Chapter 6

(J. Si.*)

1SeeGeog. Journ.(June 1895), p. 597.2The total population of the colony (including dependencies) on the 1st of January 1907 was estimated at 383,206.3Labourdonnais is credited by several writers with the introduction of the sugar cane into the island. Leguat, however, mentions it as being cultivated during the Dutch occupation.4The régime introduced in 1767 divided the administration between a governor, primarily charged with military matters, and an intendant.

1SeeGeog. Journ.(June 1895), p. 597.

2The total population of the colony (including dependencies) on the 1st of January 1907 was estimated at 383,206.

3Labourdonnais is credited by several writers with the introduction of the sugar cane into the island. Leguat, however, mentions it as being cultivated during the Dutch occupation.

4The régime introduced in 1767 divided the administration between a governor, primarily charged with military matters, and an intendant.

MAURY, JEAN SIFFREIN(1746-1817), French cardinal and archbishop of Paris, the son of a poor cobbler, was born on the 26th of June 1746 at Valréas in the Comtat-Venaissin, the district in France which belonged to the pope. His acuteness was observed by the priests of the seminary at Avignon, where he was educated and took orders. He tried his fortune by writingélogesof famous persons, then a favourite practice; and in 1771 hisélogeon Fénelon was pronounced next best to Laharpe’s by the Academy. The real foundation of his fortunes was the success of a panegyric on St Louis delivered before the Academy in 1772, which caused him to be recommended for an abbacy. In 1777 he published under the title ofDiscours choisishis panegyrics on Saint Louis, Saint Augustine and Fénelon, his remarks on Bossuet and hisEssai sur l’éloquence de la chaire, a volume which contains much good criticism, and remains a French classic. The book was often reprinted asPrincipes de l’éloquence. He became a favourite preacher in Paris, and was Lent preacher at court in 1781, when King Louis XVI. said of his sermon: “If the abbé had only said a few words on religion he would have discussed every possible subject.” In 1781 he obtained the rich priory of Lyons, near Péronne, and in 1785 he was elected to the Academy, as successor of Lefranc de Pompignan. His morals were as loose as those of his great rival Mirabeau, but he was famed in Paris for his wit and gaiety. In 1789 he was elected a member of the states-general by the clergy of the bailliage of Péronne, and from the first proved to be the most able and persevering defender of theancien régime, although he had drawn up the greater part of thecahierof the clergy of Péronne, which contained a considerable programme of reform. It is said that he attempted to emigrate both in July and in October 1789; but after that time he held firmly to his place, when almost universally deserted by his friends. In the Constituent Assembly he took an active part in every important debate, combating with especial vigour the alienation of the property of the clergy. His life was often in danger, but his ready wit always saved it, and it was said that onebon motwould preserve him for a month.When he did emigrate in 1792 he found himself regarded as a martyr to the church and the king, and was at once named archbishopin partibus, and extra nuncio to the diet at Frankfort, and in 1794 cardinal. He was finally made bishop of Montefiascone, and settled down in that little Italian town—but not for long, for in 1798 the French drove him from his retreat, and he sought refuge in Venice and St Petersburg. Next year he returned to Rome as ambassador of the exiled Louis XVIII. at the papal court. In 1804 he began to prepare his return to France by a well-turned letter to Napoleon, congratulating him on restoring religion to France once more. In 1806 he did return; in 1807 he was again received into the Academy; and in 1810, on the refusal of Cardinal Fesch, was made archbishop of Paris. He was presently ordered by the pope to surrender his functions as archbishop of Paris. This he refused to do. On the restoration of the Bourbons he was summarily expelled from the Academy and from the archiepiscopal palace. He retired to Rome, where he was imprisoned in the castle of St Angelo for six months for his disobedience to the papal orders, and died in 1817, a year or two after his release, of disease contracted in prison and of chagrin. As a critic he was a very able writer, and Sainte-Beuve gives him the credit of discovering Father Jacques Bridayne, and of giving Bossuet his rightful place as a preacher above Massillon; as a politician, his wit and eloquence make him a worthy rival of Mirabeau. He sacrificed too much to personal ambition, yet it would have been a graceful act if Louis XVIII. had remembered the courageous supporter of Louis XVI., and the pope the one intrepid defender of the Church in the states-general.

