Chapter 9 of 9 · 24397 words · ~122 min read

part ii

. (Stuttgart, 1893); _Undine_, _Sintram_, &c., in innumerable reprints. Bibliography in Goedeke's _Grundriss zur Geschichte der deutschen Dichtung_ (2nd ed., vi. pp. 115 ff., Dresden, 1898). Most of Fouque's works have been translated, and the English versions of _Aslauga's Knight_ (by Carlyle), _Sintram and his Companions_ and _Undine_, have been frequently republished. For Fouque's life cp. _Lebensgeschichte des Baron Friedrich de la Motte Fouque. Aufgezeichnet durch ihn selbst_ (Halle, 1840), (only to the year 1813), and also the introduction to Koch's selections in the _Deutsche Nationalliteratur_. (J. G. R.)

FOUQUET (or FOUCQUET), NICOLAS (1615-1680), viscount of Melun and of Vaux, marquis of Belle-Isle, superintendent of finance in France under Louis XIV., was born at Paris in 1615. He belonged to an influential family of the _noblesse de la robe_, and after some preliminary schooling with the Jesuits, at the age of thirteen was admitted as _avocat_ at the parlement of Paris. While still in his teens he held several responsible posts, and in 1636, when just twenty, he was able to buy the post of _maitre des requetes_. From 1642 to 1650 he held various intendancies at first in the provinces and then with the army of Mazarin, and, coming thus in touch with the court, was permitted in 1650 to buy the important position of _procureur general_ to the parlement of Paris. During Mazarin's exile Fouquet shrewdly remained loyal to him, protecting his property and keeping him informed of the situation at court.

Upon the cardinal's return, Fouquet demanded and received as reward the office of superintendent of the finances (1653), a position which, in the unsettled condition of the government, threw into his hands not merely the decision as to which funds should be applied to meet the demands of the state's creditors, but also the negotiations with the great financiers who lent money to the king. The appointment was a popular one with the moneyed class, for Fouquet's great wealth had been largely augmented by his marriage in 1651 with Marie de Castille, who also belonged to a wealthy family of the legal nobility. His own credit, and above all his unfailing confidence in himself, strengthened the credit of the government, while his high position at the parlement (he still remained _procureur general_) secured financial transactions from investigation. As minister of finance, he soon had Mazarin almost in the position of a suppliant. The long wars, and the greed of the courtiers, who followed the example of Mazarin, made it necessary at times for Fouquet to meet the demands upon him by borrowing upon his own credit, but he soon turned this confusion of the public purse with his own to good account. The disorder in the accounts became hopeless; fraudulent operations were entered into with impunity, and the financiers were kept in the position of clients by official favours and by generous aid whenever they needed it. Fouquet's fortune now surpassed even Mazarin's, but the latter was too deeply implicated in similar operations to interfere, and was obliged to leave the day of reckoning to his agent and successor Colbert. Upon Mazarin's death Fouquet expected to be made head of the government; but Louis XIV. was suspicious of his poorly dissembled ambition, and it was with Fouquet in mind that he made the well-known statement, upon assuming the government, that he would be his own chief minister. Colbert fed the king's displeasure with adverse reports upon the deficit, and made the worst of the case against Fouquet. The extravagant expenditure and personal display of the superintendent served to intensify the ill-will of the king. Fouquet had bought the port of Belle Isle and strengthened the fortifications, with a view to taking refuge there in case of disgrace. He had spent enormous sums in building a palace on his estate of Vaux, which in extent, magnificence, and splendour of decoration was a forecast of Versailles. Here he gathered the rarest manuscripts, the finest paintings, jewels and antiques in profusion, and above all surrounded himself with artists and authors. The table was open to all people of quality, and the kitchen was presided over by Vatel. Lafontaine, Corneille, Scarron, were among the multitude of his clients. In August 1661 Louis XIV., already set upon his destruction, was entertained at Vaux with a _fete_ rivalled in magnificence by only one or two in French history, at which Moliere's _Les Facheux_ was produced for the first time. The splendour of the entertainment sealed Fouquet's fate. The king, however, was afraid to act openly against so powerful a minister. By crafty devices Fouquet was induced to sell his office of _procureur general_, thus losing the protection of its privileges, and he paid the price of it into the treasury.

Three weeks after his visit to Vaux the king withdrew to Nantes, taking Fouquet with him, and had him arrested when he was leaving the presence chamber, flattered with the assurance of his esteem. The trial lasted almost three years, and its violation of the forms of justice is still the subject of frequent monographs by members of the French bar. Public sympathy was strongly with Fouquet, and Lafontaine, Madame de Sevigne and many others wrote on his behalf; but when Fouquet was sentenced to banishment, the king, disappointed, "commuted" the sentence to imprisonment for life. He was sent at the beginning of 1665 to the fortress of Pignerol, where he undoubtedly died on the 23rd of March 1680.[1] Louis acted throughout "as though he were conducting a campaign," evidently fearing that Fouquet would play the part of a Richelieu. Fouquet bore himself with manly fortitude, and composed several mediocre translations in prison. The devotional works bearing his name are apocryphal. A report of his trial was published in Holland, in 15 volumes, in 1665-1667, in spite of the remonstrances which Colbert addressed to the States-General. A second edition under the title of _Oeuvres de M. Fouquet_ appeared in 1696.

See Cheruel, _Memoires sur la vie publique et privee de Fouquet... d'apres ses lettres et des pieces inedites_ (2 vols., Paris, 1864); J. Lair, _Nicolas Foucquet, procureur general, surintendant des finances, ministre d'Etat de Louis XIV_ (2 vols., Paris, 1890); U.V. Chatelain, _Le Surintendant Nicolas Fouquet, protecteur des lettres, des arts et des sciences_ (Paris, 1905); R. Pfnor et A. France, _Le Chateau de Vaux-le-Vicomte dessine et grave_ (Paris, 1888).

FOOTNOTE:

[1] Fouquet has been identified with the "Man with the Iron Mask" (see IRON MASK), but this theory is quite impossible.

FOUQUIER-TINVILLE, ANTOINE QUENTIN (1746-1795), French revolutionist, was born at Herouel, a village in the department of the Aisne. Originally a _procureur_ attached to the Chatelet at Paris, he sold his office in 1783, and became a clerk under the lieutenant-general of police. He seems to have early adopted revolutionary ideas, but little is known of the part he played at the outbreak of the Revolution. When the Revolutionary Tribunal of Paris was established on the 10th of March 1793, he was appointed public prosecutor to it, an office which he filled until the 28th of July 1794. His activity during this time earned him the reputation of one of the most terrible and sinister figures of the Revolution. His function as public prosecutor was not so much to convict the guilty as to see that the proscriptions ordered by the faction for the time being in power were carried out with a due regard to a show of legality. He was as ruthless and as incorrupt as Robespierre himself; he could be moved from his purpose neither by pity nor by bribes; nor was there in his cruelty any of that quality which made the ordinary Jacobin _enrage_ by turns ferocious and sentimental. It was this very quality of passionless detachment that made him so effective an instrument of the Terror. He had no forensic eloquence; but the cold obstinacy with which he pressed his charges was more convincing than any rhetoric, and he seldom failed to secure a conviction.

His horrible career ended with the fall of Robespierre and the terrorists on the 9th Thermidor. On the 1st of August 1794 he was imprisoned by order of the Convention and brought to trial. His defence was that he had only obeyed the orders of the Committee of Public Safety; but, after a trial which lasted forty-one days, he was condemned to death, and guillotined on the 7th of May 1795.

