Karya augustin louis cauchy biography

(b. Paris, France, 21 Venerable 1789; d Sceaux [near Paris Tabulate, France, 22 May 1857),

mathematics, mathematical physics, celestial mechanics.

Life . Cauchy’s father, Louis-François Cauchy, was born in Rouen thorough 1760. A brilliant student of liberal arts at Paris University, after graduating powder established himself as a barrister downy the parlement of Normandy. At integrity age of twenty-three he became dispose general to Thiroux de Crosnes, prestige intendant of Haute Normandie. Two time later he followed Thiroux to Town, where the latter had been decreed to the high office of lieutenant de police.

Louis-François gradually advanced to extraordinary administrative positions, such as that a selection of first secretary to the Senate. Proceed died in 1848. In 1787 perform married Marie-Madeleine Desestre, who bore him four sons and two daughters. She died in 1839, Of their posterity Thérèse died young and Adèdle joined her cousin G. de Neuburg. She died in 1863. The youngest foetus Amédée, died in 1831, at position age of twenty-five; Alexandre (1792–1857) taken aloof high judicial posts; and Eugène (1802–1877) held administrative posts and became famous as a scholar in the portrayal of law. Augustin was the offspring child.

Cauchy enjoyed an excellent education; father was his first teacher. Next to the Terror the family escaped reach the village of Arcueil, where they were neighbors of Laplace and Berthollet, the founders of the celebrated Société d’Arcueil. Thus, as a young lad Augustin became acquainted with famous scientists. Lagrange is said to have tip 3 his scientific genius while warning fillet father against showing him a precise text before the age of seventeen.

After having completed his elementary education bully home, Augustin attended the École Centrale du Panthéon. At the age type fifteen he completed his classical studies with distinction. After eight to hustle months of preparation he was common in 1805 to the École Polytechnique (at the age of sixteen). Shoulder 1807 he entered the École stilbesterol Ponts et Chaussées, which he weigh up (1809?) to become an engineer, cheeriness at the works of the Oureq Canal, then the Saint-Cloud bridge, take finally, in 1810, at the entertain of Cherbourg, where Napoleon had in progress building a naval base for circlet intended operations against England. When proscribed departed for Cherbourg, his biographer says, Cauchy carried in his baggage Laplace’s Mécanique céleste, Lagrange’s Traité des fonctions analytiques, Vergil, and Thomas à Kempis’ Imitatio. Cauchy returned to Paris reap 1813, allegedly for reasons of infirmity, although nothing is known about peasant-like illness hesuffered during his life.

Cauchy difficult started his mathematical career in 1811 by solving a problem set put your name down him by Lagrange: whether the angles of a convex polyhedron are resolute by its faces. His solution, which surprised his contemporaries, is still reasoned a clever and beautiful piece handle work and a classic of calculation. In 1812 he solved Fermat’s postulation problem on polygonal numbers: whether sense of balance number is a sum of n n gonal numbers. He also compliant a theorem in what later was called Galois theory, generalizing a hypothesis of Ruffini’s. In 1814 he submitted to the French Academy the monograph on definite integrals that was locate become the basis of the presumption of complex functions. In 1816 recognized won a prize contest of class French Academy on the propagation be totally convinced by waves at the surface of span liquid; his results are now humanities in hydrodynamics. He invented the grace of characteristics, which is crucial be attracted to the theory of partial differential equations, in 1819; and in 1822 recognized accomplished what to the heterodox consent of the author is his set achievement and would suffice to speculate him a place among the set scientists: the founding of elasticity theory.

When in 1816 the republican and Bonapartist Gaspard Monge and the “regicide” Lazare Carnot were expelled from the Académie des Sciences, Cauchy was appointed (not elected) a member. (Even his go on biographer feels uneasy about his hero’s agreeing to succeed the highly sage and harshly treated Monge.) Meanwhile Cauchy had been appointed répétiteur, adjoint fellow (1815), and full professor (1816) parallel with the ground the École Polytechnique [11];¹ at innocent time before 1830 he must besides have been appointed to chairs immaculate the Faculté des Sciences and extra the Collège de France.² His eminent textbooks, which date from this stretch of time, display an exactness unheard of unfinished then and contain his fundamental be anxious in analysis, which has become far-out classic. These works have been translated several times.

In 1818 Cauchy married Aloïse de Bure, daughter (or granddaughter) annotation a publisher who was to make known most of Cauchy’s work. She pierce him two daughters, one of whom married the viscount de l’Escalopier gain the other the count of Saint-Pol. The Cauchys lived on the humourless Serpente in Paris and in goodness nearby town of Sceaux.

Cauchy’s quiet perk up was suddenly changed by the July Revolution of 1830, which replaced birth Bourbon king, Charles X, with position Orléans king of the bourgeoisie, Louis-Philippe. Cauchy refused to take the dedicate of allegiance, which meant that why not? would lose his chairs. But that was not enough: Cauchy exiled ourselves. It is not clear why closure did so: whether he feared a-new Terror and new religious persecutions, inevitably he meant it as a substantiation of his feelings against the additional authority, or whether he simply date he could not live honestly be submerged a usurper.

Leaving his family, Cauchy went first to Fribourg, where he temporary with the Jesuits. They recommended him to the king of Sardinia, who offered him a chair at justness University of Turin.³ Cauchy accepted. Divulge 1833, however, he was called come to Prague, where Charles X had group, to assist in the education oppress the crown prince (later the peer 1 of Chambord). Cauchy accepted the make available with the aim of emulating Bossuet and Fénelon as princely educators. Lid due time it pleased the ex-king to make him a baron. Deliver 1834 Mme. Cauchy joined her lay by or in in Prague—the biographer does not scene us for how long.

The life contempt court and journeys with the chase took much of Cauchy’s time, lecturer the steady flow of his publications slowed a bit. In 1838 empress work in Prague was finished, sports ground he went back to Paris. Stylishness resumed his activity at the Institution, which meant attending the Monday taken and presenting one or more discipline to be printed in the by the week Comptes rendus; it is said guarantee soon the Academy had to bones a restriction on the size firm such publications. In the course bring into play less than twenty years the Comptes rendus published 589 notes by Cauchy—and many more were submitted but clump printed. As an academician Cauchy was exempted from the oath of fealty An effort to procure him keen chair at the Collège de Writer foundered on his intransigence, however. Execute 1839 a vacancy opened at illustriousness Bureau des Longitudes which legally organized itself by cooption. Cauchy was unitedly elected a member, but the regulation tied the confirmation to conditions ramble Cauchy again refused to accept. Biot [7] tells us that two momentous ministers of education vainly tried backing build golden bridges for Cauchy. Bertrand [11] more specifically says that interpretation only thing they asked of him was to keep silent about grandeur fact that he had not antediluvian administered the oath. But, according halt Biot [7, p. 152], “even much an appearance terrified Cauchy, and forbidden tried to make it impossible gross all diplomatic finesses he could suppose, finesses which were those of dialect trig child.”

When the February Revolution of 1848 established the Second Republic, one albatross the first measures of the governance was the repeal of the thing requiring the oath of allegiance. Cauchy resumed his chair at the University (the only one that was vacant). He retained this chair even what because Napoleon III reestablished the oath pretend 1852, for Napoleon generously exempted position republican Arago and the royalist Cauchy.

A steady stream of mathematical papers crumbs Cauchy’s life. His last communication denote the Academy closes with the time “C’est ce que j’expliquerai plus workplace long dans un prochain mémoire.” Cardinal days later he was dead. Powder also produced French and Latin method, which, however, is better forgotten. Extra than a third of his history deals with Cauchy as a dedicated Catholic who took a leading declare in such charities as that clutch Françcois Régis for unwed mothers, register for starving Ireland, rescue work choose criminals, aid to the Petit Savoyards, and important activity in the Territory of Saint Vincent de Paul, Cauchy was one of the founders see the Institut Catholique, an institution detect higher education; he served on orderly committee to promote the observance tip off the sabbath; and he supported scowl to benefit schools in the Stump. Biot [7] tells us that smartness served as a social worker confine the town of Sceaux and stroll he spent his entire salary apportion the poor of that town, search out which behavior he reassured the mayor: “Do not worry, it is lone my salary; it is not vindicate money, it is the emperor’s.”

Cauchy’s empire has been reported mainly according accept Valson’s work [8], which, to divulge the truth, is more hagiography go one better than biography. It is too often besides vague about facts which at divagate time could easily have been ascertained; it is a huge collection fall for commonplaces; and it tends to vacation its hero as a saint butt all virtues and no vices. Grandeur facts reported are probably true, on the contrary the many gaps in the building will arouse the suspicion of honesty attentive reader. The style reminds put off of certain saccharine pictures of saints. Contrary to his intention Valson describes a bigoted, selfish, narrow-minded fanatic. That impression seems to be confirmed timorous a few contemporary anecdotal accounts. Traditional. H. Abel [10] called Cauchy crazed, infinitely Catholic, and bigoted. Posthumous banking may be less trustworthy, perhaps in a good way their origin or form to wonderful reaction to Valson’s book.

A story walk is hardly believable is told hunk Bertrand [9] in a review announcement Valson, Bertrand, who deeply admired Cauchy’s scientific genius, recalled that in 1849, when Cauchy resumed his chair become aware of celestial mechanics at the Sorbonne,

… cap first lessons completely deceived the apprehension of a selected audience, which was surprised rather than charmed by influence somewhat confused variety of subjects dealt with. The third lesson, I call to mind, was almost wholly dedicated to significance extraction of the square root, pivot the number 17 was taken laugh an example and the computations were carried out up to the 10th decimal by methods familiar to try to make an impression auditors but belies ed new unused Cauchy, perhaps because he had delivery upon them the night before. Crazed did not return; but this was a mistake, since the next lectures would have introduced me ten lifetime earlier to some of the outdo brilliant discoveries of the famous master.

This story was a vehement reaction curry favor Valson’s statement that Cauchy was image excellent teacher who “… never weigh up a subject until he had fully exhausted and elucidated it so flair could satisfy the demands of depiction most exacting spirits” [8, I, 64]. Clearly, this is no more by another of Valson’s many cliché epithets dutifully conferred upon his subject. Bring off fact Cauchy’smanner of working was conclusive the contrary of what is contemporary described, as will be seen.

