Carl Gustav Jacob Jacobi

"The best approach to Jacobi is perhaps through his beautiful lectures on dynamics ("Vorlesungen über Dynamik"), published in 1866 after lecture notes from 1842-43."

"It is true that M. Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of number is as important as a question of the system of the world."

"Any progress in the theory of partial differential equations must also bring about a progress in Mechanics."

"Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case."

"History knew a midnight, which we may estimate at about the year 1000 A.D., when the human race lost the arts and sciences even to the memory. The last twilight of paganism was gone, and the new day had not yet begun. Whatever was left of culture in the world was found only in the Saracens, and a Pope eager to learn studied in disguise in their universities, and so became the wonder of the West. At last Christendom, tired of praying to the dead bones of the martyrs, flocked to the tomb of the Saviour Himself, only to find for a second time that the grave was empty and that Christ was risen from the dead. Then mankind too rose from the dead. It returned to the activities and the business of life; there was a feverish revival in the arts and in the crafts. The cities flourished, a new citizenry was founded. Cimabue rediscovered the extinct art of painting; Dante, that of poetry. Then it was, also, that great courageous spirits like Abelard and Saint Thomas Aquinas dared to introduce into Catholicism the concepts of Aristotelian logic, and thus founded scholastic philosophy. But when the Church took the sciences under her wing, she demanded that the forms in which they moved be subjected to the same unconditioned faith in authority as were her own laws. And so it happened that scholasticism, far from freeing the human spirit, enchained it for many centuries to come, until the very possibility of free scientific research came to be doubted. At last, however, here too daylight broke, and mankind, reassured, determined to take advantage of its gifts and to create a knowledge of nature based on independent thought. The dawn of the day in history is know as the Renaissance or the Revival of Learning."

"His [Lagrange's] lectures on differential calculus form the basis of his Theorie des fonctions analytiques which was published in 1797. ...its object is to substitute for the differential calculus a group of theorems based upon the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in his Residual Analysis... Lagrange believed that he could... get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. ...Another treatise in the same lines was his Leçons sur le calcul des fonctions, issued in 1804. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass."

"Jacobi's most celebrated investigations are those on elliptic functions, the modern notation in which is substantially due to him, and the theory of which he established simultaneously with Abel, but independently of him. Jacobi's results are given in his treatise on elliptic functions, published in 1829, and in some later papers in Crelle's Journal; they are earlier than Weierstrass's researches... The correspondence between Legendre and Jacobi on elliptic functions has been reprinted in the first volume of Jacobi's collected works. Jacobi, like Abel, recognised that elliptic functions were not merely a group of theorems on integration, but that they were types of a new kind of function, namely, one of double periodicity; hence he paid particular attention to the theory of the theta function."

"I ought also to mention his papers on Abelian transcendants; his investigations on the theory of numbers... his important memoirs on the theory of differential equations, both ordinary and partial; his development of the calculus of variations; and his contributions to the problem of three bodies, and other particular dynamical problems. Most of the results of the researches last named are included in his Vorlesungen über Dynamik."

"The most important of Legendre's works is his Functions elliptiques, issued in two volumes in 1825 and 1826. He took up the subject where Euler, Landen, and Lagrange had left it, and for forty years was the only one to cultivate this new branch of analysis, until at last Jacobi and Abel stepped in with admirable new discoveries."

"The theory of determinants was studied by Hoëné Wronski in Italy and J. Binet in France; but they were forestalled by the great master of this subject, Cauchy. In a paper (Jour. de l'ecole Polyt., IX., 16) Cauchy developed several general theorems. He introduced the name determinant a term previously used by Gauss in the functions considered by him. In 1826 Jacobi began using this calculus, and he gave brilliant proof of its power. In 1841 he wrote extended memoirs on determinants in Crelle's Journal, which rendered the theory easily accessible."

"Cauchy made some researches on the calculus of variations. This subject is now in its essential principles the same as when it came from the hands of Lagrange. Recent studies pertain to the variation of a double integral when the limits are also variable, and to variations of multiple integrals in general. ...In 1837 Jacobi published a memoir, showing that the difficult integrations demanded by the discussion of the second variation, by which the existence of a maximum or minimum can be ascertained, are included in the integrations of the first variation, and thus are superfluous. This important theorem, presented with great brevity by Jacobi, was elucidated and extended by V. A. Lebesgue, C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Clebsch. ...In 1852 G. Mainardi attempted to exhibit a new method of discriminating maxima and minima, and extended Jacobi's theorem to double integrals. Mainardi and F. Brioschi showed the value of determinants in exhibiting the terms of the second variation."

"Gauss' researches on the theory of numbers were the starting-point for a school of writers, among the earliest of whom was Jacobi. The latter contributed to Crelle's Journal an article on cubic residues, giving theorems without proofs. After the publication of Gauss' paper on biquadratic residues, giving the law of biquadratic reciprocity, and his treatment of complex numbers, Jacobi found a similar law for cubic residues. By the theory of elliptical functions, he was led to beautiful theorems on the representation of numbers by 2, 4, 6, and 8 squares."

"The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843."

"Advances in theoretical mechanics, bearing on the integration and the alteration in form of dynamical equations, were made since Lagrange by Poisson, William Rowan Hamilton, Jacobi, Madame Kowalevski, and others. Lagrange had established the "Lagrangian form" of the equations of motion. He had given a theory of the variation of the arbitrary constants which, however, turned out to be less fruitful in results than a theory advanced by Poisson. ...Hamilton's method of integration was freed by Jacobi of an unnecessary complication, and was then applied by him to the determination of a geodetic line on the general ellipsoid. With aid of elliptic coordinates Jacobi integrated the partial differential equation and expressed the equation of the geodetic in form of a relation between two Abelian integrals. Jacobi applied to differential equations of dynamics the theory of the ultimate multiplier. The differential equations of dynamics are only one of the classes of differential equations considered by Jacobi. Dynamic investigations along the lines of Lagrange, Hamilton, and Jacobi were made by Liouville, A. Desboves, Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin, Brioschi, leading up to the development of the theory of a system of canonical integrals."

"Among standard works on mechanics are Jacobi's Vorlesungen über Dynamik, edited by Clebsch, 1866."

"C. J. Jacobi was especially distressed because on several occasions when he came to Gauss to relate some new discoveries the latter pulled out from his desk drawer some papers that contained the very same discoveries. Jacobi resolved to get even. ...Gauss opened his desk drawer and pulled out some papers ...Jacobi then remarked, "It is a pity that you did not publish this work since you have published so may poorer papers.""

"Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants."

"In 1829, the year that Abel died, Carl Gustav Jacob Jacobi published his "Fundamenta nova theoriae functionum ellipticarum." Jacobi based his theory of elliptic functions on four functions defined by infinite series and called theta functions. ...The addition theorems of elliptic functions can also be considered as special applications of Abel's theorem on the sum of integrals of algebraic equations. The question now arose whether hyper-elliptic integrals could be inverted in the way elliptic integrals had been inverted to yield elliptic functions. The solution was found by Jacobi in 1832 when he published his result that the inversion could be performed with functions of more than one variable. Thus the theory of Abelian functions of p variables was born, which became an important branch of Nineteenth century mathematics."

"Sylvester has given the name "Jacobian" to the functional determinant in order to pay respect to Jacobi's work on algebra and elimination theory. The best known of Jacobi's papers on this subject is his "De formatione et proprietatibus determinantium" (1841), which made the theory of determinants the common good of the mathematicians."

"Das einzige Ziel der Wissenschaft [...] ist die Ehre des menschlichen Geistes."