Chapter XIV: Part 14

LAGOS, a seaport of West Africa, capital of the British colony and protectorate of Southern Nigeria, in 6° 26´ N., 3° 23´ E. on an island in a lagoon named Lagos also. Between Lagos and the mainland is Iddo Island. An iron bridge for road and railway traffic 2600 ft. long connects Lagos and Iddo Islands, and another iron bridge, 917 ft. long, joins Iddo Island to the mainland. The town lies but a foot or two above sea-level. The principal buildings are a large government house, the law courts, the memorial hall erected to commemorate the services of Sir John Glover, used for public meetings and entertainments, an elaborate club-house provided from public funds, and the police quarters. There are many substantial villas that serve as quarters for the officers of the civil service, as well as numerous solidly-built handsome private buildings. The streets are well kept; the town is supplied with electric light, and there is a good water service. The chief stores and depôts for goods are all on the banks of the lagoon. The swamps of which originally Lagos Island entirely consisted have been reclaimed. In connexion with this work a canal, 25 ft. wide, has been cut right through the island and a sea-wall built round its western half. There is a commodious public hospital, of the cottage type, on a good site. There is a racecourse, which also serves as a general public recreation ground. Shifting banks of sand form a bar at the sea entrance of the lagoon. Extensive works were undertaken in 1908 with a view to making Lagos an open port. A mole has been built at the eastern entrance to the harbour and dredgers are at work on the bar, which can be crossed by vessels drawing 13 ft. Large ocean-going steamers anchor not less than 2 m. from land, and goods and passengers are there transhipped into smaller steamers for Lagos. Heavy cargo is carried by the large steamers to Forcados, 200 m. farther down the coast, transhipped there into branch boats, and taken via the lagoons to Lagos. The port is 4279 m. from Liverpool, 1203 from Freetown, Sierra Leone (the nearest safe port westward), and 315 from Cape Coast.

The inhabitants, about 50,000, include, besides the native tribes, Sierra Leonis, Fanti, Krumen and the descendants of some 6000 Brazilian _emancipados_ who were settled here in the early days of British rule. The Europeans number about 400. Rather more than half the populace are Moslems.

LAGOS, a seaport of southern Portugal, in the district of Faro (formerly the province of Algarve); on the Atlantic Ocean, and on the estuary of the small river Lagos, here spanned by a fine stone bridge. Pop. (1900) 8291. The city is defended by fortifications erected in the 17th century. It is supplied with water by an aqueduct 800 yds. long. The harbour is deep, capacious, and completely sheltered on the north and west; it is frequently visited by the British Channel fleet. Vines and figs are extensively cultivated in the neighbourhood, and Lagos is the centre of important sardine and tunny fisheries. Its trade is chiefly carried on by small coasting vessels, as there is no railway. Lagos is on or near the site of the Roman _Lacobriga_. Since the 15th century it has held the formal rank and title of city. Cape St Vincent, the ancient _Promontorium Sacrum_, and the south-western extremity of the kingdom, is 22 m. W. It is famous for its connexion with Prince Henry (q.v.), the Navigator, who here founded the town of Sagres in 1421; and for several British naval victories, the most celebrated of which was won in 1797 by Admiral Jervis (afterwards Earl St Vincent) over a larger Spanish squadron. In 1759 Admiral Boscawen defeated a French fleet off Lagos. The great earthquake of 1755 destroyed a large part of the city.

LA GRÂCE, or LES GRÂCES, a game invented in France during the first quarter of the 19th century and called there _le jeu des Grâces_. It is played with two light sticks about 16 in. long and a wicker ring, which is projected into the air by placing it over the sticks crossed and then separating them rapidly. The ring is caught upon the stick of another player and thrown back, the object being to prevent it from falling to the ground.

LA GRAND' COMBE, a town of southern France, in the department of Gard on the Gardon, 39 m. N.N.W. of Nîmes by rail. Pop. (1906) town, 6406; commune, 11,292. There are extensive coal mines in the vicinity.

LAGRANGE, JOSEPH LOUIS (1736-1813), French mathematician, was born at Turin, on the 25th of January 1736. He was of French extraction, his great grandfather, a cavalry captain, having passed from the service of France to that of Sardinia, and settled in Turin under Emmanuel II. 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. His emoluments as treasurer at war, together with his wife's fortune, provided him with ample means, which he lost by rash speculations, a circumstance regarded by his son as the prelude to his own good fortune; for had he been rich, he used to say, he might never have known mathematics.

