Arch. Hist. Exact Sci. (2007) 61:343–424
DOI 10.1007/s00407-007-0003-9
The acoustic origins of harmonic analysis
Olivier Darrigol
Received: 20 December 2006 / Published online: 6 April 2007
© Springer-Verlag 2007
In music, harmony refers to a pleasant combination of sounds. In mathematics, a harmonic function is a sine function obtained by projecting a circular
motion on a diameter, and harmonic analysis is the theory of the development of periodic functions into harmonic components, or the theory of similar
developments. The occurrence of the same word in musical and mathematical
contexts is neither a coincidence nor a purely metaphorical effect. As is well
known, in the mid-nineteenth century Hermann Helmholtz connected the two
meanings through the idea that the ear functions as a harmonic analyzer in a
physico-mathematical sense.1
This formulation of Helmholtz’s achievement suggests that harmonic analysis preceded its acoustic and musical application. So does too the fact that
Joseph Fourier’s foundation of this kind of analysis had to do with heat rather
1 Cf. Steven Turner, “The Ohm-Seebeck dispute, Hermann von Helmholtz, and the origins of
physiological acoustics,” BJHS, 10 (1977), 1–24.
Communicated by J.Z. Buchwald.
The following abbreviations are used: AHES, Archive for the history of exact sciences; BJHS,
British journal for the history of science; CAP, Academia Scientiarum Imperialis Petropolitana,
Commentarii; EOm:n, Leonhard Euler, Opera omnia (Leipzig, 1911 on), series m, vol. n; FOn,
Joseph Fourier, Oeuvres, 2 vols. (1888–1890), vol. n; HAB, Académie Royale des Sciences et des
Belles-Lettres de Berlin, Histoire; HAS, Académie Royale des Sciences, Histoire; JEP, Journal de
l’Ecole Polytechnique; LOn, Joseph Louis Lagrange, Oeuvres, 14 vols. (Paris, 1867–1892), vol. n;
MAS, Académie (Royale) des Sciences, Mémoires; MT, Miscellanea Taurinensia; NCAP,
Academia Scientiarum Imperialis Petropolitana, Novi commentarii. Anachronistic
notation is
used for sums and integrals (Lagrange was first to systematically use the
notations for discrete
sums, and Fourier the first to indicate the limits of summation or integration in the modern way).
O. Darrigol (B)
CNRS: Rehseis, 83 rue Broca, 75013 Paris, France
e-mail: darrigol@paris7.jussieu.fr
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O. Darrigol
than sound and preceded Ohm’s and Helmholtz’s acoustic studies. In reality,
historians of eighteenth-century mechanics know that the first roots of harmonic analysis are to be found in Daniel Bernoulli’s much earlier work on the
theory of vibrations, which crucially depended on previous acoustic knowledge.
The aim of the present paper is to show that acoustic theories for the emission,
perception, and propagation of sound constantly bridged musical and mathematical harmonics, from Bernoulli’s earliest intuitions to Fourier’s perennial
foundation of harmonic analysis.
Much valuable commentary has been written both on the early theory of
vibrating bodies and on Fourier’s theory of heat. In particular, the present study
largely benefited from histories of the former theory by Heinrich Burckhardt
and by Clifford Truesdell, and from studies of the latter theory by Ivor GrattanGuinness, John Herivel, Jean Dhombres, and Jean-Bernard Robert. However,
relatively little attention has been paid to the persisting connections between
music theory, acoustics, and harmonic analysis. Eighteenth-century works on
vibrating strings have been studied independently of music theory, even though
their authors were deeply involved in it; and the genesis of Fourier’s theory of
heat has been studied independently of its acoustic antecedents, even though
Fourier was quite aware of them. In both cases, proper attention to acoustic
and musical contexts leads to valuable insights.2
Firstly, it appears that important aspects of work on vibrating strings by
Jean le Rond d’Alembert, Leonhard Euler, and Joseph Louis Lagrange have
been misunderstood or neglected. For instance, d’Alembert’s instance that
Bernoulli’s mixtures of sine curves could not explain the hearing of simultaneous harmonics from a single vibrating string has been regarded as erroneous
reasoning, whereas it is perfectly consistent with d’Alembert’s ideas on the emission of sound. More important, Lagrange’s objections to Bernoulli’s mixtures
have been mistaken for a rejection of the mathematical validity of trigonometric
series whereas they only concerned the physical interpretation of the harmonic
components. With this understanding, it becomes clear that Lagrange’s second
memoir on sound contains much of harmonic analysis. In modern terms, Lagrange had the general notion of the development of a function over the eigenfunctions of a Hermitian differential operator, was aware of the mutual orthogonality of the eigenfunctions, and used this property to express the coefficients
of the development. In the simplest case in which the operator is d2 /dx2 , this
2 Heinrich Burckhardt, “Entwicklungen nach oscillirenden Functionen und Integration der Differ-
entialgleichungen der mathematischen Physik,” Deutsche Mathematiker-Vereinigung, Jahresbericht, 10 (1901–1908), 1800 pages; Clifford Truesdell, “The rational mechanics of flexible and elastic
bodies. 1638–1788,” EO2:11(2); Ivor Grattan-Guinness in collaboration with Jerome Ravetz, Joseph
Fourier 1768–1830. A survey of his life and work, based on a critical edition of his monograph on
the propagation of heat, presented at the Institut de France in 1807 (Cambridge, 1972); John Herivel,
Joseph Fourier, the man and the physicist (Oxford, 1975); Jean Dhombres and Jean-Bernard Robert,
Joseph Fourier, 1768–1830: Créateur de la physique-mathématique (Paris, 2000).
The acoustic origins of harmonic analysis
345
expression contains Fourier’s theorem (without any proof of the completeness
of the eigenfunctions, however).
Secondly, the opinions of eighteenth-century string-theorists about the generality and physical meaning of Bernoulli’s trigonometric sums and their ideas
on the foundations of music are both seen to depend on their ideas on the
physical nature of (musical) sound. Euler, d’Alembert, and Lagrange all agreed
that musical sounds (tones) should be regarded as periodic repetitions of pulses
whose precise shape did not matter much. Accordingly, they denied any physical meaning to harmonic analysis. With regard to music theory, Euler and
Lagrange defined the harmony of two sounds through the frequent coincidence of the periodic pulses corresponding to two different tones; d’Alembert
adopted Jean-Philippe Rameau’s idea of founding musical harmony on the
simultaneous hearing of harmonics from a single sonorous body, but regarded
this fact as irreducible to the periodicities of the physical motions associated
with sound. Against these three geometers, Bernoulli understood sound as a
mixture of sinusoidal motions with separate physical existence. Accordingly,
he regarded trigonometric series as a general, physically meaningful way of
describing any vibration. About musical harmony, he approved Rameau’s system, but explained the importance of harmonics in the definition of harmony
by considerations of periodicity.
It may seem strange that d’Alembert and Bernoulli both supported Rameau’s system of music and yet disagreed on the significance of harmonic
analysis, or that Euler and d’Alembert joined forces against harmonic analysis,
and yet defended conflicting systems of music. The reason for these anomalies
is that their assessment of the connection between harmonic analysis and music
depended on their understanding of the perception of sound and also on their
general views on the relations between mathematics, physics, and music. About
hearing, Euler and Lagrange maintained the old scheme according to which the
aural nerves detect the beating of the eardrum by incoming pulses; Bernoulli
anticipated the modern idea of the ear as a harmonic analyzer; d’Alembert
professed an ignorabimus on the matter. In their broader philosophy, Euler
and Lagrange anchored physics and music on mathematics; Bernoulli rather
adjusted mathematics to physics and music; d’Alembert defended a partial
autonomy of physics with respect to mathematics, and ridiculed the reduction
of music to mathematics.
A third conclusion of the present approach is that Fourier’s heat theory
depended on earlier, acoustics-driven harmonic analysis to a much larger extent
than is usually assumed. It is in the context of a discussion of Lagrange’s general
theory of vibrations that Fourier first expressed his enthusiasm for harmonic
analysis, ten years before he began to work on heat. His first discrete model of
heat propagation mimicked Lagrange’s theory of the discretely loaded string,
of which he was aware. His main argument for the generality of trigonometric series resembled Lagrange’s old reliance on the limit of a discrete model,
although he may not have read Lagrange’s relevant memoir of 1759. Lastly,
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O. Darrigol
Fourier was prompt to apply his new analysis to the vibrating string, with proper
homage to Bernoulli’s anticipations.3
In this light, it becomes probable that Lagrange’s reaction to Fourier’s memoir did not amount to the full-fledged rejection of Fourier series suspected by
many commentators. There is clear evidence that Lagrange believed that any
function could be in some sense developed in a trigonometric series over a
given interval. What he truly objected was the physical existence of the sine
components of string (and heat) motion, and presumably some of Fourier’s
alleged proofs of convergence. In essence his position remained the same as
in his old memoirs on sound, and his attitude toward Fourier’s theory must
have resembled his earlier attitude toward Bernoulli’s mixtures. Although he
acknowledged the mathematical validity of trigonometric developments, he
rejected their physical import. So to say, Fourier combined Bernoulli’s physical understanding of partial modes with Lagrange’s algebraic understanding
of their mathematical properties to obtain a far-reaching and better-founded
harmonic analysis.
Section 1 of this paper is devoted to seventeenth-century notions of harmonics and harmony in music, and to vibration theory in the first half of the
eighteenth-century. Section 2 revisits the quarrel over vibrating strings, with a
focus on the status of Bernoulli’s mixtures of sine curves. Section 3 analyzes
Lagrange’s memoirs on sound and their impact on this quarrel. Section 4 contains a summary of the views of the main string theorists on musical theory, in
connection with their views on harmonic analysis. Section 5 is devoted to the
genesis of Fourier’s heat theory in the light of earlier acoustics.
1 Harmony in early acoustics
1.1 Harmonic sounds
The earliest musicians must have been aware of the existence of harmonious
musical intervals. The expression of these intervals by ratios of small integers is a
Pythagorean notion, probably based on the corresponding lengths of the vibrating string on a monochord. The interpretation of sound as a vibration also goes
back to antiquity, although its more precise expression waited early-modern
mechanical philosophy.4
In the seventeenth century, Galileo Galilei and Marin Mersenne popularized
the correspondence between pitch and frequency that Giovanni Battista Bened3 About the origin of Fourier’s trigonometric solutions, Grattan-Guinness (Ref. 2, 441) writes: “It
first arose by accident…(and also perhaps from Fourier’s knowledge of eighteenth-century superposition of special solutions).” Dhombres and Robert (Ref. 2, 171) note Fourier’s early familiarity
with the quarrel of vibrating strings and with eighteenth-century simple-mode analysis.
4 Cf. Hendrik Floris Cohen, Quantifying music: The science of music at the first stage of the scientific
revolution, 1580–1650 (Dordrecht, 1984), Chaps. 1–4; Sigalia Dostrovsky, “Early vibration theory:
Physics and music in the seventeenth century,” AHES, 14 (1975), 169–218, on 169–174; Patrice
Bailhache, Une histoire de l’acoustique musicale (Paris, 2001), Chaps. 1–3; Truesdell, Ref. 2, 15–16.
The acoustic origins of harmonic analysis
347
etti, Galileo’s father Vincenzo, and Isaac Beeckman had earlier articulated.
Accordingly, a musical tone is a periodic succession of pulses transmitted by the
air to the eardrum. The number of pulses in a time unit determines the pitch of
the tone. Two tones are said to be consonant if and only if their frequency ratio
is reducible to the ratio of two small integers. Galileo justified this definition by
arguing that the frequent coincidence of the pulses striking the eardrum was a
source of delight:5
The length of strings is not the direct and immediate reason behind the
forms of musical intervals, nor is their tension, nor their thickness, but
rather, the ratio of the numbers of vibrations and impacts of air waves
on our eardrum, which likewise vibrates according to the same measure
of time. This point established, we may perhaps assign a very congruous
reason why it comes about that among sounds different in pitch, some
pairs are received in our sensorium with great delight, others with less,
and some strike us with great irritation. Hence the first and most welcome
consonance is the octave, in which for every impact that the lower string
delivers to the eardrum, the higher gives two [and so forth].
Marin Mersenne’s monumental Harmonie universelle of 1636 relied on a similar explanation of consonance as the coincidence of pulses, although Mersenne
realized that the simplicity of a frequency ratio did not necessarily correspond
to consonance as agreed by musicians. Nearly a century elapsed before Rameau, Euler, and Tartini truly improved on this explanation. In his discussion
of string instruments, Mersenne nonetheless described one the basic facts of
Rameau’s later theory: that under quiet condition and with proper experience
he and many musicians could hear at least four tones at a time from the thickest
strings of a viola bass: the natural tone, the octave above, the twelfth, and the
seventeenth:6
The string struck and sounded freely makes at least five sounds at the same
time, the first of which is the natural sound of the string and serves as the
foundation for the rest….These tones follow the ratio of the numbers 1, 2,
3, 4, 5…. They follow the same progression as the jumps of the trumpet.
This phenomenon puzzled Mersenne, for he could not see any counterpart in
the observed motion of strings:
5 Galileo Galilei, Discorsi e dimostrazioni matematiche: Intorno à due nuoue scienze attenenti alla
mecanica i movimenti locali(Leiden, 1638), English by Stillman Drake, Two new sciences (Madison,
1974), 104. Cf. Cohen, Ref. 4, Chaps. 3–4; Dostrovsky, Ref. 4, 174–183; Bailhache, Ref. 4, 76–89.
Mersenne inaugurated the use of the word “frequency” in this context. I use “pulse” as a uniform
translation of the French “battement” or “coup” and of the Latin “pulsus” or “ictus.” While most
authors understood sound propagation in analogy with water waves, Beeckman traced it to the
production of numerous air corpuscles at each maximum of the velocity of the sonorous body: cf.
Cohen, Ref. 4, 120–121.
6 Marin Mersenne, Harmonie Universelle. Traité des instrumens à chordes (Paris, 1636), book 4,
Proposition 11, quoted in Truesdell, Ref. 2, 32. On the shortcomings of the coincidence theory, cf.
Cohen, Ref. 4, 95.
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O. Darrigol
[Since the string] produces five or six tones…, it seems that it is entirely
necessary that it beat the air five, four, three, and two times at the same
time, which is impossible to imagine unless one says that half of the string
beats the air twice, one third beats it three times, etc. while the whole
strings beats it once. This picture runs against experience, which clearly
shows that all parts of the string make the same number of returns in the
same time, because the continuous string has a single motion, even though
parts near the bridge move more slowly.
Mersenne went on to speculate than some peculiar reaction of the air to the
beats of the string might explain the higher tones. More eloquent was his moral
corollary:7
If the sound of each string is the more harmonious and agreeable as a
higher number of different sounds is emitted at the same time and if it
is permitted to compare the moral actions to the natural ones and to
transpose physics to human actions, we may say that each action is the
more harmonious and agreeable to God as it is accompanied with a larger
number of motivations as long as those are all good.
At the turn of the seventeenth and eighteenth centuries, Joseph Sauveur
argued for the very picture of harmonic emission that Mersenne had judged
impossible:
While meditating on the phenomena of sound, I was made to observe that
especially at night one may hear from long strings not only the principal
sound, but also other small sounds, a twelfth and a seventeenth above….
I concluded that the string in addition to the undulations it makes in its
entire length so as to form the fundamental sound may divide itself in
two, three, four, etc. undulations which form the octave, the twelfth, the
fifteenth of this sound.
Sauveur obtained the additional modes of vibration by plucking a monochord
after placing a light obstacle at a simple fraction of the length of the string.
To his surprise, the string did not move appreciably for a sequence of equidistant points which he called “nodes” in allusion to the theory of the moon
(see Fig. 1). The number of nodes determined the order of the overtone, which
Sauveur called “harmonic” since it was harmonious with the fundamental. As
he soon found out, John Wallis had already obtained the higher modes of a
string by resonance with another string tuned to sound one of the harmonics of
the former. Sauveur was nonetheless first to exploit these modes to elucidate
Mersenne’s mystery of simultaneous overtones.8
7 Mersenne, Ref. 6, 210–211, partially quoted in Dostrovsky, Ref. 4, 194.
8 Joseph Sauveur, “Systême général des intervalles des sons, et son application à tous les systêmes
et à tous les instrumens de musique,” MAS (1701), 403–498, on 405–406. Cf. Truesdell, Ref. 2,
121–122; Dostrovsky, Ref. 4, 206–209. The sort of resonance discovered by Wallis has much smaller
amplitude than resonance excited by a sub-multiple of the natural frequency of the resonating
The acoustic origins of harmonic analysis
349
Fig. 1 Sauveur’s observation
of the higher modes of a
vibrating string (Ref. 8, 477)
Mersenne and Sauveur of course knew that flutes, trumpets and other blown
instruments produced overtones when blown stronger, as this is the way in
which higher notes are obtained on these instruments. In this case too, Sauveur
believed that the corresponding aerial motion had nodes of negligible motion.
In his opinion, harmonics could be heard and obtained from any musical instrument, or any “resonating and harmonious bodies.” The restriction to “harmonious” bodies probably means that he was aware that the overtones of most
vibrating bodies, for instance those of a vibrating blade, do not form a harmonic
sequence.9
That Sauveur discovered the higher modes of a vibrating string while meditating on simultaneous harmonics is a clear indication that he regarded the
actual motion of a string as some sort of superposition of the fundamental and
the higher modes. According to Bernard de Fontenelle, Sauveur conceived the
matter as follows:10
A harpsichord string being plucked, a good trained ear may hear, besides
the sound that corresponds to its length, thickness, and tension, sounds
higher than that of the whole string, produced by some of its parts and
somehow emerging from the principal vibration to form particular vibrations. This complication of vibrations can be conceived through the example of a loose rope attached by its extremities, as the rope of dancers.
Indeed, as the rope-dancer gives a big shake to the rope, with his two
hands he can give two separate shakes to the two halves of the rope; the
two halves being thus determined, one can still give a shake to each one,
etc. Thus every, half, third, and quarter of a string of an instrument has its
own vibrations during the total vibration of the whole string.
string, which explains why it was not discovered earlier. The multiple modes of a vibrating string
occurred in a then well-known string instrument called Trompette marine, which may have inspired
Sauveur’s experiments: cf. Dostrovsky, Ref. 4, 194–196, 205; Sauveur, Ref. 8, 406, 483–484. Here
and elsewhere, I use the words “mode” and “overtone” anachronistically, just to avoid long circumlocutions.
9 Sauveur, Ref. 8, 482–483.
10 Fontenelle (after Sauveur), “Sur l’application des sons harmoniques aux jeux d’orgues,” HAS
(1702), 119–122, on 120–121.
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O. Darrigol
The existence of harmonic overtones played a central role in Sauveur’s
conception of musical harmony. In his view, two combined tones were most
harmonious when one was a harmonic of the other. For other combinations he
agreed with Galileo and Mersenne that harmony had to do with the frequency
of the coincidences of the periodic pulses of the two tones. When the ratio of the
frequencies was not a simple one, this frequency could be so low as to produce
the sensation of a modulated amplitude, a phenomenon that organ-makers had
long used to tune their pipes, under the name of “beats.” Sauveur explained
dissonance by the unpleasant character of beats. Moreover, he showed that the
beat rate was the difference of the frequencies of the superposed tones and used
this result to perform some of the first measurements of the absolute frequency
of musical tones.11
In a memoir of 1702 on organ theory, Sauveur showed than organs were built
to imitate the harmonics produced by musical instruments:
By the mixture of its stops, the organ does nothing but imitating the
harmony that nature observes in the sounding bodies that are called harmonious; for one can identify the harmonics 1. 2. 3. 4. 5. 6. in bells and in
the long strings of a harpsichord at nighttime.
A stop is the name organ makers give to a series of pipes that can be selectively
connected to a keyboard of the organ by pulling a stop (literally). There are
simple stops in which each key controls only one pipe and all pipes are of a similar kind with length varying according to the pitch. There are compound stops
in which a single key controls several pipes whose characteristic frequencies are
harmonics of the same frequency. Moreover, the organist can pull several stops
that are harmonically related to each other. The previous quotation shows that
Sauveur had a notion of timbre, or sound color, corresponding to the harmonic
composition of the sound. He compared the organist to a painter mixing colors
on his palette to reproduce natural hues or to a chef, “who likes his stews milder
or spicier.”12
Sauveur’s clear and concise memoirs defined the basis of eighteenth-century
acoustics, a word he coined to name “a superior science of music…having as its
object sound in general, while music has as its object sound to the extent that
it is pleasant to the ear.” He systematized the known correspondence between
frequency and pitch by denoting tones through their relation to a reference
frequency. He introduced the logarithmic scale of intervals. As we saw, he deepened the understanding of harmonic overtones, he explained beats and used
them to explain dissonance, and he related timbre to harmonic composition.13
11 Fontenelle (after Sauveur), “Sur la détermination d’un son fixe,” HAS (1700), 182–195, on
194–195; “Sur un nouveau systême de musique,” HAS (1701), 159–180, on 159–162. Beeckman
anticipated both the explanation of beats and their relation to dissonance in his unpublished journal: cf. Cohen, Ref. 4, 144–145.
12 Sauveur, “Application des sons harmoniques à la composition des jeux d’orgue,” MAS (1702),
424–451, on 450–451, 445–446. Cf. Dostrovsky, Ref. 4, 208–209.
13 Sauveur, Ref. 8, 404. Cf. Dostrovsky, Ref. 4, 201–204, 206–209. As we know, the observation on
bells could not have been accurate.
The acoustic origins of harmonic analysis
Fig. 2 Superposition of
fundamental and octave in the
pulse-coincidence view
351
amplitude
time
There are striking similarities between Sauveur’s acoustics and the modern acoustics based on Fourier analysis. For Sauveur, the sounds produced by
musical instruments correspond to complex periodic vibrations that are mixtures of simpler periodic vibrations whose frequencies are multiples of the
fundamental frequency. The ear is able to analyze these mixtures into their
components. The fundamental frequency defines the pitch of the sound, and
the harmonic composition defines the timbre. What is missing is the sine character of the simple modes of vibration. Sauveur, like Galileo and Mersenne,
regarded sound as a succession of pulses whose precise shape mattered little to
the ear. What mattered was the order or disorder of the succession of pulses.
To him a simple sound (pure tone) was a succession of similar pulses, whereas
a compound sound involved pulses of different sizes. Figure 2 represents the
kind of motion he may have had in mind for the superposition of a simple tone
and its octave. In this view, the harmonic structure of the compound vibration
is apparent, just as the partial oscillations of the rope of Sauveur’s dancer. The
ear’s ability to separate the harmonics of a sound simply corresponds to the
ear’s ability to detect successive pulses and their magnitude, as would happen
with the beats of a drum.
1.2 Harmonic motion
The first intimation that harmonic (sine-like) motion plays a basic role in acoustics is found in Christiaan Huygens’s theory of musical strings. In his celebrated
Horologium Oscillatorium of 1673, Huygens showed that the pendulous motion
of a body sliding down on a cycloid was harmonic and isochronous. Around that
time, he also understood that the force responsible for this motion was proportional to the distance traveled by the body from the point of equilibrium. Probably noticing that a similar circumstance held in the case of a tense, weightless
elastic string loaded with one mass in the middle, he derived the oscillation
frequency as a function of tension, length, and mass. The reasoning implied
harmonic oscillations for the loaded string. He also sketched a generalization
to a string loaded with several masses, in which he assumed all the masses to
perform harmonic oscillations of the same frequency and phase.14
14 Christiaan Huygens, Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (Paris, 1673), part 2, Proposition 25; MS fragments, in Oeuvres
complètes, vol. 18, 489–495. Cf. Truesdell, Ref. 2, 47–49.
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O. Darrigol
Harmonic oscillations also occur in Isaac Newton’s derivation of the velocity
of sound. Like Huygens, Newton relied on analogy with pendulous oscillations,
although any other assumption on the form of the oscillations would have led to
the same propagation velocity. In substance, he computed the compression of a
slice of elastic fluid owing to the pendulous oscillations of the delimiting planes,
and adjusted the phase difference (associated with propagation) so that the
resulting pressure gradient would agree with the restoring force of a pendulum
mimicking the oscillation of a material plane of the fluid.15
More relevantly, in 1713 Brook Taylor gave the first theory of the fundamental mode of a vibrating string. He first showed that the resultant of the tensions
acting on the extremities of an element of the string was proportional to the curvature of this element and directed along its normal. By dubious reasoning, he
then predicted that for small vibrations the restoring force was approximately
directed toward the axis and proportional to the distance from the axis. Granted
that Newton’s acceleration law applies to mass elements, this implies that all
the elements of the string perform synchronous harmonic oscillations. The proportionality between restoring force and distance from the axis can only be true
if the string has the shape of a sine at any time, since in this approximation
the curvature is given by the second derivative of the distance from the axis as
a function of the abscissa along it. Taylor overlooked the possibility of higher
modes, despite Wallis’s earlier description of them. Worse, he believed that the
simple mode of motion was the only possible motion, apparently misled by the
observation that all the points of a vibrating string reach the axis at the same
time. Of course, he could not have written the differential equation of vibrating
strings, since the calculus of partial differentials did not exist yet. Instead he
relied on the pendulum analogy and on some intuition of the desired motion.
