Euler (1 777).
The young Ampere, however, soon resumed his Latin lessons, to enable him to master the works of Euler and Bernouilli.
The complete analytical treatment was first given by Leonhard Euler.
Bonnet, Euler, Haller, Schmid and others " suppose miracles to be already implanted in nature.
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
a3 a 2 0 b 2 b1 0 a 3 0 0 b2 This is Euler's method.
For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.
The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers), which Riemann read, mastered and returned within a few days.
At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with "isoperimetrical" problems, known later as the Calculus of Variations.
He made his first appearance in public as the critic of Newton, and the arbiter between d'Alembert and Euler.
The essential point in his advance on Euler's mode of investigating curves of maximum or minimum consisted in his purely analytical conception of the subject.
He is thus justly regarded as the inventor of the "method of variations" - a name supplied by Euler in 1766.
Euler's eulogium was enhanced by his desire to quit Berlin, d'Alembert's by his dread of a royal command to repair thither; and the result was that an invitation, conveying the wish of the "greatest king in Europe" to have the "greatest mathematician" at his court, was sent to Turin.
This is especially the case between Lagrange and Euler on the one side, and between Lagrange and Laplace on the other.
The calculus of variations lay undeveloped in Euler's mode of treating isoperimetrical problems. The fruitful method, again, of the variation of elements was introduced by Euler, but adopted and perfected by Lagrange, who first recognized its supreme importance to the analytical investigation of the planetary movements.
To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.
The first, second and third sections of this publication comprise respectively the papers communicated by him to the Academies of Sciences of Turin, Berlin and Paris; the fourth includes his miscellaneous contributions to other scientific collections, together with his additions to Euler's Algebra, and his Lecons elementaires at the Ecole Normale in 1795.
His development of the equation x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.
Euler and J.
In this they were completely successful, for they obtained general solutions for the equations ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b.
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.
These equations were found by d'Alembert from two principles - that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium, and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His ingenious method, published in 1752, in his Essai sur la resistance des fluides, was brought to perfection in his Opuscules mathematiques, and was adopted by Leonhard Euler.
The resolution of the questions concerning the motion of fluids was effected by means of Euler's partial differential coefficients.
This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.
Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler.
Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0, dT F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (FG - x 3 y 3), (16) _Ft (FG _x 323 Frdx3 (17) F x3 X3 elliptic integrals of the third kind.
A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and fractional values being faulty, and was finally given by Niels Heinrik Abel.
He was as keen in his resentments as he was ardent in his friendships; fondly attached to his family, he yet disliked a deserving son; he gave full praise to Leibnitz and Leonhard Euler, yet was blind to the excellence of Sir Isaac Newton.
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 first, for a memoir on the construction of a clepsydra for measuring time exactly at sea, he gained at the age of twenty-four; the second, for one on the physical cause of the inclination of the planetary orbits, he divided with his father; and the third, for a communication on the tides, he shared with Euler, Colin Maclaurin and another competitor.
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.
In 1774 he published a French translation of Leonhard Euler's Elements of Algebra.
He was tragically drowned while bathing in the Neva in July 1789, a few months after his marriage with a daughter of Albert Euler, son of Leonhard Euler.
His memoir (1775) on the rotatory motion of a body contains (as the author was aware) conclusions at variance with those arrived at by Jean le Rond, d'Alembert and Leonhard Euler in their researches on the same subject.
Tartaglia, Nova Scientia (1537) Galileo (1638); Robins, New Principles of Gunnery (1743); Euler (trans.
In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessel's functions are particular cases of Laplace's functions.
As far as the circlesquaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries 1 See Euler, ” Annotationes in locum quendam Cartesii," in Nov.
Leonhard Euler took up the subject several times during his life, effecting mainly improvements in the theory of the various series.
In these last Mayer sought to explain the magnetic action of the earth by a modification of Euler's hypothesis, and made the first really definite attempt to establish a mathematical theory of magnetic action (C. Hansteen, Magnetismus der Erde, i.
Euler, who added to it a critical commentary of his own.
Leonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him.
Leonhard Euler in 1747 had suggested that achromatism might be obtained by the combination of glass and water lenses.
John Dollond, to whom the Copley medal of the Royal Society had been the first inventor of the achromatic telescope; but it was ruled by Lord Mansfield that" it was not the person who locked his invention in his scrutoire that ought to profit for such invention, but he who brought it forth for the benefit of mankind."3 In 1747 Leonhard Euler communicated to the Berlin Academy of Sciences a memoir in which he endeavoured to prove the possibility of correcting both the chromatic and.