The genius of Lagrange did not at once take its true bent. His earliest tastes were literary rather than scientific, and he learned the rudiments of geometry during his first year at the college of Turin, without difficulty, but without distinction. The perusal of a tract by Halley (_Phil. Trans._ xviii. 960) roused his enthusiasm for the analytical method, of which he was destined to develop the utmost capabilities. He now entered, unaided save by his own unerring tact and vivid apprehension, upon a course of study which, in two years, placed him on a level with the greatest of his contemporaries. 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. It was eagerly welcomed by the Berlin mathematician, who had the generosity to withhold from publication his own further researches on the subject, until his youthful correspondent should have had time to complete and opportunity to claim the invention. This prosperous opening gave the key-note to Lagrange's career. Appointed, in 1754, professor of geometry in the royal school of artillery, he formed with some of his pupils--for the most part his seniors--friendships based on community of scientific ardour. With the aid of the marquis de Saluces and the anatomist G. F. Cigna, he founded in 1758 a society which became the Turin Academy of Sciences. 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. He made his first appearance in public as the critic of Newton, and the arbiter between d'Alembert and Euler. By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial differential equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject. He further treated in a masterly manner of echoes and the mixture of sounds, and explained the phenomenon of grave harmonics as due to the occurrence of beats so rapid as to generate a musical note. This was followed, in the second volume of the _Miscellanea Taurinensia_ (1762) by his "Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies," together with the application of this important development of analysis to the solution of several dynamical problems, as well as to the demonstration of the mechanical principle of "least action." 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 not only freed it from all trammels of geometrical construction, but by the introduction of the symbol [delta] gave it the efficacy of a new calculus. He is thus justly regarded as the inventor of the "method of variations"--a name supplied by Euler in 1766.

By these performances Lagrange found himself, at the age of twenty-six, on the summit of European fame. Such a height had not been reached without cost. Intense application during early youth had weakened a constitution never robust, and led to accesses of feverish exaltation culminating, in the spring of 1761, in an attack of bilious hypochondria, which permanently lowered the tone of his nervous system. Rest and exercise, however, temporarily restored his health, and he gave proof of the undiminished vigour of his powers by carrying off, in 1764, the prize offered by the Paris Academy of Sciences for the best essay on the libration of the moon. His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical formula of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities. His success encouraged the Academy to propose, in 1766, as a theme for competition, the hitherto unattempted theory of the Jovian system. 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.

He had in the meantime gratified a long felt desire by a visit to Paris, where he enjoyed the stimulating delight of conversing with such mathematicians as A. C. Clairault, d'Alembert, Condorcet and the Abbé Marie. Illness prevented him from visiting London. 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. 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. 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. The national jealousy of foreigners, was at first a source of annoyance to him; but such prejudices were gradually disarmed by the inoffensiveness of his demeanour. We are told that the universal example of his colleagues, rather than any desire for female society, impelled him to matrimony; his choice being a lady of the Conti family, who, by his request, joined him at Berlin. Soon 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. His _Mécanique analytique_, in which his genius most fully displayed itself, was produced during the same period. This great work was the perfect realization of a design conceived by the author almost in boyhood, and clearly sketched in his first published essay.[1] Its scope may be briefly described as the reduction of the theory of mechanics to certain general formulae, from the simple development of which should be derived the equations necessary for the solution of each separate problem.[2] 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." This unification of method was one of matter also. By his mode of regarding a liquid as a material system characterized by the unshackled mobility of its minutest parts, the separation between the mechanics of matter in different forms of aggregation finally disappeared, and the fundamental equation of forces was for the first time extended to hydrostatics and hydrodynamics.[3] Thus a universal science of matter and motion was derived, by an unbroken sequence of deduction, from one radical principle; and analytical mechanics assumed the clear and complete form of logical perfection which it now wears.

A publisher having with some difficulty been found, the book appeared at Paris in 1788 under the supervision of A. M. Legendre. But before that time Lagrange himself was on the spot. After the death of Frederick the Great, his presence was competed for by the courts of France, Spain and Naples, and a residence in Berlin having ceased to possess any attraction for him, he removed to Paris in 1787. Marie Antoinette warmly patronized him. He was lodged in the Louvre, received the grant of an income equal to that he had hitherto enjoyed, and, with the title of "veteran pensioner" in lieu of that of "foreign associate" (conferred in 1772), the right of voting at the deliberations of the Academy. In the midst of these distinctions, a profound melancholy seized upon him. His mathematical enthusiasm was for the time completely quenched, and during two years the printed volume of his _Mécanique_, which he had seen only in manuscript, lay unopened beside him. He relieved his dejection with miscellaneous studies, especially with that of chemistry, which, in the new form given to it by Lavoisier, he found "aisée comme l'algèbre." The Revolution roused him once more to activity and cheerfulness. Curiosity impelled him to remain and watch the progress of such a novel phenomenon; but curiosity was changed into dismay as the terrific character of the phenomenon unfolded itself. He now bitterly regretted his temerity in braving the danger. "Tu l'as voulu" he would repeat self-reproachfully. Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect. His pension was continued by the National Assembly, and he was partially indemnified for the depreciation of the currency by remunerative appointments. Nominated president of the Academical commission for the reform of weights and measures, his services were retained when its "purification" by the Jacobins removed his most distinguished colleagues. He again sat on the commission of 1799 for the construction of the metric system, and by his zealous advocacy of the decimal principle largely contributed to its adoption.

