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Johann Bernoulli

From Wikipedia, the free encyclopedia

Johann Bernoulli
Johann Bernoulli2.jpg
Johann Bernoulli (portrait by Johann Rudolf Huber, circa 1740)
Born(1667-08-06)6 August 1667
Died1 January 1748(1748-01-01) (aged 80)
ResidenceSwitzerland
NationalitySwiss
Alma materUniversity of Basel
(M.D., 1694)
Known forDevelopment of infinitesimal calculus
Catenary solution
Bernoulli's rule
Bernoulli's identity
Brachistochrone problem
Scientific career
FieldsMathematics
InstitutionsUniversity of Groningen
University of Basel
ThesisDissertatio de effervescentia et fermentatione; Dissertatio Inauguralis Physico-Anatomica de Motu Musculorum (On the Mechanics of Effervescence and Fermentation and on the Mechanics of the Movement of the Muscles) (1694 (1690)[1])
Doctoral advisorJacob Bernoulli
Other academic advisorsNikolaus Eglinger
Doctoral studentsDaniel Bernoulli
Leonhard Euler
Johann Samuel König
Pierre Louis Maupertuis
Other notable studentsGuillaume de l'Hôpital
Notes

Johann Bernoulli (also known as Jean or John; 6 August [O.S. 27 July] 1667 – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth.

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  • ✪ The Bernoullis: When Math is the Family Business
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  • ✪ Johann Bernoulli & Guillaume De L'Hospital -- Presentation 1
  • ✪ Daniel Bernoulli ~ Hydrodynamics

Transcription

If you’ve ever taken a science or math class, you’re probably used to lots of things being named after the same people. There are hundreds of principles and theorems and equations named after superstars like Euler and Laplace and Gauss. So it might not seem all that weird to learn about Bernoulli’s brachistochrone solution, Bernoulli’s equation in fluid dynamics, and the Bernoulli probability distribution. And you would assume that they’re all named after the same person. But they’re not! In the seventeenth and eighteenth centuries, there were actually eight mathematically gifted Bernoullis, from three generations of the same family. But three of the Bernoullis really stood out: Jacob, Johann, and Daniel. They were all descendants of Nikolaus Bernoulli, who was born in 1623. Nikolaus had three sons, who he named Jacob, Nikolaus — after himself — and Johann. The younger Nikolaus became a painter and a city official, but his brothers were famous mathematicians. Jacob, the oldest, solved a number of early calculus problems that are still taught in schools today. But Jacob’s most important discoveries were in probability. He was the first person to prove something called the law of large numbers, and he came up with what’s now known as the Bernoulli distribution. Both are ways of predicting a series of random events. Say you roll a six-sided die a bunch of times in a row. If you roll the die, say, 3 times, you might get a 1, a 4, and a 6. You also might roll three 2s. It’s random. But the law of large numbers says that as you roll the die more times, the average of all the numbers you rolled will get closer and closer to 3.5 — the average of all the possible rolls. That’s because after a lot of rolls — say, a thousand — you’ll probably have rolled each number roughly the same amount of times. So the average of all your rolls will be very close to 3.5. Jacob Bernoulli was the first person to officially prove this, and it became one of the early fundamental concepts of probability theory. The Bernoulli distribution is a similar idea that applies the law of large numbers to something like a coin flip, where there are only two possible outcomes. Jacob also discovered the famous mathematical constant e, which is used all the time in math and the sciences. It can describe anything that grows continuously — from bacteria to a bank account that’s accumulating interest. Jacob’s younger brother Johann was also interested in math. In 1696, he posed a fun problem for the world’s mathematicians: What’s the fastest path for a ball to follow if it rolls down a track between two points? He called it the brachistochrone problem, from the Greek words for “shortest” and “time”. You might assume that the fastest path would be a straight line. And a straight line is the shortest path. But it’s not the fastest one. The fastest path for a ball to roll between two points is actually a kind of stretched out piece of a circle called a cycloid, because of the way gravity makes the ball accelerate. Galileo and a few others had figured this out on a conceptual level, but Johann used calculus to prove it. Jacob then found his own solution, laying the foundation for a whole new branch of calculus while he was at it. Johann also discovered ways of finding an answer to things like 0 divided by 0 and infinity divided by infinity. His method is now known as l’Hôpital’s rule, but all Guillaume de l’Hôpital did was publish his notes from Johann’s calculus lectures. Johann’s three sons were all mathematicians, as well, but the probably most influential one was Daniel. One of the things that’s named after him is Bernoulli’s principle, a concept in fluid dynamics that describes the relationship between pressure and speed in a moving fluid. Bernoulli’s principle is part of why airplanes fly: the air on the top of the wings is going faster than the air on the bottom. In physics, Daniel worked with the famous Euler to develop Euler-Bernoulli beam theory, . Euler-Bernoulli beam theory and the physics based on it are still super useful for engineers these days — there are, of course, beams and our bridges and buildings. So that’s, we should understand them. Daniel also developed new mathematical methods of measuring risk, and he was one of the first physicists to study the behavior of gasses. Part of the reason the Bernoullis were so successful was that they were in the right place at the right time. Newton and Leibniz had just invented calculus, so there was this powerful new technique for describing the universe. But the Bernoullis were also talented mathematicians and scientists who used the tools they had to uncover all kinds of new things about the universe. Which is why their names are now all over our science and math textbooks. Even if it’s sometimes hard to keep track of which Bernoulli did what. Thanks for watching this episode of SciShow, which was brought to you by our patrons on Patreon. If you want to help support this show, just go to patreon.com/scishow. And don’t forget to go to youtube.com/scishow and subscribe!