Dollond admitted the accuracy of Euler's analysis, but disputed his hypothesis on the grounds that it was purely a theoretical assumption, that the theory was opposed to the results of Newton's experiments on the refrangibility of light, and that it was impossible to determine a physical law from analytical reasoning alone (Phil.
In 1754 Euler communicated to the Berlin Academy a further memoir, in - which, starting from the hypothesis that light consists of vibrations excited in an elastic fluid by luminous bodies, and that the difference of colour of light is due to the greater or less frequency of these vibrations in a given time, he deduced his previous results.
Argand had been led to deny that such an expression as i 2 could be expressed in the form A+Bi, - although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of --7112.
Hamilton at once found that the Law of the Norms holds, - not being aware that Euler had long before decomposed the product of two sums of four squares into this very set of four squares.
It was shown by Euler (1776) that any displacement in which one point 0 of the body is fixed is equivalent to a pure, rotation about some axis through 0.
The question was first discussed by Euler (1750); the geometrical representation to be given is due to Poinsot (1851).
The phenomenon is known as the Eulerian nutalion, since it is supposed to come under the free rotations first discussed by Euler.
These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated.
It will be suftlcient to take the case of motion about a fixed point 0; the angular co-ordinates 0, ~, i,l of Euler may then be used for the purpose.
James Gregory and Leonhard Euler arrived at the correct view from a false conception of the achromatism of the eye; this was determined by Chester More Hall in 1728, Klingenstierna in 1754 and by Dollond in 1757, who constructed the celebrated achromatic telescopes.
Similarly the continued fraction given by Euler as equivalent to 1(e - 1) (e being the base of Napierian logarithms), viz.
qn= k l a2, a3,, an The theory of continuants is due in the first place to Euler.
-u,-1+un' a result given by Euler.
Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's Algebra (1795).
Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of; (a+b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii.
Competent judges have compared him to Leonhard Euler for his range, analytical power and introduction of new and fertile theories.
The problem set was "to perfect in one important point the theory of the movement of a solid body round an immovable point," and her solution added a result of the highest interest to those transmitted to us by Leonhard Euler and J.
Euler and G.
LEONHARD EULER (1707-1783), Swiss mathematician, was born at Basel on the 15th of April 5707, his father Paul Euler, who had considerable attainments as a mathematician, being Calvinistic pastor of the neighbouring village of Riechen.
Having taken his degree as master of arts in 1723, Euler applied himself, at his father's desire, to the study of theology and the Oriental languages with the view of entering the church, but, with his father's consent, he soon returned to geometry as his principal pursuit.
In 1727, on the invitation of Catherine I., Euler took up his residence in St Petersburg, and was made an associate of the Academy of Sciences.
In 1735 a problem proposed by the academy, for the solution of which several eminent mathematicians had demanded the space of some months, was solvecdby Euler in three days,but the effort threw him into a fever which endangered his life and deprived him of the use of his right eye.
In 1741 Euler accepted the invitation of Frederick the Great to Berlin, where he was made a member of the Academy of Sciences and professor of mathematics.
On its being ascertained that the farm belonged to Euler, the general immediately ordered compensation to be paid, and the empress Elizabeth sent an additional sum of four thousand crowns.
In 1766 Euler with difficulty obtained permission from the king of Prussia to return to St Petersburg, to which he had been originally invited by Catherine II.
Theory, however, is frequently unsoundly applied in it, and it is to be observed generally that Euler's strength lay rather in pure than in applied mathematics.
In 1755 Euler had been elected a foreign member of the Academy of Sciences at Paris, and some time afterwards the academical prize was adjudged to three of his memoirs Concerning the Inequalities in the Motions of the Planets.
Euler, assisted by his eldest son Johann Albert, was a competitor for these prizes, and obtained both.
The inherent difficulties of this task were immensely enhanced by the fact that Euler was virtually blind, and had to carry all the elaborate computations it involved in his memory.
Some time after this an operation restored Euler's sight; but a too harsh use of the recovered faculty, along with some carelessness on the part of the surgeons, brought about a relapse.
Euler's knowledge was more general than might have been expected in one who had pursued with such unremitting ardour mathematics and astronomy as his favourite studies.
Euler's constitution was uncommonly vigorous, and his general health was always good.
Euler's genius was great and his industry still greater.
The works which Euler published separately are: Dissertatio physica de sono (Basel, 1727, in 4to); Mechanica, sive motus scientia analytice exposita (St Petersburg, 1736, in 2 vols.
See Rudio, Leonhard Euler (Basel, 1884); M.
The value of the constant known as Euler's, and the Bernoullian numbers up to the 62nd, he worked out to an unimagined degree of accuracy.
