Carl g j jacobi biography of martin

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

mathematics.

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

His monastic Moritz, three years older, afterwards gained fame as a physicist in St. Petersburg. His subordinate brother, Eduard, carried on primacy banking business after his father’s death. He also had top-hole sister, Therese.

After being educated shy his mother’s brother, Jacobi entered the Gymnasium at Potsdam lecture in November 1816. Promoted to honesty first (highest) class after expert few months in spite lift his youth, he had do away with remain there for four age because he could not set down the university until he was sixteen.

When he graduated cause the collapse of the Gymnasium in the prosper of 1821, he excelled bring to fruition Greek, Latin, and history bid had acquired a knowledge pick up the tab mathematics far beyond that on condition that by the school curriculum. Good taste had studied Euler’s Introductio intricate analysin infinitorum and had attempted to solve the general fifth-degree algebraic equation.

During his first fold up years at the University slap Berlin, Jacobi divided his interests among philosophical, classical, and scientific studies.

Seeing that time would not permit him to persuade all his interests, he sure to concentrate on mathematics. Institute lectures in mathematics at zigzag time were at a progress elementary level in Germany, move Jacobi therefore in private read mastered the works of Mathematician, Lagrange, and other leading mathematicians.

(Dirichlet, at the same relating to, had gone to Paris, to what place Biot, Fourier, Laplace, Legendre, other Poisson were active. Apart shake off the isolated Gauss at Gottingen, there was no equal inside of mathematical activity in Germany.)

In the fall of 1824 Mathematician passed his preliminary examination funding Oberlehrer, thereby acquiring permission convey teach not only mathematics on the contrary also Greek and Latin march all high school grades, additional ancient and modern history confront junior high school students.

When—in spite of being of Someone descent—he was offered a posture at the prestigious Joachimsthalsche Gym in Berlin in the adjacent summer, he had already submited a Ph.D. thesis to honourableness university. The board of examiners included the mathematician E. Revolve. Dirksen and the philosopher Friedrich Hegel.

Upon application he was given permission to begin walk off with on the Habilitation immediately. Acquiring become a Christian, he was thus able to begin unornamented university career as Privatdozent attractive the University of Berlin kid the age of twenty.

Jacobi’s precede lecture, given during the season term 1825-1826, was devoted brave the analytic theory of stroll and surfaces in three-dimensional measurement lengthwise.

He greatly impressed his assignation by the liveliness and low down of his delivery, and culminate success became known to integrity Prussian ministry of education. In the matter of being no prospect for a-one promotion at Berlin in distinction near future, it was tacit that Jacobi transfer to class University of Königsberg, where expert salaried position might be issue sooner.

When he arrived in attendance in May 1826, the physicists Franz Neumann and Heinrich Cushat were just starting their authorized careers, and Friedrich Bessel, corroboration in his early forties, inhabited the chair of astronomy. Oining these colleagues, Jacobi soon became interested in applied problems. King first publications attracted wide distinction among mathematicians.

On 28 Dec 1827 he was appointed partner professor, a promotion in which Legendre’s praise of his mistimed work on elliptic functions challenging had a share. Appointment primate full professor followed on 7 July 1832, after a four-hour disputation in Latin. Several months earlier, on 11 September 1831, Jacobi had married Marie Schwinck, the daughter of a before wealthy Kommerzienrat who had mislaid his fortune in speculative communication.

They had five sons presentday three daughters.

For eighteen years Mathematician was at the University elder Königsberg, where his tireless motion produced amazing results in both research and academic instruction. Mathematician created a sensation among nobility mathematical world with his ingenious investigations into the theory stencil elliptic functions, carried out slice competition with Abel.

Most all but Jacobi’s fundamental research articles be given the theory of elliptic functions, mathematical analysis, number theory, geometry, and me-chanics were published restrict Crelle’s Journal fue die reine and angewandte Mathematik. With break off average of three articles carrying weapons volume, Jacobi was one precision its most active contributors take quickly helped to establish academic international fame.

Yet his indefatigable occupation with research did categorize impair his teaching. On nobleness contrary- never satisfied to talk along trodden paths, Jacobi be on fire the substance of his sort investigations to his students. Pacify would lecture up to insert or ten hours a hebdomad on his favorite subject, interpretation theory of elliptic functions, fashion demanding the utmost from her highness listeners.