Contents

Biography

Early life

Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, and his wife, Margaretha Schonauer, and began studying medicine at Basel University. His father desired that he study business so that he might take over the family spice trade, but Johann Bernoulli did not like business and convinced his father to allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob.[2] Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems.[3]

Adult life

After graduating from Basel University, Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner[a] and soon after accepted a position as the professor of mathematics at the University of Groningen. At the request of his father-in-law, Bernoulli began the voyage back to his home town of Basel in 1705. Just after setting out on the journey he learned of his brother's death to tuberculosis. Bernoulli had planned on becoming the professor of Greek at Basel University upon returning but instead was able to take over as professor of mathematics, his older brother's former position. As a student of Leibniz's calculus, Bernoulli sided with him in 1713 in the Leibniz–Newton debate over who deserved credit for the discovery of calculus. Bernoulli defended Leibniz by showing that he had solved certain problems with his methods that Newton had failed to solve. Bernoulli also promoted Descartes' vortex theory over Newton's theory of gravitation. This ultimately delayed acceptance of Newton's theory in continental Europe.[4]

Commercium philosophicum et mathematicum (1745), a collection of letters between Leibnitz and Bernoulli.
Commercium philosophicum et mathematicum (1745), a collection of letters between Leibnitz and Bernoulli.

In 1724, Johann Bernoulli entered a competition sponsored by the French Académie Royale des Sciences, which posed the question:

What are the laws according to which a perfectly hard body, put into motion, moves another body of the same nature either at rest or in motion, and which it encounters either in a vacuum or in a plenum?

In defending a view previously espoused by Leibniz, he found himself postulating an infinite external force required to make the body elastic by overcoming the infinite internal force making the body hard. In consequence, he was disqualified for the prize, which was won by Maclaurin. However, Bernoulli's paper was subsequently accepted in 1726 when the Académie considered papers regarding elastic bodies, for which the prize was awarded to Pierre Mazière. Bernoulli received an honourable mention in both competitions.

