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.
背景和历史背景
伯努利家族最初来自荷兰，由于西班牙的宗教迫害，他们被迫离开家园，最终定居在瑞士巴塞尔。这个家族成为数学和科学史上最具影响力的王朝之一，延续了几代人。他们的工作为许多现代数学概念奠定了基础，尤其是在微积分、概率和物理学方面。伯努利家族生活在数学快速发展的时代，牛顿和莱布尼茨发展了微积分。他们是最早应用和扩展这些新思想的人之一，为塑造科学的未来做出了重大贡献。
关于作者
伯努利家族中最杰出的成员包括雅各布·伯努利、他的兄弟约翰·伯努利以及年轻一代如丹尼尔·伯努利。雅各布·伯努利是应用微积分解决复杂问题的先驱，而约翰·伯努利则以其教学和进一步发展微积分符号和方法而闻名。年轻一代中最著名的丹尼尔·伯努利，对流体动力学和气体动力学理论做出了开创性的贡献。他们的著作不仅是数学的，而且与物理学和自然哲学有着深刻的联系，反映了启蒙运动时期科学探究的跨学科性质。
详细解释和意义
伯努利家族的工作在许多领域具有奠基性作用：
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微积分和分析： 雅各布·伯努利是最早理解无穷小微积分的力量的人之一。他引入了“积分”一词，并致力于构建积分微积分，这对于理解曲线下的面积和求解微分方程至关重要。
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概率论： 在他的著作《推测术》中，雅各布·伯努利奠定了概率的基本原理，这对于统计学、风险评估和决策至关重要。
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物理学和力学： 丹尼尔·伯努利的《流体动力学》介绍了解释流体流动和能量守恒的原理。他对气体动力学理论的研究有助于解释气体定律，这在化学和物理学中是基础性的。
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数学符号： 约翰·伯努利对微积分中使用的符号做出了贡献，例如使用 φ(x) 表示函数，这种方法至今仍在沿用。
这些贡献不仅仅是历史事实；它们构成了许多科学和工程学科的骨干。
给学生的教训和启发
研究伯努利家族的故事和著作提供了几个宝贵的教训：
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毅力和热情： 伯努利家族对数学和科学充满热情。尽管存在个人和职业冲突，但他们的奉献精神表明了在学习和发现中坚持不懈的重要性。
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跨学科思维： 他们的工作结合了数学、物理学和哲学，鼓励学生开阔思路，连接不同的知识领域。
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创新和应用： 他们展示了抽象的数学思想如何应用于解决现实世界的问题，激励学生寻求他们所学知识的实际用途。
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伦理和协作： 虽然一些家庭成员之间存在冲突，但总体的遗产突出了分享知识和共同努力推进科学的重要性。
学生如何应用这些见解
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在学习中： 效仿伯努利家族的好奇心，超越课本进行探索。尝试理解公式和理论背后的“为什么”，并将它们应用于解决问题。
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在日常生活中： 在日常决策中使用逻辑思维和解决问题的技能。例如，理解概率可以帮助做出明智的选择。
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在社交互动中： 伯努利家族的故事也教会了我们谦逊和尊重在合作中的价值。认识到他人的贡献可以带来更好的团队合作。
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培养积极的态度： 培养终身学习和韧性的心态。伯努利家族面临挑战，但他们继续创新，这对于面临学业或个人困难的学生来说是一个很好的例子。
鼓励伯努利家族的精神
为了培养伯努利家族的精神，学生们应该：
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积极参与具有挑战性的科目，如数学和科学，将它们视为理解世界的工具。
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参与讨论、辩论和协作项目，以培养沟通和团队合作技能。
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反思科学工作的伦理维度，重视诚实和正直。
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探索科学家的历史故事，以欣赏发现的人性一面，使学习更具关联性和启发性。
通过研究伯努利家族，学生们不仅获得知识，还学会了态度和技能，这些将服务于他们生活的许多领域，从学术到个人成长和社会关系。