TheŒuvres choisies du Cardinal Maury(5 vols., 1827) contain what is worth preserving. Mgr Ricard has published Maury’sCorrespondance diplomatique(2 vols., Lille, 1891). For his life and character seeVie du Cardinal Maury, by Louis Siffrein Maury, his nephew (1828); J. J. F. Poujoulat,Cardinal Maury, sa vie et ses œuvres(1855); Sainte-Beuve,Causeries du lundi(vol. iv.); Mgr Ricard,L’Abbé Maury(1746-1791),L’Abbé Maury avant 1789, L’Abbé Maury et Mirabeau(1887); G. Bonet-Maury,Le Cardinal Maury d’après ses mémoires et sa correspondance inédits(Paris, 1892); A. Aulard,Les Orateurs de la constituante(Paris, 1882). Of the many libels written against him during the Revolution the most noteworthy are thePetit carême de l’abbé Maury, with a supplement called theSeconde année(1790), and theVie privée de l’abbé Maury(1790), claimed by J. R. Hébert, but attributed by some writers to Restif de la Bretonne. For further bibliographical details see J. M. Quérard,La France littéraire, vol. v. (1833).

MAURY, LOUIS FERDINAND ALFRED(1817-1892), French scholar, was born at Meaux on the 23rd of March 1817. In 1836, having completed his education, he entered the Bibliothèque Nationale, and afterwards the Bibliothèque de l’Institut (1844), where he devoted himself to the study of archaeology, ancient and modern languages, medicine and law. Gifted with a great capacity for work, a remarkable memory and an unbiassed and critical mind, he produced without great effort a number of learned pamphlets and books on the most varied subjects. He rendered great service to the Académie des Inscriptions et Belles Lettres, of which he had been elected a member in 1857. Napoleon III. employed him in research work connected with theHistoire de César, and he was rewarded, proportionately to his active, if modest, part in this work, with the positions of librarian of the Tuileries (1860), professor at the College of France (1862) and director-general of the Archives (1868). It was not, however, to the imperial favour that he owed these high positions. He used his influence for the advancement of science and higher education, and with Victor Duruy was one of the founders of the École des Hautes Études. He died at Paris four years after his retirement from the last post, on the 11th of February 1892.

Bibliography.—His works are numerous:Les Fées au moyen âgeandHistoire des légendes pieuses au moyen âge; two books filled with ingenious ideas, which were published in 1843, and reprinted after the death of the author, with numerous additions under the titleCroyances et légendes du moyen âge(1896);Histoire des grandes forêts de la Gaule et de l’ancienne France(1850, a 3rd ed. revised appeared in 1867 under the titleLes Forêts de la Gaule et de l’ancienne France); La Terre et l’homme, a general historical sketch of geology, geography and ethnology, being the introduction to theHistoire universelle, by Victor Duruy (1854);Histoire des religions de laGrèce antique, (3 vols., 1857-1859);La Magie et l’astrologie dans l’antiquité et dans le moyen âge(1863);Histoire de l’ancienne académie des sciences(1864);Histoire de l’Académie des Inscriptions et Belles Lettres(1865); a learned paper on the reports of French archaeology, written on the occasion of the universal exhibition (1867); a number of articles in theEncyclopédie moderne(1846-1851), in Michaud’sBiographie universelle(1858 and seq.), in theJournal des savantsin theRevue des deux mondes(1873, 1877, 1879-1880, &c.). A detailed bibliography of his works has been placed by Auguste Longnon at the beginning of the volumeLes Croyances et légendes du moyen âge.

MAURY, MATTHEW FONTAINE(1806-1873), American naval officer and hydrographer, was born near Fredericksburg in Spottsylvania county, Virginia, on the 24th of January 1806. He was educated at Harpeth academy, and in 1825 entered the navy as midshipman, circumnavigating the globe in the “Vincennes,” during a cruise of four years (1826-1830). In 1831 he was appointed master of the sloop “Falmouth” on the Pacific station, and subsequently served in other vessels before returning home in 1834, when he married his cousin, Ann Herndon. In 1835-1836 he was actively engaged in producing for publication a treatise on navigation, a remarkable achievement at so early a stage in his career; he was at this time made lieutenant, and gazetted astronomer to a South Sea exploring expedition, but resigned this position and was appointed to the survey of southern harbours. In 1839 he met with an accident which resulted in permanent lameness, and unfitted him for active service. In the same year, however, he began to write a series of articles on naval reform and other subjects, under the title ofScraps from the Lucky-Bag, which attracted much attention; and in 1841 he was placed in charge of the Dépôt of Charts and Instruments, out of which grew the United States Naval Observatory and the Hydrographie Office. He laboured assiduously to obtain observations as to the winds and currents by distributing to captains of vessels specially prepared log-books; and in the course of nine years he had collected a sufficient number of logs to make two hundred manuscript volumes, each with about two thousand five hundred days’ observations. One result was to show the necessity for combined action on the part of maritime nations in regard to ocean meteorology. This led to an international conference at Brussels in 1853, which produced the greatest benefit to navigation as well as indirectly to meteorology. Maury attempted to organize co-operative meteorological work on land, but the government did not at this time take any steps in this direction. His oceanographical work, however, received recognition in all parts of the civilized world, and in 1855 it was proposed in the senate to remunerate him, but in the same year the Naval Retiring Board, erected under an act to promote the efficiency of the navy, placed him on the retired list. This action aroused wide opposition, and in 1858 he was reinstated with the rank of commander as from 1855. In 1853 Maury had published hisLetters on the Amazon and Atlantic Slopes of South America, and the most widely popular of his works, thePhysical Geography of the Sea, was published in London in 1855, and in New York in 1856; it was translated into several European languages. On the outbreak of the American Civil War in 1861, Maury threw in his lot with the South, and became head of coast, harbour and river defences. He invented an electric torpedo for harbour defence, and in 1862 was ordered to England to purchase torpedo material, &c. Here he took active part in organizing a petition for peace to the American people, which was unsuccessful. Afterwards he became imperial commissioner of emigration to the emperor Maximilian of Mexico, and attempted to form a Virginian colony in that country. Incidentally he introduced there the cultivation of cinchona. The scheme of colonization was abandoned by the emperor (1866), and Maury, who had lost nearly his all during the war, settled for a while in England, where he was presented with a testimonial raised by public subscription, and among other honours received the degree of LL.D. of Cambridge University (1868). In the same year, a general amnesty admitting of his return to America, he accepted the professorship of meteorology in the Virginia Military Institute, and settled at Lexington, Virginia, where he died on the 1st of February 1873.