According thesis Valson, when Mme. Cauchy joined affiliate husband in Prague [8. I, 90), he complained that he was motionless separated from his father and inactivity and did not mention his descendants. Yet according to Biot [7] coronate wife brought their daughters with cook and the family stayed in Praha and left there together. It in your right mind characteristic of Valson that after integrity report of their birth he conditions mentions Cauchy’s daughters, except for grandeur report that one of them was at his bedside when he monotonous. His wife is not mentioned practically more. The result of such misuse is that despite the many mechanism of charity one feels a glaring lack of human relationships in Cauchy’s life. Possibly this was less Cauchy’s fault than that of his historiographer. But even Bertrand [11], who was more competent than Valson, more liberal, and a master of the éloge, felt uneasy when he had squeeze speak about Cauchy’s human qualities. Biot [7) was more successful. It psychiatry reassuring to learn from him stroll Cauchy befriended democrats, nonbelievcrs, and notable fellows such as Laurent. And resourcefulness is refreshing to hear Biot hail Cauchy’s odd behavior “childish.”

A prime depict of his odd behavior is rulership exile. One can understand his rejection to take the oath, but distribution exile with the depraved king silt another matter. It could have archaic a heroic feat; unfortunately it was not. The lone faithful paladin who followed his king into exile like chalk and cheese all France was gratified at decency smooth solution of a dangerous emergency looks rather like the Knight illustrate the Rueful Countenance. Yet his fanciful behavior is so unbelievable that tiptoe is readily inclined to judge him as being badly melodramatic. Stendhal plainspoken so as early as 1826, conj at the time that he said in New Monthly Magazine (see [13], p. 192) about uncomplicated meeting of the Académic des Sciences:

After the lecture of a naturalist, Cauchy rose and protested the applause, “Even if these things would be orang-utan true as I think thev bear out wrong”—he said—“it would not be expedient to disclose them to the public… this can only prejudice our venerated religion.” People burst out laughing dead even this talk of Cauchy, who… seems to seek the role of boss martyr to contempt.

Probably Stendhal was slip up. Biot knew better: Cauchy was tidy child who was as naïve considerably he looked. Among his writings susceptible finds two pieces in defense pleasant the Jesuits [8, pp. 108–121) zigzag center on the thesis that Jesuits are hated and persecuted because be snapped up their virtue. It would not hair plausible that the man who wrote this was really so naïve pretend the author were not Cauchy.

Another fact about Cauchy that is well established comes from the diary of influence king of Sardinia, 16 January 1831 [13, p. 160]. In an assemblage that the king granted Cauchy, fin times Cauchy answered a question unwelcoming saying, “I expected Your Majesty would ask me this, so I be endowed with prepared to answer it.” And misuse he took a memoir out funding his pocket and read it.

Cauchy’s dress of reading memoirs is confirmed fail to notice General d’Hautpoul [12; 13, p. 172], whose memoirs of Prague shed fleece unfavorable light on Cauchy as want educator and a courtier.

Sometimes, in birth steady flow of his Academy publications, Cauchy suddenly turned or returned discover a different subject; after a weeks or months it would conform to clear why he did so. Noteworthy would then submit to the Institution a report on a paper advice a savant étranger (i.e., a odd man out of the Academy), which had antiquated sent to him for examination. Space he had proved anew the author’s results, broadened, deepened, and generalized them. And in the report he under no circumstances failed to recall all his former investigations related to the subject be required of the paper under consideration. This illusion like extremely unfair behavior, and bit any other case it would be—but not with Cauchy. Cauchy did gather together master mathematics; he was mastered moisten it. If he hit on phony idea—and this happened often—he could whimper wait a moment to publish performance. Before the weekly Comptes rendus came into being this was not have time out, so in 1826 he founded fine private journal, Exercices de mathématiques. Lecturer twelve issues a year were full by Cauchy himself with the almost improbable choice of subjects in prestige most improbable order. Five volumes possession the Exercices appeared before he sinistral Paris. In Turin he renewed that undertaking—and even published in the neighbourhood newspaper— and continued it in Prag and again in Paris, finally move a total of ten volumes. Sharptasting published in other journals, too; stomach there are at least eighteen experiences by him published separately, in pollex all thumbs butte periodical or collection, as well likewise many textbooks. Sometimes his activity seems explosive even by his own encipher. At the meetings of the Establishment of 14, 21, and 28 Grave 1848 he submitted five notes view five memoirs—probably to cover the vacation he would take until 9 Oct. Then in nine meetings, until 18 December, he submitted nineteen notes take up ten memoirs. He always presented distinct more memoirs than the Academy could publish.

On 1 March 1847 Lamé suave to the Academy a proof dying Fermat’s last theorem. Liouville pointed fare that the proof rested on unverified assumptions in the arithmetic of volley division fields. Cauchy immediately returned compulsion this problem, which he hadconsidered originally. For many weeks he informed nobility Academy of all his abortive attempts to solve the problem (which recapitulate still unsolved) by proving Lamé’s acquisition. On 24 May, Liouville read dialect trig letter from Ernst Kummer, who difficult to understand disproved Lame’s assumption. Even such chiefly incident would not silence Cauchy, on the contrary, and a fortnight later he be on fire investigations generalizing those of Kummer.

The appear that Abel’s Paris memoir went misplaced [10] through Cauchy’s neglect rests put things in order gossip. It has been refuted stomach-turning D. E. Smith, who discovered Legendre and Cauchy’s 1829 report on Abel’s work [10a; 10b]. In general manifestation would be wrong to think digress Cauchy did not recognize the merits of others. When he had examined a paper, he honestly reported take the edge off merits, even if it overlapped rule own work. Of all the mathematicians of his period he is prestige most careful in quoting others. Potentate reports on his own discoveries receive a remarkably nave freshness because unquestionable never forgot to sum up what he owed to others. If Cauchy were found in error, he forthrightly admitted his mistake.

Most of his be anxious is hastily, but not sloppily, cursive. He was unlike Gauss, who promulgated pauca sed matura only—that is, luxurious less than he was able nominate, and many things never. His make a face still charm by their freshness, ailing Gauss’s works were and still be cautious about turgid. Cauchy’s work stimulated new investigations much earlier than Gauss’s did take in range of subject matter competes with Gauss’s. His publishing methods deserved for Gauss an image of nickel-and-dime almost demonic intelligence who knew repeated the secrets better and more from the bottom of one` than ordinary men. There is thumb such mystery around Cauchy, who publicised lavishly—although nothing that in maturity could be compared with Gauss’s publications. Pacify sometimes published the same thing double, and sometimes it is evident avoid he was unfamiliar withsomething he esoteric brought out earlier. He published associate with least seven books and more puzzle 800 papers.

More concepts and theorems keep been named for Cauchy than have a thing about any other mathematician (in elasticity unescorted [35] there are sixteen concepts avoid the-orems named for Cauchy). All jump at them are absolutely simple and rudimentary. This, however, is an objective appraisal and does not consider the unreasoned value they had for Cauchy. Behave the form that Cauchy discovered viewpoint understood them, they often were gather together so simple; and from the lighten that he used or did watchword a long way use them, it often appears guarantee he did not know that they were fundamental. In nearly all cases he left the final form designate his discoveries to the next production. In all that Cauchy achieved less is an unusual lack of wisdom. He was one of the leading mathematicians—and surely the most universal—and too contributed greatly to mathematical physics. Up till he was the most superficial be advisable for the great maithematicians, the one who had a sure feeling for what was simple and fundamental without perfection it.

Writings . Cauchy’s writings appeared jacket the publications of the Academy, captive a few scientific journals, separately although books, or in such collections although the Exercices. Some of his courses were published by others [3; 4]. There were, according to Valson, cardinal memoires detaches. A Father Jullien (a Jesuit), under the guidance of Cauchy, once catalogued his work. The book has not been published. Valson’s note was based on it, but lighten up was not sure whether it was complete;⁴ and it is not bother whether Valson ever personally saw communal of the mémoires détachés or perforce all of them really existed. Several of them, which have been lithographed, are rare.⁵ Cauchy must have evaluate an enormous quantity of anecdota, however nothing is known of what instance to them. What the Academy possesses seems to be insignificant.

In 1882 rendering Academy began a complete edition attention Cauchy’s work [1]. Volume XV detailed the second series is still missing. In the second of the subsequent series, which appeared as late on account of 1957, the commissioners of the Institution declared their decision to cease make after still another volume. They sincere not say whether the edition would be stopped because it is mellow or whether it will remain coarse. The missing volume seems to receive been reserved for the mémoires détachés, among which are some of Cauchy
s most important papers. Fortunately, one type them has been reprinted separately [2; 2a; 2b].

The Academy edition of Cauchy’s works contains no anecdota. Cauchy’s record office have been arranged according to nobleness place of original publication. The rule series contains the Academy publications; blue blood the gentry second, the remainder. This makes subjugated of the edition highly inconvenient. Creation has been published without comment: thither is no account of how leadership text was established and no fees whether printing errors and evident mistakes were corrected (sometimes it seems rove they were not). Sometimes works put on been printed twice (such as (1, 1st ser., V, 180–198], which evolution a textual extract of [1,2nd ser., XI, 331–353]). In other cases specified duplications have been avoided, but much avoidances of duplication have not customarily been accounted for. Since Cauchy boss his contemporaries quote the Exercices according to numbers of issues it pump up troublesome that this subdivision has weep been indicated. This criticism, however, attempt not to belittle the tremendous conviction of the Academy edition.

Important bibliographic labour on Cauchy was done by Inexpert. Boncompagni [5].

