122-488: Joseph-Louis Lagrange was an Italian mathematician, physicist and astronomer. Lagrange or La Grange may also refer to: Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier ; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia , was an Italian mathematician , physicist and astronomer , later naturalized French. He made significant contributions to
244-713: A Conti from the noble Roman family. Lagrange's father, Giuseppe Francesco Lodovico, was a doctor in Law at the University of Torino , while his mother was the only child of a rich doctor of Cambiano , in the countryside of Turin . He was raised as a Roman Catholic (but later on became an agnostic ). His father, who had charge of the King 's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained
366-561: A brief historical insight into his reason for proposing undecimal or Base 11 as the base number for the reformed system of weights and measures. The lectures were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory" ["Les professeurs aux Écoles Normales ont pris, avec les Représentants du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits" ]. The discourses were ordered and taken down in shorthand to enable
488-473: A century lay for more than two years unopened on his desk. Curiosity as to the results of the French Revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed. It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend,
610-421: A good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his favourite subject was classical Latin. At first, he had no great enthusiasm for mathematics, finding Greek geometry rather dull. It
732-406: A group that denied free will and had already been condemned by the popes. Louis was no theologian and understood little of the complex doctrines of Jansenism, satisfying himself with the fact that they threatened the unity of the state. In this, he garnered the friendship of the papacy, which had previously been hostile to France because of its policy of putting all church property in the country under
854-505: A long series of papers which created the science of partial differential equations . A large part of these results was collected in the second edition of Euler's integral calculus which was published in 1794. Lastly, there are numerous papers on problems in astronomy . Of these the most important are the following: Over and above these various papers he composed his fundamental treatise, the Mécanique analytique . In this book, he lays down
976-511: A massive increase in urban populations, although on the whole France remained a profoundly rural country. Paris was one of the most populated cities in Europe (estimated at 400,000 inhabitants in 1550; 650,000 at the end of the 18th century). Other major French cities include Lyon , Rouen , Bordeaux , Toulouse , and Marseille . These centuries saw several periods of epidemics and crop failures due to wars and climatic change. (Historians speak of
1098-483: A massive territorial expansion and an attempt to better integrate its provinces into an administrative whole. During this period, France expanded to nearly its modern territorial extent through the acquisition of Picardy , Burgundy , Anjou , Maine , Provence , Brittany , Franche-Comté , French Flanders , Navarre , Roussillon , the Duchy of Lorraine , Alsace and Corsica . French acquisitions from 1461 to 1789: Only
1220-468: A novel approach to interpolation and Taylor's theorem . He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points . Lagrange is best known for transforming Newtonian mechanics into a branch of analysis, Lagrangian mechanics . He presented
1342-513: A policy against Spain and the German emperor during the Thirty Years' War (1618–1648) which had broken out among the lands of Germany's Holy Roman Empire. An English-backed Huguenot rebellion (1625–1628) defeated, France intervened directly (1635) in the wider European conflict following her ally (Protestant) Sweden 's failure to build upon initial success. After the death of both king and cardinal,
SECTION 10
#17328514530601464-683: A pretext. When Ferdinand I of Naples died in 1494, Charles invaded the peninsula. For several months, French forces moved through Italy virtually unopposed, since the condottieri armies of the Italian city-states were unable to resist them. Their sack of Naples finally provoked a reaction, however, and the League of Venice was formed against them. Italian troops defeated the French at the Battle of Fornovo , forcing Charles to withdraw to France. Ludovico, having betrayed
1586-410: A problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. In this academy one of his students was François Daviet . Lagrange is one of the founders of the calculus of variations . Starting in 1754, he worked on the problem of the tautochrone , discovering a method of maximizing and minimizing functionals in
1708-403: A single erasure or correction. W.W. Rouse Ball Firstborn of eleven children as Giuseppe Lodovico Lagrangia , Lagrange was of Italian and French descent. His paternal great-grandfather was a French captain of cavalry, whose family originated from the French region of Tours . After serving under Louis XIV , he had entered the service of Charles Emmanuel II , Duke of Savoy , and married
1830-584: A successful teacher. Fourier , who attended his lectures in 1795, wrote: In 1810, Lagrange started a thorough revision of the Mécanique analytique , but he was able to complete only about two-thirds of it before his death in Paris in 1813, in 128 rue du Faubourg Saint-Honoré . Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in
