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.
Предпосылки и исторический контекст
Семья Бернулли, родом из Нидерландов, была вынуждена покинуть родину из-за религиозных преследований со стороны Испании и в конечном итоге обосновалась в Базеле, Швейцария. Эта семья стала одной из самых влиятельных династий в истории математики и науки, охватывающей несколько поколений. Их работы заложили основу для многих современных математических концепций, особенно в исчислении, теории вероятностей и физике. Бернулли жили в эпоху, когда математика быстро развивалась, с развитием исчисления Ньютоном и Лейбницем. Они были одними из первых, кто применил и расширил эти новые идеи, внеся значительный вклад, который сформировал будущее науки.
Об авторах
Наиболее выдающимися членами семьи Бернулли являются Джеймс (Якоб) Бернулли, его брат Иоганн Бернулли и младшее поколение, такое как Даниил Бернулли. Джеймс Бернулли был пионером в применении исчисления для решения сложных задач, в то время как Иоганн Бернулли был известен своей преподавательской деятельностью и дальнейшей разработкой обозначений и методов исчисления. Даниил Бернулли, самый известный из младших Бернулли, внес новаторский вклад в динамику жидкостей и кинетическую теорию газов. Их работы были не только математическими, но и глубоко связаны с физикой и натурфилософией, отражая междисциплинарный характер научных исследований в эпоху Просвещения.
Подробное объяснение и значимость
Работа семьи Бернулли является основополагающей во многих областях:
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Исчисление и анализ: Джеймс Бернулли был одним из первых, кто понял силу бесконечно малого исчисления. Он ввел термин «интеграл» и работал над построением интегрального исчисления, которое необходимо для понимания площадей под кривыми и решения дифференциальных уравнений.
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Теория вероятностей: В своей книге «Ars Conjectandi» Джеймс Бернулли заложил основополагающие принципы теории вероятностей, которые имеют решающее значение для статистики, оценки рисков и принятия решений.
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Физика и механика: «Гидродинамика» Даниила Бернулли представила принципы, объясняющие течение жидкости и сохранение энергии. Его работа по кинетической теории газов помогла объяснить газовые законы, которые являются фундаментальными в химии и физике.
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Математические обозначения: Иоганн Бернулли внес вклад в обозначения, используемые в исчислении, например, использование φ(x) для обозначения функций, которое используется и сегодня.
Эти вклады — не просто исторические факты; они составляют основу многих научных и инженерных дисциплин.
Уроки и вдохновение для студентов
Изучение истории и работ семьи Бернулли предлагает несколько ценных уроков:
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Настойчивость и страсть: Бернулли были глубоко увлечены математикой и наукой. Их преданность делу, несмотря на личные и профессиональные конфликты, показывает важность настойчивости в обучении и открытиях.
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Междисциплинарное мышление: Их работа объединила математику, физику и философию, побуждая студентов мыслить широко и связывать различные области знаний.
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Инновации и применение: Они продемонстрировали, как абстрактные математические идеи можно применять для решения реальных проблем, вдохновляя студентов искать практическое применение своим знаниям.
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Этика и сотрудничество: Хотя у некоторых членов семьи были конфликты, общее наследие подчеркивает важность обмена знаниями и совместной работы для развития науки.
Как студенты могут применить эти идеи
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В обучении: Подражайте любопытству Бернулли, изучая материал за пределами учебников. Постарайтесь понять «почему» формул и теорий и примените их для решения задач.
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В повседневной жизни: Используйте логическое мышление и навыки решения проблем в повседневных решениях. Понимание вероятности, например, может помочь в принятии обоснованных решений.
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В социальных взаимодействиях: История Бернулли также учит ценности смирения и уважения в сотрудничестве. Признание вклада других может привести к улучшению командной работы.
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Развитие позитивного отношения: Развивайте мышление непрерывного обучения и устойчивости. Бернулли сталкивались с трудностями, но продолжали внедрять инновации, что является отличным примером для студентов, сталкивающихся с академическими или личными трудностями.
Поощрение духа Бернулли
Чтобы воспитать дух семьи Бернулли, студенты должны:
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Активно заниматься сложными предметами, такими как математика и естественные науки, рассматривая их как инструменты для понимания мира.
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Участвовать в дискуссиях, дебатах и совместных проектах для развития навыков общения и командной работы.
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Размышлять об этических аспектах научной работы, ценя честность и порядочность.
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Изучать исторические истории ученых, чтобы оценить человеческую сторону открытий, делая обучение более понятным и вдохновляющим.
Изучая Бернулли, студенты не только получают знания, но и учатся отношениям и навыкам, которые пригодятся им во многих областях жизни, от учебы до личностного роста и социальных отношений.
