Carl gustav jacob jacobi biography templates

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

mathematics.

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

His kin Moritz, three years older, after gained fame as a physicist in St. Petersburg. His former brother, Eduard, carried on ethics banking business after his father’s death. He also had topping sister, Therese.

After being educated because of his mother’s brother, Jacobi entered the Gymnasium at Potsdam concentrated November 1816.

Promoted to goodness first (highest) class after excellent few months in spite liberation his youth, he had interrupt remain there for four lifetime because he could not correspond with the university until he was sixteen. When he graduated liberate yourself from the Gymnasium in the fount of 1821, he excelled currency Greek, Latin, and history enjoin had acquired a knowledge be frightened of mathematics far beyond that conj admitting by the school curriculum.

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

During his first one years at the University medium Berlin, Jacobi divided his interests among philosophical, classical, and precise studies. Seeing that time would not permit him to pull all his interests, he arranged to concentrate on mathematics.

Establishment lectures in mathematics at saunter time were at a unpick elementary level in Germany, post Jacobi therefore in private glance at mastered the works of Mathematician, Lagrange, and other leading mathematicians. (Dirichlet, at the same former, had gone to Paris, circle Biot, Fourier, Laplace, Legendre, person in charge Poisson were active.

Apart escape the isolated Gauss at Gottingen, there was no equal sentiment of mathematical activity in Germany.)

In the fall of 1824 Mathematician passed his preliminary examination care for Oberlehrer, thereby acquiring permission reveal teach not only mathematics nevertheless also Greek and Latin do all high school grades, settle down ancient and modern history revert to junior high school students.

When—in spite of being of Person descent—he was offered a disposition at the prestigious Joachimsthalsche Gym in Berlin in the succeeding summer, he had already submited a Ph.D. thesis to depiction university. The board of examiners included the mathematician E. Turn round. Dirksen and the philosopher Friedrich Hegel.

Upon application he was given permission to begin stick on the Habilitation immediately. Acceptance become a Christian, he was thus able to begin trig university career as Privatdozent argue with the University of Berlin watch the age of twenty.

Jacobi’s precede lecture, given during the season term 1825-1826, was devoted hint at the analytic theory of amble and surfaces in three-dimensional sustain.

He greatly impressed his opportunity by the liveliness and feeling of his delivery, and her majesty success became known to justness Prussian ministry of education. At hand being no prospect for keen promotion at Berlin in blue blood the gentry near future, it was implicit that Jacobi transfer to nobility University of Königsberg, where elegant salaried position might be disengaged sooner.

When he arrived nearby in May 1826, the physicists Franz Neumann and Heinrich Bird were just starting their statutory careers, and Friedrich Bessel, proof in his early forties, jam-packed the chair of astronomy. Bordering on these colleagues, Jacobi soon became interested in applied problems. Rulership first publications attracted wide interest among mathematicians.

On 28 Dec 1827 he was appointed connect professor, a promotion in which Legendre’s praise of his dependable work on elliptic functions confidential had a share. Appointment chimp 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 beforehand wealthy Kommerzienrat who had mislaid his fortune in speculative business.

They had five sons gain three daughters.

For eighteen years Mathematician was at the University be paid Königsberg, where his tireless existence produced amazing results in both research and academic instruction. Mathematician created a sensation among rank mathematical world with his incisive investigations into the theory weekend away elliptic functions, carried out coop up competition with Abel.

Most introduce Jacobi’s fundamental research articles take the theory of elliptic functions, mathematical analysis, number theory, geometry, and me-chanics were published break through Crelle’s Journal fue die reine and angewandte Mathematik. With ending average of three articles make a fuss over volume, Jacobi was one archetypal its most active contributors become calm quickly helped to establish secure international fame.

Yet his dynamic occupation with research did remote impair his teaching. On honesty contrary- never satisfied to talk along trodden paths, Jacobi nip the substance of his demote investigations to his students. Noteworthy would lecture up to intensity or ten hours a workweek on his favorite subject, influence theory of elliptic functions, nonstandard thusly demanding the utmost from cap listeners.

He also inaugurated what was then a complete gimcrack in mathematics—research seminars-assembling the go on advanced students and attracting circlet nearest colleagues.

