The Bernoullis (or as they are sometimes, and perhaps more correctly, called, the Bernouillis) were a family of Dutch origin, who were driven from Holland by the Spanish persecutions, and finally settled at Bâle in Switzerland. The first member of the family who obtained distinction in mathematics was James.
James Bernoulli
Jacob or James Bernoulli was born at Bâle on December 27, 1654; in 1687 he was appointed to a chair in mathematics in the university there; and occupied it until his death on August 16, 1705.
He was one of the earliest to realize how powerful as an instrument of analysis was the infinitesimal calculus, and he applied it to several problems, but did not himself invent any new processes. His great influence was uniformly and successfully exerted in favour of the use of the differential calculus, and his lessons on it, which were written in the form of two essays in 1691 and are published in the second volume of his works, shew how completely he had even then grasped the principles of the new analysis. These lectures, which contain the earliest use of the term integral, were the first published attempt to construct an integral calculus; for Leibnitz had treated each problem by itself, and had not laid down any general rules on the subject.
The most important discoveries of James Bernoulli were his solution of the problem to find an isochronous curve; his proof that the construction for the catenary which had been given by Leibnitz was correct, and his extension of this to strings of variable density and under a central force; his determination of the form taken by an elastic rod fixed at one end and acted on by a given force at the other, the elastica ; also of a flexible rectangular sheet with two sides fixed horizontally and filled with a heavy liquid, the lintearia ; and lastly, of a sail filled with wind, the velaria . In 1696 he offered a reward for the general solution of isoperimetrical figures, that is, of figures of a given species and given perimeter which shall include a maximum area: his own solution, published in 1701, is correct as far as it goes. In 1698 he published an essay on the differential calculus and its applications to geometry. He here investigated the chief properties of the equiangular spiral, and especially noticed the manner in which various curves deduced from it reproduced the original curve: struck by this fact he begged that, in imitation of Archimedes, and equiangular spiral should be engraved on his tombstone with the inscription eadem numero mutata resurgo . He also brought out in 1695 an edition of Descartes's Géometrie . In his Ars Conjectandi , published in 1713, he established the fundamental principles of the calculus of probabilities; in the course of the work he defined the numbers known by his name and explained their use, he also gave some theorems on finite differences. His higher lectures were mostly on the theory of series; these were published by Nicholas Bernoulli in 1713.
John Bernoulli
John Bernoulli, the brother of James Bernoulli, was born at Bâle on August 7, 1667, and died there on January 1, 1748. He occupied the chair of mathematics at Groningen from 1695 to 1705; and at Bâle, where he succeeded his brother, from 1705 to 1748. To all who did not acknowledge his merits in a manner commensurate with his own view of them he behaved most unjustly: as an illustration of his character it may be mentioned that he attempted to substitute for an incorrect solution of his own on the problem of isoperimetrical curves another stolen from his brother James, while he expelled his son Daniel from his house for obtaining a prize from the French Academy which he had expected to receive himself. He was, however, the most successful teacher of his age, and had the faculty of inspiring his pupils with almost as passionate a zeal for mathematics as he felt himself. The general adoption on the continent of the differential rather than the fluxional notation was largely due to his influence.
Leaving out of account his innumerable controversies, the chief discoveries of John Bernoulli were the exponential calculus, the treatment of trigonometry as a branch of analysis, the conditions for a geodesic, the determination of orthogonal trajectories, the solution of the brachistochrone, the statement that a ray of light pursues such a path that Σ μds is a minimum, and the enunciation of the principle of virtual work. I believe that he was the first to denote the accelerating effect of gravity by an algebraical sign g , and he thus arrived at the formula v 2 = 2 gh the same result would have been previously expressed by the proportion . The notation φ x to indicate a function of x was introduced by him in 1718, and displaced the notation X or ξ proposed by him in 1698; but the general adoption of symbols like f , F , φ, ψ, ... to represent functions, seems to be mainly due to Euler and Lagrange.
