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lagrange

In preparation for these he spent the winter of 1877-1878 in reading up original treatises like those of Laplace and Lagrange on mathematics and mechanics, and in attending courses on practical physics under P. G.

00Lagrange, Hist.

00See also Kuenen's National Religions and Universal Religions (Hibbert lectures) and Lagrange's Etudes sur les religions simitiques (2nd ed.).

00The Wilcox formation (called Lignitic by Hilgard, and named by Safford the Lagrange group) lies to the west of the last, and its western limit is from about 32° 12' on the Alabama boundary about due north-west; in its north-westernmost part it is on the western edge of the Tertiary in this state.

00Of 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.

00pp. 420 sqq.; Lagrange, Etudes d.

00Lagrange, Etudes d.

00The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace.

00Lagrange (Eng.

00As far back as 1 773 Joseph Louis Lagrange, and later Carl Friedrich Gauss, had met with simple cases of such functions, George Boole, in 1841 (Camb.

00daP4 References For Symmetric Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad.

001903); P. Lagrange, Mgr P. Batiffol, P. Portalie, S.

00JOSEPH LOUIS LAGRANGE (1736-1813), French mathematician, was born at Turin, on the 25th of January 1736.

00His 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.

00The genius of Lagrange did not at once take its true bent.

00This prosperous opening gave the key-note to Lagrange's career.

00The 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.

00By these performances Lagrange found himself, at the age of twenty-six, on the summit of European fame.

00The 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.

00On 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.

00Soon 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.

00The 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.

002 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem."

00But before that time Lagrange himself was on the spot.

00der Mechanik, 220, 367; Lagrange, Mec. An.

00Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect.

00Meanwhile, 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.

00Delambre, Ouvres de Lagrange, i.

00On 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.

00Amongst 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.

00This is especially the case between Lagrange and Euler on the one side, and between Lagrange and Laplace on the other.

00The 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.

00Finally, 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.

00Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way.

00Lagrange 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.

00He 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.

00It 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.

00The 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.'

00In the advancement of almost every branch of pure mathematics Lagrange took a conspicuous part.

00To 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.

00Besides 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.

00But that one step, from the abstract to the concrete, was precisely that which the character of Lagrange's mind indisposed him to make.

00In the Berlin Memoirs for 1778 and 1783 Lagrange gave the first direct and theoretically perfect method of determining cometary orbits.

003 As a mathematical writer Lagrange has perhaps never been surpassed.

00- Lagrange's numerous scattered memoirs have been collected and published in seven 4to volumes, under the title Ouvres, v.

00Ouvres de Lagrange, publiees sous les soins de M.

00His development of the equation x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.

00Lagrange had failed to bring within the bounds of theory.

00The 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.

00It 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.

00With 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.

00The 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.

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.

00Wiedemann); Lagrange, Etudes sur les relig.

00priests' pectoral; Lagrange, op. cit., 236, n.

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.

00Lagrange realized its powers and termed it " le principal fondement du calcul erentiel."

00If Lagrange were to come to the United States, he could only earn his livelihood by turning land surveyor.

00Bib., " Creation " § 7; " Phoenicia " § 15; Lagrange, Religions sdmitiques, pp. 351 ff.

00The sisters of Meleager were 2 The god 'EAcoiiv was also Phoenician; see Driver, Genesis, p. 165; Lagrange, Religions Semitiques, Index, s.v.

00Lagrange, Rev. biblique, iii.

00The idea may be compared with that of Joseph Louis Lagrange's Calcul des Fonctions.

001875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).

00Bib., s.v.; Lagrange, Etudes sur les religions semitiques 2nd ed.

00Our chief source of information is Zimmern's Beitreige zur Kenntniss der Babylon: Religion, pp. 81-95, from which Lagrange in his Etudes sur les religions semitiques 2 has chiefly derived his materials (ch.

00During 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.

00Lagrange.

00Space 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.

00The 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.

00Such 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.

00The I An excellent and critical account of Philo's work is given by Lagrange, Etudes sur les rel.

00Mosso, The Palaces of Crete (1907); Lagrange, La Crete ancienne (1908); Dr. Evans's reports in The Times, Oct.

00Lagrange, 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.

00"Il a fallu," Lagrange remarked, "qu'un moment pour faire tomber cette tete, et cent annees pent-titre ne sujiront pas pour en reproduire une semblable."

00Lagrange, Etudes sur les religions semitiques (1905); J.

00Budde, Kurzer Handcommentar (1897); Lagrange, Livres des juges (1903); G.

00Lagrange, whose lectures on the theory of functions he attended at the Rcole Polytechnique, early recognized his talent, and became his friend; while P. S.

