the acoustic origins of harmonic...

Arch. Hist. Exact Sci. (2007) 61:343–424DOI 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 mathemat-ics, a harmonic function is a sine function obtained by projecting a circularmotion on a diameter, and harmonic analysis is the theory of the develop-ment of periodic functions into harmonic components, or the theory of similardevelopments. The occurrence of the same word in musical and mathematicalcontexts is neither a coincidence nor a purely metaphorical effect. As is wellknown, in the mid-nineteenth century Hermann Helmholtz connected the twomeanings through the idea that the ear functions as a harmonic analyzer in aphysico-mathematical sense.1

This formulation of Helmholtz’s achievement suggests that harmonic anal-ysis preceded its acoustic and musical application. So does too the fact thatJoseph 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 ofphysiological 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 desBelles-Lettres de Berlin, Histoire; HAS, Académie Royale des Sciences, Histoire; JEP, Journal del’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 isused 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, Francee-mail: [email protected]

344 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 har-monic analysis are to be found in Daniel Bernoulli’s much earlier work on thetheory 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 math-ematical harmonics, from Bernoulli’s earliest intuitions to Fourier’s perennialfoundation of harmonic analysis.

Much valuable commentary has been written both on the early theory ofvibrating bodies and on Fourier’s theory of heat. In particular, the present studylargely benefited from histories of the former theory by Heinrich Burckhardtand by Clifford Truesdell, and from studies of the latter theory by Ivor Grattan-Guinness, John Herivel, Jean Dhombres, and Jean-Bernard Robert. However,relatively little attention has been paid to the persisting connections betweenmusic theory, acoustics, and harmonic analysis. Eighteenth-century works onvibrating strings have been studied independently of music theory, even thoughtheir authors were deeply involved in it; and the genesis of Fourier’s theory ofheat has been studied independently of its acoustic antecedents, even thoughFourier was quite aware of them. In both cases, proper attention to acousticand musical contexts leads to valuable insights.2

Firstly, it appears that important aspects of work on vibrating strings byJean le Rond d’Alembert, Leonhard Euler, and Joseph Louis Lagrange havebeen misunderstood or neglected. For instance, d’Alembert’s instance thatBernoulli’s mixtures of sine curves could not explain the hearing of simulta-neous harmonics from a single vibrating string has been regarded as erroneousreasoning, whereas it is perfectly consistent with d’Alembert’s ideas on the emis-sion of sound. More important, Lagrange’s objections to Bernoulli’s mixtureshave been mistaken for a rejection of the mathematical validity of trigonometricseries whereas they only concerned the physical interpretation of the harmoniccomponents. With this understanding, it becomes clear that Lagrange’s secondmemoir on sound contains much of harmonic analysis. In modern terms, Lag-range had the general notion of the development of a function over the eigen-functions of a Hermitian differential operator, was aware of the mutual orthog-onality of the eigenfunctions, and used this property to express the coefficientsof 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, Jahresbe-richt, 10 (1901–1908), 1800 pages; Clifford Truesdell, “The rational mechanics of flexible and elasticbodies. 1638–1788,” EO2:11(2); Ivor Grattan-Guinness in collaboration with Jerome Ravetz, JosephFourier 1768–1830. A survey of his life and work, based on a critical edition of his monograph onthe 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 completenessof the eigenfunctions, however).

Secondly, the opinions of eighteenth-century string-theorists about the gen-erality and physical meaning of Bernoulli’s trigonometric sums and their ideason the foundations of music are both seen to depend on their ideas on thephysical nature of (musical) sound. Euler, d’Alembert, and Lagrange all agreedthat musical sounds (tones) should be regarded as periodic repetitions of pulseswhose precise shape did not matter much. Accordingly, they denied any phys-ical meaning to harmonic analysis. With regard to music theory, Euler andLagrange defined the harmony of two sounds through the frequent coinci-dence of the periodic pulses corresponding to two different tones; d’Alembertadopted Jean-Philippe Rameau’s idea of founding musical harmony on thesimultaneous hearing of harmonics from a single sonorous body, but regardedthis fact as irreducible to the periodicities of the physical motions associatedwith sound. Against these three geometers, Bernoulli understood sound as amixture of sinusoidal motions with separate physical existence. Accordingly,he regarded trigonometric series as a general, physically meaningful way ofdescribing any vibration. About musical harmony, he approved Rameau’s sys-tem, but explained the importance of harmonics in the definition of harmonyby considerations of periodicity.

It may seem strange that d’Alembert and Bernoulli both supported Ra-meau’s system of music and yet disagreed on the significance of harmonicanalysis, or that Euler and d’Alembert joined forces against harmonic analysis,and yet defended conflicting systems of music. The reason for these anomaliesis that their assessment of the connection between harmonic analysis and musicdepended on their understanding of the perception of sound and also on theirgeneral views on the relations between mathematics, physics, and music. Abouthearing, Euler and Lagrange maintained the old scheme according to which theaural nerves detect the beating of the eardrum by incoming pulses; Bernoullianticipated the modern idea of the ear as a harmonic analyzer; d’Alembertprofessed an ignorabimus on the matter. In their broader philosophy, Eulerand Lagrange anchored physics and music on mathematics; Bernoulli ratheradjusted mathematics to physics and music; d’Alembert defended a partialautonomy of physics with respect to mathematics, and ridiculed the reductionof music to mathematics.

A third conclusion of the present approach is that Fourier’s heat theorydepended on earlier, acoustics-driven harmonic analysis to a much larger extentthan is usually assumed. It is in the context of a discussion of Lagrange’s generaltheory of vibrations that Fourier first expressed his enthusiasm for harmonicanalysis, ten years before he began to work on heat. His first discrete model ofheat propagation mimicked Lagrange’s theory of the discretely loaded string,of which he was aware. His main argument for the generality of trigonomet-ric 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,

346 O. Darrigol

Fourier was prompt to apply his new analysis to the vibrating string, with properhomage to Bernoulli’s anticipations.3

In this light, it becomes probable that Lagrange’s reaction to Fourier’s mem-oir did not amount to the full-fledged rejection of Fourier series suspected bymany commentators. There is clear evidence that Lagrange believed that anyfunction could be in some sense developed in a trigonometric series over agiven interval. What he truly objected was the physical existence of the sinecomponents of string (and heat) motion, and presumably some of Fourier’salleged proofs of convergence. In essence his position remained the same asin his old memoirs on sound, and his attitude toward Fourier’s theory musthave resembled his earlier attitude toward Bernoulli’s mixtures. Although heacknowledged the mathematical validity of trigonometric developments, herejected their physical import. So to say, Fourier combined Bernoulli’s phys-ical understanding of partial modes with Lagrange’s algebraic understandingof their mathematical properties to obtain a far-reaching and better-foundedharmonic analysis.

Section 1 of this paper is devoted to seventeenth-century notions of har-monics and harmony in music, and to vibration theory in the first half of theeighteenth-century. Section 2 revisits the quarrel over vibrating strings, with afocus on the status of Bernoulli’s mixtures of sine curves. Section 3 analyzesLagrange’s memoirs on sound and their impact on this quarrel. Section 4 con-tains a summary of the views of the main string theorists on musical theory, inconnection with their views on harmonic analysis. Section 5 is devoted to thegenesis 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 harmoniousmusical intervals. The expression of these intervals by ratios of small integers is aPythagorean notion, probably based on the corresponding lengths of the vibrat-ing string on a monochord. The interpretation of sound as a vibration also goesback to antiquity, although its more precise expression waited early-modernmechanical philosophy.4

In the seventeenth century, Galileo Galilei and Marin Mersenne popularizedthe correspondence between pitch and frequency that Giovanni Battista Bened-

3 About the origin of Fourier’s trigonometric solutions, Grattan-Guinness (Ref. 2, 441) writes: “Itfirst arose by accident…(and also perhaps from Fourier’s knowledge of eighteenth-century super-position of special solutions).” Dhombres and Robert (Ref. 2, 171) note Fourier’s early familiaritywith 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 scientificrevolution, 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; PatriceBailhache, Une histoire de l’acoustique musicale (Paris, 2001), Chaps. 1–3; Truesdell, Ref. 2, 15–16.


etti, Galileo’s father Vincenzo, and Isaac Beeckman had earlier articulated.Accordingly, a musical tone is a periodic succession of pulses transmitted by theair to the eardrum. The number of pulses in a time unit determines the pitch ofthe tone. Two tones are said to be consonant if and only if their frequency ratiois reducible to the ratio of two small integers. Galileo justified this definition byarguing that the frequent coincidence of the pulses striking the eardrum was asource of delight:5

The length of strings is not the direct and immediate reason behind theforms of musical intervals, nor is their tension, nor their thickness, butrather, the ratio of the numbers of vibrations and impacts of air waveson our eardrum, which likewise vibrates according to the same measureof time. This point established, we may perhaps assign a very congruousreason why it comes about that among sounds different in pitch, somepairs are received in our sensorium with great delight, others with less,and some strike us with great irritation. Hence the first and most welcomeconsonance is the octave, in which for every impact that the lower stringdelivers to the eardrum, the higher gives two [and so forth].

Marin Mersenne’s monumental Harmonie universelle of 1636 relied on a sim-ilar explanation of consonance as the coincidence of pulses, although Mersennerealized that the simplicity of a frequency ratio did not necessarily correspondto consonance as agreed by musicians. Nearly a century elapsed before Ra-meau, Euler, and Tartini truly improved on this explanation. In his discussionof string instruments, Mersenne nonetheless described one the basic facts ofRameau’s later theory: that under quiet condition and with proper experiencehe and many musicians could hear at least four tones at a time from the thickeststrings of a viola bass: the natural tone, the octave above, the twelfth, and theseventeenth:6

The string struck and sounded freely makes at least five sounds at the sametime, the first of which is the natural sound of the string and serves as thefoundation 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 inthe observed motion of strings:

5 Galileo Galilei, Discorsi e dimostrazioni matematiche: Intorno à due nuoue scienze attenenti allamecanica 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 uniformtranslation of the French “battement” or “coup” and of the Latin “pulsus” or “ictus.” While mostauthors understood sound propagation in analogy with water waves, Beeckman traced it to theproduction 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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[Since the string] produces five or six tones…, it seems that it is entirelynecessary that it beat the air five, four, three, and two times at the sametime, which is impossible to imagine unless one says that half of the stringbeats the air twice, one third beats it three times, etc. while the wholestrings beats it once. This picture runs against experience, which clearlyshows that all parts of the string make the same number of returns in thesame time, because the continuous string has a single motion, even thoughparts near the bridge move more slowly.

Mersenne went on to speculate than some peculiar reaction of the air to thebeats of the string might explain the higher tones. More eloquent was his moralcorollary:7

If the sound of each string is the more harmonious and agreeable as ahigher number of different sounds is emitted at the same time and if itis permitted to compare the moral actions to the natural ones and totranspose physics to human actions, we may say that each action is themore harmonious and agreeable to God as it is accompanied with a largernumber of motivations as long as those are all good.

At the turn of the seventeenth and eighteenth centuries, Joseph Sauveurargued for the very picture of harmonic emission that Mersenne had judgedimpossible:

While meditating on the phenomena of sound, I was made to observe thatespecially at night one may hear from long strings not only the principalsound, but also other small sounds, a twelfth and a seventeenth above….I concluded that the string in addition to the undulations it makes in itsentire length so as to form the fundamental sound may divide itself intwo, three, four, etc. undulations which form the octave, the twelfth, thefifteenth of this sound.

Sauveur obtained the additional modes of vibration by plucking a monochordafter 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 equi-distant 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, whichSauveur called “harmonic” since it was harmonious with the fundamental. Ashe soon found out, John Wallis had already obtained the higher modes of astring by resonance with another string tuned to sound one of the harmonics ofthe former. Sauveur was nonetheless first to exploit these modes to elucidateMersenne’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 systêmeset 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 smalleramplitude than resonance excited by a sub-multiple of the natural frequency of the resonating


Fig. 1 Sauveur’s observationof the higher modes of avibrating string (Ref. 8, 477)

Mersenne and Sauveur of course knew that flutes, trumpets and other blowninstruments produced overtones when blown stronger, as this is the way inwhich higher notes are obtained on these instruments. In this case too, Sauveurbelieved that the corresponding aerial motion had nodes of negligible motion.In his opinion, harmonics could be heard and obtained from any musical instru-ment, or any “resonating and harmonious bodies.” The restriction to “harmo-nious” bodies probably means that he was aware that the overtones of mostvibrating bodies, for instance those of a vibrating blade, do not form a harmonicsequence.9

That Sauveur discovered the higher modes of a vibrating string while med-itating on simultaneous harmonics is a clear indication that he regarded theactual motion of a string as some sort of superposition of the fundamental andthe higher modes. According to Bernard de Fontenelle, Sauveur conceived thematter as follows:10

A harpsichord string being plucked, a good trained ear may hear, besidesthe sound that corresponds to its length, thickness, and tension, soundshigher than that of the whole string, produced by some of its parts andsomehow emerging from the principal vibration to form particular vibra-tions. This complication of vibrations can be conceived through the exam-ple 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 twohands he can give two separate shakes to the two halves of the rope; thetwo 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 itsown vibrations during the total vibration of the whole string.

string, which explains why it was not discovered earlier. The multiple modes of a vibrating stringoccurred in a then well-known string instrument called Trompette marine, which may have inspiredSauveur’s experiments: cf. Dostrovsky, Ref. 4, 194–196, 205; Sauveur, Ref. 8, 406, 483–484. Hereand elsewhere, I use the words “mode” and “overtone” anachronistically, just to avoid long circum-locutions.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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The existence of harmonic overtones played a central role in Sauveur’sconception of musical harmony. In his view, two combined tones were mostharmonious when one was a harmonic of the other. For other combinations heagreed with Galileo and Mersenne that harmony had to do with the frequencyof the coincidences of the periodic pulses of the two tones. When the ratio of thefrequencies was not a simple one, this frequency could be so low as to producethe sensation of a modulated amplitude, a phenomenon that organ-makers hadlong used to tune their pipes, under the name of “beats.” Sauveur explaineddissonance by the unpleasant character of beats. Moreover, he showed that thebeat rate was the difference of the frequencies of the superposed tones and usedthis result to perform some of the first measurements of the absolute frequencyof musical tones.11

In a memoir of 1702 on organ theory, Sauveur showed than organs were builtto imitate the harmonics produced by musical instruments:

By the mixture of its stops, the organ does nothing but imitating theharmony that nature observes in the sounding bodies that are called har-monious; for one can identify the harmonics 1. 2. 3. 4. 5. 6. in bells and inthe long strings of a harpsichord at nighttime.

A stop is the name organ makers give to a series of pipes that can be selectivelyconnected to a keyboard of the organ by pulling a stop (literally). There aresimple stops in which each key controls only one pipe and all pipes are of a sim-ilar kind with length varying according to the pitch. There are compound stopsin which a single key controls several pipes whose characteristic frequencies areharmonics of the same frequency. Moreover, the organist can pull several stopsthat are harmonically related to each other. The previous quotation shows thatSauveur had a notion of timbre, or sound color, corresponding to the harmoniccomposition of the sound. He compared the organist to a painter mixing colorson his palette to reproduce natural hues or to a chef, “who likes his stews milderor spicier.”12

Sauveur’s clear and concise memoirs defined the basis of eighteenth-centuryacoustics, a word he coined to name “a superior science of music…having as itsobject sound in general, while music has as its object sound to the extent thatit is pleasant to the ear.” He systematized the known correspondence betweenfrequency and pitch by denoting tones through their relation to a referencefrequency. He introduced the logarithmic scale of intervals. As we saw, he deep-ened the understanding of harmonic overtones, he explained beats and usedthem 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, on194–195; “Sur un nouveau systême de musique,” HAS (1701), 159–180, on 159–162. Beeckmananticipated both the explanation of beats and their relation to dissonance in his unpublished jour-nal: 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 onbells could not have been accurate.


Fig. 2 Superposition offundamental and octave in thepulse-coincidence view

amplitude

time

There are striking similarities between Sauveur’s acoustics and the mod-ern acoustics based on Fourier analysis. For Sauveur, the sounds produced bymusical instruments correspond to complex periodic vibrations that are mix-tures of simpler periodic vibrations whose frequencies are multiples of thefundamental frequency. The ear is able to analyze these mixtures into theircomponents. The fundamental frequency defines the pitch of the sound, andthe harmonic composition defines the timbre. What is missing is the sine char-acter of the simple modes of vibration. Sauveur, like Galileo and Mersenne,regarded sound as a succession of pulses whose precise shape mattered little tothe 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, whereasa compound sound involved pulses of different sizes. Figure 2 represents thekind of motion he may have had in mind for the superposition of a simple toneand its octave. In this view, the harmonic structure of the compound vibrationis apparent, just as the partial oscillations of the rope of Sauveur’s dancer. Theear’s ability to separate the harmonics of a sound simply corresponds to theear’s ability to detect successive pulses and their magnitude, as would happenwith the beats of a drum.

1.2 Harmonic motion

The first intimation that harmonic (sine-like) motion plays a basic role in acous-tics is found in Christiaan Huygens’s theory of musical strings. In his celebratedHorologium Oscillatorium of 1673, Huygens showed that the pendulous motionof a body sliding down on a cycloid was harmonic and isochronous. Around thattime, he also understood that the force responsible for this motion was propor-tional to the distance traveled by the body from the point of equilibrium. Prob-ably noticing that a similar circumstance held in the case of a tense, weightlesselastic string loaded with one mass in the middle, he derived the oscillationfrequency as a function of tension, length, and mass. The reasoning impliedharmonic oscillations for the loaded string. He also sketched a generalizationto a string loaded with several masses, in which he assumed all the masses toperform harmonic oscillations of the same frequency and phase.14

14 Christiaan Huygens, Horologium oscillatorium sive de motu pendulorum ad horologia apta-to demonstrationes geometricae (Paris, 1673), part 2, Proposition 25; MS fragments, in Oeuvrescomplètes, vol. 18, 489–495. Cf. Truesdell, Ref. 2, 47–49.

