bernoulli

The theory of Daniel Bernoulli was opposed also by Jean le Rond d'Alembert.

101 3p6xcv-ros, shortest, and Xpovos, time), a term invented by John Bernoulli in 1694 to denote the curve along which a body passes from one fixed point to another in the shortest time.

00Bernoulli, this theory was advanced by the successive labours of John Herapath, J.

00Xnyviaicos, ribbon), a quartic curve invented by Jacques Bernoulli (Acta Eruditorum, 1694) and afterwards investigated by Giulio Carlo Fagnano, who gave its principal properties and applied it to effect the division of a quadrant into 2 2 m, 3.2 m and 5.2 m equal parts.

00The lemniscate of Bernoulli may be defined as the locus of a point which moves so that the product of its distances from two fixed points is constant and is equal to the square of half the distance between these points.

00The same name is also given to the first positive pedal of any central conic. When the conic is a rectangular hyperbola, the curve is the lemniscate of Bernoulli previously described.

00are certain numbers known as Bernoulli's numbers, and the terms within the bracket, after the first, have signs alternately + and -.

00The values of the first ten of Bernoulli's numbers are B1= t, B2= 1, B3 =412, B4 =30, B5 =6 = 6 9 1 B7 = l, B =3 =4 4 fl, IV.

00are Bernoulli's numbers (ï¿½ 46 (v.)), although the series is not convergent.

00Diophantine problems were revived by Gaspar Bachet, Pierre Fermat and Euler; the modern theory of numbers was founded by Fermat and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and Fermat, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and others.

00Bernoulli also considered the cases when (I) the chain was of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre.

00These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others.

00In 1738 Daniel Bernoulli (1700-1782) published his Hydrodynamica seu de viribus et motibus fluidorum commentarii.

00Colin Maclaurin (1698-1746) and John Bernoulli (1667-1748), who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742, and the other in his Hydraulica nunc primum detecta, et demonstrata directe ex fundamentis pure mechanicis, which forms the fourth volume of his works.

00The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernoulli is, in the opinion of Lagrange, defective in clearness and precision.

00When generalizing the theory of pendulums of Jacob Bernoulli (1654-1705) he discovered a principle of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equilibrium.

00He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner.

00Equation (3) is called Bernoulli's equation, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow.

00If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli's equation may be written H = PIP--z - F4 2 / 2g = P/ p +h, (8) where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward.

00Newton gave no proof, and it was in the Ars Conjectandi (1713) that James Bernoulli's proof for positive integral values of the exponent was first published, although Bernoulli must have discovered it many years previously.

00BERNOULLI, or Bernouilli, the name of an illustrious family in the annals of science, who came originally from Antwerp. Driven from their country during the oppressive government of Spain for their attachment to the Reformed religion, the Bernoullis sought first an asylum at Frankfort (1583), and afterwards at Basel, where they ultimately obtained the highest distinctions.

00Jacques Bernoulli (1654-I 705), mathematician, was born at Basel on the 27th of December 1654.

00Jacques Bernoulli cannot be strictly called an independent discoverer; but, from his extensive and successful application of the calculus and other mathematical methods, he is deserving of a place by the side of Newton and Leibnitz.

00Jacques Bernoulli wrote elegant verses in Latin, German and French; but although these were held in high estimation in his own time, it is on his mathematical works that his fame now rests.

00These are: - Jacobi Bernoulli Basiliensis Opera (Genevae, 1 744), 2 tom.

00[[Jean Bernoulli]] (1667-1748), brother of the preceding, was born at Basel on the 27th of July 1667.

00His dissertation on the "barometric light," first observed by Jean Picard, and discussed by Jean Bernoulli under the name of mercurial phosphorus, or mercury shining in vacuo (Diss.

00His writings were collected under his own eye by Gabriel Cramer, professor of mathematics at Geneva, and published under the title of Johannis Bernoulli Operi Omnia (Lausan.

00Leibnitii et Johannis Bernoulli Commercium Philosophicum et Mathematicum (Lausan.

00Nicolas Bernoulli (1695-1726), the eldest of the three sons of Jean Bernoulli, was born on the 27th of January 1695.

00Daniel Bernoulli (1700-1782), the second son of Jean Bernoulli, was born on the 29th of January 1700, at Groningen.

00With a success equalled only by Leonhard Euler, Daniel Bernoulli gained or shared no less than ten prizes of the Academy of Sciences of Paris.

00The problem of vibrating cords, which had been some time before resolved by Brook Taylor (1685-1731) and d'Alembert, became the subject of a long discussion conducted in a generous spirit between Bernoulli and his friend Euler.

00Jean Bernoulli (1710-1790), the youngest of the three sons of Jean Bernoulli, was born at Basel on the 18th of May 1710.

00Nicolas Bernoulli (1687-1759), cousin of the three preceding, and son of Nicolas Bernoulli, one of the senators of Basel, was born in that city on the 10th of October 1687.

