# Lagrange Sentence Examples

The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of

**Lagrange**and Laplace.His father, Joseph Louis

**Lagrange**, married Maria Theresa Gros, only daughter of a rich physician at Cambiano, and had by her eleven children, of whom only the eldest (the subject of this notice) and the youngest survived infancy.See also Kuenen's National Religions and Universal Religions (Hibbert lectures) and

**Lagrange**'s Etudes sur les religions simitiques (2nd ed.).Of this school, which had

**Lagrange**for its professor of mathematics, we have an amusing account in the life of Gilbert Elliot, 1st earl of Minto, who with his brother Hugh, afterwards British minister at Berlin, there made the acquaintance of Mirabeau.The genius of

**Lagrange**did not at once take its true bent.AdvertisementThis prosperous opening gave the key-note to

**Lagrange**'s career.The first volume of its memoirs,' published in the following year, contained a paper by

**Lagrange**entitled Recherches sur la nature et la propagation du son, in which the power of his analysis and his address in its application were equally conspicuous.By these performances

**Lagrange**found himself, at the age of twenty-six, on the summit of European fame.The prize was again awarded to

**Lagrange**; and he earned the same distinction with essays on the problem of three bodies in 1772, on the secular equation of the moon in 1774, and in 1778 on the theory of cometary perturbations.On the 6th of November 1766,

**Lagrange**was installed in his new position, with a salary of 6000 francs, ample leisure for scientific research, and royal favour sufficient to secure him respect without exciting envy.AdvertisementSoon after marriage his wife was attacked by a lingering illness, to which she succumbed,

**Lagrange**devoting all his time, and a considerable store of medical knowledge, to her care.The long series of memoirs - some of them complete treatises of great moment in the history of science - communicated by

**Lagrange**to the Berlin Academy between the years 1767 and 1787 were not the only fruits of his exile.But before that time

**Lagrange**himself was on the spot.Even from revolutionary tribunals, however, the name of

**Lagrange**uniformly commanded respect.Meanwhile, on the 31st of May 1792 he married Mademoiselle Lemonnier, daughter of the astronomer of that name, a young and beautiful girl, whose devotion ignored disparity of years, and formed the one tie with life which

**Lagrange**found it hard to break.AdvertisementOn the establishment of the Institute,

**Lagrange**was placed at the head of the section of geometry; he was one of the first members of the Bureau des Longitudes; and his name appeared in 1791 on the list of foreign members of the Royal Society.Amongst the brilliant group of mathematicians whose magnanimous rivalry contributed to accomplish the task of generalization and deduction reserved for the 18th century,

**Lagrange**occupies an eminent place.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.Finally, of the grand series of researches by which the stability of the solar system was ascertained, the glory must be almost equally divided between

**Lagrange**and Laplace.AdvertisementLaplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until

**Lagrange**showed him the way.**Lagrange**saw in the problems of nature so many occasions for analytical triumphs; Laplace regarded analytical triumphs as the means of solving the problems of nature.He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but

**Lagrange**, by the happy artifice of transferring the origin of coordinates from the centre of the sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally.It deserves to be recorded as one of the numerous coincidences of discovery that Laplace, on being made acquainted by

**Lagrange**with his new method, produced analogous expressions, to which his independent researches had led him.The final achievement of

**Lagrange**in this direction was the extension of the method of the variation of arbitrary constants, successfully used by him in the investigation of periodical as well as of secular inequalities, to any system whatever of mutually interacting bodies.'AdvertisementIn the advancement of almost every branch of pure mathematics

**Lagrange**took a conspicuous part.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.Besides this most important contribution to the general fabric of dynamical science, we owe to

**Lagrange**several minor theorems of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.But that one step, from the abstract to the concrete, was precisely that which the character of

**Lagrange**'s mind indisposed him to make.In the Berlin Memoirs for 1778 and 1783

**Lagrange**gave the first direct and theoretically perfect method of determining cometary orbits.His development of the equation x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis

**Lagrange**.**Lagrange**had failed to bring within the bounds of theory.The discordance of their results incited Laplace to a searching examination of the whole subject of planetary perturbations, and his maiden effort was rewarded with a discovery which constituted, when developed and completely demonstrated by his own further labours and those of his illustrious rival