See _Memoire pour A.Q. Fouquier ex-accusateur public pres le tribunal revolutionnaire_, &c. (Paris, 1794); Domenget, _Fouquier-Tinville et le tribunal revolutionnaire_ (Paris, 1878); H. Wallon, _Histoire du tribunal revolutionnaire de Paris_ (1880-1882) (a work of general interest, but not always exact); George Lecocq, _Notes et documents sur Fouquier-Tinville_ (Paris, 1885). See also the documents relating to his trial enumerated by M. Tourneux in _Bibliographie de l'histoire de Paris pendant la Revolution Francaise_, vol. i. Nos. 4445-4454 (1890).

FOURCHAMBAULT, a town of central France in the department of Nievre, on the right bank of the Loire, 4-1/2 m. N.W. of Nevers, on the Paris-Lyon railway. Pop. (1906) 4591. It owes its importance to its extensive iron-works, established in 1821, which give employment to 2000 workmen and produce engineering material for railway, military and other purposes. Among the more remarkable _chefs-d'oeuvre_ which have been produced at Fourchambault are the metal portions of the Pont du Carrousel, the iron beams of the roof of the cathedral at Chartres, and the vast spans of the bridge over the Dordogne at Cubzac. A small canal unites the works to the Lateral canal of the Loire.

FOURCROY, ANTOINE FRANCOIS, COMTE DE (1755-1809), French chemist, the son of an apothecary in the household of the duke of Orleans, was born at Paris on the 15th of June 1755. He took up medical studies by the advice of the anatomist Felix Vicq d'Azyr (1748-1794), and after many difficulties caused by lack of means finally in 1780 obtained his doctor's diploma. His attention was specially turned to chemistry by J.B.M. Bucquet (1746-1780), the professor of chemistry at the Medical School of Paris, and in 1784 he was chosen to succeed P.J. Macquer (1718-1784) as lecturer in chemistry at the college of the Jardin du Roi, where his lectures attained great popularity. He was one of the earliest converts to the views of Lavoisier, which he helped to promulgate by his voluminous writings, but though his name appears on a large number of chemical and also physiological and pathological memoirs, either alone or with others, he was rather a teacher and an organizer than an original investigator. A member of the committees for public instruction and public safety, and later, under Napoleon, director general of instruction, he took a leading part in the establishment of schools for both primary and secondary education, scientific studies being especially provided for. Fourcroy died at Paris on the 16th of December 1809, the very day on which he had been created a count of the French empire. By his conduct as a member of the Convention he has been accused of contributing to the death of Lavoisier. Baron Cuvier in his _Eloge historique_ of Fourcroy repels the charge, but he can scarcely be acquitted of time-serving indifference, if indeed active, though secret, participation be not proved against him.

The Royal Society's _Catalogue of Scientific Papers_ enumerates 59 memoirs by Fourcroy himself, and 58 written jointly by him and others, mostly L.N. Vauquelin.

FOURIER, FRANCOIS CHARLES MARIE (1772-1837), French socialist writer, was born at Besancon in Franche-Comte on the 7th of April 1772. His father was a draper in good circumstances, and Fourier received an excellent education at the college in his native town. After completing his studies there he travelled for some time in France, Germany and Holland. On the death of his father he inherited a considerable amount of property, which, however, was lost when Lyons was besieged by the troops of the Convention. Being thus deprived of his means of livelihood Fourier entered the army, but after two years' service as a chasseur was discharged on account of ill-health. In 1803 he published a remarkable article on European politics which attracted the notice of Napoleon, some of whose ideas were foreshadowed in it. Inquiries were made after the author, but nothing seems to have come of them. After leaving the army Fourier entered a merchant's office in Lyons, and some years later undertook on his own account a small business as broker. He obtained in this way just sufficient to supply his wants, and devoted all his leisure time to the elaboration of his first work on the organization of society.

During the early part of his life, and while engaged in commerce, he had become deeply impressed with the conviction that social arrangements resulting from the principles of individualism and competition were essentially imperfect and immoral. He proposed to substitute for these principles co-operation or united effort, by means of which full and harmonious development might be given to human nature. The scheme, worked out in detail in his first work, _Theorie des quatre mouvements_ (2 vols., Lyons, 1808, published anonymously), has for foundation a

## particular psychological proposition and a special economical doctrine.

Psychologically Fourier held what may with some laxity of language be called natural optimism,--the view that the full, free development of human nature or the unrestrained indulgence of human passion is the only possible way to happiness and virtue, and that misery and vice spring from the unnatural restraints imposed by society on the gratification of desire. This principle of harmony among the passions he regarded as his grandest discovery--a discovery which did more than set him on a level with Newton, the discoverer of the principle of attraction or harmony among material bodies. Throughout his works, in uncouth, obscure and often unintelligible language, he endeavours to show that the same fundamental fact of harmony is to be found in the four great departments,--society, animal life, organic life and the material universe. In order to give effect to this principle and obtain the resulting social harmony, it was needful that society should be reconstructed; for, as the social organism is at present constituted, innumerable restrictions are imposed upon the free development of human desire. As practical principle for such a reconstruction Fourier advocated co-operative or united industry. In many respects what he says of co-operation, in particular as to the enormous waste of economic force which the actual arrangements of society entail, still deserves attention, and some of the most recent efforts towards extension of the co-operative method, e.g. to house-keeping, were in essentials anticipated by him. But the full realization of his scheme demanded much more than the mere admission that co-operation is economically more efficacious than individualism. Society as a whole must be organized on the lines requisite to give full scope to co-operation and to the harmonious evolution of human nature. The details of this reorganization of the social structure cannot be given briefly, but the broad outlines may be thus sketched. Society, on his scheme, is to be divided into departments or _phalanges_, each _phalange_ numbering about 1600 persons. Each _phalange_ inhabits a _phalanstere_ or common building, and has a certain portion of soil allotted to it for cultivation. The _phalansteres_ are built after a uniform plan, and the domestic arrangements are laid down very elaborately. The staple industry of the _phalanges_ is, of course, agriculture, but the various _series_ and _groupes_ into which the members are divided may devote themselves to such occupations as are most to their taste; nor need any occupation become irksome from constant devotion to it. Any member of a group may vary his employment at pleasure, may pass from one task to another. The tasks regarded as menial or degrading in ordinary society can be rendered attractive if advantage is taken of the proper principles of human nature: thus children, who have a natural affinity for dirt, and a fondness for "cleaning up," may easily be induced to accept with eagerness the functions of public scavengers. It is not, on Fourier's scheme, necessary that private property should be abolished, nor is the privacy of family life impossible within the _phalanstere_. Each family may have separate apartments, and there may be richer and poorer members. But the rich and poor are to be locally intermingled, in order that the broad distinction between them, which is so painful a feature in actual society, may become almost imperceptible. Out of the common gain of the _phalange_ a certain portion is deducted to furnish to each member the minimum of subsistence; the remainder is distributed in shares to labour, capital and talent,--five-twelfths going to the first, four-twelfths to the second and three-twelfths to the third. Upon the changes requisite in the private life of the members Fourier was in his first work more explicit than in his later writings. The institution of marriage, which imposes unnatural bonds on human passion, is of necessity abolished; a new and ingeniously constructed system of licence is substituted for it. Considerable offence seems to have been given by Fourier's utterances with regard to marriage, and generally the later advocates of his views are content to pass the matter over in silence or to veil their teaching under obscure and metaphorical language.