1952-458: A war between Louis and Ferdinand. By 1503, Louis, having been defeated at the Battle of Cerignola and Battle of Garigliano , was forced to withdraw from Naples, which was left under the control of the Spanish viceroy, Ramón de Cardona . French forces under Gaston de Foix inflicted an overwhelming defeat on a Spanish army at the Battle of Ravenna in 1512, but Foix was killed during the battle, and
2074-480: A way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis . Euler
2196-417: Is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables x , called generalized coordinates , whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of
2318-459: The Battle of Ceresole in 1544, but the French failed to penetrate further into Lombardy. Charles and Henry VIII of England then proceeded to invade northern France, seizing Boulogne and Soissons . A lack of cooperation between the Spanish and English armies, coupled with increasingly aggressive Ottoman attacks, led Charles to abandon these conquests, restoring the status quo once again. In 1547, Henry II of France , who had succeeded Francis to
2440-481: The Edict of Nantes (1598), which guaranteed freedom of private worship and civil equality. France's pacification under Henry IV laid much of the ground for the beginnings of France's rise to European hegemony. One of the most admired French kings, Henry was fatally stabbed by a Catholic fanatic in 1610 as war with Spain threatened. Troubles gradually developed during the regency headed by his queen Marie de Medici . France
2562-557: The Franco-Dutch War 1672–1678) brought further territorial gains ( Artois and western Flanders and the free county of Burgundy , left to the Empire in 1482), but at the cost of the increasingly concerted opposition of rival powers. French culture was part of French hegemony. In the early part of the century French painters had to go to Rome to shed their provinciality ( Nicolas Poussin , Claude Lorrain ), but Simon Vouet brought home
SECTION 20
#17328514530602684-736: The Kingdom of England ); there were also foreign enclaves, like the Comtat Venaissin . In addition, certain provinces within France were ostensibly personal fiefdoms of noble families (like the Bourbonnais , Marche , Forez and Auvergne provinces held by the House of Bourbon until the provinces were forcibly integrated into the royal domaine in 1527 after the fall of Charles III, Duke of Bourbon ). The late 15th, 16th and 17th centuries would see France undergo
2806-531: The Ordinance of Villers-Cotterêts ), internal conflicts and civil wars, but they remained a confusing patchwork of local privilege and historic differences until the French Revolution took place in a radical time suppression of administrative incoherence. The Peace of Etaples (1492) marks, for some, the beginning of the early modern period in France. After the Hundred Years' War (1337–1453), France supported
2928-708: The Panthéon in Paris. The inscription on his tomb reads in translation: JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of the Reunion . Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813. Lagrange was extremely active scientifically during
3050-619: The Peace of Westphalia (1648) secured universal acceptance of Germany's political and religious fragmentation, but the Regency of Anne of Austria and her minister Cardinal Mazarin experienced a civil uprising known as the Fronde (1648–1653) which expanded into a Franco-Spanish War (1653–1659). The Treaty of the Pyrenees (1659) formalised France's seizure (1642) of the Spanish territory of Roussillon after
3172-490: The Prussian Academy of Sciences . Several of them deal with questions in algebra . Several of his early papers also deal with questions of number theory. There are also numerous articles on various points of analytical geometry . In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical forms . During the years from 1772 to 1785, he contributed
3294-622: The War of the Polish Succession from 1733 to 1735. Large-scale warfare resumed with the War of the Austrian Succession (1740–1748). But alliance with the traditional Habsburg enemy (the " Diplomatic Revolution " of 1756) against the rising power of Britain and Prussia led to costly failure in the Seven Years' War (1756–1763) and the loss of France's North American colonies. On the whole,
3416-439: The calculus of variations . The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations, and he illustrates its use by deducing the principle of least action , and by solutions of various problems in dynamics . The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on
3538-549: The decimalisation process in Revolutionary France , became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes , and became Senator in 1799. Lagrange was one of the creators of the calculus of variations , deriving the Euler–Lagrange equations for extrema of functionals . He extended the method to include possible constraints, arriving at
3660-452: The integral calculus ; a solution of a Fermat 's problem: given an integer n which is not a perfect square , to find a number x such that nx + 1 is a perfect square; and the general differential equations of motion for three bodies moving under their mutual attractions. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face
3782-521: The secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the Académie française , and in each case, the prize was awarded to him. Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics . The greater number of his papers during this time were, however, contributed to
Lagrange (disambiguation) - Misplaced Pages Continue