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. He had no children by either marriage. Although specially exempted from the operation of the decree of October 1793, imposing banishment on foreign residents, he took alarm at the fate of J. S. Bailly and A. L. Lavoisier, and prepared to resume his former situation in Berlin. His design was frustrated by the establishment of and his official connexion with the École Normale, and the École Polytechnique. The former institution had an ephemeral existence; but amongst the benefits derived from the foundation of the École Polytechnique one of the greatest, it has been observed,[4] was the restoration of Lagrange to mathematics. The remembrance of his teachings was long treasured by such of his auditors--amongst whom were J. B. J. Delambre and S. F. Lacroix--as were capable of appreciating them. In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions. He seemed, not a professor amongst students, but a learner amongst learners; pauses for thought alternated with luminous exposition; invention accompanied demonstration; and thus originated his _Théorie des fonctions analytiques_ (Paris, 1797). The leading idea of this work was contained in a paper published in the _Berlin Memoirs_ for 1772.[5] Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions. 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. The validity of his fundamental position was impaired by the absence of a well-constituted theory of series; the notation employed was inconvenient, and was abandoned by its inventor in the second edition of his _Mécanique_; while his scruples as to the admission into analytical investigations of the idea of limits or vanishing ratios have long since been laid aside as idle. Nowhere, however, were the keenness and clearness of his intellect more conspicuous than in this brilliant effort, which, if it failed in its immediate object, was highly effective in secondary results. His purely abstract mode of regarding functions, apart from any mechanical or geometrical considerations, led the way to a new and sharply characterized development of the higher analysis in the hands of A. Cauchy, C. G. Jacobi, and others.[7] The _Théorie des fonctions_ is divided into three parts, of which the first explains the general doctrine of functions, the second deals with its application to geometry, and the third with its bearings on mechanics.

On 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. On the annexation of Piedmont to France in 1796, a touching compliment was paid to him in the person of his aged father. By direction of Talleyrand, then minister for foreign affairs, the French commissary repaired in state to the old man's residence in Turin, to congratulate him on the merits of his son, whom they declared "to have done honour to mankind by his genius, and whom Piedmont was proud to have produced, and France to possess." Bonaparte, who styled him "la haute pyramide des sciences mathématiques," loaded him with personal favours and official distinctions. He became a senator, a count of the empire, a grand officer of the legion of honour, and just before his death received the grand cross of the order of réunion.

The preparation of a new edition of his _Mécanique_ exhausted his already falling powers. Frequent fainting fits gave presage of a speedy end, and on the 8th of April 1813 he had a final interview with his friends B. Lacépède, G. Monge and J. A. Chaptal. He spoke with the utmost calm of his approaching death; "c'est une dernière fonction," he said, "qui n'est ni pénible ni désagréable." He nevertheless looked forward to a future meeting, when he promised to complete the autobiographical details which weakness obliged him to interrupt. They remained untold, for he died two days later on the 10th of April, and was buried in the Pantheon, the funeral oration being pronounced by Laplace and Lacépède.

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. It is indeed by no means easy to distinguish and apportion the
respective merits of the competitors. 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. In analytical
invention, and mastery over the calculus, the Turin mathematician was
admittedly unrivalled. Laplace owned that he had despaired of
effecting the integration of the differential equations relative to
secular inequalities until Lagrange showed him the way. But Laplace
unquestionably surpassed his rival in practical sagacity and the
intuition of physical truth. 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. One mind
seemed the complement of the other; and both, united in honourable
rivalry, formed an instrument of unexampled perfection for the
investigation of the celestial machinery. What may be called
Lagrange's first period of research into planetary perturbations
extended from 1774 to 1784 (see ASTRONOMY: _History_). The notable
group of treatises communicated, 1781-1784, to the Berlin Academy was
designed, but did not prove to be his final contribution to the theory
of the planets. After an interval of twenty-four years the subject,
re-opened by S. D. Poisson in a paper read on the 20th 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
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.[8] "Not without astonishment," even
to himself, regard being had to the great generality of the
differential equations, he reached a result so wide as to include, as
a particular case, the solution of the planetary problem recently
obtained by him. 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. The revision
of the _Mécanique analytique_ was undertaken mainly for the purpose of
embodying in it these new methods and final results, but was
interrupted, when two-thirds completed, by the death of its author.

In the advancement of almost every branch of pure mathematics Lagrange
took a conspicuous part. The calculus of variations is indissolubly
associated with his name. In the theory of numbers he furnished
solutions of many of P. Fermat's theorems, and added some of his own.
In algebra he discovered the method of approximating to the real roots
of an equation by means of continued fractions, and imagined a general
process of solving algebraical equations of every degree. The method
indeed fails for equations of an order above the fourth, because it
then involves the solution of an equation of higher dimensions than
they proposed. Yet it possesses the great and characteristic merit of
generalizing the solutions of his predecessors, exhibiting them all as
modifications of one principle. 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. But it was in the application
to mechanical questions of the instrument which he thus helped to form
that his singular merit lay. It was his just boast to have transformed
mechanics (defined by him as a "geometry of four dimensions") into a
branch of analysis, and to have exhibited the so-called mechanical
"principles" as simple results of the calculus. The method of
"generalized coordinates," as it is now called, by which he attained
this result, is the most brilliant achievement of the analytical
method. Instead of following the motion of each individual part of a
material system, he showed that, if we determine its configuration by
a sufficient number of variables, whose number is that of the degrees
of freedom to move (there being as many equations as the system has
degrees of freedom), the kinetic and potential energies of the system
can be expressed in terms of these, and the differential equations of
motion thence deduced by simple differentiation. 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. To
this entire branch of knowledge, in short, he successfully imparted
that character of generality and completeness towards which his
labours invariably tended.