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Harmonic oscillations also occur in Isaac Newton’s derivation of the velocityof sound. Like Huygens, Newton relied on analogy with pendulous oscillations,although any other assumption on the form of the oscillations would have led tothe same propagation velocity. In substance, he computed the compression of aslice of elastic fluid owing to the pendulous oscillations of the delimiting planes,and adjusted the phase difference (associated with propagation) so that theresulting pressure gradient would agree with the restoring force of a pendulummimicking the oscillation of a material plane of the fluid.15

More relevantly, in 1713 Brook Taylor gave the first theory of the fundamen-tal mode of a vibrating string. He first showed that the resultant of the tensionsacting on the extremities of an element of the string was proportional to the cur-vature of this element and directed along its normal. By dubious reasoning, hethen predicted that for small vibrations the restoring force was approximatelydirected toward the axis and proportional to the distance from the axis. Grantedthat Newton’s acceleration law applies to mass elements, this implies that allthe elements of the string perform synchronous harmonic oscillations. The pro-portionality between restoring force and distance from the axis can only be trueif the string has the shape of a sine at any time, since in this approximationthe curvature is given by the second derivative of the distance from the axis asa function of the abscissa along it. Taylor overlooked the possibility of highermodes, despite Wallis’s earlier description of them. Worse, he believed that thesimple mode of motion was the only possible motion, apparently misled by theobservation that all the points of a vibrating string reach the axis at the sametime. Of course, he could not have written the differential equation of vibratingstrings, since the calculus of partial differentials did not exist yet. Instead herelied on the pendulum analogy and on some intuition of the desired motion.This sufficed to yield the fundamental mode as well as the correct frequencyformula

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 loadedwith equidistant, discrete masses, presumably to avoid the then intractableproblem of differentials involving two continuous variables. Unfortunately, hefollowed Taylor in assuming that the restoring forces were proportional to thedistances 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 matteris 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 dis-tance 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.


Daniel was first, in 1733, to cash the benefits of discretization in the problemof the small oscillations of a vertically suspended chain. Unlike his father, herealized that complex oscillations were possible with no well-defined frequency.He nonetheless selected the initial conditions so that the restoring force on eachmass should be proportional to its distance from the vertical axis. In the caseof a finite number of masses hanging on a weightless inextensible thread, hefound as many simple modes as there were masses. He then proceeded to thecontinuous, uniform limit in which the shape of the chain is given by our Besselfunctions. In this case, there are infinitely many simple modes with incommen-surable frequencies increasing with the number of intersections with the verticalaxis (Sauveur’s nodes). Lastly, Daniel Bernoulli understood that the case of achain of infinite length formally agreed with that of a musical string and hepointed to the experimental confirmation of higher modes in this case.17

In 1742 Daniel Bernoulli solved the more difficult problem of vibrating elas-tic bands, and again found a series of simple modes with incommensurablefrequencies growing with the number of nodes. In one of the experiments heperformed with a clamped band, he found that the sounds of two differentmodes were heard simultaneously:

Both sounds exist at once and are very distinctly perceived…. This is nowonder, since neither oscillation helps or hinder the other; indeed, whenthe band is curved by reason of one oscillation, it may always be con-sidered as straight in respect to another oscillation, since the oscillationsare virtually infinitely small. Therefore oscillations of any kind may occur,whether the band be destitute of all other oscillation or executing othersat 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 thesuperposition of modes already assumed by Sauveur in the case of vibratingstrings. The argument was necessarily more physical than mathematical, sinceno partial differential equation of motion could yet be written. Clearly, it is thehearing of the sounds of several modes that prompted Bernoulli to imaginesuperposition. That he did not reach the same idea in his earlier discussion ofsimple modes is not surprising: in the suspended-chain case, the visually per-ceived motion is indecipherably complex; in the related vibrated-string case thehearing of simultaneous overtones is more difficult than in the elastic-band casefor which the overtones are dissonant.18

17 Johann Bernoulli, “Theoremata selecta, pro conservatione virium virarum demonstranda etexperimentis 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 oscillationibuscorporum 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. Eulerobtained 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

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At that stage of the history of acoustics, Daniel Bernoulli had in hand thenotion that bodies vibrate according to discrete modes of harmonic motion, orto a superposition of such modes. The frequencies of these harmonic modesare generally incommensurable. In the special case of vibrating strings, they areintegral multiples of the same fundamental frequency. It is not clear whetherBernoulli already believed the most general motion to result from a superpo-sition of discrete modes, since at that time no one had yet solved the generalinitial-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 resultedfrom 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 stringssimply by combining Taylor’s expression of the restoring force with Newton’sacceleration law. For the mathematics, he relied on the notion of differentialform for a function of several variables, earlier developed by Leonhard Eulerand Alexis Fontaine. Treating the ordinate y of a string element as a functionof the time t and the curvilinear abscissa s, d’Alembert formed the successivedifferentials:

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 approx-imate force acting on a string element is Tβds, its mass σds, and its accelerationα. The resulting equation of motion is σα = Tβ, or

σ∂2y∂t2

= T∂2y∂s2 (3)

in modern notation. Having chosen the unit of time so that α = β, d’Alembertastutely 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 functionof 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 con-firmatae,” 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 thoroughbut 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.


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 theordinate y constantly vanishes at the ends s = 0, l of the string implies therelations

= −�, and �(t + l) = �(t − l) at any t. (7)

Therefore, the solution depends on a single generating function �, which mustbe periodic of period 2l and defined over the whole real axis. The initial condi-tions

y(0, s) = Y(s) and y(0, s) = V(s) (8)

determine the function � (up to a constant). Namely, it must the periodicfunction of period 2l such that

�(±s) = ±12

Y(s)+ 12

∫

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 willbe true if the function Y(s) is twice differentiable and the function V(s) is oncedifferentiable. At s = 0, agreement of the right and left derivatives requiresV(0) = 0 and Y ′′(0) = 0. Lastly, agreement of the right derivatives at s = l andthe left derivatives at s = −l requires V(l) = 0 and Y ′′(l) = 0. These condi-tions are automatically satisfied if the boundary conditions and the equation ofmotion hold true.21

Besides these legitimate restrictions, d’Alembert required that the functiony(s, t) should be given by a single equation, valid even for the non-physicalvalues of s. In doing so he was merely following the Leibnizian tradition ofrestricting mathematical analysis to “continuous functions” in the old sense,that is, functions that can be expressed by a single equation (algebraic or tran-scendent) whose range of validity defines the maximal range of the variable.These functions roughly correspond to our analytic functions, since the knowl-edge of their value for any small, finite range of the variable determines theirvalue for the whole range of the variable. Exceptions to this correspondence arefunctions 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 envibration,” 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 tol-erated infinite derivatives and he seems to have confused conditions that only applied to the caseV = 0 with the more general conditions of consistency.

356 O. Darrigol

sequel, quotations marks around “continuous” are used to indicate continuityin the old sense.22

Through the formula (6), Leibnizian continuity implies that the functionsY(s) and V(s) should be periodic odd function of period 2l. This restrictiondid not annihilate d’Alembert’s main conclusion that the problem of vibratingstrings admitted infinitely more solutions than that given by Taylor. After show-ing that Taylor’s solution was a particular case of his for which the generatingfunction was a sine function, d’Alembert rejected Taylor’s claim that any initialcondition 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 pluckedstrings, that is, a triangular shape with no initial velocity. In this case, he sug-gested approximating the continuous string by a massless elastic thread loadedwith a large number of discrete massless string.23

After reading d’Alembert’s memoir, Euler hurried to publish a theory basedon the same wave equation and the same general integral (6), without therestriction that the generating curve and the initial curve of the string should begiven by a single equation. In the case of a string initially at rest, he argued thatthe generating function could easily be constructed from the initial curve of thestring, even if this curve was “irregular and mechanical.” In modern terms, hetolerated any continuous curve with piecewise continuous slope and curvature.The non-existence of the partial derivatives entering the wave equation did notworry him as long as the relation between solution and generating functionhad a well-defined geometric meaning. He later prided himself of covering thecase of the plucked string, for which the initial shape is triangular. We mayretrospectively judge that d’Alembert abusively restricted the solutions to beanalytic, while Euler tolerated solutions for which the differentials in the waveequation did not acquire a well-defined meaning until modern distribution the-ory. Elements of the ensuing quarrel will only be mentioned to the extent thatthey 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 Thelogic of personal knowledge: Essays presented to Michael Polyani on his seventeenth birthday (Lon-don, 1961), 71–88; Jesper Lützen, “Euler’s vision of a general partial differential calculus for ageneralized kind of function, Mathematics magazine, 56 (1983), 299–306; Adolf Yushkevich, “Theconcept of function up to the middle of the nineteenth century,” AHES, 16 (1976), 37–85. Analyticfunctions in the modern sense must have a Taylor development at every point of their domain ofanalyticity.23 D’Alembert, Ref. 20, 226–230, 244–247.24 Euler, “De vibratione chordarum exercitatio,” Nova acta eruditorum (1749), 512–517, alsoin 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; Grattan-Guinness, 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 complexnumber analysis from Euler to Weierstrass(New York, 1986), 21–33. On the plucked string, see Eulerto 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.


As Euler noted, the periodicity of the generating function implies that themotion 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 analternating sequence of n similar parts separated by n − 1 nodes. A particular,analytic case of this kind of solution is the product sin(nπx/l) cos nπνt, whichcorresponds to the simple modes already described by Daniel Bernoulli. Euleralso gave the superposition

y =∑

an sin(nπx/l) cos nπνt, (10)

which he believed to be analytic (whether the sum be finite or not) and thereforeunable to represent the general solution.25

2.2 Bernoulli’s response

In 1753 Daniel Bernoulli published a lengthy, polite, but angry reaction tod’Alembert’s and Euler’s contributions to the problem of vibrating string. Inhis opinion, he had indicated the true physical solution years earlier and thesophisticated mathematics of his competitors had only obscured the subject:26

I saw at once that one could admit this multitude of curves [for the vibrat-ing string according to d’Alembert and Euler] only in a sense altogetherimproper. I do not less admire the calculations of Messrs. d’Alembert andEuler, which certainly include what is most profound and most advancedin all of analysis, but which show at the same time that an abstract analysis,if heeded without any synthetic examination of the question proposed, ismore likely to surprise than enlighten. It seems to me that giving attentionto the nature of the vibrations or strings suffices to foresee without anycalculation all that these great geometers have found by the most difficultand abstract calculations that the analytic mind has yet conceived.

Bernoulli argued that according to both experience and theory any sonorousbody could vibrate in a series of simple modes with a well-defined frequency ofoscillation. In the particular case of the vibrating strings, the various modes wereobtained by juxtaposition of Taylor modes, and their frequency was a multipleof 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 cordesexposées 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 dansun 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 argumentationdepended on two assertions:

1. D’Alembert’s and Euler’s supposedly new solutions are nothing but “mix-tures” of simple modes.

2. The “aggregation” of these partial modes in a single formula is incompatiblewith 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 periodicityproperties of the solutions of d’Alembert and Euler. He also abundantly arguedthat for a massless string loaded with a finite number of point masses, any initialconfiguration of the masses could be obtained by the superposition of simplemodes.27

In favor of the reification of partial modes evoked in assertion (2), Bernoulliadduced that the sounds produced by partial vibrations could be heard sepa-rately. Namely, “any musician agrees” that long musical strings emit the soundscorresponding to the four first modes of vibration. Also, different sounds cantravel through the same space and still be heard distinctly, as happens when welisten to an orchestra. In conformity with this analyzability of complex sounds,Bernoulli believed that the superposition of simple modes had an observablefine structure. In the extreme case of a fundamental superposed with its thou-sandth harmonic, he described the resultant corrugation and the thousand oscil-lations it made during a single oscillation of the string as a whole. He also arguedthat owing to the faster decay of higher modes, only Taylor’s fundamental modecould be seen in actual experiments. Lastly, he emphasized that d’Alembert’sand Euler’s periodic solutions with arbitrary curvature only existed if the fre-quencies of the simple modes were commensurable. As he knew, this conditionis not even met for strings of uneven thickness. Whenever it is broken, the onlyway 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 thesuperposition of simple modes in part depended on the wide-spread opinionthat pure harmonic “isochronous” oscillations were simpler than any others,could be produced separately, and enjoyed unique physical properties. Theyalso implied four unproven assumptions: that the vibrations of a continuumhave the same basic structure as the vibrations of a discrete system; that the sen-sorial analysis of a phenomenon into separate components reveals an intrinsicstructure of the phenomenon; that the cases of commensurable and incommen-surable simple modes are on the same footing; and that the superposition ofsimple modes preserved some structural features of the individual modes evenin 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 similarproblem of the suspended chain he had earlier believed that only the simple modes displayed someregularity. He now perceived the harmonic simplicity behind the apparent complexity of the moregeneral motions.28 Ibid., 152 (harmonics), 153 (orchestra), 155–156 (corrugation), 158 (decay of higher modes), 159(incommensurable frequencies).


that the sums of sine curves were not the most general possible curves of astring, Daniel Bernoulli admitted “that he was not yet quite enlightened on thispoint” since he had no mathematical proof of the contrary.29

Far from confining himself to the problem of vibrating strings, Bernoulligeneralized his considerations to the motion of any mechanical system in thevicinity of (stable) equilibrium:

What I have just said on the nature of the vibrations of bodies attached toa stretched string I do no hesitate to extend to all small reciprocal motionsthat 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 tendtoward that point with a force proportional to the small distance from thepoint of rest: and then if we suppose an arbitrary system of bodies, eachbody will be able to form as many simple regular vibrations as there arebodies in the system, and these simple vibrations will be able to coexist atthe same time in the given system.

Bernoulli made this superposition of harmonic components a basic principle ofnature:

It seems that Nature very often acts by the mere principles of the imper-ceptible, isochronous [i.e., harmonic], and infinitely diversified vibrationsto produce a great number of phenomena.

In a contemporary letter, he similarly wrote:

I admire…the hidden physical treasure that natural motions which seemsubject to no law may be reduced to the simple isochronous motions whichit seems to me Nature uses in most of its operations. I am convinced eventhat 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 infi-nite number of simple, isochronous vibrations that never fuse together nortrouble each other. One can thus conceive that the same ray of light mayprimitively include all possible colors; because the different colors prob-ably are nothing but different perceptions in the organ of sight, causedby 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 vibrationscan be formed at the same time, very different from each other, each ofwhich 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 differentvivacities, and all other phenomena indicated by Mr. Newton on primitivecolors. But this is so rich a matter that it can only be treated in the occasionof another theory.

2.3 The controversy over Bernoulli’s mixtures

In his prompt reaction to Bernoulli’s memoirs, Euler praised his colleague forhaving best developed the “physical part” of the problem of vibrating stringsbut denied the generality and superiority of the multi-modes solution. Hisarguments were of two kinds. Firstly, he rejected Bernoulli’s claim that thesuperposition of simple modes always preserved the structure of the partialmodes:

When the number of terms becomes infinite, it seems doubtful that thecurve is composed of an infinite number of sine curves: The infinitity seemsto destroy the nature of the composition.

To illustrate this point, Euler gave the particular case

y =∞∑

n=1

αn sin(nπx/l) = 12i

∞∑

n=0

(αneiπnx/l − αne−iπnx/l)

= α sin(πx/l)1 − 2α cos(πx/l)+ α2 , (11)

in which the last expression “gives a much simpler idea of the curve than wewould 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 synthesisis able to produce a much greater variety of functions than appears at a firstglance. Euler himself admitted that “if Mr. Bernoulli’s consideration providedall the curves that may occur in the motion of strings, it would certainly beinfinitely preferable to our method, which could then only be regarded as anextremely thorny detour to reach a solution so easy to find.”31

Euler went on to argue that the superposition of sine curves could not pro-duce the most general string shape:

All the curves comprised in this equation [Bernoulli’s], even though thenumber of terms is increased to infinity, have certain characters that dis-tinguish them from all other curves. For if we take a negative abscissa x,the ordinate also becomes negative and equal to that which correspondsto the positive abscissa x; similarly, the ordinate that corresponds to theabscissa 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, alsoin EO2:10, 232–254, on 234–235. Cf. Burkhardt, Ref. 2, 20–24; Truesdell, Ref. 2, 259–261 (with aslip in the formula p. 260).


abscissa x. Therefore, if the curve that has been given to the string atthe beginning does not have these properties, it is certain that it is notcomprised in the said equation. Now no algebraic curve can have theseproperties, which must therefore all be excluded from this equation; thereis no doubt that an infinite number of transcendental curves must also beexcluded.

Euler was evidently right in asserting that Bernoulli’s equation yielded periodicand odd functions. More surprisingly for the modern reader, he believed thatthe coincidence of such a function with the initial string curve, which is onlydefined over the interval 0 ≤ x ≤ l, required this curve to enjoy the sameproperties. The reason is that in his eyes a sum of sine functions was necessarilyanalytic, so that coincidence with another analytic function within the interval0 ≤ x ≤ l necessarily implied full identity. Implicitly but surely, he lent to thelimit of a sequence of functions the analyticity properties of the general termof this sequence.32

In addition to this purely mathematical objection, Euler criticized some ofBernoulli’s physical arguments. To the claim that the hearing of harmonics jus-tified harmonic decomposition, he opposed that the nth harmonic of a stringwas not the signature of the sine curve sin(nπx/l). It could just as well be pro-duced by any curve made of a succession of similar loops separated by nodes.To Bernoulli’s evocation of the differential damping of modes, he answeredthat no extraneous circumstance should be introduced in judging the mathe-matical character of the solution of an ideal problem. To Bernoulli’s claim thatthe problem of the non-uniform vibrating string only admitted the multi-modesolution, he replied that the lack of a solution of the kind he had provided forthe uniform string should only be imputed to the imperfection of contemporaryanalysis.33

D’Alembert’s first response to Bernoulli’s views is found in the article “Fon-damental” he wrote in 1757 for the Encyclopédie. There he acknowledged thefacts regarding the hearing and making of harmonic sounds, and he brieflyexplained how the excellent Jean Philippe Rameau had based a theory of har-mony on them. He next gave two reasons to reject Bernoulli’s exploitation ofthese facts as justifying the multi-mode analysis of vibrating strings. Firstly, themathematical superposition of a higher mode with the fundamental could notbe regarded as a physical superposition, because the nodes associated with thehigher 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 contra-dict his tolerance for non-analytic string curves. What was at stake was the analyticity of trigono-metric sums. As we will see in a moment (below, p. 1428–1439), in the same period Euler becameaware of trigonometric series with periodic discontinuities that should have forced him to drop theobjection.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 O. Darrigol

Fig. 3 The superpositiony = sin x cos t + 1

2 sin 2x cos 2tfor x = 1 (solid line) and itstwo harmonic components(dotted lines). Half of theextrema of the superpositiondo not coincide with extremaof the harmonics

642

1.5

1.0

0.5

–0.5

–1.0

0.0

t/π

Secondly, d’Alembert claimed that harmonics could be heard even when thestring oscillated in its fundamental mode.34

Both arguments require some unpacking. The motion of nodes in the super-position of the fundamental with a higher mode was no news to Bernoulli, whohimself wrote that the fundamental mode was a moving axis for the highermode. However, he and d’Alembert differed in their perception of this fact.For Bernoulli, the ambient air and the ear responded separately to the primaryvibration of the axis and to the secondary vibration around the axis. For d’Al-embert, the displacements of the ambient air followed those of the string, withreversals 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 maydepend on the abscissa, they cannot explain the hearing of the correspondingharmonic (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 fromBernoulli’s. Whereas for the latter simplicity meant harmonicity, for d’Alem-bert (as for Euler) simplicity meant the absence of nodes and the simultaneouspassing of every point of the string through its axis. Under this criterion, amusical string plucked in the usual manner performs a simple oscillation.