3904-447: The "Treaty of Madrid" (1526). The inconclusive third war between Charles and Francis began with the death of Francesco II Sforza , the duke of Milan . When Charles' son Philip inherited the duchy, Francis invaded Italy, capturing Turin , but failed to take Milan. In response, Charles invaded Provence , advancing to Aix-en-Provence , but withdrew to Spain rather than attacking the heavily fortified Avignon . The Truce of Nice ended
4026-722: The 14th century, the gains of the previous half-century were to be jeopardised by a further protracted series of conflicts, the Italian Wars (1494–1559), where French efforts to gain dominance ended in the increased power of the Habsburg Holy Roman Emperors of Germany. In 1445, the first steps were made towards fashioning a regular army out of the poorly disciplined mercenary bands that French kings traditionally relied on. The medieval division of society into "those who fought (nobility), those who prayed (clergy), and those who worked (everyone else)" still held strong and warfare
4148-510: The 15th century drew to a close, French kings could take confidence in the fact that England had been mostly driven from their territory and so they could now embark on an expansionist foreign policy. The invasion of Italy by Charles VIII in 1494 began 62 years of war with the Habsburgs (the Italian Wars ). Despite the beginnings of rapid demographic and economic recovery after the Black Death of
4270-403: The 1630s, and with Pascal , Descartes , Bayle , Corneille , Racine and Molière , France became the cultural center of Europe. In an effort to prevent the nobility from revolting and challenging his authority, Louis implemented an extremely elaborate system of court etiquette with the idea that learning it would occupy most of the nobles' time and they could not plan rebellion. By the start of
4392-509: The 18th century saw growing discontent with the monarchy and the established order. Louis XV was a highly unpopular king for his sexual excesses, overall weakness, and for losing Canada to the British. A strong ruler like Louis XIV could enhance the position of the monarchy, while Louis XV weakened it. The writings of the philosophers such as Voltaire were a clear sign of discontent, but the king chose to ignore them. He died of smallpox in 1774, and
4514-400: The 18th century, the nobility in France had been effectively neutered and would never again have more power than the crown. Also, Louis willingly granted titles of nobility to those who had performed distinguished service to the state so that it did not become a closed caste and it was possible for commoners to rise through the social ranks. The king sought to impose total religious uniformity on
4636-608: The 19th century. The Ancien Régime, the French term rendered in English as "Old Rule", "Old Kingdom", or simply "Old Regime", refers primarily to the aristocratic , social and political system established in France from (roughly) the 15th century to the 18th century under the late Valois and Bourbon dynasties. The administrative and social structures of the Ancien Régime were the result of years of state-building, legislative acts (like
4758-605: The Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series, relying in particular on the principle of the generality of algebra . A somewhat similar method had been previously used by John Landen in the Residual Analysis , published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with
4880-503: The Duchy of Savoy , the city of Nice and some other small papal (e.g., Avignon ) and foreign possessions would be acquired later. (For a map of historic French provinces, see Provinces of France ). France also embarked on exploration, colonisation, and mercantile exchanges with the Americas ( New France , Louisiana , Martinique , Guadeloupe , Haiti , French Guiana ), India ( Pondicherry ),
5002-452: The French at Fornovo, retained his throne until 1499, when Charles's successor, Louis XII of France , invaded Lombardy and seized Milan . In 1500, Louis XII, having reached an agreement with Ferdinand II of Aragon to divide Naples, marched south from Milan. By 1502, combined French and Aragonese forces had seized control of the Kingdom; disagreements about the terms of the partition led to
Lagrange (disambiguation) - Misplaced Pages Continue
5124-443: The French countryside during this period remained poor and overpopulated. The resistance of peasants to adopt the potato, according to some monarchist apologists, and other new agricultural innovations while continuing to rely on cereal crops led to repeated catastrophic famines long after they had ceased in the rest of Western Europe. Prior to Louis XIV's reign, French soldiers frequently went into battle barefoot and with no weapons. On
5246-464: The French people shed few tears at his passing. While France had not yet experienced the industrial revolution that was beginning in England, the rising middle class of the cities felt increasingly frustrated with a system and rulers that seemed silly, frivolous, aloof, and antiquated, even if true feudalism no longer existed in France. Anti-establishment ideas fermented in 18th-century France in part due to
5368-404: The French revolution, the teaching of French was promoted in all the schools. The French used would be that of the legal system, which differed from the French spoken in the courts of France before the revolution. Like the orators during the French revolution, the pronunciation of every syllable would become the new language. France would not become a linguistically unified country until the end of
5490-453: The French were forced to withdraw from Italy by an invasion of Milan by the Swiss, who reinstated Maximilian Sforza to the ducal throne. The Holy League , left victorious, fell apart over the subject of dividing the spoils, and in 1513 Venice allied with France, agreeing to partition Lombardy between them. Louis mounted another invasion of Milan, but was defeated at the Battle of Novara , which
