Mathematics for Jakob–Calculus, Geometry, Probability Theory

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Jakob Bernoulli (1654–1705), the most distinguished of the family, made discoveries in calculus, which he used to solve minimization problems, and he contributed to geometry and the theory of probabilities. Jakob’s Ars Conjectandi (The Art of Conjecturing, 1713) was one of the most important early contributions to probability theory.(1) He gave the Bernoulli law of large numbers, the first precise mathematical theorem to interpret probabilities in terms of observable relative frequencies–basic to all modern sampling theory. One area where this law applies is simple games involving coins, cards, dice, and roulette wheels where the outcome of a given trial cannot be predicated with certainty, although the collective results of a large number of trials display some regularity. Another example is actuarial statements about the life expectancy for persons of a certain age, which describe the collective experience of a large number of individuals but do not purport to say what will happen to any particular person. The Art of Conjecturing also contains his theory of permutations and combinations and the so-called Bernoulli numbers, by which he derived the exponential series.(2)

Mathematics for Johann–Calculus

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Jakob’s brother Johann Bernoulli (1667–1748) also contributed to differential and integral calculus, exceeding his elder brother in the number of contributions made to mathematics.

Together, Jakob and Johann Bernoulli, enthusiastic followers of Leibniz, became the major force in disseminating and promoting the calculus throughout Europe. As an example, the Bernoulli brothers were the first to give public lectures on calculus. Their efforts, perhaps as much as those of Leibniz himself, gave the subject the flavor and appearance that it retains to this day.(3)

Bio

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Johann and Jakob were professors of mathematics at Basle. Daniel Bernoulli (1700–1782), son of Johann, was professor of mathematics at St. Petersburg and then held successively the chairs of botany, physiology, and physics at Basle.

For Daniel–Hydrodynamics Including Bernoulli’s Principle, Mathematical Physics

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Although Daniel Bernoulli’s original studies were in medicine, his greatest contributions were to hydrodynamics and mathematical physics.(4) Daniel’s reputation was established in 1738 with Hydrodynamica (Hydrodynamics), in which he considered the properties of basic importance in fluid flow, particularly pressure, density, and velocity, and set forth their fundamental relationship. He put forward what is called Bernoulli’s principle, which states that the pressure in a fluid decreases as its velocity increases. Note, this as an example of a function, or the relation, of two variables, namely pressure and velocity. In this case, pressure has an inverse relationship to velocity.

Daniel Bernoulli also established the basis for the kinetic theory of gases and heat by demonstrating that the impact of molecules on a surface would explain pressure and that, assuming the constant, random motion of molecules, pressure and motion increase with temperature.(5) Finally, Daniel first derived what is called Bernoulli’s theorem, in fluid dynamics, relating the pressure, velocity, and elevation in a moving fluid (liquid or gas). His theorem is the basis for many engineering applications, such as aircraft-wing design. The air flowing over the upper curved surface of an aircraft wing moves faster than the air beneath the wing, so that the pressure underneath is greater than that on the top of the wing, causing lift. This is the fundamental basis of how an airplane is able to fly.(6)

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Footnotes:

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(1) Encyclopaedia Britannica, Macropaedia, Volume 26, 1993, 15th Edition, p. 135.

(2) Encyclopaedia Britannica, Micropaedia, Volume 2, 1993, 15th Edition, p. 154.

(3) William Dunham, Journey Through Genius - The Great Theorems of Mathematics (New York, 1990), p. 190.

(4) Judy Pearsall and Bill Trumble (editors), The Oxford Encyclopedic English Dictionary (New York, 1996), p. 135.

(5) Encyclopaedia Britannica, Micropaedia, Volume 2, 1993, 15th Edition, p. 153.

(6) Ibid., p. 154.

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