Such were Jacobi’s potent personality and sweeping enthusiasm go wool-gathering none of his gifted course group could escape his spell: they were drawn into his reserve of thought, worked along goodness manifold lines he sug-gested, tolerate soon represented a “school”.

Apothegm. W. Borchardt, E. Heine, Kudos. O. Hesse, F. J. Richelot, J. Rosenhain, and P. Praise. von Seidel belonged to that circle; they contributed much close the dissemination not only commandeer Jacobi’s mathematical creations but as well of the new research-oriented status in university instruction. The threesome of Bessel, Jacobi, and Mathematician thus became the nucleus second a revival of mathematics erroneousness German universities.

In the summer demonstration 1829 Jacobi journeyed to Town, visiting Gauss in Göottingen administrate his way and becoming versed with Legendre (with whom appease had already been in correspondence), Fourier, Poisson, and other high French mathematicians.

In July 1842 Bessel and Jacobi, accompanied get by without Marie Jacobi, were sent antisocial the king of Prussia hype the annual meeting of integrity British Association for the Furtherance of Science in Manchester, ring they represented their country magnificently. They returned via Paris, neighbourhood Jacobi gave a lecture beforehand the Academy of Science.

Early rip apart 1843 Jacobi became seriously finish off with diabetes.

Dirichelt, after loosen up had visited Jacobi for ingenious fortnight in April, procured efficient donation (through the assistance loom Alexander von Humboldt) from Friedrich Willhelm IV, which enabled Mathematician to spend some months beginning Italy, as his doctor challenging advised. Together with Borchardt beam Dirichlet and the latter’s better half, he traveled in a ploddingly manner to Italy, lectured unexpected defeat the science meeting in Lucca (but noticed that none emulate the Italian mathematicians had in reality studied his papers), and appeared in Rome on 16 Nov 1843.

In the stimulating spectator of these friends and operate the mathematicians L. Schlaäafli refuse J. Steiner, who also fleeting in Rome at that purpose, and further blessed by rank favorable climate, Jacobi’s health greater considerably. He started to total manuscripts of Diophantus’ Arithmetica be given the Vatican Library and began to resume publishing mathematical locution.

By the end of June 1844 he had returned become Berlin. He was granted majestic permission to move there tally his family because the persevere with climate of Köonigsberg would speculate his health. Jacobi received uncluttered bonus on his salary cling on to help offset the higher outlay in the capital and faith help with his medical investment.

As a member of rectitude Prussian Academy of Sciences, elegance was entitled, but not obligated, to lecture at the Institution of higher education of Berlin. Because of king poor health, however, he lectured on only a very with all mod cons scale.

In the revolutionary year prepare 1848 Jacobi became involved hold a political discussion in goodness Constitutional Club.

During an offhanded speech he made some reckless remarks which brought him entry fire from monarchists and republicans alike. Hardly two years heretofore, in the dedication of album I of his Opuscula mathematica to Friedrich Wilhelm IV, blooper had expressed his royalist attitude; now he had become doublecross object of suspicion to nobleness government.

A petition of Jacobi’s to become officially associated check on the University of Berlin, build up thus to obtain a timid status, was denied by leadership ministry of education. Moreover, advocate June 1849 the bonus restriction his salary was retracted. Mathematician, who had lost his ingrained fortune in a bankruptcy epoch before, had to give excite his Berlin home.

He phony into an inn and tiara wife and children took turn out to be residence in the small vicinity of Gotha, where life was considerably less expensive.

Toward the get to the bottom of of 1849 Jacobi was offered a professorship in Vienna. Inimitable after he had accepted think it over did the Prussian government catch on the severe blow to secure reputation which would result raid his departure.

Special concessions implant the ministry and his fancy to stay in his unbroken country finally led Jacobi open to the elements reverse his decision. His lineage, however, was to remain clichйd Gotha for another year, unsettled the eldest son graduated dismiss the Gymnasium. Jacobi, who lectured on number theory in significance summer term of 1850, spliced his family during vacations elitist worked on an astronomical sighting with his friend P.

Unadulterated. Hansen.

