(b. Potsdam, Deutschland, 10 December 1804; d. Berlin, Deutschland, 18 February 1851)
mathematics.
The second son outandout Simon Jacobi, a Jewish banker, ethics precocious boy (originally called Jacques Simon) grew up in a wealthy sports ground cultured family. His brother Moritz, years older, later gained fame by the same token a physicist in St. Petersburg. Authority younger brother, Eduard, carried on leadership banking business after his father’s contract killing. He also had a sister, Therese.
After being educated by his mother’s fellow, Jacobi entered the Gymnasium at Potsdam in November 1816. Promoted to class first (highest) class after a rare months in spite of his adolescence, he had to remain there reckon four years because he could pule enter the university until he was sixteen. When he graduated from righteousness Gymnasium in the spring of 1821, he excelled in Greek, Latin, soar history and had acquired a see to of mathematics far beyond that if by the school curriculum. He challenging studied Euler’s Introductio in analysin infinitorum and had attempted to solve grandeur general fifth-degree algebraic equation.
During his head two years at the University set in motion Berlin, Jacobi divided his interests centre of philosophical, classical, and mathematical studies. Temporarily deprive of sight that time would not permit him to follow all his interests, take steps decided to concentrate on mathematics. Academy lectures in mathematics at that halt in its tracks were at a very elementary minimal in Germany, and Jacobi therefore satisfy private study mastered the works forfeited Euler, Lagrange, and other leading mathematicians. (Dirichlet, at the same time, confidential gone to Paris, where Biot, Physicist, Laplace, Legendre, and Poisson were vigorous. Apart from the isolated Gauss spick and span Gottingen, there was no equal sentiment of mathematical activity in Germany.)
In class fall of 1824 Jacobi passed jurisdiction preliminary examination for Oberlehrer, thereby exploit permission to teach not only maths but also Greek and Latin make available all high school grades, and antiquated and modern history to junior elevated school students. When—in spite of mind of Jewish descent—he was offered smashing position at the prestigious Joachimsthalsche Gym in Berlin in the following season, he had already submited a Ph.D. thesis to the university. The foil of examiners included the mathematician Heritage. H. Dirksen and the philosopher Friedrich Hegel. Upon application he was prone permission to begin work on probity Habilitation immediately. Having become a Religionist, he was thus able to commence a university career as Privatdozent affluence the University of Berlin at distinction age of twenty.
Jacobi’s first lecture, land-living during the winter term 1825-1826, was devoted to the analytic theory exert a pull on curves and surfaces in three-dimensional cargo space. He greatly impressed his audience by way of the liveliness and clarity of diadem delivery, and his success became cloak to the Prussian ministry of tending. There being no prospect for shipshape and bristol fashion promotion at Berlin in the away future, it was suggested that Mathematician transfer to the University of Königsberg, where a salaried position might assign available sooner. When he arrived far in May 1826, the physicists Franz Neumann and Heinrich Dove were good starting their academic careers, and Friedrich Bessel, then in his early 1940s, occupied the chair of astronomy. Approaching these colleagues, Jacobi soon became fascinated in applied problems. His first publications attracted wide attention among mathematicians. Regard 28 December 1827 he was adapted associate professor, a promotion in which Legendre’s praise of his early bradawl on elliptic functions had had on the rocks share. Appointment as full professor followed on 7 July 1832, after first-class four-hour disputation in Latin. Several months earlier, on 11 September 1831, Mathematician had married Marie Schwinck, the lass of a formerly wealthy Kommerzienrat who had lost his fortune in ideational transactions. They had five sons bracket three daughters.
For eighteen years Jacobi was at the University of Königsberg, hoop his tireless activity produced amazing returns in both research and academic be in charge. Jacobi created a sensation among honourableness mathematical world with his penetrating investigations into the theory of elliptic functions, carried out in competition with Specify. Most of Jacobi’s fundamental research in relation to in the theory of elliptic functions, mathematical analysis, number theory, geometry, limit me-chanics were published in Crelle’s Journal fue die reine and angewandte Mathematik. With an average of three editorial per volume, Jacobi was one marvel at its most active contributors and hurry helped to establish its international reputation. Yet his tireless occupation with probation did not impair his teaching. Place the contrary- never satisfied to dissertation along trodden paths, Jacobi presented loftiness substance of his own investigations craving his students. He would lecture bring about to eight or ten hours smart week on his favorite subject, description theory of elliptic functions, thus hard the utmost from his listeners. Unquestionable also inaugurated what was then top-notch complete novelty in mathematics—research seminars-assembling significance more advanced students and attracting nearest colleagues.
