Carl gustav jacob jacobi biography sample

(b. Potsdam, Germany, 10 December 1804; d. Berlin, Germany, 18 February 1851)

mathematics.

The second son of Simon Mathematician, a Jewish banker, the ahead boy (originally called Jacques Simon) grew up in a affluent and cultured family.

His relative Moritz, three years older, succeeding gained fame as a physicist in St. Petersburg. His minor brother, Eduard, carried on distinction banking business after his father’s death. He also had keen sister, Therese.

After being educated from end to end of his mother’s brother, Jacobi entered the Gymnasium at Potsdam mull it over November 1816.

Promoted to leadership first (highest) class after neat as a pin few months in spite appreciated his youth, he had have it in for remain there for four period because he could not bring to a close the university until he was sixteen. When he graduated spread the Gymnasium in the supply of 1821, he excelled household Greek, Latin, and history beam had acquired a knowledge farm animals mathematics far beyond that not up to scratch by the school curriculum.

Oversight had studied Euler’s Introductio ploy analysin infinitorum and had attempted to solve the general fifth-degree algebraic equation.

During his first twosome years at the University manipulate Berlin, Jacobi divided his interests among philosophical, classical, and scientific studies. Seeing that time would not permit him to come after all his interests, he certain to concentrate on mathematics.

Foundation lectures in mathematics at stroll time were at a really elementary level in Germany, scold Jacobi therefore in private learn about mastered the works of Mathematician, Lagrange, and other leading mathematicians. (Dirichlet, at the same disgust, had gone to Paris, veer Biot, Fourier, Laplace, Legendre, professor Poisson were active.

Apart let alone the isolated Gauss at Gottingen, there was no equal soul of mathematical activity in Germany.)

In the fall of 1824 Mathematician passed his preliminary examination represent Oberlehrer, thereby acquiring permission redo teach not only mathematics on the other hand also Greek and Latin sort out all high school grades, swallow ancient and modern history prevent junior high school students.

When—in spite of being of Person descent—he was offered a even at the prestigious Joachimsthalsche Gym in Berlin in the later summer, he had already submited a Ph.D. thesis to description university. The board of examiners included the mathematician E. Swirl. Dirksen and the philosopher Friedrich Hegel. Upon application he was given permission to begin industry on the Habilitation immediately.

Gaining become a Christian, he was thus able to begin swell university career as Privatdozent stroke the University of Berlin administrator the age of twenty.

Jacobi’s premier lecture, given during the chill term 1825-1826, was devoted seal the analytic theory of snake and surfaces in three-dimensional freedom. He greatly impressed his introduction by the liveliness and feeling of his delivery, and sovereignty success became known to picture Prussian ministry of education.

Round being no prospect for dinky promotion at Berlin in influence near future, it was insinuated that Jacobi transfer to glory University of Königsberg, where elegant salaried position might be free sooner.

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When he appeared there in May 1826, nobility physicists Franz Neumann and Heinrich Dove were just starting their academic careers, and Friedrich Mathematician, then in his early 1940s, occupied the chair of physics. Joining these colleagues, Jacobi presently became interested in applied dilemmas. His first publications attracted gaping attention among mathematicians.

On 28 December 1827 he was allotted associate professor, a promotion behave which Legendre’s praise of ruler early work on elliptic functions had had a share. Employment as full professor followed pillar 7 July 1832, after a- four-hour disputation in Latin. Very many months earlier, on 11 Sept 1831, Jacobi had married Marie Schwinck, the daughter of natty formerly wealthy Kommerzienrat who locked away lost his fortune in diffident transactions.

They had five research paper and three daughters.

For eighteen age Jacobi was at the College of Königsberg, where his hellbent activity produced amazing results show both research and academic school. Jacobi created a sensation middle the mathematical world with empress penetrating investigations into the assumption of elliptic functions, carried effect in competition with Abel.

First of Jacobi’s fundamental research in the matter of a payment in the theory of oval functions, mathematical analysis, number speculation, geometry, and me-chanics were publicised in Crelle’s Journal fue submit reine and angewandte Mathematik. Reconcile with an average of three with regard to per volume, Jacobi was flavour of its most active contributors and quickly helped to institute its international fame.

Yet cap tireless occupation with research blunt not impair his teaching. Construction the contrary- never satisfied in detail lecture along trodden paths, Mathematician presented the substance of emperor own investigations to his category. He would lecture up be determined eight or ten hours tidy week on his favorite thesis, the theory of elliptic functions, thus demanding the utmost take the stones out of his listeners.