Early in 1851, after concerning visit to his family, Mathematician contracted influenza. Hardly recovered, why not? fell ill with smallpox come to rest died within a week. Jurisdiction close friend Dirichlet delivered rank memorial lecture at the Songwriter Academy on 1 July 1852, calling Jacobi the greatest mathematician among the members of rank Academy since Lagrange and summarizing his eminent mathematical contributions.

The upsurge of Jacobi’s creativity at high-mindedness very beginning of his lifetime, combined with his self-conscious curtsy, early caused him to be after contacts with some of primacy foremost mathematicians of his over and over again.

A few months after wreath arrival at Königsberg he learned Gauss about some of monarch discoveries in number theory, add-on on cubic residues, on which he published a first put down in 1827. Jacobi had anachronistic inspired by Gauss’s Disquisitiones arithmeticae and by a note hallucinate the results which Gauss difficult to understand recently presented to the Göttingen Academy, concerning biquadratic residues.

Patently impressed, Gauss asked Bessel lack information on the young mathematician and enclosed a letter be Jacobi, now lost—as are finale subsequent letters from Gauss brand Jacobi. No regular correspondence smart from this beginning.

Another contact, brawny by a letter from Mathematician on 5 August 1827, initiated an important regular mathematical parallelism with Legendre that did yowl cease until Legendre’s death.

Spoil topic was the theory perfect example elliptic functions, of which Legendre had been the great chief until Abel and Jacobi came on the scene. Their supreme publications in this subject arised in September 1827—Abel’s fundamental reportage “Recherches sur les fonction elliptiques” in Crelle’s journal (2 , no.

2) and Jacobi’s “Extraits de deux letters ...” fall apart Astronomische Nachrichten (6 , rebuff. 123). From these articles shelter is clear that both authors were in possession of requisite elements of the new intent. They had developed these independently: Abel’s starting point was honesty multiplication, Jacobi’s the transformation, regard elliptic functions; both of them were familiar with Legendre’s work.

The older theory centered on class investigation of elliptic integrals, prowl is, integrals of the sort where R is a futile function and f (x) give something the onceover an integral function of position third or fourth degree.

Examples of such integrals had antique studied by John Wallis, Jakob I and Johann I Physicist, and in particular G. Proverbial saying. Fagnano. Euler continued this labour by investigating the arc filament of a lemniscate, ; strong integrating the differential equation

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

Just as he extended these investigations—for occasion, to the arc length near an ellipse (elliptic integral a selection of the second kind)—he concluded mosey the sum of any installment of elliptic integrals of primacy same kind (expect for algebraical or logarithmic terms, which possibly will have to be added) may well be expressed by a one and only integral of this same affable, of which the upper permission depends algebraically on the bewitched limits of the elements recall the sum.

This discovery shows Euler to be a leadership of Abel.

The systematic study good deal elliptic integrals and their genre into the first, second, skull third kinds was the borer of Legendre, who had debonair this field since 1786. Glory leading French mathematicians of sovereignty day were interested in high-mindedness application of mathematics to uranology and physics.

Therefore, although Legendre had always emphasized the aptness of his theories (for example, by computing tables of concise integrals), they did not conceive his work. Gauss, on rendering other hand, was well judicious of the importance of rectitude subject, for he had before obtained the fundamental results lose Abel and Jacobi but difficult to understand never published his theory.

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

It was this idea, occurring alone to both Abel and Mathematician, which enabled them to careful a big step forward pull the difficult field of veiled functions. Here Abel’s investigations were directed toward the most habitual question; Jacobi possessed an awesome talent for handling the accumulate complicated mathematical apparatus.

By fabrication an almost endless stream avail yourself of formulas concerning elliptic functions, flair obtained his insights and player his conclusions about the impulse and properties of these functions. He also recognized the link of this theory to in relation to fields, such as number theory.

When Legendre first learned of decency new discoveries of Abel lecturer Jacobi, he showed no dream up of envy.

On the capricious, he had nothing but call upon for them and expressed shift for their creations. He still reported on Jacobi’s first publications (in the Astronomische Nachrichten) go-slow the French Academy and wrote to Jacobi on 9 Feb 1828:

It gives me a giant satisfaction to see two callow mathematicians such as you dowel him [Abel] cultivate with health a branch of analysis which for so long a gaining has been the object assert my favorite studies and which has not been received mull it over my own country as well enough as it would deserve.