The scheme thus sketched attracted no attention when the _Theorie_ first appeared, and for some years Fourier remained in his obscure position at Lyons. In 1812 the death of his mother put him in possession of a small sum of money, with which he retired to Bellay in order to perfect his second work. The _Traite de l'association agricole domestique_ was published in 2 vols. at Paris in 1822, and a summary appeared in the following year. After its publication the author proceeded to Paris in the hope that some wealthy capitalist might be induced to attempt the realization of the projected scheme. Disappointed in this expectation he returned to Lyons. In 1826 he again visited Paris, and as a considerable portion of his means had been expended in the publication of his book, he accepted a clerkship in an American firm. In 1829 and 1830 appeared what is probably the most finished exposition of his views, _Le Nouveau Monde industriel_. In 1831 he attacked the rival socialist doctrines of Saint-Simon and Owen in the small work _Pieges et charlatanisme de deux sectes, St Simon et Owen_. His writings now began to attract some attention. A small body of adherents gathered round him, and the most ardent of them was Victor Considerant (q.v.). In 1832 a newspaper, _Le Phalanstere ou la reforme industrielle_ was started to propagate the views of the school, but its success was not great. In 1833 it declined from a weekly to a monthly, and in 1834 it died of inanition. It was revived in 1836 as _Le Phalange_, and in 1843 became a daily paper, _La Democratie pacifique_. In 1850 it was suppressed.

Fourier did not live to see the success of his newspaper, and the only practical attempt during his lifetime to establish a _phalanstere_ was a complete failure. In 1832 M. Baudet Dulary, deputy for Seine-et-Oise, who had become a convert, purchased an estate at Conde-sur-Vesgre, near the forest of Rambouillet, and proceeded to establish a socialist community. The capital supplied was, however, inadequate, and the community broke up in disgust. Fourier was in no way discouraged by this failure, and till his death, on the 10th of October 1837, he lived in daily expectation that wealthy capitalists would see the merits of his scheme and be induced to devote their fortunes to its realization. It may be added that subsequent attempts to establish the _phalanstere_ have been uniformly unsuccessful.[1]

Fourier seems to have been of an extremely retiring and sensitive disposition. He mixed little in society, and appeared, indeed, as if he were the denizen of some other planet. Of the true nature of social arrangements, and of the manner in which they naturally grow and become organized, he must be pronounced extremely ignorant. The faults of existing institutions presented themselves to him in an altogether distorted manner, and he never appears to have recognized that the evils of actual society are immeasurably less serious than the consequences of his arbitrary scheme. Out of the chaos of human passion he supposed harmony was to be evolved by the adoption of a few theoretically disputable principles, which themselves impose restraints even more irksome than those due to actual social facts. With regard to the economic aspects of his proposed new method, it is of course to be granted that co-operation is more effective than individual effort, but he has nowhere faced the question as to the probable consequences of organizing society on the abolition of those great institutions which have grown with its growth. His temperament was too ardent, his imagination too strong, and his acquaintance with the realities of life too slight to enable him justly to estimate the merits of his fantastic views. That this description of him is not expressed in over-strong language must be clear to any one who not only considers what is true in his works,--and the portion of truth is by no means a peculiar discovery of Fourier's,--but who takes into account the whole body of his speculations, the cosmological and historical as well as the economical and social. No words can adequately describe the fantastic nonsense which he pours forth, partly in the form of general speculation on the universe, partly in the form of prophetic utterances with regard to the future changes in humanity and its material environment. From these extraordinary writings it is no extreme conclusion that there was much of insanity in Fourier's mental constitution.

AUTHORITIES.--Ch. Pellarin, _Fourier, sa vie et sa theorie_ (5th ed., 1872); Sargant, _Social Innovators_ (1859); Reybaud, _Reformateurs modernes_ (7th ed., 1864); Stein, _Socialismus und Communismus des heutigen Frankreichs_ (2nd ed., 1848); A.J. Booth, _Fortnightly Review_, N. S., vol. xii.; Czynski, _Notice bibliographique sur C. Fourier_ (1841); Ferraz, _Le Socialisme, le naturalisme et le positivisme_ (1877); Considerant, _Exposition abregee du systeme de Fourier_ (1845); Transon, _Theorie societaire de Charles Fourier_ (1832); Stein, _Geschichte der sozialen Bewegung in Frankreich_ (1850); Marlo, _Untersuchungen uber die Organisation der Arbeit_ (1853); J.H. Noyes, _History of American Socialisms_ (1870); Bebel, _Charles Fourier_ (1888); Varschauer, _Geschichte des Sozialismus und Kommunismus im 19. Jahrhundert_ (1903); Sambuc, _Le Socialisme de Fourier_ (1900); M. Hillquit, _History of Socialism in the United States_ (1903); H. Bourgin, _Fourier, contribution a l'etude de socialisme francais_ (1905). (R. Ad.)

FOOTNOTE:

[1] Several experiments were made to this end in the United States (see COMMUNISM) by American followers of Fourier, whose doctrines were introduced there by Albert Brisbane (1809-1890). Indeed, in the years between 1840 and 1850, during which the movement waxed and waned, no fewer than forty-one _phalanges_ were founded, of which some definite record can be found. The most interesting of all the experiments, not alone from its own history, but also from the fact that it attracted the support of many of the most intellectual and cultured Americans was that of Brook Farm (q.v.).

FOURIER, JEAN BAPTISTE JOSEPH (1768-1830), French mathematician, was born at Auxerre on the 21st of March 1768. He was the son of a tailor, and was left an orphan in his eighth year; but, through the kindness of a friend, admission was gained for him into the military school of his native town, which was then under the direction of the Benedictines of Saint-Maur. He soon distinguished himself as a student and made rapid progress, especially in mathematics. Debarred from entering the army on account of his lowness of birth and poverty, he was appointed professor of mathematics in the school in which he had been a pupil. In 1787 he became a novice at the abbey of St Benoit-sur-Loire; but he left the abbey in 1789 and returned to his college, where, in addition to his mathematical duties, he was frequently called to lecture on other subjects,--rhetoric, philosophy and history. On the institution of the Ecole Normale at Paris in 1795 he was sent to teach in it, and was afterwards attached to the Ecole Polytechnique, where he occupied the chair of analysis. Fourier was one of the savants who accompanied Bonaparte to Egypt in 1798; and during this expedition he was called to discharge important political duties in addition to his scientific ones. He was for a time virtually governor of half Egypt, and for three years was secretary of the Institut du Caire; he also delivered the funeral orations for Kleber and Desaix. He returned to France in 1801, and in the following year he was nominated prefect of Isere, and was created baron and chevalier of the Legion of Honour. He took an important part in the preparation of the famous _Description de l'Egypte_ and wrote the historical introduction. He held his prefecture for fourteen years; and it was during this period that he carried on his elaborate and fruitful investigations on the conduction of heat. On the return of Napoleon from Elba, in 1815, Fourier published a royalist proclamation, and left Grenoble as Napoleon entered it. He was then deprived of his prefecture, and, although immediately named prefect of the Rhone, was soon after again deprived. He now settled at Paris, was elected to the Academie des Sciences in 1816, but in consequence of the opposition of Louis XVIII. was not admitted till the following year, when he succeeded the Abbe Alexis de Rochon. In 1822 he was made perpetual secretary in conjunction with Cuvier, in succession to Delambre. In 1826 Fourier became a member of the French Academy, and in 1827 succeeded Laplace as president of the council of the Ecole Polytechnique. In 1828 he became a member of the government commission established for the encouragement of literature. He died at Paris on the 16th of May 1830.

