عائلة برنولي - حساب موجز لتاريخ الرياضيات بقلم دبليو دبليو راوس بول

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.

الخلفية والسياق التاريخي

أُجبرت عائلة برنولي، التي تنحدر في الأصل من هولندا، على مغادرة وطنها بسبب الاضطهاد الديني الإسباني واستقرت في النهاية في بازل، سويسرا. أصبحت هذه العائلة واحدة من أكثر السلالات تأثيرًا في تاريخ الرياضيات والعلوم، والتي امتدت على مدى أجيال عديدة. أرست أعمالهم الأساس للعديد من المفاهيم الرياضية الحديثة، وخاصة في حساب التفاضل والتكامل والاحتمالات والفيزياء. عاش آل برنولي في وقت كانت فيه الرياضيات تتطور بسرعة، مع تطوير حساب التفاضل والتكامل بواسطة نيوتن وليبنيز. كانوا من بين أوائل من طبقوا هذه الأفكار الجديدة ووسّعوها، وقدموا مساهمات كبيرة شكلت مستقبل العلوم.

عن المؤلفين

يشمل أبرز أفراد عائلة برنولي جيمس (جاكوب) برنولي، وشقيقه جون برنولي، والجيل الأصغر مثل دانيال برنولي. كان جيمس برنولي رائدًا في تطبيق حساب التفاضل والتكامل لحل المشكلات المعقدة، بينما اشتهر جون برنولي بتدريسه ومواصلة تطوير تدوين وأساليب حساب التفاضل والتكامل. قدم دانيال برنولي، وهو الأكثر شهرة بين آل برنولي الأصغر سنًا، مساهمات رائدة في ديناميكيات الموائع والنظرية الحركية للغازات. لم تكن أعمالهم رياضية فحسب، بل كانت مرتبطة أيضًا ارتباطًا وثيقًا بالفيزياء والفلسفة الطبيعية، مما يعكس الطبيعة متعددة التخصصات للبحث العلمي خلال عصر التنوير.

شرح وتوضيح مفصل

تعتبر أعمال عائلة برنولي أساسية في العديد من المجالات:

• حساب التفاضل والتكامل والتحليل: كان جيمس برنولي من بين أوائل من فهموا قوة حساب التفاضل والتكامل المتناهي في الصغر. قدم مصطلح "التكامل" وعمل على بناء حساب التفاضل والتكامل المتكامل، وهو أمر ضروري لفهم المساحات الموجودة أسفل المنحنيات وحل المعادلات التفاضلية.

• نظرية الاحتمالات: في كتابه "فن التخمين"، وضع جيمس برنولي المبادئ الأساسية للاحتمالات، والتي تعتبر حاسمة للإحصاء وتقييم المخاطر واتخاذ القرار.

• الفيزياء والميكانيكا: قدم كتاب "ديناميكية الموائع" لدانيال برنولي مبادئ تشرح تدفق الموائع والحفاظ على الطاقة. ساعد عمله على النظرية الحركية للغازات في شرح قوانين الغازات، وهي أساسية في الكيمياء والفيزياء.

• الترميز الرياضي: ساهم جون برنولي في الترميز المستخدم في حساب التفاضل والتكامل، مثل استخدام φ(x) لتدوين الدوال، والذي لا يزال قيد الاستخدام حتى اليوم.

هذه المساهمات ليست مجرد حقائق تاريخية؛ بل تشكل العمود الفقري للعديد من التخصصات العلمية والهندسية.

الدروس والإلهام للطلاب

يوفرت دراسة قصة وأعمال عائلة برنولي العديد من الدروس القيمة:

• المثابرة والشغف: كان آل برنولي شغوفين بشدة بالرياضيات والعلوم. إن تفانيهم، على الرغم من الصراعات الشخصية والمهنية، يوضح أهمية المثابرة في التعلم والاكتشاف.

• التفكير متعدد التخصصات: جمعت أعمالهم بين الرياضيات والفيزياء والفلسفة، مما شجع الطلاب على التفكير على نطاق واسع وربط مجالات المعرفة المختلفة.

• الابتكار والتطبيق: لقد أظهروا كيف يمكن تطبيق الأفكار الرياضية المجردة لحل المشكلات الواقعية، مما ألهم الطلاب للبحث عن الاستخدامات العملية لتعلمهم.

• الأخلاق والتعاون: في حين أن بعض أفراد الأسرة كانت لديهم صراعات، فإن الإرث العام يسلط الضوء على أهمية تبادل المعرفة والعمل معًا للنهوض بالعلوم.

كيف يمكن للطلاب تطبيق هذه الأفكار

• في التعلم: حاول محاكاة فضول آل برنولي من خلال الاستكشاف خارج الكتب المدرسية. حاول أن تفهم "لماذا" وراء الصيغ والنظريات، وقم بتطبيقها لحل المشكلات.

• في الحياة اليومية: استخدم مهارات التفكير المنطقي وحل المشكلات في القرارات اليومية. على سبيل المثال، يمكن أن يساعد فهم الاحتمالات في اتخاذ خيارات مستنيرة.

• في التفاعلات الاجتماعية: تعلمنا قصة آل برنولي أيضًا عن قيمة التواضع والاحترام في التعاون. يمكن أن يؤدي التعرف على مساهمات الآخرين إلى عمل جماعي أفضل.

• تطوير المواقف الإيجابية: قم بتنمية عقلية التعلم المستمر والمرونة. واجه آل برنولي تحديات لكنهم استمروا في الابتكار، وهو مثال رائع للطلاب الذين يواجهون صعوبات أكاديمية أو شخصية.

تشجيع روح آل برنولي

لتعزيز روح عائلة برنولي، يجب على الطلاب:

• الانخراط بنشاط في الموضوعات الصعبة مثل الرياضيات والعلوم، والنظر إليها على أنها أدوات لفهم العالم.

• المشاركة في المناقشات والمناظرات والمشاريع التعاونية لتطوير مهارات الاتصال والعمل الجماعي.

• التفكير في الأبعاد الأخلاقية للعمل العلمي، وتقييم الصدق والنزاهة.

• استكشاف القصص التاريخية للعلماء لتقدير الجانب الإنساني للاكتشاف، مما يجعل التعلم أكثر صلة وإلهامًا.

من خلال دراسة آل برنولي، لا يكتسب الطلاب المعرفة فحسب، بل يتعلمون أيضًا المواقف والمهارات التي ستخدمهم في العديد من مجالات الحياة، من الأكاديميين إلى النمو الشخصي والعلاقات الاجتماعية.