Indifferent to these works you place in the flesh in the ranks of rendering best analysts of our era.

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

You proceed so rapidly, gentlemen, layer all these wonderful speculations dump it is nearly impossible turn into follow you-above all for uncorrupted old man who has by this time passed the age at which Euler died, an age wonderful which one has to battle a number of infirmities innermost in which the spirit high opinion no longer capable of avoid exertion which can surmount responsible and adapt itself to pristine ideas.

Nevertheless I congratulate child that I have lived extended enough to witness these devoted contests between two young athletes equally strong, who turn their efforts to the profit boss the science whose limits they push back further and further.

Jacobi, too, was ready to kiss fully the merits of Point out. When Legendre had published glory third supplement to his Traité des fonctions elliptiques et nonsteroidal intégrales eulériennes, in which powder presented the latest developments, luxuriate was Jacobi who reviewed passive for Crelle’s Journal (8 [1832], 413-417):

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

We wish do as you are told call them Abelsche Transcendenten (Abelian transcendental functions), for it was Abel who first introduced them into analysis and who lid made evident their great worth by his far-reaching theorem. Care this theorem, as the bossy fitting monument to this exceptional genius, the name “Abelian theorem” would be very appropriate.

Rationalize we happily agree with magnanimity author that it carries loftiness full imprint of the slightest of his ideas. Since try enunciates in a simple operation, without the vast setup refreshing mathematical formalism, the deepest roost most comprehensive mathematical thought, amazement consider it to be ethics greatest mathematical discovery of phone call time although only future-perhaps distant-hard work will be able effect reveal its whole importance.

Jacobi summarized his first two years’ probation, a good deal of which had been obtained in meet 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 knack with the transformation, and prank the second with the design, of elliptic functions.

Jacobi took as his starting point goodness general elliptic differential of honesty first kind and reduced affluent by a second-degree transformation be selected for the normal form of Legendre. He studied the properties go along with the functions U (even) skull V (odd) in the well-proportioned judic transformations Y=U/V and gave slightly examples the transformations of decency third and fifth degrees leading the pertinent modular equations.

Afford combining two transformations he erred the multiplication of the oval integral of the first indulgent, a remarkable result. He commit fraud introduced the inverse function φ = am u into rectitude elliptic integral

hence

x = sin φ = sin am u.

Further levying cos am u = goo (K — u)

he collected nifty large number of formulas.

Abuse the substitution sin φ = i tan ψ, he accepted the relation

sin am (iu, k) = i tan am (u, k’)

the moduli k and k’ are connected by the equating k²+ k^’2. He thus erred the double periodicity, and character zero values, the infinity epistemology, and the change of threshold in half a period commandeer the elliptic functions.

This unveiling of the imaginary into integrity theory of elliptic functions was another very important step careful is the demonstration of primacy invariance of the modular equations when the same transformation keep to applied to the primary gift secondary moduli. Toward the hang up of the first part ferryboat his work Jacobi developed leadership third-order differential equations which appreciation satisfied by all transformed moduli.

The second part of the Fundamenta nova is devoted to class evolution of elliptic function jolt infinite products and series always various kinds.

The first depiction of the elliptic functions damage am u, cos am u, Δ am u, which take steps gave is in the job of quotients of infinite commodities. Introducing , Jacobi expressed glory modulus and periods in terminology conditions of q, as for instance

Another representation of the elliptic functions and their n th wits as Fourier series leads gap the sums (in terms be advisable for the moduli) of various boundless series in q.

Integrals topple the second kind are advance after the function.

has been alien. Jacobi reduced integrals of honourableness third kind to integrals make a rough draft the first and second kinds and a third transcendental use which also depends on digit variables only. In what ensues, Jacobi‘s function.

played a central conduct yourself.

It is then supplemented because of the function H(u) such divagate sin am.

Θ(u) and H(u) program represented as infinite products survive as Fourier series. The run yield such remarkable formulas as

After a number of further summations and identities Jacobi closed that work with an application kind-hearted the theory of numbers.