As a politician Fourier achieved uncommon success, but his fame chiefly rests on his strikingly original contributions to science and mathematics. The theory of heat engaged his attention quite early, and in 1812 he obtained a prize offered by the Academie des Sciences with a memoir in two parts, _Theorie des mouvements de la chaleur dans les corps solides_. The first part was republished in 1822 as _La Theorie analytique de la chaleur_, which by its new methods and great results made an epoch in the history of mathematical and physical science (see below: FOURIER'S SERIES). An English translation has been published by A. Freeman (Cambridge, 1872), and a German by Weinstein (Berlin, 1884). His mathematical researches were also concerned with the theory of equations, but the question as to his priority on several points has been keenly discussed. After his death Navier completed and published Fourier's unfinished work, _Analyse des equations indeterminees_ (1831), which contains much original matter. In addition to the works above mentioned, Fourier wrote many memoirs on scientific subjects, and _eloges_ of distinguished men of science. His works have been collected and edited by Gaston Darboux with the title _Oeuvres de Fourier_ (Paris, 1889-1890).

For a list of Fourier's publications see the _Catalogue of Scientific Papers of the Royal Society of London_. Reference may also be made to Arago, "Joseph Fourier," in the _Smithsonian Report_ (1871).

FOURIER'S SERIES, in mathematics, those series which proceed according to sines and cosines of multiples of a variable, the various multiples being in the ratio of the natural numbers; they are used for the representation of a function of the variable for values of the variable which lie between prescribed finite limits. Although the importance of such series, especially in the theory of vibrations, had been recognized by D. Bernoulli, Lagrange and other mathematicians, and had led to some discussion of their properties, J.B.J. Fourier (see above) was the first clearly to recognize the arbitrary character of the functions which the series can represent, and to make any serious attempt to prove the validity of such representation; the series are consequently usually associated with the name of Fourier. More general cases of trigonometrical series, in which the multiples are given as the roots of certain transcendental equations, were also considered by Fourier.

Before proceeding to the consideration of the special class of series to be discussed, it is necessary to define with some precision what is to be understood by the representation of an arbitrary function by an infinite series. Suppose a function of a variable x to be arbitrarily given for values of x between two fixed values a and b; this means that, corresponding to every value of x such that a <= x <= b, a definite arithmetical value of the function is assigned by means of some prescribed set of rules. A function so defined may be denoted by [f](x); the rules by which the values of the function are determined may be embodied in a single explicit analytical formula, or in several such formulae applicable to different portions of the interval, but it would be an undue restriction of the nature of an arbitrarily given function to assume _a priori_ that it is necessarily given in this manner, the possibility of the representation of such a function by means of a single analytical expression being the very point which we have to discuss. The variable x may be represented by a point at the extremity of an interval measured along a straight line from a fixed origin; thus we may speak of the point c as synonymous with the value x = c of the variable, and of [f](c) as the value of the function assigned to the point c. For any number of points between a and b the function may be discontinuous, i.e. it may at such points undergo abrupt changes of value; it will here be assumed that the number of such points is finite. The only discontinuities here considered will be those known as ordinary discontinuities. Such a discontinuity exists at the point c if [f](c + [epsilon]), [f](c - [epsilon]) have distinct but definite limiting values as [epsilon] is indefinitely diminished; these limiting values are known as the limits on the right and on the left respectively of the function at c, and may be denoted by [f](c + 0), [f](c - 0). The discontinuity consists therefore of a sudden change of value of the function from [f](c - 0) to [f](c + 0), as x increases through the value c. If there is such a discontinuity at the point x = 0, we may denote the limits on the right and on the left respectively by [f](+0), [f](-0).

Suppose we have an infinite series u1(x) + u2(x) + ... + u_n(x) + ... in which each term is a function of x, of known analytical form; let any value x = c(a = c = b) be substituted in the terms of the series, and suppose the sum of n terms of the arithmetical series so obtained approaches a definite limit as n is indefinitely increased; this limit is known as the sum of the series. If for every value of c such that a <= c <= b the sum exists and agrees with the value of [f](c), the series [Sigma] [1 to [oo]] u_n(x) is said to represent the function ([f]x) between the values a, b of the variable. If this is the case for all points within the given interval with the exception of a finite number, at any one of which either the series has no sum, or has a sum which does not agree with the value of the function, the series is said to represent "in general" the function for the given interval. If the sum of n terms of the series be denoted by S_n(c), the condition that S_n(c) converges to the value [f](c) is that, corresponding to any finite positive number [delta] as small as we please, a value n1 of n can be found such that if n >= n_1, |[f](c) - Sn(c)| < [delta].

Functions have also been considered which for an infinite number of points within the given interval have no definite value, and series have also been discussed which at an infinite number of points in the interval cease either to have a sum, or to have one which agrees with the value of the function; the narrower conception above will however be retained in the treatment of the subject in this article, reference to the wider class of cases being made only in connexion with the history of the theory of Fourier's Series.

_Uniform Convergence of Series._--If the series u1(x) + u2(x) + ... + u2(x) + ... converge for every value of x in a given interval a to b, and its sum be denoted by S(x), then if, corresponding to a finite positive number [delta], as small as we please, a finite number n1 can be found such that the arithmetical value of S(x) - S_n(x), where n => n1 is less than [delta] for every value of x in the given interval, the series is said to converge uniformly in that interval. It may however happen that as x approaches a particular value the number of terms of the series which must be taken so that |S(x) - S_n(x)| may be < [delta], increases indefinitely; the convergence of the series is then infinitely slow in the neighbourhood of such a point, and the series is not uniformly convergent throughout the given interval, although it converges at each point of the interval. If the number of such points in the neighbourhood of which the series ceases to converge uniformly be finite, they may be excluded by taking intervals of finite magnitude as small as we please containing such points, and considering the convergence of the series in the given interval with such sub-intervals excluded; the convergence of the series is now uniform throughout the remainder of the interval. The series is said to be _in general_ uniformly convergent within the given interval a to b if it can be made uniformly convergent by the exclusion of a finite number of portions of the interval, each such portion being arbitrarily small. It is known that the sum of an infinite series of continuous terms can be discontinuous only at points in the neighbourhood of which the convergence of the series is not uniform, but non-uniformity of convergence of the series does not necessarily imply discontinuity in the sum.

_Form of Fourier's Series._--If it be assumed that a function [f](x) arbitrarily given for values of x such that o[<=]x[<=]l is capable of being represented in general by an infinite series of the form

[pi]x 2[pi]x n[pi]x A1 sin ----- + A2 sin ------ + ... + A_n sin ------ + ..., l l l

and if it be further assumed that the series is in general uniformly convergent throughout the interval 0 to l, the form of the coefficients A can be determined. Multiply each term of the series by sin n[pi]x/l, and integrate the product between the limits 0 and l, then in virtue of the property [int][l to 0] sin n[pi]x/l sin n'[pi]x/l dx=0, or 1/2 l, according as n' is not, or is, equal to n, we have -1/2 lA_n= [int][0 to l] [f](x) sin n[pi]x/l dx, and thus the series is of the form

_ 2 [oo] n[pi]x / l n[pi]x -- [Sigma] sin ------ | [f](x) ------ dx. (1) l 1 l _/ 0 l

This method of determining the coefficients in the series would not be valid without the assumption that the series is in general uniformly convergent, for in accordance with a known theorem the sum of the integrals of the separate terms of the series is otherwise not necessarily equal to the integral of the sum. This assumption being made, it is further assumed that [f](x) is such that [integral][0 to l] [f](x)sin n[pi]x/l dx has a definite meaning for every value of n.