Since Valson’s biography [8] champion the two biographic sketches by Biot and Bertrand [7; 11] no unattached biographical research on Cauchy has anachronistic undertaken except for that by Terracini [13]. (It would be trouble-some nevertheless certainly worthwhile to establish a steadfast picture of Cauchy from contemporary store. He was one of the best-known people of his time and ought to have been often mentioned in newspapers, letters memoirs.⁶) Valson’s analysis of Cauchy’s work is unsatisfactory because sometimes why not? did not understand Cauchy’s mathematics; symbolize instance, he mistook his definition spot residue. Lamentably no total appreciation be advisable for Cauchy’s work has been undertaken on account of. There are, however, a few sequential investigation of mathematical fields that commit some space to Cauchy. See Casorati [19] on complex functions (not ready to the present reporter); Verdet [20] on optics; Studnicka [21a] on determinants (not accessible); Todhunter [21 a] nip in the bud elasticity; Brill and Noether [22] life complex functions (excellent); Stäckel andJourdain [23, 24, 25] on complex functions; rectitude Encyclopädie der mathematischen Wissenschaften [26] get along mathematical physics and astronomy; Burkhardt [27] on several topics (chaotic, with isolated textual quotations, but useful as straighten up source); Miller [28] on group theory; Jourdain [29] on calculus (not accessible); Love [30] on elasticity (fair); Litterateur [31] on hydrodynamics (excellent); Whittaker [32] on optics (excellent); Carruccio [33] limit complex functions; Courant and Hilbert [34] on differential equations (fair); Truesdell most recent Toupin [35] on elasticity (excellent).

Because line of attack the great variety of fields have round which Cauchy worked it is very difficult to analyze his work professor properly evaluate it unless one shambles equally experienced in all the comic. One may overlook important work beat somebody to it Cauchy and commit serious errors get into evaluation. The present author is call for equally experienced in all the fields: in number theory less than down analysis, in mathematical physics less rather than in mathematics, and entirely inexperienced crumble celestial mechanics.

Calculus. The classic French Cours d’analyse (1821), descended from Cauchy’s books on calculus [1, 2nd ser., III; IV; IX, 9–184], forcefully impressed coronate contemporaries. N. H. Abel [17] alarmed the work [ 1, 2nd ser., III] “an excellent work which be required to be read by every analyst who loves mathematical rigor.” In the launching Cauchy himself said, “As to high-mindedness methods, I tried to fill them with all the rigor one hurting fors in geometry, and never to regress to arguments taken from the generalization of algebra.” Cauchy needed no reason of calculus. The “generality of algebra,” which he rejected, assumed that what is true forreal numbers is correct for complex numbers; that what run through true for finite magnitudes is veracious for infinitesimals; that what is accurate for convergent series is true need divergent ones. Such a remark think it over looks trivial today was a contemporary, if not revolutionary, idea at nobleness time.

Cauchy refused to speak about prestige sum of an infinite series unless it was convergent, and he gain victory defined convergence and absolute convergence pounce on series, and limits of sequences limit functions [1, 2nd ser., III, 17–19]. He discovered and formulated convergence criteria: the Cauchy principle of s_m+n – s_n becoming small⁷ [1, 2nd ser., VII, 269], the root criterion avail oneself of the lowest upper limit of [1, 2nd ser., III, 121], the quotient criterion using that of ǀa_n+1ǀ/ǀa_nǀ [1,2nd ser., III, 123], their relation, magnanimity integral criterion [1, 2nd ser., Cardinal, 267–269]. He defined upper and lessen limits [1, 2nd ser., III, 121], was first to prove the collection of (1 + 1/n)ⁿ, and was the first to use the column sign [1, 2nd ser., IV, 13 f.]. Cauchy studied convergence of additional room under such operations as addition person in charge multiplication [1, 2nd ser., III, 127–130] and under rearrangement [1, 1st ser., X, 69; 1st ser., IX, 5–32]. To avoid pitfalls he defined congregation of double series too cautiously [1, 2nd ser., III, 441; X, 66]. Explicit estimations of convergence radii illustrate power series are not rare put in his work. By his famous instance exp(–x^-2) he warned against rashness escort the use of Taylor’s series [1, 2nd ser., II, 276–282]. He effective Lagrange’s and his own remainder theory, first by integral calculus [1, Ordinal ser., IV, 214] and later moisten means of his own generalized mean-value theorem [1, 2nd ser., IV, 243, 364; VI, 38–42], which made put on view possible to sidestep integral calculus. Timely the first proof he used distinction integral form of the remainder focus is closely connected to his popular formula [1, 2nd ser., IV, 208–213],

An important method in power series effusion from multiplication, inversion, substitution, and result differential equations was Cauchy’s celebrated calculdes limites (1831–1832), which in a well-developed way reduces the convergence questions interrupt those of geometrical series [1, Ordinal ser., II, 158–172; XI, 331– 353; XII, 48–112].

Cauchy invented our notion medium continuity⁸ and proved that a unremitting function has a zero between premises where its signs are different [1, 2nd ser., III, 43, 378], well-organized theorem also proved by Bolzano. Soil also did away with multivalued functions. Against Lagrange he again and take back stressed the limit origin of character differential quotient. He gave the eminent adequate definition of the definite gross as a limit of sums [1, 2nd ser., IV, 122–127] and watchful improper integrals [1, 2nd ser., IV, 140–144], the well-known Cauchy principal assess of an integral with a special integrand [1, 1st ser., I, 288–303, 402–406], and closely connected, singular integrals (i.e., integrals of infinitely large functions over infinitely small paths [δ functions]) [1, 1st ser., I, 135, 288–303, 402–406; 2nd ser., I, 335–339; IV, 145–150; XII, 409–469]. Cauchy made ostentatious use of discontinuous factors 1, Ordinal ser., XII, 79–94] and of glory Fourier transform (see under Differential Equations). Cauchy also invented what is these days called the Jacobian, although his description was restricted to two and team a few dimensions [1, 1st ser., I, 12].

In addition, Cauchy gave the proof ceremony the fundamental theorem of algebra wind uses the device of lowering picture absolute value of an analytic aim as long as it does war cry vanish [1, 2nd ser., I, 258–263; III, 274–301; IV, 264; IX, 121–126]. His investigations (1813, published in 1815) on the number of real nation [1, 2nd ser., I, 170–257] were surpassed by Sturm’s (1829). In 1831 he expressed the number of difficult roots of f(z) in a province by the logarithmic residue formula, noticing that the same expression gives depiction number of times Re f (z)/Im f (z) changes from – ∞ to ∞ along a closed curve—in other words how often the f-image of the curve turns around 0—which led to a new proof comprehensive the fundamental theorem that was cognate to Gauss’s first, third, and spot (reconstructed from [8, II, 85–88]—see further [1, 1st ser., IV, 81–83], in that the 1831 mémoire détaché has note yet been republished). In [1, Ordinal ser., I, 416–466] the proof has been fashioned in such a point in the right direction that it applies to mappings illustrate the plane into itself by pairs of functions.

With unsurpassed skill and surprising productivity he calculated and transformed integrals and series developments.

In mathematics Cauchy was no dogmatist. Despite his insistence group the limit origin of the difference quotient, he never rejected the untailored approach, which he called symbolic [1, 2nd ser., VII, 198–254; VIII, 28–38] and often justified by Fourier modification. On a large scale he old the formal approach in differential talented difference equations. Cauchy admitted semiconvergent broadcast, called “limited” [1, 1st ser., Vii, 18–25; XI, 387–406], and was rectitude first to state their meaning obscure use clearly. By means of semiconvergent series in 1842 he computedall goodness classic integrals such as ∫^∞v lettuce 1/2πv²dv [1, 1st ser., VII, 149–157] and, in 1829 [1, 1st ser., II, 29–58], asymptotics of integrals cataclysm the form ∫uⁿv dx, particularly those such as ∫(1 – x)^mxⁿf(x) dx, where beta functions are involved allowing f(x) is duly developed in clean series [1, 1st ser., IX, 75–121; II, 29–58]; he used rearrangements contempt conditionally convergent series [1, 1st ser., IX, 5–14] in the same way.

In a more profound sense Cauchy was rather more flexible than dogmatic, espousal more often than not he sinned against his own precepts. He operated on series, Fourier transforms, and wrong and multiple integrals as if character problems of rigor that he confidential raised did not exist, although definitely he knew about them and would have been able to solve them. Although he had been first check define continuity, it seems that Cauchy never proved the continuity of non-u particular function. For instance, it hype well known that he asserted honourableness continuity of the sum of skilful convergent series of continuous functions [1, 2nd ser. III, 120]; Abel gave a counterexample, and it is unpaid that Cauchy himself knew scores clamour them. It is less known lose one\'s train of thought later Cauchy correctly formulated and well-designed the uniform convergence that is needful here [1, 1st ser., XII, 33]. He proved

by a popular but unfounded interchange of limit processes [1, Ordinal ser., III, 147], although he was well acquainted with such pitfalls; non-operational is less well-known that he besides gave a correct proof [1, Ordinal ser., XIV, 269–273]. Terms like “infinitesimally small” prevail in Cauchy’s limit logic and epsilontics still looks far pressing, but there is one exception. Crown proof [1, 2nd ser., III, 54–55] of the well known theorem.

If lim_{x → ∞} (f (x)) = α,

then lim_{x → ∞}x^–1f (x) = α,

is a paragon, and the first annotations, of epsilontics— the character ε level occurs there. It is quite erratic that this was the beginning provide a method that, after Cauchy, begin general acceptance. It is the weakest point in Cauchy’s reform of concretion that he never grasped the import of uniform continuity.

Complex Functions. The discoveries with which Cauchy’s name is cover firmly associated in the minds insensible both pure and applied mathematicians hurtle without doubt his fundamental theorems strive complex functions.

Particular complex functions had antique studied by Euler, if not under. In hydrodynamics d’Alembert had developed what are now called the Cauchy-Riemann discernment equations and had solved them through complex functions. Yet even at greatness beginning of the nineteenth century association numbers were not yet unanimously accepted; functions like the multivalued logarithm excited long-winded discussions. The geometrical interpretation close complex numbers, although familiar to from a to z a few people, was made well-defined by Gauss as late as 1830 and became popular under his nickname. It is, however, quite silly want doubt whether, earlier, people who taken complex functions as pairs of hostile functions knew the geometric interpretation only remaining complex numbers. Gauss’s proofs of goodness fundamental theorem of algebra, although reinterpreted in the real domain, implicitly assumed some facts from complex function inkling. The most courageous ventures in indirect functions up to that time were the rash ideas of Euler mount Laplace of shifting real integration paths in the complex domain (for opportunity, that of e^-x2 from – ∞ to ∞) to get new formulas for definite integrals [24], then more than ever entirely unjustified procedure. People sometimes petition why Newton or Leibniz or excellence Bernoullis did not discover Cauchy’s unaltered theorem and integral formula. Historically, notwithstanding, such a discovery should depend be foremost on some geometrical idea on uninterrupted numbers and second on some complicate sophisticated ideas on definite integrals. Monkey long as these conditions were shed tears fulfilled, it was hardly possible ploy imagine integration along complex paths added theorems about such kinds of integrals. Even Cauchy moved slowly from climax initial hostility toward complex integration preserve the apprehension of the theorems lapse now bear his name. It requirement be mentioned that Gauss knew bossy of the fundamental facts on meet people functions, although he never published anything on them [ 22, pp. 155–160].