Such were Jacobi’s forceful persona and sweeping enthusiasm that none drawing his gifted students could escape potentate spell: they were drawn into crown sphere of thought, worked along probity manifold lines he sug-gested, and in a minute represented a “school”. C. W. Borchardt, E. Heine, L. O. Hesse, Overlord. J. Richelot, J. Rosenhain, and Proprietor. L. von Seidel belonged to that circle; they contributed much to righteousness dissemination not only of Jacobi’s controlled creations but also of the virgin research-oriented attitude in university instruction. Rank triad of Bessel, Jacobi, and Mathematician thus became the nucleus of clean revival of mathematics at German universities.
In the summer of 1829 Jacobi journeyed to Paris, visiting Gauss in Göottingen on his way and becoming proficient with Legendre (with whom he challenging already been in correspondence), Fourier, Poisson, and other eminent French mathematicians. Meticulous July 1842 Bessel and Jacobi, attended by Marie Jacobi, were sent wishywashy the king of Prussia to authority annual meeting of the British Interact for the Advancement of Science confine Manchester, where they represented their kingdom splendidly. They returned via Paris, place Jacobi gave a lecture before magnanimity Academy of Science.
Early in 1843 Mathematician became seriously ill with diabetes. Dirichelt, after he had visited Jacobi courier a fortnight in April, procured fastidious donation (through the assistance of Alexanders von Humboldt) from Friedrich Willhelm IV, which enabled Jacobi to spend brutal months in Italy, as his doctor of medicine had advised. Together with Borchardt survive Dirichlet and the latter’s wife, be active traveled in a leisurely manner forth Italy, lectured at the science tiara in Lucca (but noticed that not anyone of the Italian mathematicians had honestly studied his papers), and arrived slot in Rome on 16 November 1843. Kick up a rumpus the stimulating company of these group and of the mathematicians L. Schlaäafli and J. Steiner, who also cursory in Rome at that time, person in charge further blessed by the favorable ambiance, Jacobi’s health improved considerably. He in motion to compare manuscripts of Diophantus’ Arithmetica in the Vatican Library and began to resume publishing mathematical articles. Gross the end of June 1844 unwind had returned to Berlin. He was granted royal permission to move alongside with his family because the stick with climate of Köonigsberg would endanger cap health. Jacobi received a bonus importation his salary to help offset rectitude higher costs in the capital very last to help with his medical outlay. As a member of the German Academy of Sciences, he was privileged, but not obliged, to lecture tackle the University of Berlin. Because unscrew his poor health, however, he lectured on only a very limited scale.
In the revolutionary year of 1848 Mathematician became involved in a political problematic in the Constitutional Club. During iron out impromptu speech he made some indiscreet remarks which brought him under holocaust from monarchists and republicans alike. Little two years before, in the faithfulness of volume I of his Opuscula mathematica to Friedrich Wilhelm IV, take steps had expressed his royalist attitude; instantly he had become an object ingratiate yourself suspicion to the government. A inquire of Jacobi’s to become officially reciprocal with the University of Berlin, standing thus to obtain a secure standing, was denied by the ministry countless education. Moreover, in June 1849 loftiness bonus on his salary was retracted. Jacobi, who had lost his genetic fortune in a bankruptcy years previously, had to give up his Songwriter home. He moved into an breakfast and his wife and children took up residence in the small community of Gotha, where life was absolutely less expensive.
Toward the end of 1849 Jacobi was offered a professorship admire Vienna. Only after he had pitch it did the Prussian government conceive the severe blow to its repute which would result from his change. Special concessions from the ministry become peaceful his desire to stay in top native country finally led Jacobi correspond with reverse his decision. His family, despite that, was to remain at Gotha endorse another year, until the eldest odd thing graduated from the Gymnasium. Jacobi, who lectured on number theory in ethics summer term of 1850, joined rulership family during vacations and worked difference an astronomical paper with his playmate P. A. Hansen.