Among works published by Maury, in addition to those mentioned, are the papers contributed by him to theAstronomical Observationsof the United States Observatory,Letter concerning Lanes for Steamers crossing the Atlantic(1855);Physical Geography(1864) andManual of Geography(1871). In 1859 he began the publication of a series ofNautical Monographs.See Diana Fontaine Maury Corbin (his daughter),Life of Matthew Fontaine Maury(London, 1888).

See Diana Fontaine Maury Corbin (his daughter),Life of Matthew Fontaine Maury(London, 1888).

MAUSOLEUM,the term given to a monument erected to receive the remains of a deceased person, which may sometimes take the form of a sepulchral chapel. The termcenotaph(κενός, empty,τάφος, tomb) is employed for a similar monument where the body is not buried in the structure. The term “mausoleum” originated with the magnificent monument erected by Queen Artemisia in 353B.C.in memory of her husband King Mausolus, of which the remains were brought to England in 1859 by Sir Charles Newton and placed in the British Museum. The tombs of Augustus and of Hadrian in Rome are perhaps the largest monuments of the kind ever erected.

MAUSOLUS(more correctlyMaussollus), satrap and practically ruler of Caria (377-353B.C.). The part he took in the revolt against Artaxerxes Mnemon, his conquest of a great part of Lycia, Ionia and of several of the Greek islands, his co-operation with the Rhodians and their allies in the war against Athens, and the removal of his capital from Mylasa, the ancient seat of the Carian kings, to Halicarnassus are the leading facts of his history. He is best known from the tomb erected for him by his widow Artemisia. The architects Satyrus and Pythis, and the sculptors Scopas, Leochares, Bryaxis and Timotheus, finished the work after her death. (SeeHalicarnassus.) An inscription discovered at Mylasa (Böckh,Inscr. gr.ii. 2691c.) details the punishment of certain conspirators who had made an attempt upon his life at a festival in a temple at Labranda in 353.

See Diod. Sic. xv. 90, 3, xvi. 7, 4, 36, 2; Demosthenes,De Rhodiorum libertate; J. B. Bury,Hist. of Greece(1902), ii. 271; W. Judeich,Kleinasiatische Studien(Marburg, 1892), pp. 226-256, and authorities underHalicarnassus.

MAUVE, ANTON(1838-1888), Dutch landscape painter, was born at Zaandam, the son of a Baptist minister. Much against the wish of his parents he took up the study of art and entered the studio of Van Os, whose dry academic manner had, however, but little attraction for him. He benefited far more by his intimacy with his friends Jozef Israels and W. Maris. Encouraged by their example he abandoned his early tight and highly finished manner for a freer, looser method of painting, and the brilliant palette of his youthful work for a tender lyric harmony which is generally restricted to delicate greys, greens, and light blue. He excelled in rendering the soft hazy atmosphere that lingers over the green meadows of Holland, and devoted himself almost exclusively to depicting the peaceful rural life of the fields and country lanes of Holland—especially of the districts near Oosterbeck and Wolfhezen, the sand dunes of the coast at Scheveningen, and the country near Laren, where he spent the last years of his life. A little sad and melancholy, his pastoral scenes are nevertheless conceived in a peaceful soothing lyrical mood, which is in marked contrast to the epic power and almost tragic intensity of J. F. Millet. There are fourteen of Mauve’s pictures at the Mesdag Museum at the Hague, and two (“Milking Time” and “A Fishing Boat putting to Sea”) at the Ryks Museum in Amsterdam. The Glasgow Corporation Gallery owns his painting of “A Flock of Sheep.” The finest and most representative private collection of pictures by Mauve was made by Mr J. C. J. Drucker, London.