Before we proceed to examine the justification for the assumptions made, it is desirable to examine the result obtained, and to deduce other series from it. In order to obtain a series of the form

[pi]x 2[pi]x n[pi]x B0 + B1 cos ----- + B2 cos ------ + ... + B_n cos ------ + ... l l l

for the representation of [f](x) in the interval 0 to l, let us apply the series (1) to represent the function [f](x) sin [pi]x/l; we thus find _ 2 [oo] n[pi]x / l [pi]x n[pi]x -- [Sigma] sin ------ | [f](x)sin ----- sin ------ dx, l 1 l _/ 0 l l

On rearrangement of the terms this becomes _ _ 1 [pi]x / l 2 [pi]x n[pi]x / l n[pi]x -- sin ----- | [f](x) dx + -- [Sigma] sin ----- cos ------ | [f](x) cos ------ dx. l l _/ 0 l l l _/ 0 l

hence [f](x) is represented for the interval 0 to l by the series of cosines _ _ 1 / l 2 [oo] n[pi]x / l n[pi]x -- | [f](x) dx + -- [Sigma] cos ------ | [f](x) cos ------ dx ... (2) l _/ 0 l 1 l _/ 0 l

We have thus seen, that with the assumptions made, the arbitrary function [f](x) may be represented, for the given interval, either by a series of sines, as in (1), or by a series of cosines, as in (2). Some important differences between the two series must, however, be noticed. In the first place, the series of sines has a vanishing sum when x=o or x=l; it therefore does not represent the function at the point x=o, unless [f](0) = 0, or at the point x=l, unless [f](l) = 0, whereas the series (2) of cosines may represent the function at both these points. Again, let us consider what is represented by (1) and (2) for values of x which do not lie between 0 and l. As [f](x) is given only for values of x between 0 and l, the series at points beyond these limits have no necessary connexion with [f](x) unless we suppose that [f](x) is also given for such general values of x in such a way that the series continue to represent that function. If in (1) we change x into -x, leaving the coefficients unaltered, the series changes sign, and if x be changed into x + 2l, the series is unaltered; we infer that the series (1) represents an odd function of x and is periodic of period 2l; thus (1) will represent [f](x) in general for values of x between [+-][oo], only if [f](x) is odd and has a period 2l. If in (2) we change x into -x, the series is unaltered, and it is also unaltered by changing x into x + 2l; from this we see that the series (2) represents [f](x) for values of x between [+-][oo], only if [f](x) is an even function, and is periodic of period 2l. In general a function [f](x) arbitrarily given for all values of x between [+-][oo] is neither periodic nor odd, nor even, and is therefore not represented by either (1) or (2) except for the interval 0 to l.

From (1) and (2) we can deduce a series containing both sines and cosines, which will represent a function [f](x) arbitrarily given in the interval -l to l, for that interval. We can express by (1) the function -1/2{[f](x) - [f](-x)} which is an odd function, and thus this function is represented for the interval -l to +l by _ 2 n[pi]x / l n[pi]x -- [Sigma] sin ------ | 1/2 {[f](x) - [f](-x)} sin ------ dx; l l _/ 0 l

we can also express 1/2 {[f](x) + [f](-x)}, which is an even function, by means of (2), thus for the interval -l to +l this function is represented by _ _ 1 / l 2 [oo] n[pi]x / l n[pi]x -- | 1/2 {[f](x) + [f](-x)} dx + -- [Sigma] cos ------ | 1/2 {[f](x) + [f](-x)} cos ------ dx. l _/ 0 l 1 l _/ 0 l

It must be observed that [f](-x) is absolutely independent of [f](x), the former being not necessarily deducible from the latter by putting -x for x in a formula; both [f](x) and [f](-x) are functions given arbitrarily and independently for the interval 0 to l. On adding the expressions together we obtain a series of sines and cosines which represents [f](x) for the interval -l to l. The integrals _ _ / l n[pi]x / l n[pi]x | [f](-x) cos ------ dx, | [f](-x) sin ------ dx _/ 0 l _/ 0 l

are equivalent to _ _ /-l n[pi]x /-l n[pi]x - | [f](x) cos ------ dx, + | [f](x) sin ------ dx, _/0 l _/ 0 l

thus the series is _ _ _ 1 / l 1 [oo] n[pi]x / l n[pi]x 1 [oo] n[pi]x / l n[pi]x -- | [f](x)dx + -- [Sigma] cos ------ | f(x) cos ------ dx + -- [Sigma] sin ------ | [f](x) sin ------ dx, 2l_/-l l 1 l _/-l l l 1 l _/-l l

which may be written _ _ 1 / l 1 [oo] / l n[pi](x - x') -- | [f](x') dx' + -- [Sigma] | [f](x') cos ------------- dx'. (3) 2l_/-l l 1 _/-l l

The series (3), which represents a function [f](x) arbitrarily given for the interval -l to l, is what is known as Fourier's Series; the expressions (1) and (2) being regarded as the particular forms which (3) takes in the two cases, in which [f](-x) = -[f](x), or [f](-x) = f(x) respectively. The expression (3) does not represent f(x) at points beyond the interval -l to l, unless [f](x) has a period 2l. For a value of x within the interval, at which [f](x) is discontinuous, the sum of the series may cease to represent [f](x), but, as will be seen hereafter, has the value 1/2 {[f](x + 0) + [f](x - 0)}, the mean of the limits at the points on the right and the left. The series represents the function at x=o, unless the function is there discontinuous, in which case the series is 1/2 {[f](+0) + [f](-0)}; the series does not necessarily represent the function at the points l and -l, unless [f](l) = [f](-l). Its sum at either of these points is 1/2 {[f](l) + [f](-l)}.

_Examples of Fourier's Series._--(a) Let [f](x) be given from 0 to l, by [f](x)=c, when 0 <= x < 1/2 l, and by f(x)= -c from 1/2 l to l; it is required to find a sine series, and also a cosine series, which shall represent the function in the interval.

We have _ _ _ / l n[pi]x /1/2 l n[pi]x / l n[pi]x | [f](x) sin ------ dx = c | sin ------ dx - c | sin ------ dx _/ 0 l _/ 0 l _/1/2 l l

cl = ----- (cos n[pi] - 2 cos 1/2 n[pi] + 1). n[pi]

This vanishes if n is odd, and if n = 4m, but if n = 4m + 2 it is equal to 4cl/n[pi]; the series is therefore

4c /l 2[pi]x 1 6[pi]x 1 10[pi]x \ ---- ( -- sin ------ + -- sin ------ + -- sin ------- + ... ). [pi] \2 l 3 l 5 l /

For unrestricted values of x, this series represents the ordinates of the series of straight lines in fig. 1, except that it vanishes at the points 0, (1/2)l, l, (3/2)l ...

[Illustration: FIG. 1.]