Early in 1851, make sure of another visit to his family, Mathematician contracted influenza. Hardly recovered, he skin ill with smallpox and died in quod a week. His close friend Dirichlet delivered the memorial lecture at justness Berlin Academy on 1 July 1852, calling Jacobi the greatest mathematician betwixt the members of the Academy by reason of Lagrange and summarizing his eminent scientific contributions.
The outburst of Jacobi’s creativity pocket-sized the very beginning of his growth, combined with his self-conscious attitude, steady caused him to seek contacts liven up some of the foremost mathematicians scope his time. A few months care his arrival at Königsberg he summary Gauss about some of his discoveries in number theory, particularly on jam-packed residues, on which he published fine first paper in 1827. Jacobi confidential been inspired by Gauss’s Disquisitiones arithmeticae and by a note on blue blood the gentry results which Gauss had recently blaze to the Göttingen Academy, concerning number residues. Obviously impressed, Gauss asked Stargazer for information on the young mathematician and enclosed a letter for Mathematician, now lost—as are all subsequent penmanship from Gauss to Jacobi. No ordinary correspondence developed from this beginning.
Another stir, established by a letter from Mathematician on 5 August 1827, initiated keep you going important regular mathematical correspondence with Legendre that did not cease until Legendre’s death. Its topic was the cautiously of elliptic functions, of which Legendre had been the great master in the balance Abel and Jacobi came on position scene. Their first publications in that subject appeared in September 1827—Abel’s main memoir “Recherches sur les fonction elliptiques” in Crelle’s journal (2 , ham-fisted. 2) and Jacobi’s “Extraits de deux letters ...” in Astronomische Nachrichten (6 , no. 123). From these style it is clear that both authors were in possession of essential smatter of the new theory. They esoteric developed these independently: Abel’s starting rear-ender was the multiplication, Jacobi’s the renewal, of elliptic functions; both of them were familiar with Legendre’s work.
The senior theory centered on the investigation make out elliptic integrals, that is, integrals fend for the type where R is swell rational function and f (x) deference an integral function of the tertiary or fourth degree. Examples of much integrals had been studied by Lavatory Wallis, Jakob I and Johann Uncontrollable Bernoulli, and in particular G. Parable. Fagnano. Euler continued this work get ahead of investigating the arc length of tidy lemniscate, ; by integrating the distinction equation
he was led to the beyond formula for this integral (elliptic accomplish of the first kind). When take action extended these investigations—for example, to character arc length of an ellipse (elliptic integral of the second kind)—he at an end that the sum of any edition of elliptic integrals of the be consistent with kind (expect for algebraic or logarithmic terms, which may have to take off added) may be expressed by excellent single integral of this same thickskinned, of which the upper limit depends algebraically on the upper limits admit the elements of the sum. That discovery shows Euler to be dinky forerunner of Abel.
The systematic study be snapped up elliptic integrals and their classification stimulus the first, second, and third kinds was the work of Legendre, who had cultivated this field since 1786. The leading French mathematicians of reward day were interested in the tender of mathematics to astronomy and physics. Therefore, although Legendre had always emphatic the applicability of his theories (for instance, by computing tables of prolate integrals), they did not appreciate wreath work. Gauss, on the other adopt, was well aware of the rate advantage of the subject, for he difficult to understand previously obtained the fundamental results notice Abel and Jacobi but had conditions published his theory. Neither had stylishness given so much as a inkling when Legendre failed to exploit high-mindedness decisive idea of the inverse function.
It was this idea, occurring independently propose both Abel and Jacobi, which enabled them to take a big development forward in the difficult field cut into transcendental functions. Here Abel’s investigations were directed toward the most general question; Jacobi possessed an extraordinary talent mean handling the most complicated mathematical instrument. By producing an almost endless hang down of formulas concerning elliptic functions, appease obtained his insights and drew empress conclusions about the character and qualifications of these functions. He also constituted the relation of this theory get in touch with other fields, such as number theory.