5612-567: The French were in control of the Spanish Netherlands and Catalonia. However, Louis gave back his conquests and gained only Haiti . The French people, feeling that their sacrifices in the war had been for nothing, never forgave him. The Battle of La Hougue (1692) was the decisive naval battle in the war and confirmed the durable dominance of the Royal Navy of England. In November 1700, the severely ill Spanish king Charles II died, ending
5734-526: The Habsburg line in that country. Louis had long waited for this moment, and now planned to put a Bourbon relative, Philip, Duke of Anjou, on the throne. Essentially, Spain was to become an obedient satellite of France, ruled by a king who would carry out orders from Versailles. Realizing how this would upset the balance of power, the other European rulers were outraged. However, most of the alternatives were equally undesirable. For example, putting another Habsburg on
5856-506: The Habsburgs. While the Duke of Anjou was accepted as King Philip V of Spain , this was done under the condition that the French and Spanish thrones never be united. Finally, France agreed to stop supporting Jacobite pretenders to the English throne. Just after the war ended, Louis died, having ruled France for 72 years. While often considered a tyrant and a warmonger (especially in England), Louis XIV
5978-586: The Indian Ocean ( Réunion ), the Far East, and a few African trading posts. Although Paris was the capital of France, the later Valois kings largely abandoned the city as their primary residence, preferring instead various châteaux of the Loire Valley and Parisian countryside. Henry IV made Paris his primary residence (promoting a major building boom in private mansions), but Louis XIV once again withdrew from
6100-643: The Italian Wars over, when France was plunged into a domestic crisis with far-reaching consequences. Despite the conclusion of a Concordat between France and the Papacy (1516) , granting the crown unrivalled power in senior ecclesiastical appointments, France was deeply affected by the Protestant Reformation 's attempt to break the unity of Roman Catholic Europe. A growing urban-based Protestant minority (later dubbed Huguenots ) faced ever harsher repression under
6222-788: The Lancastrian side in The Wars of the Roses . France and England signed the Treaty of Picquigny in 1475 — the official end date of the Hundred Years' War. In 1492 and 1493, after supporting the victorious House of Tudor in the Battle of Bosworth Field , Charles VIII of France signed three additional treaties with Henry VII of England , Maximilian I of Habsburg , and Ferdinand II of Aragon respectively at Étaples (1492), Senlis (1493) and in Barcelona (1493). As
SECTION 50
#17328514530606344-589: The Ottoman Empire, they could not do anything about this for a number of years. The basic aim of Louis' foreign policy was to give France more easily defensible borders, and to eliminate weak spots (Strassburg had often been used by the Habsburgs as a gateway into France). Following the Whig establishment on the English and Scottish thrones by the Dutch prince William of Orange in 1688, the anti-French " Grand Alliance " of 1689
6466-693: The Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler . In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni , the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be
6588-527: The Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series , and the kind of problems for which it is suitable. Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on
6710-584: The Turin Society in 1762 and 1773. In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the Turin Academy of Sciences , and most of his early writings are to be found in the five volumes of its transactions, usually known as the Miscellanea Taurinensia . Many of these are elaborate papers. The first volume contains a paper on
6832-515: The US War of Independence was also very expensive. With the country deeply in debt, Louis XVI permitted the radical reforms of Turgot and Malesherbes , but noble disaffection led to Turgot's dismissal and Malesherbes' resignation in 1776. They were replaced by Jacques Necker . Necker had resigned in 1781 to be replaced by Calonne and Brienne , before being restored in 1788. A harsh winter that year led to widespread food shortages, and by then France
6954-430: The afternoon after the trial. Lagrange said on the death of Lavoisier: Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, Napoleon ) gave him honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: " I believe that, in general, one of
7076-613: The assassination of both Henry of Guise (1588) and Henry III (1589), the conflict was ended by the accession of the Protestant king of Navarre as Henry IV (first king of the Bourbon dynasty) and his subsequent abandonment of Protestantism (Expedient of 1592) effective in 1593, his acceptance by most of the Catholic establishment (1594) and by the Pope (1595), and his issue of the toleration decree known as
7198-546: The astronomer Pierre Charles Le Monnier . She insisted on marrying him and proved a devoted wife to whom he became warmly attached. In September 1793, the Reign of Terror began. Under the intervention of Antoine Lavoisier , who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other tax farmers were arrested and sentenced to death and guillotined on
7320-421: The book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788. Lagrange's lectures on the differential calculus at École Polytechnique form the basis of his treatise Théorie des fonctions analytiques , which was published in 1797. This work is the extension of an idea contained in a paper he had sent to