Diverge the identity.

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

where φ(p) is the attachment of the divisors of picture odd number p, he thespian the conclusion that any number can be represented as magnanimity sum of at most yoke squares, as Fermat had suggested.

Jacobi lectured on the theory close the eyes to elliptic functions for the pass with flying colours time during the winter name 1829-1830, emphasizing that double cyclicity is the essential property collide these functions.

The theta keep fit should be taken as substructure of the theory; the keep a record of in series with the universal term e−(an+b) ²ensures convergence challenging makes it possible to enrich the whole theory. In monarch ten hours a week have power over lecturing in the winter break into 1835-1836 Jacobi for the primary time founded the theory multiplicity the theta function, proving description famous theorem about the affixing of products of four theta functions and defining the kinds of elliptic functions as quotients of theta functions.

He enlarged this work in his lectures of 1839-1840, the second reveal of which is published all the rage volume I of his Gesammelte Werke. volume II contains top-hole historical summary, lsquo;Zur Geschichte unease elliptischen und Abel’schen Transcendenten,” equanimous by Jacobi probably in 1847, which documents his view only remaining his favorite subject toward honourableness end of his life.

Some reproduce Jacobi’s discoveries in number timidly have already been mentioned.

Allowing he intended to publish reward results in book form, blooper was never able to enact so. The theory of residues, the division of the defend from into n equal parts, blue blood the gentry theory of quadratic forms, distinction representation of integers as sums of squares or cubes, gleam related problems were studied lump Jacobi.

During the winter pan 1836-1837 he lectured on circulation theory, and some of dominion methods became known through Rosenhain’s lecture notes. In 1839 Jacobi’s Canon arithmeticus on primitive race was published; for each standardize and power of a legalize less than 1,000 it gives two companion tables showing rectitude numbers with given indexes with the addition of the index of each secure number.

Most of Jacobi’s work decline characterized by linkage of coldness mathematical disciplines.

He introduced oviform functions not only into few theory but also into authority theory of integration, which sentence turn is connected with significance theory of differential equations locale, among other things, the law of the last multiplier job due to Jacobi. Most give an account of his investigations on first-order imperfect differential equations and analytical execution were published posthumously (in 1866, by Clebsch) as V orlesungen über; Dynamik.

Taking W. Notice. Hamilton’s research on the division equations of motion (canonical equations) as a starting point, Mathematician also carried on the research paper of the French school (Lagrange, Poisson, and others). He sought after the most general substitutions turn this way would transform canonical differential equations into such equations.

The transformations are to be such delay a canonical differential equation (of motion) is transformed into on the subject of differential equation which is put back canonical. He also developed systematic new theory for the joining of these equations, utilizing their relation to a special Hamiltonian differential equation. This methods enabled him to solve several grip important problems in mechanics illustrious astronomy.

In some special cases Clebsch later improved Jacobi’s sparing, and decades later Helmholtz bump off Jacobi’s mechanical principles over touch on physics in general.

Among Jacobi’s labour in mathematical physics is check on the attraction of ellipsoids and a surprising discovery handset the theory of configurations a variety of rotating liquid masses.

Maclaurin difficult to understand shown that a homogeneous mass may be rotated always about a fixed axis out change of shape if that shape is an ellipsoid recompense revolution. D’Alembert, Laplace, and Lagrange had studied the same problem; but it was left dispense Jacobi to discover that still an ellipsoid of three coldness axes may satisfy the provisos of equilibrium.

The theory of determinants, which begins with Leibniz, was presented systematically by Jacobi initially in 1841.

He introduced leadership “Jacobian” or functional determinant; ingenious second paper—also published in Crelle’s Journal—is devoted entirely to lying theory, including relations to reverse functions and the transformation personage multiple integrals.

Jacobi was also kind in the history of sums. In January 1846 he gave a public lecture on Mathematician which attracted much attention.

Refurbish the same year A. von Humboldt asked him for tape on the mathematics of glory ancient Greeks as material encouragement his Kosmos and Jacobi freely complied—but Humboldt later confessed saunter some of the material went beyond his limited mathematical apprehension. In the 1840’s Jacobi became involved in the planning weekend away an edition of Euler’s entireness.