This sufficed to yield the fundamental mode as well as the correct frequency
formula
ν = (1/2l) T/σ ,
(1)
where l is the length of the string, T its tension, σ its mass per unit length.16
In 1727 Johann Bernoulli pioneered the study of a weightless string loaded
with equidistant, discrete masses, presumably to avoid the then intractable
problem of differentials involving two continuous variables. Unfortunately, he
followed Taylor in assuming that the restoring forces were proportional to the
distances from the axis and in overlooking higher modes of motion. His son
15 Cf. Dostrovsky, Ref. 4, 211–218; Truesdell, “The theory of aerial sound, 1687–1788,” EO2:13,
XIX-LXXII, on XXXII-XXXIII. The observation that the form of the oscillation does not matter
is found in Lagrange’s second memoir on sound, discussed below.
16 Brook Taylor, “De motu nervi tensii,” Royal Society of London, Philosophical transactions,
28 (1713), 26–32; Methodus incrementorum directa et inversa (London, 1715), 88–93. Ibid. on 90,
Taylor fallaciously reasoned that any departure of the curvature from proportionality with the distance from the axis would promptly be corrected by the resulting excess or defect of acceleration.
Cf. Truesdell, Ref. 2, 129–132; John Cannon and Sigalia Dostrovsky, The evolution of dynamics:
Vibration theory from 1687 to 1742 (New York, 1981), 15–20.
The acoustic origins of harmonic analysis
353
Daniel was first, in 1733, to cash the benefits of discretization in the problem
of the small oscillations of a vertically suspended chain. Unlike his father, he
realized that complex oscillations were possible with no well-defined frequency.
He nonetheless selected the initial conditions so that the restoring force on each
mass should be proportional to its distance from the vertical axis. In the case
of a finite number of masses hanging on a weightless inextensible thread, he
found as many simple modes as there were masses. He then proceeded to the
continuous, uniform limit in which the shape of the chain is given by our Bessel
functions. In this case, there are infinitely many simple modes with incommensurable frequencies increasing with the number of intersections with the vertical
axis (Sauveur’s nodes). Lastly, Daniel Bernoulli understood that the case of a
chain of infinite length formally agreed with that of a musical string and he
pointed to the experimental confirmation of higher modes in this case.17
In 1742 Daniel Bernoulli solved the more difficult problem of vibrating elastic bands, and again found a series of simple modes with incommensurable
frequencies growing with the number of nodes. In one of the experiments he
performed with a clamped band, he found that the sounds of two different
modes were heard simultaneously:
Both sounds exist at once and are very distinctly perceived…. This is no
wonder, since neither oscillation helps or hinder the other; indeed, when
the band is curved by reason of one oscillation, it may always be considered as straight in respect to another oscillation, since the oscillations
are virtually infinitely small. Therefore oscillations of any kind may occur,
whether the band be destitute of all other oscillation or executing others
at the same time. In free bands, whose oscillations we shall now examine,
I have often perceived three or four sounds at the same time.
For the first time, Daniel Bernoulli here gave a theoretical justification for the
superposition of modes already assumed by Sauveur in the case of vibrating
strings. The argument was necessarily more physical than mathematical, since
no partial differential equation of motion could yet be written. Clearly, it is the
hearing of the sounds of several modes that prompted Bernoulli to imagine
superposition. That he did not reach the same idea in his earlier discussion of
simple modes is not surprising: in the suspended-chain case, the visually perceived motion is indecipherably complex; in the related vibrated-string case the
hearing of simultaneous overtones is more difficult than in the elastic-band case
for which the overtones are dissonant.18
17 Johann Bernoulli, “Theoremata selecta, pro conservatione virium virarum demonstranda et
experimentis confirmanda, excerpta ex epistolis datis ad filium Danielem, 11 Oct. and 20 Dec.
(stil. nov.) 1727,” CAP, 2 (1727), 200–207, also in Opera omnia, vol. 3, 124–130; Daniel Bernoulli,
“Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae,”
CAP, 6 (1732–1733), 108–122 [1740]; “Demonstrationes theorematum suorum de oscillationibus
corporum filo flexibili connexorum et catenae verticaliter suspensae,” CAP, 7 (1734–1735), 162–173
[1740]. Cf. Truesdell, Ref. 2, 132–136, 154–162; Cannon and Dostrovsky, Ref. 16, 47–51, 53–69. Euler
obtained similar results soon after Daniel Bernoulli: cf. Truesdell, Ref. 2, 162–165.
18 D. Bernoulli, “De vibrationibus et sono laminarum elasticarum commentationes physico-
mathematicae,” CAP, 13 (1741–1743), 105–120 [1751]; “De sonis multifariis quos laminae elasticae
354
O. Darrigol
At that stage of the history of acoustics, Daniel Bernoulli had in hand the
notion that bodies vibrate according to discrete modes of harmonic motion, or
to a superposition of such modes. The frequencies of these harmonic modes
are generally incommensurable. In the special case of vibrating strings, they are
integral multiples of the same fundamental frequency. It is not clear whether
Bernoulli already believed the most general motion to result from a superposition of discrete modes, since at that time no one had yet solved the general
initial-condition problem for a vibrating system (not even in the two-body case).
As we will see in a moment, Bernoulli’s later claim for such generality resulted
from competition with other investigators of the vibrating string.
2 The quarrel over vibrating strings
2.1 D’Alembert’s and Euler’s breakthroughs
In 1746 Jean le Rond d’Alembert obtained the equation of vibrating strings
simply by combining Taylor’s expression of the restoring force with Newton’s
acceleration law. For the mathematics, he relied on the notion of differential
form for a function of several variables, earlier developed by Leonhard Euler
and Alexis Fontaine. Treating the ordinate y of a string element as a function
of the time t and the curvilinear abscissa s, d’Alembert formed the successive
differentials:
dy = pdt + qds,
dp = αdt + νds,
dq = νdt + βds.
(2)
If T denotes the tension of the string and σ its mass per unit length, the approximate force acting on a string element is Tβds, its mass σ ds, and its acceleration
α. The resulting equation of motion is σ α = Tβ, or
σ
∂ 2y
∂ 2y
=
T
∂t2
∂s2
(3)
in modern notation. Having chosen the unit of time so that α = β, d’Alembert
astutely formed
dp + dq = (α + ν)(dt + ds) and
dp − dq = (α − ν)(dt − ds),
(4)
whence he concluded that p + q was a function of t + s only, and p − q a function
of t − s only.19 Equivalently, he had
p = (t + s) + (t − s) and
q = (t + s) − (t − s).
(5)
diversimode edunt disquisiones mechanico-geometricae experimentis acusticis illustratae et confirmatae,” ibid., 167–196, on 173–174, cited in Truesdell, Ref. 2, 197. Cf. ibid. 192–199; Burkhardt,
Refs. 2, 6; Cannon and Dostrovsky, Ref. 16, 83–92. A year later, Euler provided a more thorough
but less physical treatment of the same problem: cf. Truesdell, Ref. 2, 219–222.
19 I have corrected a probable slip in d’Alembert’s paper.
The acoustic origins of harmonic analysis
355
Integrating dy = pdt + qds then leads to the general solution20
y = (t + s) + (t − s).
(6)
The functions and are restricted by the boundary conditions. That the
ordinate y constantly vanishes at the ends s = 0, l of the string implies the
relations
= −,
and
(t + l) = (t − l)
at any t.
(7)
Therefore, the solution depends on a single generating function , which must
be periodic of period 2l and defined over the whole real axis. The initial conditions
y(0, s) = Y(s)
and
ẏ(0, s) = V(s)
(8)
determine the function (up to a constant). Namely, it must the periodic
function of period 2l such that
1
1
(±s) = ± Y(s) +
2
2
Vds
for 0 ≤ s ≤ l.
(9)
The former considerations require the functions to be twice differentiable.
In the intervals 0 < s < l and −l < s < 0 and in all similar intervals, this will
be true if the function Y(s) is twice differentiable and the function V(s) is once
differentiable. At s = 0, agreement of the right and left derivatives requires
V(0) = 0 and Y (0) = 0. Lastly, agreement of the right derivatives at s = l and
the left derivatives at s = −l requires V(l) = 0 and Y (l) = 0. These conditions are automatically satisfied if the boundary conditions and the equation of
motion hold true.21
Besides these legitimate restrictions, d’Alembert required that the function
y(s, t) should be given by a single equation, valid even for the non-physical
values of s. In doing so he was merely following the Leibnizian tradition of
restricting mathematical analysis to “continuous functions” in the old sense,
that is, functions that can be expressed by a single equation (algebraic or transcendent) whose range of validity defines the maximal range of the variable.
These functions roughly correspond to our analytic functions, since the knowledge of their value for any small, finite range of the variable determines their
value for the whole range of the variable. Exceptions to this correspondence are
functions such as x2/3 that do not have a Taylor development everywhere. In the
20 Jean le Rond d’Alembert, “Recherches sur la courbe que forme une corde tendue mise en
vibration,” HAB (1747), 214–249. Cf. Burkhardt, Ref. 2, 11–13; Truesdell, Ref. 2, 237–244.
21 This is a drastic simplification of d’Alembert’s considerations. In his lengthy discussion, he tolerated infinite derivatives and he seems to have confused conditions that only applied to the case
V = 0 with the more general conditions of consistency.
356
O. Darrigol
sequel, quotations marks around “continuous” are used to indicate continuity
in the old sense.22
Through the formula (6), Leibnizian continuity implies that the functions
Y(s) and V(s) should be periodic odd function of period 2l. This restriction
did not annihilate d’Alembert’s main conclusion that the problem of vibrating
strings admitted infinitely more solutions than that given by Taylor. After showing that Taylor’s solution was a particular case of his for which the generating
function was a sine function, d’Alembert rejected Taylor’s claim that any initial
condition would promptly lead to the sine form of the string. At the same time,
he admitted that his method did not allow for the initial condition of plucked
strings, that is, a triangular shape with no initial velocity. In this case, he suggested approximating the continuous string by a massless elastic thread loaded
with a large number of discrete massless string.23
After reading d’Alembert’s memoir, Euler hurried to publish a theory based
on the same wave equation and the same general integral (6), without the
restriction that the generating curve and the initial curve of the string should be
given by a single equation. In the case of a string initially at rest, he argued that
the generating function could easily be constructed from the initial curve of the
string, even if this curve was “irregular and mechanical.” In modern terms, he
tolerated any continuous curve with piecewise continuous slope and curvature.
The non-existence of the partial derivatives entering the wave equation did not
worry him as long as the relation between solution and generating function
had a well-defined geometric meaning. He later prided himself of covering the
case of the plucked string, for which the initial shape is triangular. We may
retrospectively judge that d’Alembert abusively restricted the solutions to be
analytic, while Euler tolerated solutions for which the differentials in the wave
equation did not acquire a well-defined meaning until modern distribution theory. Elements of the ensuing quarrel will only be mentioned to the extent that
they have to do with the status of harmonic solutions.24
22 Cf. Andreas Speiser, “Über die diskontinuierlichen Kurven,” EO1:25 (1952), XXII–XXIV;
Truesdell, Ref. 2, 247–248; Jerome Ravetz, “Vibrating strings and arbitrary functions,” in The
logic of personal knowledge: Essays presented to Michael Polyani on his seventeenth birthday (London, 1961), 71–88; Jesper Lützen, “Euler’s vision of a general partial differential calculus for a
generalized kind of function, Mathematics magazine, 56 (1983), 299–306; Adolf Yushkevich, “The
concept of function up to the middle of the nineteenth century,” AHES, 16 (1976), 37–85. Analytic
functions in the modern sense must have a Taylor development at every point of their domain of
analyticity.
23 D’Alembert, Ref. 20, 226–230, 244–247.
24 Euler, “De vibratione chordarum exercitatio,” Nova acta eruditorum (1749), 512–517, also
in EO2:10, 50–62. Cf. Burkhardt, Ref. 2, 13–14; Truesdell, Ref. 2, 244–250. On the quarrel, cf.
Burkhardt, Ref. 2, 14–18; Rudolf Langer, “Fourier’s series: The genesis and evolution of a theory,”
The American mathematical monthly, 54 (II), 1–86; Truesdell, Ref. 2, 273–281, 286–295; GrattanGuinness, The development of the foundations of mathematical analysis from Euler to Riemann
(Cambridge, 1970), chap. 1; Umberto Bottazini, The higher calculus: A history of real and complex
number analysis from Euler to Weierstrass(New York, 1986), 21–33. On the plucked string, see Euler
to d’Alembert, 20 Dec 1763, in EO4A:5, and Euler, “De chordis vibrantibus disquisitio ulterior,”
NCAP, 17 (1772), 381–409, also in EO2:11, 62–80, commented in Truesdell, Ref. 2, 289–290.
The acoustic origins of harmonic analysis
357
As Euler noted, the periodicity of the generating function implies that
√ the
motion of the string is a periodic function of time with the period 2l σ/T
already given by Taylor. The frequency ν of the oscillation is multiplied by n,
Euler went on, in the singular cases in which the curve of the string is made of an
alternating sequence of n similar parts separated by n − 1 nodes. A particular,
analytic case of this kind of solution is the product sin(nπ x/l) cos nπ νt, which
corresponds to the simple modes already described by Daniel Bernoulli. Euler
also gave the superposition
y=
an sin(nπ x/l) cos nπ νt,
(10)
which he believed to be analytic (whether the sum be finite or not) and therefore
unable to represent the general solution.25
2.2 Bernoulli’s response
In 1753 Daniel Bernoulli published a lengthy, polite, but angry reaction to
d’Alembert’s and Euler’s contributions to the problem of vibrating string. In
his opinion, he had indicated the true physical solution years earlier and the
sophisticated mathematics of his competitors had only obscured the subject:26
I saw at once that one could admit this multitude of curves [for the vibrating string according to d’Alembert and Euler] only in a sense altogether
improper. I do not less admire the calculations of Messrs. d’Alembert and
Euler, which certainly include what is most profound and most advanced
in all of analysis, but which show at the same time that an abstract analysis,
if heeded without any synthetic examination of the question proposed, is
more likely to surprise than enlighten. It seems to me that giving attention
to the nature of the vibrations or strings suffices to foresee without any
calculation all that these great geometers have found by the most difficult
and abstract calculations that the analytic mind has yet conceived.
Bernoulli argued that according to both experience and theory any sonorous
body could vibrate in a series of simple modes with a well-defined frequency of
oscillation. In the particular case of the vibrating strings, the various modes were
obtained by juxtaposition of Taylor modes, and their frequency was a multiple
of the fundamental frequency. As he had earlier indicated, these modes could be
25 Euler, Ref. 24, 61–62. At that time, calculus with trigonometric functions was still a novelty,
to which Euler much contributed, cf. Victor Katz, “The calculus of the trigonometric functions,”
Historia mathematica, 14 (1987), 311–324.
26 Daniel Bernoulli, “Réflexions et éclaircissements sur les nouvelles vibrations des cordes
exposées dans les mémoires de l’académie de 1747 & 1748,” HAB, 9 (1753), 147–172, on 148;
“Sur le mêlange de plusieurs espèces de vibrations simples isochrones, qui peuvent coexister dans
un même système de corps,” ibid., 173–195. Cf. Burkhardt, Ref. 2, 17–19; Truesdell, Ref. 2, 254–259
(citation on 255).
358
O. Darrigol
superposed to produce more complex vibrations. The rest of his argumentation
depended on two assertions:
1. D’Alembert’s and Euler’s supposedly new solutions are nothing but “mixtures” of simple modes.
2. The “aggregation” of these partial modes in a single formula is incompatible
with the physical character of the decomposition.
Bernoulli did not have a direct mathematical proof of the first point. Instead,
he showed that with his superposition he could reproduce the basic periodicity
properties of the solutions of d’Alembert and Euler. He also abundantly argued
that for a massless string loaded with a finite number of point masses, any initial
configuration of the masses could be obtained by the superposition of simple
modes.27
In favor of the reification of partial modes evoked in assertion (2), Bernoulli
adduced that the sounds produced by partial vibrations could be heard separately. Namely, “any musician agrees” that long musical strings emit the sounds
corresponding to the four first modes of vibration. Also, different sounds can
travel through the same space and still be heard distinctly, as happens when we
listen to an orchestra. In conformity with this analyzability of complex sounds,
Bernoulli believed that the superposition of simple modes had an observable
fine structure. In the extreme case of a fundamental superposed with its thousandth harmonic, he described the resultant corrugation and the thousand oscillations it made during a single oscillation of the string as a whole. He also argued
that owing to the faster decay of higher modes, only Taylor’s fundamental mode
could be seen in actual experiments. Lastly, he emphasized that d’Alembert’s
and Euler’s periodic solutions with arbitrary curvature only existed if the frequencies of the simple modes were commensurable. As he knew, this condition
is not even met for strings of uneven thickness. Whenever it is broken, the only
way to form the general solution is the superposition of simple modes.28
To sum up, Daniel Bernoulli’s belief in the generality and superiority of the
superposition of simple modes in part depended on the wide-spread opinion
that pure harmonic “isochronous” oscillations were simpler than any others,
could be produced separately, and enjoyed unique physical properties. They
also implied four unproven assumptions: that the vibrations of a continuum
have the same basic structure as the vibrations of a discrete system; that the sensorial analysis of a phenomenon into separate components reveals an intrinsic
structure of the phenomenon; that the cases of commensurable and incommensurable simple modes are on the same footing; and that the superposition of
simple modes preserved some structural features of the individual modes even
in the case of an infinite number of modes. As for Euler’s conflicting contention
27 Daniel Bernoulli, Ref. 26, 151–152, 173–187. Ibid., on 189, he admitted that for the similar
problem of the suspended chain he had earlier believed that only the simple modes displayed some
regularity. He now perceived the harmonic simplicity behind the apparent complexity of the more
general motions.
28 Ibid., 152 (harmonics), 153 (orchestra), 155–156 (corrugation), 158 (decay of higher modes), 159
(incommensurable frequencies).
The acoustic origins of harmonic analysis
359
that the sums of sine curves were not the most general possible curves of a
string, Daniel Bernoulli admitted “that he was not yet quite enlightened on this
point” since he had no mathematical proof of the contrary.29
Far from confining himself to the problem of vibrating strings, Bernoulli
generalized his considerations to the motion of any mechanical system in the
vicinity of (stable) equilibrium:
What I have just said on the nature of the vibrations of bodies attached to
a stretched string I do no hesitate to extend to all small reciprocal motions
that can occur in Nature, providing these are set up by a permanent cause.
For every body that is somewhat displaced from its point of rest will tend
toward that point with a force proportional to the small distance from the
point of rest: and then if we suppose an arbitrary system of bodies, each
body will be able to form as many simple regular vibrations as there are
bodies in the system, and these simple vibrations will be able to coexist at
the same time in the given system.
Bernoulli made this superposition of harmonic components a basic principle of
nature:
It seems that Nature very often acts by the mere principles of the imperceptible, isochronous [i.e., harmonic], and infinitely diversified vibrations
to produce a great number of phenomena.
In a contemporary letter, he similarly wrote:
I admire…the hidden physical treasure that natural motions which seem
subject to no law may be reduced to the simple isochronous motions which
it seems to me Nature uses in most of its operations. I am convinced even
that the inequalities in the motions of the heavenly bodies consist in two,
three, or more simple reciprocal motions.
Bernoulli dared include light in this cosmic vision:30
A mass of luminous matter is a system made of an infinite number of parts,
or globules, and each globule can be subjected at the same time to an infinite number of simple, isochronous vibrations that never fuse together nor
trouble each other. One can thus conceive that the same ray of light may
primitively include all possible colors; because the different colors probably are nothing but different perceptions in the organ of sight, caused
by the different simple vibrations of the celestial [i.e., ethereal] globules.
It is certain that in the same mass of air a great number of vibrations
can be formed at the same time, very different from each other, each of
which separately causes a different sound in the organ of hearing. This
29 Ibid., 157. Bernoulli used the word “isochronous” as synonymous to pendulous or harmonic,
which prompted misunderstanding on the part of his readers.
30 Ibid., 173, 187–188, 188–189 (light); Daniel to Johann III Bernoulli, undated, cited in Truesdell,
Ref. 2, 257–258.
360
O. Darrigol
idea seems to me very fit to explain the different refractions, the different
vivacities, and all other phenomena indicated by Mr. Newton on primitive
colors. But this is so rich a matter that it can only be treated in the occasion
of another theory.
2.3 The controversy over Bernoulli’s mixtures
In his prompt reaction to Bernoulli’s memoirs, Euler praised his colleague for
having best developed the “physical part” of the problem of vibrating strings
but denied the generality and superiority of the multi-modes solution. His
arguments were of two kinds. Firstly, he rejected Bernoulli’s claim that the
superposition of simple modes always preserved the structure of the partial
modes:
When the number of terms becomes infinite, it seems doubtful that the
curve is composed of an infinite number of sine curves: The infinitity seems
to destroy the nature of the composition.
To illustrate this point, Euler gave the particular case
y=
∞
n=1
∞
α n sin(nπ x/l) =
1 n iπ nx/l
(α e
− α n e−iπ nx/l )
2i
α sin(π x/l)
=
,
1 − 2α cos(π x/l) + α 2
n=0
(11)
in which the last expression “gives a much simpler idea of the curve than we
would have if we saw it as composed of an infinity of Taylorian sine curves.”
A modern reader would rather take this as an indication that Fourier synthesis
is able to produce a much greater variety of functions than appears at a first
glance. Euler himself admitted that “if Mr. Bernoulli’s consideration provided
all the curves that may occur in the motion of strings, it would certainly be
infinitely preferable to our method, which could then only be regarded as an
extremely thorny detour to reach a solution so easy to find.”31
Euler went on to argue that the superposition of sine curves could not produce the most general string shape:
All the curves comprised in this equation [Bernoulli’s], even though the
number of terms is increased to infinity, have certain characters that distinguish them from all other curves. For if we take a negative abscissa x,
the ordinate also becomes negative and equal to that which corresponds
to the positive abscissa x; similarly, the ordinate that corresponds to the
abscissa l + x is negative and equal to that which corresponds to the
31 Euler, “Remarques sur les mémoires précédens de M. Bernoulli,” HAB, 9 (1753), 196–222, also
in EO2:10, 232–254, on 234–235. Cf. Burkhardt, Ref. 2, 20–24; Truesdell, Ref. 2, 259–261 (with a
slip in the formula p. 260).
The acoustic origins of harmonic analysis
361
abscissa x. Therefore, if the curve that has been given to the string at
the beginning does not have these properties, it is certain that it is not
comprised in the said equation. Now no algebraic curve can have these
properties, which must therefore all be excluded from this equation; there
is no doubt that an infinite number of transcendental curves must also be
excluded.
Euler was evidently right in asserting that Bernoulli’s equation yielded periodic
and odd functions. More surprisingly for the modern reader, he believed that
the coincidence of such a function with the initial string curve, which is only
defined over the interval 0 ≤ x ≤ l, required this curve to enjoy the same
properties. The reason is that in his eyes a sum of sine functions was necessarily
analytic, so that coincidence with another analytic function within the interval
0 ≤ x ≤ l necessarily implied full identity. Implicitly but surely, he lent to the
limit of a sequence of functions the analyticity properties of the general term
of this sequence.32
In addition to this purely mathematical objection, Euler criticized some of
Bernoulli’s physical arguments. To the claim that the hearing of harmonics justified harmonic decomposition, he opposed that the nth harmonic of a string
was not the signature of the sine curve sin(nπ x/l). It could just as well be produced by any curve made of a succession of similar loops separated by nodes.
To Bernoulli’s evocation of the differential damping of modes, he answered
that no extraneous circumstance should be introduced in judging the mathematical character of the solution of an ideal problem. To Bernoulli’s claim that
the problem of the non-uniform vibrating string only admitted the multi-mode
solution, he replied that the lack of a solution of the kind he had provided for
the uniform string should only be imputed to the imperfection of contemporary
analysis.33
D’Alembert’s first response to Bernoulli’s views is found in the article “Fondamental” he wrote in 1757 for the Encyclopédie. There he acknowledged the
facts regarding the hearing and making of harmonic sounds, and he briefly
explained how the excellent Jean Philippe Rameau had based a theory of harmony on them. He next gave two reasons to reject Bernoulli’s exploitation of
these facts as justifying the multi-mode analysis of vibrating strings. Firstly, the
mathematical superposition of a higher mode with the fundamental could not
be regarded as a physical superposition, because the nodes associated with the
higher mode are not at rest as they would be if the higher mode existed alone.
32 Euler, Ref. 31, 236–237. Contrary to a well-spread misreading, Euler’s objection did not contradict his tolerance for non-analytic string curves. What was at stake was the analyticity of trigonometric sums. As we will see in a moment (below, p. 1428–1439), in the same period Euler became
aware of trigonometric series with periodic discontinuities that should have forced him to drop the
objection.
33 Euler, Ref. 31, 252, 236, 254. In 1762, Euler provided the desired sort of solution for a non-
uniform strings with simple departures form uniformity: see below, p. 1540–1545.