In a letter published in 1758 in the Journal des sçavans, Bernoulli defendedhis sums of sine curves, mostly against Euler contention that they could notreproduce every proposed curve. Owing to the infinite number of adjustablecoefficients in the sum, he argued, “one may cause the final curve to pass throughas many given points as one wishes and thus identify this curve with the oneproposed, to any degree of precision.” He did not worry about Euler’s remarkthat sums of sine curves were analytic whereas the proposed string curve ingeneral was not. In his view, such utterances derived from the doubtful appli-

34 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’sstatement about moving nodes.35 D’Alembert, Opuscules mathématiques, 1 (Paris, 1761), 58–60, 63–65; 4 (1768), 154. D’Alembertreasoned in the case of two harmonic components only. He later generalized the objection to anynumber of harmonics.


cation of mathematical continuity to a physical problem. He was satisfied thathis method worked rigorously in the case of a weightless string loaded with afinite number of mass points.36

In a nasty footnote to his memoir of 1762 on organ pipes, Bernoulli accusedhis critics of espousing a “metaphysics…in which nx · x

n is not xx in the casewhen n is absolutely zero” and traced their objections to their ignorance of thefact that “physical beings cannot be composed of absolutely vanishing parts.”In the main text, he emphasized that organ pipes, like vibrating strings, couldemit several harmonics at the same time, in conformity with his view that themost general motion was a superposition of sine curves. He also insisted thatonly one tone was heard when only one mode of oscillation occurred in thetube 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 Opus-cules, published in 1761, the former approved the latter’s criticism of Bernoulliand added that a sum of sine curves necessarily had continuous curvature andtherefore could not reproduce the most general curve. Indeed the two rivalsshared the conviction that the limit of a sequence of functions had the sameproperties of continuity and analyticity as the general term of the sequence.D’Alembert acknowledged that this argument only made Bernoulli’s solutionsless general than Euler’s and that it still allowed that his own solutions, beinganalytic, odd, and periodic functions, could all be represented as sums of sinecurves. He nonetheless excluded this possibility, for he was convinced that hisown “reentering curves” could not generally be expressed as sums of sines. Atmost, he was willing to admit trigonometric series as “approximations” not as“geometrical, exact, and rigorous solutions.” In later years, he came to regardsin5/3(πx/l) as a possible equation for the initial curve of a vibrating string, andbrandished it as a non-Bernoullian solution. For he believed that every trigono-metric series had a power series development (since its terms did separately),whereas sin5/3(πx/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 vibrationsdes cordes tendues,” Journal des sçavans (March 1758), 157–166. Cf. Truesdell, Ref. 2, 262. In hisresponse to Bernoulli, Euler (Ref. 31, 236) had conceded that the infinite number of indeterminatecoefficients in Bernoulli’s sums of sine curves seemed to enable them to reproduce an arbitrarycurve.37 Daniel Bernoulli, “Recherches physiques, mécaniques et analytiques, sur le son et sur les tonsdes 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 (con-tinuous curvature), 47 (approximations); “Nouvelles réflexions sur les vibrations des cordes so-nores,” ibid., 4 (1768), 128–155, on 153–155 (more against Bernoulli); “Premier supplément aumémoire précédent,” ibid., 156–179, on 169 (introducing sin5/3); “Second supplément au mémoireprécédent,” 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 beexpressed as a trigonometric sum: Euler, “De serierum determinatione seu nova methodus in-veniendi terminos generales serierum,” NCAP, 3 (1750–1751), 36–85, also in EO1:14, 463–515, on470–473; 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 thefinite to the infinite”: the true generality of Bernoulli’s solution in the discretecase did not imply its generality in the continuous case. Secondly, d’Alembertrejected Bernoulli’s physical interpretation of this solution as a “mixture” ofreal, partial vibrations. In conformity with his earlier criticism in the “Fonda-mental” article, he argued at great length that the partial vibrations “generallyreferred to a curved axis” determined by other modes and thus lost the peri-odicity properties they had when produced in isolation. As he wrote in a laterduplication of this argument, “The claimed Taylorian multiple vibrations onlyexist 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 toBernoulli’s defense of the passage from the discrete to the continuous, by argu-ing that the admission of nx · x

n = xx for n = 0 amounted to the confusionbetween a rectangle and an infinite line. D’Alembert was less suspicious oflimiting processes when they served his polemic purposes. In what he hopedto be the final blow to Bernoulli’s theory, he argued that the incommensura-bility of the frequencies of the simple modes in the discrete case excluded theϕ(x + ct) + ϕ(x − ct) form of the solution even when the number of discretemasses 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 simi-lar problem of a longitudinally vibrating string could provide a model for thepropagation of sound. The following year, Euler published a brilliant memoirbased on a discrete version of this idea. The basic model is a tense elastic cordloaded 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 thiskind—read:

xk = α2(xk+1 − 2xk + xk−1), (12)

where xk is the displacement of the kth mass from its equilibrium position, α2isthe ratio of the elasticity constant of the cord to the mass of the loads, andx0 = xn+1 = 0 holds at the boundaries. Euler first sought solutions of the form

xk = ak cosωt. (13)

39 D’Alembert, “Recherches,” Ref. 38, 45, 58–61; “Extrait de différentes lettres de M. d’Alembertà 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’sidiosyncratic use of “isochronous” (meaning harmonic).40 D’Alembert, “Nouvelles réflexions,” Ref. 38, 154, 175–178.


The equation of motion then implies the relations

(2α2 − ω2)ak = α2(ak+1 + ak−1), (14)

which reminded Euler of the trigonometric identity

2 sin kϕ cosϕ = sin(k + 1)ϕ + sin(k − 1)ϕ. (15)

Taking into account the boundary condition a0 = 0, the solution can be writtenas

ak = sin kϕ, with 2α2 − ω2 = 2α2 cosϕ, or ω = 2α sinϕ

2. (16)

The boundary condition an+1 = 0 further requires

ϕ = rπn + 1

, with r = 1, 2, . . . n. (17)

This means that the loaded string has n simple modes of motion with the fre-quencies

ωr = 2α sinrπ

2(n + 1). (18)

Euler obtained the general solution of the equations of motion by superposingsimple modes in the manner:41

xk =n∑

r=1

cr sinkrπ

n + 1cosωrt. (19)

The generality of this solution depends on the possibility of reproducing anyinitial set of values Xk of the displacements by a suitable choice of the coeffi-cients cr. As Euler was only interested in the propagation of a pulse for whichall the Xk

′s vanish except X1, he only sought the coefficients in this case. Byinduction from the cases n = 1, 2, 3 he found

cr = X12

n + 1sin

rπn + 1

(20)

as well as the identity

n∑

k=1

sinkrπ

n + 1sin

kπn + 1

= n + 12

δr1 (21)

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 condi-tion xk(0) = X1δk1.

Euler’s remarkable finding easily generalizes to any initial condition. Aproper generalization of identity (21),

n∑

k=1

sinkrπ

n + 1sin

ksπn + 1

= n + 12

δrs, (22)

which we would now describe as the orthogonality of the simple modes, leadsto the expression

cr = 2n + 1

n∑

s=1

Xs sinsrπ

n + 1(23)

of the coefficients.For the abovementioned pulses, Euler inferred a value of the speed of prop-

agation in the cases n = 2, 3. He did not investigate the continuum limit, for hewanted 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 conclusionsinterestingly contradicted Daniel Bernoulli’s acoustics:

While the motion of one particle [only one load] is oscillatory, the motionof two or more particles no longer is and differs more and more from itas the number of particles increases; consequently, sound cannot at all beunderstood as propagated through the air as some able men would haveit when they assert that when a string or other sounding instrument isset in motion there are in the air particles of this sort which take on anoscillatory motion and excite the organ of hearing.

Euler thus refused to regard a propagating pulse as a superposition of simplemodes of oscillation, even though he was first to mathematically derive propa-gation through a superposition of oscillation modes. His comment also revealsantipathy with Bernoulli’s idea that hearing involves harmonic analysis. As wewill see in a moment, he adhered to the old concept of hearing as the percep-tion of rhythms in series of pulses, in conformity with his paper’s emphasis onpulses.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 motionof a vibrating string for which only one part of the string departs from equilibrium. Euler may alsohave been targeting d’Ortous de Mairan’s theory of sound propagation (1720), according to whichthe particles of air contained as many resonant strings as there were sounds to be propagated.


3.2 Lagrange’s first memoir on sound

In 1759, the young and ambitious Joseph Louis Lagrange published a long mem-oir on the nature and propagation of sound whose brilliance impressed bothd’Alembert and Euler. While Lagrange seems to have been unaware of Euler’smemoir on the propagation of pulses, he had carefully studied the memoirs onvibrating strings by d’Alembert, Euler, and the Bernoullis. He was especiallyreceptive to two of d’Alembert’s suggestions, that sound propagation could betreated in analogy with the problem of vibrating strings, and that for arbitrary(non-analytic) initial conditions the latter problem could only be treated bytaking the limit of a loaded massless string when the number of loads reachesinfinity.43

The mathematical core of Lagrange’s memoir was the solution of the generalinitial value problem for the uniformly and discretely loaded string or for theequivalent problem of a discrete succession of elastic air slices. He recoveredEuler’s equations of motion (12), now applied to the successive ordinates yk ofthe loads:

yk = α2(yk+1 − 2yk + yk−1), (24)

with the boundary conditions y0 = yn+1 = 0. Whereas Euler directly based hissolution on simple sine modes and on the orthogonality of these modes, Lag-range applied a general method of resolution of a system of linear differentialequation of any order that he borrowed from d’Alembert.44

In modern terms, the method consists in replacing the original system bya higher number of linear differential equations of first order and in diago-nalizing the implied linear operator. In Lagrange’s notation, this means that heformed linear combinations M1y1+M2y2+· · ·+Mnyn that were proportional totheir second time derivative. Once the corresponding differential equations aresolved for given initial conditions, there remains to determine the yk

′s from thevarious combinations. As he did not anticipate the orthogonality of the variouschoices of the vectors (M1, M2, . . .Mn), Lagrange relied on some abstruse trigo-nometry. His result nonetheless agrees with the one Euler would have obtainedfor arbitrary initial conditions. In the case for which the initial velocities vanishand 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 asa plausible approximation to the real problem. Lagrange nonetheless read him as suggesting thatthe 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. Inhis 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 methodfor integrating systems of linear differential equations, which can be traced to the par. 101 of hisTraité de dynamique(Paris, 1743).

368 O. Darrigol

yk = 2n + 1

n∑

r=1

n∑

s=1

Ys sinsrπ

n + 1sin

krπn + 1

cosωrt. (25)

Lagrange discussed the behavior of this solution as a superposition of sim-ple modes of incommensurable frequencies, and praised Daniel Bernoulli foranticipating the results:45

One cannot overestimate the sagacity and penetration of this celebratedGeometer, who, through a purely synthetic investigation of the proposedquestion, succeeded in reducing to simple and general laws motions thatseemed 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 inverselyproportional to this spacing, Lagrange assumed

ωr = 2α sinrπ

2(n + 1)→ ωr = rπ

lc, (26)

where c is a constant with the dimension of velocity (√

T/σ , if T denotes thetension of the string and σ its mass per unit length). The limit of the solution(25) then reads

y(x) = 2l

∞∑

r=1

l∫

0

dX Y(X) sinrπX

lsin

rπxl

cosrπct

l. (27)

Lagrange rather wrote

y(x) = 2l

l∫

0

dX Y(X)∞∑

r=1

sinrπX

lsin

rπxl

cosrπct

l, (28)

no doubt because his subsequent considerations involved an evaluation of thepurely trigonometric sum appearing in this expression.46

The formula (27), which Lagrange did not exactly write, gives the motion ofthe string as an infinite sum of simple modes. For t = 0, it degenerates into Fou-rier’s celebrated formula for the sine-series development of a function definedover 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 regardedthe 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 opinion of earlier commentators, I will argue that Lagrange truly had in handboth the simple-mode analysis of the vibrations of a continuum and the Fourierdecomposition of a not-well defined but intently large class of functions, surelyimplying 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 performedan illicit permutation of sum and integral when passing from Eq. (27) to(28). This criticism is based on an anachronistic reading of the latter equa-tion. Lagrange’s subsequent developments clearly show that by this formula hemeant what we would now write as

y(x) = limn→∞

⎛

⎝2l

l∫

0

dX Y(X)n∑

r=1

sinrπX

lsin

rπxl

cosrπct

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 onlyused Eq. (28) as an intermediate, insignificant step in reasoning meant to justifyEuler’s solution to the vibrating-string problem, not to confirm Bernoulli’s sim-ple-mode analysis. Surely, Lagrange did not dwell on this step and failed to seeits potentialities for a new analysis à la Fourier. It remains true, however, thathe believed in the mathematical truth of the relevant equation including thespecial case t = 0 that gives Fourier’s fundamental formula, for Y functions asgeneral as Euler’s string curves. Otherwise, his whole train of reasoning wouldfall apart.

Granted that Lagrange believed in the infinite superposition (27), one maystill wonder whether he actually proved it. D’Alembert and later commentatorsargued that the substitution (26) for the frequenciesωr was illegitimate, since forvalues of r that are close to n the argument rπ/2(n + 1) of the sine function didnot vanish when n became infinite. Lagrange’s reply to this criticism consistedin remarking that the equation of the discrete problem of which the frequenciesωr are the solutions tends toward an equation of which the numbers rπc/l arethe solutions. This reply fails to address the proper question, which is whetherthe limit of the discrete formula (25) truly yields the continuous formula (27).However, Lagrange’s questionable step does not interfere with the derivationof Fourier’s formula, since the latter only requires the case t = 0. Moreover, itcan be shown that the terms of the series for which r/n is non-negligible actuallydo not contribute to the sum of the series.49

47 Lagrange nevertheless excluded polygonal string shapes for which the wave equation (3) doesnot hold at the origin of time: “Addition aux premières recherches sur la nature et la propagationdu 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 mayretain 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 inLagrange’s derivation of Eq. (28): the substitution of integrals for the discretesums. The typical error committed in replacing a sum of n values of a functionwith the corresponding integral is of the order of 1/n. As there are as manysimple modes as there are discrete values of the X variable, the net error couldwell be finite. Retrospectively, we can tell that when proper restrictions (forinstance the Dirichlet conditions) are applied to the function Y(X), this neterror vanishes. The intuitive reason is that owing to the oscillations of the in-tegrands, the typical error for the r-mode is 1/rn, which yields a net error of theorder 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

sinrπx

lcos

rπctl

= 12

(

sinrπ(x + ct)

l+ sin

rπ(x − ct)l

)

, (30)

this equation can be rewritten as

y(x, t) = ψ(x + ct)+ ψ(x − ct) (31)

if

ψ(ξ) = limn→∞

1l

l∫

0

Y(X)�n(X, ξ)dX (32)

wherein

�n(X, ξ) =n∑

r=1

sinrπX

lsin

rπξl

. (33)

Through some laborious trigonometry, Lagrange found50

�n = sin (n+1)πXl sin nπξ

l − sin nπXl sin (n+1)πξ

l

2(

cos πXl − cos πξl

) . (34)

with rπc/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).


Implicitly retreating to the discrete case, he took (n + 1)X/l to be an integer,which yields the simpler formula

�n = (−1)(n+1)X/l sin πXl sin (n+1)πξ

l

2(

cos πXl − cos πξl

) . (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 multipleof 2l. In the special case X = ξ , and remembering that for Lagrange (n + 1)X/lis an integer, this expression reaches the value n + 1. Again Lagrange retreatedto the discrete problem, and identified the element dX in Eq. (21) with l/(n+1).Consequently, for 0 ≤ ξ ≤ l, the equalityψ(ξ) = Y (ξ) holds. Outside this inter-val, the values of X that contribute to the integral differ from ξ or from −ξ by amultiple of 2l. The sign of this contribution is such that the function ψ(ξ) endsup being the extension Y(X) of the function Y(X) that is odd and periodic ofperiod 2l. Combined with Eq. (31), this remark leads to Lagrange’s final resultfor the case of zero initial velocity:

y(x, t) = 12[Y(x + ct)+ Y(x − ct)]. (36)

Lagrange could have reached this result much faster by noting that by con-struction 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. Alasthe simplest route is rarely the first that comes to mind. Instead Lagrange com-puted theψ integrals in the above manner, which perplexed most of his readers.D’Alembert soon noted the absurdity of the statement that for infinite n thequantity (n + 1)ξ/l = (n + 1)X/l ± (n + 1)ct/l was always an integer. Lagrangereplied that by construction (n + 1)X/l always was an integer and that a (con-tinuous) function of time was perfectly determined if its value for multiples ofl/(n + 1)c were known for arbitrarily large n. He thus completed his retreat tothe discrete model. In substance, his exploration of the consequences of Eq. (28)only was a duplication of its derivation through the discrete model, togetherwith the remark that for commensurable eigenfrequencies the quantities x ± ctbecome the natural arguments.51

In the final Eq. (36) Lagrange immediately recognized the d’Alembert-Eulersolution of the problem of vibrating strings, without the restrictions imposed byd’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 alldoubt and established upon direct and clear principles that rest in no wayon the law of continuity [i.e., analyticity] which Mr. d’Alembert requires;there, moreover, is how it can happen that the same formula that hasserved to support and prove the theory of Mr. Bernoulli on the mixture ofisochronous [i.e., harmonic] vibrations when the number of moving bodiesis finite shows us the insufficiency of this theory when the number of thesebodies becomes infinite. Indeed the change that this formula undergoes inpassing from one case to the other is such that the simple motions whichmade up the absolute motions of the whole system destroy each otherfor the most part, and those which remain are so disfigured and alteredas to become absolutely unrecognizable. It is truly annoying that so inge-nious a theory…is shown false in the principal case, to which all the smallreciprocal motions occurring in nature may be related.

Quite diplomatically, Lagrange managed to praise each of the protagonistsof the quarrel over vibrating strings: d’Alembert for confining his and Euler’smethod of resolution to analytic functions, Euler for giving the most generalsolution, and Bernoulli for illuminating the discrete case. At the same time hepromoted himself as the geometer who first rigorously proved the pertinenceof Euler’s solution as well as the failure of Bernoulli’s intuitions in the con-tinuous case. His proud announcement, in a contemporary letter to Euler, of“the complete fall of the Bernoullian theory” has usually been mistaken fora flat rejection of trigonometric series. Careful reading of the above citationrather indicates that Lagrange regarded the formula (28) for the simple-modeanalysis of string motion as perfectly valid and even made it the very source ofthe falsity of Bernoulli’s physical interpretation of this analysis. In the limit ofan infinite number of simple modes, he figured that what we would now calldestructive interference deprived the resultant oscillation of any resemblancewith the partial oscillations.53

As was earlier argued, Lagrange’s reasoning did contain the simple-modeanalysis of the vibrations of a continuum as well as Fourier’s fundamentalformula. However, Lagrange’s failure to provide a cogent proof and his simul-taneous denial of the physical relevance of simple-mode analysis for a vibratingcontinuum was not without consequence. He avoided trigonometric series inmost of his works on partial differential equations, and made a sparing use ofthem in his work on celestial mechanics. None of his readers, not even Euleror Bernoulli, saw that his memoir contained the integral expression of thecoefficients.