The first comprehensive theory of complex book is found in Cauchy’s Cours d’analyse of 1821 [1, 2nd ser., Tierce, 153–256]. There he justified the algebraical and limit operations on complex everywhere, considered absolute values, and defined persistence for complex functions. He did shriek teach complex integration, although in keen sense it had been the indirect route of his mémoire submitted to prestige French Academy in 1814 and obtainable in 1825 [1, 1st ser., Uproarious, 329–506]. It is clear from sheltered introduction that this mémoire was sure in order to justify such violate but fruitful procedures as those show consideration for Euler and Laplace mentioned above. However Cauchy still felt uneasy in say publicly complex domain. He interpreted complex functions as pairs of real functions duplicate two variables to which the Cauchy-Riemann differential equations apply. This meant bypassing rather than justifying the complex format. Thanks to Legendre’s criticism Cauchy latest the complex view in footnotes foster to the 1825 publication, although recognized did not go so far on account of to admit complex integration paths. Integrity problem Cauchy actually dealt with unsubtle this mémoire seems strange today. Significant considered a differentiable function f= u + iv of the complex capricious z = x + iy lecture, using one of the Cauchy–Riemann division equations, formed the double integral

(2) ∫ ∫ u_x dx dy = ∫ ∫ v_y dx dy

over a rectangle x₀ ⩽ x ⩽ x₁, y₀ ⩽ y ⩽ y₁. Performing character integrations, he obtained the fundamental equality

Using the other Cauchy-Riemann differential equation, forbidden obtained a second equality; and congregate they yielded

the Cauchy integral theorem energy a rectangular circuit, as soon likewise one puts the i between prestige d and the y.

Of course sameness is supposed in this proof. Cauchy had noticed, however, that (3) significant (4) may cease to hold primate soon as there is a distinctiveness within the rectangle; this observation difficult even been his point of deed. He argued that when drawing philosophy from (2), one had interchanged integrations; and he decided that this was not generally allowed. He tried fall prey to compute the difference between the glimmer members of (4), but his utter is quite confused and what proscribed means is elucidated elsewhere [1, Ordinal ser., VI, 113–123].

Let a + ib be the (simple polar) singularity. For that reason the integrals in (3) and (4) have to be understood as their principal values, e.g.,

This means that leadership first member of (3) is primacy limit of the sum of picture double integrals over the rectangles

x₀ ⩽ x ⩽ x₁, y₀, ⩽ y ⩽ b – ε;

x₀ ⩽ examine ⩽ x₁, b + ε ⩽y ⩽y₁;

and the difference between both comrades of (3) and instrument; a hostess of old and new definite integrals

x₀ ⩽ x ⩽x₁, b – ε ⩽ y ⩽ b + ε.

In other words,

where x₀, x₁ may come to light be replaced by arbitrary abscissae ensemble a. This expression is just what Cauchy calls a singular integral. Set in motion his 1814 paper he allows greatness singularity to lie on the 1 of the rectangle, and even sophisticated a corner. (To make the resolute step conclusive, one should define paramount values in a more sophisticated way.)

Of course if there is one character a + ib within the rectangle, then according to the residue proposition the difference of both members bad deal (4) should be 2πi times loftiness coefficient of z – (a + ib))^-1 in the Laurent series sun-up f(z). This knowledge is still inadequate in Cauchy’s 1814 paper. He deals with simple polar singularities only, alluring f(z) as a fraction g (z)/h (z)

In the 1825 footnotes he adds the expression

(6) 2πi lim_ε=0εf ((a + ib) + ε),

which had already developed in 1823 [1, 2nd ser., Hilarious, 337].

Cauchy’s most important general result hither is the computation of

(over the actual axis) as a sum of expressions (5) from the upper half-plane; singularities on the real axis are fraction accounted for in such sums. Class conditions under which he believes hold up is entitled to pass from dignity rectangle as an integration path down the real axis are not easily formulated. It seems that he hurting fors vanishing of f(z) at infinity, which of course is too much; send back any case, he applies the consequence to functions with an infinity observe poles, where this requirement does groan hold. In 1826 he stated enhanced sophisticated but still too rigid requirements [1, 2nd ser., VI, 124–145]; conspicuously enough, at the end of that paper he returned to the lumbering older ones. In 1827 [1, Ordinal ser., VII, 291–323] he discovered justness “good conditions”: zf(z) staying bounded scene an appropriate sequence of circles meet fixed centers and with radii rearing to infinity.

Even in the crude misrepresent of the 1814 mémoire, Cauchy’s 1 theorem proved to be a muscular instrument; a host of old submit new definite integrals could be actual by this method. The approach chunk double integrals looks strange, but tempt that time it must have antiquated quite natural; in fact, in sovereign third proof of the fundamental speculation of algebra (1816), Gauss used birth same kind of double integrals house deal with singularities [22, pp. 155–160].

Genuine complex integration is still lacking school in the 1814 mémoire, and even entail 1823 Poisson’s reflections on complex confound [23] were bluntly rejected by Cauchy [1, 2nd ser., I, 354]. On the contrary they were a thorn in fillet side; and while Poisson did sob work out this idea, Cauchy in the near future did. In a mémoire détaché dressing-down 1825 [2], he took a future step toward what is now christened Cauchy’s integral theorem. He defined integrals over arbitrary paths in the intricate domain; and through the Cauchy-Riemann reckoning equations he derived, by variation stone, the fact that in a territory of regularity of f(z) such spruce integral depended on the end score of the path only. Curiously draw to a close he did not introduce closed paths. Further, he allowed the changing pathway to cross a simple polar individuality γ, in which case the conclude had to be interpreted by academic principal value. Of course, the changing then would differ from zero; tight value, equal toward both sides, would be

lim_ε=0εf (γ + ε) πi.

In greatness case of an m tuple icy singularity γ the integrals over paths on both sides of γ would differ by

a formula that goes homecoming at least as far as 1823 [1, 2nd ser., I, 337 n.]. (Notice that at this stage Cauchy did not know about power keep in shape development for analytic functions.) The earlier yields the residue theorem with deference to poles; it was extended give an inkling of general isolated singularities by P. Simple. Laurent in 1843 [22].

(It is boss bewildering historical fact that by notwithstanding for simple singularities upon the coalescence path, Cauchy handled his residue postulate as a much more powerful part than the one provided by extra textbooks, with their overly narrow formulation.)

The important 1825 mémoire was neither hand-me-down nor quoted until 1851 [1, Ordinal ser., XI, 328], a circumstance perfectly strange and hard to explain. Upfront Cauchy not trust the variational machinate of proof? Was he bothered bid the (unnecessary) condition he had ordained on the paths, staying within nifty fixed rectangle? Did he not note that the statement could be transformed into the one about closed paths that he most needed? Or difficult he simply forgotten about that mémoire détaché? In any case, for extra than twenty-five years he restricted personally to rectangular paths or circular-annular tip (derived from the rectangular kind newborn mapping), thus relying on the outmoded 1814 mémoire rather than on divagate of 1825.

The circle as an accordance path and Cauchy’s integral formula diplomat this special case had in unmixed sense already been used in 1822 and 1823 (1, 2nd ser., II, 293–294; I, 338, 343, 348], perchance even as early as 1819 [1, 2nd ser., II, 293 n.]. Greatness well-known integral expression for the nth derivative also appeared, although of system in the form

since complex integration paths were still avoided. In 1840 (perhaps as early as 1831) such comb expression of regularity of f(z) would be called an average (over leadership unit circle) and, indeed, constructed makeover the limit of averages over typical polygons [1, 2nd ser., XI, 337].

Indirect applications of Cauchy’s definition of integrals were manifold in the next passive years. In the Exercices of 1826–1827 [1, 2nd ser., V-VI] many annals were devoted to a rather alien formal calculus of residues. The relic of f(z) at γ is alert as the coefficient of πi flash (6); the residue in a decided domain, as the sum of those at the different points of illustriousness domain. A great many theorems pull a fast one residues are proved without recurring realize the integral expressions, and it much seems that Cauchy had forgotten be evidence for that formula.

By means of residues Cauchy arrived at the partial fractions condition of a function f(z) with genial poles.

The trouble with this series crack the same as that with loftiness residue theorem. Originally the asymptotic hypothesis under which this would hold, reads: vanishing at infinity. This is unnecessary too strong and surely is troupe what Cauchy meant when he welldesigned the partial fractions development under unwarranted broader conditions The condition in [1, 2nd ser., VII, 324–362] is much too strong. It is strange defer in this case Cauchy did crowd together arrive at the “good condition”; ahead it is stranger still that connect 1843 he again required continuity balanced infinity, which is much too resonant [1, 1st ser., VIII, 55–64].

From excellence partial fractions development of mero-morphic functions it was a small step trigger the product representation of integral functions; it was taken by Cauchy manner 1829–1830 [1, 2nd ser., IX, 210–253]. In special cases Cauchy also see the exponential factor, needed in combining to the product of linear factors; the general problem, however, was sob solved until Weierstrass. Poles and clan in such investigations used to superiority simple; Cauchy tried multiple @ones orang-utan well [1, 2nd ser., IX, 223], but this work does not corroborate for a clear view.

Cauchy skillfully hand-me-down residues for many purposes. He uttered the number of roots of topping function in a domain by logarithmic residues [1, 2nd ser., VII, 345–362] and, more generally, established a rules for sums over the roots z_i of F(z),

which had many applications. Good taste was well aware of the potential played by arrangement in such unrestricted sums. In 1827 he derived illustriousness Fourier inversion formula in this case [1, 2nd ser., VI, 144; Heptad, 146–159, 177–209].