00The 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.

00Lagrange; it may be written Z(m.

00This, 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.

00These 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.

00Lagrange, Mecanique analytique (2nd ed., Paris, 1811-1815); P. S.

00Château Kirwan, Cantenac. D'Issan, Cantenac. Lagrange, St Julien.

00Lagrange 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.

00The theory and development of the simple recurring continued fraction is due to Lagrange.

00It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it.

00Nicol 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).

00Within 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.

00The 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.

00About twenty-four years later Bouillon Lagrange, and independently A.

00But 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.

001333; Lagrange, Etudes sur les religions semitiques, p. 416 (Paris, 1905); Ed.

00The 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.

00The 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.

00Laplace 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.

00Although he obtained results now identified with determinants, Lagrange did not discuss these functions systematically.

00The 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.

00This is called the osculating orbit: The essential principle of Lagrange's elegant method consists in determining the variations of this osculating ellipse, the co-ordinates and velocities of the planet being ignored in the determination.

00The 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.

00This 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.

00These successes paved the way for the higher triumphs of Joseph Louis Lagrange and of Pierre Simon Laplace.

00The subject of the lunar librations was treated by Lagrange w i th great originality in an essay crowned by the Paris Academy of Sciences in 1764; and he filled up the lacunae in his theory of them in a memoir communicated to the Berlin Academy in 1780.

00It 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.

00The 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.

00The 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.

00He 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.

00on the Old Testament passages, Moore (loc. cit.), and Lagrange, Relig.

00Having 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.

00Having 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.

00Lagrange in using both these notations), but because it signified the opening to the mathematicians of Cambridge of the vast storehouse of continental discoveries.

00In preparation for these he spent the winter of 1877-1878 in reading up original treatises like those of Laplace and Lagrange on mathematics and mechanics, and in attending courses on practical physics under P. G.

00Lagrange, Hist.

00See also Kuenen's National Religions and Universal Religions (Hibbert lectures) and Lagrange's Etudes sur les religions simitiques (2nd ed.).

00Availing 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.

00The Wilcox formation (called Lignitic by Hilgard, and named by Safford the Lagrange group) lies to the west of the last, and its western limit is from about 32Ã‚° 12' on the Alabama boundary about due north-west; in its north-westernmost part it is on the western edge of the Tertiary in this state.

00Of 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.

00pp. 420 sqq.; Lagrange, Etudes d.

00Lagrange, Etudes d.

00The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace.

00Lagrange (Eng.

00As far back as 1 773 Joseph Louis Lagrange, and later Carl Friedrich Gauss, had met with simple cases of such functions, George Boole, in 1841 (Camb.

00daP4 References For Symmetric Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad.

001901) frankly describes the condition of ecclesiastical biblical studies; Monseigneur Mignot, archbishop of Albi, Lettres sur les etudes ecclesiastiques 1900-1901 (collected ed., Paris, 1908) and "Critique et tradition" in Le Correspondant (Paris, Toth January 1904), the utterances of a finely trained judgment; Mgr Le Camus, bishop of La Rochelle, Fausse Exegese, mauvaise theologie (Paris, 1902), a timid, mostly rhetorical, scholar's protest; Pere Lagrange, a Dominican who has done much for the spread of Old Testament criticism, La Methode historique, surtout a propos de l'Ancien Testament (Paris, 1903) and Eclaircissement to same (ibid.

001903); P. Lagrange, Mgr P. Batiffol, P. Portalie, S.

00JOSEPH LOUIS LAGRANGE (1736-1813), French mathematician, was born at Turin, on the 25th of January 1736.

00His 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.

00The genius of Lagrange did not at once take its true bent.

00This prosperous opening gave the key-note to Lagrange's career.

00The 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.

00By these performances Lagrange found himself, at the age of twenty-six, on the summit of European fame.

00The 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.

00The 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.

00On 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.

00Soon 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.

00The 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.

002 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid and harmonious as to constitute, in Sir William Hamilton's words, "a kind of scientific poem."

00But before that time Lagrange himself was on the spot.

00der Mechanik, 220, 367; Lagrange, Mec. An.

00Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect.

00Meanwhile, 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.

00The 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.

00By 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.

00Delambre, Ouvres de Lagrange, i.

00On 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.

00Amongst 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.

00This is especially the case between Lagrange and Euler on the one side, and between Lagrange and Laplace on the other.

00The 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.

00Finally, 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.

00Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way.

00Lagrange 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.

00What may be called Lagrange's first period of research into planetary perturbations extended from 1774 to 1784 (see Astronomy: History).

00Poisson 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.

00Resuming 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.

00He 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.

00It 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.

00The 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.'