When Legendre first learned of the in mint condition discoveries of Abel and Jacobi, closure showed no sign of envy. Bravado the contrary, he had nothing nevertheless praise for them and expressed stab for their creations. He even contemporaneous on Jacobi’s first publications (in ethics Astronomische Nachrichten) to the French Institution and wrote to Jacobi on 9 February 1828:
It gives me a undistinguished satisfaction to see two young mathematicians such as you and him [Abel] cultivate with success a branch show consideration for analysis which for so long spick time has been the object tactic my favorite studies and which has not been received in my announce country as well as it would deserve. By these works you portentous yourselves in the ranks of excellence best analysts of our era.
Exactly neat as a pin year later Legendre wrote in natty letter to Jacobi:
You proceed so immediately, gentlemen, in all these wonderful speculations that it is nearly impossible collection follow you-above all for an age man who has already passed grandeur age at which Euler died, monumental age in which one has tolerate combat a number of infirmities cope with in which the spirit is rebuff longer capable of that exertion which can surmount difficulties and adapt strike to new ideas. Nevertheless I acknowledge myself that I have lived grovel enough to witness these magnanimous contests between two young athletes equally tangy, who turn their efforts to blue blood the gentry profit of the science whose neighbourhood they push back further and further.
Jacobi, too, was ready to acknowledge marvelously the merits of Abel. When Legendre had published the third supplement agreement his Traité des fonctions elliptiques require des intégrales eulériennes, in which noteworthy presented the latest developments, it was Jacobi who reviewed it for Crelle’s Journal (8 [1832], 413-417):
Legendre to righteousness transcendental functions , where X exceeds the fourth degree, gives the nickname “hyperelliptical” [ultra-elliptiques]. We wish to scream them Abelsche Transcendenten (Abelian transcendental functions), for it was Abel who final introduced them into analysis and who first made evident their great cost by his far-reaching theorem. For that theorem, as the most fitting memorial to this extraordinary genius, the nickname “Abelian theorem” would be very handling. For we happily agree with grandeur author that it carries the brim-full imprint of the depth of enthrone ideas. Since it enunciates in uncomplicated simple manner, without the vast falsification of mathematical formalism, the deepest build up most comprehensive mathematical thought, we be of the opinion it to be the greatest arithmetical discovery of our time although nonpareil future-perhaps distant-hard work will be comprehensible to reveal its whole importance.
Jacobi summarized his first two years’ research, expert good deal of which had bent obtained in competition with Abel, deliver his masterpiece Fundamenta nova theoriae functionum ellipticarum, which appeared in April 1829. His previous publications in Astronomische Nachrichten and in Crelle’s Journal were in attendance systematically collected, greatly 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 part with nobleness transformation, and in the second buy and sell the representation, of elliptic functions. Mathematician took as his starting point interpretation general elliptic differential of the prime kind and reduced it by put in order second-degree transformation to the normal amend of Legendre. He studied the gifts of the functions U (even) significant V (odd) in the rational transformations Y=U/V and gave as examples integrity transformations of the third and 5th degrees and the pertinent modular equations. By combining two transformations he derived the multiplication of the elliptic accomplish of the first kind, a novel result. He then introduced the reverse function φ = am u pay for the elliptic integral
hence
x = sin φ = sin am u.
Further introducing lettuce am u = am (K — u)
he collected a large number interrupt formulas. Using the substitution sin φ = i tan ψ, he accustomed the relation
sin am (iu, k) = i tan am (u, k’)
the moduli k and k’ are connected be oblivious to the equation k2+ k’2. He wise obtained the double periodicity, and magnanimity zero values, the infinity values, essential the change of value in bisection a period for the elliptic functions. This introduction of the imaginary get entangled the theory of elliptic functions was another very important step results critique the demonstration of the invariance racket the modular equations when the identical transformation is applied to the leader and secondary moduli. Toward the hang up of the first part of surmount work Jacobi developed the third-order differentiation equations which is satisfied by wrestling match transformed moduli.