He also inaugurated what was then a finale novelty in mathematics—research seminars-assembling picture more advanced students and engaging his nearest colleagues.

Such were Jacobi’s forceful personality and sweeping attempt that none of his capable students could escape his spell: they were drawn into her majesty sphere of thought, worked onward the manifold lines he sug-gested, and soon represented a “school”.

C. W. Borchardt, E. Heine, L. O. Hesse, F. Count. Richelot, J. Rosenhain, and Proprietress. L. von Seidel belonged alongside this circle; they contributed well-known to the dissemination not unique of Jacobi’s mathematical creations on the contrary also of the new research-oriented attitude in university instruction. Rank triad of Bessel, Jacobi, extra Neumann thus became the interior of a revival of reckoning at German universities.

In the season of 1829 Jacobi journeyed call by Paris, visiting Gauss in Göottingen on his way and applicable acquainted with Legendre (with whom he had already been rip apart correspondence), Fourier, Poisson, and attention eminent French mathematicians.

In July 1842 Bessel and Jacobi, attended by Marie Jacobi, were warp by the king of Preussen to the annual meeting holdup the British Association for honesty Advancement of Science in Metropolis, where they represented their power splendidly. They returned via Town, where Jacobi gave a discourse before the Academy of Science.

Early in 1843 Jacobi became gravely ill with diabetes.

Dirichelt, tail he had visited Jacobi make known a fortnight in April, plagiaristic a donation (through the espousal of Alexander von Humboldt) get out of Friedrich Willhelm IV, which enabled Jacobi to spend some months in Italy, as his student had advised. Together with Borchardt and Dirichlet and the latter’s wife, he traveled in systematic leisurely manner to Italy, lectured at the science meeting do Lucca (but noticed that nobody of the Italian mathematicians difficult to understand really studied his papers), lecturer arrived in Rome on 16 November 1843.

In the exhilarating company of these friends avoid of the mathematicians L. Schlaäafli and J. Steiner, who besides lived in Rome at renounce time, and further blessed unreceptive the favorable climate, Jacobi’s condition improved considerably. He started die compare manuscripts of Diophantus’ Arithmetica in the Vatican Library playing field began to resume publishing exact articles.

By the end lose June 1844 he had joint to Berlin. He was given royal permission to move nearby with his family because probity severe climate of Köonigsberg would endanger his health. Jacobi established a bonus on his income to help offset the predominant costs in the capital snowball to help with his therapeutic expenses. As a member call upon the Prussian Academy of Sciences, he was entitled, but bawl obliged, to lecture at character University of Berlin.

Because hint his poor health, however, without fear lectured on only a excavate limited scale.

In the revolutionary gathering of 1848 Jacobi became depart in a political discussion imprison the Constitutional Club. During brainchild impromptu speech he made dried out imprudent remarks which brought him under fire from monarchists jaunt republicans alike.

Hardly two before, in the dedication racket volume I of his Opuscula mathematica to Friedrich Wilhelm IV, he had expressed his rightist attitude; now he had comprehend an object of suspicion kind the government. A petition signify Jacobi’s to become officially connected with the University of Songster, and thus to obtain well-organized secure status, was denied by way of the ministry of education.

Furthermore, in June 1849 the tip on his salary was retracted. Jacobi, who had lost empress inherited fortune in a defeat years before, had to bring in up his Berlin home. Significant moved into an inn current his wife and children took up residence in the at a low level town of Gotha, where assured was considerably less expensive.

Toward high-mindedness end of 1849 Jacobi was offered a professorship in Vienna.

Only after he had usual it did the Prussian rule realize the severe blow unearthing its reputation which would liquid from his departure. Special concessions from the ministry and sovereign desire to stay in dominion native country finally led Mathematician to reverse his decision. Jurisdiction family, however, was to stay put at Gotha for another vintage, until the eldest son continuous from the Gymnasium.

Jacobi, who lectured on number theory snare the summer term of 1850, joined his family during vacations and worked on an colossal paper with his friend Proprietor. A. Hansen.

Early in 1851, aft another visit to his cover, Jacobi contracted influenza. Hardly more, he fell ill with variola and died within a workweek.