362
Fig. 3 The superposition
y = sin x cos t + 12 sin 2x cos 2t
for x = 1 (solid line) and its
two harmonic components
(dotted lines). Half of the
extrema of the superposition
do not coincide with extrema
of the harmonics
O. Darrigol
1.5
1.0
0.5
0.0
2
4
t/π
6
–0.5
–1.0
Secondly, d’Alembert claimed that harmonics could be heard even when the
string oscillated in its fundamental mode.34
Both arguments require some unpacking. The motion of nodes in the superposition of the fundamental with a higher mode was no news to Bernoulli, who
himself wrote that the fundamental mode was a moving axis for the higher
mode. However, he and d’Alembert differed in their perception of this fact.
For Bernoulli, the ambient air and the ear responded separately to the primary
vibration of the axis and to the secondary vibration around the axis. For d’Alembert, the displacements of the ambient air followed those of the string, with
reversals occurring at each extremum of the ordinate of a part of the string.
Since these extrema do not have the periodicity of the higher modes and may
depend on the abscissa, they cannot explain the hearing of the corresponding
harmonic (see Fig. 3).35
As for d’Alembert’s hearing of harmonics even for “simple oscillations,”
it is easily understood by noting that his concept of simplicity differed from
Bernoulli’s. Whereas for the latter simplicity meant harmonicity, for d’Alembert (as for Euler) simplicity meant the absence of nodes and the simultaneous
passing of every point of the string through its axis. Under this criterion, a
musical string plucked in the usual manner performs a simple oscillation.
In a letter published in 1758 in the Journal des sçavans, Bernoulli defended
his sums of sine curves, mostly against Euler contention that they could not
reproduce every proposed curve. Owing to the infinite number of adjustable
coefficients in the sum, he argued, “one may cause the final curve to pass through
as many given points as one wishes and thus identify this curve with the one
proposed, to any degree of precision.” He did not worry about Euler’s remark
that sums of sine curves were analytic whereas the proposed string curve in
general was not. In his view, such utterances derived from the doubtful appli34 D’Alembert, “Fondamental,” in d’Alembert and Denis Diderot (eds.), Encyclopédie, 7 (1757).
Owing to his usual antipathy to d’Alembert, Truesdell (Ref. 2, 262) misinterpreted d’Alembert’s
statement about moving nodes.
35 D’Alembert, Opuscules mathématiques, 1 (Paris, 1761), 58–60, 63–65; 4 (1768), 154. D’Alembert
reasoned in the case of two harmonic components only. He later generalized the objection to any
number of harmonics.
The acoustic origins of harmonic analysis
363
cation of mathematical continuity to a physical problem. He was satisfied that
his method worked rigorously in the case of a weightless string loaded with a
finite number of mass points.36
In a nasty footnote to his memoir of 1762 on organ pipes, Bernoulli accused
his critics of espousing a “metaphysics…in which nx · nx is not xx in the case
when n is absolutely zero” and traced their objections to their ignorance of the
fact that “physical beings cannot be composed of absolutely vanishing parts.”
In the main text, he emphasized that organ pipes, like vibrating strings, could
emit several harmonics at the same time, in conformity with his view that the
most general motion was a superposition of sine curves. He also insisted that
only one tone was heard when only one mode of oscillation occurred in the
tube or string, probably in answer to d’Alembert’s contradictory claim.37
D’Alembert and Euler stuck to their guns. In the first volume of his Opuscules, published in 1761, the former approved the latter’s criticism of Bernoulli
and added that a sum of sine curves necessarily had continuous curvature and
therefore could not reproduce the most general curve. Indeed the two rivals
shared the conviction that the limit of a sequence of functions had the same
properties of continuity and analyticity as the general term of the sequence.
D’Alembert acknowledged that this argument only made Bernoulli’s solutions
less general than Euler’s and that it still allowed that his own solutions, being
analytic, odd, and periodic functions, could all be represented as sums of sine
curves. He nonetheless excluded this possibility, for he was convinced that his
own “reentering curves” could not generally be expressed as sums of sines. At
most, he was willing to admit trigonometric series as “approximations” not as
“geometrical, exact, and rigorous solutions.” In later years, he came to regard
sin5/3 (π x/l) as a possible equation for the initial curve of a vibrating string, and
brandished it as a non-Bernoullian solution. For he believed that every trigonometric series had a power series development (since its terms did separately),
whereas sin5/3 (π x/l) does not have any around x = 0.38
36 Daniel Bernoulli, “Lettre de Monsieur Daniel Bernoulli, de l’Académie Royale des Sciences,
à M. Clairaut de la même Académie, au sujet des nouvelles découvertes faites sur les vibrations
des cordes tendues,” Journal des sçavans (March 1758), 157–166. Cf. Truesdell, Ref. 2, 262. In his
response to Bernoulli, Euler (Ref. 31, 236) had conceded that the infinite number of indeterminate
coefficients in Bernoulli’s sums of sine curves seemed to enable them to reproduce an arbitrary
curve.
37 Daniel Bernoulli, “Recherches physiques, mécaniques et analytiques, sur le son et sur les tons
des tuyaux d’orgues différemment construits,” MAS (1762), 431–485, on 442n, 441–442.
38 D’Alembert, “Recherches sur les vibrations des cordes sonores,” Opuscules mathématiques,
1 (1761), 1–64, 65–73 (supplément), on 38 (reentering curves), 42 (approving Euler), 46 (continuous curvature), 47 (approximations); “Nouvelles réflexions sur les vibrations des cordes sonores,” ibid., 4 (1768), 128–155, on 153–155 (more against Bernoulli); “Premier supplément au
mémoire précédent,” ibid., 156–179, on 169 (introducing sin5/3 ); “Second supplément au mémoire
précédent,” ibid., 180–199, on 192 (sin5/3 against Bernoulli). On the latter point, cf. Truesdell, Ref.
2, 287. Both Euler and Lagrange had given proofs that any analytic, periodic function could be
expressed as a trigonometric sum: Euler, “De serierum determinatione seu nova methodus inveniendi terminos generales serierum,” NCAP, 3 (1750–1751), 36–85, also in EO1:14, 463–515, on
470–473; Lagrange, Ref. 64, 514–516; cf. Burkhardt, Ref. 2, 48–50, and below p. 1248–1253.
364
O. Darrigol
D’Alembert traced Bernoulli’s alleged error to two basic misconceptions.
Firstly, he reproached Bernoulli with “having too lightly concluded from the
finite to the infinite”: the true generality of Bernoulli’s solution in the discrete
case did not imply its generality in the continuous case. Secondly, d’Alembert
rejected Bernoulli’s physical interpretation of this solution as a “mixture” of
real, partial vibrations. In conformity with his earlier criticism in the “Fondamental” article, he argued at great length that the partial vibrations “generally
referred to a curved axis” determined by other modes and thus lost the periodicity properties they had when produced in isolation. As he wrote in a later
duplication of this argument, “The claimed Taylorian multiple vibrations only
exist in idea and have no more reality than they would in a string at rest.”39
In the fourth volume of his Opuscules (1768), d’Alembert replied to
Bernoulli’s defense of the passage from the discrete to the continuous, by arguing that the admission of nx · nx = xx for n = 0 amounted to the confusion
between a rectangle and an infinite line. D’Alembert was less suspicious of
limiting processes when they served his polemic purposes. In what he hoped
to be the final blow to Bernoulli’s theory, he argued that the incommensurability of the frequencies of the simple modes in the discrete case excluded the
ϕ(x + ct) + ϕ(x − ct) form of the solution even when the number of discrete
masses reached infinity.40
3 From the discrete to the continuous
3.1 Euler’s discretely loaded cord
In his study of 1747 on vibrating strings, d’Alembert indicated that the similar problem of a longitudinally vibrating string could provide a model for the
propagation of sound. The following year, Euler published a brilliant memoir
based on a discrete version of this idea. The basic model is a tense elastic cord
loaded with n equal point masses evenly spaced between its fixed extremities.
The general equations of motion—the first to be written for a problem of this
kind—read:
ẍk = α 2 (xk+1 − 2xk + xk−1 ),
(12)
where xk is the displacement of the kth mass from its equilibrium position, α 2 is
the ratio of the elasticity constant of the cord to the mass of the loads, and
x0 = xn+1 = 0 holds at the boundaries. Euler first sought solutions of the form
xk = ak cos ωt.
(13)
39 D’Alembert, “Recherches,” Ref. 38, 45, 58–61; “Extrait de différentes lettres de M. d’Alembert
à M. de la Grange,” HAB, 19 (1763), 235–255 [dated 11 Jun 1769]. On the latter, cf. Truesdell,
Ref. 2, 288. Some of d’Alembert’s objections resulted from his misunderstanding of Bernoulli’s
idiosyncratic use of “isochronous” (meaning harmonic).
40 D’Alembert, “Nouvelles réflexions,” Ref. 38, 154, 175–178.
The acoustic origins of harmonic analysis
365
The equation of motion then implies the relations
(2α 2 − ω2 )ak = α 2 (ak+1 + ak−1 ),
(14)
which reminded Euler of the trigonometric identity
2 sin kϕ cos ϕ = sin(k + 1)ϕ + sin(k − 1)ϕ.
(15)
Taking into account the boundary condition a0 = 0, the solution can be written
as
ak = sin kϕ,
with 2α 2 − ω2 = 2α 2 cos ϕ, or ω = 2α sin
ϕ
.
2
(16)
The boundary condition an+1 = 0 further requires
ϕ=
rπ
,
n+1
with r = 1, 2, . . . n.
(17)
This means that the loaded string has n simple modes of motion with the frequencies
ωr = 2α sin
rπ
.
2(n + 1)
(18)
Euler obtained the general solution of the equations of motion by superposing
simple modes in the manner:41
xk =
n
cr sin
r=1
krπ
cos ωr t.
n+1
(19)
The generality of this solution depends on the possibility of reproducing any
initial set of values Xk of the displacements by a suitable choice of the coefficients cr . As Euler was only interested in the propagation of a pulse for which
all the Xk s vanish except X1 , he only sought the coefficients in this case. By
induction from the cases n = 1, 2, 3 he found
rπ
2
sin
n+1
n+1
(20)
kπ
n+1
krπ
sin
=
δr1
n+1
n+1
2
(21)
cr = X1
as well as the identity
n
k=1
sin
41 D’Alembert, Ref. 20, 248. Euler, “De propagatione pulsuum per medium elasticum,” NCAP,
1 (1747–1748), 67–105, also in EO2:10, 98–131. Cf. Burkhardt, Ref. 2, 24–27; Truesdell, Ref. 2,
229–234.
366
O. Darrigol
following which the injection of (20) into (19) leads to the desired initial condition xk (0) = X1 δk1 .
Euler’s remarkable finding easily generalizes to any initial condition. A
proper generalization of identity (21),
n
k=1
sin
ksπ
n+1
krπ
sin
=
δrs ,
n+1
n+1
2
(22)
which we would now describe as the orthogonality of the simple modes, leads
to the expression
2 srπ
Xs sin
n+1
n+1
n
cr =
(23)
s=1
of the coefficients.
For the abovementioned pulses, Euler inferred a value of the speed of propagation in the cases n = 2, 3. He did not investigate the continuum limit, for he
wanted to find a result different from Newton’s value for the speed of sound,
which had long known to be 15% below the truth. One of Euler’s conclusions
interestingly contradicted Daniel Bernoulli’s acoustics:
While the motion of one particle [only one load] is oscillatory, the motion
of two or more particles no longer is and differs more and more from it
as the number of particles increases; consequently, sound cannot at all be
understood as propagated through the air as some able men would have
it when they assert that when a string or other sounding instrument is
set in motion there are in the air particles of this sort which take on an
oscillatory motion and excite the organ of hearing.
Euler thus refused to regard a propagating pulse as a superposition of simple
modes of oscillation, even though he was first to mathematically derive propagation through a superposition of oscillation modes. His comment also reveals
antipathy with Bernoulli’s idea that hearing involves harmonic analysis. As we
will see in a moment, he adhered to the old concept of hearing as the perception of rhythms in series of pulses, in conformity with his paper’s emphasis on
pulses.42
42 Euler, Ref. 41, 99, cited in Truesdell, Ref. 2, 230. This comment anticipates Euler’s later argument
(see below, p. 1301–1308) that trigonometric series could not properly represent states of motion
of a vibrating string for which only one part of the string departs from equilibrium. Euler may also
have been targeting d’Ortous de Mairan’s theory of sound propagation (1720), according to which
the particles of air contained as many resonant strings as there were sounds to be propagated.
The acoustic origins of harmonic analysis
367
3.2 Lagrange’s first memoir on sound
In 1759, the young and ambitious Joseph Louis Lagrange published a long memoir on the nature and propagation of sound whose brilliance impressed both
d’Alembert and Euler. While Lagrange seems to have been unaware of Euler’s
memoir on the propagation of pulses, he had carefully studied the memoirs on
vibrating strings by d’Alembert, Euler, and the Bernoullis. He was especially
receptive to two of d’Alembert’s suggestions, that sound propagation could be
treated in analogy with the problem of vibrating strings, and that for arbitrary
(non-analytic) initial conditions the latter problem could only be treated by
taking the limit of a loaded massless string when the number of loads reaches
infinity.43
The mathematical core of Lagrange’s memoir was the solution of the general
initial value problem for the uniformly and discretely loaded string or for the
equivalent problem of a discrete succession of elastic air slices. He recovered
Euler’s equations of motion (12), now applied to the successive ordinates yk of
the loads:
ÿk = α 2 (yk+1 − 2yk + yk−1 ),
(24)
with the boundary conditions y0 = yn+1 = 0. Whereas Euler directly based his
solution on simple sine modes and on the orthogonality of these modes, Lagrange applied a general method of resolution of a system of linear differential
equation of any order that he borrowed from d’Alembert.44
In modern terms, the method consists in replacing the original system by
a higher number of linear differential equations of first order and in diagonalizing the implied linear operator. In Lagrange’s notation, this means that he
formed linear combinations M1 y1 +M2 y2 +· · ·+Mn yn that were proportional to
their second time derivative. Once the corresponding differential equations are
solved for given initial conditions, there remains to determine the yk s from the
various combinations. As he did not anticipate the orthogonality of the various
choices of the vectors (M1 , M2 , . . . Mn ), Lagrange relied on some abstruse trigonometry. His result nonetheless agrees with the one Euler would have obtained
for arbitrary initial conditions. In the case for which the initial velocities vanish
and the initial ordinates have the values Ys , the solution is
43 Joseph Louis Lagrange, “Recherches sur la nature et la propagation du son,” MT, 1 (1759),
1–112, also in LO1, 39–148, on 44n; d’Alembert, Ref. 20, 246–248. Cf. Burkhardt, Ref. 2, 27–34;
Truesdell, Ref. 2, 263–271. D’Alembert only regarded the densely and discretely loaded string as
a plausible approximation to the real problem. Lagrange nonetheless read him as suggesting that
the limit of infinitely many loads would yield the exact solution.
44 Lagrange, Ref. 43, 72–90; d’Alembert, “Recherches sur le calcul intégral. Quatrième partie:
Méthodes pour intégrer quelques équations différentielles,” HAB(1747), 275–291, on 283–291. In
his memoir on vibrating strings (Ref. 20, 247), d’Alembert had given the equations of motion (22)
for the case n = 2, and indicated that the general problem could be solved by his general method
for integrating systems of linear differential equations, which can be traced to the par. 101 of his
Traité de dynamique(Paris, 1743).
368
O. Darrigol
2 srπ
krπ
Ys sin
sin
cos ωr t.
n+1
n+1
n+1
n
yk =
n
(25)
r=1 s=1
Lagrange discussed the behavior of this solution as a superposition of simple modes of incommensurable frequencies, and praised Daniel Bernoulli for
anticipating the results:45
One cannot overestimate the sagacity and penetration of this celebrated
Geometer, who, through a purely synthetic investigation of the proposed
question, succeeded in reducing to simple and general laws motions that
seemed to resist it by their very nature.
Lagrange next took the limit in which the number n of loads reaches infinity.
In this limit, the ratio kl/(n + 1) gives the abscissa x of a point of the string,
and the spacing l/(n + 1) gives the element dx. As the coefficient α is inversely
proportional to this spacing, Lagrange assumed
ωr = 2α sin
rπ
rπ
→ ωr =
c,
2(n + 1)
l
(26)
√
where c is a constant with the dimension of velocity ( T/σ , if T denotes the
tension of the string and σ its mass per unit length). The limit of the solution
(25) then reads
∞
2
y(x) =
l
l
dX Y(X) sin
r=1 0
rπ X
rπ x
rπ ct
sin
cos
.
l
l
l
(27)
rπ x
rπ ct
rπ X
sin
cos
,
l
l
l
(28)
Lagrange rather wrote
2
y(x) =
l
l
dX Y(X)
0
∞
r=1
sin
no doubt because his subsequent considerations involved an evaluation of the
purely trigonometric sum appearing in this expression.46
The formula (27), which Lagrange did not exactly write, gives the motion of
the string as an infinite sum of simple modes. For t = 0, it degenerates into Fourier’s celebrated formula for the sine-series development of a function defined
over the interval [0, l] and vanishing at the extremities of this interval. Against
45 Lagrange, Ref. 43, 95.
46 In a contemporary paper, Euler took the same limit in an incorrect manner: cf. Truesdell, Ref. 2,
271–273. Lagrange systematically wrote dx instead of dX in Eq. (28), probably because he regarded
the relevant integral as the limit of a sum rather than the inverse of a derivation. In his comment,
he made clear that X was the true integration variable.
The acoustic origins of harmonic analysis
369
the opinion of earlier commentators, I will argue that Lagrange truly had in hand
both the simple-mode analysis of the vibrations of a continuum and the Fourier
decomposition of a not-well defined but intently large class of functions, surely
implying non-analytic functions and functions with non-continuous derivatives
(in the modern sense).47
Firstly, I must brush away the wide-spread objection that Fourier performed
an illicit permutation of sum and integral when passing from Eq. (27) to
(28). This criticism is based on an anachronistic reading of the latter equation. Lagrange’s subsequent developments clearly show that by this formula he
meant what we would now write as
⎛
2
y(x) = lim ⎝
n→∞
l
l
dX Y(X)
0
n
r=1
⎞
rπ x
rπ ct ⎠
rπ X
sin
cos
sin
.
l
l
l
(29)
This expression is trivially identical to Fourier’s formula for t = 0.48
Another widespread objection to Lagrange’s priority is that Lagrange only
used Eq. (28) as an intermediate, insignificant step in reasoning meant to justify
Euler’s solution to the vibrating-string problem, not to confirm Bernoulli’s simple-mode analysis. Surely, Lagrange did not dwell on this step and failed to see
its potentialities for a new analysis à la Fourier. It remains true, however, that
he believed in the mathematical truth of the relevant equation including the
special case t = 0 that gives Fourier’s fundamental formula, for Y functions as
general as Euler’s string curves. Otherwise, his whole train of reasoning would
fall apart.
Granted that Lagrange believed in the infinite superposition (27), one may
still wonder whether he actually proved it. D’Alembert and later commentators
argued that the substitution (26) for the frequencies ωr was illegitimate, since for
values of r that are close to n the argument rπ/2(n + 1) of the sine function did
not vanish when n became infinite. Lagrange’s reply to this criticism consisted
in remarking that the equation of the discrete problem of which the frequencies
ωr are the solutions tends toward an equation of which the numbers rπ c/l are
the solutions. This reply fails to address the proper question, which is whether
the limit of the discrete formula (25) truly yields the continuous formula (27).
However, Lagrange’s questionable step does not interfere with the derivation
of Fourier’s formula, since the latter only requires the case t = 0. Moreover, it
can be shown that the terms of the series for which r/n is non-negligible actually
do not contribute to the sum of the series.49
47 Lagrange nevertheless excluded polygonal string shapes for which the wave equation (3) does
not hold at the origin of time: “Addition aux premières recherches sur la nature et la propagation
du son,” MT, 2 (1760–1761), also in LO1, 317–332, on 331–332.
48 See e.g. Burkhardt, Ref. 2, 28.
49 Lagrange, Ref. 47, 319–322. Lagrange seems to have understood the latter point in later years:
see his Mécanique analytique,
2nd edn., 2 vols. (Paris, 1811), LO11, 422. For instance, one may
√
retain only the first n terms of the series. The error committed in this partial sum by confusing ωr
370
O. Darrigol
Lagrange’s contemporary readers failed to detect a more genuine flaw in
Lagrange’s derivation of Eq. (28): the substitution of integrals for the discrete
sums. The typical error committed in replacing a sum of n values of a function
with the corresponding integral is of the order of 1/n. As there are as many
simple modes as there are discrete values of the X variable, the net error could
well be finite. Retrospectively, we can tell that when proper restrictions (for
instance the Dirichlet conditions) are applied to the function Y(X), this net
error vanishes. The intuitive reason is that owing to the oscillations of the integrands, the typical error for the r-mode is 1/rn, which yields a net error of the
order ln n/n . As Lagrange had no inkling of such considerations, his Eq. (28)
lacked a respectable proof even according to contemporary standards.
I now return to Lagrange’s use of the infinite superposition expressed in Eq.
(26). Thanks to the trigonometric identity
sin
rπ ct
1
rπ(x + ct)
rπ(x − ct)
rπ x
cos
=
sin
+ sin
,
l
l
2
l
l
(30)
this equation can be rewritten as
y(x, t) = ψ(x + ct) + ψ(x − ct)
(31)
if
1
ψ(ξ ) = lim
n→∞ l
l
Y(X)n (X, ξ )dX
(32)
0
wherein
n (X, ξ ) =
n
r=1
sin
rπ ξ
rπ X
sin
.
l
l
(33)
Through some laborious trigonometry, Lagrange found50
n =
sin
sin nπl ξ − sin nπl X sin
2 cos πlX − cos πlξ
(n+1)π X
l
(n+1)π ξ
l
.
(34)
√
with rπ c/l reaches zero for infinite n since the error in each term is of the order (1/ n)2 = 1/n;
and so does of course the sum of the remaining terms if the series is convergent.
50 I use the more general formula that Lagrange introduces in a later section of his memoir (Ref.
43, 112).
The acoustic origins of harmonic analysis
371
Implicitly retreating to the discrete case, he took (n + 1)X/l to be an integer,
which yields the simpler formula
ξ
sin π X sin (n+1)π
l
.
n = (−1)(n+1)X/l l
2 cos πlX − cos πlξ
(35)
Alleging that for infinite n the quantity (n + 1)ξ/l was always an integer,
Lagrange concluded that ∞ vanished whenever X ± ξ was not a multiple
of 2l. In the special case X = ξ , and remembering that for Lagrange (n + 1)X/l
is an integer, this expression reaches the value n + 1. Again Lagrange retreated
to the discrete problem, and identified the element dX in Eq. (21) with l/(n+1).
Consequently, for 0 ≤ ξ ≤ l, the equality ψ(ξ ) = Y (ξ ) holds. Outside this interval, the values of X that contribute to the integral differ from ξ or from −ξ by a
multiple of 2l. The sign of this contribution is such that the function ψ(ξ ) ends
up being the extension Ȳ(X) of the function Y(X) that is odd and periodic of
period 2l. Combined with Eq. (31), this remark leads to Lagrange’s final result
for the case of zero initial velocity:
y(x, t) =
1
[Ȳ(x + ct) + Ȳ(x − ct)].
2
(36)
Lagrange could have reached this result much faster by noting that by construction the integral ψ(ξ ) is odd and 2l-periodic. The equation
y(x, t) = ψ(x + ct) + ψ(x − ct)
(31)
and the initial condition y(x, 0) = Y(x) then give the desired result. Alas
the simplest route is rarely the first that comes to mind. Instead Lagrange computed the ψ integrals in the above manner, which perplexed most of his readers.
D’Alembert soon noted the absurdity of the statement that for infinite n the
quantity (n + 1)ξ/l = (n + 1)X/l ± (n + 1)ct/l was always an integer. Lagrange
replied that by construction (n + 1)X/l always was an integer and that a (continuous) function of time was perfectly determined if its value for multiples of
l/(n + 1)c were known for arbitrarily large n. He thus completed his retreat to
the discrete model. In substance, his exploration of the consequences of Eq. (28)
only was a duplication of its derivation through the discrete model, together
with the remark that for commensurable eigenfrequencies the quantities x ± ct
become the natural arguments.51
In the final Eq. (36) Lagrange immediately recognized the d’Alembert-Euler
solution of the problem of vibrating strings, without the restrictions imposed by
d’Alembert on the initial conditions:52
51 D’Alembert, “Recherches” (Ref. 38, supplement), 65–73. Lagrange, Ref. 47, 322.
52 Lagrange, Ref. 43, 107, cited in Truesdell, Ref. 2, 270.
372
O. Darrigol
There, then, is the theory of this great geometer [Euler] placed beyond all
doubt and established upon direct and clear principles that rest in no way
on the law of continuity [i.e., analyticity] which Mr. d’Alembert requires;
there, moreover, is how it can happen that the same formula that has
served to support and prove the theory of Mr. Bernoulli on the mixture of
isochronous [i.e., harmonic] vibrations when the number of moving bodies
is finite shows us the insufficiency of this theory when the number of these
bodies becomes infinite. Indeed the change that this formula undergoes in
passing from one case to the other is such that the simple motions which
made up the absolute motions of the whole system destroy each other
for the most part, and those which remain are so disfigured and altered
as to become absolutely unrecognizable. It is truly annoying that so ingenious a theory…is shown false in the principal case, to which all the small
reciprocal motions occurring in nature may be related.