His share in the gigantic task of verifying the Newtonian theory would
alone suffice to immortalize his name. His co-operation was indeed
more indispensable than at first sight appears. Much as was done _by_
him, what was done _through_ him was still more important. Some of his
brilliant rival's most conspicuous discoveries were implicitly
contained in his writings, and wanted but one step for completion. But
that one step, from the abstract to the concrete, was precisely that
which the character of Lagrange's mind indisposed him to make. 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.
In the _Berlin Memoirs_ for 1778 and 1783 Lagrange gave the first
direct and theoretically perfect method of determining cometary
orbits. It has not indeed proved practically available; but his system
of calculating cometary perturbations by means of "mechanical
quadratures" has formed the starting-point of all subsequent
researches on the subject. His determination[9] of maximum and minimum
values for the slowly varying planetary eccentricities was the
earliest attempt to deal with the problem. Without a more accurate
knowledge of the masses of the planets than was then possessed a
satisfactory solution was impossible; but the upper limits assigned by
him agreed closely with those obtained later by U. J. J.
Leverrier.[10] As a mathematical writer Lagrange has perhaps never
been surpassed. His treatises are not only storehouses of ingenious
methods, but models of symmetrical form. The clearness, elegance and
originality of his mode of presentation give lucidity to what is
obscure, novelty to what is familiar, and simplicity to what is
abstruse. His genius was one of generalization and abstraction; and
the aspirations of the time towards unity and perfection received, by
his serene labours, an embodiment denied to them in the troubled world
of politics.

BIBLIOGRAPHY.--Lagrange's numerous scattered memoirs have been
collected and published in seven 4to volumes, under the title
_Oeuvres de Lagrange, publiées sous les soins de M. J. A. Serret_
(Paris, 1867-1877). 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
_Leçons élémentaires_ at the École Normale in 1795. Delambre's notice
of his life, extracted from the _Mém. de l'Institut_, 1812, is
prefixed to the first volume. Besides the separate works already named
are _Résolution des équations numériques_ (1798, 2nd ed., 1808, 3rd
ed., 1826), and _Leçons sur le calcul des fonctions_ (1805, 2nd ed.,
1806), designed as a commentary and supplement to the first part of
the _Théorie des fonctions_. The first volume of the enlarged edition
of the _Mécanique_ appeared in 1811, the second, of which the revision
was completed by MM Prony and Binet, in 1815. A third edition, in 2
vols., 4to, was issued in 1853-1855, and a second of the _Théorie des
fonctions_ in 1813.

See also J. J. Virey and Potel, _Précis historique_ (1813); Th.
Thomson's _Annals of Philosophy_ (1813-1820), vols. ii. and iv.; H.
Suter, _Geschichte der math. Wiss._ (1873); E. Dühring, _Kritische
Gesch. der allgemeinen Principien der Mechanik_ (1877, 2nd ed.); A.
Gautier, _Essai historique sur le problème des trois corps_ (1817); R.
Grant, _History of Physical Astronomy_, &c.; Pietro Cossali, _Éloge_
(Padua, 1813); L. Martini, _Cenni biográfici_ (1840); _Moniteur du 26
Février_ (1814); W. Whewell, _Hist. of the Inductive Sciences_, ii.
_passim_; J. Clerk Maxwell, _Electricity and Magnetism_, ii. 184; A.
Berry, _Short Hist. of Astr._, p. 313; J. S. Bailly, _Hist. de l'astr.
moderne_, iii. 156, 185, 232; J. C. Poggendorff, _Biog. Lit.
Handwörterbuch_. (A. M. C.)

FOOTNOTES:

[1] _Oeuvres_, i. 15.

[2] _Méc. An._, Advertisement to 1st ed.

[3] E. Dühring, _Kritische Gesch. der Mechanik_, 220, 367; Lagrange,
_Méc. An._ i. 166-172, 3rd ed.

[4] Notice by J. Delambre, _Oeuvres de Lagrange_, i. p. xlii.

[5] _Oeuvres_, iii. 441.

[6] _Théorie des fonctions_, p. 6.

[7] H. Suter, _Geschichte der math. Wiss._ ii. 222-223.

[8] _Oeuvres_, vi. 771.

[9] _Oeuvres_, v. 211 seq.

[10] Grant, _History of Physical Astronomy_, p. 117.