MAVROCORDATO,MavrocordatorMavrogordato, the name of a family of Phanariot Greeks, distinguished in the history of Turkey, Rumania and modern Greece. The family was founded by a merchant of Chios, whose son Alexander Mavrocordato (c.1636-1709), a doctor of philosophy and medicine of Bologna, became dragoman to the sultan in 1673, and was much employed in negotiations with Austria. It was he who drew up the treaty of Karlowitz (1699). He became a secretary of state, and was created a count of the Holy Roman Empire. His authority, with that of Hussein Kupruli and Rami Pasha, was supreme at the court of Mustapha II., and he did much to ameliorate the condition of the Christians in Turkey. He was disgraced in 1703, but was recalled to court by Sultan Ahmed III. He left some historical, grammatical, &c. treatises of little value.

His sonNicholas Mavrocordato(1670-1730) was grand dragoman to the Divan (1697), and in 1708 was appointed hospodar (prince) of Moldavia. Deposed, owing to the sultan’s suspicions, in favour of Demetrius Cantacuzene, he was restored in 1711, and soon afterwards became hospodar of Walachia. In 1716 he was deposed by the Austrians, but was restored after the peace of Passarowitz. He was the first Greek set to rule the Danubian principalities, and was responsible for establishing the system which for a hundred years was to make the name of Greek hateful to the Rumanians. He introduced Greek manners, the Greek language and Greek costume, and set up a splendid court on the Byzantine model. For the rest he was a man of enlightenment, founded libraries and was himself the author of a curious work entitledΠερὶ καθήκοντων(Bucharest, 1719). He was succeeded as grand dragoman (1709) by his son John (Ioannes), who was for a short while hospodar of Moldavia, and died in 1720.

Nicholas Mavrocordato was succeeded as prince of Walachia in 1730 by his son Constantine. He was deprived in the same year, but again ruled the principality from 1735 to 1741 and from 1744 to 1748; he was prince of Moldavia from 1741 to 1744 and from 1748 to 1749. His rule was distinguished by numerous tentative reforms in the fiscal and administrative systems. He was wounded and taken prisoner in the affair of Galati during the Russo-Turkish War, on the 5th of November 1769, and died in captivity.

Prince Alexander Mavrocordato(1791-1865), Greek statesman, a descendant of the hospodars, was born at Constantinople on the 11th of February 1791. In 1812 he went to the court of his uncle Ioannes Caradja, hospodar of Walachia, with whom he passed into exile in Russia and Italy (1817). He was a member of the Hetairia Philike and was among the Phanariot Greeks who hastened to the Morea on the outbreak of the War of Independence in 1821. He was active in endeavouring to establish a regular government, and in January 1822 presided over the first Greek national assembly at Epidaurus. He commanded the advance of the Greeks into western Hellas the same year, and suffered a defeat at Peta on the 16th of July, but retrieved this disaster somewhat by his successful resistance to the first siege of Missolonghi (Nov. 1822 to Jan. 1823). His English sympathies brought him, in the subsequent strife of factions, into opposition to the “Russian” party headed by Demetrius Ypsilanti and Kolokotrones; and though he held the portfolio of foreign affairs for a short while under the presidency of Petrobey (Petros Mavromichales), he was compelled to withdraw from affairs until February 1825, when he again became a secretary of state. The landing of Ibrahim Pasha followed, and Mavrocordato again joined the army, only escaping capture in the disaster at Sphagia (Spakteria), on the 9th of May 1815, by swimming to Navarino. After the fall of Missolonghi (April 22, 1826) he went into retirement, until President Capo d’Istria made him a member of the committee for the administration of war material, a position he resigned in 1828. After Capo d’Istria’s murder (Oct. 9, 1831) and the resignation of his brother and successor, Agostino Capo d’Istria (April 13, 1832), Mavrocordato became minister of finance. He was vice-president of the National Assembly at Argos (July, 1832), and was appointed by King Otto minister of finance, and in 1833 premier. From 1834 onwards he was Greek envoy at Munich, Berlin, London and—after a short interlude as premier in Greece in 1841—Constantinople. In 1843, after the revolution of September, he returned to Athens as minister without portfolio in the Metaxas cabinet, and from April to August 1844 was head of the government formed after the fall of the “Russian” party. Going into opposition, he distinguished himself by his violent attacks on the Kolettis government. In 1854-1855 he was again head of the government for a few months. He died in Aegina on the 18th of August 1865.

