figure 9.4 ). Euler became aware of the “ quite well-

known problem ” whether someone could take a walk that would return to its starting

point after crossing each of the seven bridges only once. His letters reveal that, even

in 1736, he considered the problem “ banal ” because its solution “ bears little relationship

to mathematics, and I do not understand why you expect a mathematician to produce

it, rather than any one else, for the solution is based on reason alone, and its discovery

does not depend on any mathematical principle. ”^38 Euler ’ s distancing of this problem

from what he considered “ mathematics ” helps clarify the new step he made by consid-

ering it (as he puts it in his 1736 paper) an example of a branch of geometry “ that has

been almost unknown up to now; Leibniz spoke of it first, calling it the ‘ geometry of

position ’ [ geometria situs ]. This branch of geometry deals with relations dependent on

Box 9.2

Euler and the harmonic series

Following the ancient definition of a harmonic mean (box 1.1), Oresme proved that the

harmonic series^111

2

1

3

1

k= 1 k^4

∞

∑ =+ + +^ diverges. Euler proved that

11

k= 1 kpp^11

∞

∑ =∏ −() , where

the product on the right-hand side goes over all prime numbers p > 1. Euler also showed that

in general ζ

−

()s

k kps p s

==

= ()

∞

∑ ∏

11

1 11

, whose name ζ ( s ) came from Bernhard Riemann, who

brought the properties of this “ Riemann zeta function ” to the center stage of mathematics. These

expressions relate the sum over the reciprocal of each number 1, 2, 3, ... (raised to the power

s > 1 ) to the product only over the prime numbers, indicating the deep structure whereby the

primes underlie all other numbers.