Euler (Ber., 97, 30, 1989) by distilling the addition compound of methyl iodide and 2 3 5-trimethylpyrollidine with caustic potash.
They were accordingly taken up anew by a band of continental inquirers, primarily by three men of untiring energy and vivid genius, Leonhard Euler, Alexis Clairault, and Jean le Rond d'Alembert.
But the apparent anomaly disappeared under Euler's powerful treatment in 1749, and his result was shortly afterwards still further assured by Clairault.
Euler devised in 1753 a new method, that of the " variation of parameters," for their investigation, and applied it to unravel some of the earth's irregularities in a memoir crowned by the French Academy in 1756; while in 1757, Clairault estimated the masses of the moon and Venus by their respective disturbing effects upon terrestrial movements.
iv.); and in the same year expanded Euler's adumbrated method of the variation of parameters into a highly effective engine of perturbational research.
He executed besides a chart and forty drawings of the moon (published at GÃ¶ttingen in 1881), and calculated lunar tables from a skilful development of Euler's theory, for which a reward of boo() was in 1765 paid to his widow by the British government.
Euler conceived the idea of starting with a preliminary solution of the problem in which the orbit of the moon should be supposed to lie in the ecliptic, and to have no eccentricity, while that of the sun was circular.
In a series of remarkable papers published in1877-1888Hill improved Euler's method, and worked it out with much more rigour and fullness than Euler had been able to do.
Euler, Theoria motuum lunae nova methodo pertractata (Petropolis, 1772); G.
A connexion between the number of faces, vertices and edges of regular polyhedra was discovered by Euler, and the result, which assumes the form E + 2' = F ± V, where E, F, V are the number of edges, faces and vertices, is known as Euler's theorem on polyhedra.
Euler, Sur la cause de la lumiere zodiacale (1746); Mairan, Sur la cause de la lumiere zodiacale (1747); R.
But if, instead of rotating around PP, it rotates around some other axis, RR, making a small angle, POR, with the axis of figure PP; then it has been known since the time of Euler that the axis of rotation RR, if referred to the spheroid regarded as fixed, will gradually rotate round the axis of figure PP in a period defined in the following way: - If we put C = the moment of momentum of the spheroid around the axis of figure, and A = the corresponding moment around an axis passing through the equator EQ, then, calling one day the period of rotation of the spheroid, the axis RR will make a revolution around PP in a number of days represented by the fraction C/(C - A).
This was afterwards disproved by Leonhard Euler for the case when n= 5.
This paper offers a glimpse of how Euler used infinitesimals and infinite series to compute differentials for the elementary functions.
In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not the title of President.
mathematical physics had led Euler to a wide study of differential equations.
Euler's work on fluid mechanics is also quite remarkable.
Integrating the automatic layout of Euler Diagrams with the layout of graphs and less structured diagrammatic notations.
For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=Ã¯¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
The post of director of the mathematical department of the Berlin Academy (of which he had been a member since 1759) becoming vacant by the removal of Euler to St Petersburg, the latter and d'Alembert united to recommend Lagrange as his successor.
Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see Circle), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions.
The theoretical assumptions of Newton and Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the quadratic or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values,, t, cos 7 7, and sec r t, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.
NUMBERS This mathematical subject, created by Euler, though relating essentially to positive integer numbers, is scarcely regarded as a part of the Theory of Numbers (see Number).
Euler (see Differential Equations).
In his optical researches, Optiska Undersiikningar, presented to the Stockholm Academy in 1853, he not only pointed out that the electric spark yields two superposed spectra, one from the metal of the electrode and the other from the gas in which it passes, but deduced from Euler's theory of resonance that an incandescent gas emits luminous rays of the same refrangibility as those which it can absorb.
Gregory's series and the identities 7 r /4 =5 tan1 + + 2 tan-',A (Euler, 1779), 7r/4 = tani ++2 tan-' s (Hutton, 1776), neither of which was nearly so advantageous as several found by Charles Hutton, calculated 7r correct to 136 places."
He executed besides a chart and forty drawings of the moon (published at GÃƒ¶ttingen in 1881), and calculated lunar tables from a skilful development of Euler's theory, for which a reward of boo() was in 1765 paid to his widow by the British government.
to describe a circle touching three given circles, which he discovered in 1805, his generalization of Euler's theorem on polyhedra in 1811, and in several other elegant problems. More important is his memoir on wave-propagation which obtained the Grand Prix of the Institut in 1816.
Block diagrams illustrate the conversion of an airplane motion parameters from body to earth axis for Euler angles and for the quaternion parameters.
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