Quite diplomatically, Lagrange managed to praise each of the protagonists
of the quarrel over vibrating strings: d’Alembert for confining his and Euler’s
method of resolution to analytic functions, Euler for giving the most general
solution, and Bernoulli for illuminating the discrete case. At the same time he
promoted himself as the geometer who first rigorously proved the pertinence
of Euler’s solution as well as the failure of Bernoulli’s intuitions in the continuous case. His proud announcement, in a contemporary letter to Euler, of
“the complete fall of the Bernoullian theory” has usually been mistaken for
a flat rejection of trigonometric series. Careful reading of the above citation
rather indicates that Lagrange regarded the formula (28) for the simple-mode
analysis of string motion as perfectly valid and even made it the very source of
the falsity of Bernoulli’s physical interpretation of this analysis. In the limit of
an infinite number of simple modes, he figured that what we would now call
destructive interference deprived the resultant oscillation of any resemblance
with the partial oscillations.53
As was earlier argued, Lagrange’s reasoning did contain the simple-mode
analysis of the vibrations of a continuum as well as Fourier’s fundamental
formula. However, Lagrange’s failure to provide a cogent proof and his simultaneous denial of the physical relevance of simple-mode analysis for a vibrating
continuum was not without consequence. He avoided trigonometric series in
most of his works on partial differential equations, and made a sparing use of
them in his work on celestial mechanics. None of his readers, not even Euler
or Bernoulli, saw that his memoir contained the integral expression of the
coefficients.
The rest of Lagrange’s memoir concerned the application of his calculations
to the one-dimensional propagation of sound. For this purpose, he considered
the superposition of simple modes for which, in the discrete case, the initial
ordinate Yk at the abscissa X = kl/(n + 1) is the only one that does not vanish.
53 Lagrange to Euler, 2 Oct 1759, LO14, 162–164.
The acoustic origins of harmonic analysis
373
In the continuous limit, he found
y(x, t) ∝
∞
sin
r=1
rπ x
rπ ct
rπ X
sin
cos
.
l
l
l
(37)
By his earlier reasoning on such sums, they only take non-zero values if x ± ct
differs from X by a multiple of 2l. From this result, Lagrange inferred that
sound propagated with the velocity predicted by Newton and echoed on solid
walls situated at x = 0 and x = l. He also argued that the linearity of his equations of motion allowed various pulses to cross each other without alteration,
so that “the air is able to transmit to the ear the impressions of several different
sounds without confusion.” He condemned Daniel Bernoulli’s explanation of
the same fact in terms of the superposition of simple modes, because the sort
of oscillations it implied was totally lacking in the case of pulse propagation.
As we will see, the propagation of pulses long remained a major obstacle to
Bernoulli’s views.54
3.3 Lagrange’s second memoir on sound
In a sequel published the following year, Lagrange acknowledged the
“extremely laborious and confusing” character of his passage from the finite to
the infinite, and proposed a “simpler method” intended to dissolve the doubts
that d’Alembert and Bernoulli had expressed in letters to him. The new method
consisted in a direct application of d’Alembert’s procedure of linear combination for solving systems of linear differential equations.55
Remember that in the discrete case Lagrange formed linear combinations
M1 y1 + M2 y2 + · · · + Mn yn such that the second-order finite differences occurring in the equations of motion would be proportional to their second timederivative. In the continuous case, he similarly formed the integrals
l
s=
M(x)y(x)dx
(38)
0
and chose the functions M(x) so that s would obey an equation of the type
s̈ + ω2 s = 0.
(39)
54 Lagrange, Ref. 43, 126–139, 141 (quote). Lagrange considered pulses of velocity rather than the
displacement pulses of my simplified account.
55 Lagrange, “Nouvelles recherches sur la nature et la propagation du son,” MT, 2 (1760–1761),
also in LO1, 149–316, on 159. Cf. Burkhardt, Ref. 2, 35–37. In his first letter to Lagrange (27 Sept
1759; LO13, 3–4), d’Alembert wrote: “I find it hard to believe that this solution necessarily requires
a so extensive apparatus of calculation.”
374
O. Darrigol
Since the equation of motion
2
∂ 2y
2∂ y
−
c
=0
∂t2
∂x2
(40)
implies
l
s̈ − c2
M(x)
0
∂ 2y
dx = 0,
∂x2
(41)
the desired behavior of s occurs if and only if
l
M(x)
0
∂ 2y
dx = −κ 2 s,
∂x2
with κc = ω.
(42)
Granted that y(0) = y(l) = 0 and M(0) = M(l) = 0, a double integration by
parts transforms this condition into
l
(M + κ 2 M)ydx = 0.
(43)
0
It is therefore sufficient that
M + κ 2 M = 0.
(44)
Mκ = sin κx
(45)
The solution
of this equation satisfies the boundary conditions M(0) = M(l) = 0 if κ is a
whole number r times π/l. Calling sr the corresponding value of s, and assuming that the velocity of the string vanishes at the origin of time, Eq. (39) now
implies
l
sr (t) ≡
rπ ct
rπ x
dx = cos
y(x, t) sin
l
l
0
l
Y sin
rπ x
dx,
l
(46)
0
which Lagrange transformed into
1
sr (t) =
2
l
0
1
rπ(x − ct)
dx +
Y sin
l
2
l
Y sin
0
rπ(x + ct)
dx.
l
(47)
The acoustic origins of harmonic analysis
375
Calling Ȳ(x) the odd, 2l-periodic function of x that coincides with Y(x) for
0 ≤ x ≤ l, we have
1
sr (t) =
4
=
=
1
4
1
2
l
−l
rπ(x − ct)
1
Ȳ(x) sin
dx +
l
4
l
Ȳ(x + ct) sin
−l
rπ x
1
dx +
l
4
l
Ȳ(x) sin
−l
rπ(x + ct)
dx
l
l
Ȳ(x − ct) sin
−l
l
[Ȳ(x + ct) + Ȳ(x − ct)] sin
rπ x
dx
l
rπ x
dx
l
(48)
0
Consequently, the function y(x, t) has the same sr coefficients as the function
1
2 [Ȳ(x + ct) + Ȳ(x − ct)]. Implicitly admitting that the list of these coefficients
(our Fourier coefficients) completely determine a function of x over the interval
[0, l], Lagrange thus recovered the Euler-d’Alembert solution of the problem
of vibrating string.56
Even though he now started with the partial differential equation (40), which
involves a second-order spatial derivative, Lagrange believed his derivation to
include “irregular” (non-analytic) initial conditions because multiplication by
M(x) of this equation and integration by parts absorbed the spatial derivations
of y. He thus anticipated a basic feature of modern distribution theory: the
introduction of test functions that absorb differential operators. Yet he did not
venture so far as tolerating a polygonal initial shape of the string, for he believed
the equation of motion to lose any meaning in this case.57
In the following sections of his memoir, Lagrange extended his method to
differential operators more complex than ∂ 2 /∂x2 . In modern terms, he determined the eigenfunctions and spectrum of a variety of differential operators,
and he solved equations of motion involving them by determining the projections of the solution on these eigenfunctions. He was only satisfied when he
could in the end obtain a “construction” of the differential equations of motion,
that is, a generic expression of the solution that was no longer reminiscent of this
decomposition, as in the Euler-d’Alembert solution. He noted the considerable
difficulty of the latter process, which he called the elimination of κ (the wave
numbers of the Fourier components).58
Lagrange’s new method eluded the expression of a function in terms of
its projections (generalized Fourier-coefficients) on the eigenfunctions of the
differential operator. Yet the difficult last section of his memoir, devoted to the
56 Lagrange, Ref. 55, 158–168. Lagrange’s reasoning is slightly more complex, owing to his failure
to introduce Ȳ, and also because he does not require the initial velocity to vanish.
57 Lagrange, Ref. 47, 331–332. Lagrange argued that an initial polygonal shape would promptly
evolve into one amenable to his mathematical analysis.
58 Lagrange, Ref. 55, 180: “This operation involves more considerable difficulties.”
376
O. Darrigol
theory of wind instruments, contained a solution of this problem including a
generalization of Fourier’s future theorem. Unfortunately, Lagrange’s idiosyncratic use of M multipliers and the additional complexity brought by the higher
dimensionality have so far prevented commentators from appreciating the profundity and originality of this section. In order to shed light on this matter, I will
restrict Lagrange’s argument to one dimension only, in which case the equation
of motion and the boundary conditions are the same as for a vibrating string.59
The basic problem is that of the motion of air in a one-dimensional cavity of
length l. Lagrange characterized the displacement y(x) of a particle of air from
its rest position x through the coefficients
l
sκ =
Mκ (x)y(x)dx,
with κ = π/l, 2π/l, 3π/l, . . .
(49)
0
In analogy with the discrete problem for which the number of different values
of κ is equal to the number of values that x can take, he assumed the existence
of the inverse linear relation
y(x) =
sκ Pκ (x).
(50)
κ
In order to determine the coefficients Pκ (x), Lagrange injected this formula
into the former one, which gives
sκ =
λ
l
Mκ (x)Pλ (x)dx.
sλ
(51)
0
This identity holds for any value of the sequence (sκ )κ if and only if
l
Mκ (x)Pλ (x)dx = δκλ .
(52)
0
Lagrange then showed that the function Pλ (x) such that
Pλ
l
2
+ λ Pλ = 0,
Pλ (0) = Pλ (l) = 0,
Pλ (x)Mλ (x) = 1
and
(53)
0
59 Ibid., Chap. 6: “Réflexions sur la théorie des instruments à vent,” 298–316, esp. 312–316, on
303–304 (orthogonality of eigenfunctions), 314–316 (generalized Fourier theorem).
The acoustic origins of harmonic analysis
377
met this condition. Indeed, the two first conditions and the definition (44) of
Mκ imply that
l
(κ 2 − λ2 )
l
Mκ (x)Pλ (x)dx =
0
Mκ (x)Pλ (x)dx −
0
l
Mκ (x)Pλ (x)dx = 0.
0
(54)
Although Lagrange conceived the M s as multipliers and the P s as simple
modes—and therefore noted them differently, he was well aware that the conditions (52) determined the P s uniquely as
Pλ (x) =
2
2
sin λx = Mλ (x).
l
l
(55)
Consequently, the form
y(x, t) =
κ
sκ (t)Pκ (x) =
κ
l
Pk (x)
Mκ (ξ )Y(ξ )dξ cos cκt
(56)
0
of the general solution of the wave equation (resulting from Eqs. 49 and 50) is
the one given by modern Fourier analysis.
The formula Lagrange actually wrote instead of (56) was more general,
since it covered any sound-wave motion in a three-dimensional cavity with
non-degenerate simple modes. The particular case t = 0 yields Fourier’s theorem in the case of one-dimension, and a more general theorem of harmonic
decomposition in higher dimensions. In modern terms, we would say that Lagrange exploited the mutual orthogonality of the simple modes. He derived this
property from what we would now call the Hermitian character of the spatial
operator in the wave equation. The only element lacking was a rigorous proof
of completeness for the system of eigenfunctions.60
At the end of these powerful developments, Lagrange again deplored the
physical impotence of the simple-mode analysis expressed in Eqs. (56) or (27):
“This construction is hardly of any use for knowing the motion of the particles of
air.” He gave an obscure reason for this failure: the terms of the infinite series
“did not converge or at least could not be regarded as convergent” because
the coefficients of the simple modes depended on the initial conditions, which
should be arbitrary.61
From the similar but more detailed argument later found in the second
edition of the Mécanique analytique, it becomes clear that Lagrange meant
Bernoulli’s composition of modes to be meaningful only if the first few modes
60 Ibid., 312–316.
61 Ibid., 316.
378
O. Darrigol
had amplitude much higher than the sum of the amplitudes of the remaining
modes. Presumably, he had computed the Fourier coefficients for a plucked
string (triangular shape) and found their sum to be divergent (they behave like
1/n for large n). He emphasized that in this case and for most initial conditions,
the theoretical motion of the string, as given by the d’Alembert-Euler formula,
did not at all resemble the motion obtained by superposing a couple of simple
modes.62
Lagrange nonetheless ended his second memoir on sound propagation with
the following praise of Bernoulli’s analysis:
This method [projection over simple modes] serves to demonstrate the
beautiful proposition of Mr. Daniel Bernoulli: When a [finite or infinite]
system of bodies undergoes infinitely small oscillations, the motion of each
body can be regarded as composed of several partial motions synchronous
to those of simple pendulums.
The contradiction with Lagrange’s previous considerations disappears if we
keep in mind that he distinguished between the mathematical truth and the
physical relevance of the superposition of simple modes.63
3.4 Lagrange’s later reflections on vibrating strings
In 1764 Lagrange obtained a new analysis of the problem of vibrating strings as
a particular case of the general method he had earlier invented for studying the
vibrations of an arbitrary system of bodies. He seized this opportunity to simplify his treatment of the discretely loaded string (thus getting closer to Euler’s
approach of 1748) and to argue, by intricate trigonometry, that the general
solution (25) could be transformed into a form similar to the d’Alembert-Euler
form for the continuous string:
kl
kl
− ct + ϕ
+ ct
n+1
n+1
k−1
k+1
l − ct − ψ
l − ct
+ψ
n+1
n+1
k−1
k+1
l + ct − ψ
l + ct
+ψ
n+1
n+1
yk = ϕ
(57)
Lagrange proudly announced to d’Alembert:
I have just completed a calculation which seems to me to shed great light
on this question [of vibrating strings]. I have found a way to construct, in
62 Lagrange, Ref. 49, 424.
63 Lagrange, Ref. 55, 316.
The acoustic origins of harmonic analysis
379
a general manner, the formula [in the discrete case] of my first Recherches
sur le son, and this construction is such that it degenerates into that of Mr.
Euler when the number of moving bodies is infinite.
As Clifford Truesdell rightly noted, the “construction” is flawed because the
coefficients of Lagrange’s power-series expressions for the Fourier coefficients
of ϕ and ψ also depend on time. However, in the continuous limit the ψ terms
disappear and the ϕ terms reproduce the d’Alembert-Euler solution because
their hidden time-dependence becomes negligible. Lagrange thus circumvented
the difficulties he had encountered in taking the continuous limit on the sum
of simple modes before doing the trigonometry that generates the x ± ct arguments.64
The reason for Lagrange thus avoiding infinite trigonometric sums may have
been Euler’s and d’Alembert’s contention that a trigonometric sum should be
analytic whereas the initial curve of a string needed not to be so. In 1759,
Lagrange had already expressed his sympathy for Euler’s judgment that sums
of sine curves could not possibly represent the most general shape of a tense
string, “owing to certain properties that seem to distinguish [such sums] from
other imaginable curves.” In a letter to d’Alembert of 20 March 1765, he found
it “hard to believe that Mr. Bernoulli’s solution… should be the only one that
occurs in nature.” He added that the phenomena of sound propagation, as
explored in his second dissertation, necessarily implied “discontinuous” [nonanalytic] functions. As we will see in a moment, Euler had made the same point
in memoirs he had recently sent to Lagrange for the Miscellanea Taurinensia.65
At the same time, Lagrange still believed that the trigonometric series
obtained in the discretely loaded string problem retained some validity in the
limiting case of a continuous string. Following his new derivation of Euler’s
construction, he introduced the sibylline distinction between requiring two
curves to be identical and requiring their difference to be smaller than any
given quantity. He believed that the initial string curve and the corresponding
trigonometric series were related in this second manner only:66
It is clear that, whatever is the initial curve [of the string], one can always
pass a curve of the form y = α sin π x + β sin 3π x + · · · through infinitely
many points which are infinitely near to this initial curve in such a manner that the difference between the two curves be as small as one wishes,
although this difference can only be absolutely zero in the case for which
the initial curve has also the same form; in any other case this initial curve
will only be a sort of asymptote which the generating curve [the odd,
2l-periodic extension of the initial curve] will approach at infinity without
ever completely coinciding with it.
64 Lagrange, “Solutions de différents problèmes de calcul intégral,” MT, 3 (1765), also in LO1,
471–668, on 534–551; Lagrange to d’Alembert, 1 Sep 1764, LO13, 12–14, Truesdell, Ref. 2, 278.
65 Lagrange Ref. 42, 70; Lagrange to d’Alembert, 20 Mar 1765, LO13, 37–38; Euler to Lagrange,
16 Feb 1765, LO14, 205–207.
66 Lagrange, Ref. 64, 552.
380
O. Darrigol
There followed a suspicious argument based on the identity
krπ
2 srπ
sin
,
Ys sin
n+1
n+1
n+1
n
Yk =
n
(58)
r=1 s=1
which results from the simple-mode analysis for t = 0 (compare with Eq. 25).
Lagrange inferred from it that the continuous function
2 srπ
f (x) =
Ys sin
sin rπ x
n+1
n+1
n
n
(59)
r=1 s=1
was a continuous interpolation of the discrete function k/(n + 1) → Yk . He
next showed that for any given function α(x)
n
n
1 srπ
2
s
sin
sin rπ x
(60)
g(x) =
Ys α
n + 1 α(x)
n+1
n+1
r=1 s=1
was another interpolation of the same discrete function. Lagrange meant this
trivial non-uniqueness of continuous interpolations of discrete data to justify his
statement that trigonometric series could only “asymptotically” approach the
function they purport to represent. The argument boils down to the following
platitude: for any finite value of n, the trigonometric sum that coincides with a
given function for a sequence of n equidistant values of the variable has zero
probability to represent this function exactly on the relevant interval.67
More interesting is the notation that Lagrange used in this context. Instead
of the familiar interpolation formula (59), he wrote
y = 2 Y sin Xπ dX sin xπ + 2 Y sin 2Xπ dX sin 2xπ + · · ·
+2 Y sin nXπ dX sin nxπ
(59 )
with the strange convention dX = 1/(n + 1), X = s/(n + 1) (the pseudointegrals are taken between 0 and 1). Clearly, he had in mind the limit n → ∞
for which discrete sums become integrals. Yet he shied off taking this limit since
he stopped the series at finite n. Otherwise he would have reached Fourier’s
fundamental formula in a very simple (but non-demonstrative) manner. His
reluctance to send n to infinity probably resulted from his suspicion that something funny happened in this limit. At a time when conditions of integrability
and the distinction between uniform and non-uniform limits did not exist, Lagrange could only remain in the murk. Moreover, his algebraic conception of
67 Ibid., 552–554.
The acoustic origins of harmonic analysis
381
calculus deprived him of the tools that we now know to be necessary to discuss
the pertinence of trigonometric series.68
There is no mention of Lagrange’s special notion of asymptotic convergence
in his later writings. In 1769, answering one of d’Alembert’s many letters on
vibrating strings, he described the “violent revulsion” he had developed for this
subject as a consequence of working too long on it. Plausibly, his reticence also
derived from his desire to preserve his friendship with both d’Alembert and
Euler. The rivalry between the two geometers had soured so much that it had
become impossible to agree with one without irritating the other. Lagrange
waited the second edition of his Mécanique analytique, published in 1811, for a
last significant return to the vibrating string. Surprisingly, he reproduced his first
treatment of this problem, arguing that it was “not out of place in this treatise,
because it led directly to the rigorous solution of one of the most interesting
questions of mechanics.” Perhaps he also wanted to set the record straight after
Fourier had used similar considerations in the domain of heat propagation.69
Lagrange’s ultimate judgment on the value of simple-mode analysis agreed
with his earliest pronouncements on this matter:
Although [the formula (27)] rigorously gives the motion of the string at
any time t, the infinite series that enter this formula prevent it to represent
this motion in a clear and sensible manner.
Again, he explained that the harmonic sounds heard from a single vibrating
string could not be traced to partial modes:
The series that could give the different sounds disappears from the formula
when the number of bodies is infinite, and the result is, for every point of
the string, a simple and uniform law of isochronism which immediately
and simply depends on the initial state.
To sum up, from his first theory of sound propagation to the end of his life,
Lagrange kept an ambivalent stance about simple-mode superposition and
trigonometric series. On the one hand, he admitted their mathematical ability to represent the most general motions and functions, if only in a mysterious
“asymptotic” manner. On the other hand, he denied them any physical meaning
in the continuum case.70
68 As we will see in a moment, Alexis Clairaut already had the trigonometric interpolation formula
in 1757 as well as its continuous limit. Poisson, in his “Second mémoire sur la propagation de la chaleur dans les solides,” JEP, 19 (1823), 249–509, on 444–446, ignored Lagrange’s prevention to take
the continuous limit and judged that Lagrange’s interpolation formula (59 ) was “the first general
formula” for the trigonometric development of an arbitrary function. On Lagrange’s style, cf., e.g.,
Craig Fraser, “Joseph Louis Lagrange’s algebraic version of the calculus,” Historia mathematica, 14
(1987), 38–53.
69 Lagrange to Euler, 15 Jul 1769, LO13, 138; Lagrange, Ref. 49, 440–441. Cf. Truesdell, Ref. 2,
289, 295n. In the first edition, Mécanique analytique (Paris, 1788), Lagrange gave a new derivation
of the solution for the discretely loaded string (pp. 300–314), and briefly the equation and the
d’Alembert-Euler solution for the continuous string (p. 334).
70 Lagrange, Ref. 49, 425, 436.
382
O. Darrigol
3.5 Lagrange and other string theorists
D’Alembert’s reception of Lagrange’s first dissertation on sound started a long
friendship spiced up with some disagreement on vibrating strings. In his letter of
thanks, d’Alembert assorted his praise of Lagrange’s treatment of the discrete
case with a rejection of the limiting process that led to the continuous string: “As
for the method through which you pass from an indefinite number of vibrating
bodies to an infinite number, I do not find it as demonstrative as you claim.”
D’Alembert detailed his criticism two years later, in the first volume of his
Opuscules. As was already mentioned, his two main objections concerned the
limit of the frequency of the simple modes and Lagrange’s cancellation of any
cosine that had an infinite argument. On the physical side, d’Alembert doubted
that the limit of the discrete case should yield the truly observed behavior of a
uniform string. Mathematics could only predict the latter behavior in the case
when the initial shape was given by an equation, and “the rest should be left to
physics.”71
As Lagrange’s reply failed to satisfy him, d’Alembert persisted in his rejection of Lagrange’s limiting process. He was slightly more receptive to Lagrange’s
“extremely complicated” treatment of 1764, for it required that all the derivatives dn y/dxn should be finite. Lagrange had indeed noted that his power-series
expressions for the Fourier coefficients of φ and ψ in his solution (57) for the discretely loaded string only converged (in the limit of an infinite number of loads)
under this condition. D’Alembert believed Lagrange’s condition to imply the
analyticity he required in his solutions to the continuous problem. Moreover,
he fancied that the equation of vibrating strings
∂ 2 y/∂x2 − ∂ 2 y/∂t2 = 0
(61)
∂ n+2 y/∂xn+2 − ∂ n+2 y/∂t2 ∂ n x = 0,
(62)
implied
and therefore required indefinite differentiability with respect to x. He rejoiced
that he and Lagrange “now agreed on vibrating strings,” even though Lagrange
never truly conceded that non-analytic solutions should be excluded. In the
earlier cited letter to d’Alembert of March 1765, Lagrange asserted that propagation phenomena required “discontinuous” functions. In his old age, he noted
that “the principle of the discontinuity of the functions is now received for the
integrals of all differential equations.”72
71 D’Alembert to Lagrange, 27 Sep 1759, LO13, 3–4; d’Alembert, “Recherches,” Ref. 38, 65–73
(criticizing Lagrange), 39–40 (rest to physics).