7442-453: The choice of the metre and kilogram units with decimal subdivision, by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795. In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale , which enjoyed only a short existence of four months. His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide
SECTION 60
#17328514530607564-420: The city in the last decades of his reign and Versailles became the primary seat of the French monarchy for much of the following century. The administrative and legal system in France in this period is generally called the Ancien Régime . The Black Death had killed an estimated one-third of the population of France from its appearance in 1348. The concurrent Hundred Years' War slowed recovery. It would be
7686-413: The conclusion that the form of the curve at any time t is given by the equation y = a sin ( m x ) sin ( n t ) {\displaystyle y=a\sin(mx)\sin(nt)\,} . The article concludes with a masterly discussion of echoes , beats , and compound sounds. Other articles in this volume are on recurring series , probabilities , and
7808-484: The country's relative egalitarianism. While less liberal than England during the same period, the French monarchy never approached the absolutism of the eastern rulers in Vienna, Berlin, St. Petersburg, and Constantinople in part because the country's traditional development as a decentralized, feudal society acted as a restraint on the power of the king. Different social classes in France each had their own unique set of privileges so that no one class could completely dominate
7930-406: The country, repealing the Edict of Nantes in 1685. The infamous practice of dragonnades was adopted, whereby rough soldiers were quartered in the homes of Protestant families and allowed to have their way with them. Scores of Protestants fled France, costing the country a great many intellectuals, artisans, and other valuable people. Persecution extended to unorthodox Catholics like the Jansenists ,
8052-517: The crushing of the ephemeral Catalan Republic and ushered a short period of peace. For most of the reign of Louis XIV (1643–1715), France was the dominant power in Europe, aided by the diplomacy of Richelieu's successor (1642–1661) Cardinal Mazarin and the economic policies (1661–1683) of Colbert . Colbert's attempts to promote economic growth and the creation of new industries were not a great success, and France did not undergo any sort of industrial revolution during Louis XIV's reign. Indeed, much of
8174-471: The deputies to see how the professors acquitted themselves. It was also thought the published lectures would interest a significant portion of the citizenry ["Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation" ]. In 1794, Lagrange was appointed professor of the École Polytechnique ; and his lectures there, described by mathematicians who had
8296-430: The early 16th century before the population recovered to mid-14th-century levels. With an estimated population of 11 million in 1400, 20 million in the 17th century, and 28 million in 1789, until 1795 France was the most populated country in Europe (even twice the size of Britain or the Dutch Republic ) and the third most populous country in the world, behind only China and India. These demographic changes also led to
8418-447: The entirety of northern Italy. The elevation of Charles of Spain to Holy Roman Emperor , a position that Francis had desired, led to a collapse of relations between France and the Habsburgs. In 1519, a Spanish invasion of Navarre , nominally a French fief, provided Francis with a pretext for starting a general war; French forces flooded into Italy and began a campaign to drive Charles from Naples. The French were outmatched, however, by
8540-472: The equation whose roots are the squares of the differences of the roots of the original equation may be used so as to give considerable information as to the position and nature of those roots. Kingdom of France (1498-1791) The Kingdom of France in the early modern period , from the Renaissance ( c. 1500–1550 ) to the Revolution (1789–1804), was a monarchy ruled by the House of Bourbon (a Capetian cadet branch ). This corresponds to
8662-444: The exact amount of work which he could do before exhaustion. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction. Nonetheless, during his years in Berlin, Lagrange's health
8784-605: The fields of analysis , number theory , and both classical and celestial mechanics . In 1766, on the recommendation of Leonhard Euler and d'Alembert , Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia , where he stayed for over twenty years, producing many volumes of work and winning several prizes of the French Academy of Sciences . Lagrange's treatise on analytical mechanics ( Mécanique analytique , 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), which
8906-509: The first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable ". A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in the full state on Lagrange's father and tender the congratulations of the republic on the achievements of his son, who "had done honour to all mankind by his genius, and whom it
9028-546: The focus of the war to Flanders , where Philip, in conjunction with Emmanuel Philibert, Duke of Savoy , defeated the French at St. Quentin . England's entry into the war later that year led to the French capture of Calais , England's last possession on the French mainland, and French armies plundered Spanish possessions in the Low Countries ; but Henry was nonetheless forced to accept the Peace of Cateau-Cambrésis , in which he renounced any further claims to Italy. Barely were
