JOHANN I (1667–1748)

But already he had been diligently learning Mathematics from his brother Jacob, and from 1687 onwards, the two brothers were busily exploring Calculus. Notwithstanding such diversions, Johann qualified Licencié in Medicine in 1690, with a thesis entitled De Effervescentia et Fermentatione, which, following standard practice, was given immediate publication. At this point he took a break from his Medical Studies and spent most of the following year teaching Mathematics in Geneva.

In May of that year, in the Acta Eruditorum, Jacob issued the challenge for the problem of the catenary and it was immediately solved by Johann, whose solution was published that same year. This feat elevated Johann to a recognition shared only by the acknowledged masters of the time—Leibniz, Newton, and Huygens. Such recognition was yet but barely accorded to Jacob—no matter how well it was his due. These developments led to a tension between the brothers—a tension not mollified by time. Competition rather than collaboration would characterize their relationship henceforth.

By the Fall of 1691 Johann was in Paris where he was welcomed into the circle of Malebranche. This entry into the learned Society of Paris has been attributed by Spiess to Johann's solution of the catenary. Here he made the acquaintance of l'Hospital, "the Grandseigneur of the Science of Mathematics in France", who was, however, deficient in his knowledge of Calculus. Johann was now commissioned to teach Calculus to l'Hospital—a commission that continued by correspondence after Bernoulli returned to Basel. This evolved into the first—and very successful—textbook on the Calculus, Analyse des Infiniment Petits, published by l'Hospital in 1696 in Paris. The subsequent claim by Johann Bernoulli—that most of the results presented in the work were his intellectual property—was received with widespread scepticism for over two centuries. Vindication had to wait until the twentieth century when the publication of Johann Bernoulli's original lectures (edited by Schafheitlin) in 1923 and of his correspondence with l'Hospital (edited by Spiess) in 1955 proved that the marquis had indeed unjustly downplayed Bernoulli's role by mentioning him merely as one among many contributors.

Bernoulli made the acquaintance of Varignon (1692) and Leibniz (1693), with both of whom he formed a friendship. In the case of Leibniz, rightfully considered one of the great correspondents of his time, his correspondence with Johann far exceeds that with any other. In these early years the chief mathematical results of Johann were his solution of the problem of the velaria (1692) and the analysis of the series which since bear his name (1694): moreover, it has been shown by Feigenbaum that he was the first to explore the development later known as Taylor's Series. But already he was working on a more finished solution for "Bernoulli's Differential Equation", named for Jacob who had posed the problem together with a solution that did not quite satisfy Johann.

Meanwhile after his return to Basel he had resumed his Medical Studies which he completed in 1694 when he received his doctorate with a thesis De Motu Musculorum in which he demonstrated the applicability of Mathematics to problems in Medicine and Physiology. At this time also he married Dorothea Falkner, the daughter of an Alderman on the City Council of Basel. The following year he accepted an invitation—apparently instigated by Huygens—to go as Professor of Mathematics in Groningen where he assumed his duties in the Fall of 1695. As his motive in accepting the Chair was mainly the opportunity it offered for collaboration with Huygens, it came as a deep shock to him, on arrival at Groningen, to learn of the recent sudden demise of his friend.

In the following year he issued, in the Acta Eruditorum, a public challenge for the solution to the problem he termed the "Brachistochrone"—the name by which it is since known. Some months later he re-issued the challenge on a printed leaflet directed to "Acutissimis ... in toto orbe mathematicis". As already noted it was Jacob alone who recognized that the problem opened a new area—the Calculus of Variations—to be explored. Independent solutions were provided by Leibniz and Newton. A solution was submitted by l'Hospital—but that had been arrived at only with help directly from Johann. But it was Johann himself who gave the cleverest solution by putting the question into direct correspondence with a problem in the theory of Optics, already solved by Fermat. This was the first occasion in which was shown the reciprocal formal analogy between Mechanics and Optics which was to prove so fruitful during the nineteenth century, and would be further explored in the twentieth century and continues to bear the potential for further development.

Over twenty years were to elapse before there was any sign of recognition on the part of Johann of the value of Jacob's work in the new area he had explored: this came in the form of the publication of the Calculus of Variations 1718. While fundamentally the work of Jacob, it was clear that Johann had the edge when it came to organising a mathematical subject in a neat and transparent form. Meantime, Johann had been directing his attention to other questions such as the Geometry of Curves, where he recognised the characteristics of the geodetic curves on convex surfaces (1698) and the development of the Integral Calculus for the class of rational functions (1699–1701).