72 D’Alembert, “Nouvelles réflexions,” Ref. 38, 128 (on Lagrange’s limit), 152 (“complicated”);
Lagrange, Ref. 64, 554 (“My construction…is only exact…if dn y/dxn makes no jump in the initial curve”); Lagrange to d’Alembert, 1 Sep 1764, LO13, 12–14 (same restriction); d’Alembert to
Lagrange, 12 Jan 1765, LO13, 23–29, on 24 (derivatives and analyticity; agreement); d’Alembert,
“Nouvelles réflexions,” Ref. 38, 192–193 (derivating the wave equation); Lagrange to d’Alembert,
The acoustic origins of harmonic analysis
383
Lagrange’s persistent admission of non-analytic string curves was important
in his denial that trigonometric series could “geometrically” reproduce every
string curve. For he believed to have proved that any analytic periodic function was a trigonometric series. The proof was based on integrating the linear
differential equation of infinite order obtained through the Taylor development
of f (x + 1) − f (x) = 0. D’Alembert rejected this proof in his Opuscules, arguing that the implied series could fail to converge. At any rate, “continuous”
periodic functions in his sense did not necessarily admit a Taylor development
everywhere. As was already mentioned, d’Alembert tolerated the string curve
y = sin5/3 (π x/l), whose second derivative is infinite when x is a multiple of
π . As another example of a “continuous” periodic function that did not meet
Lagrange’s criteria and therefore could not be represented by a trigonometric
series, d’Alembert gave the cycloid x = 1 − cos u, y = u ± sin u for which
y ∼ x2/3 around the corner u = 0. Lagrange’s interesting reply involved trigonometric series (with unspecified coefficients) for the algebraic functions x, x2 ,
and x2/3 , which implies that by 1768 at least he did not believe trigonometric
series needed to be analytic over the whole real axis.73
The following year, d’Alembert sent to Lagrange his last piece on vibrating
strings, in which he repeated that curves with corners as well as y = sin5/3 (π x/l)
could not be represented by trigonometric series. He also confirmed that
Bernoulli’s partial vibrations did not have any physical reality, even in the
discrete case for which Bernoulli’s solution was not to be doubted. Lagrange
congratulated d’Alembert for these “decisive” remarks, although it is not clear
which one he truly approved. He went on to express the “violent revulsion”
he had developed for the vibrating string problem. As was already mentioned,
many years elapsed before he returned to it.74
In 1759 Euler privately thanked Lagrange for having sheltered his solution
of the problem of vibrating strings from any chicanery. He publicly saluted
his friend’s “indecipherable” but “prodigious” calculations, and proceeded to
extend the discussion of sound propagation. Far from claiming any priority
for the propagation of pulses in the discrete case—which he had first studied in
1748—he directly studied the continuous case through the equation of vibrating
strings. His general solution for this equation, he now saw, was perfectly compatible with the propagation of pulses obtained by Lagrange as a limiting case
20 Mar 1765, EO2:13, 36–38; Lagrange, Ref. 49, 441. D’Alembert (“Nouvelles réflexions,” Ref.
38, 144) doubted that sound propagation could be expressed analytically. D’Alembert’s proof that
the continuity of all derivatives implied analyticity was based on Taylor developments. Lagrange
rejected it in his letter to d’Alembert, 26 Jan 1765, LO13, 29–32. D’Alembert persisted in his
2
reply, 2 Mar 1765, LO13, 32–35. Neither of them saw that functions such as e−1/x are indefinitely
differentiable and yet do not agree with their Taylor development around x = 0.
73 Lagrange, Ref. 64, 514–516; Lagrange to d’Alembert, 26 Jan and 20 Mar 1765, LO13, 30, 37
(announcing the proof); D’Alembert, “Nouvelles réflexions,” Ref. 38, 191; “Sur la manière de
déterminer certaines fonctions,” Opuscules, 4 (Paris, 1768), 343–348, on 344–345; Lagrange to
d’Alembert, 15 Aug 1768, LO13, 114–119.
74 D’Alembert, Ref. 73; Lagrange to d’Alembert, 15 Jul 1769, LO13, 138. Cf. Truesdell, Ref. 2,
288–289.
384
O. Darrigol
of the discretely loaded cord. One just had to choose a generating function that
vanished everywhere except on a small interval. Euler welcomed the striking
non-analyticity of this case. Quite generally, he asserted that the integral of a
partial differential equation should involve an arbitrary “discontinuous” (nonanalytic) function as a natural generalization of the set of integration constants
for a system of ordinary differential equations.75
To this memoir of 1759, Euler added generalizations to two- and threedimensional propagation by means of the linearized version of his general
equations of motion for an elastic fluid. In 1765, he discussed the propagation
and reflection of waves on a vibrating string by the same method. This new
emphasis on propagation brought him further apart from Daniel Bernoulli. In
May 1764, he wrote to the latter’s nephew:
I do not wish to deny absolutely that the equation composed of an infinity
of sines includes the solution to this question, since it contains arbitrary
constants which it would be possible to determine in such a way that in
putting the time = 0, [this form] would produce exactly the curve impressed
upon the string at the beginning. But your uncle will not disagree that this
operation would be infinitely troublesome and even impossible to execute,
because of the infinity of coefficients one would have to determine.
Euler went on to assert that it was “at least permissible to doubt” that a pulse
could be represented by a trigonometric series. In the memoir published a few
months later he categorically denied this possibility, and confirmed his earlier opinion that Bernoulli’s solution was “only very particular.” To Bernoulli’s
earlier remark that a trigonometric series could be made to pass through an
unlimited number of given points and therefore could reproduce any curve, he
replied that by the same argument any curve could be represented by a power
series, which evidently was not the case.76
Daniel Bernoulli admitted his incapacity to explain propagation with his
mixtures of oscillatory modes. Yet he remained unshaken by Lagrange’s and
Euler’s criticism. In his last memoir on vibrating strings, published in 1774, he
reasserted the generality and physical pertinence of his solution:
If you suspect any restriction in my solution to the problem of the vibrations of tense strings initially curved according to a given law, this restriction consists necessarily in an insufficient enumeration of the simple
Taylorian vibrations of which the absolute vibrations are composed.
75 Euler to Lagrange, 23 Oct 1759, LO14, 164–170; Euler, “De la propagation du son,” HAB(1759),
EO3:1, 428–483, on 428 (citation), 431. Cf. Truesdell, Ref. 15, XXVIII-XL. Lagrange independently
obtained similar results in his second memoir on sound, Ref. 55: cf. Truesdell, Ref. 15, IL-LIV.
76 Euler, “Supplément” and “Continuation” to “De la propagation du son,” HAB (1759), EO3:1,
452–484, 484–507; “Eclaircissements sur le mouvement des cordes vibrantes,” MT, 3 (1762–1765),
1–26, also in EO2:10, 377–396, on 385; “Sur le mouvement d’une corde qui au commencement n’a
été ébranlée que dans une partie,” HAB, 21 (1765), 307–334; EO 2:10, 426–450; Euler to Johann III
Bernoulli, 27 May 1764, cited in Truesdell, Ref. 2, 277n. Cf. Truesdell, ibid., 281–286.
The acoustic origins of harmonic analysis
385
The following year he similarly wrote to Nicolas Fuss: “I am still convinced that
my method gives every possible case in abstracto.” Unfortunately, the mathematical proof of this assertion was still lacking. Bernoulli never bothered to
determine the coefficients of the simple modes and ignored the formulas that
Lagrange had given for this purpose. While his physical intuitions pointed to
a remote future, his mathematics belonged to the first half of the eighteenth
century.77
To summarize, Bernoulli, d’Alembert, Euler, and Lagrange held different
positions regarding the permissible string curves, the curves that trigonometric
series could represent over a finite interval, and the status of partial vibrations.
Bernoulli admitted any string curve for which both the ordinate and the radius
of curvature remained very small compared to the length of the string;78 he
believed that trigonometric series could represent any such curve; he asserted
the physical existence of the partial vibrations. D’Alembert required the string
curve to be analytic and close to the axis; he believed that any “discontinuous” curve and even some “continuous” curves could not be represented by
trigonometric series; he regarded partial vibrations as mathematical fictions.
Euler admitted any continuous string curve with piecewise continuous slope
and curvature, and with small ordinate and slope; he denied that trigonometric
series could represent non-analytic curves, at least those which coincide with
segments of the axis (pulses); he ascribed some physical reality to partial vibrations of non-necessarily sine form. Lagrange originally admitted the same string
curves as Euler (except for polygonal curves), but came to believe that his passage from the discrete to the continuous required that all derivatives should be
finite; he believed that trigonometric series could in some “asymptotic” sense
represent any curve, but sometimes denied perfect identity between the series
and the curve when the latter was non-analytic or pulse-like; like d’Alembert,
he denied any physical reality of the partial vibrations.
3.6 Fourier coefficients before Fourier
In the context of vibrating strings, Lagrange long remained the only geometer
who had a formula for calculating the coefficients of the trigonometric series
for a given curve. He had it as an implicit component of the Eq. (28) of his first
memoir on sound, and as a special case of his generalization of Eq. (56). Yet he
was not the first geometer to propose such formulas. They occurred repeatedly
77 Daniel to Johann III Bernoulli, 25 Jul 1765, cited in Truesdell, Ref. 2, 285n (on propagation);
Daniel Bernoulli, “Commentatio physico-mechanica generalior principii de coexistentia vibrationum simplicium haud pertubatarum in systemate composito,” NCAP, 19 (1774), 239–259, on
258–259; Daniel Bernoulli to Nicolas Fuss (circa 1775), in Paul Heinrich Fuss, Correspondance
mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, vol. 2 (Saint Petersburg,
1843), 661–663.
78 Bernoulli believed that infinite curvature (as in polygons) would contradict the assumption of
infinitely small motion: see Daniel Bernoulli, “Mémoire sur les vibrations des cordes d’une épaisseur inégale,” HAB21 (1765), 281–306, on 283–284; Ref. 77, 247; also his letters to Johann III of 24
May 1764 and 25 July 1765, discussed in Truesdell, Ref. 2, 282n.
386
O. Darrigol
in the context of celestial mechanics, well before his wrote his first memoir on
sound.
In his prize-winning memoir of 1748 on the equalities of the motion of Saturn
and Jupiter, Euler encountered the function
f (ϕ) = F(cos ϕ) = (1 − α cos ϕ)−β ,
(63)
and its primitive f (ϕ)dϕ, of which he needed to know a rapidly converging
expansion. A power-series development with respect to α could not do, because
this coefficient was too close to one. Instead Euler used the trigonometric development
f (ϕ) =
∞
ar cos rϕ,
(64)
r=0
which thus made its first entry in the history of celestial mechanics. Euler determined the two first coefficients of this development through the approximations
1 a0 =
(F(sk ) + F(−sk )),
2p
p
k=1
1
a1 =
(sk F(sk ) − sk F(−sk )),
p
p
(65)
k=1
wherein sk = sin[(2k − 1)π/4p], and p is the number of “divisions of the right
angle,” which increases with the goodness of the approximation. He did not
indicate how he had reached these expressions. One possibility is that he had
already solved the above-mentioned problem of the discretely loaded elastic
cord. In this case, he could use the identity (21) to determine the first coefficient
of a sum of n sines that coincides with a given function for n equidistant values
of the argument over the interval [0, π ]. The formulas (65) solve the similar
problem of a discrete cosine development over the interval [−π/2, π/2]. Euler
did not take the limit of infinite p, for he did not know how to calculate the
resulting integral. Instead, he used the approximation p = 10 in his numerical
calculations. As for the higher coefficients (r ≥ 2), he computed them through a
recurrence relation that holds when the developed function is (1−α cos ϕ)−3/2 .79
D’Alembert adopted Euler’s trigonometric development of (1−α cos ϕ)−β in
his own works on celestical mechanics. His Recherches of 1754 contain the
formulas
79 Euler, Recherches sur la question des inégalités du mouvement de Saturne et de Jupiter, sujet
proposé pour le prix de l’année 1748, par l’Académie Royale des Sciences de Paris (Paris, 1749), also
in EO2: 25, 1–44, on 30. Cf. Louise and Ronald Golland, “Euler’s troublesome series: An early
example of the use of trigonometric series,” Historia mathematica, 20 (1993), 54–67, which contains an interesting reconstruction of Euler’s formulas, independent of the discrete interpolation
formula.
The acoustic origins of harmonic analysis
1
a0 =
π
π
(1 − α cos ϕ)
−β
0
387
2
dϕ, a1 =
π
π
(1 − α cos ϕ)−β cos ϕ dϕ,
(66)
0
for the two first coefficients. Although he based his derivation of these formulas
on the identities
π
π
cos nϕ dϕ = π δ0n ,
0
cos ϕ cos nϕ dϕ =
π
δ1n ,
2
(67)
0
he probably reached them by taking the limit of infinite p in Euler’s approximation (65). Indeed this limit yields the similar formulas80
1
a0 =
π
π/2
[F(sin ϕ) + F(− sin ϕ)]dϕ,
0
1
a1 =
2π
π/2
[sin ϕF(sin ϕ) − sin ϕF(− sin ϕ)]dϕ.
(68)
0
In an attempt of 1757 to reconstruct Euler’s formulas (65), Alexis Clairaut
determined the coefficient of a sum of cosines agreeing with an arbitrary function f (ϕ)for n evenly spaced values of φ between 0 and 2π . More exactly, he
sought the numbers ar such that
f (2π k/n) =
n−1
ar cos 2π kr/n,
for 1 ≤ k ≤ n.
(69)
r=0
Being aware of the orthogonality relations
n
cos 2π kr/n cos 2π ks/n =
k=1
n
n
δrs ,
cos 2π kr/n = 0 for 1 ≤ r ≤ n − 1,
2
(70)
k=1
he obtained the formulas
1
a0 =
f (2π k/n),
n
n
k=1
1 ar =
f (2π k/n) cos 2π kr/n
2n
n
for r ≥ 1 (71)
k=1
(it should be 2/n instead of 1/2n in the second formula). Clairaut noted that
these formulas also applied “in the case when the law of the function would not
even be given algebraically, in cases for which the curve that expresses it would
80 D’Alembert, Recherches sur différens points importants du système du monde, 3 vols. (Paris,
1754), vol. 2, 66–67.
388
O. Darrigol
be given only at several points.” He then took the limit of infinite n and wrote
the expressions
1
a0 =
2π
2π
f (ϕ)dϕ,
1
ar =
4π
0
2π
f (ϕ) cos rϕdϕ
(for r ≥ 1)
(72)
0
of the coefficients of the development
f (ϕ) =
∞
ar cos rϕ
(73)
r=0
(again the true expression for ar differs by a factor 4). Clairaut failed to note
that for a function defined over the interval [0, 2π ] this development was only
possible for functions such that f (ϕ) = f (2π − ϕ). However, this condition was
automatically met for the expressions (1−α cos ϕ)−β he was aiming to develop.81
Twenty years later and in the same context of celestial perturbations, Euler
gave the variant
1
a0 =
π
π
0
2
f (ϕ) dϕ, ar =
π
π
f (ϕ) cos rϕ dϕ
(for r ≥ 1)
(74)
0
of Clairaut’s formulas (72) for the coefficients of the cosine development (73) of
any function of cos ϕ. He obtained this result directly through the orthogonality
property
π
0
π
cos mϕ cos nϕdϕ = δmn ,
2
π
cos nϕdϕ = 0
for n ≥ 1
(75)
0
perhaps in analogy with his earlier use of the identity (22) in the discrete case
or with d’Alembert’s use of Eqs. (67).82
Clairaut, d’Alembert, and Euler only used the integral expression of the
coefficients of a trigonometric development for functions of the form
(1 − α cos ϕ)−β , for which the existence of the development is not in doubt
since it can be obtained algebraically by first developing in powers of cos ϕ
and then expressing cosn ϕ as linear combinations of cos nϕ, cos(n − 2)ϕ, etc.
In fact, d’Alembert and Euler favored the latter procedure and treated the
81 Alexis Clairaut, “Mémoire sur l’orbite apparente du soleil autour de la terre, en ayant égard
aux perturbations produites par les actions de la lune et des planètes principales,” MAS (1754),
521–564, on 545–549.
82 Euler, “Disquisitio ulterior super seriebus secundum multipla cujusdam anguli progredientibus,”
NCAP, 11 (1793), 114–132 [1798] (read on 29 May 1777). Cf. Langer, Ref. 24, 27–29; Grattan-Guinness, Ref. 24, 19n.
The acoustic origins of harmonic analysis
389
former as a mathematical curiosity. So did too Lagrange when he encountered
(1 − α cos ϕ)−β in his works on celestial mechanics, even though he knew the
integral expression of the coefficients from his work on vibrating strings.83
Explicit trigonometric developments were not only known for analytic and
periodic functions such as (1 − α cos ϕ)−β . In the “Subsidium calculi sinuum” he
read in 1753 (EO1:14, 582–584), Euler obtained trigonometric series that could
not possibly be analytic through more than one period. His starting point was
the identity
y=
∞
∞
α n cos nx =
n=0
1 n inx
1 − α cos x
(α e + α n e−inx ) =
,
2
1 − 2α cos x + α 2
(76)
n=0
which is the cosine counterpart of the identity (11) he had used against Bernoulli.
He then dared to take α = 1 and α = −1, which gives
∞
cos nx = −
n=1
1
2
and
∞
n=1
1
(−1)n cos nx = − .
2
(77)
For the second series, term by term integration yields,
∞
(−1)n
n=1
n
x
sin nx = − ,
2
∞
(−1)n
n=1
n2
cos nx =
π2
x2
−
, etc.,
4
12
(78)
wherein the integration constants result from the known values of the series for
x = 0.
Euler did not say anything on the convergence of these series, or on the range
of validity of the simple polynomial expressions of their sums. As the series are
evidently periodic, he could not expect these expressions to be valid for |x| ≥ π .
Possibly, he regarded these series as limiting cases of series involving the convergence factor α n (with |α| < 1), in which case the intermediate series (77)
make sense and the singularities when x is an odd multiple of π only appear in
the limit α → ±1.
In 1771 Daniel Bernoulli obtained similar series starting directly from the
divergent sums (77). To justify this procedure, he relied on the then common
idea that divergent series could converge in some sense if the grouping of several consecutive terms led to a convergent series. Unlike Euler, he identified
the singular points and specified in which interval the series were valid. For
instance, he gave
83 D’Alembert, Ref. 80, 67: “[méthode] plus curieuse et plus géométrique que commode pour le
calcul.” Lagrange, Recherches sur les inégalités des satellites de Jupiter causées par leurs attractions
mutuelles (Paris, 1766), also in LO6, 62–225, on 87–89; “Sur le problème de Kepler,” Académie
Royale des Sciences de Berlin, Mémoires(1771), also in LO3, 113–138, on 121–123; “Remarques
sur les équations séculaires des mouvements des nœuds des inclinaisons des orbites des planètes,”
MAS (1774), also in LO6, 633–709, on 645–646.
390
O. Darrigol
∞
π
1
1
sin nx = (2p + 1) − x for 2pπ < x < 2(p + 1)π .
n
2
2
(79)
n=1
Interestingly, he noted that the two series obtained by integrating the latter
series two and four times were “fully applicable to the theory of vibrating
strings,” for they shared the initial string curve’s property of vanishing at x = 0
and x = 2π .84
We saw that in a letter of 1768, Lagrange had no qualms developing the funcMoreover, in his memoir on sound
tions x, x2 , and x2/3 in trigonometric series.
of 1759, he had argued that the sum ∞
n=1 cos nx equaled −1/2 for any value
of x that was not a multiple of 2π , in agreement with Euler’s result of 1753.
D’Alembert was the only one of our string theorists to condemn such series, as
he did in his criticism of Lagrange’s memoir.85
Euler’s and Lagrange’s awareness of trigonometric series that coincided with
algebraic functions over one period of the variable seems to contradict some of
the limitations they saw in these series. Remember that in his reply to Bernoulli’s
memoir of 1753, Euler had flatly rejected the possibility that a trigonometric
series could represent a string curve that did not have a periodic (and odd) analytic continuation. This belief is clearly inconsistent with his giving (around the
same time) a trigonometric series for the functions x and x2 . He probably himself
realized this conflict, since he never repeated the argument. Instead he asserted,
from 1759 on, that trigonometric series were not able to represent pulses that
vanished everywhere except on a small segment. Lagrange agreed, in 1765, that
in such cases trigonometric series could only be “asymptotic approximations”
to the true function. Hence we must conclude that Euler and Lagrange only
required trigonometric series to be analytic over the interval in which they were
used to represent a given function. Whereas pulses and polygons failed to meet
this criterion, algebraic functions did, with some necessary singularities beyond
the interval of representation.
One may still wonder why Euler and Lagrange never used the ClairautLagrange formulas to produce the kind of trigonometric series they tolerated.
One possible reason is that these formulas were less convenient than more
direct, algebraic methods. Another possible reason is that these formulas did
not in themselves contain a proof of the existence of the series, whereas the
more algebraic procedures seemed to do so at a time when convergence considerations were usually neglected.
84 Daniel Bernoulli, “De indole singulari serierum infinitarum, quas sinus vel cosinus angulorum
arithmetice progredientium formant, earumque summatione et usu,” NCAP, 17 (1772), 3–23, also
in Werke, 2 (1982), 119–134, on 133. Ibid. on 134, Bernoulli judged that the corresponding solutions
of the equation of vibrating strings were physically unacceptable, for reasons I was not able to
understand. One may also wonder why he had the string end at x = 2π and not at x = π . The series
(79) for p = 0 already appeared in Euler to Goldbach, 7 Jul 1744 (Fuss, Ref. 77, vol. 1, 279).
85 Lagrange, Ref. 43, 110–112; d’Alembert, “Recherches,” Ref. 38, 65–73. Lagrange’s obtained his
result for cos nx by setting cos nx − cos(n + 1)x to zero for n → ∞ in his expression for the sum
of the n first terms of the series.
The acoustic origins of harmonic analysis
391
Fig. 4 Euler’s representation
of coincidences between the
pulses of consonant sounds
(Ref. 86, Table 1)
4 Musical theory
4.1 Euler’s Tentamen
Among the eminent string theorists, Euler was the one most deeply involved in
music theory. In his Basel period, he published his Dissertatio physica de sono,
which contained the first quantitative theory of sounding pipes, based on the
analogy with vibrating strings. At that time, he already planned a major treatise
on music which appeared only in 1739 as the Tentamen novae theoriae musicae
ex certissimis harmoniae principiis.In agreement with Mersenne’s and Galileo’s
views, he defined sound as “nothing but the perception of successive pulses
[ictus] occurring in the particles of air that surround the organ of hearing.” In
the full picture, the pulses caused by sonorous bodies are gradually transmitted
through the elastic “globules” of the air, and then strike the eardrum which
excites the aural nerves. The pitch of the sound corresponds to the frequency
of the pulses (when it is well-defined). Consonance corresponds to the frequent
coincidence of the pulses from different sounds (see Fig. 4). Euler attributed
the satisfaction given by consonance to the mind’s predilection for order. While
he fully understood the subjective character of this definition, he made it the
basis of a rigorous arithmetic of music.86
86 Euler, Dissertatio physica de sono(Basel, 1727), also in EO3:1, 181–196; Tentamen novae theoriae
musicae ex certissimis harmoniae principiis(Petropolis, 1739), also in EO3:1, 197–427, on 201–211
(quote from 209), 224 (order and harmony). Cf. Hermann Richard Busch, Leonhard Eulers Beitrag
zur Musiktheorie (Regensburg, 1970); Truesdell, Ref. 15, XXIV-XXIX, Cannon and Dostrovsky,
Ref. 16, 43–46 on the Dissertatio; Bailhache, Ref. 4, 112–130 on the Tentamen.
392
O. Darrigol
For any combination of sounds with commensurable frequencies, Euler
defined the “degree of suavity” (suavitatis gradus) as a measure of the ease
with which the corresponding ratio could be perceived. To the ratio 1:1 he
gave the degree one; to the next simple ratio 1:2, the degree 2. To the ratio
1: p, where p is a prime number, the degree p. To the ratio 1:4, he gave the
same degree (3) as to the ratio 1:3, because, 4 being the double of 2, this ratio
“seems almost as easy to recognize as the ratio 1:2.” More generally, to the
α
ratio 1 : p1 1 pα2 2 · · · pαnn he gave the degree ni=1 (αi pi − αi ) + 1. Lastly, to the
multiple ratio of a sequence of integers, he gave the degree of the ratio between
the smallest common multiple of these integers and the unit 1. In this manner,
he could precisely appreciate the suavity of any chord. His theory of harmony
directly followed from this notion.87
Although Euler mentioned the production of harmonic sounds by music
instruments and referred to Sauveur’s experiments on the multiple modes of
vibrating strings, he said nothing about the coexistence of harmonic sounds or
about resonance. These notions were indeed foreign to his concept of harmony,
which only required the idea of the coincidence of pulses. As we saw, in 1753
he rejected Bernoulli’s idea that the simultaneous hearing of harmonics proved
the reality of partial sine modes. In his opinion, the harmonics could result from
any motion involving a succession of similar loops separated by nodes, and the
sine shape did not have the privileged status that Taylor and Bernoulli accorded
to it. In a letter to Lagrange of 23 October 1759, he departed even further from
Bernoulli’s view by agreeing with Lagrange that the vibrating string was not the
true source of the harmonics: “For the sounds of Music, I perfectly agree with
you, Sir, that the consonant sounds that Mr. Rameau claims to be hearing from
the same string come from other shaken bodies.”88
As we will see in a moment, the simultaneous hearing of harmonics played
an important role in the foundations of Jean-Philippe Rameau’s theory of harmony. Euler’s letter to Lagrange continues with a rejection of this view: “I do
not see why this phenomenon should be regarded as the principle of Music
rather than the true proportions on which it is based.” In one of his letters to
a German princess, Euler similarly rejected Rameau’s appeal to resonance by
consonance. For Euler, the commensurability of the frequencies of two sounds
was the common cause of resonance and harmony; resonance did not cause
harmony.89
In the early 1760s, Euler worked on the daunting problem of the non-uniform
vibrating string. He was able to solve two cases, one for which the density of the
string varies as the function (1 + αs)4 of its curvilinear abscissa s, the other in
which the string is made of two uniform parts of different density. In the latter
case, he proved the existence of simple sine modes with incommensurable fre87 Euler, Tentamen, Ref. 86, 223–236, 230 (quote).
88 Ibid., 221 (Sauveur); Euler insisted on the more general multi-nodes motions in “Éclaircisse-
ments,” Ref. 76, 395; Euler to Lagrange, 23 Oct 1759, LO14, 164–170, on 168–169.