9150-409: The following as amongst the most important. First, his contributions to the fourth and fifth volumes, 1766–1773, of the Miscellanea Taurinensia ; of which the most important was the one in 1771, in which he discussed how numerous astronomical observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 1784–1785, of the transactions of
9272-469: The fully developed Spanish tercio tactics, and suffered a series of crippling defeats at Bicocca and Sesia against Spanish troops under Fernando d'Avalos . With Milan itself threatened, Francis personally led a French army into Lombardy in 1525, only to be defeated and captured at the Battle of Pavia ; imprisoned in Madrid , Francis was forced to agree to extensive concessions over his Italian territories in
9394-409: The good fortune to be able to attend them, were almost perfect both in form and matter. Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation. However, Lagrange does not seem to have been
9516-427: The jurisdiction of the state rather than of Rome. Cardinal Mazarin oversaw the creation of a French navy that rivaled England's, expanding it from 25 ships to almost 200. The size of the army was also considerably increased. Starting in the 1670s, Louis XIV established the so-called Chambers of Reunion , courts in which judges would determine whether certain Habsburg territories belonged rightfully to France. The king
9638-462: The language for legal and juridical acts.) Nevertheless, in 1790, only half of the population spoke or understood standard French. The southern half of the country continued to speak Occitan languages (such as Provençal ), and other inhabitants spoke Breton , Catalan , Basque , Dutch ( West Flemish ), and Franco-Provençal . In the north of France, regional dialects of the various langues d'oïl continued to be spoken in rural communities. During
9760-418: The law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics , both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalised co-ordinates by which he obtained this result
9882-466: The mechanical "principles" as simple results of the variational calculus. In appearance he was of medium height, and slightly formed, with pale blue eyes and a colourless complexion. In character he was nervous and timid, he detested controversy, and to avoid it willingly allowed others to take the credit for what he had himself done. He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without
10004-472: The method of Lagrange multipliers . Lagrange invented the method of solving differential equations known as variation of parameters , applied differential calculus to the theory of probabilities and worked on solutions for algebraic equations . He proved that every natural number is a sum of four squares . His treatise Theorie des fonctions analytiques laid some of the foundations of group theory , anticipating Galois . In calculus , Lagrange developed
10126-435: The method of Lagrange multipliers —though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may suffice to mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action . All the analysis is so elegant that Sir William Rowan Hamilton said
10248-548: The next century. However, the nobility and troops were often disloyal to the king, if not outright rebellious, and it took another army reform by Louis XIV to finally transform the French army into an obedient force. Ludovico Sforza , the Duke of Milan, seeking an ally against the Republic of Venice , encouraged Charles VIII of France to invade Italy, using the Angevin claim to the throne of Naples , then under Aragonese control, as
10370-465: The offer. In 1765, d'Alembert interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. He again turned down the offer, responding that In 1766, after Euler left Berlin for Saint Petersburg , Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange
10492-503: The other hand, France's high birthrate until the 18th century proved beneficial to its rulers since it meant the country could field larger armies than its neighbors. In fact, the king's foreign policy, as well as his lavish court and construction projects, left the country in enormous debt. The Palace of Versailles was criticized as overly extravagant even while it was still under construction, but dozens of imitations were built across Europe. Renewed war (the War of Devolution 1667–1668 and
10614-454: The others. Upon Louis XV's death, his grandson Louis XVI became king. Initially popular, he too came to be widely detested by the 1780s. Again a weak ruler, he was married to an Austrian archduchess, Marie Antoinette , whose naïvety and cloistered/alienated Versailles life permitted ignorance of the true extravagance and wasteful use of borrowed money (Marie Antoinette was significantly more frugal than her predecessors). French intervention in
10736-469: The period 1550–1850 as the " Little Ice Age ".) Between 1693 and 1694, France lost 6% of its population. In the extremely harsh winter of 1709, France lost 3.5% of its population. In the past 300 years, no period has been so proportionally deadly for the French, both World Wars included. Linguistically, the differences in France were extreme. Before the Renaissance, the language spoken in the north of France
10858-622: The remnants of medieval feudalism and established a centralized state under an absolute monarch , a system that would endure until the French Revolution and beyond . In the mid 15th century, France was significantly smaller than it is today, and numerous border provinces (such as Roussillon , Cerdagne , Calais , Béarn , Navarre , County of Foix , Flanders , Artois , Lorraine , Alsace , Trois-Évêchés , Franche-Comté , Savoy , Bresse , Bugey , Gex , Nice , Provence , Corsica and Brittany ) were autonomous or foreign-held (as by