**Lagrange**, the most important advance made in physical astronomy since the time of Newton.It was followed by a series of profound investigations, in which

**Lagrange**and Laplace alternately surpassed and supplemented each other in assigning limits of variation to the several elements of the planetary orbits.With

**Lagrange**, on the other hand, he always remained on the best of terms. Laplace left a son, Charles Emile Pierre Joseph Laplace (1789-1874), who succeeded to his title, and rose to the rank of general in the artillery.The first formal proof of

**Lagrange**'s theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality.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.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.**Lagrange**realized its powers and termed it " le principal fondement du calcul erentiel."If

**Lagrange**were to come to the United States, he could only earn his livelihood by turning land surveyor.The sisters of Meleager were 2 The god 'EAcoiiv was also Phoenician; see Driver, Genesis, p. 165;

**Lagrange**, Religions Semitiques, Index, s.v.The idea may be compared with that of Joseph Louis

**Lagrange**'s Calcul des Fonctions.During forty years the resources of analysis, even in the hands of d'Alembert,

**Lagrange**and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached.Space forbids any attempt to sketch here the special growth of criticism in other countries, such as France, where the brilliant genius of Renan was in part devoted to the Old Testament, or within the Roman Catholic Church, which possesses in Pere

**Lagrange**, for example, a deservedly influential critical scholar, and in the Revue Biblique an organ which devotes much attention to the critical study of the Old Testament.The mathematical treatment of the subject from this point of view by

**Lagrange**(1736-1813) and others has afforded the most important forms of statement of the theory of the motion of a system that are available for practical use.Such general statements of the theory of motion as that of

**Lagrange**, while releasing us from the rather narrow and strained view of the subject presented by detailed analysis of motion in terms of force, have also suggested a search for other forms which a statement of elementary principles might equally take as the foundation of a logical scheme.The I An excellent and critical account of Philo's work is given by

**Lagrange**, Etudes sur les rel.**Lagrange**, in 1813, Poinsot was elected to his place in the Academie des Sciences; and in 1840 he became a member of the superior council of public instruction.The memoir is remarkable inasmuch as it roused

**Lagrange**, after an interval of inactivity, to compose in his old age one of the greatest of his memoirs, viz.This, theorem, also due to

**Lagrange**, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances.These equations are due to

**Lagrange**, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated.**Lagrange**used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a+1, put x=a+I/y and form the equation in y; if the equation in y has a root between b and b+i, put y = b + I /z, and so on.The theory and development of the simple recurring continued fraction is due to

**Lagrange**.It is of no interest or importance, though both Lambert and

**Lagrange**devoted some attention to it.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).Within two years and a half he had mastered all the subjects prescribed for examination, and a great deal more, and, on going up for examination at Toulouse, he astounded his examiner by his knowledge of

**Lagrange**.The Cartesian equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis

**Lagrange**; it may be expressed in theform 1(4,2_ a2) (x 2+ y2) - 2a 2 cx - a 2 c 2 1 3 = 2 7 a4c2y2 (x2 + y2 - c2)2, where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle.But they contain what is far more valuable still, the greatest addition which dynamical science had received since the grand strides made by Sir Isaac Newton and Joseph Louis

**Lagrange**.The notion is very probably older, but it is at any rate to be found in

**Lagrange**'s Theorie des fonctions analytiques (1798); it is there remarked that the equation obtained by the elimination of the parameter a from an equationf (x,y,a) = o and the derived equation in respect to a is a curve, the envelope of the series of curves represented by the equation f (x,y,a) = o in question.The business of drawing up the new calendar was entrusted to the president of the committee of public instruction, Charles Gilbert Romme (1750-1795), who was aided in the work by the mathematicians Gaspard Monge and Joseph Louis

**Lagrange**, the poet Fabre d'Eglantine and others.Laplace developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis

**Lagrange**, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.Although he obtained results now identified with determinants,

**Lagrange**did not discuss these functions systematically.The development of the science by the successors of Newton, especially Laplace and

**Lagrange**, may be classed among the most striking achievements of the human intellect.The modern methods of celestial mechanics may be considered to begin with Joseph Louis

**Lagrange**, whose theory of the variation of elements is developed in his Mecanique analytique.This work contains a clear and excellent resume of the methods which have been devised by the leading investigators from the time of

