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.
Latar Belakang dan Konteks Sejarah
Keluarga Bernoulli, yang berasal dari Belanda, terpaksa meninggalkan tanah air mereka karena penganiayaan agama oleh Spanyol dan akhirnya menetap di Basel, Swiss. Keluarga ini menjadi salah satu dinasti paling berpengaruh dalam sejarah matematika dan sains, yang berlangsung selama beberapa generasi. Karya mereka meletakkan dasar bagi banyak konsep matematika modern, terutama dalam kalkulus, probabilitas, dan fisika. Keluarga Bernoulli hidup pada masa ketika matematika berkembang pesat, dengan pengembangan kalkulus oleh Newton dan Leibniz. Mereka adalah di antara yang pertama menerapkan dan mengembangkan ide-ide baru ini, memberikan kontribusi signifikan yang membentuk masa depan sains.
Tentang Penulis
Anggota keluarga Bernoulli yang paling menonjol termasuk James (Jacob) Bernoulli, saudaranya John Bernoulli, dan generasi yang lebih muda seperti Daniel Bernoulli. James Bernoulli adalah pelopor dalam menerapkan kalkulus untuk memecahkan masalah yang kompleks, sementara John Bernoulli dikenal karena pengajaran dan pengembangan lebih lanjut dari notasi dan metode kalkulus. Daniel Bernoulli, yang paling terkenal dari generasi Bernoulli yang lebih muda, memberikan kontribusi terobosan dalam dinamika fluida dan teori kinetik gas. Karya mereka tidak hanya bersifat matematis tetapi juga sangat terkait dengan fisika dan filsafat alam, yang mencerminkan sifat interdisipliner dari penyelidikan ilmiah selama Pencerahan.
Penjelasan Rinci dan Signifikansi
Karya keluarga Bernoulli sangat mendasar di banyak bidang:
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Kalkulus dan Analisis: James Bernoulli adalah salah satu yang pertama memahami kekuatan kalkulus infinitesimal. Ia memperkenalkan istilah "integral" dan mengerjakan konstruksi kalkulus integral, yang penting untuk memahami luas di bawah kurva dan memecahkan persamaan diferensial.
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Teori Probabilitas: Dalam bukunya "Ars Conjectandi," James Bernoulli meletakkan prinsip-prinsip dasar probabilitas, yang sangat penting untuk statistik, penilaian risiko, dan pengambilan keputusan.
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Fisika dan Mekanika: "Hydrodynamique" karya Daniel Bernoulli memperkenalkan prinsip-prinsip yang menjelaskan aliran fluida dan konservasi energi. Karyanya tentang teori kinetik gas membantu menjelaskan hukum gas, yang mendasar dalam kimia dan fisika.
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Notasi Matematika: John Bernoulli berkontribusi pada notasi yang digunakan dalam kalkulus, seperti menggunakan φ(x) untuk menunjukkan fungsi, yang masih digunakan hingga saat ini.
Kontribusi ini bukan hanya fakta sejarah; mereka membentuk tulang punggung dari banyak disiplin ilmu dan teknik.
Pelajaran dan Inspirasi untuk Siswa
Mempelajari kisah dan karya keluarga Bernoulli menawarkan beberapa pelajaran berharga:
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Ketekunan dan Semangat: Keluarga Bernoulli sangat bersemangat tentang matematika dan sains. Dedikasi mereka, meskipun ada konflik pribadi dan profesional, menunjukkan pentingnya ketekunan dalam belajar dan penemuan.
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Pemikiran Interdisipliner: Karya mereka menggabungkan matematika, fisika, dan filsafat, mendorong siswa untuk berpikir luas dan menghubungkan berbagai bidang pengetahuan.
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Inovasi dan Aplikasi: Mereka menunjukkan bagaimana ide-ide matematika abstrak dapat diterapkan untuk memecahkan masalah dunia nyata, menginspirasi siswa untuk mencari penggunaan praktis untuk pembelajaran mereka.
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Etika dan Kolaborasi: Meskipun beberapa anggota keluarga memiliki konflik, warisan keseluruhan menyoroti pentingnya berbagi pengetahuan dan bekerja sama untuk memajukan sains.
Bagaimana Siswa Dapat Menerapkan Wawasan Ini
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Dalam Pembelajaran: Meniru rasa ingin tahu keluarga Bernoulli dengan menjelajahi di luar buku teks. Cobalah untuk memahami 'mengapa' di balik rumus dan teori, dan menerapkannya untuk memecahkan masalah.
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Dalam Kehidupan Sehari-hari: Gunakan pemikiran logis dan keterampilan memecahkan masalah dalam keputusan sehari-hari. Memahami probabilitas, misalnya, dapat membantu dalam membuat pilihan yang tepat.
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Dalam Interaksi Sosial: Kisah keluarga Bernoulli juga mengajarkan tentang nilai kerendahan hati dan rasa hormat dalam kolaborasi. Mengenali kontribusi orang lain dapat mengarah pada kerja tim yang lebih baik.
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Mengembangkan Sikap Positif: Kembangkan pola pikir pembelajaran dan ketahanan seumur hidup. Keluarga Bernoulli menghadapi tantangan tetapi terus berinovasi, contoh yang bagus bagi siswa yang menghadapi kesulitan akademis atau pribadi.
Mendorong Semangat Bernoulli
Untuk memupuk semangat keluarga Bernoulli, siswa harus:
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Terlibat secara aktif dengan mata pelajaran yang menantang seperti matematika dan sains, melihatnya sebagai alat untuk memahami dunia.
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Berpartisipasi dalam diskusi, debat, dan proyek kolaboratif untuk mengembangkan keterampilan komunikasi dan kerja tim.
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Merefleksikan dimensi etika dari karya ilmiah, menghargai kejujuran dan integritas.
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Jelajahi kisah-kisah sejarah para ilmuwan untuk menghargai sisi manusia dari penemuan, membuat pembelajaran lebih mudah dipahami dan menginspirasi.
Dengan mempelajari keluarga Bernoulli, siswa tidak hanya mendapatkan pengetahuan tetapi juga mempelajari sikap dan keterampilan yang akan melayani mereka di banyak bidang kehidupan, dari akademisi hingga pertumbuhan pribadi dan hubungan sosial.