The rest of Lagrange’s memoir concerned the application of his calculationsto the one-dimensional propagation of sound. For this purpose, he consideredthe superposition of simple modes for which, in the discrete case, the initialordinate 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.


In the continuous limit, he found

y(x, t) ∝∞∑

r=1

sinrπX

lsin

rπxl

cosrπct

l. (37)

By his earlier reasoning on such sums, they only take non-zero values if x ± ctdiffers from X by a multiple of 2l. From this result, Lagrange inferred thatsound propagated with the velocity predicted by Newton and echoed on solidwalls situated at x = 0 and x = l. He also argued that the linearity of his equa-tions 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 differentsounds without confusion.” He condemned Daniel Bernoulli’s explanation ofthe same fact in terms of the superposition of simple modes, because the sortof 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 toBernoulli’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 tothe infinite, and proposed a “simpler method” intended to dissolve the doubtsthat d’Alembert and Bernoulli had expressed in letters to him. The new methodconsisted in a direct application of d’Alembert’s procedure of linear combina-tion for solving systems of linear differential equations.55

Remember that in the discrete case Lagrange formed linear combinationsM1y1 + M2y2 + · · · + Mnyn such that the second-order finite differences occur-ring in the equations of motion would be proportional to their second time-derivative. In the continuous case, he similarly formed the integrals

s =l∫

0

M(x)y(x)dx (38)

and chose the functions M(x) so that s would obey an equation of the type

s + ω2s = 0. (39)

54 Lagrange, Ref. 43, 126–139, 141 (quote). Lagrange considered pulses of velocity rather than thedisplacement 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 Sept1759; LO13, 3–4), d’Alembert wrote: “I find it hard to believe that this solution necessarily requiresa so extensive apparatus of calculation.”

374 O. Darrigol

Since the equation of motion

∂2y∂t2

− c2 ∂2y∂x2 = 0 (40)

implies

s − c2

l∫

0

M(x)∂2y∂x2 dx = 0, (41)

the desired behavior of s occurs if and only if

l∫

0

M(x)∂2y∂x2 dx = −κ2s, with κc = ω. (42)

Granted that y(0) = y(l) = 0 and M(0) = M(l) = 0, a double integration byparts transforms this condition into

l∫

0

(M′′ + κ2M)ydx = 0. (43)

It is therefore sufficient that

M′′ + κ2M = 0. (44)

The solution

Mκ = sin κx (45)

of this equation satisfies the boundary conditions M(0) = M(l) = 0 if κ is awhole number r times π/l. Calling sr the corresponding value of s, and assum-ing that the velocity of the string vanishes at the origin of time, Eq. (39) nowimplies

sr(t) ≡l∫

0

y(x, t) sinrπx

ldx = cos

rπctl

l∫

0

Y sinrπx

ldx, (46)

which Lagrange transformed into

sr(t) = 12

l∫

0

Y sinrπ(x − ct)

ldx + 1

2

l∫

0

Y sinrπ(x + ct)

ldx. (47)


Calling Y(x) the odd, 2l-periodic function of x that coincides with Y(x) for0 ≤ x ≤ l, we have

sr(t) = 14

l∫

−l

Y(x) sinrπ(x − ct)

ldx + 1

4

l∫

−l

Y(x) sinrπ(x + ct)

ldx

= 14

l∫

−l

Y(x + ct) sinrπx

ldx + 1

4

l∫

−l

Y(x − ct) sinrπx

ldx

= 12

l∫

0

[Y(x + ct)+ Y(x − ct)] sinrπx

ldx (48)

Consequently, the function y(x, t) has the same sr coefficients as the function12 [Y(x + ct) + Y(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 problemof vibrating string.56

Even though he now started with the partial differential equation (40), whichinvolves a second-order spatial derivative, Lagrange believed his derivation toinclude “irregular” (non-analytic) initial conditions because multiplication byM(x) of this equation and integration by parts absorbed the spatial derivationsof y. He thus anticipated a basic feature of modern distribution theory: theintroduction of test functions that absorb differential operators. Yet he did notventure so far as tolerating a polygonal initial shape of the string, for he believedthe equation of motion to lose any meaning in this case.57

In the following sections of his memoir, Lagrange extended his method todifferential operators more complex than ∂2/∂x2. In modern terms, he deter-mined the eigenfunctions and spectrum of a variety of differential operators,and he solved equations of motion involving them by determining the pro-jections of the solution on these eigenfunctions. He was only satisfied when hecould 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 thisdecomposition, as in the Euler-d’Alembert solution. He noted the considerabledifficulty of the latter process, which he called the elimination of κ (the wavenumbers of the Fourier components).58

Lagrange’s new method eluded the expression of a function in terms ofits projections (generalized Fourier-coefficients) on the eigenfunctions of thedifferential 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 failureto introduce Y, 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 promptlyevolve 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 ageneralization of Fourier’s future theorem. Unfortunately, Lagrange’s idiosyn-cratic use of M multipliers and the additional complexity brought by the higherdimensionality have so far prevented commentators from appreciating the pro-fundity and originality of this section. In order to shed light on this matter, I willrestrict Lagrange’s argument to one dimension only, in which case the equationof 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 oflength l. Lagrange characterized the displacement y(x) of a particle of air fromits rest position x through the coefficients

sκ =l∫

0

Mκ(x)y(x)dx, with κ = π/l, 2π/l, 3π/l, . . . (49)

In analogy with the discrete problem for which the number of different valuesof κ is equal to the number of values that x can take, he assumed the existenceof the inverse linear relation

y(x) =∑

κ

sκPκ(x). (50)

In order to determine the coefficients Pκ(x), Lagrange injected this formulainto the former one, which gives

sκ =∑

λ

sλ

l∫

0

Mκ(x)Pλ(x)dx. (51)

This identity holds for any value of the sequence (sκ)κ if and only if

l∫

0

Mκ(x)Pλ(x)dx = δκλ. (52)

Lagrange then showed that the function Pλ(x) such that

P′′λ + λ2Pλ = 0, Pλ(0) = Pλ(l) = 0, and

l∫

0

Pλ(x)Mλ(x) = 1 (53)

59 Ibid., Chap. 6: “Réflexions sur la théorie des instruments à vent,” 298–316, esp. 312–316, on303–304 (orthogonality of eigenfunctions), 314–316 (generalized Fourier theorem).


met this condition. Indeed, the two first conditions and the definition (44) ofMκ imply that

(κ2 − λ2)

l∫

0

Mκ(x)Pλ(x)dx =l∫

0

Mκ(x)P′′λ(x)dx −

l∫

0

M′′κ(x)Pλ(x)dx = 0.

(54)

Although Lagrange conceived the M′s as multipliers and the P′s as simplemodes—and therefore noted them differently, he was well aware that the con-ditions (52) determined the P′s uniquely as

Pλ(x) = 2l

sin λx = 2l

Mλ(x). (55)

Consequently, the form

y(x, t) =∑

κ

sκ(t)Pκ(x) =∑

κ

Pk(x)

l∫

0

Mκ(ξ)Y(ξ)dξ cos cκt (56)

of the general solution of the wave equation (resulting from Eqs. 49 and 50) isthe 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 withnon-degenerate simple modes. The particular case t = 0 yields Fourier’s the-orem in the case of one-dimension, and a more general theorem of harmonicdecomposition in higher dimensions. In modern terms, we would say that Lag-range exploited the mutual orthogonality of the simple modes. He derived thisproperty from what we would now call the Hermitian character of the spatialoperator in the wave equation. The only element lacking was a rigorous proofof completeness for the system of eigenfunctions.60

At the end of these powerful developments, Lagrange again deplored thephysical 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 ofair.” 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” becausethe coefficients of the simple modes depended on the initial conditions, whichshould be arbitrary.61

From the similar but more detailed argument later found in the secondedition of the Mécanique analytique, it becomes clear that Lagrange meantBernoulli’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 remainingmodes. Presumably, he had computed the Fourier coefficients for a pluckedstring (triangular shape) and found their sum to be divergent (they behave like1/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 simplemodes.62

Lagrange nonetheless ended his second memoir on sound propagation withthe following praise of Bernoulli’s analysis:

This method [projection over simple modes] serves to demonstrate thebeautiful proposition of Mr. Daniel Bernoulli: When a [finite or infinite]system of bodies undergoes infinitely small oscillations, the motion of eachbody can be regarded as composed of several partial motions synchronousto those of simple pendulums.

The contradiction with Lagrange’s previous considerations disappears if wekeep in mind that he distinguished between the mathematical truth and thephysical 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 asa particular case of the general method he had earlier invented for studying thevibrations of an arbitrary system of bodies. He seized this opportunity to sim-plify his treatment of the discretely loaded string (thus getting closer to Euler’sapproach of 1748) and to argue, by intricate trigonometry, that the generalsolution (25) could be transformed into a form similar to the d’Alembert-Eulerform for the continuous string:

yk = ϕ

(kl

n + 1− ct

)

+ ϕ

(kl

n + 1+ ct

)

+ψ(

k + 1n + 1

l − ct)

− ψ

(k − 1n + 1

l − ct)

+ψ(

k + 1n + 1

l + ct)

− ψ

(k − 1n + 1

l + ct)

(57)

Lagrange proudly announced to d’Alembert:

I have just completed a calculation which seems to me to shed great lighton this question [of vibrating strings]. I have found a way to construct, in

62 Lagrange, Ref. 49, 424.63 Lagrange, Ref. 55, 316.


a general manner, the formula [in the discrete case] of my first Recherchessur 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 thecoefficients of Lagrange’s power-series expressions for the Fourier coefficientsof ϕ and ψ also depend on time. However, in the continuous limit the ψ termsdisappear and the ϕ terms reproduce the d’Alembert-Euler solution becausetheir hidden time-dependence becomes negligible. Lagrange thus circumventedthe difficulties he had encountered in taking the continuous limit on the sumof simple modes before doing the trigonometry that generates the x ± ct argu-ments.64

The reason for Lagrange thus avoiding infinite trigonometric sums may havebeen Euler’s and d’Alembert’s contention that a trigonometric sum should beanalytic 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 sumsof sine curves could not possibly represent the most general shape of a tensestring, “owing to certain properties that seem to distinguish [such sums] fromother imaginable curves.” In a letter to d’Alembert of 20 March 1765, he foundit “hard to believe that Mr. Bernoulli’s solution… should be the only one thatoccurs in nature.” He added that the phenomena of sound propagation, asexplored in his second dissertation, necessarily implied “discontinuous” [non-analytic] functions. As we will see in a moment, Euler had made the same pointin memoirs he had recently sent to Lagrange for the Miscellanea Taurinensia.65

At the same time, Lagrange still believed that the trigonometric seriesobtained in the discretely loaded string problem retained some validity in thelimiting case of a continuous string. Following his new derivation of Euler’sconstruction, he introduced the sibylline distinction between requiring twocurves to be identical and requiring their difference to be smaller than anygiven quantity. He believed that the initial string curve and the correspondingtrigonometric series were related in this second manner only:66

It is clear that, whatever is the initial curve [of the string], one can alwayspass a curve of the form y = α sin πx + β sin 3πx + · · · through infinitelymany points which are infinitely near to this initial curve in such a man-ner 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 whichthe initial curve has also the same form; in any other case this initial curvewill only be a sort of asymptote which the generating curve [the odd,2l-periodic extension of the initial curve] will approach at infinity withoutever 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

Yk = 2n + 1

n∑

r=1

n∑

s=1

Ys sinsrπ

n + 1sin

krπn + 1

, (58)

which results from the simple-mode analysis for t = 0 (compare with Eq. 25).Lagrange inferred from it that the continuous function

f (x) = 2n + 1

n∑

r=1

n∑

s=1

Ys sinsrπ

n + 1sin rπx (59)

was a continuous interpolation of the discrete function k/(n + 1) → Yk. Henext showed that for any given function α(x)

g(x) = 2n + 1

1α(x)

n∑

r=1

n∑

s=1

Ys α

(s

n + 1

)

sinsrπ

n + 1sin rπx (60)

was another interpolation of the same discrete function. Lagrange meant thistrivial non-uniqueness of continuous interpolations of discrete data to justify hisstatement that trigonometric series could only “asymptotically” approach thefunction they purport to represent. The argument boils down to the followingplatitude: for any finite value of n, the trigonometric sum that coincides with agiven function for a sequence of n equidistant values of the variable has zeroprobability to represent this function exactly on the relevant interval.67

More interesting is the notation that Lagrange used in this context. Insteadof 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 pseudo-integrals 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 sincehe stopped the series at finite n. Otherwise he would have reached Fourier’sfundamental formula in a very simple (but non-demonstrative) manner. Hisreluctance to send n to infinity probably resulted from his suspicion that some-thing funny happened in this limit. At a time when conditions of integrabilityand the distinction between uniform and non-uniform limits did not exist, Lag-range could only remain in the murk. Moreover, his algebraic conception of

67 Ibid., 552–554.


calculus deprived him of the tools that we now know to be necessary to discussthe pertinence of trigonometric series.68

There is no mention of Lagrange’s special notion of asymptotic convergencein his later writings. In 1769, answering one of d’Alembert’s many letters onvibrating strings, he described the “violent revulsion” he had developed for thissubject as a consequence of working too long on it. Plausibly, his reticence alsoderived from his desire to preserve his friendship with both d’Alembert andEuler. The rivalry between the two geometers had soured so much that it hadbecome impossible to agree with one without irritating the other. Lagrangewaited the second edition of his Mécanique analytique, published in 1811, for alast significant return to the vibrating string. Surprisingly, he reproduced his firsttreatment 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 interestingquestions of mechanics.” Perhaps he also wanted to set the record straight afterFourier had used similar considerations in the domain of heat propagation.69

Lagrange’s ultimate judgment on the value of simple-mode analysis agreedwith his earliest pronouncements on this matter:

Although [the formula (27)] rigorously gives the motion of the string atany time t, the infinite series that enter this formula prevent it to representthis motion in a clear and sensible manner.

Again, he explained that the harmonic sounds heard from a single vibratingstring could not be traced to partial modes:

The series that could give the different sounds disappears from the formulawhen the number of bodies is infinite, and the result is, for every point ofthe string, a simple and uniform law of isochronism which immediatelyand 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 andtrigonometric series. On the one hand, he admitted their mathematical abil-ity to represent the most general motions and functions, if only in a mysterious“asymptotic” manner. On the other hand, he denied them any physical meaningin the continuum case.70

68 As we will see in a moment, Alexis Clairaut already had the trigonometric interpolation formulain 1757 as well as its continuous limit. Poisson, in his “Second mémoire sur la propagation de la chal-eur dans les solides,” JEP, 19 (1823), 249–509, on 444–446, ignored Lagrange’s prevention to takethe continuous limit and judged that Lagrange’s interpolation formula (59′) was “the first generalformula” 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 derivationof the solution for the discretely loaded string (pp. 300–314), and briefly the equation and thed’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 longfriendship spiced up with some disagreement on vibrating strings. In his letter ofthanks, d’Alembert assorted his praise of Lagrange’s treatment of the discretecase with a rejection of the limiting process that led to the continuous string: “Asfor the method through which you pass from an indefinite number of vibratingbodies 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 hisOpuscules. As was already mentioned, his two main objections concerned thelimit of the frequency of the simple modes and Lagrange’s cancellation of anycosine that had an infinite argument. On the physical side, d’Alembert doubtedthat the limit of the discrete case should yield the truly observed behavior of auniform string. Mathematics could only predict the latter behavior in the casewhen the initial shape was given by an equation, and “the rest should be left tophysics.”71

As Lagrange’s reply failed to satisfy him, d’Alembert persisted in his rejec-tion of Lagrange’s limiting process. He was slightly more receptive to Lagrange’s“extremely complicated” treatment of 1764, for it required that all the deriva-tives dny/dxn should be finite. Lagrange had indeed noted that his power-seriesexpressions for the Fourier coefficients of φ andψ in his solution (57) for the dis-cretely loaded string only converged (in the limit of an infinite number of loads)under this condition. D’Alembert believed Lagrange’s condition to imply theanalyticity he required in his solutions to the continuous problem. Moreover,he fancied that the equation of vibrating strings

∂2y/∂x2 − ∂2y/∂t2 = 0 (61)

implied

∂n+2y/∂xn+2 − ∂n+2y/∂t2∂nx = 0, (62)

and therefore required indefinite differentiability with respect to x. He rejoicedthat he and Lagrange “now agreed on vibrating strings,” even though Lagrangenever truly conceded that non-analytic solutions should be excluded. In theearlier cited letter to d’Alembert of March 1765, Lagrange asserted that propa-gation phenomena required “discontinuous” functions. In his old age, he notedthat “the principle of the discontinuity of the functions is now received for theintegrals 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 dny/dxn makes no jump in the ini-tial curve”); Lagrange to d’Alembert, 1 Sep 1764, LO13, 12–14 (same restriction); d’Alembert toLagrange, 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,


Lagrange’s persistent admission of non-analytic string curves was importantin his denial that trigonometric series could “geometrically” reproduce everystring curve. For he believed to have proved that any analytic periodic func-tion was a trigonometric series. The proof was based on integrating the lineardifferential equation of infinite order obtained through the Taylor developmentof f (x + 1) − f (x) = 0. D’Alembert rejected this proof in his Opuscules, argu-ing that the implied series could fail to converge. At any rate, “continuous”periodic functions in his sense did not necessarily admit a Taylor developmenteverywhere. As was already mentioned, d’Alembert tolerated the string curvey = sin5/3(πx/l), whose second derivative is infinite when x is a multiple ofπ . As another example of a “continuous” periodic function that did not meetLagrange’s criteria and therefore could not be represented by a trigonometricseries, d’Alembert gave the cycloid x = 1 − cos u, y = u ± sin u for whichy ∼ x2/3 around the corner u = 0. Lagrange’s interesting reply involved trigo-nometric series (with unspecified coefficients) for the algebraic functions x, x2,and x2/3, which implies that by 1768 at least he did not believe trigonometricseries needed to be analytic over the whole real axis.73

The following year, d’Alembert sent to Lagrange his last piece on vibratingstrings, in which he repeated that curves with corners as well as y = sin5/3(πx/l)could not be represented by trigonometric series. He also confirmed thatBernoulli’s partial vibrations did not have any physical reality, even in thediscrete case for which Bernoulli’s solution was not to be doubted. Lagrangecongratulated d’Alembert for these “decisive” remarks, although it is not clearwhich 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 solutionof the problem of vibrating strings from any chicanery. He publicly salutedhis friend’s “indecipherable” but “prodigious” calculations, and proceeded toextend the discussion of sound propagation. Far from claiming any priorityfor the propagation of pulses in the discrete case—which he had first studied in1748—he directly studied the continuous case through the equation of vibratingstrings. His general solution for this equation, he now saw, was perfectly com-patible 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 thatthe continuity of all derivatives implied analyticity was based on Taylor developments. Lagrangerejected it in his letter to d’Alembert, 26 Jan 1765, LO13, 29–32. D’Alembert persisted in his

reply, 2 Mar 1765, LO13, 32–35. Neither of them saw that functions such as e−1/x2are 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 dedéterminer certaines fonctions,” Opuscules, 4 (Paris, 1768), 343–348, on 344–345; Lagrange tod’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 thatvanished everywhere except on a small interval. Euler welcomed the strikingnon-analyticity of this case. Quite generally, he asserted that the integral of apartial differential equation should involve an arbitrary “discontinuous” (non-analytic) function as a natural generalization of the set of integration constantsfor a system of ordinary differential equations.75

To this memoir of 1759, Euler added generalizations to two- and three-dimensional propagation by means of the linearized version of his generalequations of motion for an elastic fluid. In 1765, he discussed the propagationand reflection of waves on a vibrating string by the same method. This newemphasis on propagation brought him further apart from Daniel Bernoulli. InMay 1764, he wrote to the latter’s nephew:

I do not wish to deny absolutely that the equation composed of an infinityof sines includes the solution to this question, since it contains arbitraryconstants which it would be possible to determine in such a way that inputting the time = 0, [this form] would produce exactly the curve impressedupon the string at the beginning. But your uncle will not disagree that thisoperation 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 pulsecould be represented by a trigonometric series. In the memoir published a fewmonths later he categorically denied this possibility, and confirmed his ear-lier opinion that Bernoulli’s solution was “only very particular.” To Bernoulli’searlier remark that a trigonometric series could be made to pass through anunlimited number of given points and therefore could reproduce any curve, hereplied that by the same argument any curve could be represented by a powerseries, which evidently was not the case.76

Daniel Bernoulli admitted his incapacity to explain propagation with hismixtures of oscillatory modes. Yet he remained unshaken by Lagrange’s andEuler’s criticism. In his last memoir on vibrating strings, published in 1774, hereasserted the generality and physical pertinence of his solution:

If you suspect any restriction in my solution to the problem of the vibra-tions of tense strings initially curved according to a given law, this restric-tion consists necessarily in an insufficient enumeration of the simpleTaylorian 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 independentlyobtained 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’aété ébranlée que dans une partie,” HAB, 21 (1765), 307–334; EO 2:10, 426–450; Euler to Johann IIIBernoulli, 27 May 1764, cited in Truesdell, Ref. 2, 277n. Cf. Truesdell, ibid., 281–286.


