134 Chapter 9
larger musical forms. Euler ’ s interest in music encompassed many aspects of contemporary
composition and practical music-making, not only its mathematical elements. In his early
manuscripts, notes on musical theory precede any material referring to his second printed
work, “ Physical Dissertation on Sound ” (1726), indicating the path that led him from
music to the mathematical physics of sound.^3
In this work, Euler addresses all kinds of sources, from musical instruments to thunder
and snapping twigs, whose sounds all arise “ from the sudden restitution of compressed
air, and as a stronger percussion of the air. ” Especially, he addresses wind instruments such
as the flute, “ since no one up to the present has given anything of substance concerning
these instruments. ”^4 He extends Newtonian methods by considering an air column to
vibrate “ following the amplitude of expansions and contractions in the same manner as
strings, and thus I can consider that same air column as a bundle of air strings with the
tension given by the weight of the atmosphere. ” Here he faces the inherent mathematical
difficulties of the vibrations of a cylindrical pipe, for which his treatment is only a begin-
ning.^5 Still, he expresses satisfaction in his general result that “ the sounds of flutes will
be sharpest in pitch with the maximum heat, and the air the least dense, but to be lowest
pitch with the maximum cold and the most dense [air]. This difference of sounds is espe-
cially observed by musicians and organists. But since all flutes have the same change in
place equally, the melody is not changed. ”
Euler puts forward his work “ to be examined along with the distinguished candidates, ”
showing that he considered his treatment of sound a calling card demonstrating his skill
as he searched for a position. Euler ’ s work on sound led to further problems in mechanics
and thence to questions regarding the relative stability of ships with varying heights of
masts and sizes of sails. His first foray into nautical science forms the pendant to his work
on sound, which informs the mechanics of masts and winds. The juxtaposition of these
diverse topics shows their interconnection in his mind: in the study of sound, practical and
theoretical concerns unite mechanics, metaphysics, and nautical engineering.^6
During the same period as he was preparing his work on sound in music and physics,
Euler was also working on a more speculative, larger work, his Tentamen novae theorae
musicae ex certissimis harmoniae principiis dilucide expositae ( Essay on a New Theory
of Music Based on the Most Certain Principles of Harmony and Clearly Expounded ,
written 1730, published 1739).^7 Not able to find a job in his native city, Euler left Basel
in 1727 to move to St. Petersburg, where he obtained the chair of natural philosophy in
1730, the year he completed his Tentamen. By devoting so much of his attention to this
work during the crucial period in which he needed to establish himself in a permanent
position, Euler showed how integral he considered music to be to mathematics and natural
philosophy. Writing in 1731 to Daniel Bernoulli, a fellow pioneer of the mathematical
study of vibrating strings, Euler clarified his larger intent: “ My main purpose was that I
should study music as a part of mathematics and deduce, in an orderly manner, from
correct principles, everything that can make a fitting together and mingling of tones