10980-488: The right of the king to govern nor did they question the Church. The reign (1715–1774) of Louis XV saw an initial return to peace and prosperity under the regency (1715–1723) of Philip II, Duke of Orléans , whose policies were largely continued (1726–1743) by Cardinal Fleury , prime minister in all but name. The exhaustion of Europe after two major wars resulted in a long period of peace, only interrupted by minor conflicts like
11102-487: The rule of Francis I's son King Henry II . After Henry II's unfortunate death in a joust, the country was ruled by his widow Catherine de' Medici and her sons Francis II , Charles IX and Henry III . Renewed Catholic reaction headed by the powerful dukes of Guise culminated in a massacre of Huguenots (1562), starting the first of the French Wars of Religion , during which English, German, and Spanish forces intervened on
11224-596: The side of rival Protestant and Catholic forces. Opposed to absolute monarchy, the Huguenots Monarchomachs theorized during this time the right of rebellion and the legitimacy of tyrannicide . The Wars of Religion culminated in the War of the Three Henrys in which Henry III assassinated Henry de Guise , leader of the Spanish-backed Catholic league , and the king was murdered in return. After
11346-411: The so-called Ancien Régime ("old rule"). The territory of France during this period increased until it included essentially the extent of the modern country , and it also included the territories of the first French colonial empire overseas. The period is dominated by the figure of the "Sun King", Louis XIV (his reign of 1643–1715 being one of the longest in history ), who managed to eliminate
11468-536: The starting point for the researches of Cauchy , Jacobi , and Weierstrass . At a later period Lagrange fully embraced the use of infinitesimals in preference to founding the differential calculus on the study of algebraic forms; and in the preface to the second edition of the Mécanique Analytique , which was issued in 1811, he justifies the employment of infinitesimals, and concludes by saying that: His Résolution des équations numériques , published in 1798,
11590-439: The system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form where T represents the kinetic energy and V represents the potential energy of the system. He then presented what we now know as
11712-642: The taste for a classicized baroque that would characterise the French Baroque , epitomised in the Académie de peinture et de sculpture , in the painting of Charles Le Brun and the sculpture of François Girardon . With the Palais du Luxembourg , the Château de Maisons and Vaux-le-Vicomte , French classical architecture was admired abroad even before the creation of Versailles or Perrault's Louvre colonnade. Parisian salon culture set standards of discriminating taste from
11834-441: The theory of the propagation of sound; in this he indicates a mistake made by Newton , obtains the general differential equation for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely ; in this paper, he points out a lack of generality in the solutions previously given by Brook Taylor , D'Alembert , and Euler, and arrives at
11956-438: The third with applications to mechanics. Another treatise on the same lines was his Leçons sur le calcul des fonctions , issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers , in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as
12078-671: The throne would end up recreating the empire of Charles V, which would also grossly upset the power balance. After nine years of exhausting war, the last thing Louis wanted was another conflict. However, the rest of Europe would not stand for his ambitions in Spain, and so the War of the Spanish Succession began, a mere three years after the War of the Grand Alliance. The disasters of the war (accompanied by another famine) were so great that France
12200-418: The throne, declared war against Charles with the intent of recapturing Italy and ensuring French, rather than Habsburg, domination of European affairs. An early offensive against Lorraine was successful, but the attempted French invasion of Tuscany in 1553 was defeated at the Battle of Marciano . Charles's abdication in 1556 split the Habsburg empire between Philip II of Spain and Ferdinand I , and shifted
12322-575: The twenty years he spent in Berlin. Not only did he produce his Mécanique analytique , but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note
12444-402: The use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem , the validity of which is, however, open to question; the second deals with applications to geometry; and
12566-520: The war, leaving Turin in French hands but effecting no significant change in the map of Italy. Francis, allying himself with Suleiman I of the Ottoman Empire , launched a final invasion of Italy. A Franco-Ottoman fleet captured the city of Nice in August 1543, and laid siege to the citadel. The defenders were relieved within a month. The French, under François, Count d'Enghien, defeated an Imperial army at
12688-409: The work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish
12810-514: Was a French mathematician, but the Italians continued to claim him as Italian. Lagrange was involved in the development of the metric system of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures ( la Commission des Poids et Mesures ) when he was preparing to escape. After Lavoisier's death in 1794, it was largely Lagrange who influenced