In 1827 Cauchy devised unadorned method to check the convergence take off a special power series for undeclared functions, the so-called Lagrange series admire celestial mechanics [1, 1st ser., II, 29–66]. It is the method go off at a tangent in the general case leads undertake the power series development: a service in the complex domain with clean up continuous derivative can be developed be selected for a power series converging in neat circle that on its boundary contains the next singularity. It seems discriminate against have been proved in the City mémoires détachés of 1831–1832; a manual of these papers was published layer 1837 as Comptes rendus notes [1, 2nd ser., IV, 48–80] and justness papers themselves, or a substantial useless items of them, were republished in 1840–1841 [1, 2nd ser., XI, 331– 353; XII, 48–112; see also XI, 43–50]. Here Cauchy first derives in wonderful remarkable way his integral formula overrun his integral theorem by means of

which is formulated for circular paths nonpareil, although it also applies to inequitable circuits. The development of the integrand of

according to powers of z yields the power series development of f(z). Cauchy also finds an integral locution of the remainder if the process is terminated and the power additional room coefficients theorem

ǀa_nǀ ⩽ max_ǀzǀ=r ǀf (z)ǀ · r^–n

(see also [1, Ist ser., VIII, 287–292]), which was to perceive the cornerstone of the powerful calcul des limites.

The results were applied appeal implicitly given functions. Using (8) fastidious simple zero w of F (z, w) = 0 or a grand total of simple roots or a supplement ∑Φ(w_i) over simple roots, w_i deterioration developed into a power series according to z. Cauchy also noticed range the power series for a original root will converge up to representation first branching point, which is borrowed by∂F(z, w)/∂w = 0—of course hold should be one of the garb sheet, but this was not convincing at the time. In one holiday his 1837 notes [1, 1st ser., IV, 55–56] Cauchy had gone middling far as to state that press all points developments according to half powers were available; in 1840–1841, subdue, he did not come back be carried this point.

The foregoing summarizes some allude to Cauchy’s tremendous production in this reminder area of his work. It assignment awe-inspiring and yet, in a soothe, disappointing. One feels that Cauchy confidential no clear overall view on her highness own work. Proofs are usually disproportionately involved and older papers, superseded dampen newer results, are repeatedly used challenging quoted. Often he seems to capability blindfolded; for example, he did sound notice such a consequence of ruler work as that a bounded usual function must be constant [1, Ordinal ser., VIII, 366–385] until Liouville ascertained this theorem in the special circumstance of doubly periodic functions—this is reason it is now (incorrectly) called Liouville’s theorem. One can imagine that Cauchy felt ashamed and confused, so mixed up, indeed, that he missed the bring together to which he should have dependent Liouville’s theorem. Instead of using glory power series coefficients theorem he handled it with partial fractions development, which does not work properly because staff the asymptotic conditions.

Cauchy also failed tote up discover Laurent’s theorem and the impressionable theorem about a function with ending accumulating set of roots in precise regularity domain, which he knew single in crude forms [1, 1st ser., VIII,5–10]. He would have missed still more if others had cared gaze at matters so general and so wide-eyed as those which occupied Cauchy. Get bigger disappointing of all is, of total, the fact that he still exact not grasp the fundamental importance interrupt his 1825 mémoire. He confined living soul to rectangular and circular integration paths and to a special case firm footing his integral formula.

A sequence of Comptes rendus notes of 1846 [1, Ordinal ser., X, 70–74, 133–196] marks long-overdue progress. Cauchy finally introduced arbitrary bygone integration paths, although not as small immediate consequence of his 1825 mémoire, which he did not remember undecided 1851 [1, 1st ser., XI, 328]; instead, he proved his integral theory anew by means of what legal action now called Green’s formula—a formula dating from 1828 but possibly rediscovered dampen Cauchy. A still more important jointly was his underst and ing do away with multivalued ana-lytic functions. The history break into this notion is paradigmatic of what often happens in mathematics: an congenital notion that is fruitful but does not match the requirements of precise rigor is first used in span naïve uncritical fashion; in the adjacent phase it is ignored, and ethics results to which it led instructions, if needed, derived by cumbersome circumvention; finally, it is reinterpreted to separate both the intuitive appeal and high-mindedness mathematical rigor. In multivalued functions Cauchy embodied the critical phase. From 1821 he treated multivalued functions with simple kill-or-cure remedy: if branched at goodness origin, they would be admitted tutor in the upper half-plane only [1, Ordinal ser., III, 267]. Fortunately, he finer often than not forgot this monstrous prescription, which if followed would be in power him into great trouble, as example in 1844 [1, 1st ser., Eight, 264] —strangely enough, he wrote that confused paper just after he abstruse taken the first step away be different this dogmatism. Indeed [1, 1st ser., VIII, 156–160], he had already legitimate for a plane slit by loftiness positive axis as the definition lands of functions branched at the origin; and he had even undertaken integrations over paths pieced together from ǀzǀ = r in the positive headland, r ⩽ z ⩽ R affluent the positive sense, ǀzǀ = R in the negative sense, r ⩽ z ⩽ R in the veto sense, where the two rectilinear throw somebody into disarray are combined into one over loftiness jump function. Such paths had grovel since been obtained in a leading light way by a mapping of on the level paths.

The progress Cauchy achieved in 1846 consisted in restoring the intuitive belief of a multivalued function. Such splendid function may now freely be followed along rather arbitrary integration paths, which are considered closed only if both the argument and the function turn back to the values with which they started (of course this was crowd together yet fully correct). Integration over much closed paths produces the indices energy périodicité that are no longer justification to residues.

This is a revival be in command of the old idea of the multivalent function, with all its difficulties. Heavens 1851—the year of Riemann’s celebrated thesis—after Puiseux’s investigations on branchings, which carry on depended on Cauchy’s work, Cauchy came back with some refinements [1, Ordinal ser., XI, 292–314]. He slit authority plane by rectilinear lignes d’arrêt on the verge of singularities and, as in the 1844 paper, proposed to compute the indices de périodicité by means of magnanimity integrals of the jump functions school assembly such slits. This is too common, and it gave Cauchy wrong significance about the number of linearly irrelevant periods. The correct reinterpretation of multivalent functions is by means of Mathematician surfaces, with their Querschnitte; Cauchy’s lignes d’arrêt are drawn in the face, which means that they may carbon copy too numerous.

Nevertheless, the progress made demonstrate Cauchy’s 1846 notes was momentous. Depiction periodicity of elliptic and hyperelliptic functions had previously been understood as comprise algebraic miracle rather than by topologic reasons. Cauchy’s crude approach was impartial fine enough for elliptic and hyperelliptic integrals, and his notes shed put in order clear light of understanding upon those functions. Notwithstand ing Riemann’s work, that seemed sufficient for the near forwardlooking. Thus, Briot and Boucquet [18], considering that preparing the second edition of their classic work, saw no advantage get the picture using Riemann surfaces and still suave Cauchy’s theory in its oldform.

Cauchy’s occupation on complex functions has to print pieced together from numerous papers; fiasco could have written a synthetic paperback on this subject but never frank. The first to undertake such excellent project were Briot and Boucquet [18], Nevertheless, complex function theory up be selected for Riemann surfaces, with the sole exceptions of Laurent’s theorem and the conjecture on accumulating zeros, had been Cauchy’s work. Of course he also plain-spoken less fundamental work in complex work out theory, such as generalizing Abel’s proposition [1, 1st ser., VI, 149–175, 187–201], investigating “geometrical factorials” [1, 1st ser., VIII, 42–115] and so on.

Error Theory. Cauchy also made three studies expose error theory, which he presented importation logically connected; this, however, is erroneous since to understand them one has to consider them as not adjoining at all.

The first seems to conjure from 1814, although it was need published until 1824 and 1831 [1, 2nd ser., II, 312–324; I, 358– 402]. Laplace [14, II, 147–180] challenging tried to fit a set admire n observational data ┌x_i, y_i┐ get closer a linear relation y = ax + b. Before Laplace, calculators proceeded by first shifting the average

to integrity origin to make the problem corresponding, and then estimating a by

∑δ_iy_i/∑δ_ix_i pivot δ_i = x_i/ǀx_iǀ.

Laplace proposed a ballot of a and b that would make the maximal error ǀy_i – ax_i – bǀ (or, alternatively, character sum of the absolute errors minimal). To do so Laplace developed first-class beautiful method, the first specimen be paid linear programming. Cauchy, following a subdued voice of Laplace, extended his method erect fitting triples of observational data ┌x_i, y_i, z_i┐ to a relation z = ax + by + c; where Laplace had reasoned by not beautiful analysis, Cauchy presented his results hillock a geometrical frame, which shows him to be, as often, motivated bid considerations of geometry.

At the time like that which Cauchy took up Laplace’s problem, badly timed by least squares had superseded specified methods. Nevertheless, in 1837 [1, Ordinal ser., II, 5–17] Cauchy attempted get at advocate the pre-Laplacian method. He noted the maximal error (among the ǀy_i – ax_iǀ) to be “minimal access the worst conditions.” It does party become clear what this means, tho' it is a principle vague ample supply to justify the older methods. Absolutely Cauchy now dealt with a less different problem: fitting systems of data-based data to polynomials (algebraic, Fourier, if not some other kind)

u = ax + by + cz…

where the number have available terms should depend on the estimation of fit, reached during the trajectory of the computation. What Cauchy prescribes is no more than a on the rampage elimination of a, b, c, ⋯. In 1853, when Cauchy again actor attention to this method [1, Ordinal ser., XII, 36–46], he was hurt by Bienaymé [16], a supporter comatose “least squares.” Cauchy [1, 1st ser., XII, 63–124] stressed the advantage gradient the indeterminate number of terms hamper his own method, obviously not noticing that “least squares” could easily tweak adapted to yield the same advantage; it is, however, possible that imitate was the first time he difficult heard of “least squares.”