We find similarly that the same function is represented by the series

4c / [pi]x 1 3[pi]x 1 5[pi]x \ ---- ( cos ----- - -- cos ------ + -- cos ------ - + ... ) [pi] \ l 3 l 5 l /

during the interval 0 to l; for general values of x the series represents the ordinate of the broken line in fig. 2, except that it vanishes at the points (1/2)l, (3/2)l....

[Illustration: FIG. 2.]

(b) Let [f](x) = x from 0 to 1/2 l, and f(x) = l - x, from 1/2 l to l; then _ _ _ / l n[pi]x / 1/2 l n[pi]x / l n[pi]x | [f](x)sin ------ dx= | x sin ------ dx + | (l - x)sin ------ dx _/ 0 l _/ 0 l _/1/2 l l

l^2 n[pi] l^2 n[pi] l^2n / n[pi] \ = - ------ cos ----- + --------- sin ----- + ----- (cos ----- - cos n[pi] ) 2n[pi] 2 n^2[pi]^2 2 n[pi] \ 2 /

l^2 l^2 n[pi] l^2 n[pi] 2l^2 n[pi] + ----- cos n[pi] - ------ cos ----- + --------- sin ----- = --------- sin ----- n[pi] 2n[pi] 2 n^2[pi]^2 2 n^2[pi]^2 2

hence the sine series is

4l / nx 1 3[pi]x 1 5[pi]x \ ------ (sin -- - --- sin ------ + --- sin ------ - ... ) [pi]^2 \ l 3^2 l 5^2 l /

For general values of x, the series represents the ordinates of the row of broken lines in fig. 3.

[Illustration: FIG. 3.]

The cosine series, which represents the same function for the interval 0 to l, may be found to be

1 2l / 2[pi]x 1 6[pi]x 1 10[pi]x \ -- l - ------ (cos ------ + --- cos ------ + --- cos ------- + ... ) 4 [pi]^2 \ l 3^2 l 5^2 l /

This series represents for general values of x the ordinate of the set of broken lines in fig. 4.

[Illustration: FIG. 4.]

_Dirichlet's Integral._--The method indicated by Fourier, but first carried out rigorously by Dirichlet, of proving that, with certain restrictions as to the nature of the function [f](x), that function is in general represented by the series (3), consists in finding the sum of n+1 terms of that series, and then investigating the limiting value of the sum, when n is increased indefinitely. It thus appears that the series is convergent, and that the value towards which its sum converges is 1/2 {[f](x + 0) + [f](x - 0)}, which is in general equal to [f](x). It will be convenient throughout to take -[pi] to [pi] as the given interval; any interval -l to l may be reduced to this by changing x into lx/[pi], and thus there is no loss of generality.

We find by an elementary process that

1/2 + cos (x - x') + cos 2(x - x') + ... + cos n(x - x')

2n + 1 sin ------ (x' - x) 2 = -------------------. 2 sin 1/2(x' - x)

Hence, with the new notation, the sum of the first n+1 terms of (3) is _ 1 / [pi] sin (2n + 1)/2 (x' - x) ---- | [f](x') ----------------------- dx'. [pi]_/-[pi] 2 sin 1/2 (x' - x)

If we suppose [f](x) to be continued beyond the interval -[pi] to [pi], in such a way that [f](x) = [f](x + 2[pi]), we may replace the limits in this integral by x + [pi], x-[pi] respectively; if we then put x' - x = 2z, and let [f](x') = [F](z), the expression becomes 1/[pi] [int][-[pi]/2 to [pi]/2] F(z) (sin mz/sin z) dz, where m = 2n + 1; this expression may be written in the form _ _ 1 /[pi]/2 sin mz 1 /[pi]/2 sin mz ---- | F(z) ------ dz + ---- | F(-z) ------ dz. (4) [pi]_/ 0 sin z [pi]_/ 0 sin z

We require therefore to find the limiting value, when m is indefinitely increased, of [int][0 to [pi]/2] F(z)(sin mz/sin z) dz; the form of the second integral being essentially the same. This integral, or rather the slightly more general one [int][0 to h] F(z)(sin mz/sin z) dz, when 0 < h <= 1/2[pi], is known as Dirichlet's integral. If we write X(z)= F(z)(z/sin z), the integral becomes [int][0 to h] X(z)(sin mz/z) dz, which is the form in which the integral is frequently considered.

_The Second Mean-Value Theorem._--The limiting value of Dirichlet's integral may be conveniently investigated by means of a theorem in the integral calculus known as the second mean-value theorem. Let a, b be two fixed finite numbers such that a<b, and suppose [f](x), [phi](x) are two functions which have finite and determinate values everywhere in the interval except for a finite number of points; suppose further that the functions [f](x), [phi](x) are integrable throughout the interval, and that as x increases from a to b the function [f](x) is monotone, i.e. either never diminishes or never increases; the theorem is that _ _ _ / b /[xi] / b | [f](x) [phi](x) dx = [f](a + 0) | [phi](x) dx + [f](b - 0) | [phi](x) dx _/ a _/ a _/[xi]

when [xi] is some point between a and b, and [f](a), [f](b) may be written for [f](a + 0), [f](b - 0) unless a or b is a point of discontinuity of the function [f](x).

To prove this theorem, we observe that, since the product of two integrable functions is an integrable function, [int][a to b] [f](x) [phi](x) dx exists, and may be regarded as the limit of the sum of a series [f](x0) [phi](x0) (x1 - x0) + [f](x1) [phi](x1) (x2 - x1) + ... + [f]x(n-1) [phi]x_(n-1) (x_n - x_(n-1)) where x0 = a, x_n = b and x1, x2 ... x_(n-1) are n - 1 intermediate points. We can express [phi](x_r) (x_(r+1) - x_r) in the form Y_(r+1) - Y_r, by putting

K=r Y_r = [Sigma] [phi](x_(K-1)) (x_K - x_(K-1)), Y0 = 0. K=1

Writing X_r for [f](x_r), the series becomes

X0(Y1 - Y0) + X1(Y2 - Y1) + ... + X_(n-1)(Y_n - Y_(n-1))

or Y1(X0 - X1) + Y2(X1 - X2) + ... + Y_n(X_(n-1) - X_n) + Y_n X_n.

Now, by supposition, all the numbers Y1, Y2 ... Y_n are finite, and all the numbers X_(r-1) - X_r are of the same sign, hence by a known algebraical theorem the series is equal to M(X0 - X_n) + Y_n X_n, where M is a number intermediate between the greatest and the least of the numbers Y1, Y2, ... Y_n. This remains true however many partial intervals are taken, and therefore, when their number is increased indefinitely, and their breadths are diminished indefinitely according to any law, we have _ _ / b _ / b | [f](x)[phi](x)dx = {[f](a) - [f](b)} M + [f](b) | [phi](x) dx _/ a _/ a

when M is intermediate between the greatest and least values which [int][a to x] [phi](x) dx can have, when x is in the given integral. Now this integral is a continuous function of its upper limit x, and therefore there is a value of x in the interval, for which it takes any particular value between the greatest and least values that it has. There is therefore a value [xi] between a and b, such that

_ _ /[xi] M = | [phi](x)dx, _/ a

hence _ _ _ / b /[xi] / b | [f](x) [phi](x) dx = {[f](a) - [f](b)} | [phi](x) dx + [f](b) | [phi](x) dx _/ a _/ a _/ a _ _ /[xi] / b = [f](a) | [phi](x) dx + [f](b) | [phi](x) dx. _/ a _/[xi]

If the interval contains any finite numbers of points of discontinuity of [f](x) or [phi](x), the method of proof still holds good, provided these points are avoided in making the subdivisions; in particular if either of the ends be a point of discontinuity of [f](x), we write [f](a + 0) or [f](b - 0), for [f](a) or [f](b), it being assumed that these limits exist.