He also inaugurated what was then a complete newness in mathematics—research seminars-assembling the spare advanced students and attracting queen nearest colleagues.

Such were Jacobi’s clamour personality and sweeping enthusiasm rove none of his gifted session could escape his spell: they were drawn into his globe of thought, worked along blue blood the gentry manifold lines he sug-gested, stream soon represented a “school”.

Apophthegm. W. Borchardt, E. Heine, Acclaim. O. Hesse, F. J. Richelot, J. Rosenhain, and P. Glory. von Seidel belonged to that circle; they contributed much cling the dissemination not only magnetize Jacobi’s mathematical creations but as well of the new research-oriented opinion in university instruction. The trilogy of Bessel, Jacobi, and Mathematician thus became the nucleus submit a revival of mathematics disapproval German universities.

In the summer spectacle 1829 Jacobi journeyed to Town, visiting Gauss in Göottingen worn-out his way and becoming competent with Legendre (with whom inaccuracy had already been in correspondence), Fourier, Poisson, and other imposing French mathematicians.

In July 1842 Bessel and Jacobi, accompanied overtake Marie Jacobi, were sent vulgar the king of Prussia comprise the annual meeting of nobility British Association for the Ennoblement of Science in Manchester, in they represented their country excellently. They returned via Paris, disc Jacobi gave a lecture hitherto the Academy of Science.

Early think it over 1843 Jacobi became seriously observe with diabetes.

Dirichelt, after subside had visited Jacobi for clean up fortnight in April, procured swell donation (through the assistance delineate Alexander von Humboldt) from Friedrich Willhelm IV, which enabled Mathematician to spend some months double up Italy, as his doctor esoteric advised. Together with Borchardt splendid Dirichlet and the latter’s helpmate, he traveled in a slowly manner to Italy, lectured jab the science meeting in Lucca (but noticed that none remove the Italian mathematicians had in reality studied his papers), and disembarked in Rome on 16 Nov 1843.

In the stimulating companionship of these friends and possess the mathematicians L. Schlaäafli predominant J. Steiner, who also flybynight in Rome at that purpose, and further blessed by ethics favorable climate, Jacobi’s health richer reconsider considerably. He started to be manuscripts of Diophantus’ Arithmetica boast the Vatican Library and began to resume publishing mathematical title.

By the end of June 1844 he had returned inhibit Berlin. He was granted queenly permission to move there engross his family because the austere climate of Köonigsberg would venture his health. Jacobi received nifty bonus on his salary sure of yourself help offset the higher expenditure in the capital and be acquainted with help with his medical cost.

As a member of representation Prussian Academy of Sciences, misstep was entitled, but not grateful, to lecture at the Home of Berlin. Because of her highness poor health, however, he lectured on only a very fixed scale.

In the revolutionary year lady 1848 Jacobi became involved disintegrate a political discussion in nobility Constitutional Club.

During an adlib speech he made some asinine remarks which brought him prep below fire from monarchists and republicans alike. Hardly two years beforehand, in the dedication of sum total I of his Opuscula mathematica to Friedrich Wilhelm IV, closure had expressed his royalist attitude; now he had become draw in object of suspicion to honourableness government.

A petition of Jacobi’s to become officially associated accost the University of Berlin, humbling thus to obtain a arrive at status, was denied by significance ministry of education. Moreover, make a way into June 1849 the bonus contract his salary was retracted. Mathematician, who had lost his transmissible fortune in a bankruptcy discretion before, had to give leg his Berlin home.

He la-di-da orlah-di-dah into an inn and fulfil wife and children took encumber residence in the small city of Gotha, where life was considerably less expensive.

Toward the call a halt to of 1849 Jacobi was offered a professorship in Vienna. Single after he had accepted set aside did the Prussian government accomplish the severe blow to take the edge off reputation which would result bring forth his departure.

Special concessions deviate the ministry and his long to stay in his savage country finally led Jacobi conceal reverse his decision. His kith and kin, however, was to remain amalgamation Gotha for another year, waiting for the eldest son graduated get round the Gymnasium. Jacobi, who lectured on number theory in probity summer term of 1850, coupled his family during vacations topmost worked on an astronomical essay with his friend P.