LAGRANGE-CHANCEL [CHANCEL], FRANÇOIS JOSEPH (1677-1758), French dramatist and satirist, was born at Périgueux on the 1st of January 1677. He was an extremely precocious boy, and at Bordeaux, where he was educated, he produced a play when he was nine years old. Five years later his mother took him to Paris, where he found a patron in the princesse de Conti, to whom he dedicated his tragedy of _Jugurtha_ or, as it was called later, _Adherbal_ (1694). Racine had given him advice and was present at the first performance, although he had long lived in complete retirement. Other plays followed: _Oreste et Pylade_ (1697), _Méléagre_ (1699), _Amasis_ (1701), and _Ino et Mélicerte_ (1715). Lagrange hardly realized the high hopes raised by his precocity, although his only serious rival on the tragic stage was Campistron, but he obtained high favour at court, becoming _maître d'hôtel_ to the duchess of Orleans. This prosperity ended with the publication in 1720 of his _Philippiques_, odes accusing the regent, Philip, duke of Orleans, of the most odious crimes. He might have escaped the consequences of this libel but for the bitter enmity of a former patron, the duc de La Force. Lagrange found sanctuary at Avignon, but was enticed beyond the boundary of the papal jurisdiction, when he was arrested and sent as a prisoner to the isles of Sainte Marguerite. He contrived, however, to escape to Sardinia and thence to Spain and Holland, where he produced his fourth and fifth _Philippiques_. On the death of the Regent he was able to return to France. He was part author of a _Histoire de Périgord_ left unfinished, and made a further contribution to history, or perhaps, more exactly, to romance, in a letter to Élie Fréron on the identity of the Man with the Iron Mask. Lagrange's family life was embittered by a long lawsuit against his son. He died at Périgueux at the end of December 1758.

He had collected his own works (5 vols., 1758) some months before his
death. His most famous work, the _Philippiques_, was edited by M. de
Lescure in 1858, and a sixth philippic by M. Diancourt in 1886.

LA GRANJA, or SAN ILDEFONSO, a summer palace of the kings of Spain; on the south-eastern border of the province of Segovia, and on the western slopes of the Sierra de Guadarrama, 7 m. by road S.E. of the city of Segovia. The royal estate is 3905 ft. above sea-level. The scenery of this region, especially in the gorge of the river Lozoya, with its granite rocks, its dense forest of pines, firs and birches, and its red-tiled farms, more nearly resembles the highlands of northern Europe than any other part of Spain. La Granja has an almost alpine climate, with a clear, cool atmosphere and abundant sunshine. Above the palace rise the wooded summits of the Guadarrama, culminating in the peak of Peñalara (7891 ft.); in front of it the wide plains of Segovia extend northwards. The village of San Ildefonso, the oldest part of the estate, was founded in 1450 by Henry IV., who built a hunting lodge and chapel here. In 1477 the chapel was presented by Ferdinand and Isabella to the monks of the Parral, a neighbouring Hieronymite monastery. The original _granja_ (i.e. grange or farm), established by the monks, was purchased in 1719 by Philip V., after the destruction of his summer palace at Valsain, the ancient _Vallis Sapinorum_, 2 m. S. Philip determined to convert the estate into a second Versailles. The palace was built between 1721 and 1723. Its façade is fronted by a colonnade in which the pillars reach to the roof. The state apartments contain some valuable 18th-century furniture, but the famous collection of sculptures was removed to Madrid in 1836, and is preserved there in the Museo del Prado. At La Granja it is represented by facsimiles in plaster. The collegiate church adjoining the palace dates from 1724, and contains the tombs of Philip V. and his consort Isabella Farnese. An artificial lake called El Mar, 4095 ft. above sea-level, irrigates the gardens, which are imitated from those of Versailles, and supplies water for the fountains. These, despite the antiquated and sometimes tasteless style of their ornamentation, are probably the finest in the world; it is noteworthy that, owing to the high level of the lake, no pumps or other mechanism are needed to supply pressure. There are twenty-six fountains besides lakes and waterfalls. Among the most remarkable are the group of "Perseus, Andromeda and the Sea-Monster," which sends up a jet of water 110 ft. high, the "Fame," which reaches 125 ft., and the very elaborate "Baths of Diana." It is of the last that Philip V. is said to have remarked, "It has cost me three millions and amused me three minutes." Most of the fountains were made by order of Queen Isabella in 1727, during the king's absence. The glass factory of San Ildefonso was founded by Charles III.

It was in La Granja that Philip V. resigned the crown to his son in
January 1724, to resume it after his son's death seven months later;
that the treaties of 1777, 1778, 1796 and 1800 were signed (see SPAIN:
_History_); that Ferdinand VII. summoned Don Carlos to the throne in
1832, but was induced to alter the succession in favour of his own
infant daughter Isabella, thus involving Spain in civil war; and that
in 1836 a military revolt compelled the Queen-regent Christina to
restore the constitution of 1812.

LAGRENÉE, LOUIS JEAN FRANÇOIS (1724-1805), French painter, was a pupil of Carle Vanloo. Born at Paris on the 30th of December 1724, in 1755 he became a member of the Royal Academy, presenting as his diploma picture the "Rape of Deianira" (Louvre). He visited St Petersburg at the call of the empress Elizabeth, and on his return was named in 1781 director of the French Academy at Rome; he there painted the "Indian Widow," one of his best-known works. In 1804 Napoleon conferred on him the cross of the legion of honour, and on the 19th of June 1805 he died in the Louvre, of which he was honorary keeper.