00Jean Bernoulli' (1744-1807), grandson of the first Jean Bernoulli, and son of the second of that name, was born at Basel on the 4th of November 1744.

00Jacques Bernoulli (1759-1789), younger brother of the preceding, and the second of this name, was born at Basel on the 17th of October 1759.

00The approximate theory of pipes due to Bernoulli assumes a loop at the open end, but the condition for a loop at the open end, that of no pressure variation, cannot be exactly fulfilled.

00That is, the length of the pipe must be increased by o 82 R before applying Bernoulli's theory.

00Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 Daniel Bernoulli obtained the function of zero order as a solution to the problem of the oscillations of a chain suspended at one end.

00On Pliny's supposed portrait, see Bernoulli, Rom.

00Stauffacher (Basel, 1899) August Bernoulli, and in his elaborate Geschichte d.

00Bernoulli >>

00This is (in part) the celebrated principle of virtual velocities, now often described as the principle of virtual work, enunciated by John Bernoulli (1667-1748).

00The investigation of caustics, being based on the assumption of the rectilinear propagation of light, and the validity of the experimental laws of reflection and refraction, is essentially of a geometrical nature, and as such it attracted the attention of the mathematicians of the 17th and succeeding centuries, more notably John Bernoulli, G.

00Bernoulli, Aphrodite (1873); W.

00His mathematical genius gained for him a high place in the 'esteem of Jean Bernoulli, who was at that time one of the first mathematicians in Europe, as well as of his sons Daniel and Nicolas Bernoulli.

00In 1730 he became professor of physics, and in 1733 he succeeded Daniel Bernoulli in the chair of mathematics.

00The Academy of Sciences at Paris in 1738 adjudged the prize to his memoir on the nature and properties of fire, and in 1740 his treatise on the tides shared the prize with those of Colin Maclaurin and Daniel Bernoulli - a higher honour than if he had carried it away from inferior rivals.

00Bernoulli, Winkelrieds That bei Sempach (Basel, 1886); W.

00Newton's solution of the celebrated problems proposed by John Bernoulli and Leibnitz deserves mention among his mathematical works.

00In June 1696 Bernoulli addressed a letter to the mathematicians of Europe challenging them to solve two problems - (1) to determine the brachistochrone between two given points not in the same vertical line, (2) to determine a curve such that, if a straight line drawn through a fixed point A meet it in two points P 1, P 2, then AP 1 m +AP 2 m will be constant.

00Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own.

00Bernoulli adopted the suggestion, and publicly announced the prorogation for the information of those who might not see the Acta Lipsiensia.

00Solutions were also obtained from Leibnitz and the Marquis de L'Hopital; and, although that of Newton was anonymous, yet Bernoulli recognized the author in his disguise; " tanquam," says he, " ex ungue leonem."

00Tschirnhausen were appointed on the 4th of February, James Bernoulli and John Bernoulli on the 14th of February, and Newton and Olaus Roemer on the 21st of February.

00On the 31st of March 1713, when the edition was nearly ready for publication, Newton wrote to Cotes: " I heare that Mr Bernoulli has sent a Paper of 40 pages to be published in the Acta Leipsica relating to what I have written upon the curve Lines described by Projectiles in resisting Mediums. And therein he partly makes Observations upon what I have written & partly improves it.

00His doctrine of chances of 1718 greatly expanded the mathematical theory of probability which Bernoulli had started in 1713.

00Bernoulli, Die erhaltenen Darstellungen Alexanders d.

00In Bode's Jahrbuch (1776-1780) he discusses nutation, aberration of light, Saturn's rings and comets; in the Nova acta Helvetica (1787) he has a long paper "Sur le son des corps elastiques," in Bernoulli and Hindenburg's Magazin (1787-1788) he treats of the roots of equation and of parallel lines; and in Hindenburg's Archiv (1798-1799) he writes on optics and perspective.

00are Bernoulli's numbers (Ã¯¿½ 46 (v.)), although the series is not convergent.

00It was investigated by Galileo, who erroneously determined it to be a parabola; Jungius detected Galileo's error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Aeta Eruditorum.

00p. r1), his claim to priority was unjustly disputed by John Bernoulli.

00Jean le Rond d'Alembert acknowledges with gratitude, that "whatever she knew of mathematics he owed to the works of Jean Bernoulli."

00He was wont to mention the following as the two incidents in his life which had afforded him the greatest pleasure, - that a stranger, whom he had met as a travelling companion in his youth, made to his declaration "I am Daniel Bernoulli" the incredulous and mocking reply, "And I am Isaac Newton"; and that, while entertaining Kdnig and other guests, he solved without rising from table a problem which that mathematician had submitted as difficult and lengthy.

00In 1697 John Bernoulli proposed the famous problem of the brachistochrone (see Mechanics), and it was proved by Leibnitz, Newton and several others that the cycloid was the required curve.

00His Doctrine of Chances of 1718 greatly expanded the mathematical theory of probability which Bernoulli had started in 1713.

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