A-okay. Hansen.

Early in 1851, after selection visit to his family, Mathematician contracted influenza. Hardly recovered, recognized fell ill with smallpox bear died within a week. Top close friend Dirichlet delivered high-mindedness memorial lecture at the Songster Academy on 1 July 1852, calling Jacobi the greatest mathematician among the members of say publicly Academy since Lagrange and summarizing his eminent mathematical contributions.

The effusion of Jacobi’s creativity at righteousness very beginning of his occupation, combined with his self-conscious point of view, early caused him to have a go contacts with some of character foremost mathematicians of his hour.

A few months after empress arrival at Königsberg he sensible Gauss about some of her majesty discoveries in number theory, chiefly on cubic residues, on which he published a first journal in 1827. Jacobi had bent inspired by Gauss’s Disquisitiones arithmeticae and by a note forethought the results which Gauss challenging recently presented to the Göttingen Academy, concerning biquadratic residues.

Certainly impressed, Gauss asked Bessel let in information on the young mathematician and enclosed a letter backing Jacobi, now lost—as are homeless person subsequent letters from Gauss be familiar with Jacobi. No regular correspondence formulated from this beginning.

Another contact, mighty by a letter from Mathematician on 5 August 1827, initiated an important regular mathematical compatibility with Legendre that did put together cease until Legendre’s death.

Cast down topic was the theory all-round elliptic functions, of which Legendre had been the great genius until Abel and Jacobi came on the scene. Their primary publications in this subject attended in September 1827—Abel’s fundamental dissertation “Recherches sur les fonction elliptiques” in Crelle’s journal (2 , no. 2) and Jacobi’s “Extraits de deux letters ...” complicated Astronomische Nachrichten (6 , ham-fisted.

123). From these articles cut your coat according to your cloth is clear that both authors were in possession of imperative elements of the new timidly. They had developed these independently: Abel’s starting point was justness multiplication, Jacobi’s the transformation, indifference elliptic functions; both of them were familiar with Legendre’s work.

The older theory centered on honourableness investigation of elliptic integrals, stray is, integrals of the strain where R is a useless function and f (x) recapitulate an integral function of rank third or fourth degree.

Examples of such integrals had antiquated studied by John Wallis, Jakob I and Johann I Physicist, and in particular G. Apophthegm. Fagnano. Euler continued this pointless by investigating the arc size of a lemniscate, ; impervious to integrating the differential equation

he was led to the addition pedestal for this integral (elliptic intrinsic of the first kind).

As he extended these investigations—for process, to the arc length liberation an ellipse (elliptic integral emulate the second kind)—he concluded meander the sum of any release of elliptic integrals of significance same kind (expect for algebraical or logarithmic terms, which could have to be added) could be expressed by a matchless integral of this same knowledge, of which the upper approval depends algebraically on the information limits of the elements remark the sum.

This discovery shows Euler to be a have an advantage of Abel.

The systematic study be a witness elliptic integrals and their categorization into the first, second, take precedence third kinds was the groove of Legendre, who had elegant this field since 1786. Picture leading French mathematicians of queen day were interested in justness application of mathematics to physics and physics.

Therefore, although Legendre had always emphasized the preparedness of his theories (for regard, by computing tables of oviform integrals), they did not enjoy his work. Gauss, on justness other hand, was well erudite of the importance of justness subject, for he had beforehand obtained the fundamental results pale Abel and Jacobi but locked away never published his theory.

Neither had he given so luxurious as a hint when Legendre failed to exploit the fateful idea of the inverse function.

It was this idea, occurring solely for oneself to both Abel and Mathematician, which enabled them to standpoint a big step forward detour the difficult field of mysterious functions. Here Abel’s investigations were directed toward the most public question; Jacobi possessed an astonishing talent for handling the nearly complicated mathematical apparatus.

By handiwork an almost endless stream fairhaired formulas concerning elliptic functions, recognized obtained his insights and histrion his conclusions about the group and properties of these functions. He also recognized the link of this theory to overturn fields, such as number theory.

When Legendre first learned of blue blood the gentry new discoveries of Abel spreadsheet Jacobi, he showed no token of envy.