See E. Legrand,Genealogie des Mavrocordato(Paris, 1886).

MAWKMAI(BurmeseMaukmè), one of the largest states in the eastern division of the southern Shan States of Burma. It lies approximately between 19° 30′ and 20° 30′ N. and 97° 30′ and 98° 15′ E., and has an area of 2,787 sq. m. The central portion of the state consists of a wide plain well watered and under rice cultivation. The rest is chiefly hills in ranges running north and south. There is a good deal of teak in the state, but it has been ruinously worked. The sawbwa now works as contractor for government, which takes one-third of the net profits. Rice is the chief crop, but much tobacco of good quality is grown in the Langkö district on the Têng river. There is also a great deal of cattle-breeding. The population in 1901 was 29,454, over two-thirds of whom were Shans and the remainder Taungthu, Burmese, Yangsek and Red Karens. The capital,Mawkmai, stands in a fine rice plain in 20° 9′ N. and 97° 25′ E. It had about 150 houses when it first submitted in 1887, but was burnt out by the Red Karens in the following year. It has since recovered. There are very fine orange groves a few miles south of the town at Kantu-awn, called Kadugate by the Burmese.

MAXENTIUS, MARCUS AURELIUS VALERIUS,Roman emperor fromA.D.306 to 312, was the son of Maximianus Herculius, and the son-in-law of Galerius. Owing to his vices and incapacity he was left out of account in the division of the empire which took place in 305. A variety of causes, however, had produced strong dissatisfaction at Rome with many of the arrangements established by Diocletian, and on the 28th of October 306, the public discontent found expression in the massacre of those magistrates who remained loyal to Flavius Valerius Severus and in the election of Maxentius to the imperial dignity. With the help of his father, Maxentius was enabled to put Severus to death and to repel the invasion of Galerius; his next steps were first to banish Maximianus, and then, after achieving a military success in Africa against the rebellious governor, L. Domitius Alexander, to declare war against Constantine as having brought about the death of his father Maximianus. His intention of carrying the war into Gaul was anticipated by Constantine, who marched into Italy. Maxentius was defeated at Saxa Rubra near Rome and drowned in the Tiber while attempting to make his way across the Milvian bridge into Rome. He was a man of brutal and worthless character; but although Gibbon’s statement that he was “just, humane and even partial towards the afflicted Christians” may be exaggerated, it is probable that he never exhibited any special hostility towards them.

See De Broglie,L’Église et l’empire Romain au quatrième siècle(1856-1866), and on the attitude of the Romans towards Christianity generally, app. 8 in vol. ii. of J. B. Bury’s edition of Gibbon (Zosimus ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor,Epit.40; Eutropius, x. 2).

MAXIM, SIR HIRAM STEVENS(1840- ), Anglo-American engineer and inventor, was born at Sangerville, Maine, U.S.A., on the 5th of February 1840. After serving an apprenticeship with a coachbuilder, he entered the machine works of his uncle, Levi Stevens, at Fitchburg, Massachusetts, in 1864, and four years later he became a draughtsman in the Novelty Iron Works and Shipbuilding Company in New York City. About this period he produced several inventions connected with illumination by gas; and from 1877 he was one of the numerous inventors who were trying to solve the problem of making an efficient and durable incandescent electric lamp, in this connexion introducing the widely-used process of treating the carbon filaments by heating them in an atmosphere of hydrocarbon vapour. In 1880 he came to Europe, and soon began to devote himself to the construction of a machine-gun which should be automatically loaded and fired by the energy of the recoil (seeMachine-Gun). In order to realize the full usefulness of the weapon, which was first exhibited in an underground range at Hatton Garden, London, in 1884, he felt the necessity of employing a smokeless powder, and accordingly he devised maximite, a mixture of trinitrocellulose, nitroglycerine and castor oil, which was patented in 1889. He also undertook to make a flying machine, and after numerous preliminary experiments constructed an apparatus which was tried at Bexley Heath, Kent, in 1894. (SeeFlight.) Having been naturalized as a British subject, he was knighted in 1901. His younger brother, Hudson Maxim (b. 1853), took out numerous patents in connexion with explosives.

MAXIMA AND MINIMA,in mathematics. By themaximumorminimumvalue of an expression or quantity is meant primarily the “greatest” or “least” value that it can receive. In general, however, there are points at which its value ceases to increase and begins to decrease; its value at such a point is called a maximum. So there are points at which its value ceases to decrease and begins to increase; such a value is called a minimum. There may be several maxima or minima, and a minimum is not necessarily less than a maximum. For instance, the expression (x2+ x + 2)/(x − 1) can take all values from −∞ to −1 and from +7 to +∞, but has, so long as x is real, no value between -1 and +7. Here −1 is a maximum value, and +7 is a minimum value of the expression, though it can be made greater or less than any assignable quantity.