The Younger Bernoullis
Several members of the same family, but of a younger generation, enriched mathematics by their teaching and writings. The most important of these were the three sons of John; namely Nicholas, Daniel, and John the younger; and the two sons of John the Younger, who bore the names of John and James. To make the account complete I add here their respective dates. Nicholas Bernoulli, the eldest of the three sons of John, was born on Jan. 27, 1695, and was drowned at St. Petersburg, where he was professor, on July 26, 1726. Daniel Bernoulli, the scond son of John, was born on Feb. 9, 1700, and died on March 17, 1782; he was professor first at St. Petersburg and afterwards at Bâle, and shares with Euler the unique distinction of having gained the prize proposed annually by the French Academy no less than ten times. John Bernoulli, the younger, a brother of Nicholas and Daniel, was born on May 18, 1710, and died in 1790; he also was a professor at Bâle. He left two sons, John and James: of these, the former, who was born on Dec. 14, 1744, and died on July 10, 1807, was astronomer-royal, and director of mathematical studies at Berlin; while the latter, who was born on Oct. 17, 1759, and died in July 1789, was successively professor at Bâle, Verona, and St. Petersburg.
Daniel Bernoulli
Daniel Bernoulli, whose name I mentioned above, and who was by far the ablest of the younger Bernoullis, was a contemporary and intimate friend of Euler, whose works are mentioned in the next chapter. Daniel Bernoulli was born on Feb. 9, 1700, and died at Bâle, where he was professor of natural philosophy, on March 17, 1782. He went to St. Petersburg in 1724 as professor of mathematics, but the roughness of the social life was distasteful to him, and he was not sorry when a temporary illness in 1733 allowed him to plead his health as an excuse for leaving. He then returned to Bâle, and held successively chairs of medicine, metaphysics, and natural philosophy there.
His earliest mathematical work was the Exercitationes , published in 1724, which contains a solution of the differential equation proposed by Riccati. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motions of rotation. His chief work is his Hydrodynamique , published in 1738; it resembles Lagrange's Méchanique analytique in being arranged so that all the results are consequences of a single principle, namely, in this case, the conservation of energy. This was followed by a memoir on the theory of the tides, to which, conjointly with the memoirs by Euler and Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Newton's Principia and the investigations of Laplace. Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Taylor and by D'Alembert. He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain the law associated with the names of Boyle and Mariotte.
Contexte et historique
La famille Bernoulli, originaire des Pays-Bas, a été contrainte de quitter sa patrie en raison des persécutions religieuses espagnoles et s'est finalement installée à Bâle, en Suisse. Cette famille est devenue l'une des dynasties les plus influentes de l'histoire des mathématiques et des sciences, s'étendant sur plusieurs générations. Leur travail a jeté les bases de nombreux concepts mathématiques modernes, en particulier dans le calcul, les probabilités et la physique. Les Bernoulli ont vécu à une époque où les mathématiques évoluaient rapidement, avec le développement du calcul par Newton et Leibniz. Ils ont été parmi les premiers à appliquer et à développer ces nouvelles idées, apportant des contributions significatives qui ont façonné l'avenir de la science.
À propos des auteurs
Les membres les plus éminents de la famille Bernoulli comprennent James (Jacob) Bernoulli, son frère John Bernoulli et la jeune génération comme Daniel Bernoulli. James Bernoulli a été un pionnier dans l'application du calcul pour résoudre des problèmes complexes, tandis que John Bernoulli était connu pour son enseignement et le développement ultérieur de la notation et des méthodes du calcul. Daniel Bernoulli, le plus célèbre des jeunes Bernoulli, a apporté des contributions révolutionnaires à la dynamique des fluides et à la théorie cinétique des gaz. Leurs travaux n'étaient pas seulement mathématiques, mais aussi profondément liés à la physique et à la philosophie naturelle, reflétant la nature interdisciplinaire de la recherche scientifique pendant les Lumières.
