Bernoulli Sentence Examples
In 1738 Daniel Bernoulli (1700-1782) published his Hydrodynamica seu de viribus et motibus fluidorum commentarii.
The theory of Daniel Bernoulli was opposed also by Jean le Rond d'Alembert.
Jean Bernoulli (1710-1790), the youngest of the three sons of Jean Bernoulli, was born at Basel on the 18th of May 1710.
The 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.
The 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.Advertisement
The 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.
Diophantine 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.
Bernoulli 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.
These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others.
Colin 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.Advertisement
The 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.
When 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.
He made use of the same suppositions as Daniel Bernoulli, though his calculus was established in a very different manner.
Equation (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.
If 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.Advertisement
Newton 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.
Jacques Bernoulli (1654-I 705), mathematician, was born at Basel on the 27th of December 1654.
Jacques 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.
Jacques 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.
Nicolas Bernoulli (1695-1726), the eldest of the three sons of Jean Bernoulli, was born on the 27th of January 1695.Advertisement
Daniel Bernoulli (1700-1782), the second son of Jean Bernoulli, was born on the 29th of January 1700, at Groningen.
Six 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.
Bernoulli adopted the suggestion, and publicly announced the prorogation for the information of those who might not see the Acta Lipsiensia.
Solutions 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."
Tschirnhausen 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.Advertisement
His doctrine of chances of 1718 greatly expanded the mathematical theory of probability which Bernoulli had started in 1713.
Bernoulli, Die erhaltenen Darstellungen Alexanders d.
In 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.
It 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.
Jean le Rond d'Alembert acknowledges with gratitude, that "whatever she knew of mathematics he owed to the works of Jean Bernoulli."
He 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.
In 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.
His Doctrine of Chances of 1718 greatly expanded the mathematical theory of probability which Bernoulli had started in 1713.
With a success equalled only by Leonhard Euler, Daniel Bernoulli gained or shared no less than ten prizes of the Academy of Sciences of Paris.
The 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.
Nicolas 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.
Jean 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.
Jacques Bernoulli (1759-1789), younger brother of the preceding, and the second of this name, was born at Basel on the 17th of October 1759.
The 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.
That is, the length of the pipe must be increased by o 82 R before applying Bernoulli's theory.
Although 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.
On Pliny's supposed portrait, see Bernoulli, Rom.
This is (in part) the celebrated principle of virtual velocities, now often described as the principle of virtual work, enunciated by John Bernoulli (1667-1748).
His 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.
In 1730 he became professor of physics, and in 1733 he succeeded Daniel Bernoulli in the chair of mathematics.
The 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.
Newton's solution of the celebrated problems proposed by John Bernoulli and Leibnitz deserves mention among his mathematical works.
In 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.