The following year he similarly wrote to Nicolas Fuss: “I am still convinced thatmy method gives every possible case in abstracto.” Unfortunately, the math-ematical proof of this assertion was still lacking. Bernoulli never bothered todetermine the coefficients of the simple modes and ignored the formulas thatLagrange had given for this purpose. While his physical intuitions pointed toa remote future, his mathematics belonged to the first half of the eighteenthcentury.77

To summarize, Bernoulli, d’Alembert, Euler, and Lagrange held differentpositions regarding the permissible string curves, the curves that trigonometricseries could represent over a finite interval, and the status of partial vibrations.Bernoulli admitted any string curve for which both the ordinate and the radiusof curvature remained very small compared to the length of the string;78 hebelieved that trigonometric series could represent any such curve; he assertedthe physical existence of the partial vibrations. D’Alembert required the stringcurve to be analytic and close to the axis; he believed that any “discontinu-ous” curve and even some “continuous” curves could not be represented bytrigonometric series; he regarded partial vibrations as mathematical fictions.Euler admitted any continuous string curve with piecewise continuous slopeand curvature, and with small ordinate and slope; he denied that trigonometricseries could represent non-analytic curves, at least those which coincide withsegments of the axis (pulses); he ascribed some physical reality to partial vibra-tions of non-necessarily sine form. Lagrange originally admitted the same stringcurves as Euler (except for polygonal curves), but came to believe that his pas-sage from the discrete to the continuous required that all derivatives should befinite; he believed that trigonometric series could in some “asymptotic” senserepresent any curve, but sometimes denied perfect identity between the seriesand 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 geometerwho had a formula for calculating the coefficients of the trigonometric seriesfor a given curve. He had it as an implicit component of the Eq. (28) of his firstmemoir on sound, and as a special case of his generalization of Eq. (56). Yet hewas 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 vibra-tionum simplicium haud pertubatarum in systemate composito,” NCAP, 19 (1774), 239–259, on258–259; Daniel Bernoulli to Nicolas Fuss (circa 1775), in Paul Heinrich Fuss, Correspondancemathématique et physique de quelques célèbres géomètres du XVIIIesiècle, vol. 2 (Saint Petersburg,1843), 661–663.78 Bernoulli believed that infinite curvature (as in polygons) would contradict the assumption ofinfinitely small motion: see Daniel Bernoulli, “Mémoire sur les vibrations des cordes d’une épais-seur inégale,” HAB21 (1765), 281–306, on 283–284; Ref. 77, 247; also his letters to Johann III of 24May 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 onsound.

In his prize-winning memoir of 1748 on the equalities of the motion of Saturnand Jupiter, Euler encountered the function

f (ϕ) = F(cosϕ) = (1 − α cosϕ)−β , (63)

and its primitive∫

f (ϕ)dϕ, of which he needed to know a rapidly convergingexpansion. A power-series development with respect to α could not do, becausethis coefficient was too close to one. Instead Euler used the trigonometric devel-opment

f (ϕ) =∞∑

r=0

ar cos rϕ, (64)

which thus made its first entry in the history of celestial mechanics. Euler deter-mined the two first coefficients of this development through the approximations

a0 = 12p

p∑

k=1

(F(sk)+ F(−sk)), a1 = 1p

p∑

k=1

(skF(sk)− skF(−sk)), (65)

wherein sk = sin[(2k − 1)π/4p], and p is the number of “divisions of the rightangle,” which increases with the goodness of the approximation. He did notindicate how he had reached these expressions. One possibility is that he hadalready solved the above-mentioned problem of the discretely loaded elasticcord. In this case, he could use the identity (21) to determine the first coefficientof a sum of n sines that coincides with a given function for n equidistant valuesof the argument over the interval [0, π ]. The formulas (65) solve the similarproblem of a discrete cosine development over the interval [−π/2, π/2]. Eulerdid not take the limit of infinite p, for he did not know how to calculate theresulting integral. Instead, he used the approximation p = 10 in his numericalcalculations. As for the higher coefficients (r ≥ 2), he computed them through arecurrence relation that holds when the developed function is (1−α cosϕ)−3/2.79

D’Alembert adopted Euler’s trigonometric development of (1−α cosϕ)−β inhis own works on celestical mechanics. His Recherches of 1754 contain theformulas

79 Euler, Recherches sur la question des inégalités du mouvement de Saturne et de Jupiter, sujetproposé pour le prix de l’année 1748, par l’Académie Royale des Sciences de Paris (Paris, 1749), alsoin EO2: 25, 1–44, on 30. Cf. Louise and Ronald Golland, “Euler’s troublesome series: An earlyexample of the use of trigonometric series,” Historia mathematica, 20 (1993), 54–67, which con-tains an interesting reconstruction of Euler’s formulas, independent of the discrete interpolationformula.


a0 = 1π

π∫

0

(1 − α cosϕ)−β dϕ, a1 = 2π

π∫

0

(1 − α cosϕ)−β cosϕ dϕ, (66)

for the two first coefficients. Although he based his derivation of these formulason the identities

π∫

0

cos nϕ dϕ = πδ0n,

π∫

0

cosϕ cos nϕ dϕ = π

2δ1n, (67)

he probably reached them by taking the limit of infinite p in Euler’s approxi-mation (65). Indeed this limit yields the similar formulas80

a0 = 1π

π/2∫

0

[F(sin ϕ)+ F(− sin ϕ)]dϕ,

a1 = 12π

π/2∫

0

[sin ϕF(sin ϕ)− sin ϕF(− sin ϕ)]dϕ. (68)

In an attempt of 1757 to reconstruct Euler’s formulas (65), Alexis Clairautdetermined the coefficient of a sum of cosines agreeing with an arbitrary func-tion f (ϕ)for n evenly spaced values of φ between 0 and 2π . More exactly, hesought the numbers ar such that

f (2πk/n) =n−1∑

r=0

ar cos 2πkr/n, for 1 ≤ k ≤ n. (69)

Being aware of the orthogonality relations

n∑

k=1

cos 2πkr/n cos 2πks/n = n2δrs,

n∑

k=1

cos 2πkr/n = 0 for 1≤r≤n − 1, (70)

he obtained the formulas

a0 = 1n

n∑

k=1

f (2πk/n), ar = 12n

n∑

k=1

f (2πk/n) cos 2πkr/n for r ≥ 1 (71)

(it should be 2/n instead of 1/2n in the second formula). Clairaut noted thatthese formulas also applied “in the case when the law of the function would noteven 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 wrotethe expressions

a0 = 12π

2π∫

0

f (ϕ)dϕ, ar = 14π

2π∫

0

f (ϕ) cos rϕdϕ (for r ≥ 1) (72)

of the coefficients of the development

f (ϕ) =∞∑

r=0

ar cos rϕ (73)

(again the true expression for ar differs by a factor 4). Clairaut failed to notethat for a function defined over the interval [0, 2π ] this development was onlypossible for functions such that f (ϕ) = f (2π − ϕ). However, this condition wasautomatically met for the expressions (1−α cosϕ)−β he was aiming to develop.81

Twenty years later and in the same context of celestial perturbations, Eulergave the variant

a0 = 1π

π∫

0

f (ϕ)dϕ, ar = 2π

π∫

0

f (ϕ) cos rϕ dϕ (for r ≥ 1) (74)

of Clairaut’s formulas (72) for the coefficients of the cosine development (73) ofany function of cosϕ. He obtained this result directly through the orthogonalityproperty

π∫

0

cos mϕ cos nϕdϕ = π

2δmn,

π∫

0

cos nϕdϕ = 0 for n ≥ 1 (75)

perhaps in analogy with his earlier use of the identity (22) in the discrete caseor with d’Alembert’s use of Eqs. (67).82

Clairaut, d’Alembert, and Euler only used the integral expression of thecoefficients of a trigonometric development for functions of the form(1 − α cosϕ)−β , for which the existence of the development is not in doubtsince it can be obtained algebraically by first developing in powers of cosϕand 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 égardaux 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-Guin-ness, Ref. 24, 19n.


former as a mathematical curiosity. So did too Lagrange when he encountered(1 − α cosϕ)−β in his works on celestial mechanics, even though he knew theintegral expression of the coefficients from his work on vibrating strings.83

Explicit trigonometric developments were not only known for analytic andperiodic functions such as (1−α cosϕ)−β . In the “Subsidium calculi sinuum” heread in 1753 (EO1:14, 582–584), Euler obtained trigonometric series that couldnot possibly be analytic through more than one period. His starting point wasthe identity

y =∞∑

n=0

αn cos nx = 12

∞∑

n=0

(αneinx + αne−inx) = 1 − α cos x1 − 2α cos x + α2 , (76)

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

∞∑

n=1

cos nx = −12

and∞∑

n=1

(−1)n cos nx = −12

. (77)

For the second series, term by term integration yields,

∞∑

n=1

(−1)n

nsin nx = −x

2,

∞∑

n=1

(−1)n

n2 cos nx = x2

4− π2

12, etc., (78)

wherein the integration constants result from the known values of the series forx = 0.

Euler did not say anything on the convergence of these series, or on the rangeof validity of the simple polynomial expressions of their sums. As the series areevidently 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 con-vergence factor αn (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 inthe limit α → ±1.

In 1771 Daniel Bernoulli obtained similar series starting directly from thedivergent sums (77). To justify this procedure, he relied on the then commonidea that divergent series could converge in some sense if the grouping of sev-eral consecutive terms led to a convergent series. Unlike Euler, he identifiedthe singular points and specified in which interval the series were valid. Forinstance, he gave

83 D’Alembert, Ref. 80, 67: “[méthode] plus curieuse et plus géométrique que commode pour lecalcul.” Lagrange, Recherches sur les inégalités des satellites de Jupiter causées par leurs attractionsmutuelles (Paris, 1766), also in LO6, 62–225, on 87–89; “Sur le problème de Kepler,” AcadémieRoyale des Sciences de Berlin, Mémoires(1771), also in LO3, 113–138, on 121–123; “Remarquessur les équations séculaires des mouvements des nœuds des inclinaisons des orbites des planètes,”MAS (1774), also in LO6, 633–709, on 645–646.

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∞∑

n=1

1n

sin nx = (2p + 1)π

2− 1

2x for 2pπ < x < 2(p + 1)π . (79)

Interestingly, he noted that the two series obtained by integrating the latterseries two and four times were “fully applicable to the theory of vibratingstrings,” for they shared the initial string curve’s property of vanishing at x = 0and x = 2π .84

We saw that in a letter of 1768, Lagrange had no qualms developing the func-tions x, x2, and x2/3 in trigonometric series. Moreover, in his memoir on soundof 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, ashe did in his criticism of Lagrange’s memoir.85

Euler’s and Lagrange’s awareness of trigonometric series that coincided withalgebraic functions over one period of the variable seems to contradict some ofthe limitations they saw in these series. Remember that in his reply to Bernoulli’smemoir of 1753, Euler had flatly rejected the possibility that a trigonometricseries could represent a string curve that did not have a periodic (and odd) ana-lytic continuation. This belief is clearly inconsistent with his giving (around thesame time) a trigonometric series for the functions x and x2. He probably himselfrealized this conflict, since he never repeated the argument. Instead he asserted,from 1759 on, that trigonometric series were not able to represent pulses thatvanished everywhere except on a small segment. Lagrange agreed, in 1765, thatin such cases trigonometric series could only be “asymptotic approximations”to the true function. Hence we must conclude that Euler and Lagrange onlyrequired trigonometric series to be analytic over the interval in which they wereused to represent a given function. Whereas pulses and polygons failed to meetthis criterion, algebraic functions did, with some necessary singularities beyondthe interval of representation.

One may still wonder why Euler and Lagrange never used the Clairaut-Lagrange formulas to produce the kind of trigonometric series they tolerated.One possible reason is that these formulas were less convenient than moredirect, algebraic methods. Another possible reason is that these formulas didnot in themselves contain a proof of the existence of the series, whereas themore algebraic procedures seemed to do so at a time when convergence con-siderations were usually neglected.

84 Daniel Bernoulli, “De indole singulari serierum infinitarum, quas sinus vel cosinus angulorumarithmetice progredientium formant, earumque summatione et usu,” NCAP, 17 (1772), 3–23, alsoin Werke, 2 (1982), 119–134, on 133. Ibid. on 134, Bernoulli judged that the corresponding solutionsof the equation of vibrating strings were physically unacceptable, for reasons I was not able tounderstand. 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 hisresult 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.


Fig. 4 Euler’s representationof coincidences between thepulses 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 inmusic theory. In his Basel period, he published his Dissertatio physica de sono,which contained the first quantitative theory of sounding pipes, based on theanalogy with vibrating strings. At that time, he already planned a major treatiseon music which appeared only in 1739 as the Tentamen novae theoriae musicaeex certissimis harmoniae principiis.In agreement with Mersenne’s and Galileo’sviews, he defined sound as “nothing but the perception of successive pulses[ictus] occurring in the particles of air that surround the organ of hearing.” Inthe full picture, the pulses caused by sonorous bodies are gradually transmittedthrough the elastic “globules” of the air, and then strike the eardrum whichexcites the aural nerves. The pitch of the sound corresponds to the frequencyof the pulses (when it is well-defined). Consonance corresponds to the frequentcoincidence of the pulses from different sounds (see Fig. 4). Euler attributedthe satisfaction given by consonance to the mind’s predilection for order. Whilehe fully understood the subjective character of this definition, he made it thebasis of a rigorous arithmetic of music.86

86 Euler, Dissertatio physica de sono(Basel, 1727), also in EO3:1, 181–196; Tentamen novae theoriaemusicae 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 Beitragzur 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.

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For any combination of sounds with commensurable frequencies, Eulerdefined the “degree of suavity” (suavitatis gradus) as a measure of the easewith which the corresponding ratio could be perceived. To the ratio 1:1 hegave the degree one; to the next simple ratio 1:2, the degree 2. To the ratio1: p, where p is a prime number, the degree p. To the ratio 1:4, he gave thesame 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 theratio 1 : pα1

1 pα22 · · · pαn

n he gave the degree∑n

i=1 (αipi − αi)+ 1. Lastly, to themultiple ratio of a sequence of integers, he gave the degree of the ratio betweenthe 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 harmonydirectly followed from this notion.87

Although Euler mentioned the production of harmonic sounds by musicinstruments and referred to Sauveur’s experiments on the multiple modes ofvibrating strings, he said nothing about the coexistence of harmonic sounds orabout 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 1753he rejected Bernoulli’s idea that the simultaneous hearing of harmonics provedthe reality of partial sine modes. In his opinion, the harmonics could result fromany motion involving a succession of similar loops separated by nodes, and thesine shape did not have the privileged status that Taylor and Bernoulli accordedto it. In a letter to Lagrange of 23 October 1759, he departed even further fromBernoulli’s view by agreeing with Lagrange that the vibrating string was not thetrue source of the harmonics: “For the sounds of Music, I perfectly agree withyou, Sir, that the consonant sounds that Mr. Rameau claims to be hearing fromthe same string come from other shaken bodies.”88

As we will see in a moment, the simultaneous hearing of harmonics playedan important role in the foundations of Jean-Philippe Rameau’s theory of har-mony. Euler’s letter to Lagrange continues with a rejection of this view: “I donot see why this phenomenon should be regarded as the principle of Musicrather than the true proportions on which it is based.” In one of his letters toa German princess, Euler similarly rejected Rameau’s appeal to resonance byconsonance. For Euler, the commensurability of the frequencies of two soundswas the common cause of resonance and harmony; resonance did not causeharmony.89

In the early 1760s, Euler worked on the daunting problem of the non-uniformvibrating string. He was able to solve two cases, one for which the density of thestring varies as the function (1 + αs)4 of its curvilinear abscissa s, the other inwhich the string is made of two uniform parts of different density. In the lattercase, he proved the existence of simple sine modes with incommensurable fre-

87 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 princessed’Allemagne sur divers sujets de physique et de philosophie (Paris, 1842).


quencies, as Bernoulli had long ago predicted. In the same period, he found outthat bells also had incommensurable modes against the wide-spread belief tothe contrary. As he explained in the introduction of his paper on non-uniformstrings, these findings as well as his earlier work on vibrating bands confirmed hisview that the cause of harmony should not be sought in nature. Vibrating bodiesgenerally do not emit mutually consonant sounds. Only musical instruments doso.90

This additional objection to Rameau’s theory went along with a partial returnto Bernoulli’s interpretation of harmonics:

The remarkable phenomenon that several sounds following the ratio ofthe numbers, 1, 2, 3, 4 etc. can be heard from the same string, which themost celebrated Daniel Bernoulli explained in the most felicitous man-ner, only occurs…in strings of uniform thickness. Although non-uniformstrings and other kinds of vibrating bodies can also emit several sounds atthe 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 theprinciple on which the highest artist of music de Rameau bases universalharmony 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 bringhim closer to accept Bernoulli’s mixtures of simple modes as the most generalsolution of vibration problems. Although the only periodic solutions he couldidentify in dissonant cases were sine functions of time, he still believed that therecould be other sorts of solution. Besides, he remained confident that trigono-metric sums were “continuous” whereas the general solution ought to dependon arbitrary “discontinuous” functions representing the initial conditions.91

To summarize, Euler’s pulse-based theory of musical harmony predisposedhim to ignore Bernoulli’s tracing of any sound to the mixture of simple modesof the sonorous body. He was nonetheless willing to admit that the hearing ofsimultaneous sounds of different pitch from the same source corresponded toa complex vibration of the source, involving a superposition of vibrations atdifferent frequencies. What he could not accept was the restriction or reductionof these partial vibrations to Bernoulli’s simple modes, whose amplitude couldonly be a sine function of time.