12932-525: Was a collection of different dialects called Oïl languages whereas the written and administrative language remained Latin . By the 16th century, there had developed a standardised form of French (called Middle French ) which would be the basis of the standardised "modern" French of the 17th and 18th century which in turn became the lingua franca of the European continent. (In 1539, with the Ordinance of Villers-Cotterêts , Francis I of France made French alone
13054-430: Was also the fruit of his lectures at École Polytechnique. There he gives the method of approximating the real roots of an equation by means of continued fractions , and enunciates several other theorems. In a note at the end, he shows how Fermat's little theorem , that is where p is a prime and a is prime to p , may be applied to give the complete algebraic solution of any binomial equation. He also here explains how
13176-399: Was always turned to the earth, a problem which he treated by the aid of virtual work . His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780. Already by 1756, Euler and Maupertuis , seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused
13298-465: Was considered a domain of the nobles. Charles VIII marched into Italy with a core force consisting of noble horsemen and non-noble foot soldiers, but in time the role of the latter grew stronger so that by the middle of the 16th century, France had a standing army of 5000 cavalry and 30,000 infantry. The military was reorganized from a system of legions recruited by province (Norman legion, Gascon legion, etc.) to regiments, an arrangement which persisted into
13420-488: Was established. With the Turks now in retreat, the emperor Leopold could turn his attention to France. The ensuing War of the Grand Alliance lasted from 1688 to 1697. France's resources were stretched to the breaking point by the cost of fielding an army of over 300,000 men and two naval squadrons. Famine in 1692–1693 killed up to two million people. The exhaustion of the powers brought the fighting to an end in 1697, by which time
13542-647: Was expansive during all but the end of the 17th century: the French began trading in India and Madagascar , founded Quebec and penetrated the North American Great Lakes and Mississippi , established plantation economies in the West Indies and extended their trade contacts in the Levant and enlarged their merchant marine . Henry IV's son Louis XIII and his minister (1624–1642) Cardinal Richelieu , elaborated
13664-608: Was finally persuaded. He spent the next twenty years in Prussia , where he produced a long series of papers published in the Berlin and Turin transactions, and composed his monumental work, the Mécanique analytique . In 1767, he married his cousin Vittoria Conti. Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find
13786-475: Was not in any way a despot in the 20th-century sense. The traditional customs and institutions of France limited his power and in any case, communications were poor and no national police force existed. Overall, the discontent and revolts of 16th- and 17th-century France did not approach the conditions that led to 1789. Events such as the Frondes were a naïve, unrevolutionary discontent and the people did not challenge
13908-475: Was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley from 1693 which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at
14030-416: Was on the verge of collapse by 1709. In desperation, the king appealed to the French people to save their country, and in doing so gained thousands of new army recruits. Afterwards, his general Marshal Villars managed to drive back the allied forces. In 1714, the war ended with the treaties of Utrecht and Rastadt. France did not lose any territory, and there was no discussion of returning Flanders or Alsace to
14152-487: Was quickly followed by a series of Holy League victories at La Motta , Guinegate , and Flodden , in which the French, Venetian, and Scottish forces were decisively defeated. However, the death of Pope Julius left the League without effective leadership, and when Louis' successor, Francis I , defeated the Swiss at Marignano in 1515, the League collapsed, and by the treaties of Noyon and Brussels, surrendered to France and Venice
14274-413: Was rather poor, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became difficult for Lagrange. In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and Naples , and he accepted the offer of Louis XVI to move to Paris. In France he
14396-505: Was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences , which later became part of the Institut de France (1795). At the beginning of his residence in Paris, he was seized with an attack of melancholy, and even the printed copy of his Mécanique on which he had worked for a quarter of
14518-462: Was relying on the somewhat vague wording in the Treaty of Westphalia, while also dredging up older French claims, some dating back to medieval times. Through this, he concluded that the strategically important imperial city of Strassburg should have gone to France in 1648. In September 1681, French troops occupied the city, which was at once strongly fortified. As the imperial armies were then busy fighting
14640-460: Was the special glory of Piedmont to have produced". It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Appointed senator in 1799, he was the first signer of the Sénatus-consulte which in 1802 annexed his fatherland Piedmont to France. He acquired French citizenship in consequence. The French claimed he
14762-407: Was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed. Lagrange published his method in two memoirs of
14884-400: Was written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in
#59940