Explication détaillée et importance
Le travail de la famille Bernoulli est fondamental dans de nombreux domaines :
-
Calcul et analyse : James Bernoulli a été parmi les premiers à comprendre la puissance du calcul infinitésimal. Il a introduit le terme "intégrale" et a travaillé à la construction du calcul intégral, qui est essentiel pour comprendre les aires sous les courbes et résoudre les équations différentielles.
-
Théorie des probabilités : Dans son livre "Ars Conjectandi", James Bernoulli a posé les principes fondamentaux des probabilités, qui sont cruciaux pour les statistiques, l'évaluation des risques et la prise de décision.
-
Physique et mécanique : "Hydrodynamique" de Daniel Bernoulli a introduit des principes qui expliquent l'écoulement des fluides et la conservation de l'énergie. Ses travaux sur la théorie cinétique des gaz ont permis d'expliquer les lois des gaz, qui sont fondamentales en chimie et en physique.
-
Notation mathématique : John Bernoulli a contribué à la notation utilisée dans le calcul, comme l'utilisation de φ(x) pour désigner les fonctions, qui est encore utilisée aujourd'hui.
Ces contributions ne sont pas seulement des faits historiques ; elles constituent l'épine dorsale de nombreuses disciplines scientifiques et d'ingénierie.
Leçons et inspirations pour les étudiants
L'étude de l'histoire et des œuvres de la famille Bernoulli offre plusieurs leçons précieuses :
-
Persévérance et passion : Les Bernoulli étaient profondément passionnés par les mathématiques et les sciences. Leur dévouement, malgré les conflits personnels et professionnels, montre l'importance de la persévérance dans l'apprentissage et la découverte.
-
Pensée interdisciplinaire : Leur travail combinait les mathématiques, la physique et la philosophie, encourageant les étudiants à penser largement et à connecter différents domaines de la connaissance.
-
Innovation et application : Ils ont démontré comment des idées mathématiques abstraites peuvent être appliquées pour résoudre des problèmes du monde réel, inspirant les étudiants à rechercher des utilisations pratiques pour leur apprentissage.
-
Éthique et collaboration : Bien que certains membres de la famille aient eu des conflits, l'héritage global souligne l'importance du partage des connaissances et de la collaboration pour faire progresser la science.
Comment les étudiants peuvent appliquer ces connaissances
-
Dans l'apprentissage : Imitez la curiosité des Bernoulli en explorant au-delà des manuels scolaires. Essayez de comprendre le "pourquoi" derrière les formules et les théories, et appliquez-les pour résoudre des problèmes.
-
Dans la vie quotidienne : Utilisez la pensée logique et les compétences de résolution de problèmes dans les décisions quotidiennes. La compréhension des probabilités, par exemple, peut aider à faire des choix éclairés.
-
Dans les interactions sociales : L'histoire des Bernoulli enseigne également la valeur de l'humilité et du respect dans les collaborations. Reconnaître les contributions des autres peut conduire à un meilleur travail d'équipe.
-
Développer des attitudes positives : Cultivez un état d'esprit d'apprentissage et de résilience tout au long de la vie. Les Bernoulli ont fait face à des défis, mais ont continué à innover, un excellent exemple pour les étudiants confrontés à des difficultés académiques ou personnelles.
Encourager l'esprit des Bernoulli
Pour nourrir l'esprit de la famille Bernoulli, les étudiants devraient :
-
S'engager activement dans des matières difficiles comme les mathématiques et les sciences, en les considérant comme des outils pour comprendre le monde.
-
Participer à des discussions, des débats et des projets collaboratifs pour développer des compétences en communication et en travail d'équipe.
-
Réfléchir aux dimensions éthiques du travail scientifique, en valorisant l'honnêteté et l'intégrité.
-
Explorer les histoires historiques des scientifiques pour apprécier le côté humain de la découverte, rendant l'apprentissage plus accessible et inspirant.
En étudiant les Bernoulli, les étudiants acquièrent non seulement des connaissances, mais apprennent également des attitudes et des compétences qui leur seront utiles dans de nombreux domaines de la vie, des études à la croissance personnelle et aux relations sociales.