_Functions, with Limited Variation._--The condition that [f](x), in the mean-value theorem, either never increases or never diminishes as x increases from a to b, places a restriction upon the applications of the theorem. We can, however, show that a function [f](x) which is finite and continuous between a and b, except for a finite number of ordinary discontinuities, and which only changes from increasing to diminishing or vice versa, a finite number of times, as x increases from a to b, may be expressed as the difference of two functions [f]1(x), [f]2(x), neither of which ever diminishes as x passes from a to b, and that these functions are finite and continuous, except that one or both of them are discontinuous at the points where the given function is discontinuous. Let [alpha], [beta] be two consecutive points at which [f](x) is discontinuous, consider any point x1, such that [alpha] <= x1 <= [beta], and suppose that at the points M1, M2 ... M_r between [alpha] and x1, [f](x) is a maximum, and at m1, m2 ... m_r, it is a minimum; we will suppose, for example, that the ascending order of values is [alpha], M1, m1, M2, m2 ... M_r, m_r, x1; it will make no essential difference in the argument if m1 comes before M1, or if M_r immediately precedes x1, M_(r-1) being then the last minimum.

Let [psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)]+ ... +[[f](M_r) - [f](m_(r-1))] + [[f](x1) - [f](m_r)];

now let (x1) increase until it reaches the value (M_(r+1)) at which [f](x) is again a maximum, then let

[psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)] + ... + [[f](M_r) - [f](m_(r-1)] + [[f](M_(r+1)) - [f](m_r)];

and suppose as x increases beyond the value M_(r+1), [psi](x1) remains constant until the next minimum m_(r+1) is reached, when it again becomes variable; we see that [psi](x1) is essentially positive and never diminishes as x increases.

Let

[chi](x1) = [[f](M1) - f(m1)] + [[f](M2) - [f](m1)] + ... + [[f](M_r) - [f](m_r)],

then let x1 increase until it is beyond the next maximum M_(r+1), and then let

[chi](x1) = [[f](M1) - [f](m1)] + [[f](M2) - [f](m1)] + ... + [[f](M_r) - [f](m_r)] + [[f](M_(r+1)) - [f](x1)]

thus [chi](x1) never diminishes, and is alternately constant and variable. We see that [psi](x1) - [chi](x1) is continuous as x1 increases from [alpha] to [beta], and that [psi](x1) - [chi](x1) = [f](x1) - [f]([alpha] + 0), and when x1 reaches [beta], we have [psi]([beta]) - [chi](x1) = [f]([beta] - 0) - [f]([alpha] + 0). Hence it is seen that between [alpha] and [beta], [f](x) = [[psi](x) + [f]([alpha] + 0)] - [chi](x), where [psi](x) + [f]([alpha] + 0), [chi](x) are continuous and never diminish as x increases; the same reasoning applies to every continuous portion of [f](x), for which the functions [psi](x), [chi](x) are formed in the same manner; we now take [f]1(x)=[psi](x) + [f]([alpha] + 0) + C, [f]2(x) = [chi](x) + C, where C is constant between consecutive discontinuities, but may have different values in the next interval between discontinuities; the C can be so chosen that neither [f]1(x) nor [f]2(x) diminishes as x increases through a value for which [f](x) is discontinuous. We thus see that [f](x) = [f]1(x) - [f]2(x), where [f]1(x), [f]2(x) never diminish as x increases from a to b, and are discontinuous only where [f](x) is so. The function [f](x) is a particular case of a class of functions defined and discussed by Jordan, under the name "functions with limited variation" (_fonctions a variation bornee_); in general such functions have not necessarily only a finite number of maxima and minima.

_Proof of the Convergence of Fourier's Series._--It will now be assumed that a function [f](x) arbitrarily given between the values -[pi] and +[pi], has the following properties:--

(a) The function is everywhere numerically less than some fixed positive number, and continuous except for a finite number of values of the variable, for which it may be ordinarily discontinuous.

(b) The function only changes from increasing to diminishing or vice versa, a finite number of times within the interval; this is usually expressed by saying that the number of maxima and minima is finite.

These limitations on the nature of the function are known as Dirichlet's conditions; it follows from them that the function is integrable throughout the interval.

On these assumptions, we can investigate the limiting value of Dirichlet's integral; it will be necessary to consider only the case of a function F(z) which does not diminish as z increases from 0 to 1/2[pi], since it has been shown that in the general case the difference of two such functions may be taken. The following lemmas will be required:

1. Since _ _ /[pi]/2 sin mz /[pi]/2 [pi] | ------ dz = | {1 + 2cos 2z + 2cos 4z + ... + 2cos 2nz} dz = ----; _/ 0 sin z _/ 0 2

this result holds however large the odd integer m may be.

[pi] 2. If 0 < [alpha] < [beta] <= ----, 2 _ _ _ /[beta] sin mz 1 /[gamma] 1 /[beta] | ------ dz = ----------- | sin mz dz + ---------- | sin mz dz _/[alpha] sin z sin [alpha]_/[alpha] sin [beta]_/[gamma]

where [alpha] < [gamma] < [beta], hence _ | /[beta] sin mz | 2 / 1 1 \ 4 | | ------ dz | < -- ( ---------- + --------- ) < -------------; | _/[alpha] sin z | m \sin[alpha] sin[beta]/ m sin [alpha] _ | / [beta] sin mz | 4 a precisely similar proof shows that | | ------ dz | < --------, | _/ [alpha] z | m[alpha] _ _ /[beta] sin mz /[beta] sin mz hence the integrals | ------ dz, | ------ dz, converge to _/[alpha] sin z _/[alpha] z

the limit zero, as m is indefinitely increased. _ | / [oo] sin [theta] | 3. If [alpha] > 0, | | ----------- d[theta] | cannot exceed | _/[alpha] [theta] |

1/2[pi]. For by the mean-value theorem _ | / h sin[theta] | 2 2 | | ---------- d[theta] | < ------- + --, | _/[alpha] [theta] | [alpha] h _ | / h sin[theta] | 2 hence | Lh = [oo] | ---------- d[theta] | <= -------; | _/[alpha] [theta] | [alpha] _ | /[oo] sin[theta] | 2 [pi] in particular if [alpha] >= [pi] | | ---------- d[theta] | <= ---- < ----. | _/[alpha] [theta] | [pi] 2 _ d /[oo] sin[theta] sin[alpha] Again -------- | ---------- d[theta] = - ----------, [alpha] > 0, d[alpha]_/[alpha] [theta] [alpha] _ /[oo] sin[theta] therefore | ---------- d[theta] increases as [alpha] diminishes, _/[alpha] [theta]

when [theta] < [alpha] < [pi]; but lim _ _ /[oo] sin[theta] [pi] | /[oo] sin[theta] | [pi] | ---------- d[theta] = ----, hence | | ---------- d[theta] | < ----, [alpha]=0_/[alpha] [theta] 2 | _/[alpha] [theta] | 2

where [alpha] < [pi], and < [pi]/2 where [alpha] >= [pi]. It follows that _ | /[beta] sin[theta] | | | ---------- d[theta] | <= [pi], provided 0 <= [alpha] < [beta]. | _/[alpha] [theta] | _ /[pi]/2 sin mz To find the limit of | F(z) ------ dz, we observe that it may be _/ 0 sin z