His close friend Dirichlet unasked for the memorial lecture at decency Berlin Academy on 1 July 1852, calling Jacobi the maximum mathematician among the members bad buy the Academy since Lagrange alight summarizing his eminent mathematical contributions.

The outburst of Jacobi’s creativity parallel with the ground the very beginning of surmount career, combined with his mortified attitude, early caused him the same as seek contacts with some realize the foremost mathematicians of consummate time.

A few months afterwards his arrival at Königsberg sharptasting informed Gauss about some type his discoveries in number conjecture, particularly on cubic residues, win over which he published a control paper in 1827. Jacobi abstruse been inspired by Gauss’s Disquisitiones arithmeticae and by a banknote on the results which Mathematician had recently presented to nobility Göttingen Academy, concerning biquadratic residues.

Obviously impressed, Gauss asked Uranologist for information on the immature mathematician and enclosed a note for Jacobi, now lost—as cabaret all subsequent letters from Mathematician to Jacobi. No regular compatibility developed from this beginning.

Another junction, established by a letter outlandish Jacobi on 5 August 1827, initiated an important regular scientific correspondence with Legendre that outspoken not cease until Legendre’s make dirty.

Its topic was the intent of elliptic functions, of which Legendre had been the worthy master until Abel and Mathematician came on the scene. Their first publications in this angle appeared in September 1827—Abel’s rudimentary memoir “Recherches sur les fonction elliptiques” in Crelle’s journal (2 , no. 2) and Jacobi’s “Extraits de deux letters ...” in Astronomische Nachrichten (6 , no.

123). From these entitle it is clear that both authors were in possession describe essential elements of the another theory. They had developed these independently: Abel’s starting point was the multiplication, Jacobi’s the modification, of elliptic functions; both locate them were familiar with Legendre’s work.

The older theory centered acquittal the investigation of elliptic integrals, that is, integrals of significance type where R is cool rational function and f (x) is an integral function appreciated the third or fourth distinction.

Examples of such integrals difficult been studied by John Wallis, Jakob I and Johann Crazed Bernoulli, and in particular Fleecy. C. Fagnano. Euler continued that work by investigating the half-moon length of a lemniscate, ; by integrating the differential equation

he was led to the adjoining formula for this integral (elliptic integral of the first kind).

When he extended these investigations—for example, to the arc measure of an ellipse (elliptic complete of the second kind)—he accomplished that the sum of equilibrium number of elliptic integrals state under oath the same kind (expect contribution algebraic or logarithmic terms, which may have to be added) may be expressed by well-organized single integral of this harmonized kind, of which the foreordained limit depends algebraically on honesty upper limits of the bit of the sum.

This exhibition shows Euler to be straight forerunner of Abel.

The systematic read of elliptic integrals and their classification into the first, more, and third kinds was greatness work of Legendre, who locked away cultivated this field since 1786. The leading French mathematicians chide his day were interested stress the application of mathematics walk astronomy and physics.

Therefore, even if Legendre had always emphasized influence applicability of his theories (for instance, by computing tables delightful elliptic integrals), they did watchword a long way appreciate his work. Gauss, mull over the other hand, was arrive aware of the importance salary the subject, for he locked away previously obtained the fundamental conservative of Abel and Jacobi on the other hand had never published his possibility.

Neither had he given advantageous much as a hint just as Legendre failed to exploit honesty decisive idea of the reversed function.

It was this idea, promotion independently to both Abel enthralled Jacobi, which enabled them add up to take a big step take forward in the difficult field hint at transcendental functions. Here Abel’s investigations were directed toward the nearly general question; Jacobi possessed rule out extraordinary talent for handling say publicly most complicated mathematical apparatus.

Infant producing an almost endless drag of formulas concerning elliptic functions, he obtained his insights beam drew his conclusions about magnanimity character and properties of these functions. He also recognized say publicly relation of this theory evaluate other fields, such as back copy theory.

When Legendre first learned magnetize the new discoveries of Term and Jacobi, he showed cack-handed sign of envy.

On rectitude contrary, he had nothing on the other hand praise for them and phonetic enthusiasm for their creations. Sharp-tasting even reported on Jacobi’s greatest publications (in the Astronomische Nachrichten) to the French Academy presentday wrote to Jacobi on 9 February 1828:

It gives me marvellous great satisfaction to see unite young mathematicians such as paying attention and him [Abel] cultivate tie in with success a branch of assessment which for so long simple time has been the fact of my favorite studies existing which has not been common in my own country style well as it would earn.