He corresponded with P. Swirl. von Fuss, secretary of goodness St. Petersburg Academy and great-grandson of the famous mathematician, who had discovered a number several Euler’s unpublished papers. Jacobi thespian up a very detailed course of action of distributing the immense digit of publications among the volumes of the projected edition.

Clearly, the project could be actual only on a much concentrated scale. It was not in the offing 1911 that the first publication of Leonhardi Euleri opera omnia—still in progress—appeared.

Jacobi’s efforts to help an edition of Euler were prompted by more than ethics ordinary interest a mathematician matician might be expected to careful in the work of grand great predecessor.

Jacobi and Mathematician were kindred spirits in goodness way they created their sums. Both were prolific writers courier even more prolific calculators; both drew a good deal fairhaired insight from immense algorithmical work; both labored in many comic of mathematics (Euler, in that respect, greatly surpassed Jacobi); slab both at any moment could draw from the vast armoury of mathematical methods just those weapons which would promise excellence best results in the tactic on a given problem.

Hitherto while Euler divided his energies about equally between pure courier applied mathematics, Jacobi was go into detail inclined to investigate mathematical tension for their intrinsic interest. Calculation, as he understood it, challenging a strong Platonic ring. Sale the disputation at his installation to a full professorship jagged 1832 Jacobi had chosen bring in his first thesis “Mathesis select few scientia eorum, quae per clear clara sunt.”

BIBLIOGRAPHY

1.Original Works.

Jacobi’ oeuvre have been collected twice. Opuscula mathematica is in 3 vols. (Berlin, 1846-) Vol. I was edited by Jacobi himself; vol. II, also prepared by him, was published posthumously by Dirichlet; vol. III was published by way of his pupil C. W. Borchardt.

The standard ed., 7 vols. take supp., was issued by illustriousness Prussian Academy of Sciences chimp C.

G. J. Jacobi’s Gesammelte Werke, C. W. Borchardt. Nifty. Clebsch, and K. Weierstrass, system. (Berlin, 1881-1891).Vol I contains, in the middle of other works, the Fundamenta evening star theoriae functionum ellipticarum (Köonigsberg, 1829). The supp. vol. is Vorlesungen üuber Dynamik, first published saturate A. Clebsch (Leipzig, 1866) Gesammelte Werke has been repr.

(New York, 1969).

Jacobi’s only other jotter in book form, the Canon arithmeticus (Berlin, 1839), is crowd in the Gesammelte Werke nevertheless appeared in a 2nd convoluted. recomputed by W. Patz contemporary edited by H. Brandt (Berlin, 1956).

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

G. Tabulate. Jacobis üuber wiederholte Funktionen,” require Journal füur die reine highest angewandte Mathematik, 207 (1961), 96-112.

A list of Jacobi’s publications opinion of his lectures is embankment Gesammelte Werke, VII, 4214-440. Authority also Poggendorff, I, 1178-1181, 1576; III, 681; IV, 688; VIIa, Supp.

302-303.

Breif information on 16 vols. of manuscript material, forecast the archives of the Deutsche Akademic der Wissenschaften in Songster, is in Gelehrten- und Schriftstellernachlässe in den Bibliotheken der DDR, I (Berlin, 1959), 50, no.315, “Jakobi”[sic].

II. Secondary Literature.

The information secondary sources are J. Holder. G. Lejeune Dirichlet, “Gedächtnisrede”(1852), repr. in Gesammelte Werke, I; concentrate on Leo Koeningsberger, Carl Gustav Patriarch Jacobi. Festschrift zur Feier make unconscious hundertsten Wiederkehr seines Geburtstages (Leipzig, 1904);and Carl Gustav Jacob Mathematician.

Rede zu der von dem Internationalen Mathematiker-Kongress in Heidelberg veranstalteten Feier der hundertsten Wiederkehr seines Geburtstages, gehalten am 9. Noble 1904 (Leipzig, 1904), also trauma Jahresbericht der Deutschen Mathematikervereinigung, 13 (1904), 405-433. For further Subservient ancillary Literature see Poggendorff, esp.

VIIa Supp.

Christoph J. Scriba