4.2 D’Alembert’s Rameau

In 1752, d’Alembert published his own Elémens de musique théorique et pra-tique 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, alsoin 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 deniedany originality: “Nothing is mine except the order, and the errors that might befound.” Yet the task of extracting the essence of Rameau’s prolix and at timesobscure 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 bemade more familiar, more luminous, and thus more useful to the many….He gave me solace by adding to the solidity of my principles a simplicitywhich I suspected but could not have reached without much effort andmaybe in a less felicitous manner…. Sciences and arts would haste eachother’s progress if, favoring truth over self-esteem, some Authors had themodesty of accepting aid, others the generosity of offering it.

In his Nouveau systême de musique théorique, published in 1726 as a phys-ics-based introduction to his Traité de l’harmonie of 1722, Rameau founded histheory of harmony on the hearing of the three first harmonics of the soundsproduced by various instruments. He possibly borrowed the idea of relatingharmony to harmonics from Sauveur, whom he cited together with Mersenne.He noted that harmonics were heard slightly after the fundamental, and thatthey were more easily distinguished if they were imagined first. In d’Alembert’sinterpretation, Rameau’s system rests on three basic facts of experience. Thefirst is the hearing of the first three harmonics of the fundamental of the soundemitted by a sonorous body, most easily for the thickest string of a violon-cello. The second fact is that, in Sauveur’s terms, bodies tuned at a harmonicof another sounding body “resonate” as a whole and bodies tuned at a sub-harmonic of the sounding body “quiver” (frémissent) with a number of nodesdepending on the order of the exciting harmonic. The third fact is the similarityanyone perceives between a tone and its octave.93

As a consequence of the two first facts, the most perfect chord should be thatobtained 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 17thmay be replaced with the 5th and the 3rd without much loss of harmony. Theresult is a major perfect chord, for instance C, E, G, C. A little less naturally, wemay combine a tone with its three subharmonics, that is, the major 8th, 12th, and17th 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 justificationsof traditional chords, Rameau and d’Alembert added the ingenious notion ofthe “fundamental bass,” namely, a succession of tones separated by a fifth (ora 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 detoutes 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 threefacts 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.


(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 hadreached the principle of the fundamental bass by mere inspection of contempo-rary music and justified it only later through the experience of harmonics. Heregarded 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 harmonyand scales. For instance, he declared that the perfect chord was “the immediatework of nature.” In the introduction to the second edition of his Elémens, pub-lished in 1779, he condemned “those geometers who, fancying themselves asmusicians, pile numbers over numbers in their writings, perhaps imagining thatthis apparatus is necessary to art.” The unnamed target of this criticism probablywas his old competitor Euler. Against the latter’s arithmetic of harmony, d’Al-embert favored a “physical” approach, based on “analogy” and “convenience.”He flatly rejected the “gratuitous” idea that the orderly coincidence of pulseswas the ultimate source of musical pleasure, and instead defended the conceptof musical harmony as an analogy with harmonies naturally found in sonorousbodies.95

D’Alembert did not worry that his and Rameau’s starting point was hardlyless gratuitous than Euler’s. Why should art imitate nature? Why should natu-ral harmonies be perceptible and preferable? Perhaps aware of this difficulty,Rameau suggested a theory of hearing following which harmony would bedirectly perceptible:

What has been said of sonorous bodies should be applied equally to thefibers which carpet the bottom of the ear’s cochlea [le fond de la conque del’oreille]; these fibers are so many sonorous bodies, to which the air trans-mits its vibrations, and from which the perception of sounds and harmonyis carried to the soul.

As no one before Helmholtz elaborated on this suggestion, Rameau’s basicfact of experience remained vulnerable to Euler’s criticism that the harmoniccharacter of overtones was the exception rather than the rule in nature. Mostsonorous bodies, even bells, emit overtones that are not harmonics of their fun-damental. It thus appears that musicians of all times have artificially selected orinvented the few sonorous bodies for with the overtones are truly consonant.Some higher principle must have presided to this selection, and Euler found itin 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) thatthere 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’sview that the distinction between consonance and dissonance had a naturalorigin. In his Elémens he gave the etymology of “dissonance” as “soundingtwice”: the tones of a dissonant chord are heard separately whereas the soundsof a consonant chord are usually perceived as a whole—a fact already assertedby Euclid. As Mersenne, Sauveur, and Rameau had noted, it requires specialattention, preparation, and training to hear the partial sounds in the latter case.While d’Alembert did not list this empirical difference between dissonanceand consonance among his basic facts, he surely included resonance inducedby harmonic and subharmonic overtones. The harmonic character of the over-tones is here essential, against Euler’s claim that nature has no preference forharmonics.97

Although d’Alembert supported a theory of music based on the harmonicscontent of musical sounds, he strongly opposed Daniel Bernoulli’s recourseto harmonic analysis in the theory of vibrations. Just as Euler defined a toneas 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 treatmentof the problem of vibrating strings, he refused to accord a privileged role tosine-shaped motions. And he denied that Bernoulli’s solutions to the problemof vibrating strings had the sort of multiple periodicity required for a physicalinterpretation of the partial vibrations:

The theory of Mr. Bernoulli cannot explain the multiplicity of sounds thatis given by observation. We should therefore acknowledge that all thesefacts are an enigma that we cannot elucidate. Indeed can we pretend tohave explained them by regarding the points of the string as composedof several other points, by assuming fictitious loops and moving nodes?There is nothing, it seems to me, that we could not justify by so arbitrarya method.

As we earlier saw, d’Alembert claimed that harmonic sounds were heard evenfor a simple vibration (meaning one loop only). At any rate, Bernoulli’s the-ory could not explain why the 12th and the 17th were the only harmonics thatcould easily be heard. D’Alembert pushed this point, for he knew that Ra-meau’s theory of harmony would entirely collapse if higher harmonics wereadmitted.98

At first glance, d’Alembert’s opposition to Euler’s theory of music and hisopposition to Bernoulli’s recourse to musical experience seem to be of a differ-ent nature. In the former case he seems to be protecting music from mathemat-ics; in the latter he seems to be protecting mathematics from music. However,both attitudes derive from the same desire to police the border between mathe-matics on the one hand, and physics and art on the other. D’Alembert believedthat art and a good part of physics were not amenable to mathematical treat-

97 D’Alembert, Ref. 92, 11.98 D’Alembert, Ref. 34 (definition of tone); Ref. 38, 61 (quote), 60 (12th and 17th). On the latterpoint, cf. Bailhache, Ref. 4, 135.


ment and ought to remain autonomous fields. Hence came his disdain for Euler’sarithmetic of music and his rejection of non-analytic solutions to the problemof vibrating strings. Symmetrically, d’Alembert rejected recourse to physical ormusical intuition in deriving mathematical results. Hence came his rejection ofBernoulli’s appeal to the hearing of harmonics in justifying mixtures of simplemodes.

In order to fully understand d’Alembert’s mind-set, it must be recalled that inthe eighteenth century physics was still often defined as the non-mathematicalstudy of inanimate nature, as opposed to the mathematical sciences of mechan-ics, geometrical optics, geodesy, fortification theory, etc. The extent to whichphysics should ultimately be subsumed under mathematics was a controver-sial issue. Whereas Euler, Bernoulli, and Lagrange shared the Newtonian andCartesian ideal of an entirely mathematized or geometrized physics, d’Alembertwished to preserve the non-mathematical essence of art and of some parts ofphysics, as well as the non-physical essence of mathematical demonstration.He equally disliked Euler’s mathematical music and Bernoulli’s musical math-ematics.99

4.3 Bernoulli and Lagrange between physics and music

Unlike Euler and d’Alembert, Daniel Bernoulli never dwelt on musical the-ory proper. In his long response to d’Alembert’s and Euler’s theories of thevibrating string, he briefly condemned Rameau’s physical foundation of musicalharmony:

If one holds a steel rod by the middle and strikes it, one hears a confusedmixture of several sounds, which are judged extremely disharmonious bya skilled musician, so that a contest of vibrations is formed that neverbegin or finish at the same time except through a great chance. Hence onesees that the harmony of the sounds heard simultaneously from the samesonorous body is not essential to this matter and should not serve as aprinciple for the systems of music.

This extract also documents Bernoulli’s approval of Euler’s recourse to thecoincidence theory of harmony, since it refers to the lack of coincidences ofthe vibrations of dissonant sounds. Elsewhere Bernoulli wrote approvingly ofRameau’s principles, despite his rejection of the physical foundation that Ra-meau purported to give them.100

99 On physics versus mixed mathematics, cf. Thomas Kuhn, “Mathematical versus experimentaltraditions 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 longitudinequam crassitie, ab invicem diversis, compositarum,” NCAP, 16 (1771), 257–280, on 268 (approvingRameau).

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Although Bernoulli did not produce his own musical theory, he greatly con-tributed to acoustics through his numerous publications on the theory of vibra-tions and through his correspondence. When Euler published his Tentamennovae theoriae musicae, Bernoulli confided to him:

I am waiting very eagerly for your theoriam musicam, for I have myselfmuch meditated and experimented over these matters. The experimentsconfirm my theoriam de sono fistularum quite beautifully.

Bernoulli’s work on flutes and other sounding pipes was indeed important, boththeoretically and experimentally, although most of it was only published witha long delay (in 1762) after he had seen Lagrange’s competing work. Bernoulliwas conversant on many other aspects of music theory, as appears in the seriesof comments he wrote to Euler after reading his treatise. He reproached Eul-er with insufficient knowledge of Mersenne’s Harmonie universelle and with afrequent lack of clarity in the foundations. He judged that Euler’s new temper-ament had little chance to be accepted by musicians, and he rejected Euler’sbasic idea of founding harmony on the mind’s predilection for perfection andorder.101

As we saw, the perception of natural and musical sounds played a significantrole in shaping Bernoulli’s theory of vibrations. His idea of a mixture of sim-ple modes most likely resulted from his ability of hear the harmonics of gravemusical sounds. Similarly, he justified his general principle of superposition byour ability to distinguish the sounds from the various instruments of an orches-tra. One may wonder why he was not satisfied with a derivation of this principlefrom Hooke’s law regarding the linear character of elastic forces. The reason isthat unlike d’Alembert, Euler, or Lagrange, Bernoulli never wrote equationsof motion for arbitrary initial conditions. Instead he investigated the pendulousmotion of simple modes, without being originally aware that any other motioncould be obtained by superposition of such modes.

The importance of the perception of harmonics in Bernoulli’s physics ofvibrations raises the question of the view he might have had regarding thehearing process. What peculiarity of the ear enables it to analyze a sound intoits harmonics? The old theory of the coincidence of pulses could only providea limited answer, in cases for which only few harmonics were present. In theletter to Euler that contained his criticism of the Tentamen, Bernoulli vaguelysuggested a new theory of hearing:

As you explain hearing in a physiological manner [through beats on theeardrum], I have thought again about a conjecture, namely whether themembrane of the eardrum should not be in unison with the perceivedsound [unisona cum sono percepto], which duty the musculi can performwith extraordinary speed and wherefrom very many phenomena could bededuced.

101 Daniel Bernoulli to Euler, 5 Nov 1740, in Fuss, Ref. 77, 461–465; 28 Jan 1741, ibid., 466–472; 7Mar 1739, ibid., 453–457 (rejecting Euler’s temperament).


Presumably, Bernoulli had in mind a kind of resonance that would justify theear’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 simplevibrations and treated the eye and the ear as harmonic analyzers:102

The different colors probably are nothing but different perceptions in theorgan 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 greatnumber of vibrations can be formed at the same time, very different fromeach other, each of which separately causes a different sound in the organof hearing.

No more than Bernoulli did Lagrange produce his own system of music. Histheoretical writings on sound are nevertheless full of remarks of musical inter-est. Like Euler, he conservatively defined a (musical) sound as a periodic seriesof pulses, and he related musical pleasure to order:

If the sonorous body is such that the vibrations of its parts begin and endalways at the same time, the ear will be struck by several taps succeedingeach other by equal time intervals, and this uniformity of impressions willproduce the pleasant feeling that is called sound.

Lagrange also reduced harmony to “the concurrence of vibrations,” which inhis view was the common foundation of every theory of harmony, includingRameau’s and Giuseppe Tartini’s. Indeed Lagrange believed that Rameau’snatural foundations of harmony (the hearing of harmonics and resonance)could be reduced to resonance only, which is explained by the concurrence ofvibrations. Similarly, he believed that the low-pitch combination tones on whichTartini founded his theory of harmony could be explained by the concurrenceof vibrations. Both cases require some explanation.103

Although Lagrange did not deny the hearing of harmonics, he believed theseadditional sounds to result from the resonance of nearby bodies. He confirmedd’Alembert’s observation that harmonics were heard even when the motionseemed most simple:

Having examined the oscillation of tensed strings with all the attentionthat I can, I have always found them simple and unique in all their exten-sion, whence it seems impossible to me to conceive how different tonescan be engendered simultaneously…. I therefore incline to believe thatthese sounds are produced by other bodies that resonate to the principalsound.

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, hemay have meant the muscles that are attached to the ossicles of the ear.103 Lagrange, Ref. 43, 144–145.

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Lagrange went on to call for a crucial experiment in which the oscillating stringwould be isolated from any other body, in which case he expected no harmonicto be heard.104

As for the Tartini tones, which d’Alembert had described in great detailsin his “Fundamental” article, Lagrange explained them by means of the beatphenomenon, which had long been understood to result from the coincidenceof vibrations. The frequency of these “third tones” is indeed the difference ofthe frequency of the two combined tones, just as is expected from beats. Forinstance, when a tone and its exact fifth are sounded on a violin, the loweroctave 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 inacoustics. In his researches on sound, Lagrange not only gave a full account ofSauveur’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 aremore frequent and hence remain less sensible.” Lagrange went on to argue thatthis explanation of harmony equally derived from the principle of concurrentvibrations, since beats somewhat measure the degree of the concurrence.106

Lagrange’s reduction of every theory of harmony to the concurrence of vibra-tions and his concomitant rejection of harmonics from musical strings squaredwell 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 partialoscillations that Bernoulli imagined and heard. As Lagrange believed to haveproved in taking the limit of the discretely loaded string, the various oscillatoryterms of Bernoulli’s solution interfered with each other to produce motionsthat no longer reflected their oscillatory character. In the second edition of hisMécanique analytique, he insisted that Bernoulli’s formula could not explainRameau’s simultaneous harmonics and added:

The series that could give the different modes disappears from the formulawhen the number of bodies is infinite, and the result is, for every point ofthe string, a simple and uniform law of isochronism which depends imme-diately on the initial state.Again, Lagrange had in mind that the physical character of the superpo-sition of simples modes disappeared in the continuous limit and left placeto d’Alembert’s and Euler’s simple “construction.”107

To sum up, Euler’s, d’Alembert’s, Bernoulli’s, and Lagrange’s diverging viewson the status of Bernoulli’s mixtures of simple modes were related to their viewson musical harmony and theory. The reason for this interrelation is that bothkinds of considerations depended on the status they accorded to the percep-tion 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.


theories of vibrations. Daniel Bernoulli regarded the perception of harmon-ics as a strong indication of the reality of the mixture of pendulous simplemodes. Euler mostly agreed that this perception revealed the superpositionof partial vibrations, but refused to restrict the partial vibrations to Taylorianoscillations. D’Alembert and Lagrange did not even accept the reality of par-tial vibrations, though for different reasons. While d’Alembert argued that theperceived harmonics eluded physico-mathematical analysis, Lagrange arguedthat harmonics truly corresponded to the multiple resonance of surroundingbodies.

Let us now consider the extent to which the perception of harmonics con-ditioned the views of our four geometers regarding musical theory. For DanielBernoulli, this perception could not be the basis of musical harmony, becausesonorous bodies in nature usually produce dissonant overtones rather than theharmonics heard in musical sounds. Euler approved this argument, the more sobecause his own theory of music rested on a quantified version of the old princi-ple of the coincidence of pulses. D’Alembert ignored this difficulty and insteadused the perception of harmonics as a means to directly base music on physics,without Euler’s dubious psycho-mathematical connection. Lagrange denied theexistence of intrinsic harmonics, and traced any perception of harmonics andany harmony to the coincidence of pulses.

Despite their linking through the problematic of the perception of harmon-ics, there is no straightforward correspondence between views on musical har-mony and views on simple-mode mixing. Euler, Lagrange, and presumablyBernoulli all supported the reduction of harmony to the coincidence of pulses,and yet differed in their assessment of simple-mode superposition: Bernoulliregarded it as real mixture, Lagrange as mathematical fiction, and Euler as anillegitimately restricted case of a broader phenomenon. D’Alembert adoptedRameau’s principle of harmonics and yet rejected Bernoullian mixtures. Thereason for this delusively complex pattern is that opinions on the perceptionof harmonics could not by themselves determine the views of our four geom-eters regarding music and the physics of vibrations. Equally important wastheir broader understanding of the relation between mathematics, physics, psy-chology, and art. Bernoulli pleaded for a physical and musical mathematics,Euler and Lagrange for a thoroughly mathematized physics and a psycho-mathematical music, d’Alembert for an autonomous mathematics, a selectivemathematical 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 begansupporting the teaching of Lagrange and Gaspard Monge at the newly foundedEcole Polytechnique. It contained an influential demonstration of the principleof virtual velocities, as well as a few remarks on the nature of equilibrium. After

402 O. Darrigol

reminding his reader that the stability of equilibrium depended on the nature ofsmall motions around equilibrium, Fourier referred to Lagrange’s and DanielBernoulli’s relevant works:

This question has been subjected to a very elegant analysis by the illus-trious author of the Mécanique analytique. One can further prove, by theresults of this solution, an important proposition that Daniel Bernoullihas known first and proved in several particular cases: namely, that thesmall oscillations of bodies are made of simple oscillations that occur atthe same time without troubling each other.