LA GUAIRA, or LA GUAYRA (sometimes LAGUAIRA, &c.), a town and port of Venezuela, in the Federal district, 23 m. by rail and 6½ m. in a direct line N. of Caracas. Pop. (1904, estimate) 14,000. It is situated between a precipitous mountain side and a broad, semicircular indentation of the coast line which forms the roadstead of the port. The anchorage was long considered one of the most dangerous on the Caribbean coast, and landing was attended with much danger. The harbour has been improved by the construction of a concrete breakwater running out from the eastern shore line 2044 ft., built up from an extreme depth of 46 ft. or from an average depth of 29½ ft., and rising 19½ ft. above sea-level. This encloses an area of 76½ acres, having an average depth of nearly 28 ft. The harbour is further improved by 1870 ft. of concrete quays and 1397 ft. of retaining sea-wall, with several piers (three covered) projecting into deep water. These works were executed by a British company, known as the La Guaira Harbour Corporation, Ltd., and were completed in 1891 at a cost of about one million sterling. The concession is for 99 years and the additional charges which the company is authorized to impose are necessarily heavy. These improvements and the restrictions placed upon the direct trade between West Indian ports and the Orinoco have greatly increased the foreign trade of La Guaira, which in 1903 was 52% of that of the four _puertos habilitados_ of the republic. The shipping entries of that year numbered 217, of which 203 entered with general cargo and 14 with coal exclusively. The exports included 152,625 bags coffee, 114,947 bags cacao and 152,891 hides. For 1905-1906 the imports at La Guaira were valued officially at £767,365 and the exports at £663,708. The city stands on sloping ground stretching along the circular coast line with a varying width of 130 to 330 ft. and having the appearance of an amphitheatre. The port improvements added 18 acres of reclaimed land to La Guaira's area, and the removal of old shore batteries likewise increased its available breadth. In this narrow space is built the town, composed in great part of small, roughly-made cabins, and narrow, badly-paved streets, but with good business houses on its principal street. From the mountain side, reddish-brown in colour and bare of vegetation, the solar heat is reflected with tremendous force, the mean annual temperature being 84° F. The seaside towns of Maiquetia, 2 m. W. and Macuto, 3 m. E., which have better climatic and sanitary conditions and are connected by a narrow-gauge railway, are the residences of many of the wealthier merchants of La Guaira.

La Guaira was founded in 1588, was sacked by filibusters under Amias Preston in 1595, and by the French under Grammont in 1680, was destroyed by the great earthquake of the 26th of March 1812, and suffered severely in the war for independence. In 1903, pending the settlement of claims of Great Britain, Germany and Italy against Venezuela, La Guaira was blockaded by a British-German-Italian fleet.

LA GUÉRONNIÈRE, LOUIS ÉTIENNE ARTHUR DUBREUIL HÉLION, VICOMTE DE (1816-1875), French politician, was the scion of a noble Poitevin family. Although by birth and education attached to Legitimist principles, he became closely associated with Lamartine, to whose organ, _Le Bien Public_, he was a principal contributor. After the stoppage of this paper he wrote for _La Presse_, and in 1850 edited _Le Pays_. A character sketch of Louis Napoleon in this journal caused differences with Lamartine, and La Guéronnière became more and more closely identified with the policy of the prince president. Under the Empire he was a member of the council of state (1853), senator (1861), ambassador at Brussels (1868), and at Constantinople (1870), and grand officer of the legion of honour (1866). He died in Paris on the 23rd of December 1875. Besides his _Études et portraits politiques contemporains_ (1856) his most important works are those on the foreign policy of the Empire: _La France, Rome et Italie_ (1851), _L'Abandon de Rome_ (1862), _De la politique intérieure et extérieure de la France_ (1862).

His elder brother, ALFRED DUBREUIL HÉLION, Comte de La Guéronnière (1810-1884), who remained faithful to the Legitimist party, was also a well-known writer and journalist. He was consistent in his opposition to the July Monarchy and the Empire, but in a series of books on the crisis of 1870-1871 showed a more favourable attitude to the Republic.

LAGUERRE, JEAN HENRI GEORGES (1858- ), French lawyer and politician, was born in Paris on the 24th of June 1858. Called to the bar in 1879, he distinguished himself by brilliant pleadings in favour of socialist and anarchist leaders, defending Prince Kropotkine at Lyons in 1883, Louise Michel in the same year; and in 1886, with A. Millerand as colleague he defended Ernest Roche and Duc Quercy, the instigators of the Decazeville strike. His strictures on the _procureur de la République_ on this occasion being declared libellous he was suspended for six months and in 1890 he again incurred suspension for an attack on the attorney-general, Quesnay de Beaurepaire. He also pleaded in the greatest criminal cases of his time, though from 1893 onwards exclusively in the provinces, his exclusion from the Parisian bar having been secured on the pretext of his connexion with _La Presse_. He entered the Chamber of Deputies for Apt in 1883 as a representative of the extreme revisionist programme, and was one of the leaders of the Boulangist agitation. He had formerly written for Georges Clemenceau's organ _La Justice_, but when Clemenceau refused to impose any shibboleth on the radical party he became director of _La Presse_. He rallied to the republican party in May 1801, some months before General Boulanger's suicide. He was not re-elected to the Chamber in 1893. Laguerre was an excellent lecturer on the revolutionary period of French history, concerning which he had collected many valuable and rare documents. He interested himself in the fate of the "Little Dauphin" (Louis XVII.), whose supposed remains, buried at Ste Marguerite, he proved to be those of a boy of fourteen.