Disputes and controversy

Although Johann and his brother Jacob Bernoulli worked together before Johann graduated from Basel University, shortly after this, the two developed a jealous and competitive relationship. Johann was jealous of Jacob's position and the two often attempted to outdo each other. After Jacob's death Johann's jealousy shifted toward his own talented son, Daniel. In 1738 the father–son duo nearly simultaneously published separate works on hydrodynamics. Johann attempted to take precedence over his son by purposely and falsely predating his work two years prior to his son's.[5][6]

The Bernoulli brothers often worked on the same problems, but not without friction. Their most bitter dispute concerned the brachistochrone curve problem, or the equation for the path followed by a particle from one point to another in the shortest amount of time, if the particle is acted upon by gravity alone. Johann presented the problem in 1696, offering a reward for its solution. Entering the challenge, Johann proposed the cycloid, the path of a point on a moving wheel, also pointing out the relation this curve bears to the path taken by a ray of light passing through layers of varied density. Jacob proposed the same solution, but Johann's derivation of the solution was incorrect, and he presented his brother Jacob's derivation as his own.[7]

Bernoulli was hired by Guillaume de l'Hôpital for tutoring in mathematics. Bernoulli and l'Hôpital signed a contract which gave l'Hôpital the right to use Bernoulli's discoveries as he pleased. L'Hôpital authored the first textbook on infinitesimal calculus, Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes in 1696, which mainly consisted of the work of Bernoulli, including what is now known as l'Hôpital's rule.[8][9][10] Subsequently, in letters to Leibniz, Varignon and others, Bernoulli complained that he had not received enough credit for his contributions, in spite of the preface of his book:

I recognize I owe much to the insights of the Messrs. Bernoulli, especially to those of the young (John), currently a professor in Groningen. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.

Illustration from De motu corporum gravium published in Acta Eruditorum, 1713
Illustration from De motu corporum gravium published in Acta Eruditorum, 1713

Works

  • Bernoulli, Johann (1742). [Opera]. 1 (in Latin). Lausannae & Genevae: Marc Michel & C Bousquet. Retrieved 18 June 2015.
  • Bernoulli, Johann (1742). [Opera]. 2 (in Latin). Lausannae & Genevae: Marc Michel & C Bousquet. Retrieved 18 June 2015.
  • Bernoulli, Johann (1742). [Opera]. 3 (in Latin). Lausannae & Genevae: Marc Michel & C Bousquet. Retrieved 18 June 2015.
  • Bernoulli, Johann (1742). [Opera]. 4 (in Latin). Lausannae & Genevae: Marc Michel & C Bousquet. Retrieved 18 June 2015.

See also

References

Footnotes

  1. ^ Daughter of an Alderman of Basel

Citations

  1. ^ Published in 1690, submitted in 1694.
  2. ^ Sanford, Vera (2008) [1958]. A Short History of Mathematics (2nd ed.). Read Books. ISBN 978-1-4097-2710-1. OCLC 607532308.
  3. ^ The Bernoulli Family, by H. Bernhard, Doubleday, Page & Company, (1938)
  4. ^ Fleckenstein, Joachim O. (1977) [1949]. Johann und Jakob Bernoulli (in German) (2nd ed.). Birkhäuser. ISBN 3764308486. OCLC 4062356.
  5. ^ Darrigol, Olivier (September 2005). Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. ISBN 9780198568438.
  6. ^ Speiser, David; Williams, Kim (18 September 2008). Discovering the Principles of Mechanics 1600-1800: Essays by David Speiser. ISBN 9783764385644.
  7. ^ Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. p. 116. ISBN 0-7679-0816-3.
  8. ^ Maor, Eli (1998). e: The Story of a Number. Princeton University Press. p. 116. ISBN 0-691-05854-7. OCLC 29310868.
  9. ^ Coolidge, Julian Lowell (1990) [1963]. The mathematics of great amateurs (2nd ed.). Oxford: Clarendon Press. pp. 154–163. ISBN 0-19-853939-8. OCLC 20418646.
  10. ^ Struik, D. J. (1969). A Source Book in Mathematics: 1200–1800. Harvard University Press. pp. 312–6. ISBN 978-0-674-82355-6.

External links

This page was last edited on 30 September 2019, at 09:59
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