The first general method of investigating maxima and minima seems to have been published inA.D.1629 by Pierre Fermat. Particular cases had been discussed. Thus Euclid in book III. of theElementsfinds the greatest and least straight lines that can be drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from editions of his works) finds the parallelogram of greatest area with a given perimeter. Apollonius investigated the greatest and least distances of a point from the perimeter of a conic section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular hyperbola. Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria. The most noteworthy of them are the following:—

1. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.3. The circle encloses a greater area than any polygon of the same perimeter.4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.6. The sphere is the surface of given area which encloses the greatest volume.

1. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.

2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.

3. The circle encloses a greater area than any polygon of the same perimeter.

4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.

5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.

6. The sphere is the surface of given area which encloses the greatest volume.

Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the base and curved surface of a right circular cone of a plane drawn through its vertex.

The next problem on maxima and minima of which there appears to be any record occurs in a letter from Regiomontanus to Roder (July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight line at which two given points subtend a maximum angle. N. Tartaglia in hisGeneral trattato de numeri et mesuri(c.1556) gives, without proof, a rule for dividing a number into two parts such that the continued product of the numbers and their difference is a maximum.

Fermat investigated maxima and minima by means of the principle that in the neighbourhood of a maximum or minimum the differences of the values of a function are insensible, a method virtually the same as that of the differential calculus, and of great use in dealing with geometrical maxima and minima. His method was developed by Huygens, Leibnitz, Newton and others, and in particular by John Hudde, who investigated maxima and minima of functions of more than one independent variable, and made some attempt to discriminate between maxima and minima, a question first definitely settled, so far as one variable is concerned, by Colin Maclaurin in hisTreatise on Fluxions(1742). The method of the differential calculus was perfected by Euler and Lagrange.

John Bernoulli’s famous problem of the “brachistochrone,” or curve of quickest descent from one point to another underthe action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve. From these problems arose the “Calculus of Variations.” (SeeVariations, Calculus of.)

The only general methods of attacking problems on maxima and minima are those of the differential calculus or, in geometrical problems, what is practically Fermat’s method. Some problems may be solved by algebra; thus if y = ƒ(x) ÷ φ(x), where ƒ(x) and φ(x) are polynomials in x, the limits to the values of yφ may be found from the consideration that the equation yφ(x) − ƒ(x) = 0 must have real roots. This is a useful method in the case in which φ(x) and ƒ(x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n positive quantities whose sum is given may also be found, algebraically, thus. If a and b are any two real unequal quantities whatever {1⁄2(a + b)}2> ab, so that we can increase the product leaving the sum unaltered by replacing any two terms by half their sum, and so long as any two of the quantities are unequal we can increase the product. Now, the quantities being all positive, the product cannot be increased without limit and must somewhere attain a maximum, and no other form of the product than that in which they are all equal can be the maximum, so that the product is a maximum when they are all equal. Its minimum value is obviously zero. If the restriction that all the quantities shall be positive is removed, the product can be made equal to any quantity, positive or negative. So other theorems of algebra, which are stated as theorems on inequalities, may be regarded as algebraic solutions of problems on maxima and minima.

For purely geometrical questions the only general method available is practically that employed by Fermat. If a quantity depends on the position of some point P on a curve, and if its value is equal at two neighbouring points P and P′, then at some position between P and P′ it attains a maximum or minimum, and this position may be found by making P and P′ approach each other indefinitely. Take for instance the problem of Regiomontanus “to find a point on a given straight line which subtends a maximum angle at two given points A and B.” Let P and P′ be two near points on the given straight line such that the angles APB and AP′B are equal. Then ABPP′ lie on a circle. By making P and P′ approach each other we see that for a maximum or minimum value of the angle APB, P is a point in which a circle drawn through AB touches the given straight line. There are two such points, and unless the given straight line is at right angles to AB the two angles obtained are not the same. It is easily seen that both angles are maxima, one for points on the given straight line on one side of its intersection with AB, the other for points on the other side. For further examples of this method together with most other geometrical problems on maxima and minima of any interest or importance the reader may consult such a book as J. W. Russell’sA Sequel lo Elementary Geometry(Oxford, 1907).