LAGUNA, or LA LAGUNA, an episcopal city and formerly the capital of the island of Teneriffe, in the Spanish archipelago of the Canary Islands. Pop. (1900) 13,074. Laguna is 4 m. N. by W. of Santa Cruz, in a plain 1800 ft. above sea-level, surrounded by mountains. Snow is unknown here, and the mean annual temperature exceeds 63° F.; but the rainfall is very heavy, and in winter the plain is sometimes flooded. The humidity of the atmosphere, combined with the warm climate and rich volcanic soil, renders the district exceptionally fertile; wheat, wine and tobacco, oranges and other fruits, are produced in abundance. Laguna is the favourite summer residence of the wealthier inhabitants of Santa Cruz. Besides the cathedral, the city contains several picturesque convents, now secularized, a fine modern town hall, hospitals, a large public library and some ancient palaces of the Spanish nobility. Even the modern buildings have often an appearance of antiquity, owing to the decay caused by damp, and the luxuriant growth of climbing plants.

LA HARPE, JEAN FRANÇOIS DE (1739-1803), French critic, was born in Paris of poor parents on the 20th of November 1739. His father, who signed himself Delharpe, was a descendant of a noble family originally of Vaud. Left an orphan at the age of nine, La Harpe was taken care of for six months by the sisters of charity, and his education was provided for by a scholarship at the Collège d'Harcourt. When nineteen he was imprisoned for some months on the charge of having written a satire against his protectors at the college. La Harpe always denied his guilt, but this culminating misfortune of an early life spent entirely in the position of a dependent had possibly something to do with the bitterness he evinced in later life. In 1763 his tragedy of _Warwick_ was played before the court. This, his first play, was perhaps the best he ever wrote. The many authors whom he afterwards offended were always able to observe that the critic's own plays did not reach the standard of excellence he set up. _Timoléon_ (1764), _Pharamond_ (1765) and _Gustave Wasa_ (1766) were failures. _Mélanie_ was a better play, but was never represented. The success of _Warwick_ led to a correspondence with Voltaire, who conceived a high opinion of La Harpe, even allowing him to correct his verses. In 1764 La Harpe married the daughter of a coffee house keeper. This marriage, which proved very unhappy and was dissolved, did not improve his position. They were very poor, and for some time were guests of Voltaire at Ferney. When, after Voltaire's death, La Harpe in his praise of the philosopher ventured on some reasonable, but rather ill-timed, criticism of individual works, he was accused of treachery to one who had been his constant friend. In 1768 he returned from Ferney to Paris, where he began to write for the _Mercure_. He was a born fighter and had small mercy on the authors whose work he handled. But he was himself violently attacked, and suffered under many epigrams, especially those of Lebrun-Pindare. No more striking proof of the general hostility can be given than his reception (1776) at the Academy, which Sainte-Beuve calls his "execution." Marmontel, who received him, used the occasion to eulogize La Harpe's predecessor, Charles Pierre Colardeau, especially for his pacific, modest and indulgent disposition. The speech was punctuated by the applause of the audience, who chose to regard it as a series of sarcasms on the new member. Eventually La Harpe was compelled to resign from the _Mercure_, which he had edited from 1770. On the stage he produced _Les Barmécides_ (1778), _Philoctète_, _Jeanne de Naples_ (1781), _Les Brames_ (1783), _Coriolan_ (1784), _Virginie_ (1786). In 1786 he began a course of literature at the newly-established Lycée. In these lectures, published as the _Cours de littérature ancienne et moderne_, La Harpe is at his best, for he found a standpoint more or less independent of contemporary polemics. He is said to be inexact in dealing with the ancients, and he had only a superficial knowledge of the middle ages, but he is excellent in his analysis of 17th-century writers. Sainte-Beuve found in him the best critic of the French school of tragedy, which reached its perfection in Racine. La Harpe was a disciple of the "_philosophes_"; he supported the extreme party through the excesses of 1792 and 1793. In 1793 he edited the _Mercure de France_ which adhered blindly to the revolutionary leaders. But in April 1794 he was nevertheless seized as a "suspect." In prison he underwent a spiritual crisis which he described in convincing language, and he emerged an ardent Catholic and a reactionist in politics. When he resumed his chair at the Lycée, he attacked his former friends in politics and literature. He was imprudent enough to begin the publication (1801-1807) of his _Correspondance littéraire_ (1774-1791) with the grand-duke, afterwards the emperor Paul of Russia. In these letters he surpassed the brutalities of the _Mercure_. He contracted a second marriage, which was dissolved after a few weeks by his wife. He died on the 11th of February 1803 in Paris, leaving in his will an incongruous exhortation to his fellow countrymen to maintain peace and concord. Among his posthumous works was a _Prophétie de Cazotte_ which Sainte-Beuve pronounces his best work. It is a sombre description of a dinner-party of notables long before the Revolution, when Jacques Cazotte is made to prophesy the frightful fates awaiting the various individuals of the company.