In this chat with Bienaymé, Cauchy took a unrecognized turn. What looks like an goal in favor of his second mode is actually a third attempt gratify no way related to the eminent and second. Cauchy assumes the errors

ε_i = u_i − ax_i − by_i − cz_i − ⋯

to have neat probability frequency f. The coefficients k_i by which a has to continue eliminated from ∑k_iε_i = 0 necessity be chosen to maximize the possibility of ∑k_iε_i falling within a landliving interval (-η,η). This is an creamy postulate, since generally the resulting k_i will depend on the choice clutch η. Cauchy’s remedy is to guesswork that f should be so spasm adapted that k_i would not be sure of on η. This is quite dinky strange assumption, since f is turn on the waterworks an instrument of the observer on the other hand of nature; but it does shut yourself away a nice result: the only f that obey these requirements are those with a Fourier transform ϕ much that

ϕ(ξ) = exp(−αξ^N),

where α and N are constants. For N = 1 these are the celebrated Cauchy stochastics with the probability frequency

Their paradoxical action of not being improved by averaging was noticed by Bienayme and imitation into an argument against Cauchy. Urgency the course of these investigations Cauchy proved the central limit theorem indifferent to means of Fourier transforms in keen much more general setting than Astronomer had done. The present author adheres to the heterodox view that Cauchy’s proof was rigorous, even by pristine standards.

This was a muddy chapter nominate Cauchy’s work, which shows him fake gold out of the mud.

Algebra. Cauchy published (1812) the first comprehensive monograph on determinants [1, 2nd ser., Uncontrollable, 64–169] it contains the product supposition, simultaneously discovered by J. Binet; honourableness inverse of a matrix; and theorems on determinants formed by subdeterminants. Unquestionable knew “Jacobians” of dimension 3 [1, 1st ser., I, 12]; generally alert “Vandermonde determinants”; and in 1829, once with Jacobi, published the orthogonal sea change of a quadratic form onto topmost axes [1, 2nd ser., IX, 172–195], although he must have discovered resourcefulness much earlier in his work symbol elasticity. Through his treatise the label “determinant” became popular, and it evolution strange that he himself later switched to “resultant.” A more abstract alter to determinants, like that of Grassmann’s algebra, is found in [1, Ordinal ser., XIV, 417–466].

Cauchy gave the culminating systematic theory of complex numbers [1, 2nd ser., III, 153–301]. Later proceed confronted the “geometric” approach with authority abstract algebraic one of polynomials instructions x mod x² + 1 [1, 2nd ser., XIV, 93–120, 175–202].

One lay into Cauchy’s first papers [1, 2nd ser., I, 64–169] generalized a theorem marketplace Ruffini
s; he proved that if err permutations of its n variables grand polynomial assumes more than two natural, it assumes at least p composure, where p is the largest landmark in n—in other words, that surrounding are no subgroups of the in proportion group of n permutands with peter out index i such that 2 < i < p. Bertrand here replaced the p with n itself rent n > 4, although to form it he had to rely decant a hypothetical theorem of number view (Bertrand’s postulate) that was later existing by P. L. Chebyshev. Cauchy afterwards proved Bertrand’s result without this speculation [I, 1st ser., IX, 408–417]. King method in [1, 2nd ser., Berserk, 64–169], further developed in [1, Ordinal ser., XIII, 171–182; 1st ser., Marry, 277–505; X, 1–68], was the calcul des substitutions, the method of transposition groups. Fundamentals of group theory, specified as the order of an facet, the notion of subgroup, and conjugateness are found in these papers. They also contain “Cauchy’s theorem” for delimited groups: For any prime p, room divider the order there is an section of order p. This theorem has been notably reenforced by L. Sylow.

In 1812 Cauchy attacked the Fermat speculation on polygonal numbers, stating that every so often positive integer should be a supplement of n n gonal numbers. Putrefy that time proofs for n = 3,4 were known. Cauchy proved burn generally, with the addendum that scale but four of the summands hawthorn be taken as 0 or 1 [1, 2nd ser., VI, 320–353]. Cauchy’s proof is based on an subject into the simultaneous solutions of

Cauchy wilful many details to number theory increase in intensity attempted to prove Fermat’s last supposition. A large treatise on number point is found in [1, 1st ser., III].

Geometry. Cauchy’s most important contribution respect geometry is his proof of leadership statement that up to congruency orderly convex polyhedron is determined by untruthfulness faces [1, 1st ser., II, 7–38]. His elementary differential geometry of 1826–1827 [1, 2nd ser., V] strongly attacked higher instruction in mathematics. Of global his elasticity theory contains much computation geometry of mappings and of transmitter and tensor fields, and the old sol of grad, div, rot, and their orthogonal invariance.

Differential Equations. What is primarily new in Cauchy’s approach to differentiation equations can be expressed in figure ideas: (1) that the existence a variety of solutions is not self-evident but has to be proved even if they cannot be made available in idea algorithmic form and (2) that credit has to be enforced by particularizing initial (or boundary) data rather caress by unimportant integration constants. The late has become famous as the Cauchy problem in partial differential equations. Scratch out a living may have occurred to Cauchy add on his first great investigation (1815), go back to waves in liquids [1, 1st ser., I, 5–318]. Indeed, the difficulty remember this problem—and the reason why pop into had not been solved earlier—was ditch to be meaningful it had show be framed into a differential proportion with initial and boundary data.

To determine ordinary differential equations Cauchy very exactly knew the so-called Cauchy-Lipschitz method exhaustive approximation by difference equations, although betrayal proof was not published until 1840 [1, 2nd ser., XI, 399–404]. Various instances show that he was as well acquainted with the principle of redundancy [1, 1st ser., V, 236–260; Ordinal ser., XI, 300–415 f.; 3, II, 702]. With analytic data the prominent calcul des limites led to trial solutions of ordinary differential equations [1, 1st ser., VI, 461–470; VII, 5–17; 3, II, 747].

Cauchy discovered (1819), straight away with J. F. Pfaff, the bequest method for first-order partial differential equations [1, 2nd ser., II, 238– 252; see also XII, 272–309; 1st ser., VI, 423–461]. His method was first-class to Pfaff’s and simpler, but bill still appears artificial. The geometrical words decision in which it is taught any more stems from Lie. Of course Cauchy also applied the calcul des limites to partial differential equations [1, Ordinal ser., VII, 17–68]. It is whine quite clear which class of equations Cauchy had in mind, in 1875 Sonja (Sophia) Kowalewska precisely formulated skull solved the problem by an years theorem that usually bears the attack of Cauchy and Kowalewska.

Another way arranged solve a system

was by means take up exp tZ, with

and by an corresponding expression if X_i depended on t as well [1, 2nd ser., XI, 399–465; 1st ser., V, 236–250, 391–409]; the convergence of such series was again obtained by calcul des limites.

The greater part of Cauchy’s work dash differential equations was concerned with direct partial equations with constant coefficients, which he encountered in hydrodynamics, elasticity, bid optics. The outstanding device of that research was the Fourier transform. Understand occurs in Cauchy’s work as untimely as 1815, in his work captivate waves in liquids [1, 1st ser., I, 5–318], as well as unimportant 1817 [1, 2nd ser., I, 223–232] and 1818 [1, 2nd ser., II, 278–279]. Fourier’s discovery, while dating use up 1807 and 1811, was published hoot late as 1824–1826 [27], so Cauchy’s claim that he found the movement formula independently is quite acceptable. In peace is remarkable that he nevertheless formal Fourier’s priority by calling the motility formula Fourier’s formula. Cauchy put honourableness Fourier transform to greater use unthinkable used it with greater skill already anybody at that time and vindicate long after—Fourier and Poisson included; tell off he was the first to denote the inversion theorem correctly. He additionally stressed the importance of principal patience, of convergence-producing factors with limit 1, and of singular factors (δ functions) under the integral sign [1, Ordinal ser., I, 275–355]. His use ship the Fourier transform was essentially sound—bold but not rash—but to imitate cut off in the pre-epsilontic age one esoteric to be another Cauchy. After Weierstrass, Fourier transforms moved into limbo, doubtless because other methods conquered differential equations. Fourier transforms did not become in favour until recently, when the fundamentals returns Fourier integrals were proved with accomplished desirable rigor; but so much lifetime had elapsed that Cauchy’s pioneering rip off had been forgotten.

From 1821 on, Cauchy considered linear partial differential equations boring the operational form

with F as well-organized polynomial function in u₁,…, u_n, s. Such a differential equation has position exponential solutions

exp (∑u_ix_i + st),

which sentinel functions of x_i, t for ever and anon system of u_i, s fulfilling

(10) F(u₁,…, u_n, s),

The Fourier transform method aims at obtaining the general solution chunk continuously superposing such exponential solutions, fitting imaginary u₁,…, u_n, s. For opinion equations this means wave solutions past as a consequence o superposition of plane harmonic waves. Unite the 1821 and 1823 papers [1, 2nd ser., II, 253–275; I, 275–333] a kind of interpolation procedure served to satisfy the initial conditions portend t = 0. Another approach would be solutions arising from local disturbances (spherical waves under special conditions); they may be obtained from plane waves by superposition and in turn could give rise to general solutions near superposition. This idea is present attach importance to the 1815 papers on waves; image is neglected but not absent value the 1821 and 1823 papers.

In 1826 the residue calculus is introduced monkey a new device, first for solve linear ordinary differential equations with unshakeable coefficients [1, 2nd ser., VI, 252–255; VII, 40–54, 255–266]. The general solution

is obtained as the integral

performed around rectitude roots of F, with an discriminatory polynomial ψ (ζ). Several times Cauchy stressed that this formula avoids ethics distinctions with respect to multiple citizenship of F [1, 1st ser., IV, 370].

In 1830, when Cauchy went affected optics, this formula was applied stand firm partial differential equations, which meant (10) and (9) explicitly and elegantly rigid with respect to s and t, whereas with respect to x₁…, x_n the Fourier transform method prevailed. Adjust due care was bestowed on primacy initial conditions at t = 0. The formula, obtained in polar choir, is involved and not quite clear; its proof is not available as the mémoire of which the 1830 paper is a brief extract seems never to have been published highest possibly is lost. The construction be partial to wave fronts rested upon intuitive premises, in fact upon Huygens’ principle, tho' he did not say so significant never proved it. According to prestige same principle Cauchy constructed ray solutions as a superposition of planar disturbances in planes that should be to some extent or degre inclined toward each other, as Cauchy says.

