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.
背景與歷史背景
伯努利家族，最初來自荷蘭，由於西班牙的宗教迫害，被迫離開家園，最終定居在瑞士巴塞爾。這個家族成為數學和科學史上最具影響力的王朝之一，跨越了好幾代人。他們的工作為許多現代數學概念奠定了基礎，尤其是在微積分、概率和物理學方面。伯努利家族生活在數學快速發展的時代，牛頓和萊布尼茨發展了微積分。他們是最早應用和擴展這些新思想的人之一，做出了重大貢獻，塑造了科學的未來。
關於作者
伯努利家族中最傑出的成員包括雅各布·伯努利、他的兄弟約翰·伯努利以及年輕一代，如丹尼爾·伯努利。雅各布·伯努利是將微積分應用於解決複雜問題的先驅，而約翰·伯努利則以其教學和進一步發展微積分符號和方法而聞名。年輕一代中最著名的丹尼爾·伯努利，對流體動力學和氣體動力學理論做出了開創性的貢獻。他們的作品不僅僅是數學上的，而且與物理學和自然哲學密切相關，反映了啟蒙運動時期科學探究的跨學科性質。
詳細解釋和意義
伯努利家族的工作在許多領域具有奠基性作用：
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微積分和分析： 雅各布·伯努利是最早理解無窮小微積分力量的人之一。他引入了“積分”一詞，並致力於構建積分微積分，這對於理解曲線下的面積和求解微分方程至關重要。
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概率論： 在他的著作《猜測的藝術》中，雅各布·伯努利奠定了概率的基本原理，這對於統計學、風險評估和決策至關重要。
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物理學和力學： 丹尼爾·伯努利的《流體力學》介紹了解釋流體流動和能量守恆的原理。他關於氣體動力學理論的工作有助於解釋氣體定律，這在化學和物理學中是基礎性的。
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數學符號： 約翰·伯努利對微積分中使用的符號做出了貢獻，例如使用 φ(x) 表示函數，至今仍在沿用。
這些貢獻不僅僅是歷史事實；它們構成了許多科學和工程學科的骨幹。
給學生的教訓和啟發
研究伯努利家族的故事和作品提供了幾個寶貴的教訓：
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毅力和熱情： 伯努利家族對數學和科學充滿熱情。儘管存在個人和專業上的衝突，但他們的奉獻精神表明了在學習和發現中堅持不懈的重要性。
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跨學科思維： 他們的工作結合了數學、物理學和哲學，鼓勵學生廣泛思考並聯繫不同的知識領域。
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創新與應用： 他們展示了抽象的數學思想如何應用於解決現實世界的問題，激勵學生尋求將他們的學習付諸實踐。
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倫理與合作： 儘管一些家庭成員之間存在衝突，但總體遺產突出了分享知識和共同努力推進科學的重要性。
學生如何應用這些見解
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在學習中： 效仿伯努利家族的好奇心，超越教科書的範圍進行探索。嘗試理解公式和理論背後的“為什麼”，並將它們應用於解決問題。
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在日常生活中： 在日常決策中使用邏輯思維和解決問題的技能。例如，理解概率可以幫助做出明智的選擇。
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在社交互動中： 伯努利家族的故事也教導了在合作中謙遜和尊重的價值。認識到他人的貢獻可以帶來更好的團隊合作。
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培養積極的態度： 培養終身學習和韌性的心態。伯努利家族面臨挑戰，但仍不斷創新，這對於面臨學術或個人困難的學生來說是一個很好的例子。
鼓勵伯努利家族的精神
為了培養伯努利家族的精神，學生應該：
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積極參與具有挑戰性的科目，如數學和科學，將它們視為理解世界的工具。
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參與討論、辯論和合作項目，以培養溝通和團隊合作技能。
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反思科學工作的倫理層面，重視誠實和正直。
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探索科學家的歷史故事，以欣賞發現的人性一面，使學習更具關聯性和啟發性。
通過研究伯努利家族，學生不僅可以獲得知識，還可以學習到在生活的許多領域中為他們服務的態度和技能，從學術到個人成長和社會關係。