Fourier went on to describe the pendulous nature of the simple modes and todiscriminate between the periodic and multiperiodic cases for the motion ofthe whole body. In the periodic case, he noted, the vibrating body produce awell-defined tone, no matter how the vibration is started.108

At the end of his memoir, Fourier emphasized that the Bernoullian analysisonly applies in the limit for which the oscillations are infinitely small, so that “thefigure represented by the results of calculation only has an abstract existenceand can never exactly be that of the oscillating body.” He mysteriously added:“This remark gives the solution of the difficulties that have been propoundedabout the figure of vibrating strings.” Perhaps he meant that the discussions ofthe mathematically permissible figures were beside the point, since the equa-tion of vibrating strings only applied to an ideal motion. At any rate, he didnot doubt that simple-mode analysis applied to nature and explained the well-defined pitch of the sound emitted by sonorous bodies. He further noted thatthe theory implied the possibility of forming multiple vibrations:

When several [sonorous] bodies are brought in contact and when thereexist certain ratios of figure and dimension, it is enough to shake one ofthem to excite and sustain the motions of the others. Far from clashing witheach other in their particular vibrations, the system of all these bodies soonmoves symmetrically and in equal times. Thus, had not our senses alreadydone 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 areespecially noticeable in the quivering of sonorous bodies; and it is thisbranch of Integral Calculus that provides the fundamental principles ofharmony.

Fourier closed his memoir with this allusion to music theory. Clearly, he believedin Rameau’s doctrine that harmony depended on the physics of multiple vibra-

108 Joseph Fourier, “Mémoire sur la statique contenant la démonstration du principe des vitessesvirtuelles et la théorie des moments,” JEP, 5 (1798), 20–60, also in FO2, 477–521, on 507. Cf. LouisCharbonneau, “Fourier et la mécanique: Une histoire méconnue. De la mécanique à la théorie de lachaleur,” in Sciences et sociétés pendant la révolution française (actes du 114e congrès national dessociétés 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.


tions and resonance; and he shared Daniel Bernoulli’s belief in the physicalreality of harmonics.109

During the brief existence of the Ecole Normale of the year III (of the revo-lutionary calendar), Fourier attended the physics lectures of René-Juste Haüy,which had a whole chapter on acoustics and music theory. Haüy referred toWallis’ and Sauveur’s experiments on harmonic vibration and multiple reso-nance, and to Tartini’s third sound. He agreed with Mersenne and Sauveur thata 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, heagreed with Mersenne, d’Alembert, and Lagrange that the emitting string didnot undergo any multiple vibrations. As his own experiments excluded Lag-range’s suggestion that the harmonics were due to the resonance of nearbybodies, he joined Mersenne in assuming that the simple vibration of a body wassomewhat able to excite multiple vibrations in the surrounding air. In his discus-sion of musical harmony he insisted on the subjective, cultural component of theappreciation of consonance, and judged that Rameau’s and Tartini’s theoriesonly gave “more or less plausible conventions.” His discussion of wind instru-ments was based on Daniel Bernoulli’s intuitive theory, with special emphasison the principle of superposition that allows us to distinguish the sounds emittedby the various instruments of an orchestra.110

Fourier’s main source on vibrations was Lagrange’s Méchanique analitiqueof 1788, to which he referred his readers for more precise considerations. Thelast section of the relevant chapter of Lagrange’s treatise contained a full deri-vation of the simple modes of the discretely loaded string, which Fourier laterimitated in the context of heat propagation. Another of Fourier’s sources wasthe first book of Laplace’s Système du monde, which Fourier recommended forthe following praise of simple-mode analysis:

When a point of a calm surface of water is slightly agitated, circular wavesare formed and spread around it. When the surface is agitated at anotherpoint, new waves are formed and mix with the former; they travel overthe surface disturbed by the first wave as they would do over a calmsurface, so that they can be perfectly distinguished in the mixture. Whatthe eye perceives with respect to waves, the ear perceives with respect tosounds or aerial vibrations, which travel simultaneously without troublingeach other and produce very distinct impressions. The principle of thecoexistence of simple oscillations, which we owe to Daniel Bernoulli, isone of these general results that interest us through the ease with whichthey enable our imagination to represent phenomena and their multiplechanges. It can easily be derived of the analytical theory of the small oscil-

109 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 Nor-male de l’an III. Leçons de mathématiques. Edition annotée des cours de Laplace, Lagrange et Monge(Paris, 1992).

404 O. Darrigol

lations of a system. These depend of linear differential equations whosecomplete integral is the sum of particular integrals. Thus, the simple oscil-lations combine with each other to form the motion of the system just asthe particular integrals that represent them add to each other to form thecomplete integrals. It is interesting thus to retrieve in natural phenomenathe intellectual truths of analysis. This correspondence, of which the Sys-tem of the World will offer us numerous examples, is one of the greatestcharms attached to mathematical speculation.

The physical pregnancy of mathematical analysis in general and the physicalreality of partial modes in particular, here chanted by Laplace, were to becomethe leitmotivs of Fourier’s own mathematical physics.111

To sum up, by 1798 Fourier believed in the unrestricted validity of the simple-mode analysis of small oscillations, in its physical character, and in its relevanceto the theory of musical harmony. He was aware of the old quarrel about vibrat-ing strings, and implicitly endorsed Daniel Bernoulli’s solution as he assertedthe complete generality of simple-mode analysis. Having read the relevant sec-tion of Lagrange’s Méchanique analitique, he knew how to calculate the simplemodes 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 thecontinuous limit of this problem. Nor did it give the expression (21) of thecoefficient of the simple-mode superposition as a function of the initial ordi-nates of the loads in the discrete case. Most likely, Fourier was not aware ofLagrange’s early memoirs on sound, which contained the relevant analysis. Hadhe read them, it would be difficult to understand the roundabout way in whichhe arrived at his famous theorem.

5.2 The discrete model

Why and when Fourier began studying the propagation of heat is not knownexactly. As he himself acknowledged, in 1804 he received a short article onthis theme by Laplace’s protégé Jean-Baptiste Biot. All relevant manuscriptsof Fourier seem to be posterior to that date. Plausibly, Biot’s considerationsprompted Fourier’s considerations and experiments. To attempt a mathemati-cal theory of heat propagation was a typically Laplacian idea, since the famousastronomer and his disciples pursued the grand aim of subjecting all physics tomathematical analysis following the model of astronomy. Biot investigated thelimited case of a long bar heated at one extremity. He assumed Newton’s lawof cooling, according to which the quantity of heat exchanged by two bodies incontact is proportional to the difference of their temperatures. He applied thislaw both to heat radiation from the bar and to heat exchange between consec-utive 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 lesuns 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.


a given slice of the bar to the heat received from the contiguous slices minusthe 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 evidentlywrote 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 capacityper unit volume, S the section of the bar, α and h two constants characterizingthe speed of heat exchange. Developing to first order in dt and to second orderin dx, this gives

C∂θ

∂t= K

∂2θ

∂x2 − hθ , 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 isthat it yields an infinitesimal value of the constant K, whereas in reality thetemperature decreases gradually from the hot extremity of the bar. Biot latersuggested that this difficulty prevented him to publish the whole reasoning andthe equation.113

After reading Biot’s article, Fourier presumably stumbled over the same diffi-culty, and therefore decided to study a discrete model of heat transfer in whichit 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 problemanalogous to that of the discretely loaded elastic string, which Lagrange hadalready solved. As can be judged from a manuscript dating from 1805/1806, hismodel consisted in a linear arrangement of n equidistant, equal, and disjointmasses, with heat exchange occurring between two consecutive masses at a rateproportional 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. Atthe Ecole Normale of the year III, Fourier had heard Haüy lecture on heat, with clear notions ofheat capacity, conduction, and equilibrium based on the caloric-fluid concept: Haüy, in Séances desEcoles Normales, Ref. 110, vol. 2, 140–144.113 Biot, Traité de physique expérimentale et mathématique (Paris, 1916), vol. 6, 669–670, quotedin Truesdell, Ref. 112, 50.114 Fourier, “Sur la propagation de la chaleur” [draft written in 1805/6] in MS 22525 du fondfrançais 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,

406 O. Darrigol

Calling θk the temperature of the mass k, this assumption leads to the equa-tions

θk = α(θk+1 − 2θk + θk−1), (82)

where α is the rate of heat exchange between two consecutive masses dividedby the heat capacity of one mass, and the fictitious temperatures θ0 and θn+1are such that

θ0 = θ1 and θn = θn+1. (83)

Fourier then followed Lagrange’s procedure for solving the similar system

xk = α2(xk+1 − 2xk + xk−1), with x0 = xn+1 = 0. (12)

Namely, he sought the simple modes for which

θk = akeht. (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)

with

h = 2α(cosϕ − 1). (87)

Equivalently, Fourier wrote

ak = a sin kϕ + b sin(k − 1)ϕ, (88)

which is better adapted to the boundary conditions (83). The first of theseconditions leads to

b = −a and ak = a1sin kϕ − sin(k − 1)ϕ

sin ϕ. (89)

in Cahiers d’histoire et de philosophie des sciences, 42 (Paris, 1994). On the chronological order ofFourier’s researches on heat, cf. Grattan-Guinness, Ref. 2, 441–442.


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

crsin krπ/n − sin(k − 1)rπ/n

sin rπ/ne−2α[1−cos(rπ/n)]t. (91)

He managed to express the coefficients cr as a function of the initial temper-atures of the masses in the case of two and three masses only. Judging fromthe 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 Eulerand Lagrange had so successfully used in the similar problem of the discretelyloaded string. And the relative complexity of the simple modes for the presentproblem prevented him to directly invert the relevant matrix.115

This aborted approach to heat propagation was nonetheless important toFourier 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 super-position (91) decreased exponentially in time, so that the system tended to theuniformity of the mode r = 0 after some time, its departure from uniformityhaving ultimately the form sin kπ/n − sin(k − 1)π/n of the mode r = 1. Thisis the prototype of an argument Fourier often gave in favor of the reality ofsimple modes. It may have been reminiscent of Daniel Bernoulli’s reference tothe 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 decidedto return to Biot’s continuous problem. In order to avoid the infinitesimal het-erogeneity of Eq. (81), he assumed that the rate of heat exchange betweensuccessive slices of the bar was inversely proportional to their width. The jus-tifying intuition was that for an equal surface of contact, the heat exchangebetween two consecutive slices should be much stronger than that between theslice and the air, because on average the various points of a slice are muchcloser to a point of the next slice than they are to the surface of the bar. In thethree-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. Theresult 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 andthe surrounding space led Fourier to the equation117

C∂θ

∂t= K

(∂2θ

∂x2 + ∂2θ

∂y2 + ∂2θ

∂z2

)

− hθ . (92)

As a simple, yet unsolved steady-state problem, Fourier investigated thepropagation of heat along the semi-infinite rectangular lamina of Fig. 5, theedge being kept at the temperature 1 and the sides to the temperature 0.

Fig. 5 Fourier’s semi-infinitelamina

y

x

-pq = 1

q = 1 q

0 =

/2

+p/2

After convincing himself that the relevant equation of propagation should be

∂2θ

∂x2 + ∂2θ

∂y2 = 0, (93)

he attempted a simple-mode analysis presumably inspired by analogy withthe vibrating-string problem. Namely, he sought solutions of the factored formφ(x)ψ(y). Guided by the physical intuition of propagation from the heated edgeof 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 requiresκ to be an odd number if the width of the lamina is π . Superposition of thesesimple modes leads to the solution

θ =∞∑

r=0

cre−(2r+1)x cos(2r + 1)y. (95)

117 Fourier, Ref. 114, folios 122–127, reproduced in Herivel, Joseph Fourier face aux objectionscontre sa théorie de la chaleur: Lettres inédites 1808–1816 (Paris, 1980), Appendix 1.


The coefficients cr must be chosen so that the temperature

θ(0, y) =∞∑

r=0

cr cos(2r + 1)y (96)

along the edge −π/2 < y < π/2 be equal to 1 or any other impressed tempera-ture distribution.118

Before attempting the determination of these coefficients, Fourier discussedthe general behavior of his solution. He first noted that any distribution ofthe form cos(2r + 1)y propagated along the lamina without deformation andwith exponential attenuation at a rate increasing with the order r. For an arbi-trary distribution, Fourier imagined a decomposition of the heat flow into thesemodes propres et élémentaires:

One can imagine that the heat that comes at every instant from the sourcethus divides itself into distinct portions, that it propagates according toone of the said elementary laws and that all these partial motions occurwithout troubling each other.

This statement was clearly reminiscent of Bernoulli’s and Laplace’s formula-tions of the superposition principle. So to say, Fourier’s lamina was a vibratingstring with imaginary time.119

Fourier now faced the problem of developing a constant into a sum of cosineswhose angle is an odd multiple of the argument. Euler, Bernoulli, and Lagrangehad all solved similar problems. In general, they admitted trigonometric devel-opments for functions that were analytic over the interval of validity of thedevelopment. Being aware of Euler’s memoir of 1753 on this topic, Fourier didnot have to worry about transgressing the accepted limits of analysis at thatstage. His only problem was to find a way to calculate the coefficients. As hehad read neither Lagrange’s memoirs on sound nor Euler’s Disquisitio ulteriorof 1777, he was unaware of the method based on the orthogonality of simplemodes. He therefore relied on his algebraic prowess to solve the infinite systemof linear equations obtained by identifying all the derivatives of the sum ofcosines at the origin with the derivatives of the function to be developed. In thepresent case for which this function is the constant one, this gives

∞∑

r=0

cr = 1, and∞∑

r=0

(2r + 1)2ncr = 0 for n = 1, 2, 3, . . . . (97)

After several pages of complicated calculations, Fourier found

cr = 4π

(−1)r

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

π

4=

∞∑

r=0

(−1)r

2r + 1cos(2r + 1)y for − π/2 < y < π/2. (99)

Fourier next described how the series changed sign whenever the argumentpassed an odd multiple of π/2.120

Having reached this simple result by intricate means, Fourier soon found amuch simpler derivation. Remember that Euler and Bernoulli had obtainedsimilar results by integrating divergent sums of the kind

∑∞n=0 cos ny. Con-

versely, Fourier differentiated the partial sum

Sn(y) =n−1∑

k=0

(−1)k

(2k + 1)cos(2k + 1)y (100)

to get

S′n(y) =

n−1∑

k=0

(−1)k+1 sin(2k + 1)y = (−1)nsin 2nycos y

, (101)

and

Sn(x) = Sn(0)+ (−1)nIn(x), with In(x) =x∫

0

sin 2nycos y

dy. (102)

Fourier repeatedly integrated the latter integral by parts to obtain a serieswhose every term contained a negative power of n, and hence concluded thatit vanished in the limit of infinite n. In another section, he showed that theintegral remaining after p partial integrations was majored by a quantity of theorder n−p, which made I∞(x) = 0 a rigorous result. Together with Leibniz’swell-known result S∞(0) = π/4, this implies the desired identity (99).121

Fourier thus obtained the first convergence proof for a trigonometric seriesthat was not trivially convergent. In the following section of his draft, he showedthat 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. ThatFourier 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 ofthe remaining integral.


combining such formulas outside the intersection of their intervals of validityled to absurdities.122

Fourier’s next challenge was to extend to any indefinitely differentiable func-tion the algebraic method of elimination he had used to develop a constant intoa 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 developmentreads

f (x) =∞∑

k=1

ak sin kx, (103)

and the linear system to be solved is

∞∑

k=1

(−1)nk2n+1ak = f (2n+1)(0), with n = 0, 1, 2, . . . (104)

After a few pages of difficult calculations, Fourier obtained

ak = 2π

∞∑

n=0

∞∑

m=0

(−1)k+n+1 1k2n+1

πm

m! f (m+2n)(0). (105)

Using Taylor’s formula, he rewrote this as

ak = 2π

∞∑

n=0

(−1)k+n+1k−2n−1f (2n)(π), (106)

in which he recognized the result of the repeated integration by parts of123

ak = 2π

π∫

0

f (x) sin kx dx. (107)

5.4 Discreteness and generality

As he later explained in a letter to Lagrange, Fourier originally expected thisresult to hold only for the indefinitely derivable functions for which his algebraicprocedure worked. He changed his mind only after returning to the discreteproblem 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, folio75v (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, heprobably surmised that the elimination would become more tractable if thestraight 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(cosϕ − 1) (87)

and ϕ = r2π/n wherein r is an integer. If n is the odd number 2p + 1, there are2p distinct sine and cosine modes corresponding to r = 1, 2, . . . , p besides thetrivial constant mode corresponding to r = 0. The general solution then reads

θk = a0 +p∑

r=1

(ar cos kr2π/n + br sin kr2π/n)e−2α(1−cos r2π/n)t. (110)

In order to compute the coefficients from the initial temperatures, Fourierrelied on the orthogonality relations

n∑

k=1

sin 2πkr/n sin 2πks/n = n2δrs, (111)

n∑

k=1

cos 2πkr/n cos 2πks/n = n2δrs, (112)

n∑

k=1

sin 2πkr/n cos 2πks/n = 0, (113)

n∑

k=1

sin 2πkr/n =n∑

k=1

cos 2πkr/n = 0, (114)

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.


just as Euler and Lagrange had used

n∑

k=1

sinkrπ

n + 1sin

ksπn + 1

= n + 12

δrs (22)

in the elimination problem for the discretely loaded string. The result is125

a0 = 1n

n∑

k=1

θk(0), ar = 2n

n∑

k=1

θk(0) cos 2πkr/n, br = 2n

n∑

k=1

θk(0) sin 2πkr/n.

(115)

After providing a physical discussion similar to the one he had give in the caseof a rectilinear alignment, Fourier took the limit n → ∞ for which the massesform a continuous annulus. The ratio 2πk/n thus becomes a continuous anglex varying between 0 and 2π . In order that the discrete equation of motion

θk = α(θk+1 − 2θk + θk−1) (82)

reaches the continuous equation

θ = K∂2θ

∂x2 , (116)

Fourier further required

α(2π/n)2 → K. (117)

Consequently, the general solution becomes

θ(x, t) = a0 +∞∑

r=1

(ar cos rx + br sin rx)e−Kr2t, (118)

with

a0 = 12π

2π∫

0

θ(x, 0)dx,

ar = 1π

2π∫

0

θ(x, 0) cos rxdx, and br = 1π

2π∫

0

θ(x, 0) sin rxdx. (119)

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 trigonometricdevelopment of any function f (x) defined over the interval 0 ≤ x ≤ 2π :

f (x) = 12π

2π∫

0

f (ξ)dξ

+ 1π

∞∑

r=1

⎡

⎣cos rx

2π∫

0

f (ξ) cos rξdξ + sin rx

2π∫

0

f (ξ) sin rξdξ

⎤

⎦, (120)

in which he recognized an extension of the theorem he had earlier obtained inthe context of the lamina problem.126

This new derivation had important consequences. As Fourier believed that itonly involved “the ordinary principles of calculus,” he became convinced thatit applied to arbitrary functions. As he later explained to Lagrange127:

By the method of approximation I had obtained the development of afunction in sines and cosines of multiple arcs. Having next resolved thequestion of an infinity of bodies that communicate heat to each other,I recognized that this development had to apply to an arbitrary func-tion as well and I reached by an entirely different route [the equation

f (x) = 2π

∞∑r=1

sin rx∫ π

0 f (ξ) sin rξdξ ], which I had already obtained.