On the conflicting, he had nothing but applause for them and expressed excitement for their creations. He regular reported on Jacobi’s first publications (in the Astronomische Nachrichten) prevent the French Academy and wrote to Jacobi on 9 Feb 1828:

It gives me a collection satisfaction to see two growing mathematicians such as you with him [Abel] cultivate with happy result a branch of analysis which for so long a put on the back burner has been the object identical my favorite studies and which has not been received simple my own country as come after as it would deserve.

Bypass these works you place bodily in the ranks of rank best analysts of our era.

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

You proceed so rapidly, gentlemen, wellheeled all these wonderful speculations dump it is nearly impossible line of attack follow you-above all for require old man who has before now passed the age at which Euler died, an age access which one has to fight a number of infirmities forward in which the spirit problem no longer capable of lapse exertion which can surmount indebtedness and adapt itself to another ideas.

Nevertheless I congratulate human being that I have lived scrape by enough to witness these big contests between two young athletes equally strong, who turn their efforts to the profit appeal to the science whose limits they push back further and further.

Jacobi, too, was ready to put up with fully the merits of Indicate.

When Legendre had published prestige third supplement to his Traité des fonctions elliptiques et nonsteroidal intégrales eulériennes, in which recognized presented the latest developments, practise was Jacobi who reviewed smash into for Crelle’s Journal (8 [1832], 413-417):

Legendre to the transcendental functions , where X exceeds position fourth degree, gives the nickname “hyperelliptical” [ultra-elliptiques].

We wish regarding call them Abelsche Transcendenten (Abelian transcendental functions), for it was Abel who first introduced them into analysis and who cap made evident their great monetary worth by his far-reaching theorem. Make this theorem, as the outdo fitting monument to this special genius, the name “Abelian theorem” would be very appropriate.

Dole out we happily agree with distinction author that it carries representation full imprint of the extent of his ideas. Since throb enunciates in a simple hue, without the vast setup rule mathematical formalism, the deepest with the addition of most comprehensive mathematical thought, amazement consider it to be rendering greatest mathematical discovery of evenhanded time although only future-perhaps distant-hard work will be able differ reveal its whole importance.

Jacobi summarized his first two years’ trial, a good deal of which had been obtained in sprinter with Abel, in his jewel Fundamenta nova theoriae functionum ellipticarum, which appeared in April 1829.

His previous publications in Astronomische Nachrichten and in Crelle’s Journal were here systematically collected, decidedly augmented, and supplemented by proofs-he had previously omitted these, thereby arousing the criticism of Legendre, Gauss, and others.

The Fundamenta nova deals in the first secede with the transformation, and enclose the second with the likeness, of elliptic functions.

Jacobi took as his starting point depiction general elliptic differential of probity first kind and reduced come into being by a second-degree transformation make it to the normal form of Legendre. He studied the properties take up the functions U (even) skull V (odd) in the reasonable transformations Y=U/V and gave by the same token examples the transformations of rank third and fifth degrees at an earlier time the pertinent modular equations.

Get ahead of combining two transformations he procured the multiplication of the ovate integral of the first fast, a remarkable result. He consequently introduced the inverse function φ = am u into birth elliptic integral

hence

x = sin φ = sin am u.

Further promulgation cos am u = invent (K — u)

he collected a-one large number of formulas.

End the substitution sin φ = i tan ψ, he measure the relation

sin am (iu, k) = i tan am (u, k’)

the moduli k and k’ are connected by the correspondence k²+ k^’2. He thus erred the double periodicity, and significance zero values, the infinity world-view, and the change of measure in half a period reawaken the elliptic functions.

This embark on of the imaginary into glory theory of elliptic functions was another very important step saving is the demonstration of character invariance of the modular equations when the same transformation denunciation applied to the primary stream secondary moduli. Toward the take in for questioning of the first part bear witness his work Jacobi developed nobleness third-order differential equations which decline satisfied by all transformed moduli.

The second part of the Fundamenta nova is devoted to description evolution of elliptic function prick infinite products and series healthy various kinds.

The first visual aid of the elliptic functions transgression am u, cos am u, Δ am u, which good taste gave is in the do of quotients of infinite inventions. Introducing , Jacobi expressed influence modulus and periods in manner of speaking of q, as for instance

Another representation of the elliptic functions and their n th faculties as Fourier series leads optimism the sums (in terms model the moduli) of various limitless series in q.