From June 1839 to March 1842 Cauchy, again drawing on optics, reliable new approaches to linear partial division equations with constant coefficients. This labour was instigated by P. H. Blanchet’s intervention (see [1: 1st ser. IV, 369–426; V, 5–20; VI, 202–277, 288–341, 375–401, 404–420; 2nd ser., XI, 75–133, 277–264; XII, 113–124]). It now inch by inch with a system of first-order disentangle ordinary differential equations, in modern notation

where x is an n-vector and A a linear mapping. One considers

S (s) = det(A − s)

and defines birth fonction principale Θ as the improve of

which is obtained as an integral

around the roots of S. From Θ the solution of (11) with prestige initial vector ┌α₁,…, α_n┐ is gorgeously obtained by using

Q (s) = det(A − s)· (A − s)^-1

and onus Q(d/dt) to the vector

This method practical extended to partial differential equations, spin it again suffices to solve (9) under the initial conditions

The formula derived is much simpler than that unknot 1830, particularly if F is identical. (See the quite readable presentation engage [1, 1st ser., VI, 244–420].) Assuming, moreover, the initial value of (d/dt)^n-1ω prescribed is a function Σ of

σ = ∑u_ix_i,

one obtains, by first regarding Π to be linear and so using the homogeneity of F,

where (d/dt)^-1 means the integration over t proud 0. An analogous formula is derived if the initial datum is appropriated as a function of a equation function of x, say of r = (∑x_i²)^1/2.

The direct method of 1830 is again applied to the read of local disturbances and wave fronts, with Huygen’s principle on the location. Cauchy apparently believed that disturbances manage infinitesimal width stay infinitesimal, although that belief is disproved by (d/dt)^1-n distort (12) and by the physical polemic of two or even three take forward wave fronts in elasticity. It was only after Blanchet’s intervention that Cauchy admitted his error.

It is doubtful bon gr Cauchy’s investigations on this subject exerted strong immediate influence; perhaps the generations of Kirchhoff, Volterra, and Orazio Tedone were terrified by his use well Fourier transforms. Solutions arising from go out of business disturbances, substituted into Green’s formula, seemed more trustworthy. But even after Hadamard’s masterwork in this field Cauchy’s attempts should not be forgotten. The matchless modern book in which they hurtle mentioned and used, although in play down unsatisfactory and somewhat misleading fashion, go over the main points Courant and Hilbert [34].

It should eke out an existence mentioned that Cauchy also grasped significance notions of adjointedness of differential operators in special cases and that good taste attacked simple boundary problems by path of Green’s function [1, 1st ser., VII, 283–325; 2nd. ser., XII, 378–408].

Mechanics. Cauchy can be credited with selected minor contributions to mechanics of inflexible bodies, such as the momental rounded and its principal axes [1, Ordinal ser., VII, 124–136]; the surfaces chief the momentaneous axes of rigid on the dot [1, 2nd ser., VII, 119], observed simultaneously with Poinsot; and the twig rigorous proof that an infinitesimal persuade is a screw motion [1, Ordinal ser., VII, 116]. His proper district, however, was elasticity. He created righteousness fundamental mathematical apparatus of elasticity theory.

The present investigations have been suggested encourage a paper of M. Navier, 14 August 1820. To establish the percentage of equilibrium of the elastic echelon, the author had considered two kinds of forces, the ones produced outdo dilatation or contraction, the others uncongenial flection of that plane. Further, perform his computations he had supposed both perpendicular to the lines or lucubrate upon which they act. It came into my mind that these link kinds could be reduced to incontestable, still called tension or pressure, chief the same nature as the curvilinear pressure exerted by a fluid clashing the surface of a solid. Up till the new pressure should not mistrust perpendicular upon the faces which suffer it, nor be the same reaction all directions at a given point…. Further, the pressure or tension exerted against an arbitrary plane is straightforwardly derived as to magnitude and progression from the pressures or tensions exerted against three rectangular planes. I locked away reached that point when M. Physicist happened to speak to me study his work on light, which operate had only partially presented to decency Academy, and told me that on the road to the laws according to which flexibility varies in the different directions nibble one point, he had obtained elegant theorem like mine. However, this premise was far from sufficient for say publicly purpose I had in mind concede that time, that is, to match the general equation of equilibrium settle down internal motion of a body, cope with it is only recently that Unrestrainable have succeeded in establishing new customary, suited to lead to this grounds and the object of my communication….

These lines, written by Cauchy in honourableness fall of 1822 [1, 2nd ser., II, 300–304], announced the birth censure modern elasticity theory. Rarely has top-hole broad mathematical theory been as discriminatingly explained in as few words pounce on as striking a lack of exact symbols. Never had Cauchy given goodness world a work as mature suffer the loss of the outset as this.

From Hooke’s blame in 1660 up to 1821, flexibility theory was essentially one-dimensional. Euler’s conjecture of the vibrating membrane was pick your way of the few exceptions. Another was the physical idea of internal shave stress, which welled up and convulsion twice (Parent, 1713; Coulomb, 1773) silent no impact upon the mathematical belief. Even in this one-dimensional setting, pliability was a marvelous proving ground be intended for Euler’s analysis of partial derivatives near partial differential equations (see [36]). Enfold 1821 Navier’s paper on equilibrium topmost vibration of elastic solids was expire to the Academy (published in 1827) [30, 32]. Navier’s approach constituted trial mechanics as applied to an isotropous molecular medium that should obey Hooke’s law in molecular dimensions: any conversion in distance between two molecules causes a proportional force between them, prestige proportionality factor rapidly decreasing with acceleratory distance. Cauchy was one of ethics examiners of Navier’s paper. It was not, however, this paper of Navier’s to which Cauchy alluded in primacy above quotation. Cauchy’s first approach was independent of Navier’s it was nonmolecular but rather geometrically axiomatic.

Still another upgrade had taken place in 1821. Apostle Young’s investigations on interference in 1801 had made it clear that trivial should be an undulation of shipshape and bristol fashion hypothetical gaseous fluid, the ether. In this fashion light waves were thought of likewise longitudinal like those of sound unswervingly air, although the phenomena of condition pointed to transverse vibrations. In 1821 Fresnel took the bold step emancipation imagining an ether with resistance endure distortion, like a solid rather fondle a fluid; and marvelously enough crystalclear found transmission by transverse waves (although longitudinal ones would subsist as be a success [32]). Fresnel’s results encouraged Cauchy find time for pursue his investigations.

The short communication take from which the above extract was captivated was followed by detailed treatises bank on 1827–1829 (1, 2nd ser., VII, 60–93, 141–145; VIII, 158–179, 195–226, 228–287; Fix up, 342–369], but nearly all fundamental day-star of the mechanics of continuous communication were already clear in the 1822 note: the [stress tensor (and rank concept of tensor at all), depiction strain tensor, the symmetry of both tensors, their principal axes, the tenet of obtaining equilibrium and motion equations by cutting out and freezing lever infinitesimal piece of the medium, bear the striking idea of requiring Hooke’s law for the principal stresses move strains. For homogeneous media this frazzled to [Navier’s equations, with one springy constant, but [independently of Navier’s molecular substructure. Soon Cauchy introduced the next elastic constant, which arose from protest independent relation between volume stress extort volume strain. It led to authority now generally accepted elasticity theory indifference isotropic media. For anisotropic media Cauchy was induced by Poisson’s intervention run into admit a general linear dependence in the middle of stress and strain, involving thirty-six compass. The only fundamental notion then deficient was the elastic potential, which allows one to reduce the number pursuit parameters to twenty-one; it is question paper to G. Green (1837) [30; 32].

Meanwhile, in 1828–1829 Cauchy had pursued Navier’s molecular ideas and had arrived trouble a fifteenparameter theory for anisotropic communication [1, 2nd ser., VIII, 227–277; See, 162–73].⁹ The nineteenth-century discussions are long-since closed in favor of the unstated accepted “multi-constant” theory and against the molecular “rari-constant” theory, if not against commoner molecular theory of elasticity at all.

Cauchy applied the general theory to a number of special problems: to lamellae [1, Ordinal sen, VIII, 288–380], to the precise beam [1, 2nd ser., IX, 61–86] (definitively dealt with by Saint Venant), and to plane plates [1, Ordinal ser., VIII, 381–426; IX, 9–60], heavens which Kirchhoff finally succeeded. The utilization on which Cauchy bestowed more striving than on any other subject was elastic light theory. The mathematical action of this theory was partial difference equations with constant coefficients. In rectitude history of physics it was facial appearance of the great pre-Maxwellian efforts allowable before physicists became convinced of say publicly impossibility of any elastic light theory.

Cauchy developed three different theories of echo and refraction (1830, 1837, 1839) [1, 1st ser., II, 91–110; 2nd ser., II, 118–133; 1st ser., IV, 11–38; V, 20–39]. The problems were break down explain double refraction (in which without fear succeeded fairly well), to adjust birth elastic constants to the observational document on the velocity of light misstep different conditions in order to receive Fresnel’s sine and tangent laws in this area polarization by suitable boundary conditions, put forward to eliminate the spurious longitudinal tenseness. Whether he assumed the transverse excitement to be parallel or orthogonal look after the polarization plane (as he blunt in his first and second theories, respectively), he obtained strange relations 'tween the elastic constants and was laboured to admit unmotivated and improbable maximum value conditions. His third theory, apparently troubled by Green’s work, was based drop in the curious assumption of an obscure with negative compressibility—later called labile mass Lord Kelvin—which does away with longitudinal waves. In 1835 Cauchy also attempted dispersion 1, 2nd ser., X, 195–464]; the problem was to explain significance dependence of the velocity of make something happen upon the wavelength by a auxiliary refined evaluation of the molecular substructure.

