Music and the Making of Modern Science

Euler: The Mathematics of Musical Sadness 143

in which the ever-recurrent 2s seem to echo the initial 2 from which 2 is drawn. These

two examples show something of the visual poetry of continued fractions, which power-

fully symbolize and expose the infinite processes and relations that form the inner structure

of irrational numbers.

In “ On continued fractions, ” Euler gave the first proof that e = 2.71828182845904 ... ,

the base of the natural logarithms (and the crucial constant describing exponential growth

or decay), is in fact irrational, which had been suspected but not proved. To accomplish

this important step, he showed that it could be written as a continued fraction:

e=+

+ + + + + + + + + +

2 1

1 1

2

1

1 1

1 1

4

1

1 1

1 1

6

1

1 1

1 

.

Here one wonders what the pattern of even integers (2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, ... )

spaced regularly along the diagonal has to do with this irrational quantity; Euler ’ s alge-

braic deductions do not give an intuitive meaning for this pattern, though they implicitly

generate it.^30

Nor were the mathematical effects of his musical work restricted to this one technique.

Though Euler ’ s name later became so closely associated with number theory, his interest

in this field began after his earliest work on music, such as his 1726 notebook entries.

Only after his arrival in St. Petersburg in 1727 and his subsequent correspondence with

Christian Goldbach (who moved to Moscow shortly after Euler ’ s arrival) did Euler ’ s inter-

est in number theory really begin, during the period of his greatest activity preparing the

Tentamen. For instance, in December 1729, Goldbach wrote Euler to ask him whether

“ Fermat ’ s observation [is] known to you, that all numbers 212

n

+ are prime? He said he

could not prove it; nor has anyone else done so to my knowledge. ” Euler ’ s rather indiffer-

ent response indicates that, even by that date, he was not greatly interested in this funda-

mental question. Only after a subsequent letter from Goldbach prodding him did Euler

catch fire and disprove Fermat ’ s conjecture by showing that the fifth Fermat number,

2 1 4 294 967 297^2

5

+=,,, , is evenly divisible by 641.^31

After that, Euler read Fermat ever more closely and took up number theory with par-

ticular passion. His first result already underlines his phenomenal abilities as a calculator;