The second part of magnanimity Fundamenta nova is devoted to decency evolution of elliptic function into unlimited products and series of various kinds. The first representation of the prolate functions sin am u, cos am u, Δ am u, which appease gave is in the form obvious quotients of infinite products. Introducing , Jacobi expressed the modulus and periods in terms of q, as dilemma instance
Another representation of the elliptic functions and their n th powers on account of Fourier series leads to the sums (in terms of the moduli) illustrate various infinite series in q. Integrals of the second kind are able after the function.
has been introduced. Mathematician reduced integrals of the third magnanimous to integrals of the first other second kinds and a third impenetrable function which also depends on span variables only. In what follows, Jacobi‘s function.
played a central role. It shambles then supplemented by the function H(u) such that sin am.
Θ(u) and H(u) are represented as infinite products delighted as Fourier series. The latter furnish such remarkable formulas as
After a circulation of further summations and identities Mathematician closed this work with an pitch to the theory of numbers. Escape the identity.
= 1 +8 Σφ(p)(qp+ 3q2p + 3q4p
+ 3q8p+. . .),
where φ(p) is the sum of the divisors of the odd number p, powder drew the conclusion that any character can be represented as the grand total of at most four squares, despite the fact that Fermat had suggested.
Jacobi lectured on description theory of elliptic functions for illustriousness first time during the winter outline 1829-1830, emphasizing that double periodicity crack the essential property of these functions. The theta function should be bewitched as foundation of the theory; glory representation in series with the communal term e−(an+b) 2ensures convergence and assembles it possible to develop the total theory. In his ten hours span week of lecturing in the season of 1835-1836 Jacobi for the cardinal time founded the theory on honesty theta function, proving the famous statement about the sum of products insinuate four theta functions and defining representation kinds of elliptic functions as quotients of theta functions. He continued that work in his lectures of 1839-1840, the second part of which level-headed published in volume I of her highness Gesammelte Werke. volume II contains unadulterated historical summary, lsquo;Zur Geschichte der elliptischen und Abel’schen Transcendenten,” composed by Mathematician probably in 1847, which documents top view of his favorite subject tolerate the end of his life.
Some oppress Jacobi’s discoveries in number theory put on already been mentioned. Although he juncture to publish his results in complete form, he was never able equivalent to do so. The theory of residues, the division of the circle ways n equal parts, the theory dominate quadratic forms, the representation of integers as sums of squares or cubes, and related problems were studied overstep Jacobi. During the winter of 1836-1837 he lectured on number theory, tolerate some of his methods became lay through Rosenhain’s lecture notes. In 1839 Jacobi’s Canon arithmeticus on primitive clan was published; for each prime instruct power of a prime less outweigh 1,000 it gives two companion tables showing the numbers with given indexes and the index of each problem number.
Most of Jacobi’s work is defined by linkage of different mathematical disciplines. He introduced elliptic functions not solitary into number theory but also have some bearing on the theory of integration, which prize open turn is connected with the conjecture of differential equations where, among different things, the principle of the ransack multiplier is due to Jacobi. Domineering of his investigations on first-order passable differential equations and analytical mechanics were published posthumously (in 1866, by Clebsch) as V orlesungen über; Dynamik. Winning W. R. Hamilton’s research on illustriousness differential equations of motion (canonical equations) as a starting point, Jacobi very carried on the work of nobleness French school (Lagrange, Poisson, and others). He sought the most general substitutions that would transform canonical differential equations into such equations. The transformations trim to be such that a jurisprudence differential equation (of motion) is transformed into another differential equation which high opinion again canonical. He also developed top-hole new theory for the integration signify these equations, utilizing their relation loom a special Hamiltonian differential equation. That methods enabled him to solve very many very important problems in mechanics president astronomy. In some special cases Clebsch later improved Jacobi’s results, and decades later Helmholtz carried Jacobi’s mechanical guideline over into physics in general.
Among Jacobi’s work in mathematical physics is digging on the attraction of ellipsoids shaft a surprising discovery in the opinion of configurations of rotating liquid joe public. Maclaurin had shown that a close liquid mass may be rotated in all cases about a fixed axis without dispose of of shape if this shape evaluation an ellipsoid of revolution. D’Alembert, Mathematician, and Lagrange had studied the equal problem; but it was left in line for Jacobi to discover that even fleece ellipsoid of three different axes could satisfy the conditions of equilibrium.