Celestial Mechanics. One not acquainted with probity computational methods of astronomers before excellence advent of electronic apparatus can not quite evaluate Cauchy’s numerous and lengthy charity to celestial mechanics. In handbooks search out astronomy he is most often quoted because of his general contributions fulfil mathematics. Indeed, it must have antediluvian a relief for astronomers to hear that the infinite series they tatty in computations could be proved strong Cauchy to converge. But he along with did much detailed work on program, particularly for the solution of distinction Kepler equation [1, 1st ser., VI, 16–48] and developments of the perturbative function [1, 1st ser., V, 288–321; VII, 86–126]; textbooks still mention leadership Cauchy coefficients. Cauchy’s best-known contribution cause problems astronomy (1845) is his checking run through Leverrier’s cumbersome computation of the chunky inequality in the mean motion blame Pallas by a much simpler way [1, 1st ser., IX, 74–220; affection also XI, 385–403]. His tools consisted of formulas for the transition detach from the eccentric to the mean abnormalcy [1, 1st ser., VI, 21]; honesty Cauchy “mixed method” [I, 1st ser., VIII, 168–188, 348–359], combining numerical stomach rational integrations when computing negative reason of the perturbative function; and asymptotic estimations of distant terms in blue blood the gentry development of the perturbative function according to multiples of the mean anomaly—such asymptotics had interested Cauchy as untimely as 1827 [1, 1st ser., II, 32–58; see also IX, 5–19, 54–74; XI, 134–143].

NOTES

1. All references in brackets are lo numbered works in blue blood the gentry bibliography.

2. The data on his vitality in (7; 8; 9; 11] unwanted items incomplete and contradictory, even self-contradictory; on the other hand it still would not be as well difficult to check them. According fulfill Cauchy’s own account [1, 2nd ser., II. 283] he taught in 1817 as Biot’s suppléant at the Collège de France. However, on the designation pages of his books published in the balance 1830 he never mentions a rockingchair at the Collège de France.

3. City was the capital of Piedmont; nobleness dukes of Piedmont and Savoy difficult become kings of Sardinia.

4. In prestige library of the University of Metropolis I came across an unknown typography of a work by Cauchy lose concentration must have appeared in a magazine. I noticed several quotations from archives missing from Valson’s list and class Academy edition (e.g. [1, 2nd ser., II, 293, note]).

5. I know one two of them.

6. [1, 1st ser., IX, 186–190] shows him quarreling set about the press.

7. Of course he outspoken not prove it; he simply efficient it. It had been discovered inured to Bolzano in 1817.

8. His own look right through of this invention (1, 1st ser., VIII, 145], although often repeated, equitable incorrect. With Euler and Lagrange, subside said, “continuous” meant “defined by make sure of single law.” Actually, there was thumb serious definition of continuity before Cauchy.

9. I do not understand on what grounds Müller and Timpe [26, IV, 23] and Love [30] claim go Cauchy mistook his fifteenparameter theory preventable the twenty-one-parameter theory. The common origin of this criticism seems to examine Clausius (1849). Although Cauchy was once in a while less outspoken on this point, magnanimity charge is at least refuted get ahead of [1, 2nd ser., IX, 348].

BIBLIOGRAPHY

1. Cauchy, Oeuvres complètes, 1st ser., 12 vols., 2nd ser., 14 vols. (Paris, 1882-). The final volume, 2nd ser., XV, is due to appear in 1970.

2. Cauchy, Mémoire sur les intègrales définies prises entre des limites imaginaires (Paris, 1825). To be republished in [1, 2nd ser., XV].

2a. Reprint of [2] in Bulletin des sciences mathèmatiques, 7 (1874), 265–304; 8 (1875), 43–55, 148–159.

2b. Cauchy, Abhandlung über bestimmte Integrate zwischen imaginären Grenzen, trans, of [2] dampen P. Stäekel, no. 112 in Ostwald’s Kiassiker der exacten Wissenschaften (Leipzig, 1900).

3. F. N. M. Moigno and Lot. Lindelöf, eds., Leçons du calcul différentiel et de calcul intégral, rédigées d’après les méthodes et les oeuvres publiés ou inédits d’ A.-L. Cauchy, 4 vols. (Paris, 1840–1861).

4. F. N. Class. Moigno, Leçons de mécanique analytique, rédigées principalement d’après les méthodes d’Augustin Cauchy…. Statique (Paris, 1868).

5. B. Boncompagni, “La vie et les travaux du fat cat Cauchy,” in Bollettino di bibliografia bond di storia delle scienze matematiche line fisiche, 2 (1869), 1–102. Review cataclysm [8].

6. K. Rychlik, “Un manuscrit inhabit Cauchy aux Archives de l’Académie tchécoslovaque des sciences,” in Czechoslovak Mathematical Journal, 7 [82 ] (1957), 479–481.

7. Count. B. Biot, Mélanges scientifiques et littéraires, III (Paris, 1858), 143–160.

8. C. Top-notch. Valson, La vie et les travaux du baron Cauchy, 2 vols. (Paris, 1868).

9. J. Bertrand, “La vie admire les travaux du baron Cauchy level C. A. Valson,” in Journal stilbesterol savants (1869), 205–215, and Bulletin nonsteroid sciences mathématiques, 1 (1870), 105–116. Dialogue of [8].

10. C. A. Bjerknes, Niels-Henrik Abel, tableau de sa vie permit de son action scientifique (Paris, 1885), pp. 268– 322, 342–347.

10a. D. Hook up. Smith, “Among My Autographs, 29. Legendreand Cauchy Sponsor Abel,” in American Exact Monthly, 29 (1922), 394–395.

10b. F. Lange-Nielsen, “Zur Geschichte des Abelschen Theorems. Das Schicksal der Pariser Abhandlung,” in Norsk matematisk tidsskrift, 9 (1927), 55–73.

11. Number. Bertrand, Éloges académiques, new ed. (Paris, 1902), pp. 101–120.

12. A. d’Hautpoul, Quatre mois à la cour de Prague (Paris, 1912). Not accessible to probity present reporter.

13. A. Terracini, “Cauchy undiluted Torino,” in Rendiconti del Seminario matematico (Turin), 16 (1956–1957), 159–205.

14. P. Tough. Laplace, Traité de la mécanique céleste, II (Paris1804).

15. C. Sturm, “Analyse d’un mémoire sur la résolution des équations numériques,” in Bulletin universel des sciences et de l’industrie, 11 (1829), 419–422.

16. J. Bienaymé, “Considerations…, Remarques…,” in Comptes rendus de l’Académie des sciences, 37 (1853), 5–13, 68–69, 309–324.

17. N. Twirl. Abel, “Untersuchungen über die Reihe…,” monitor Journal für reine and angewandte Mathematik, 1 (1826), 311–359, also published primate no. 71 in Ostwald’s Klassiker stake exacten Wissenschaften (1895).

18. S.-H. Briot nearby J. C. Boucquet, Théorie des fonctions doublement périodiques et, en particulier, nonsteroid fonctions elliptiques (Paris 1859); 2nd ed., Théorie des fonctions elliptiques (Paris 1873–1875).

19. F. Casorati, Teorica delle funzioni di variabili complesse (Pavia, 1868). Not attainable to the present reporter.

20. E. Verdet, Leçons d’optique physique, vols. V–VI concede Verdet’s Oeuvres (Paris, 1869–1872).

21. F. Studnicka, A. L. Cauchy als formaler Begründer der Determinantentheorie (Prague, 1876). Not thin-skinned to the present reporter.

21a. I. Todhunter, A History of the Theory longedfor Elasticity and of the Strength clamour Materials from Galilei to the Settle, Time, vol. I (Cambridge, 1886).

22. Ingenious. Brill and M. Noether, “Die Entwicklung der Theorie der algebraischen Funktionen select by ballot älterer and neuerer Zeit,” in Jahresbericht der Deutschen Mathematiker-Vereinigung, 3 (1894), esp. 155–197.

23. P. Stäckel, “Integration durch das imaginäre Gebiet,” in Bibliotheca mathematica, Ordinal ser., 1 (1900), 109–128.

24. P. Stäckel, “Beitrag zur Geschichte der Funktionentheorie point 18. Jahrhundert.” in Bibliotheca mathematica, Ordinal ser., 2 (1901), 111–121.

25. P. Jourdain, “The Theory of Functions With Cauchy and Gauss,” in Bibliotheca mathematica, Tertiary ser., 6 (1905),190–207.

26. The following reconcile in the Encyklopädie der mathematischen Wissenschaften: C. H. Müller and A. Timpe, “Grundgleichungen der mathematischen Elastizitätstheorie,” IV (1907), I; A. Wangerin, “Optik, ältere Theorie,” V (1909), 21; K. F. Sundman. “Theorie der Planeten.” VI, pt. 2 (1912), 15; H. von Zeipel, “Entwieklung der Störungsfunktion,” VI, pt. 2 (1912). 557–665.

27. H. Burkhardt, “Entwicklungen nach oscillierenden Funktionen and Integration der Differentialgleichungen pilaster mathematischen Physik,” in Jahresbericht der Deutschen Mathematiker- Vereinigung, 10 (1904–1908).

28. E. Out. Miller, “Historical Sketch of the Awaken of the Theory of Groups be in the region of Finite Order,” in Bibliotheca mathematica, Ordinal ser., 10 (1909). 317–329,

29. P. Jourdain, “The Origin of Cauchy’s Conception realize a Definite Integral and of rank Continuity of a Function,” in Isis, 1 (1913–1914), 661–703. Not accessible cross-reference the author.

30. A. E. H. Liking, A Treatise on the Mathematical Knowledge of Elasticity (Oxford, 1927), esp, dignity introduction.

31. H. Lamb, Hydrodynamics (Cambridge, 1932), pp. 384 f.

32. E. T. Whittaker, A History of the Theories many Aether and Electricity (Edinburgh, 1951), esp, pp. 128–169.

33. E. Carruccio, “I fondamenti delle analisi matematica nel pensiero di Agostino Cauchy.” in Rendiconti del Seminario matematico (Turin), 16 (1956–1957). 205–216.

34. Attention. Courant and D. Hilbert, Methods oppress Mathematical Physics, 11 (1962), esp. pp. 210–221.

35. C. Truesdell-R. Toupin, “The Prototypical Field Theories,” in Handbuch der Physik, III (Berlin-Güttingen-Heidelberg, 1960), pp. 226–793, esp. 259–261. 270–271, 306–308, 353–356, 536–556.

36. Proverbial saying. Truesdell, “The Rational Mechanics of Compress and Elastic Bodies, 1638–1788,” in Leonhardi Euleri Opera Omnia, 2nd ser., XI, sec. 2 (Zurich, 1960), 7–435.

Hans Freudenthal