**Lagrange**until the present, and thus forms the most encyclopaedic treatise to which the student can refer.These successes paved the way for the higher triumphs of Joseph Louis

**Lagrange**and of Pierre Simon Laplace.It was especially adapted to the tracing out of " secular inequalities," or those depending upon changes in the orbital elements of the bodies affected by them, and hence progressing indefinitely with time; and by its means, accordingly, the mechanical stability of the solar system was splendidly demonstrated through the successive efforts of

**Lagrange**and Laplace.The proper share of each in bringing about this memorable result is not easy to apportion, since they freely imparted and profited by one another's advances and improvements; it need only be said that the fundamental proposition of the invariability of the planetary major axes laid down with restrictions by Laplace in 1773, was finally established by

**Lagrange**in 1776; while Laplace in 1784 proved the subsistence of such a relation between the eccentricities of the planetary orbits on the one hand, and their inclinations on the other, that an increase of either element could, in any single case, proceed only to a very small extent.The prize of the Berlin Academy was, in 1780, adjudged to

**Lagrange**for a treatise on the perturbations of comets; and he contributed to the Berlin Memoirs, 1781-1784, a set of five elaborate papers, embodying and unifying his perfected methods and their results.He gave the first satisfactory demonstration of equilibrium on an inclined plane, reducing it to the level by a sound and ingenious train of reasoning; while, by establishing the theory of "virtual velocities," he laid down the fundamental principle which, in the opinion of

**Lagrange**, contains the general expression of the laws of equilibrium.Having received his early education from his father Louis Francois Cauchy (1760-1848), who held several minor public appointments and counted

**Lagrange**and Laplace among his friends, Cauchy entered Ecole Centrale du Pantheon in 1802, and proceeded to the Ecole Polytechnique in 1805, and to the Ecole des Ponts et Chaussees in 1807.Having adopted the profession of an engineer, he left Paris for Cherbourg in 1810, but returned in 1813 on account of his health, whereupon

**Lagrange**and Laplace persuaded him to renounce engineering and to devote himself to mathematics.Availing himself of the admirable generalized co-ordinate system of

**Lagrange**, Maxwell showed how to reduce all electric and magnetic phenomena to stresses and motions of a material medium, and, as one preliminary, but excessively severe, test of the truth of his theory, he pointed out that (if the electromagnetic medium be that which is required for the explanation of the phenomena of light) the velocity of light in vacuo should xvii.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.The former institution had an ephemeral existence; but amongst the benefits derived from the foundation of the Ecole Polytechnique one of the greatest, it has been observed, 4 was the restoration of

**Lagrange**to mathematics.By means of this "calculus of derived functions"

**Lagrange**hoped to give to the solution of all analytical problems the utmost "rigour of the demonstrations of the ancients"; 6 but it cannot be said that the attempt was successful.Poisson in a paper read on the 10th of June 1808, was once more attacked by

**Lagrange**with all his pristine vigour and fertility of invention.Resuming the inquiry into the invariability of mean motions, Poisson carried the approximation, with

**Lagrange**'s formulae, as far as the squares of the disturbing forces, hitherto neglected, with the same result as to the stability of the system.He proposed to apply the same principles to the calculation of the disturbances produced in the rotation of the planets by external action on their equatorial protuberances, but was anticipated by Poisson, who gave formulae for the variation of the elements of rotation strictly corresponding with those found by

**Lagrange**for the variation of the elements of revolution.As notable instances may be mentioned Laplace's discoveries relating to the velocity of sound and the secular acceleration of the moon, both of which were led close up to by

**Lagrange**'s analytical demonstrations.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.But the priest belongs to the realm of religion proper, which involves a relation of dependence on the superior power, whereas the asipu belongs to the realm of magic, which is coercive and seeks " to constrain the hostile power to give way " (

**Lagrange**).Poisson's application to them in 1809 of

**Lagrange**'s theory of the variation of constants; Philippe de Pontecoulant successfully used in 1829, for the prediction of the impending return of Halley's comet, a system of " mechanical quadratures " published by**Lagrange**in the Berlin Memoirs for 1778; and in his Theorie analytique du systeme du monde (1846) he modified and refined general theories of the lunar and planetary revolutions.In the pilot episode, Kris is serving time for grand theft auto at the Camp

**LaGrange**, a juvenile detention facility.