By the beginning of the new century the recognition of Johann's eminence was evident in the invitations to fill a choice of mathematical chairsincluding Leiden and Utrecht. But life in the Independent Netherlands had become contentious for him especially since he was accused of heresy—that he had taken to Spinozism. A second pressure was operating on him—the desire of his father-in-law to have his daughter and her family near him. Accordingly in 1705 he resolved to return to Basel to accept the chair of Greek. On his way to Basel he was informed of the recent death of Jacob: Johann was immediately appointed to fill the chair left vacant by Jacob's death.

On his return he focussed—almost exclusively—on problems in Mechanics. In the Memoires de l'Academie des Sciences for the year 1710 he presented for the first time the solution of the Kepler problem in terms of angle variables—the forms in use ever since. It should be noted that, contrary to what has frequently been said, Johann Bernoulli had recognized that Kepler's Laws require a spherically symmetric gravitational field. In the following year he published the solution to a more general problem where the central force has a more general form and where resistant forces are in effect. And in 1714 he published the only book we have from his hand, Essai d'une nouvelle theorie de la manoeuvre des Vaisseaux.

When drawn into the Leibniz-Newton controversy, he published the solution for the ballistic curve under the general form for the law of resistance that removed any residual doubts about the superiority of the Leibnizian system (1719). It was under the stimulation of a paper published by Taylor, to which he felt compelled to respond, that Bernoulli returned to the isoperimetric problem: this led to his giving a systematic form to the comprehensive work of Jacob on the Calculus of Variations, already mentioned, which appeared in the Memoires de l'Academie des Sciences (1718).

Following the death of Newton (1727), Johann Bernoulli was acknowledged without question as the leading mathematician in Europe. He continued publishing on a number of topics—on the transmission of momentum (1727), on the motion of planets at aphelion (1730) and on the inclination of the orbital planes (1735). In his work on Hydraulics, Hydraulica, dated 1732, communicated to Euler in 1739–40 and published in the CP for the years 1737–38, which were put in print in 1744/47, he pioneered in the application of Newton's Laws to an infinitesimal physical element. Truesdell, in the Opera Omnia of Euler, has drawn attention to a letter of Euler (1740) wherein Euler acknowledges (to Johann): "Your theory of water in motion following the true and genuine method which you ... did apply as the first and only one for treating problems of this kind adequately''. The method introduced Newton's Second Law into Hydromechanics. It was largely from the inspiration of this work that Euler subsequently established the subject on a sound basis.

In his career in Basel, besides his teaching career lasting 38 years, he was also extremely active on the University Board. On one occasion he was given the responsibility—and the full power necessary to be effective—for the total reorganization of the Gymnasium. Many public offices were conferred on him "honoris causa".

He continued to work to his final days. In 1742–44 there was published in Geneva his works collected into four volumes (Opera Omnia) under the editorship of G. Cramer. He had the gratification of being at the center of an active circle of mathematicians who had been trained under him: among these were Clairaut, Maupertuis, Cramer, Lesage, Spleiss, his nephew Nicolaus I, as well as his three sons, Nicolaus II, Daniel and Johann II, not forgetting the most famous and accomplished of all—Leonhard Euler. It was his younger son Johann II who succeeded him as Professor of Mathematics in Basel on the retirement of Johann I in 1743. For such a complex m personality whose interests ranged widely over a long and fruitful career, and whose inclination to disputation was legendary even in his lifetime, it is not easy to make a summary assessment. Here we shall confine ourselves to quoting the impressions of one whose familiarity with the correspondence of Johann makes worthy of particular attention. On the basis of that correspondence Speiser observes:

With respect to his character and personality, two impressions stand out. One is, I need not insist, his love for disputes and fights. But one must note that almost all of his fights were directed against peers or even persons of superior standing. After all in his fight for Leibniz against Newton and the English he was the one of the three who was in the least exalted position, and therefore needed the greatest courage: the quatrain of Voltaire, who was a great admirer of Newton, testifies to this.

The other strong impression I receive is his great dedication to his students. He spent much time on them and showed them freely and generously indeed what they needed to learn. He always recognized unhesitatingly and without reservation Euler's superiority. Indeed, that he should be the teacher of one of the greatest scientists and teachers ever, was the most fitting tribute which history possibly could pay to him.

The quatrain of Voltaire mentioned above, Son esprit vit la vérité Et son coeur connut la justice Il a fait l'honneur de la Suisse Et celui de l'humanité. is included at the beginning of the Opera Omnia of Johann.

Ó Mathúna, 1999