89 Ibid., 169; Euler to a German princess, 8 Jul 1760, in Euler, Lettres de L. Euler à une princesse
d’Allemagne sur divers sujets de physique et de philosophie (Paris, 1842).
The acoustic origins of harmonic analysis
393
quencies, as Bernoulli had long ago predicted. In the same period, he found out
that bells also had incommensurable modes against the wide-spread belief to
the contrary. As he explained in the introduction of his paper on non-uniform
strings, these findings as well as his earlier work on vibrating bands confirmed his
view that the cause of harmony should not be sought in nature. Vibrating bodies
generally do not emit mutually consonant sounds. Only musical instruments do
so.90
This additional objection to Rameau’s theory went along with a partial return
to Bernoulli’s interpretation of harmonics:
The remarkable phenomenon that several sounds following the ratio of
the numbers, 1, 2, 3, 4 etc. can be heard from the same string, which the
most celebrated Daniel Bernoulli explained in the most felicitous manner, only occurs…in strings of uniform thickness. Although non-uniform
strings and other kinds of vibrating bodies can also emit several sounds at
the same time, these sounds can differ from the ratio of the numbers 1,
2, 3, 4 etc. in whatever manner. Wherefrom we may understand that the
principle on which the highest artist of music de Rameau bases universal
harmony rests on an invalid foundation.
As Euler could not deny the existence of partial vibrations in the dissonant case,
he felt compelled to admit them in the consonant case too. This did not bring
him closer to accept Bernoulli’s mixtures of simple modes as the most general
solution of vibration problems. Although the only periodic solutions he could
identify in dissonant cases were sine functions of time, he still believed that there
could be other sorts of solution. Besides, he remained confident that trigonometric sums were “continuous” whereas the general solution ought to depend
on arbitrary “discontinuous” functions representing the initial conditions.91
To summarize, Euler’s pulse-based theory of musical harmony predisposed
him to ignore Bernoulli’s tracing of any sound to the mixture of simple modes
of the sonorous body. He was nonetheless willing to admit that the hearing of
simultaneous sounds of different pitch from the same source corresponded to
a complex vibration of the source, involving a superposition of vibrations at
different frequencies. What he could not accept was the restriction or reduction
of these partial vibrations to Bernoulli’s simple modes, whose amplitude could
only be a sine function of time.
4.2 D’Alembert’s Rameau
In 1752, d’Alembert published his own Elémens de musique théorique et pratique suivant les principes de M. Rameau, a fairly short text which purported to
90 Euler, “De motu vibratorio cordarum inaequaliter crassarum,” NCAP, 9 (1763), 246–304, also
in EO2:10, 293–343, on 293–294; “Tentamen de sono campanarum,” NCAP, 10 (1764), 261–281,
also in EO2:10, 360–376, on 361, 372. Cf. Truesdell, Ref. 2, 301–307, 320–322.
91 Euler, “De motu,” Ref. 90, 295 (citation), 306 (continuity).
394
O. Darrigol
bring Rameau’s theory to the educated masses. D’Alembert modestly denied
any originality: “Nothing is mine except the order, and the errors that might be
found.” Yet the task of extracting the essence of Rameau’s prolix and at times
obscure writings was not an easy one, as Rameau himself noted:92
Mr. d’Alembert has sought in my works…truths to be simplified and to be
made more familiar, more luminous, and thus more useful to the many….
He gave me solace by adding to the solidity of my principles a simplicity
which I suspected but could not have reached without much effort and
maybe in a less felicitous manner…. Sciences and arts would haste each
other’s progress if, favoring truth over self-esteem, some Authors had the
modesty of accepting aid, others the generosity of offering it.
In his Nouveau systême de musique théorique, published in 1726 as a physics-based introduction to his Traité de l’harmonie of 1722, Rameau founded his
theory of harmony on the hearing of the three first harmonics of the sounds
produced by various instruments. He possibly borrowed the idea of relating
harmony to harmonics from Sauveur, whom he cited together with Mersenne.
He noted that harmonics were heard slightly after the fundamental, and that
they were more easily distinguished if they were imagined first. In d’Alembert’s
interpretation, Rameau’s system rests on three basic facts of experience. The
first is the hearing of the first three harmonics of the fundamental of the sound
emitted by a sonorous body, most easily for the thickest string of a violoncello. The second fact is that, in Sauveur’s terms, bodies tuned at a harmonic
of another sounding body “resonate” as a whole and bodies tuned at a subharmonic of the sounding body “quiver” (frémissent) with a number of nodes
depending on the order of the exciting harmonic. The third fact is the similarity
anyone perceives between a tone and its octave.93
As a consequence of the two first facts, the most perfect chord should be that
obtained by combining a tone with its three harmonics, that is, the major 8th,
12th, and 17th above. According to the third principle, the 12th and the 17th
may be replaced with the 5th and the 3rd without much loss of harmony. The
result is a major perfect chord, for instance C, E, G, C. A little less naturally, we
may combine a tone with its three subharmonics, that is, the major 8th, 12th, and
17th below. Adding two octaves to the first subharmonic, and three to the third,
we get a minor perfect chord, for instance A, C, E, A. To such justifications
of traditional chords, Rameau and d’Alembert added the ingenious notion of
the “fundamental bass,” namely, a succession of tones separated by a fifth (or
a third), from which they derived diatonic scales by selecting raising harmonics
92 D’Alembert, Elémens de musique théorique et pratique suivant les principes de M. Rameau
(Paris, 1752), v; Rameau, ibid., 2nd edn. (Paris, 1779), 211–212, cited by Bailhache, Ref. 4, 131.
93 Jean-Philippe Rameau, Nouveau systême de musique théorique, où l’on découvre le principe de
toutes les règles nécessaires à la pratique, pour servir d’introduction au Traité de l’harmonie (Paris,
1726), Chap 1; Traité de l’harmonie (Paris, 1722); d’Alembert, Ref. 92, 12–18. The latter’s three
facts can be found in Rameau, Génération harmonique ou traité de musique théorique et pratique
(Paris, 1737), 8–10. On Rameau, cf. Truesdell, Ref. 2, 123–124. On d’Alembert, cf. Bailhache, Ref.
2, 130–139.
The acoustic origins of harmonic analysis
395
(up to an octave) of this bass. For instance, the fundamental bass G, C, G, C, F,
C, F yields the Greek diatonic scale B, C, D, E, F, G, A. In reality, Rameau had
reached the principle of the fundamental bass by mere inspection of contemporary music and justified it only later through the experience of harmonics. He
regarded this principle as the hidden rule of all harmonious music and called it
“the compass of the ear.”94
D’Alembert emphasized the naturalness of these derivations of harmony
and scales. For instance, he declared that the perfect chord was “the immediate
work of nature.” In the introduction to the second edition of his Elémens, published in 1779, he condemned “those geometers who, fancying themselves as
musicians, pile numbers over numbers in their writings, perhaps imagining that
this apparatus is necessary to art.” The unnamed target of this criticism probably
was his old competitor Euler. Against the latter’s arithmetic of harmony, d’Alembert favored a “physical” approach, based on “analogy” and “convenience.”
He flatly rejected the “gratuitous” idea that the orderly coincidence of pulses
was the ultimate source of musical pleasure, and instead defended the concept
of musical harmony as an analogy with harmonies naturally found in sonorous
bodies.95
D’Alembert did not worry that his and Rameau’s starting point was hardly
less gratuitous than Euler’s. Why should art imitate nature? Why should natural harmonies be perceptible and preferable? Perhaps aware of this difficulty,
Rameau suggested a theory of hearing following which harmony would be
directly perceptible:
What has been said of sonorous bodies should be applied equally to the
fibers which carpet the bottom of the ear’s cochlea [le fond de la conque de
l’oreille]; these fibers are so many sonorous bodies, to which the air transmits its vibrations, and from which the perception of sounds and harmony
is carried to the soul.
As no one before Helmholtz elaborated on this suggestion, Rameau’s basic
fact of experience remained vulnerable to Euler’s criticism that the harmonic
character of overtones was the exception rather than the rule in nature. Most
sonorous bodies, even bells, emit overtones that are not harmonics of their fundamental. It thus appears that musicians of all times have artificially selected or
invented the few sonorous bodies for with the overtones are truly consonant.
Some higher principle must have presided to this selection, and Euler found it
in the coincidence of pulses.96
As far as I can tell, d’Alembert never answered Euler’s criticism of Rameau,
although he could not possibly have ignored the fact of dissonant overtones.
94 D’Alembert, Ref. 92, 21–34. Rameau, Nouveau systême, Ref. 93, vii–viii; Génération, Ref. 93, 5
(quote)
95 D’Alembert, Ref. 92, 20; 2nd edn. (1779), cited in Bailhache, Ref. 4, 132.
96 Rameau, Génération, Ref. 93, 7, Proposition 12, cited by Truesdell, Ref. 2, 125 (conque omitted).
Rameau regarded this mechanism as a natural extension of Mairan’s idea (MAS (1720), 11) that
there were as many kinds of air particles as there were sound frequencies.
396
O. Darrigol
The reason for this silence may be that other facts supported his and Rameau’s
view that the distinction between consonance and dissonance had a natural
origin. In his Elémens he gave the etymology of “dissonance” as “sounding
twice”: the tones of a dissonant chord are heard separately whereas the sounds
of a consonant chord are usually perceived as a whole—a fact already asserted
by Euclid. As Mersenne, Sauveur, and Rameau had noted, it requires special
attention, preparation, and training to hear the partial sounds in the latter case.
While d’Alembert did not list this empirical difference between dissonance
and consonance among his basic facts, he surely included resonance induced
by harmonic and subharmonic overtones. The harmonic character of the overtones is here essential, against Euler’s claim that nature has no preference for
harmonics.97
Although d’Alembert supported a theory of music based on the harmonics
content of musical sounds, he strongly opposed Daniel Bernoulli’s recourse
to harmonic analysis in the theory of vibrations. Just as Euler defined a tone
as a periodic succession of pulses, d’Alembert defined it as the repetition of
“cycles” whose precise shape did not matter. As a consequence of his treatment
of the problem of vibrating strings, he refused to accord a privileged role to
sine-shaped motions. And he denied that Bernoulli’s solutions to the problem
of vibrating strings had the sort of multiple periodicity required for a physical
interpretation of the partial vibrations:
The theory of Mr. Bernoulli cannot explain the multiplicity of sounds that
is given by observation. We should therefore acknowledge that all these
facts are an enigma that we cannot elucidate. Indeed can we pretend to
have explained them by regarding the points of the string as composed
of several other points, by assuming fictitious loops and moving nodes?
There is nothing, it seems to me, that we could not justify by so arbitrary
a method.
As we earlier saw, d’Alembert claimed that harmonic sounds were heard even
for a simple vibration (meaning one loop only). At any rate, Bernoulli’s theory could not explain why the 12th and the 17th were the only harmonics that
could easily be heard. D’Alembert pushed this point, for he knew that Rameau’s theory of harmony would entirely collapse if higher harmonics were
admitted.98
At first glance, d’Alembert’s opposition to Euler’s theory of music and his
opposition to Bernoulli’s recourse to musical experience seem to be of a different nature. In the former case he seems to be protecting music from mathematics; in the latter he seems to be protecting mathematics from music. However,
both attitudes derive from the same desire to police the border between mathematics on the one hand, and physics and art on the other. D’Alembert believed
that art and a good part of physics were not amenable to mathematical treat97 D’Alembert, Ref. 92, 11.
98 D’Alembert, Ref. 34 (definition of tone); Ref. 38, 61 (quote), 60 (12th and 17th). On the latter
point, cf. Bailhache, Ref. 4, 135.
The acoustic origins of harmonic analysis
397
ment and ought to remain autonomous fields. Hence came his disdain for Euler’s
arithmetic of music and his rejection of non-analytic solutions to the problem
of vibrating strings. Symmetrically, d’Alembert rejected recourse to physical or
musical intuition in deriving mathematical results. Hence came his rejection of
Bernoulli’s appeal to the hearing of harmonics in justifying mixtures of simple
modes.
In order to fully understand d’Alembert’s mind-set, it must be recalled that in
the eighteenth century physics was still often defined as the non-mathematical
study of inanimate nature, as opposed to the mathematical sciences of mechanics, geometrical optics, geodesy, fortification theory, etc. The extent to which
physics should ultimately be subsumed under mathematics was a controversial issue. Whereas Euler, Bernoulli, and Lagrange shared the Newtonian and
Cartesian ideal of an entirely mathematized or geometrized physics, d’Alembert
wished to preserve the non-mathematical essence of art and of some parts of
physics, as well as the non-physical essence of mathematical demonstration.
He equally disliked Euler’s mathematical music and Bernoulli’s musical mathematics.99
4.3 Bernoulli and Lagrange between physics and music
Unlike Euler and d’Alembert, Daniel Bernoulli never dwelt on musical theory proper. In his long response to d’Alembert’s and Euler’s theories of the
vibrating string, he briefly condemned Rameau’s physical foundation of musical
harmony:
If one holds a steel rod by the middle and strikes it, one hears a confused
mixture of several sounds, which are judged extremely disharmonious by
a skilled musician, so that a contest of vibrations is formed that never
begin or finish at the same time except through a great chance. Hence one
sees that the harmony of the sounds heard simultaneously from the same
sonorous body is not essential to this matter and should not serve as a
principle for the systems of music.
This extract also documents Bernoulli’s approval of Euler’s recourse to the
coincidence theory of harmony, since it refers to the lack of coincidences of
the vibrations of dissonant sounds. Elsewhere Bernoulli wrote approvingly of
Rameau’s principles, despite his rejection of the physical foundation that Rameau purported to give them.100
99 On physics versus mixed mathematics, cf. Thomas Kuhn, “Mathematical versus experimental
traditions in the development of physical science,” Journal of interdisciplinary history, 7 (1976), 1–
31; John Heilbron, “A mathematician’s mutiny, with morals,” in Paul Horwich (ed.), World changes:
Thomas Kuhn and the nature of science (Cambridge, Mass., 1993), 81–129.
100 Bernoulli, Ref. 26, 153; “De vibrationibus chordarum, ex duabus partibus, tam longitudine
quam crassitie, ab invicem diversis, compositarum,” NCAP, 16 (1771), 257–280, on 268 (approving
Rameau).
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O. Darrigol
Although Bernoulli did not produce his own musical theory, he greatly contributed to acoustics through his numerous publications on the theory of vibrations and through his correspondence. When Euler published his Tentamen
novae theoriae musicae, Bernoulli confided to him:
I am waiting very eagerly for your theoriam musicam, for I have myself
much meditated and experimented over these matters. The experiments
confirm my theoriam de sono fistularum quite beautifully.
Bernoulli’s work on flutes and other sounding pipes was indeed important, both
theoretically and experimentally, although most of it was only published with
a long delay (in 1762) after he had seen Lagrange’s competing work. Bernoulli
was conversant on many other aspects of music theory, as appears in the series
of comments he wrote to Euler after reading his treatise. He reproached Euler with insufficient knowledge of Mersenne’s Harmonie universelle and with a
frequent lack of clarity in the foundations. He judged that Euler’s new temperament had little chance to be accepted by musicians, and he rejected Euler’s
basic idea of founding harmony on the mind’s predilection for perfection and
order.101
As we saw, the perception of natural and musical sounds played a significant
role in shaping Bernoulli’s theory of vibrations. His idea of a mixture of simple modes most likely resulted from his ability of hear the harmonics of grave
musical sounds. Similarly, he justified his general principle of superposition by
our ability to distinguish the sounds from the various instruments of an orchestra. One may wonder why he was not satisfied with a derivation of this principle
from Hooke’s law regarding the linear character of elastic forces. The reason is
that unlike d’Alembert, Euler, or Lagrange, Bernoulli never wrote equations
of motion for arbitrary initial conditions. Instead he investigated the pendulous
motion of simple modes, without being originally aware that any other motion
could be obtained by superposition of such modes.
The importance of the perception of harmonics in Bernoulli’s physics of
vibrations raises the question of the view he might have had regarding the
hearing process. What peculiarity of the ear enables it to analyze a sound into
its harmonics? The old theory of the coincidence of pulses could only provide
a limited answer, in cases for which only few harmonics were present. In the
letter to Euler that contained his criticism of the Tentamen, Bernoulli vaguely
suggested a new theory of hearing:
As you explain hearing in a physiological manner [through beats on the
eardrum], I have thought again about a conjecture, namely whether the
membrane of the eardrum should not be in unison with the perceived
sound [unisona cum sono percepto], which duty the musculi can perform
with extraordinary speed and wherefrom very many phenomena could be
deduced.
101 Daniel Bernoulli to Euler, 5 Nov 1740, in Fuss, Ref. 77, 461–465; 28 Jan 1741, ibid., 466–472; 7
Mar 1739, ibid., 453–457 (rejecting Euler’s temperament).
The acoustic origins of harmonic analysis
399
Presumably, Bernoulli had in mind a kind of resonance that would justify the
ear’s ability to perform harmonic analysis. He did not elaborate on this process.
We only know that he regarded both light and sound as mixtures of simple
vibrations and treated the eye and the ear as harmonic analyzers:102
The different colors probably are nothing but different perceptions in the
organ of sight, caused by the different simple vibrations of the celestial
[i.e., ethereal] globules. It is certain that in the same mass of air a great
number of vibrations can be formed at the same time, very different from
each other, each of which separately causes a different sound in the organ
of hearing.
No more than Bernoulli did Lagrange produce his own system of music. His
theoretical writings on sound are nevertheless full of remarks of musical interest. Like Euler, he conservatively defined a (musical) sound as a periodic series
of pulses, and he related musical pleasure to order:
If the sonorous body is such that the vibrations of its parts begin and end
always at the same time, the ear will be struck by several taps succeeding
each other by equal time intervals, and this uniformity of impressions will
produce the pleasant feeling that is called sound.
Lagrange also reduced harmony to “the concurrence of vibrations,” which in
his view was the common foundation of every theory of harmony, including
Rameau’s and Giuseppe Tartini’s. Indeed Lagrange believed that Rameau’s
natural foundations of harmony (the hearing of harmonics and resonance)
could be reduced to resonance only, which is explained by the concurrence of
vibrations. Similarly, he believed that the low-pitch combination tones on which
Tartini founded his theory of harmony could be explained by the concurrence
of vibrations. Both cases require some explanation.103
Although Lagrange did not deny the hearing of harmonics, he believed these
additional sounds to result from the resonance of nearby bodies. He confirmed
d’Alembert’s observation that harmonics were heard even when the motion
seemed most simple:
Having examined the oscillation of tensed strings with all the attention
that I can, I have always found them simple and unique in all their extension, whence it seems impossible to me to conceive how different tones
can be engendered simultaneously…. I therefore incline to believe that
these sounds are produced by other bodies that resonate to the principal
sound.
102 Daniel Bernoulli to Euler, 28 Jan 1741, in Fuss, Ref. 77, 466–472; Daniel Bernoulli, Ref. 26,
188–189. What he meant by musculi is not clear; as musculus can stand for muscle in low Latin, he
may have meant the muscles that are attached to the ossicles of the ear.
103 Lagrange, Ref. 43, 144–145.
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O. Darrigol
Lagrange went on to call for a crucial experiment in which the oscillating string
would be isolated from any other body, in which case he expected no harmonic
to be heard.104
As for the Tartini tones, which d’Alembert had described in great details
in his “Fundamental” article, Lagrange explained them by means of the beat
phenomenon, which had long been understood to result from the coincidence
of vibrations. The frequency of these “third tones” is indeed the difference of
the frequency of the two combined tones, just as is expected from beats. For
instance, when a tone and its exact fifth are sounded on a violin, the lower
octave is heard as a result of 3/2 − 1 = 1/2.105
As we saw, Sauveur was the first author to give an important role to beats in
acoustics. In his researches on sound, Lagrange not only gave a full account of
Sauveur’s experiments but also reproduced his explanation of dissonance: “Mr.
Sauveur has the idea that a chord pleases the ear the more that its beats are
more frequent and hence remain less sensible.” Lagrange went on to argue that
this explanation of harmony equally derived from the principle of concurrent
vibrations, since beats somewhat measure the degree of the concurrence.106
Lagrange’s reduction of every theory of harmony to the concurrence of vibrations and his concomitant rejection of harmonics from musical strings squared
well with his hostility to Bernoulli’s solution of the vibrating string problem.
In his eyes the motion of a plucked string did not involve any of the partial
oscillations that Bernoulli imagined and heard. As Lagrange believed to have
proved in taking the limit of the discretely loaded string, the various oscillatory
terms of Bernoulli’s solution interfered with each other to produce motions
that no longer reflected their oscillatory character. In the second edition of his
Mécanique analytique, he insisted that Bernoulli’s formula could not explain
Rameau’s simultaneous harmonics and added:
The series that could give the different modes disappears from the formula
when the number of bodies is infinite, and the result is, for every point of
the string, a simple and uniform law of isochronism which depends immediately on the initial state.
Again, Lagrange had in mind that the physical character of the superposition of simples modes disappeared in the continuous limit and left place
to d’Alembert’s and Euler’s simple “construction.”107
To sum up, Euler’s, d’Alembert’s, Bernoulli’s, and Lagrange’s diverging views
on the status of Bernoulli’s mixtures of simple modes were related to their views
on musical harmony and theory. The reason for this interrelation is that both
kinds of considerations depended on the status they accorded to the perception of harmonics. Let us first consider the way this status conditioned their
104 Ibid., 146–147.
105 Ibid., 144.
106 Ibid., 144.
107 Lagrange, Ref. 49, 436.
The acoustic origins of harmonic analysis
401
theories of vibrations. Daniel Bernoulli regarded the perception of harmonics as a strong indication of the reality of the mixture of pendulous simple
modes. Euler mostly agreed that this perception revealed the superposition
of partial vibrations, but refused to restrict the partial vibrations to Taylorian
oscillations. D’Alembert and Lagrange did not even accept the reality of partial vibrations, though for different reasons. While d’Alembert argued that the
perceived harmonics eluded physico-mathematical analysis, Lagrange argued
that harmonics truly corresponded to the multiple resonance of surrounding
bodies.
Let us now consider the extent to which the perception of harmonics conditioned the views of our four geometers regarding musical theory. For Daniel
Bernoulli, this perception could not be the basis of musical harmony, because
sonorous bodies in nature usually produce dissonant overtones rather than the
harmonics heard in musical sounds. Euler approved this argument, the more so
because his own theory of music rested on a quantified version of the old principle of the coincidence of pulses. D’Alembert ignored this difficulty and instead
used the perception of harmonics as a means to directly base music on physics,
without Euler’s dubious psycho-mathematical connection. Lagrange denied the
existence of intrinsic harmonics, and traced any perception of harmonics and
any harmony to the coincidence of pulses.
Despite their linking through the problematic of the perception of harmonics, there is no straightforward correspondence between views on musical harmony and views on simple-mode mixing. Euler, Lagrange, and presumably
Bernoulli all supported the reduction of harmony to the coincidence of pulses,
and yet differed in their assessment of simple-mode superposition: Bernoulli
regarded it as real mixture, Lagrange as mathematical fiction, and Euler as an
illegitimately restricted case of a broader phenomenon. D’Alembert adopted
Rameau’s principle of harmonics and yet rejected Bernoullian mixtures. The
reason for this delusively complex pattern is that opinions on the perception
of harmonics could not by themselves determine the views of our four geometers regarding music and the physics of vibrations. Equally important was
their broader understanding of the relation between mathematics, physics, psychology, and art. Bernoulli pleaded for a physical and musical mathematics,
Euler and Lagrange for a thoroughly mathematized physics and a psychomathematical music, d’Alembert for an autonomous mathematics, a selective
mathematical physics, and a physics-based music.
5 Fourier on the harmonies of heat flow
5.1 Early belief in simple-mode analysis
Joseph Fourier’s first publication appeared in 1798, three year after he began
supporting the teaching of Lagrange and Gaspard Monge at the newly founded
Ecole Polytechnique. It contained an influential demonstration of the principle
of virtual velocities, as well as a few remarks on the nature of equilibrium. After
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O. Darrigol
reminding his reader that the stability of equilibrium depended on the nature of
small motions around equilibrium, Fourier referred to Lagrange’s and Daniel
Bernoulli’s relevant works:
This question has been subjected to a very elegant analysis by the illustrious author of the Mécanique analytique. One can further prove, by the
results of this solution, an important proposition that Daniel Bernoulli
has known first and proved in several particular cases: namely, that the
small oscillations of bodies are made of simple oscillations that occur at
the same time without troubling each other.
Fourier went on to describe the pendulous nature of the simple modes and to
discriminate between the periodic and multiperiodic cases for the motion of
the whole body. In the periodic case, he noted, the vibrating body produce a
well-defined tone, no matter how the vibration is started.108
At the end of his memoir, Fourier emphasized that the Bernoullian analysis
only applies in the limit for which the oscillations are infinitely small, so that “the
figure represented by the results of calculation only has an abstract existence
and can never exactly be that of the oscillating body.” He mysteriously added:
“This remark gives the solution of the difficulties that have been propounded
about the figure of vibrating strings.” Perhaps he meant that the discussions of
the mathematically permissible figures were beside the point, since the equation of vibrating strings only applied to an ideal motion. At any rate, he did
not doubt that simple-mode analysis applied to nature and explained the welldefined pitch of the sound emitted by sonorous bodies. He further noted that
the theory implied the possibility of forming multiple vibrations:
When several [sonorous] bodies are brought in contact and when there
exist certain ratios of figure and dimension, it is enough to shake one of
them to excite and sustain the motions of the others. Far from clashing with
each other in their particular vibrations, the system of all these bodies soon
moves symmetrically and in equal times. Thus, had not our senses already
done so, calculus would inform us of the coexistence of simple vibrations,
and, if we may so speak, of the harmonic composition of oscillations….