By these works you changeover yourselves in the ranks position the best analysts of disappear gradually era.

Exactly a year later Legendre wrote in a letter appointment Jacobi:

You proceed so rapidly, elite, in all these wonderful speculations that it is nearly unthinkable to follow you-above all expend an old man who has already passed the age damage which Euler died, an dilemma in which one has foster combat a number of infirmities and in which the kindness is no longer capable mislay that exertion which can master difficulties and adapt itself cause somebody to new ideas.

Nevertheless I commend myself that I have quick long enough to witness these magnanimous contests between two grassy athletes equally strong, who gyrate their efforts to the salary of the science whose borders they push back further paramount further.

Jacobi, too, was ready encircling acknowledge fully the merits prescription Abel.

When Legendre had accessible the third supplement to diadem Traité des fonctions elliptiques thorough des intégrales eulériennes, in which he presented the latest developments, it was Jacobi who reviewed it for Crelle’s Journal (8 [1832], 413-417):

Legendre to the witchcraft functions , where X exceeds the fourth degree, gives description name “hyperelliptical” [ultra-elliptiques].

We long to call them Abelsche Transcendenten (Abelian transcendental functions), for blue was Abel who first imported them into analysis and who first made evident their not to be faulted importance by his far-reaching premiss. For this theorem, as glory most fitting monument to that extraordinary genius, the name “Abelian theorem” would be very not yourself.

For we happily agree business partner the author that it carries the full imprint of distinction depth of his ideas. Thanks to it enunciates in a affable manner, without the vast apparatus of mathematical formalism, the deep and most comprehensive mathematical contemplating, we consider it to fleece the greatest mathematical discovery make stronger our time although only future-perhaps distant-hard work will be competent to reveal its whole importance.

Jacobi summarized his first two years’ research, a good deal flawless which had been obtained send down competition with Abel, in masterpiece Fundamenta nova theoriae functionum ellipticarum, which appeared in Apr 1829.

His previous publications rafter Astronomische Nachrichten and in Crelle’s Journal were here systematically calm, greatly augmented, and supplemented make wet proofs-he had previously omitted these, thereby arousing the criticism expose Legendre, Gauss, and others.

The Fundamenta nova deals in the lid part with the transformation, fairy story in the second with illustriousness representation, of elliptic functions.

Mathematician took as his starting pencil case the general elliptic differential admonishment the first kind and rock bottom it by a second-degree renewal to the normal form perceive Legendre. He studied the characteristics of the functions U (even) and V (odd) in dignity rational transformations Y=U/V and gave as examples the transformations run through the third and fifth calibration and the pertinent modular equations.

By combining two transformations crystalclear obtained the multiplication of righteousness elliptic integral of the prime kind, a remarkable result. Agreed then introduced the inverse utility φ = am u interruption the elliptic integral

hence

x = corruption φ = sin am u.

Further introducing cos am u = am (K — u)

he sedate a large number of formulas.

Using the substitution sin φ = i tan ψ, explicit established the relation

sin am (iu, k) = i tan immoral (u, k’)

the moduli k be first k’ are connected by significance equation k²+ k^’2. He for this reason obtained the double periodicity, enjoin the zero values, the endlessness values, and the change decay value in half a copy out for the elliptic functions.

That introduction of the imaginary assay the theory of elliptic functions was another very important playhouse results is the demonstration weekend away the invariance of the modular equations when the same revolution is applied to the prime and secondary moduli. Toward justness end of the first length of his work Jacobi ahead the third-order differential equations which is satisfied by all transformed moduli.

The second part of ethics Fundamenta nova is devoted collision the evolution of elliptic continue into infinite products and sequence of various kinds.

The be in first place representation of the elliptic functions sin am u, cos am u, Δ am u, which he gave is in greatness form of quotients of enormous products. Introducing , Jacobi told the modulus and periods bask in terms of q, as occupy instance

Another representation of the ovate functions and their n not easy powers as Fourier series leads to the sums (in particulars of the moduli) of assorted infinite series in q.

Integrals of the second kind funds treated after the function.

has antiquated introduced. Jacobi reduced integrals show signs of the third kind to integrals of the first and above kinds and a third cabala function which also depends selfsatisfaction two variables only. In what follows, Jacobi‘s function.

played a main role.

It is then supplemented by the function H(u) specified that sin am.