This is not all. Having used the orthogonality of simple modes to derive theanalogous formula in the discrete case, Fourier thought of doing the same inthe continuous case. Namely, he used the relations

π∫

0

sin rx sin sx dx = π

2δrs etc., (121)

which are the continuous counterpart of the relations (111–114), to derive hisfundamental formula a third time:

Seeking to verify the same theorem a third time, I used the procedurewhich consists in multiplying by [sin rx] the two sides of [a sine develop-ment] 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 wasquite 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.


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 takingthe limit of infinite n in the successive terms of the series for the discrete prob-lem did not necessarily converge toward the limit of its sum. As we know, theconvergence requires restrictions on the function representing the initial stringcurve or the initial temperature distribution. Far from worrying about this diffi-culty, Lagrange and Fourier believed that their limit of the discrete problemjustified the arbitrariness of this function.

Fourier’s third derivation of his theorem was even more problematic. Fourierannounced this derivation as follows:

I am now going to show that the equation f (x)= 2π

∞∑r=1

sin rx∫ π

0 f (ξ) sin rξdξ

always holds whatever be the nature of the proposed function f (x).

Hopefully, he only meant that the sine development of a function should takethis 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 theconvergence of the series nor on its ability to represent the proposed function.In modern terms, he realized that the orthogonality of the sine functions ofmultiple arcs did not imply their completeness.129

Albeit for bad reasons, Fourier’s detour through the discrete problems con-vinced him that his fundamental theorem held for arbitrary functions “whetherthe 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 con-fidently applied his theorem to any function for which he could easily computethe integrals (119) that give the trigonometric coefficients.130

His first heterodox result was the development of the function cos x as aseries of sines over the interval 0 < x < π , which violated the preconceptionthat the developed function should be odd. Most daring were his development

�(x) = 4π

∞∑

p=0

(−1)p

(2p + 1)2sin(2p + 1)x (122)

for the “triangular” function �(x) that takes the value x for 0 ≤ x ≤ π/2 andthe value π − x for π/2 ≤ x ≤ π , and his development

χa(x) = 2π

∞∑

r=1

1r(1 − cos ra) sin rx (123)

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, Appendix3, 63–64.130 Fourier (1807), Ref. 115, 225.

416 O. Darrigol

the former series accomplished what d’Alembert had declared impossible, andthe latter what Euler had declared impossible. Fourier concluded that “hisresults fully confirmed the opinion of Daniel Bernoulli.” He of course had inmind the quarrel over vibrating strings. In the memoir submitted to the Institutin 1807, he proudly announced:

The application of [our] principles to the question of the motion of vibrat-ing strings has solved all the difficulties encountered by Daniel Bernoulli’sanalysis. Indeed the solution proposed by this great geometer did not seemapplicable to the case when the initial figure of the string is a triangle ora trapeze, or such that only one part of the string is disturbed while theother parts coincide with the axis.

To be true, not even Bernoulli ventured to apply trigonometric series to thesecases.131

After this major mathematical discovery, Fourier returned to the annulus.He now took external conductivity into account, in which case the equation forheat propagation reads

∂θ

∂t= K

∂2θ

∂x2 − hθ , (124)

where K and h denote the internal and external conductivities divided by theheat capacity. Fourier first obtained the exponential solutions in the steady case,then solved the time-dependent case by superposition of simple modes. The par-tial modes decay exponentially, at a rate increasing with the order of the mode.Fourier’s discussion of this behavior resembled his discussion of propagationalong the lamina. He noted that after a sufficiently long time, only the first modesurvived, and that the higher modes “disappeared one after the other.”132

In the following months, Fourier considered the more difficult cases of thepropagation of heat within a homogenous sphere, a cylinder, and a cube. Perhapsin the context of the sphere, he realized that the equation for the heat motionwithin the body should not involve external conductivity. He introduced theconcept of heat flux through an internal surface element, and argued by purelymacroscopic, semi-empirical reasoning that this flux should be proportional tothe temperature gradient along the normal to the surface. Together with theconservation of heat, this assumption implies the fundamental equation

∂θ

∂t= K

(∂2θ

∂x2 + ∂2θ

∂y2 + ∂2θ

∂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.


to which Fourier added the boundary condition that the flux across the exter-nal surface of the body should be proportional to the difference between thetemperature at the surface and the temperature of the environment.133

Fourier’s solutions for the spherical and cylindrical cases required powerfulextensions of harmonic analysis, involving simple modes with incommensurablefrequencies, Bessel functions (for the cylinder), and transcendental equations.As Fourier himself realized, these parts of his theory of heat were the mostinnovative 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 muchmore about the nature of the connection between Fourier’s heat theory andearlier acoustics.134

More important for our purpose is Fourier’s insistence on the physical realityof simple modes. In the oral presentation of his theory to the French Academi-cians on 21 December 1807, he described the cooling of a body in the followingterms:

The system of initial temperatures can be such that the ratios originallyestablished among them persist without any alteration during the wholeprocess of cooling. This singular state that enjoys the property of subsistingonce it is formed can be compared to the figure that a sonorous string takeswhen it yields the principal sound. It can take diverse analogous forms,the ones corresponding to the subordinate sounds [harmonics] in the caseof the elastic string. Consequently, for every solid there is an infinite num-ber of simple modes according to which heat can propagate and dissipatewithout change of the initial distribution law…. Whatever be the mannerin which the different points of the body have been heated, the initial andarbitrary system [of temperatures] can be decomposed into several simpleand durable states similar to those I just described. Each of these statessubsists independently of all the others and undergoes no other changesthan those which would still occur if it were alone. The relevant decom-position is not a purely rational and analytical result; it occurs effectivelyand results from the physical properties of heat. Indeed the speed withwhich the temperatures decrease in each simple system is not the samefor every system…. To be true, these properties are not always as sensibleas the isochronism of pendulums and the multiple resonance of vibratingstrings are; but they can be established by observation and they becamemanifest in all of my experiments.

Fourier thus ascribed the same degree of physical reality to the harmonicsof a vibrating string and to the partial modes of heat propagation. In the intro-duction to his treatise of 1822, he framed this reification of harmonic analysiswithin 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 andLagrange, in Herivel, Ref. 117.

418 O. Darrigol

Mathematical analysis therefore has necessary relations with sensible phe-nomena; its object is not created by the intelligence of man, it is a pre-existing element of the universal order and has nothing contingent orfortuitous; it impregnates all of nature.

Or else:

Analytical analysis extends as much as nature itself; it defines every sen-sible relation, measures times, spaces, forces, and temperatures; this diffi-cult science forms slowly, but it conserves all the principles that it onceacquired; it grows and strengthens incessantly amidst so many variationsand 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 comparableto the harmonic resonances.

5.5 A problematic reception

Toward the end of 1807, Fourier submitted to the French Academy a longmemoir entitled Théorie de la propagation de la chaleur dans les solides. Thismemoir contained the various results he had reached so far, rearranged in anorder differing from the chronological order of discovery: the discretized barand the discretized annulus, the general equations for a continuous body, thelamina, Fourier’s theorem by elimination, the same theorem by orthogonalityof simple modes, the continuous annulus, the transition from the discrete to thecontinuous annulus, the sphere, the cylinder, the cube, and experiments.136

Somewhat strangely, Fourier presented three derivations of his fundamentaltheorem instead of selecting the most direct one. One may surmise that thederivation 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 proofscomplemented each other. The proof by elimination established the existenceof the trigonometric development but only for indefinitely differentiable func-tions. The proof by orthogonality gave the form of the coefficients withoutestablishing the existence of the development. The proof by taking the limitof the discrete problem seemed to give both the existence and the form of thedevelopment for arbitrary functions, but depended on an artificial model ofheat propagation. Lastly, Fourier gave a rigorous proof of convergence in theparticular 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 1851of the Bibliothèque de l’Ecole Nationale des Ponts et Chaussées, in Herivel, Ref. 117, 53–58, on55–56 (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 four examiners of Fourier’s memoir, Lagrange, Laplace, Monge, andLacroix never wrote the expected report. Instead Laplace’s star pupil SiméonDenis Poisson published a brief summary of Fourier’s main results in the March1808 issue of the Bulletin de la Société Philomathique. The examiners must havethought that Fourier’s memoir should not be published in its present shape.Unfortunately, there is almost no written record of their objections. For themost, these must be inferred from letters that Fourier wrote to Lagrange andLaplace 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 collec-tion of Lagrange’s manuscripts at the Bibliothèque Mazarine. There Lagrangeargues that Fourier’s identity

∞∑

n=1

(−1)n−1

nsin nx = x

2for − π < x < π , (126)

leads to an absurdity. He first shifts the variable x by π to get

∞∑

n=1

1n

sin nx = π

2− 1

2x for 0 < x < 2π . (127)

Then he derives to get

∞∑

n=1

cos nx = −12

for 0 < x < 2π . (128)

Lastly, he integrates the latter identity from 0 to x to obtain

∞∑

n=1

1n

sin nx = −12

x for 0 < x < 2π , (129)

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 byproviding an improved version of his indeed rigorous proof of the identity (126).More broadly, he defended his fundamental theorem by giving a short historyof 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 ashort MS note by Fourier in ffr 22529, folio 126. In an appended comment to Lagrange’s note (MS40), Prony, Poisson, Legendre, and Lacroix accepted the series (128) but argued that its sum becameinfinite 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 hispredecessors. In particular, he or Lacroix must have told Fourier that Euler hadalready used the orthogonality of cosines to derive the expression of the Fouriercoefficients of a trigonometric radical. Fourier apologized for not having beenaware of this work, emphasized the much greater span of his own theorem, andflattered 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 elementssimilar to the ones I have used.”140

Lagrange could hardly have been satisfied with this reply. He must have feltthat Fourier had simply rediscovered a theorem he had known for himself sincehis first researches on sound. Surely, following his debate with d’Alembert theway in which trigonometric series converged toward the developed functionhad become obscure to him. But he never doubted that they converged insome sense, which he called “asymptotic.” The general strategy of projectingfunctions over eigenfunctions of certain operators belonged to him, and imme-diately gave Fourier’s theorem in the simplest case of the vibrating string. Asfar as rigor was concerned, Fourier’s only improvement was his irreproachablederivation of the trigonometric series in the simple cases of a constant or a linearfunction. His elimination strategy worked only in cases that Lagrange himselfjudged unproblematic, while his proof based on taking the limit of the discreteproblem 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 inthe second edition of his Mécanique analytique.141

From Fourier’s contemporary letter to Laplace, we learned that the lattergeometer objected to the development of the sine as a series of cosines, pre-sumably because a sine is odd whereas a cosine is even. Fourier easily answeredthat the conflict disappeared when the domain of validity of the formulas wasproperly taken into account. He added that all the series he had given wererigorously convergent. In a contemporary manuscript note, Fourier answeredLaplace’s more serious objections regarding the derivation of the fundamentalequation of heat propagation and boundary conditions. Laplace still believedthat Fourier had not solved the difficulty of differential heterogeneity resultingfrom Biot’s naïve application of Newton’s law of cooling to contiguous infinites-imal particles, and he indicated that in a proper derivation heat exchange shouldoccur between particles separated by a small finite distance. Possibly, Laplacehad only read Fourier’s first, problematic derivation, which was included in theearly 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, Fourierwrote: “On trouve dans les ouvrages de tous les géomètres des résultats et des procédés de calculanalogues à ceux que nous avons employés. Ce sont des cas particuliers d’une méthode généralequi n’était point encore formée, et qu’il devenait nécessaire d’établir pour connaître, même dans lesquestions les plus simples, les lois mathématiques de la distribution de la chaleur.” An earlier MSversion of this comment (ffr 22529, folio 34r) has “les oeuvres d’Euler, de Clairaut, de Lagrange etde Daniel Bernoulli” instead of “les ouvrages de tous les géomètres.”141 Lagrange, Ref. 49, 421–442.


note of 1808, Fourier referred to his new reasoning based on the concept of heatflux, and also introduced a few molecular considerations in a more Laplacianstyle. The stakes were high, because Fourier’s priority as the discoverer of thefundamental equation depended on the validity of the reasoning found in hismemoir of 1807. For several years this issue embittered the relations Fourierhad with Laplace and Biot.142

After reading Fourier’s memoir, Laplace provided his own derivation of theheat equation, with the comment: “I must note that Mr. Fourier has alreadyobtained these equations, whose true foundation seems to me to be those Ihave just given.” He also discovered the integral form of the solution in theone-dimensional case for which the heat equation reduces to

∂y∂t

= ∂2y∂x2 . (130)

A power-series development à la Lagrange yields the solution

y(x, t) =∞∑

n=0

tn

n!Y(2n)(x) if y(x, 0) = Y(x) for any x. (131)

This series differs from a Taylor development by having n! instead of (2n)! inthe denominators (and by not including the odd-order derivatives). Astutelycompensating for this difference by means of the well-known identity

(2n)!n! = 1√

π

+∞∫

−∞(2u)2ne−u2

du, (132)

Laplace obtained

y(x, t) = 1√π

+∞∫

−∞

∞∑

n=0

tn(2u)2n

(2n)! Y(2n)e−u2du. (133)

As the missing odd terms of the series would not contribute to the integral, thisis the Taylor development of

y(x, t) = 1√π

+∞∫

−∞

∞∑

n=0

Y(x + 2u√

t)e−u2du. (134)

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 introductionof this draft.

422 O. Darrigol

As Ivor Grattan-Guinness remarks, this form of the solution is analogous tothe d’Alembert-Euler solution of the problem of vibrating strings inasmuchas it directly relates the configuration at any time to the initial configuration.However, Laplace’s and Poisson’s preference for this type of solution overFourier’s trigonometric solution was not a simple repeat of d’Alembert’s andEuler’s opposition to Bernoulli’s solution, because Laplace and Poisson did notchallenge Fourier over the generality of his solutions.143

In response to Laplace’s contribution, Fourier extended his method to thecase of an infinite body, for which the spatial frequencies of the simple modesform a quasi-continuum. He thus obtained the Fourier-integral type of solution,which he showed to be related to the Laplacian form. Friendly discussions withLaplace also led to new developments on terrestrial temperatures and radiantheat. In 1811 Fourier reworked his memoir to include these topics and sent theresult to the Academy of sciences, which was offering a prize for the mathe-matical theory of heat and its experimental verification. The jury composed ofLagrange, Laplace, Lacroix, Malus, and Haüy selected Fourier’s memoir, withthe following comment:144

This piece contains the true differential equations for the transmission ofheat, be it inside bodies or at their surface; the novelty and importance ofthe subject determined the Class to crown this work, although it must benoted that the manner in which the author arrives at his equations is notdevoid of difficulties and that his analysis, to integrate them, still leavessomething to be desired, either regarding generality, or even regardingrigor.

This far-from-enthusiastic endorsement and the earlier lack of officialresponse from the academicians have usually been regarded as the consequenceof 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 theobjections to Fourier’s work came at least in part from other academicians,especially Laplace. Lagrange’s acceptance of Fourier’s theorem is evident in hisstatement of 1811 that the infinite superposition of simple modes “rigorouslygives the motion of [a vibrating] string at any time.” What may have antago-nized 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 pointsd’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 AcadémieRoyale 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 discussionof 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.”


anticipation of the theorem and perhaps also Fourier’s reification of the partialmodes.145

The reproach about difficulties in Fourier’s derivation of the heat equa-tion almost certainly came from Laplace, who regarded molecular action ata distance as the true foundation of any physical theory. The reproach aboutgenerality and lack of rigor in Fourier’s solutions of this equation could equallywell come from Lagrange and from Laplace. Laplace’s protégé Poisson laterrepeated it almost word for word. And it was a perfectly legitimate one, sinceat that time Fourier had not even attempted a general proof of his theorem. Asis 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 ofan infinite number of simple modes. Like Lagrange, Fourier had no qualmrewriting

f (x) = 12π

+∞∑

n=−∞

+π∫

−πdXf (X) cos n(x − X) (for − π < x < π) (135)

as

f (x) = 12π

+π∫

−πdXf (X)

+∞∑

n=−∞cos n(x − X), (136)

although in his “construction” of this identity he interpreted it as

f (x) = limN→∞

12π

+π∫

−πdXf (X)

+N∑

n=−N

cos n(x − X). (137)

Again like Lagrange, Fourier based his derivation or construction on the trigo-nometric identity

�N (ξ) =+N∑

n=−N

cos nξ = cos Nξ − cos(N + 1)ξ1 − cos ξ

= cos Nξ + sin Nξsin ξ

1 − cos ξ

(138)

and on the equivalence of this sum with a periodic series of impulsions whenN 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 dechimie, 80 (1815), 434–441, on 440. On Poisson’s attitude, cf. Herivel, Ref. 2, 126, 175–176. Thegeneral 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 modelin which the sum �N (ξ) vanishes for every permitted value of the variable ξexcept for the value ξ = 0 for which this sum is 2N. Instead, Fourier arguedthat 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 Nxand sin Nx factors in the expression (138) of �N (ξ). For the remaining integralover the tiny interval [−ε, ε], Fourier exploited the smallness of ε and againthe largeness of N to reach the approximations

+ε∫

−ε�N (ξ)f (x + ξ)dξ≈ f (x)

+ε∫

−ε�N (ξ)dξ≈ f (x)

+π∫

−π�N (ξ)dξ=2π . (139)

Altogether, we have

12π

+π∫

−πdXf (X)�N (x − X) = 1

2π

x+π∫

x−π�N (ξ)f (x + ξ)dξ

≈ 12π

+ε∫

−ε�N (ξ)f (x + ξ)dξ ≈ f (x), (140)

which leads to the desired result (137) in the limit of infinite N and infinitelysmall ε.

This is the basis of Gustave Lejeune Dirichlet’s later rigorous proof ofFourier’s theorem, which closed a long chapter of the history of trigonometricseries. From then on, Fourier analysis became as respectable as any other part ofmathematics. The stage was set for amazingly rich developments ranging fromthe foundation of mathematical analysis to numerous physical and technicalapplications. Interestingly, one of these applications was the Ohm-Helmholtztheory of consonance. Harmonic analysis thus returned to its then forgottenacoustic origins.147

147 Gustav Lejeune Dirichlet, “Sur la convergence des séries trigonométriques qui servent àreprésenter une fonction arbitraire entre des limites données,” Journal für die reine und ange-wandte Mathematik, 4 (1829), 157–169. On the Ohm-Helmholtz theory, cf. Turner, Ref. 1.

the acoustic origins of harmonic...

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