Nature produces these phenomena under the most varied forms: they are
especially noticeable in the quivering of sonorous bodies; and it is this
branch of Integral Calculus that provides the fundamental principles of
harmony.
Fourier closed his memoir with this allusion to music theory. Clearly, he believed
in Rameau’s doctrine that harmony depended on the physics of multiple vibra108 Joseph Fourier, “Mémoire sur la statique contenant la démonstration du principe des vitesses
virtuelles et la théorie des moments,” JEP, 5 (1798), 20–60, also in FO2, 477–521, on 507. Cf. Louis
Charbonneau, “Fourier et la mécanique: Une histoire méconnue. De la mécanique à la théorie de la
chaleur,” in Sciences et sociétés pendant la révolution française (actes du 114e congrès national des
sociétés savantes, section histoire des sciences et des techniques) (Paris, 1990), 97–116, on 98–105;
Dhombres and Robert, Ref. 2, 171–172. On Fourier’s years at the Polytechnique, see ibid., chap. 4;
Grattan-Guinness, Ref. 2, 4–14; Herivel, Ref. 2, 61–64.
The acoustic origins of harmonic analysis
403
tions and resonance; and he shared Daniel Bernoulli’s belief in the physical
reality of harmonics.109
During the brief existence of the Ecole Normale of the year III (of the revolutionary calendar), Fourier attended the physics lectures of René-Juste Haüy,
which had a whole chapter on acoustics and music theory. Haüy referred to
Wallis’ and Sauveur’s experiments on harmonic vibration and multiple resonance, and to Tartini’s third sound. He agreed with Mersenne and Sauveur that
a few harmonic sounds could be heard from the thickest strings of a violoncello,
and expressed his belief in the existence of the higher harmonics. However, he
agreed with Mersenne, d’Alembert, and Lagrange that the emitting string did
not undergo any multiple vibrations. As his own experiments excluded Lagrange’s suggestion that the harmonics were due to the resonance of nearby
bodies, he joined Mersenne in assuming that the simple vibration of a body was
somewhat able to excite multiple vibrations in the surrounding air. In his discussion of musical harmony he insisted on the subjective, cultural component of the
appreciation of consonance, and judged that Rameau’s and Tartini’s theories
only gave “more or less plausible conventions.” His discussion of wind instruments was based on Daniel Bernoulli’s intuitive theory, with special emphasis
on the principle of superposition that allows us to distinguish the sounds emitted
by the various instruments of an orchestra.110
Fourier’s main source on vibrations was Lagrange’s Méchanique analitique
of 1788, to which he referred his readers for more precise considerations. The
last section of the relevant chapter of Lagrange’s treatise contained a full derivation of the simple modes of the discretely loaded string, which Fourier later
imitated in the context of heat propagation. Another of Fourier’s sources was
the first book of Laplace’s Système du monde, which Fourier recommended for
the following praise of simple-mode analysis:
When a point of a calm surface of water is slightly agitated, circular waves
are formed and spread around it. When the surface is agitated at another
point, new waves are formed and mix with the former; they travel over
the surface disturbed by the first wave as they would do over a calm
surface, so that they can be perfectly distinguished in the mixture. What
the eye perceives with respect to waves, the ear perceives with respect to
sounds or aerial vibrations, which travel simultaneously without troubling
each other and produce very distinct impressions. The principle of the
coexistence of simple oscillations, which we owe to Daniel Bernoulli, is
one of these general results that interest us through the ease with which
they enable our imagination to represent phenomena and their multiple
changes. It can easily be derived of the analytical theory of the small oscil109 Fourier, Ref. 108, 510–521.
110 René-Juste Haüy, physics lectures in Séances des Ecoles Normales recueillies par des sténogra-
phes et revues par les professeurs, 2nd edn., 10 vols. (Paris, 1800–1801), vol. 5, 222–243, on 222–226
(harmonics), 227–233 (musical harmony), 233–240 (Bernoulli). Cf. Dhombres et al. L’Ecole Normale de l’an III. Leçons de mathématiques. Edition annotée des cours de Laplace, Lagrange et Monge
(Paris, 1992).
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O. Darrigol
lations of a system. These depend of linear differential equations whose
complete integral is the sum of particular integrals. Thus, the simple oscillations combine with each other to form the motion of the system just as
the particular integrals that represent them add to each other to form the
complete integrals. It is interesting thus to retrieve in natural phenomena
the intellectual truths of analysis. This correspondence, of which the System of the World will offer us numerous examples, is one of the greatest
charms attached to mathematical speculation.
The physical pregnancy of mathematical analysis in general and the physical
reality of partial modes in particular, here chanted by Laplace, were to become
the leitmotivs of Fourier’s own mathematical physics.111
To sum up, by 1798 Fourier believed in the unrestricted validity of the simplemode analysis of small oscillations, in its physical character, and in its relevance
to the theory of musical harmony. He was aware of the old quarrel about vibrating strings, and implicitly endorsed Daniel Bernoulli’s solution as he asserted
the complete generality of simple-mode analysis. Having read the relevant section of Lagrange’s Méchanique analitique, he knew how to calculate the simple
modes of a discretely loaded elastic string. It is important to recall, however,
that the first edition of this treatise did not involve any consideration of the
continuous limit of this problem. Nor did it give the expression (21) of the
coefficient of the simple-mode superposition as a function of the initial ordinates of the loads in the discrete case. Most likely, Fourier was not aware of
Lagrange’s early memoirs on sound, which contained the relevant analysis. Had
he read them, it would be difficult to understand the roundabout way in which
he arrived at his famous theorem.
5.2 The discrete model
Why and when Fourier began studying the propagation of heat is not known
exactly. As he himself acknowledged, in 1804 he received a short article on
this theme by Laplace’s protégé Jean-Baptiste Biot. All relevant manuscripts
of Fourier seem to be posterior to that date. Plausibly, Biot’s considerations
prompted Fourier’s considerations and experiments. To attempt a mathematical theory of heat propagation was a typically Laplacian idea, since the famous
astronomer and his disciples pursued the grand aim of subjecting all physics to
mathematical analysis following the model of astronomy. Biot investigated the
limited case of a long bar heated at one extremity. He assumed Newton’s law
of cooling, according to which the quantity of heat exchanged by two bodies in
contact is proportional to the difference of their temperatures. He applied this
law both to heat radiation from the bar and to heat exchange between consecutive slices of the bar. Equating the temporal variation of the heat content of
111 Lagrange, Ref. 69, part II, Sect. 5, part 3: “Du mouvement de plusieurs corps qui agissent les
uns sur les autres, soit par des forces d’attraction, soit en se tenant par des fils ou par des leviers,”
esp. 312–314. Pierre Simon de Laplace, Exposition du système du monde, vol. 1 (Paris, 1797), 171.
The acoustic origins of harmonic analysis
405
a given slice of the bar to the heat received from the contiguous slices minus
the radiated heat, he obtained “a partial differential equation of second order,”
whose general solution in the steady case was “the sum of two exponentials.”112
Although Biot announced this equation without writing it down, he evidently
wrote something like
Cθ (x, t + dt)Sdx − Cθ (x, t)Sdx
= αS[θ (x − dx, t) − θ (x, t)]dt + αS[θ (x + dx, t) − θ (x, t)]dt
−hdxθ (x, t)dt
(80)
where θ (x, t) is the temperature at time t and abscissa x, C is the heat capacity
per unit volume, S the section of the bar, α and h two constants characterizing
the speed of heat exchange. Developing to first order in dt and to second order
in dx, this gives
C
∂ 2θ
∂θ
= K 2 − hθ ,
∂t
∂x
with K = αdx.
(81)
√
√
The steady-state solution is a linear combination of e−x h/K and e+x h/K ,
in conformity with Biot’s announcement. The trouble with this reasoning is
that it yields an infinitesimal value of the constant K, whereas in reality the
temperature decreases gradually from the hot extremity of the bar. Biot later
suggested that this difficulty prevented him to publish the whole reasoning and
the equation.113
After reading Biot’s article, Fourier presumably stumbled over the same difficulty, and therefore decided to study a discrete model of heat transfer in which
it did not occur. He later indicated that this approach was the first he had tried.
It must have been clear to him from the start that it would lead to a problem
analogous to that of the discretely loaded elastic string, which Lagrange had
already solved. As can be judged from a manuscript dating from 1805/1806, his
model consisted in a linear arrangement of n equidistant, equal, and disjoint
masses, with heat exchange occurring between two consecutive masses at a rate
proportional to their temperature difference.114
112 Jean-Baptiste Biot, “Mémoire sur la propagation de la chaleur,” Bibliothèque britannique, 27
(1804), 310–329. Cf. Truesdell, The tragicomic history of thermodynamics. 1822–1854 (New York,
1980), 47–51. About Fourier having received Biot’s paper, cf. Grattan-Guinness, Ref. 2, 85, 186. At
the Ecole Normale of the year III, Fourier had heard Haüy lecture on heat, with clear notions of
heat capacity, conduction, and equilibrium based on the caloric-fluid concept: Haüy, in Séances des
Ecoles Normales, Ref. 110, vol. 2, 140–144.
113 Biot, Traité de physique expérimentale et mathématique (Paris, 1916), vol. 6, 669–670, quoted
in Truesdell, Ref. 112, 50.
114 Fourier, “Sur la propagation de la chaleur” [draft written in 1805/6] in MS 22525 du fond
français de la Bibliothèque de France, 109–149, on 113–122. Cf. Grattan-Guinness, Ref. 2, 36–38;
Herivel, Ref. 2, 149–150; 192–194. About Fourier starting with the discrete model, see FO2, 94.
About the dating of the MS, cf. Louis Charbonneau, Catalogue des manuscrits de Joseph Fourier,
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O. Darrigol
Calling θk the temperature of the mass k, this assumption leads to the equations
θ̇k = α(θk+1 − 2θk + θk−1 ),
(82)
where α is the rate of heat exchange between two consecutive masses divided
by the heat capacity of one mass, and the fictitious temperatures θ0 and θn+1
are such that
θ0 = θ1
and
θn = θn+1 .
(83)
Fourier then followed Lagrange’s procedure for solving the similar system
ẍk = α 2 (xk+1 − 2xk + xk−1 ),
with x0 = xn+1 = 0.
(12)
Namely, he sought the simple modes for which
θk = ak eht .
(84)
The resulting relation between the successive amplitudes ak ,
(h + 2α)ak = α(ak−1 + ak+1 ),
(85)
has exactly the same form as the corresponding relation for the loaded string.
The general solution found in Lagrange’s Méchanique analitique is
ak = A cos kϕ + B sin kϕ,
(86)
h = 2α(cos ϕ − 1).
(87)
ak = a sin kϕ + b sin(k − 1)ϕ,
(88)
with
Equivalently, Fourier wrote
which is better adapted to the boundary conditions (83). The first of these
conditions leads to
b = −a
and
ak = a1
sin kϕ − sin(k − 1)ϕ
.
sin ϕ
(89)
in Cahiers d’histoire et de philosophie des sciences, 42 (Paris, 1994). On the chronological order of
Fourier’s researches on heat, cf. Grattan-Guinness, Ref. 2, 441–442.
The acoustic origins of harmonic analysis
407
The second condition leads to
sin nϕ = 0,
(90)
which yields the n distinct modes ϕr = rπ/n with r = 0, 1, 2, . . . , n − 1.
Fourier then superposed the simple modes to obtain the general solution
θk =
n−1
r=0
cr
sin krπ/n − sin(k − 1)rπ/n −2α[1−cos(rπ/n)]t
e
.
sin rπ/n
(91)
He managed to express the coefficients cr as a function of the initial temperatures of the masses in the case of two and three masses only. Judging from
the first extant draft of his theory, he was unable to do so in the general case.
Most likely, he was not aware of the trigonometric identity (22) which Euler
and Lagrange had so successfully used in the similar problem of the discretely
loaded string. And the relative complexity of the simple modes for the present
problem prevented him to directly invert the relevant matrix.115
This aborted approach to heat propagation was nonetheless important to
Fourier because it consolidated his faith in the power of simple-mode analysis.
In particular, Fourier saw that the ratio of the successive modes in the superposition (91) decreased exponentially in time, so that the system tended to the
uniformity of the mode r = 0 after some time, its departure from uniformity
having ultimately the form sin kπ/n − sin(k − 1)π/n of the mode r = 1. This
is the prototype of an argument Fourier often gave in favor of the reality of
simple modes. It may have been reminiscent of Daniel Bernoulli’s reference to
the differential decay of the partial oscillations of a string.116
5.3 The lamina and a theorem
Deterred by the seeming complexity of the discrete problem, Fourier decided
to return to Biot’s continuous problem. In order to avoid the infinitesimal heterogeneity of Eq. (81), he assumed that the rate of heat exchange between
successive slices of the bar was inversely proportional to their width. The justifying intuition was that for an equal surface of contact, the heat exchange
between two consecutive slices should be much stronger than that between the
slice and the air, because on average the various points of a slice are much
closer to a point of the next slice than they are to the surface of the bar. In the
three-dimensional case, the same intuition applied to internal heat exchange
115 Fourier, Ref. 114; reproduced almost identically in Fourier, Sur la propagation de la chaleur
[1807], Bibliothèque de l’Ecole Nationale des Ponts et Chaussées, ed. by Grattan-Guinness, Ref. 2,
39–55. On folios 120v–122 of the draft (Ref. 114), Fourier began to take the limit of infinite n. The
result is not clear, for the text is crossed out and interrupted at that point.
116 Fourier, Ref. 114, also in Ref. 115, 52–55.
408
O. Darrigol
along the x, y, and z axes and heat exchange between any point of the body and
the surrounding space led Fourier to the equation117
2
∂ θ
∂θ
∂ 2θ
∂ 2θ
=K
C
+ 2 + 2 − hθ .
∂t
∂x2
∂y
∂z
(92)
As a simple, yet unsolved steady-state problem, Fourier investigated the
propagation of heat along the semi-infinite rectangular lamina of Fig. 5, the
edge being kept at the temperature 1 and the sides to the temperature 0.
Fig. 5 Fourier’s semi-infinite
lamina
y
q=1
q=0
+p /2
-p /2
x
q=1
After convincing himself that the relevant equation of propagation should be
∂ 2θ
∂ 2θ
+ 2 = 0,
2
∂x
∂y
(93)
he attempted a simple-mode analysis presumably inspired by analogy with
the vibrating-string problem. Namely, he sought solutions of the factored form
φ(x)ψ(y). Guided by the physical intuition of propagation from the heated edge
of the lamina, he retained only the solutions of the type
θ = ce−κx cos κy,
(94)
which decrease exponentially from the edge. The condition on the sides requires
κ to be an odd number if the width of the lamina is π . Superposition of these
simple modes leads to the solution
θ=
∞
cr e−(2r+1)x cos(2r + 1)y.
(95)
r=0
117 Fourier, Ref. 114, folios 122–127, reproduced in Herivel, Joseph Fourier face aux objections
contre sa théorie de la chaleur: Lettres inédites 1808–1816 (Paris, 1980), Appendix 1.
The acoustic origins of harmonic analysis
409
The coefficients cr must be chosen so that the temperature
θ (0, y) =
∞
cr cos(2r + 1)y
(96)
r=0
along the edge −π/2 < y < π/2 be equal to 1 or any other impressed temperature distribution.118
Before attempting the determination of these coefficients, Fourier discussed
the general behavior of his solution. He first noted that any distribution of
the form cos(2r + 1)y propagated along the lamina without deformation and
with exponential attenuation at a rate increasing with the order r. For an arbitrary distribution, Fourier imagined a decomposition of the heat flow into these
modes propres et élémentaires:
One can imagine that the heat that comes at every instant from the source
thus divides itself into distinct portions, that it propagates according to
one of the said elementary laws and that all these partial motions occur
without troubling each other.
This statement was clearly reminiscent of Bernoulli’s and Laplace’s formulations of the superposition principle. So to say, Fourier’s lamina was a vibrating
string with imaginary time.119
Fourier now faced the problem of developing a constant into a sum of cosines
whose angle is an odd multiple of the argument. Euler, Bernoulli, and Lagrange
had all solved similar problems. In general, they admitted trigonometric developments for functions that were analytic over the interval of validity of the
development. Being aware of Euler’s memoir of 1753 on this topic, Fourier did
not have to worry about transgressing the accepted limits of analysis at that
stage. His only problem was to find a way to calculate the coefficients. As he
had read neither Lagrange’s memoirs on sound nor Euler’s Disquisitio ulterior
of 1777, he was unaware of the method based on the orthogonality of simple
modes. He therefore relied on his algebraic prowess to solve the infinite system
of linear equations obtained by identifying all the derivatives of the sum of
cosines at the origin with the derivatives of the function to be developed. In the
present case for which this function is the constant one, this gives
∞
r=0
cr = 1,
and
∞
(2r + 1)2n cr = 0
for n = 1, 2, 3, . . . .
(97)
r=0
After several pages of complicated calculations, Fourier found
cr =
4 (−1)r
,
π 2r + 1
(98)
118 Fourier, Ref. 114, folios 128–132, nearly in Ref. 115, 134–144. Cf. Grattan-Guinness, Ref. 2,
131–182; Herivel, Ref. 2; Dhombres and Robert, Ref. 2, 522–541.
119 Fourier, Ref. 114, folio 130, cited in Grattan-Guinness, Ref. 2, 144.
410
O. Darrigol
which gives the identity
∞
(−1)r
π
=
cos(2r + 1)y
4
2r + 1
for − π/2 < y < π/2.
(99)
r=0
Fourier next described how the series changed sign whenever the argument
passed an odd multiple of π/2.120
Having reached this simple result by intricate means, Fourier soon found a
much simpler derivation. Remember that Euler and Bernoulli
had obtained
similar results by integrating divergent sums of the kind ∞
n=0 cos ny. Conversely, Fourier differentiated the partial sum
Sn (y) =
n−1
(−1)k
cos(2k + 1)y
(2k + 1)
(100)
k=0
to get
Sn (y) =
n−1
(−1)k+1 sin(2k + 1)y = (−1)n
k=0
sin 2ny
,
cos y
(101)
and
x
n
Sn (x) = Sn (0) + (−1) In (x),
with In (x) =
sin 2ny
dy.
cos y
(102)
0
Fourier repeatedly integrated the latter integral by parts to obtain a series
whose every term contained a negative power of n, and hence concluded that
it vanished in the limit of infinite n. In another section, he showed that the
integral remaining after p partial integrations was majored by a quantity of the
order n−p , which made I∞ (x) = 0 a rigorous result. Together with Leibniz’s
well-known result S∞ (0) = π/4, this implies the desired identity (99).121
Fourier thus obtained the first convergence proof for a trigonometric series
that was not trivially convergent. In the following section of his draft, he showed
that Euler’s series of 1753 could be obtained by the same method. Unlike Euler,
he carefully specified the intervals of validity of the formulas, and warned that
120 Fourier, Ref. 114, folios 133–139, on 139v (reference to Euler); also in Ref. 115, 147–159. That
Fourier did not know of Euler’s Disquisitio is clear from Fourier’s letter to Lagrange, undated
[1808], in Herivel, Ref. 114, 20–23.
121 Fourier, Ref. 114, folios 139v-140; also in Ref. 115, 159–161, 166–168. As Fourier later realized,
a simpler proof can be obtained by integrating by parts only once and majoring the integrand of
the remaining integral.
The acoustic origins of harmonic analysis
411
combining such formulas outside the intersection of their intervals of validity
led to absurdities.122
Fourier’s next challenge was to extend to any indefinitely differentiable function the algebraic method of elimination he had used to develop a constant into
a series of cosines. Some time must have elapsed before he managed to do so,
for the draft of 1805/1806 does not include it. In the sine case, the development
reads
f (x) =
∞
ak sin kx,
(103)
k=1
and the linear system to be solved is
∞
(−1)n k2n+1 ak = f (2n+1) (0),
with n = 0, 1, 2, . . .
(104)
k=1
After a few pages of difficult calculations, Fourier obtained
∞ ∞
1 π m (m+2n)
2 ak =
(−1)k+n+1 2n+1
(0).
f
π
m!
k
(105)
n=0 m=0
Using Taylor’s formula, he rewrote this as
ak =
∞
2
(−1)k+n+1 k−2n−1 f (2n) (π ),
π
(106)
n=0
in which he recognized the result of the repeated integration by parts of123
2
ak =
π
π
f (x) sin kx dx.
(107)
0
5.4 Discreteness and generality
As he later explained in a letter to Lagrange, Fourier originally expected this
result to hold only for the indefinitely derivable functions for which his algebraic
procedure worked. He changed his mind only after returning to the discrete
problem of heat propagation. As we saw, his first attempt of this kind led to
122 Fourier, Ref. 114, folios 141–142; also in Ref. 112, 162–165. The warning is in MS ffr 22529, folio
75v (note inserted in copy of the draft of 1805/1806).
123 Fourier (1807), Ref. 115, 193–213.
412
O. Darrigol
an elimination problem that he did not know how to solve. At some point, he
probably surmised that the elimination would become more tractable if the
straight alignment of discrete masses was replaced by a cyclic alignment.124
The boundary conditions thus become:
θ0 = θn
and
θn+1 = θn ,
(108)
and the simple modes have the form
θk = a cos kϕeht
or
θk = a sin kϕeht
(109)
with
h = 2α(cos ϕ − 1)
(87)
and ϕ = r2π/n wherein r is an integer. If n is the odd number 2p + 1, there are
2p distinct sine and cosine modes corresponding to r = 1, 2, . . . , p besides the
trivial constant mode corresponding to r = 0. The general solution then reads
θk = a0 +
p
(ar cos kr2π/n + br sin kr2π/n)e−2α(1−cos r2π/n)t .
(110)
r=1
In order to compute the coefficients from the initial temperatures, Fourier
relied on the orthogonality relations
n
k=1
n
k=1
n
sin 2π kr/n sin 2π ks/n =
cos 2π kr/n cos 2π ks/n =
n
δrs ,
2
n
δrs ,
2
sin 2π kr/n cos 2π ks/n = 0,
(111)
(112)
(113)
k=1
n
k=1
sin 2π kr/n =
n
cos 2π kr/n = 0,
(114)
k=1
124 Fourier to Lagrange, Ref. 115. Although the name of the addressee is not written in the MS,
Herivel has convincingly argued that it can only be Lagrange.
The acoustic origins of harmonic analysis
413
just as Euler and Lagrange had used
n
sin
k=1
ksπ
n+1
krπ
sin
=
δrs
n+1
n+1
2
(22)
in the elimination problem for the discretely loaded string. The result is125
1
θk (0),
n
n
a0 =
2
θk (0) cos 2π kr/n,
n
n
ar =
k=1
2
θk (0) sin 2π kr/n.
n
n
br =
k=1
k=1
(115)
After providing a physical discussion similar to the one he had give in the case
of a rectilinear alignment, Fourier took the limit n → ∞ for which the masses
form a continuous annulus. The ratio 2π k/n thus becomes a continuous angle
x varying between 0 and 2π . In order that the discrete equation of motion
θ̇k = α(θk+1 − 2θk + θk−1 )
(82)
reaches the continuous equation
θ̇ = K
∂ 2θ
,
∂x2
(116)
Fourier further required
α(2π/n)2 → K.
(117)
Consequently, the general solution becomes
θ (x, t) = a0 +
∞
(ar cos rx + br sin rx)e−Kr t ,
2
(118)
r=1
with
1
a0 =
2π
2π
θ (x, 0)dx,
0
1
ar =
π
2π
θ (x, 0) cos rxdx,
0
1
and br =
π
2π
θ (x, 0) sin rxdx.
(119)
0
125 Fourier (1807), Ref. 115, 55–81. Fourier’s index j for the simple modes corresponds to my k+1.
414
O. Darrigol
In the particular case t = 0, Fourier thus obtained the following trigonometric
development of any function f (x) defined over the interval 0 ≤ x ≤ 2π :
1
f (x) =
2π
2π
f (ξ )dξ
0
⎤
⎡
2π
2π
∞
1 ⎣
cos rx f (ξ ) cos rξ dξ + sin rx f (ξ ) sin rξ dξ ⎦, (120)
+
π
r=1
0
0
in which he recognized an extension of the theorem he had earlier obtained in
the context of the lamina problem.126
This new derivation had important consequences. As Fourier believed that it
only involved “the ordinary principles of calculus,” he became convinced that
it applied to arbitrary functions. As he later explained to Lagrange127 :
By the method of approximation I had obtained the development of a
function in sines and cosines of multiple arcs. Having next resolved the
question of an infinity of bodies that communicate heat to each other,
I recognized that this development had to apply to an arbitrary function as well and I reached by an entirely different route [the equation
∞
π
sin rx 0 f (ξ ) sin rξ dξ ], which I had already obtained.
f (x) = π2
r=1
This is not all. Having used the orthogonality of simple modes to derive the
analogous formula in the discrete case, Fourier thought of doing the same in
the continuous case. Namely, he used the relations
π
sin rx sin sx dx =
π
δrs etc.,
2
(121)
0
which are the continuous counterpart of the relations (111–114), to derive his
fundamental formula a third time:
Seeking to verify the same theorem a third time, I used the procedure
which consists in multiplying by [sin rx] the two sides of [a sine development] and integrating from x = 0 to x = π .