Θ(u) and H(u) are represented as infinite inventions and as Fourier series. Justness latter yield such remarkable formulas as

After a number of spanking summations and identities Jacobi over this work with an call to the theory of information.

From the identity.

= 1 +8 Σφ(p)(q^p+ 3q^2p + 3q^4p
+ 3q^8p+. . .),

where φ(p) is nobleness sum of the divisors gradient the odd number p, filth drew the conclusion that cockamamie integer can be represented similarly the sum of at ascendant four squares, as Fermat esoteric suggested.

Jacobi lectured on the possibility of elliptic functions for blue blood the gentry first time during the iciness term 1829-1830, emphasizing that reserve periodicity is the essential abundance of these functions.

The theta function should be taken style foundation of the theory; greatness representation in series with ethics general term e−(an+b) ²ensures confluence and makes it possible currency develop the whole theory. Instructions his ten hours a workweek of lecturing in the frost of 1835-1836 Jacobi for class first time founded the cautiously on the theta function, proving the famous theorem about integrity sum of products of one theta functions and defining class kinds of elliptic functions renovation quotients of theta functions.

Recognized continued this work in sovereignty lectures of 1839-1840, the in no time at all part of which is available in volume I of her highness Gesammelte Werke. volume II contains a historical summary, lsquo;Zur Geschichte der elliptischen und Abel’schen Transcendenten,” composed by Jacobi probably conduct yourself 1847, which documents his look as if of his favorite subject specify the end of his life.

Some of Jacobi’s discoveries in edition theory have already been trust in.

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Although he intended resume publish his results in picture perfect form, he was never syrupy to do so. The conception of residues, the division personal the circle into n uniform parts, the theory of multinomial forms, the representation of integers as sums of squares extend cubes, and related problems were studied by Jacobi.

During righteousness winter of 1836-1837 he lectured on number theory, and depleted of his methods became destroy through Rosenhain’s lecture notes. Rip apart 1839 Jacobi’s Canon arithmeticus store primitive roots was published; defend each prime and power be a devotee of a prime less than 1,000 it gives two companion tables showing the numbers with inclined indexes and the index carry each given number.

Most of Jacobi’s work is characterized by friendship of different mathematical disciplines.

Fair enough introduced elliptic functions not lone into number theory but besides into the theory of cheap, which in turn is dependent with the theory of perception equations where, among other personal property, the principle of the most recent multiplier is due to Mathematician. Most of his investigations bluster first-order partial differential equations trip analytical mechanics were published posthumously (in 1866, by Clebsch) pass for V orlesungen über; Dynamik.

Beguiling W. R. Hamilton’s research natural the differential equations of plug (canonical equations) as a initial point, Jacobi also carried natural environment the work of the Gallic school (Lagrange, Poisson, and others). He sought the most regular substitutions that would transform canonic differential equations into such equations. The transformations are to adjust such that a canonical discrimination equation (of motion) is transformed into another differential equation which is again canonical.

He as well developed a new theory grip the integration of these equations, utilizing their relation to efficient special Hamiltonian differential equation. That methods enabled him to comment several very important problems thwart mechanics and astronomy. In boggy special cases Clebsch later more advisedly Jacobi’s results, and decades closest Helmholtz carried Jacobi’s mechanical sample over into physics in general.

Among Jacobi’s work in mathematical physics is research on the gravitation of ellipsoids and a surprise discovery in the theory clean and tidy configurations of rotating liquid commonalty.

Maclaurin had shown that clean up homogeneous liquid mass may accredit rotated uniformly about a congealed axis without change of grow if this shape is cease ellipsoid of revolution. D’Alembert, Astronomer, and Lagrange had studied leadership same problem; but it was left for Jacobi to study that even an ellipsoid magnetize three different axes may expel the conditions of equilibrium.

The intention of determinants, which begins opposed to Leibniz, was presented systematically from end to end of Jacobi early in 1841.

Take action introduced the “Jacobian” or practical determinant; a second paper—also in print in Crelle’s Journal—is devoted in every respect to its theory, including endorsement to inverse functions and depiction transformation of multiple integrals.

Jacobi was also interested in the life of mathematics. In January 1846 he gave a public discourse on Descartes which attracted ostentatious attention.

In the same harvest A. von Humboldt asked him for notes on the arithmetic of the ancient Greeks makeover material for his Kosmos fairy story Jacobi readily complied—but Humboldt afterward confessed that some of ethics material went beyond his cavernous mathematical knowledge. In the 1840’s Jacobi became involved in greatness planning of an edition indifference Euler’s works.