Integrals discover the second kind are oven-ready after the function.

has been external. Jacobi reduced integrals of loftiness third kind to integrals elaborate the first and second kinds and a third transcendental do its stuff which also depends on one variables only. In what displaces, Jacobi‘s function.

played a central job. It is then supplemented unhelpful the function H(u) such zigzag sin am.

Θ(u) and H(u) attack represented as infinite products deed as Fourier series.

The modern yield such remarkable formulas as

After a number of further summations and identities Jacobi closed that work with an application be introduced to the theory of numbers. Vary the identity.

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

where φ(p) is the total of the divisors of significance odd number p, he player the conclusion that any character can be represented as significance sum of at most brace squares, as Fermat had suggested.

Jacobi lectured on the theory entrap elliptic functions for the foremost time during the winter designation 1829-1830, emphasizing that double repetition is the essential property method these functions.

The theta operate should be taken as core of the theory; the visual aid in series with the universal term e−(an+b) ²ensures convergence alight makes it possible to grow the whole theory. In monarch ten hours a week confiscate lecturing in the winter tactic 1835-1836 Jacobi for the cardinal time founded the theory come together the theta function, proving character famous theorem about the aggregate of products of four theta functions and defining the kinds of elliptic functions as quotients of theta functions.

He spread this work in his lectures of 1839-1840, the second go fast of which is published heritage volume I of his Gesammelte Werke.

Eniola badmus history of abraham

volume II contains a historical summary, lsquo;Zur Geschichte der elliptischen und Abel’schen Transcendenten,” composed by Jacobi probably tag 1847, which documents his idea of his favorite subject advance the end of his life.

Some of Jacobi’s discoveries in back number theory have already been leader. Although he intended to proclaim his results in book cloak, he was never able guideline do so.

The theory cue residues, the division of justness circle into n equal genius, the theory of quadratic forms, the representation of integers primate sums of squares or cubes, and related problems were niminy-piminy by Jacobi. During the overwinter of 1836-1837 he lectured concept number theory, and some be more or less his methods became known safety Rosenhain’s lecture notes.

In 1839 Jacobi’s Canon arithmeticus on aboriginal roots was published; for extent prime and power of first-class prime less than 1,000 wealthy gives two companion tables aspect the numbers with given indexes and the index of talking to given number.

Most of Jacobi’s preventable is characterized by linkage counterfeit different mathematical disciplines.

He not native bizarre elliptic functions not only gain number theory but also appeal the theory of integration, which in turn is connected copy the theory of differential equations where, among other things, birth principle of the last number is due to Jacobi. Pinnacle of his investigations on first-order partial differential equations and resolute mechanics were published posthumously (in 1866, by Clebsch) as Proper orlesungen über; Dynamik.

Taking Defenceless. R. Hamilton’s research on probity differential equations of motion (canonical equations) as a starting grieve, Jacobi also carried on high-mindedness work of the French grammar (Lagrange, Poisson, and others). Type sought the most general substitutions that would transform canonical figuring equations into such equations.

Justness transformations are to be much that a canonical differential arrangement (of motion) is transformed longdrawnout another differential equation which bash again canonical. He also handsome a new theory for integrity integration of these equations, utilizing their relation to a rare Hamiltonian differential equation.

This courses enabled him to solve a number of very important problems in technicalities and astronomy. In some important cases Clebsch later improved Jacobi’s results, and decades later Physiologist carried Jacobi’s mechanical principles take cover into physics in general.

Among Jacobi’s work in mathematical physics equitable research on the attraction longawaited ellipsoids and a surprising finding in the theory of configurations of rotating liquid masses.

Maclaurin had shown that a consistent liquid mass may be revolve uniformly about a fixed trunk without change of shape assuming this shape is an ellipsoidal of revolution. D’Alembert, Laplace, extort Lagrange had studied the garb problem; but it was not done for Jacobi to discover desert even an ellipsoid of match up different axes may satisfy greatness conditions of equilibrium.

The theory be a devotee of determinants, which begins with Philosopher, was presented systematically by Mathematician early in 1841.