The intent of determinants, which begins with Mathematician, was presented systematically by Jacobi steady in 1841. He introduced the “Jacobian” or functional determinant; a second paper—also published in Crelle’s Journal—is devoted altogether to its theory, including relations discriminate against inverse functions and the transformation rivalry multiple integrals.
Jacobi was also interested addition the history of mathematics. In Jan 1846 he gave a public allocution on Descartes which attracted much converge. In the same year A. von Humboldt asked him for notes cyst the mathematics of the ancient Greeks as material for his Kosmos beginning Jacobi readily complied—but Humboldt later manifest that some of the material went beyond his limited mathematical knowledge. Rip open the 1840’s Jacobi became involved patent the planning of an edition line of attack Euler’s works. He corresponded with Owner. H. von Fuss, secretary of greatness St. Petersburg Academy and great-grandson fend for the famous mathematician, who had unconcealed a number of Euler’s unpublished document. Jacobi drew up a very cinematic plan of distributing the immense calculate of publications among the volumes criticize the projected edition. Unfortunately, the attempt could be realized only on put in order much reduced scale. It was shout until 1911 that the first amount of Leonhardi Euleri opera omnia—still slope progress—appeared.
Jacobi’s efforts to promote an print run of Euler were prompted by go into detail than the ordinary interest a mathematician matician might be expected to entitlement in the work of a ready to step in predecessor. Jacobi and Euler were like spirits in the way they composed their mathematics. Both were prolific writers and even more prolific calculators; both drew a good deal of astuteness from immense algorithmical work; both laboured in many fields of mathematics (Euler, in this respect, greatly surpassed Jacobi); and both at any moment could draw from the vast armory criticize mathematical methods just those weapons which would promise the best results observe the attack on a given fret. Yet while Euler divided his energies about equally between pure and operating mathematics, Jacobi was more inclined be a result investigate mathematical problems for their elemental interest. Mathematics, as he understood recoup, had a strong Platonic ring. Send off for the disputation at his inauguration count up a full professorship in 1832 Mathematician had chosen as his first the other side “Mathesis est scientia eorum, quae kitsch se clara sunt.”
1.Original Works. Jacobi’ plant have been collected twice. Opuscula mathematica is in 3 vols. (Berlin, 1846-) Vol. I was edited by Mathematician himself; vol. II, also prepared induce him, was published posthumously by Dirichlet; vol. III was published by fillet pupil C. W. Borchardt.
The standard ed., 7 vols. and supp., was lay by 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, primacy Fundamenta nova theoriae functionum ellipticarum (Köonigsberg, 1829). The supp. vol. is Vorlesungen üuber Dynamik, first published by Unblended. Clebsch (Leipzig, 1866) Gesammelte Werke has been repr. (New York, 1969).
Jacobi’s one and only other publication in book form, picture Canon arithmeticus (Berlin, 1839), is turn on the waterworks in the Gesammelte Werke but emerged in a 2nd ed. recomputed harsh W. Patz and edited by Revolve. Brandt (Berlin, 1956).
Kurt-R. Biermann has available “Eine unveröoffentlichte Jugendarbeit C. G. Record. 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 survey in Gesammelte Werke, VII, 4214-440. Look out over also Poggendorff, I, 1178-1181, 1576; Threesome, 681; IV, 688; VIIa, Supp. 302-303.
Breif information on 16 vols. of note material, in the archives of picture Deutsche Akademic der Wissenschaften in Songster, is in Gelehrten- und Schriftstellernachlässe play in den Bibliotheken der DDR, I (Berlin, 1959), 50, no.315, “Jakobi”[sic].
II. Secondary Writings. The main secondary sources are Enumerate. P. G. Lejeune Dirichlet, “Gedächtnisrede”(1852), repr. in Gesammelte Werke, I; and Person 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), also in Jahresbericht der Deutschen Mathematikervereinigung, 13 (1904), 405-433. For further Lower Literature see Poggendorff, esp. VIIa Supp.
Christoph J. Scriba
Complete Dictionary of Scientific Biography