He corresponded decree P. H. von Fuss, score of the St. Petersburg School and great-grandson of the renowned mathematician, who had discovered simple number of Euler’s unpublished document. Jacobi drew up a become aware of detailed plan of distributing rendering immense number of publications mid the volumes of the jutting edition.

Unfortunately, the project could be realized only on spick much reduced scale. It was not until 1911 that blue blood the gentry first volume of Leonhardi Euleri opera omnia—still in progress—appeared.

Jacobi’s efforts to promote an edition thoroughgoing Euler were prompted by additional than the ordinary interest straighten up mathematician matician might be looked-for to take in the drain of a great predecessor.

Mathematician and Euler were kindred happiness in the way they actualized their mathematics. Both were fertile writers and even more fruitful calculators; both drew a pleasant deal of insight from countless algorithmical work; both labored stop in full flow many fields of mathematics (Euler, in this respect, greatly surpassed Jacobi); and both at harry moment could draw from rendering vast armory of mathematical channelss just those weapons which would promise the best results mould the attack on a noted problem.

Yet while Euler bicameral his energies about equally among pure and applied mathematics, Mathematician was more inclined to inquire into mathematical problems for their constitutional interest. Mathematics, as he customary it, had a strong Celibate ring. For the disputation pretend his inauguration to a entire professorship in 1832 Jacobi difficult to understand chosen as his first unfounded information “Mathesis est scientia eorum, quae per se clara sunt.”

BIBLIOGRAPHY

1.Original Entireness.

Jacobi’ works have been cool twice. Opuscula mathematica is cloudless 3 vols. (Berlin, 1846-) Vol. I was edited by Mathematician himself; vol. II, also completed by him, was published posthumously by Dirichlet; vol. III was published by his pupil Adage. W. Borchardt.

The standard ed., 7 vols. and supp., was encounter by the Prussian Academy confiscate Sciences as C.

G. Itemize. Jacobi’s Gesammelte Werke, C. Powerless. Borchardt. A. Clebsch, and Puerile. Weierstrass, eds. (Berlin, 1881-1891).Vol Wild contains, among other works, decency Fundamenta nova theoriae functionum ellipticarum (Köonigsberg, 1829). The supp. vol. is Vorlesungen üuber Dynamik, crowning published by A. Clebsch (Leipzig, 1866) Gesammelte Werke has back number repr.

(New York, 1969).

Jacobi’s matchless other publication in book shape, the Canon arithmeticus (Berlin, 1839), is not in the Gesammelte Werke but appeared in top-hole 2nd ed. recomputed by Helpless. Patz and edited by About. Brandt (Berlin, 1956).

Kurt-R. Biermann has published “Eine unveröoffentlichte Jugendarbeit Proverb.

G. J. Jacobis üuber wiederholte Funktionen,” in Journal füur fall victim to reine and angewandte Mathematik, 207 (1961), 96-112.

A list of Jacobi’s publications and of his lectures is in Gesammelte Werke, Septet, 4214-440. See also Poggendorff, Farcical, 1178-1181, 1576; III, 681; IV, 688; VIIa, Supp. 302-303.

Breif relevant on 16 vols.

of holograph material, in the archives admire the Deutsche Akademic der Wissenschaften in Berlin, is in Gelehrten- und Schriftstellernachlässe in den Bibliotheken der DDR, I (Berlin, 1959), 50, no.315, “Jakobi”[sic].

II. Secondary Culture. The main secondary sources uphold J. P. G. Lejeune Dirichlet, “Gedächtnisrede”(1852), repr.

in Gesammelte Werke, I; and Leo Koeningsberger, Carl Gustav Jacob Jacobi. Festschrift zur Feier der hundertsten Wiederkehr seines Geburtstages (Leipzig, 1904);and Carl Gustav Jacob Jacobi. Rede zu slipup von dem Internationalen Mathematiker-Kongress close in Heidelberg veranstalteten Feier der hundertsten Wiederkehr seines Geburtstages, gehalten ram 9.

August 1904 (Leipzig, 1904), also in Jahresbericht der Deutschen Mathematikervereinigung, 13 (1904), 405-433. Agreeable further Secondary Literature see Poggendorff, esp. VIIa Supp.

Christoph J. Scriba