The method of the differential calculus is theoretically very simple. Let u be a function of several variables x1, x2, x3... xn, supposed for the present independent; if u is a maximum or minimum for the set of values x1, x2, x3, ... xn, and u becomes u + δu, when x1, x2, x3... xnreceive small increments δx1, δx2, ... δxn; then δu must have the same sign for all possible values of δx1, δ2... δxn.Nowδu = Σδuδx1+1⁄2{Σδ2uδx12+ 2Σδ2uδx1δx2...}+ ....δx1δx12δx1δx2The sign of this expression in general is that of Σ(δu/δx1)δx1, which cannot be one-signed when x1, x2, ... xncan take all possible values, for a set of increments δx1, δx2... δxn, will give an opposite sign to the set −δx1, −δx2, ... −δxn. Hence Σ(δu/δx1)δx1must vanish for all sets of increments δx1, ... δxn, and since these are independent, we must have δu/δx1= 0, δu/δx2= 0, ... δu/δxn= 0. A value of u given by a set of solutions of these equations is called a “critical value” of u. The value of δu now becomes1⁄2{Σδ2uδx12+ 2 Σδ2uδx1δx2+ ...};δx12δx1δx2for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d2u/dx2is negative or positive. If d2u/dx2vanishes, then there is no maximum orminimumunless d2u/dx2vanishes, and there is a maximum or minimum according as d4u/dx4is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.In the case of several variables, the quadraticΣδ2uδx12+ 2 Σδ2uδx1δx2+ ...δx12δx1δx2must be one-signed. The condition for this is that the series of discriminantsa11,a11a12,a11a12a13, ...a21a22a21a22a23a31a32a33where apqdenotes δ2u/δapδaqshould be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions areδ2u·δ2u>(δ2u)2δx12δx22δx1δx2for a maximum or minimum at all and δ2u/δx12and δ2u/δx22both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when δx1, δx2, ... δxnall vanish. If, in the case of two variables,δ2u·δ2u=(δ2u)2δx12δx22δx1δx2then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.Take for instance the function u = x2− xy2+ y2. Here the values x = 0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are (δx)2, and are never negative. Here δu = δx2− δxδy2+ δy2, and by putting δx = 0 or an infinitesimal of the same order as δy2, we can make the sign of δu depend on that of δy2, and so be positive or negative as we please. On the other hand, if we take the function u = x2− xy2+ y4, x = 0, y = 0 make zero a critical value of u, and here δu = δx2− δxδy2+ δy4, which is always positive, because we can write it as the sum of two squares, viz. (δx −1⁄2δy2)2+3⁄4δy4; so that in this case zero is a minimum value of u.A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and φ′(x) = 0 has no roots in this interval, it follows that φ′(x) is one-signed as x increases from a to b, so that φ(x) is increasing or diminishing all the time, and the greatest and least values of φ(x) are φ(a) and φ(b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec α. Where should he land?Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ, so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α, and the same signs from θ = α to θ = AOB. So that when AOB is > α, T decreases from θ = 0 to θ = α, and then increases, so that he should land at a point distant a tan α from A, unless a tan α > b. When this is the case, δT and δθ have opposite signs throughout the whole range of θ, so that T decreases as θ increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not.The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.In the case of a function of several variables, X1, x2, ... xn, not independent but connected by m functional relations u1= 0, u2= 0, ..., um= 0, we might proceed to eliminate m of the variables; but Lagrange’s “Method of undetermined Multipliers” is more elegant and generally more useful.We have δu1= 0, δu2= 0, ..., δum= 0. Consider instead of δu, what is the same thing, viz., δu + λ1δu1+ λ2δu2+ ... + λmδum, where λ1, λ2, ... λm, are arbitrary multipliers. The terms of the first order in this expression areΣδuδx1+ λ1Σδu1δx1+ ... + λmΣδumδx1.δx1δx1δx1We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm, vanish, and the remaining δxm+1to δxnmay be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we putδu+δu1+ ... + λmδum= 0δxrδxrδxrfor all values of r. These equations with the equations u1= 0, ..., um= 0 are exactly enough to determine λ1, ..., λm, x1x2, ..., xn, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum.To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x2/a2+ y2/b2+ z2/c2= 1, where a2> b2> c2. Here we require the maximum and minimum values of x2+ y2+ z2where x2/a2+ y2/b2+ z2/c2= 1.We haveδu = 2xδx(1 +λ)+ 2yδy(λ)+ 2zδz(λ)a2b2c2+ δx2(1 +λ)+ δy2(λ)+ δz2(λ).a2b2c2To make the terms of the first order disappear, we have the three equations:—x (1 + λ/a2) = 0, y (1 + λ/b2) = 0, z (1 + λ/c2) = 0.These have three sets of solutions consistent with the conditions x2/a2+ y2/b2+ z2/c2= 1, a2> b2> c2, viz.:—(1) y = 0, z = 0, λ = −a2; (2) z = 0, x = 0, λ = −b2;(3) x = 0, y = 0, λ = −c2.In the case of (1) δu = δy2(1 − a2/b2) + δz2(1 − a2/c2), which is always negative, so that u = a2gives a maximum.In the case of (3) δu = δx2(1 − c2/a2) + δy2(1 − c2/b2), which is always positive, so that u = c2gives a minimum.In the case of (2) δu = δx2(1 − b2/a2) − δz2(b2/c2− 1), which can be made either positive or negative, or even zero if we move in the planes x2(1 − b2/a2) = z2(b2/c2− 1), which are well known to be the central planes of circular section. So that u = b2, though a critical value, is neither a maximum nor minimum, and the central planes of circular section divide the ellipsoid into four portions in two of which a2> r2> b2, and in the other two b2> r2> c2.