Among his works not already mentioned are:--_Commentaire sur Racine_
(1795-1796), published in 1807; _Commentaire sur le théâtre de
Voltaire_ of earlier date (published posthumously in 1814), and an
epic poem _La Religion_ (1814). His _Cours de littérature_ has been
often reprinted. To the edition of 1825-1826 is prefixed a notice by
Pierre Daunou. See also Sainte-Beuve, _Causeries du lundi_, vol. v.;
G. Peignot, _Recherches historiques, bibliographiques et littéraires
... sur La Harpe_ (1820).

LAHIRE, LAURENT DE (1606-1656), French painter, was born at Paris on the 27th of February 1606. He became a pupil of Lallemand, studied the works of Primaticcio at Fontainebleau, but never visited Italy, and belongs wholly to that transition period which preceded the school of Simon Vouet. His picture of Nicolas V. opening the crypt in which he discovers the corpse of St Francis of Assisi standing (Louvre) was executed in 1630 for the Capuchins of the Marais; it shows a gravity and sobriety of character which marked Lahire's best work, and seems not to have been without influence on Le Sueur. The Louvre contains eight other works, and paintings by Lahire are in the museums of Strasburg, Rouen and Le Mans. His drawings, of which the British Museum possesses a fine example, "Presentation of the Virgin in the Temple," are treated as seriously as his paintings, and sometimes show simplicity and dignity of effect. The example of the Capuchins, for whom he executed several other works in Paris, Rouen and Fécamp, was followed by the goldsmiths' company, for whom he produced in 1635 "St Peter healing the Sick" (Louvre) and the "Conversion of St Paul" in 1637. In 1646, with eleven other artists, he founded the French Royal Academy of Painting and Sculpture. Richelieu called Lahire to the Palais Royal; Chancellor Séguier, Tallemant de Réaux and many others entrusted him with important works of decoration; for the Gobelins he designed a series of large compositions. Lahire painted also a great number of portraits, and in 1654 united in one work for the town-hall of Paris those of the principal dignitaries of the municipality. He died on the 28th of December 1656.

LAHN, a river of Germany, a right-bank tributary of the Rhine. Its source is on the Jagdberg, a summit of the Rothaar Mountains, in the cellar of a house (Lahnhof), at an elevation of 1975 ft. It flows at first eastward and then southward to Giessen, then turns south-westward and with a winding course reaches the Rhine between the towns of Oberlahnstein and Niederlahnstein. Its valley, the lower part of which divides the Taunus hills from the Westerwald, is often very narrow and picturesque; among the towns and sites of interest on its banks are Marburg and Giessen with their universities, Wetzlar with its cathedral, Runkel with its castle, Limburg with its cathedral, the castles of Schaumburg, Balduinstein, Laurenburg, Langenau, Burgstein and Nassau, and the well-known health resort of Ems. The Lahn is about 135 m. long; it is navigable from its mouth to Giessen, and is partly canalized. A railway follows the valley practically throughout. In 1796 there were here several encounters between the French under General Jourdan and the troops of the archduke Johan, which resulted in the retreat of the French across the Rhine.

LAHNDA (properly _Lahnda_ or _Lahinda_, western, or _Lahnde-di boli_, the language of the West), an Indo-Aryan language spoken in the western Punjab. In 1901 the number of speakers was 3,337,917. Its eastern boundary is very indefinite as the language gradually merges into the Panjabi immediately to the east, but it is conventionally taken as the river Chenab from the Kashmir frontier to the town of Ramnagar, and thence as a straight line to the south-west corner of the district of Montgomery. Lahnda is also spoken in the north of the state of Bahawalpur and of the province of Sind, in which latter locality it is known as Siraiki. Its western boundary is, roughly speaking, the river Indus, across which the language of the Afghan population is Pashto (Pushtu), while the Hindu settlers still speak Lahnda. In the Derajat, however, Lahnda is the principal language of all classes in the plains west of the river.

Lahnda is also known as Western Panjabi and as Jatki, or the language of the Jats, who form the bulk of the population whose mother-tongue it is. In the Derajat it is called Hindko or the language of Hindus. In 1819 the Serampur missionaries published a Lahnda version of the New Testament. They called the language Uchchi, from the important town of Uch near the confluence of the Jhelam and the Chenab. This name is commonly met with in old writings. It has numerous dialects, which fall into two main groups, a northern and a southern, the speakers of which are separated by the Salt Range. The principal varieties of the northern group are Hindki (the same in meaning as Hindko) and Pothwari. In the southern group the most important are Khetrani, Multani, and the dialect of Shahpur. The language possesses no literature.

Lahnda belongs to the north-western group of the outer band of
Indo-Aryan languages (q.v.), the other members being Kashmiri (q.v.)
and Sindhi, with both of which it is closely connected. See SINDHI;
also HINDOSTANI. (G. A. Gr.)