Fourier was clearly unaware of Euler’s anterior use of the same strategy.128
A couple of critical remarks are in order. Fourier’s second derivation was
quite similar to Lagrange’s derivation of a similar formula in his first memoir
126 Fourier (1807), Ref. 115, 284–288.
127 Ibid, 286; Fourier to Lagrange, Ref. 120, 21.
128 Fourier (1807), Ref. 115, 216–217; Fourier to Lagrange, Ref. 120, 21.
The acoustic origins of harmonic analysis
415
on sound. This similarity extends to the major defect of the two derivations:
Lagrange and Fourier both ignored the fact that the series obtained by taking
the limit of infinite n in the successive terms of the series for the discrete problem did not necessarily converge toward the limit of its sum. As we know, the
convergence requires restrictions on the function representing the initial string
curve or the initial temperature distribution. Far from worrying about this difficulty, Lagrange and Fourier believed that their limit of the discrete problem
justified the arbitrariness of this function.
Fourier’s third derivation of his theorem was even more problematic. Fourier
announced this derivation as follows:
∞
π
sin rx 0 f (ξ ) sin rξ dξ
I am now going to show that the equation f (x) = π2
r=1
always holds whatever be the nature of the proposed function f (x).
Hopefully, he only meant that the sine development of a function should take
this form if it exists. Contemporary readers did not interpret him so generously,
for he later had to explain that this method did not teach anything about the
convergence of the series nor on its ability to represent the proposed function.
In modern terms, he realized that the orthogonality of the sine functions of
multiple arcs did not imply their completeness.129
Albeit for bad reasons, Fourier’s detour through the discrete problems convinced him that his fundamental theorem held for arbitrary functions “whether
the nature of the function can be expressed by the known means of analysis,
or it corresponds to a curve drawn in any, entirely arbitrary manner.” He confidently applied his theorem to any function for which he could easily compute
the integrals (119) that give the trigonometric coefficients.130
His first heterodox result was the development of the function cos x as a
series of sines over the interval 0 < x < π , which violated the preconception
that the developed function should be odd. Most daring were his development
(x) =
∞
4 (−1)p
sin(2p + 1)x
π
(2p + 1)2
(122)
p=0
for the “triangular” function (x) that takes the value x for 0 ≤ x ≤ π/2 and
the value π − x for π/2 ≤ x ≤ π , and his development
χa (x) =
∞
2 1
(1 − cos ra) sin rx
π
r
(123)
r=1
for the function χa (x) that takes the value 1 for 0 < x < a and 0 for a < x < π ,
and other developments for non-analytic functions. As Fourier perfectly knew,
129 Fourier (1807), Ref. 115, 216; note 9 (1808–1809) to this memoir, in Herivel, Ref. 117, Appendix
3, 63–64.
130 Fourier (1807), Ref. 115, 225.
416
O. Darrigol
the former series accomplished what d’Alembert had declared impossible, and
the latter what Euler had declared impossible. Fourier concluded that “his
results fully confirmed the opinion of Daniel Bernoulli.” He of course had in
mind the quarrel over vibrating strings. In the memoir submitted to the Institut
in 1807, he proudly announced:
The application of [our] principles to the question of the motion of vibrating strings has solved all the difficulties encountered by Daniel Bernoulli’s
analysis. Indeed the solution proposed by this great geometer did not seem
applicable to the case when the initial figure of the string is a triangle or
a trapeze, or such that only one part of the string is disturbed while the
other parts coincide with the axis.
To be true, not even Bernoulli ventured to apply trigonometric series to these
cases.131
After this major mathematical discovery, Fourier returned to the annulus.
He now took external conductivity into account, in which case the equation for
heat propagation reads
∂θ
∂ 2θ
= K 2 − hθ ,
∂t
∂x
(124)
where K and h denote the internal and external conductivities divided by the
heat capacity. Fourier first obtained the exponential solutions in the steady case,
then solved the time-dependent case by superposition of simple modes. The partial modes decay exponentially, at a rate increasing with the order of the mode.
Fourier’s discussion of this behavior resembled his discussion of propagation
along the lamina. He noted that after a sufficiently long time, only the first mode
survived, and that the higher modes “disappeared one after the other.”132
In the following months, Fourier considered the more difficult cases of the
propagation of heat within a homogenous sphere, a cylinder, and a cube. Perhaps
in the context of the sphere, he realized that the equation for the heat motion
within the body should not involve external conductivity. He introduced the
concept of heat flux through an internal surface element, and argued by purely
macroscopic, semi-empirical reasoning that this flux should be proportional to
the temperature gradient along the normal to the surface. Together with the
conservation of heat, this assumption implies the fundamental equation
2
∂ θ
∂ 2θ
∂ 2θ
∂θ
,
=K
+
+
∂t
∂x2
∂y2
∂z2
(125)
131 Fourier, introduction to an early draft (1806?) of his memoir of 1807, together with MS 22525,
Ref. 114, in Grattan-Guinness, Ref. 2, 182–186, on 183; Fourier (1807), Ref. 115, 250–251.
132 Ibid., 256–280, on 280. Fourier performed his first annulus experiments in 1806: cf. FO2, 69–70.
The acoustic origins of harmonic analysis
417
to which Fourier added the boundary condition that the flux across the external surface of the body should be proportional to the difference between the
temperature at the surface and the temperature of the environment.133
Fourier’s solutions for the spherical and cylindrical cases required powerful
extensions of harmonic analysis, involving simple modes with incommensurable
frequencies, Bessel functions (for the cylinder), and transcendental equations.
As Fourier himself realized, these parts of his theory of heat were the most
innovative and the most worthy of Lagrange’s and Laplace’s consideration.
There is no need to discuss them here, however, because they do not tell much
more about the nature of the connection between Fourier’s heat theory and
earlier acoustics.134
More important for our purpose is Fourier’s insistence on the physical reality
of simple modes. In the oral presentation of his theory to the French Academicians on 21 December 1807, he described the cooling of a body in the following
terms:
The system of initial temperatures can be such that the ratios originally
established among them persist without any alteration during the whole
process of cooling. This singular state that enjoys the property of subsisting
once it is formed can be compared to the figure that a sonorous string takes
when it yields the principal sound. It can take diverse analogous forms,
the ones corresponding to the subordinate sounds [harmonics] in the case
of the elastic string. Consequently, for every solid there is an infinite number of simple modes according to which heat can propagate and dissipate
without change of the initial distribution law…. Whatever be the manner
in which the different points of the body have been heated, the initial and
arbitrary system [of temperatures] can be decomposed into several simple
and durable states similar to those I just described. Each of these states
subsists independently of all the others and undergoes no other changes
than those which would still occur if it were alone. The relevant decomposition is not a purely rational and analytical result; it occurs effectively
and results from the physical properties of heat. Indeed the speed with
which the temperatures decrease in each simple system is not the same
for every system…. To be true, these properties are not always as sensible
as the isochronism of pendulums and the multiple resonance of vibrating
strings are; but they can be established by observation and they became
manifest in all of my experiments.
Fourier thus ascribed the same degree of physical reality to the harmonics
of a vibrating string and to the partial modes of heat propagation. In the introduction to his treatise of 1822, he framed this reification of harmonic analysis
within a Laplacian glorification of mathematical physics:
133 Fourier (1807), Ref. 115, 91–129. Cf. Herivel, Ref. 2, 180–189; Dhombres and Robert, Ref. 2,
480–515.
134 Fourier (1807), with comments by Grattan-Guinness, Ref. 2, 289–419; Fourier to Laplace and
Lagrange, in Herivel, Ref. 117.
418
O. Darrigol
Mathematical analysis therefore has necessary relations with sensible phenomena; its object is not created by the intelligence of man, it is a preexisting element of the universal order and has nothing contingent or
fortuitous; it impregnates all of nature.
Or else:
Analytical analysis extends as much as nature itself; it defines every sensible relation, measures times, spaces, forces, and temperatures; this difficult science forms slowly, but it conserves all the principles that it once
acquired; it grows and strengthens incessantly amidst so many variations
and errors of the human mind.
And the strikingly musical pronouncement:135
If the order that takes place in these phenomena [of heat propagation]
could be seized by our senses, it would cause us an impression comparable
to the harmonic resonances.
5.5 A problematic reception
Toward the end of 1807, Fourier submitted to the French Academy a long
memoir entitled Théorie de la propagation de la chaleur dans les solides. This
memoir contained the various results he had reached so far, rearranged in an
order differing from the chronological order of discovery: the discretized bar
and the discretized annulus, the general equations for a continuous body, the
lamina, Fourier’s theorem by elimination, the same theorem by orthogonality
of simple modes, the continuous annulus, the transition from the discrete to the
continuous annulus, the sphere, the cylinder, the cube, and experiments.136
Somewhat strangely, Fourier presented three derivations of his fundamental
theorem instead of selecting the most direct one. One may surmise that the
derivation by elimination had given him too much sweat for being left aside.
In a note added to his memoir in 1808, Fourier explained that the three proofs
complemented each other. The proof by elimination established the existence
of the trigonometric development but only for indefinitely differentiable functions. The proof by orthogonality gave the form of the coefficients without
establishing the existence of the development. The proof by taking the limit
of the discrete problem seemed to give both the existence and the form of the
development for arbitrary functions, but depended on an artificial model of
heat propagation. Lastly, Fourier gave a rigorous proof of convergence in the
particular case of the trigonometric series for a linear function.137
135 Fourier, “Extrait du mémoire sur la chaleur” (read at the Institut on 21 Dec 1807), MS 1851
of the Bibliothèque de l’Ecole Nationale des Ponts et Chaussées, in Herivel, Ref. 117, 53–58, on
55–56 (similar statements are found in FO1, 244, 528, 531); Fourier, Théorie analytique de la chaleur
(Paris, 1822), also in FO1, on 14, XXIII-XXIV. Cf. Dhombres and Robert, Ref. 2, 541–549.
136 Fourier (1807), Ref. 115.
137 Fourier, note 9, in Herivel, Ref. 117, 63–64.
The acoustic origins of harmonic analysis
419
The four examiners of Fourier’s memoir, Lagrange, Laplace, Monge, and
Lacroix never wrote the expected report. Instead Laplace’s star pupil Siméon
Denis Poisson published a brief summary of Fourier’s main results in the March
1808 issue of the Bulletin de la Société Philomathique. The examiners must have
thought that Fourier’s memoir should not be published in its present shape.
Unfortunately, there is almost no written record of their objections. For the
most, these must be inferred from letters that Fourier wrote to Lagrange and
Laplace and from justifying notes he added to his manuscript in 1808–1809.138
The only extant objection to Fourier’s theory is a two-page note in the collection of Lagrange’s manuscripts at the Bibliothèque Mazarine. There Lagrange
argues that Fourier’s identity
∞
(−1)n−1
n
n=1
sin nx =
x
2
for − π < x < π ,
(126)
leads to an absurdity. He first shifts the variable x by π to get
∞
π
1
1
sin nx = − x
n
2
2
for 0 < x < 2π .
(127)
n=1
Then he derives to get
∞
n=1
cos nx = −
1
2
for 0 < x < 2π .
(128)
Lastly, he integrates the latter identity from 0 to x to obtain
∞
1
1
sin nx = − x for 0 < x < 2π ,
n
2
(129)
n=1
which contradicts (127).139
Part of Fourier’s letter to Lagrange seems to be a reply to this objection.
As he could not accept Lagrange’s recourse to the series (128), he answered by
providing an improved version of his indeed rigorous proof of the identity (126).
More broadly, he defended his fundamental theorem by giving a short history
of the three derivations of his fundamental theorem. Another of Lagrange’s
138 Siméon Denis Poisson, “Mémoire sur la propagation de la chaleur dans les solides (extrait),”
Société philomathique de Paris, Nouveau bulletin des sciences, 1 (1808), 112–116, also in FO2,
213–221. Cf. Grattan-Guinness, Ref. 2, 442–443; Herivel, Ref. 117, 153.
139 Lagrange, MS 40 of Paquet 3, Bibliothèque de l’Institut, Paris. The argument is echoed in a
short MS note by Fourier in ffr 22529, folio 126. In an appended comment to Lagrange’s note (MS
40), Prony, Poisson, Legendre, and Lacroix accepted the series (128) but argued that its sum became
infinite for x = 0 so that the integration from 0 became meaningless.
420
O. Darrigol
objection must have been that Fourier did not refer to relevant work by his
predecessors. In particular, he or Lacroix must have told Fourier that Euler had
already used the orthogonality of cosines to derive the expression of the Fourier
coefficients of a trigonometric radical. Fourier apologized for not having been
aware of this work, emphasized the much greater span of his own theorem, and
flattered Lagrange: “If I had had to cite…a few works, it would have been yours,
which I carefully read in the past…and which contain a multitude of elements
similar to the ones I have used.”140
Lagrange could hardly have been satisfied with this reply. He must have felt
that Fourier had simply rediscovered a theorem he had known for himself since
his first researches on sound. Surely, following his debate with d’Alembert the
way in which trigonometric series converged toward the developed function
had become obscure to him. But he never doubted that they converged in
some sense, which he called “asymptotic.” The general strategy of projecting
functions over eigenfunctions of certain operators belonged to him, and immediately gave Fourier’s theorem in the simplest case of the vibrating string. As
far as rigor was concerned, Fourier’s only improvement was his irreproachable
derivation of the trigonometric series in the simple cases of a constant or a linear
function. His elimination strategy worked only in cases that Lagrange himself
judged unproblematic, while his proof based on taking the limit of the discrete
problem had exactly the same flaws as Lagrange’s own similar considerations.
Lagrange graciously refrained from directing Fourier to his relevant memoirs.
Instead he soon integrated the relevant section of his first memoir on sound in
the second edition of his Mécanique analytique.141
From Fourier’s contemporary letter to Laplace, we learned that the latter
geometer objected to the development of the sine as a series of cosines, presumably because a sine is odd whereas a cosine is even. Fourier easily answered
that the conflict disappeared when the domain of validity of the formulas was
properly taken into account. He added that all the series he had given were
rigorously convergent. In a contemporary manuscript note, Fourier answered
Laplace’s more serious objections regarding the derivation of the fundamental
equation of heat propagation and boundary conditions. Laplace still believed
that Fourier had not solved the difficulty of differential heterogeneity resulting
from Biot’s naïve application of Newton’s law of cooling to contiguous infinitesimal particles, and he indicated that in a proper derivation heat exchange should
occur between particles separated by a small finite distance. Possibly, Laplace
had only read Fourier’s first, problematic derivation, which was included in the
early version of his memoir that Fourier gave to Laplace and Biot in 1806. In his
140 Fourier to Lagrange, Ref. 120, 22. In his Théorie analytique of 1822, Ref. 138, par. 428, Fourier
wrote: “On trouve dans les ouvrages de tous les géomètres des résultats et des procédés de calcul
analogues à ceux que nous avons employés. Ce sont des cas particuliers d’une méthode générale
qui n’était point encore formée, et qu’il devenait nécessaire d’établir pour connaître, même dans les
questions les plus simples, les lois mathématiques de la distribution de la chaleur.” An earlier MS
version of this comment (ffr 22529, folio 34r) has “les oeuvres d’Euler, de Clairaut, de Lagrange et
de Daniel Bernoulli” instead of “les ouvrages de tous les géomètres.”
141 Lagrange, Ref. 49, 421–442.
The acoustic origins of harmonic analysis
421
note of 1808, Fourier referred to his new reasoning based on the concept of heat
flux, and also introduced a few molecular considerations in a more Laplacian
style. The stakes were high, because Fourier’s priority as the discoverer of the
fundamental equation depended on the validity of the reasoning found in his
memoir of 1807. For several years this issue embittered the relations Fourier
had with Laplace and Biot.142
After reading Fourier’s memoir, Laplace provided his own derivation of the
heat equation, with the comment: “I must note that Mr. Fourier has already
obtained these equations, whose true foundation seems to me to be those I
have just given.” He also discovered the integral form of the solution in the
one-dimensional case for which the heat equation reduces to
∂ 2y
∂y
= 2.
∂t
∂x
(130)
A power-series development à la Lagrange yields the solution
y(x, t) =
∞ n
t (2n)
Y (x)
n!
if y(x, 0) = Y(x) for any x.
(131)
n=0
This series differs from a Taylor development by having n! instead of (2n)! in
the denominators (and by not including the odd-order derivatives). Astutely
compensating for this difference by means of the well-known identity
(2n)!
1
=√
n!
π
+∞
2
(2u)2n e−u du,
(132)
−∞
Laplace obtained
1
y(x, t) = √
π
+∞
∞
−∞ n=0
tn (2u)2n (2n) −u2
Y e du.
(2n)!
(133)
As the missing odd terms of the series would not contribute to the integral, this
is the Taylor development of
1
y(x, t) = √
π
+∞
∞
√
2
Y(x + 2u t)e−u du.
(134)
−∞ n=0
142 Fourier to Laplace (undated), in Herivel, Ref. 117, 24–26; Fourier, MS ffr 22501, folio 76r–81r,
in Herivel, Ref. 117, 28–35. On Fourier sending a draft to Laplace and Biot, cf. Fourier to Lagrange,
Ref. 120; The text reproduced in Grattan-Guinness, Ref. 2, 182–186, probably was the introduction
of this draft.
422
O. Darrigol
As Ivor Grattan-Guinness remarks, this form of the solution is analogous to
the d’Alembert-Euler solution of the problem of vibrating strings inasmuch
as it directly relates the configuration at any time to the initial configuration.
However, Laplace’s and Poisson’s preference for this type of solution over
Fourier’s trigonometric solution was not a simple repeat of d’Alembert’s and
Euler’s opposition to Bernoulli’s solution, because Laplace and Poisson did not
challenge Fourier over the generality of his solutions.143
In response to Laplace’s contribution, Fourier extended his method to the
case of an infinite body, for which the spatial frequencies of the simple modes
form a quasi-continuum. He thus obtained the Fourier-integral type of solution,
which he showed to be related to the Laplacian form. Friendly discussions with
Laplace also led to new developments on terrestrial temperatures and radiant
heat. In 1811 Fourier reworked his memoir to include these topics and sent the
result to the Academy of sciences, which was offering a prize for the mathematical theory of heat and its experimental verification. The jury composed of
Lagrange, Laplace, Lacroix, Malus, and Haüy selected Fourier’s memoir, with
the following comment:144
This piece contains the true differential equations for the transmission of
heat, be it inside bodies or at their surface; the novelty and importance of
the subject determined the Class to crown this work, although it must be
noted that the manner in which the author arrives at his equations is not
devoid of difficulties and that his analysis, to integrate them, still leaves
something to be desired, either regarding generality, or even regarding
rigor.
This far-from-enthusiastic endorsement and the earlier lack of official
response from the academicians have usually been regarded as the consequence
of Lagrange’s hostility to Fourier’s theory. There are reasons to think, however,
that Lagrange did not deny the correctness of Fourier’s theorem and that the
objections to Fourier’s work came at least in part from other academicians,
especially Laplace. Lagrange’s acceptance of Fourier’s theorem is evident in his
statement of 1811 that the infinite superposition of simple modes “rigorously
gives the motion of [a vibrating] string at any time.” What may have antagonized him was not the theorem itself, but Fourier’s lack of reference to his own
143 Laplace, “Sur les mouvements de la lumière dans les milieux transparents,” Institut de France,
Mémoires de la classe des sciences mathématiques et physiques (1809) [pub. 1810], 300–342, on 338,
also in Œuvres, vol. 12, 265–298, on 295, and in Herivel, Ref. 117, 78–80; “Mémoire sur divers points
d’analyse,” JEP, 15 (1809), 229–264, also in LO14, 178–214, on 184–214. Cf. Grattan-Guinness, Ref.
2, 446–447.
144 Fourier, “Théorie du mouvement de la chaleur dans les corps solides” (1811), pub. in Académie
Royale des Sciences, Mémoires, 4 (1819–1820) [pub. 1824], 185–555, and ibid. 5 (1821–1822) [pub.
1826], 153–246; second part also in FO2, 1–94; Lagrange et al., jury report, quoted in FO1, vii–viii.
On the Fourier transform, cf. Grattan-Guinness, Ref. 2, 448–450; Dhombres and Robert, Ref. 2,
591–599. On terrestrical temperatures and radiant heat, cf. Grattan-Guinness, Ref. 2, 449; Herivel,
Ref. 2, 197–205. On the prize problem, cf. Grattan-Guinness, Ref. 2, 451–452. Further discussion
of Fourier’s theory is in Grattan-Guinness, Convolutions in French mathematics. 1800–1840, 3 vols.
(Basel, 1990), vol. 2, Chap. 9: “The entry of Fourier.”
The acoustic origins of harmonic analysis
423
anticipation of the theorem and perhaps also Fourier’s reification of the partial
modes.145
The reproach about difficulties in Fourier’s derivation of the heat equation almost certainly came from Laplace, who regarded molecular action at
a distance as the true foundation of any physical theory. The reproach about
generality and lack of rigor in Fourier’s solutions of this equation could equally
well come from Lagrange and from Laplace. Laplace’s protégé Poisson later
repeated it almost word for word. And it was a perfectly legitimate one, since
at that time Fourier had not even attempted a general proof of his theorem. As
is well-known, he only did so in his treatise of 1822.146
This proof has some analogy with Lagrange’s old discussion of the sum of
an infinite number of simple modes. Like Lagrange, Fourier had no qualm
rewriting
+∞ 1 f (x) =
dXf (X) cos n(x − X)
2π n=−∞
+π
(for − π < x < π )
(135)
−π
as
f (x) =
1
2π
+π
dXf (X)
+∞
cos n(x − X),
(136)
n=−∞
−π
although in his “construction” of this identity he interpreted it as
1
f (x) = lim
N→∞ 2π
+π
dXf (X)
−π
+N
cos n(x − X).
(137)
n=−N
Again like Lagrange, Fourier based his derivation or construction on the trigonometric identity
N (ξ ) =
+N
n=−N
cos nξ =
sin ξ
cos Nξ − cos(N + 1)ξ
= cos Nξ + sin Nξ
1 − cos ξ
1 − cos ξ
(138)
and on the equivalence of this sum with a periodic series of impulsions when
N becomes infinite. The main difference with Lagrange’s reasoning concerns
145 Lagrange, Ref. 49, 425.
146 Poisson, “Extrait d’un mémoire sur la distribution de la chaleur,” Journal de physique et de
chimie, 80 (1815), 434–441, on 440. On Poisson’s attitude, cf. Herivel, Ref. 2, 126, 175–176. The
general proof of Fourier’s theorem is in Fourier (1822), Ref. 135, 494–499: cf. Grattan-Guinness,
Ref. 24, Chap. 5; Ref. 2, 471–473.
424
O. Darrigol
the proof of this equivalence. Lagrange would here return to a discrete model
in which the sum N (ξ ) vanishes for every permitted value of the variable ξ
except for the value ξ = 0 for which this sum is 2N. Instead, Fourier argued
that
for any continuous (in our sense) function f and for any small ε the integral
N (ξ )f (x + ξ )dξ over the domain [x − π , −ε] ∪ [ε, x + π ] (with −π < x < π )
vanished in the limit of infinite N because of the fast oscillations of the cos Nx
and sin Nx factors in the expression (138) of N (ξ ). For the remaining integral
over the tiny interval [−ε, ε], Fourier exploited the smallness of ε and again
the largeness of N to reach the approximations
+ε
+ε
+π
N (ξ )f (x + ξ )dξ ≈ f (x) N (ξ )dξ ≈ f (x) N (ξ )dξ = 2π .
−ε
−ε
(139)
−π
Altogether, we have
1
2π
+π
−π
1
dXf (X)N (x − X) =
2π
≈
1
2π
x+π
N (ξ )f (x + ξ )dξ
x−π
+ε
N (ξ )f (x + ξ )dξ ≈ f (x),
(140)
−ε
which leads to the desired result (137) in the limit of infinite N and infinitely
small ε.
This is the basis of Gustave Lejeune Dirichlet’s later rigorous proof of
Fourier’s theorem, which closed a long chapter of the history of trigonometric
series. From then on, Fourier analysis became as respectable as any other part of
mathematics. The stage was set for amazingly rich developments ranging from
the foundation of mathematical analysis to numerous physical and technical
applications. Interestingly, one of these applications was the Ohm-Helmholtz
theory of consonance. Harmonic analysis thus returned to its then forgotten
acoustic origins.147
147 Gustav Lejeune Dirichlet, “Sur la convergence des séries trigonométriques qui servent à
représenter une fonction arbitraire entre des limites données,” Journal für die reine und angewandte Mathematik, 4 (1829), 157–169. On the Ohm-Helmholtz theory, cf. Turner, Ref. 1.