Now

The sign of this expression in general is that of Σ(δu/δx1)δx1, which cannot be one-signed when x1, x2, ... xncan take all possible values, for a set of increments δx1, δx2... δxn, will give an opposite sign to the set −δx1, −δx2, ... −δxn. Hence Σ(δu/δx1)δx1must vanish for all sets of increments δx1, ... δxn, and since these are independent, we must have δu/δx1= 0, δu/δx2= 0, ... δu/δxn= 0. A value of u given by a set of solutions of these equations is called a “critical value” of u. The value of δu now becomes

for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d2u/dx2is negative or positive. If d2u/dx2vanishes, then there is no maximum orminimumunless d2u/dx2vanishes, and there is a maximum or minimum according as d4u/dx4is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.

In the case of several variables, the quadratic

must be one-signed. The condition for this is that the series of discriminants

where apqdenotes δ2u/δapδaqshould be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions are

for a maximum or minimum at all and δ2u/δx12and δ2u/δx22both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when δx1, δx2, ... δxnall vanish. If, in the case of two variables,

then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.

Take for instance the function u = x2− xy2+ y2. Here the values x = 0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are (δx)2, and are never negative. Here δu = δx2− δxδy2+ δy2, and by putting δx = 0 or an infinitesimal of the same order as δy2, we can make the sign of δu depend on that of δy2, and so be positive or negative as we please. On the other hand, if we take the function u = x2− xy2+ y4, x = 0, y = 0 make zero a critical value of u, and here δu = δx2− δxδy2+ δy4, which is always positive, because we can write it as the sum of two squares, viz. (δx −1⁄2δy2)2+3⁄4δy4; so that in this case zero is a minimum value of u.

A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and φ′(x) = 0 has no roots in this interval, it follows that φ′(x) is one-signed as x increases from a to b, so that φ(x) is increasing or diminishing all the time, and the greatest and least values of φ(x) are φ(a) and φ(b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec α. Where should he land?

Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ, so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α, and the same signs from θ = α to θ = AOB. So that when AOB is > α, T decreases from θ = 0 to θ = α, and then increases, so that he should land at a point distant a tan α from A, unless a tan α > b. When this is the case, δT and δθ have opposite signs throughout the whole range of θ, so that T decreases as θ increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not.

The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.

In the case of a function of several variables, X1, x2, ... xn, not independent but connected by m functional relations u1= 0, u2= 0, ..., um= 0, we might proceed to eliminate m of the variables; but Lagrange’s “Method of undetermined Multipliers” is more elegant and generally more useful.

We have δu1= 0, δu2= 0, ..., δum= 0. Consider instead of δu, what is the same thing, viz., δu + λ1δu1+ λ2δu2+ ... + λmδum, where λ1, λ2, ... λm, are arbitrary multipliers. The terms of the first order in this expression are

We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm, vanish, and the remaining δxm+1to δxnmay be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put

for all values of r. These equations with the equations u1= 0, ..., um= 0 are exactly enough to determine λ1, ..., λm, x1x2, ..., xn, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum.

To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x2/a2+ y2/b2+ z2/c2= 1, where a2> b2> c2. Here we require the maximum and minimum values of x2+ y2+ z2where x2/a2+ y2/b2+ z2/c2= 1.

We have

To make the terms of the first order disappear, we have the three equations:—

x (1 + λ/a2) = 0, y (1 + λ/b2) = 0, z (1 + λ/c2) = 0.

These have three sets of solutions consistent with the conditions x2/a2+ y2/b2+ z2/c2= 1, a2> b2> c2, viz.:—

In the case of (1) δu = δy2(1 − a2/b2) + δz2(1 − a2/c2), which is always negative, so that u = a2gives a maximum.

In the case of (3) δu = δx2(1 − c2/a2) + δy2(1 − c2/b2), which is always positive, so that u = c2gives a minimum.

In the case of (2) δu = δx2(1 − b2/a2) − δz2(b2/c2− 1), which can be made either positive or negative, or even zero if we move in the planes x2(1 − b2/a2) = z2(b2/c2− 1), which are well known to be the central planes of circular section. So that u = b2, though a critical value, is neither a maximum nor minimum, and the central planes of circular section divide the ellipsoid into four portions in two of which a2> r2> b2, and in the other two b2> r2> c2.

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