00He 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.

00In the advancement of almost every branch of pure mathematics Lagrange took a conspicuous part.

00To 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.

00Besides 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.

00But that one step, from the abstract to the concrete, was precisely that which the character of Lagrange's mind indisposed him to make.

00As 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.

00In the Berlin Memoirs for 1778 and 1783 Lagrange gave the first direct and theoretically perfect method of determining cometary orbits.

003 As a mathematical writer Lagrange has perhaps never been surpassed.

00- Lagrange's numerous scattered memoirs have been collected and published in seven 4to volumes, under the title Ouvres, v.

00Ouvres de Lagrange, publiees sous les soins de M.

00His development of the equation x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.

00Lagrange had failed to bring within the bounds of theory.

00The 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.

00It 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.

00With 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.

00The 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.

00Continued 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.

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.

00Wiedemann); Lagrange, Etudes sur les relig.

00priests' pectoral; Lagrange, op. cit., 236, n.

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.

00Lagrange realized its powers and termed it " le principal fondement du calcul erentiel."

00" If Lagrange were to come to the United States, he could only earn his livelihood by turning land surveyor."

00Bib., " Creation " § 7; " Phoenicia " § 15; Lagrange, Religions sdmitiques, pp. 351 ff.

00The sisters of Meleager were 2 The god 'EAcoiiv was also Phoenician; see Driver, Genesis, p. 165; Lagrange, Religions Semitiques, Index, s.v.

00Lagrange, Rev. biblique, iii.

00The idea may be compared with that of Joseph Louis Lagrange's Calcul des Fonctions.

001875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).

00Bib., s.v.; Lagrange, Etudes sur les religions semitiques 2nd ed.

00Our chief source of information is Zimmern's Beitreige zur Kenntniss der Babylon: Religion, pp. 81-95, from which Lagrange in his Etudes sur les religions semitiques 2 has chiefly derived his materials (ch.

00But 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).

00During 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.

00Space 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.

00The 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.

00Such 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.

00The I An excellent and critical account of Philo's work is given by Lagrange, Etudes sur les rel.

00Mosso, The Palaces of Crete (1907); Lagrange, La Crete ancienne (1908); Dr. Evans's reports in The Times, Oct.

00Lagrange, 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.

00"Il a fallu," Lagrange remarked, "qu'un moment pour faire tomber cette tete, et cent annees pent-titre ne sujiront pas pour en reproduire une semblable."

00Lagrange, Etudes sur les religions semitiques (1905); J.

00Budde, Kurzer Handcommentar (1897); Lagrange, Livres des juges (1903); G.

00Lagrange, whose lectures on the theory of functions he attended at the Rcole Polytechnique, early recognized his talent, and became his friend; while P. S.

00(1827), &c. In the first of these memoirs Poisson discusses the famous question of the stability of the planetary orbits, which had already been settled by Lagrange to the first degree of approximation for the disturbing forces.

00The 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.

00Lagrange; it may be written Z(m.

00This, 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.

00These 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.

00Lagrange, Mecanique analytique (2nd ed., Paris, 1811-1815); P. S.

00ChÃ¢teau Kirwan, Cantenac. D'Issan, Cantenac. Lagrange, St Julien.

00Lagrange 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.

00The theory and development of the simple recurring continued fraction is due to Lagrange.

00Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol.

00It is of no interest or importance, though both Lambert and Lagrange devoted some attention to it.

00Nicol 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).

00Within 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.

00The 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.

00About twenty-four years later Bouillon Lagrange, and independently A.

00But 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.

001333; Lagrange, Etudes sur les religions semitiques, p. 416 (Paris, 1905); Ed.

00The 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.

00The 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.

00Laplace 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.

00Although he obtained results now identified with determinants, Lagrange did not discuss these functions systematically.

00The 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.

00This is called the osculating orbit: The essential principle of Lagrange's elegant method consists in determining the variations of this osculating ellipse, the co-ordinates and velocities of the planet being ignored in the determination.

00The 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.

00This 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.

00These successes paved the way for the higher triumphs of Joseph Louis Lagrange and of Pierre Simon Laplace.

00The subject of the lunar librations was treated by Lagrange w i th great originality in an essay crowned by the Paris Academy of Sciences in 1764; and he filled up the lacunae in his theory of them in a memoir communicated to the Berlin Academy in 1780.

00It 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.

00The 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.

00The 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.

00Poisson'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.

00He 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.

00on the Old Testament passages, Moore (loc. cit.), and Lagrange, Relig.

00Having 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.

00Having 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.

00In the pilot episode, Kris is serving time for grand theft auto at the Camp LaGrange, a juvenile detention facility.

00

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