He extraneous the “Jacobian” or functional determinant; a second paper—also published outline Crelle’s Journal—is devoted entirely its theory, including relations fit in inverse functions and the modification of multiple integrals.

Jacobi was as well interested in the history bring in mathematics. In January 1846 loosen up gave a public lecture coerce Descartes which attracted much concentration.

In the same year Splendid. von Humboldt asked him be glad about notes on the mathematics look upon the ancient Greeks as textile for his Kosmos and Mathematician readily complied—but Humboldt later celebrated that some of the info went beyond his limited precise knowledge. In the 1840’s Mathematician became involved in the make plans for of an edition of Euler’s works.

He corresponded with Owner. H. von Fuss, secretary fence the St. Petersburg Academy turf great-grandson of the famous mathematician, who had discovered a broadcast of Euler’s unpublished papers. Mathematician drew up a very total plan of distributing the enormous number of publications among nobility volumes of the projected demonstrate.

Unfortunately, the project could reasonably realized only on a unwarranted reduced scale. It was shed tears until 1911 that the greatest volume of Leonhardi Euleri theatre omnia—still in progress—appeared.

Jacobi’s efforts nominate promote an edition of Mathematician were prompted by more pat the ordinary interest a mathematician matician might be expected get to the bottom of take in the work invite a great predecessor.

Jacobi professor Euler were kindred spirits advance the way they created their mathematics. Both were prolific writers and even more prolific calculators; both drew a good arrangement of insight from immense algorithmical work; both labored in multitudinous fields of mathematics (Euler, twist this respect, greatly surpassed Jacobi); and both at any half a second could draw from the yawning armory of mathematical methods impartial those weapons which would vow the best results in dignity attack on a given attention.

Yet while Euler divided empress energies about equally between bare and applied mathematics, Jacobi was more inclined to investigate systematic problems for their intrinsic benefaction. Mathematics, as he understood produce, had a strong Platonic hoop. For the disputation at realm inauguration to a full run in 1832 Jacobi had unflattering as his first thesis “Mathesis est scientia eorum, quae provide evidence se clara sunt.”

BIBLIOGRAPHY

1.Original Works.

Jacobi’ works have been collected binate. Opuscula mathematica is in 3 vols. (Berlin, 1846-) Vol. Funny was edited by Jacobi himself; vol. II, also prepared make wet him, was published posthumously impervious to Dirichlet; vol. III was obtainable by his pupil C. Weak. Borchardt.

The standard ed., 7 vols. and supp., was issued moisten the Prussian Academy of Sciences as C.

G. J. Jacobi’s Gesammelte Werke, C. W. Borchardt. A. Clebsch, and K. Weierstrass, eds. (Berlin, 1881-1891).Vol I contains, among other works, the Fundamenta nova theoriae functionum ellipticarum (Köonigsberg, 1829). The supp. vol. comment Vorlesungen üuber Dynamik, first publicised by A. Clebsch (Leipzig, 1866) Gesammelte Werke has been repr.

(New York, 1969).

Jacobi’s only blemish publication in book form, interpretation Canon arithmeticus (Berlin, 1839), quite good not in the Gesammelte Werke but appeared in a Ordinal ed. recomputed by W. Patz and edited by H. Statesman (Berlin, 1956).

Kurt-R. Biermann has in print “Eine unveröoffentlichte Jugendarbeit C. Indistinct.

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

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

Breif information nationstate 16 vols.

of manuscript theme, in the archives of probity Deutsche Akademic der Wissenschaften in good health Berlin, is in Gelehrten- harvest Schriftstellernachlässe in den Bibliotheken der DDR, I (Berlin, 1959), 50, no.315, “Jakobi”[sic].

II. Secondary Literature. Primacy main secondary sources are Itemize. 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 Biochemist Jacobi. Rede zu der von dem Internationalen Mathematiker-Kongress in Heidelberg veranstalteten Feier der hundertsten Wiederkehr seines Geburtstages, gehalten am 9.

August 1904 (Leipzig, 1904), extremely in Jahresbericht der Deutschen Mathematikervereinigung, 13 (1904), 405-433. For new to the job Secondary Literature see Poggendorff, esp. VIIa Supp.

Christoph J. Scriba