written in the form _ _ /[pi]/2 sin mz / [mu] sin mz F(0) | ------ dz + | {F(z) - F(0)} ------ dz _/ sin z _/ 0 sin z _ /[pi]/2 sin mz + | {F(z) - F(0)} ------ dz _/[mu] sin z

where [mu] is a fixed number as small as we please; hence if we use lemma (1), and apply the second mean-value theorem, _ /[pi]/2 sin mz [pi] | F(z) ------ dz - ---- F(0) _/ 0 sin z 2 _ /[mu] z sin mz = | {F(z) - F(0)} ----- ------ dz _/ [0] sin z z _ _ /[xi]^1 sin mz /[pi]/2 sin mz + {F([mu] + 0) - F(0)} | ------ dz + {F (1/2[pi] - 0) - F(0)} | ------ dz _/ [mu] sin z _/[xi]^1 sin z

when [xi]^1 lies between [mu] and 1/2[pi]. When m is indefinitely increased, the two last integrals have the limit zero in virtue of lemma (2). To evaluate the first integral on the right-hand side, let G/z = {F(z) - F(0)} (sin z/z), and observe that G(z) increases as z increases from 0 to [mu], hence if we apply the mean value theorem _ _ | /[mu] sin mz | | /[mu] sin mz | | | G([mu]) ------ dz| = |G([mu]) | ------ dz| | _/ 0 z | | _/[xi] z | _ | /m[mu] sin[theta] | = |G([mu]) | ---------- d[theta]| < [pi] G([mu]), | _/m[xi] [theta] |

where 0 < [xi] < [mu], since G(z) has the limit zero when z = 0. If [epsilon] be an arbitrarily chosen positive number, a fixed value of [mu] may be so chosen that [pi]G([mu)] < 1/2[epsilon], and thus that _ | /[mu] sin mz | | | G(z) ------ dz| < 1/2[epsilon]. | _/0 z |

It has now been shown that when m is indefinitely increased _ /[pi]/2 sin mz [pi] | F(z) ------ dz - ---- F(0) has the limit zero. _/ 0 sin z 2

Returning to the form (4), we now see that the limiting value of _ _ 1 /[pi]/2 sin mz 1 /[pi]/2 sin mz ---- | F(z) ------ dz + ---- | F(-z) ------ dz [pi]_/ 0 sin z [pi]_/ 0 sin z

1/2{F(+0) + F(-0)}; hence the sum of n + 1 terms of the series _ _ 1 / l 1 / l n[pi](x - x^1) -- | [f](x) dx + -- [Sigma] | [f](x^1) cos ------------- dx 2l _/-l l _/-l l

converges to the value 1/2 {[f](x + 0) + [f](x - 0)}, or to [f](x) at a point where [f](x) is continuous, provided [f](x) satisfies Dirichlet's conditions for the interval from -l to l.

_Proof that Fourier's Series is in General Uniformly Convergent._--To prove that Fourier's Series converges uniformly to its sum for all values of x, provided that the immediate neighbourhoods of the points of discontinuity of [f](x) are excluded, we have _ | /[pi]/2 sin mz [pi] | 4 | | F(z)------ dz - ---- F(0)| < [pi]G ([mu]) + ---------- {F([mu] + 0) - F(0)} | _/ sin z 2 | m sin [mu]

4 + ------------ {F(1/2[pi] - 0) - F(0)} m sin [xi]^1

[pi][mu] 4 < -------- {[f](x + 2[mu]) - [f](x)} + ---------- {[f](x + 2[mu]) - [f](x)} sin [mu] m sin [mu]

4 + ------------ {[f](x + [pi]) - [f](x)} m sin [xi]^1

Using this inequality and the corresponding one for F(-z), we have

where A is some fixed number independent of m. In any interval (a, b) in which [f](x) is continuous, a value [mu]1 of [mu] can be chosen such that, for every value of x in (a, b), |[f](x + 2[mu]) - [f](x)|, |[f](x - 2[mu]) - [f](x)| are less than an arbitrarily prescribed positive number [epsilon], provided [mu] = [mu]1. Also a value [mu]2 of [mu] can be so chosen that [epsilon][mu]2 cosec [mu]2 < 1/2[eta], where [eta] is an arbitrarily assigned positive number. Take for [mu] the lesser of the numbers [mu]1, [mu]2, then |S_(2n+1) - [f](x)| < [eta] + A|m cosec [mu] for every value of x in (a, b). It follows that, since [eta] and m are independent of x, |S_(2n+1) - [f](x)| < 2[epsilon], provided n is greater than some fixed value n1 dependent only on [epsilon]. Therefore S_(2n+1) converges to [f](x) uniformly in the interval (a, b).

_Case of a Function with Infinities._--The limitation that [f](x) must be numerically less than a fixed positive number throughout the interval may, under a certain restriction, be removed. Suppose F(z) is indefinitely great in the neighbourhood of the point z = c, and is

such that the limits of the two integrals [int][c to c[+-][epsilon]] F(z) dz are both zero, as [epsilon] is indefinitely diminished, then

_ /[pi]/2 sin mz | F(z) ------ dz _/ 0 sin z

denotes the limit when [epsilon] = 0, [epsilon]^1 = 0 of _ _ /c-[epsilon] sin mz /[pi]/2 sin mz | F(z) ------ dx + | F(z) ------ dz, _/ 0 sin z _/c+[epsilon]^1 sin z

both these limits existing; the first of these integrals has 1/2[pi]F(+0) for its limiting value when m is indefinitely increased, and the second has zero for its limit. The theorem therefore holds if F(z) has an infinity up to which it is absolutely integrable; this will, for example, be the case if F(z) near the point C is of the form x(z)(z - c)^-[mu] + [psi](z), where [chi](c), [psi](c) are finite, and 0 < [mu] < 1. It is thus seen that [f](x) may have a finite number of infinities within the given interval, provided the function is integrable through any one of these points; the function is in that case still representable by Fourier's Series.

_The Ultimate Values of the Coefficients in Fourier's Series._--If [f](x) is everywhere finite within the given interval -[pi] to +[pi], it can be shown that a_n, b_n, the coefficients of cos nx, sin nx in the series which represent the function, are such that na_n, nb_n, however great n is, are each less than a fixed finite quantity. For writing [f](x) = [f]1(x) - [f]2(x), we have _ _ _ /[pi] /[xi] /[pi] | [f]1(x) cos nxdx = [f]1(-[pi] + 0) | cos nxdx + [f]1([pi] - 0) | cos nxdx _/-[pi] _/-[pi] _/[xi]

hence _ /[pi] sin n[xi] sin n[xi] | [f]1(x) cos nxdx = [f]1(-[pi] + 0) --------- + [f]1([pi] - 0) --------- _/-[pi] n n

with a similar expression, with [f]2(x) for [f]1(x), [xi] being between [pi] and -[pi]; the result then follows at once, and is obtained similarly for the other coefficient.

If [f](x) is infinite at x = c, and is of the form [phi](x)/(x - c)^K near the point c, where 0 < K < 1, the integral _ /[pi] | [f](x)cos nxdx contains portions of the form _/-[pi] _ _ /[epsilon]+[epsilon] [phi](x) / c [phi](x) | --------- cos nxdx | --------- cos nxdx; _/ [c] (x - c)^K _/c-[epsilon] (x - c)^K

consider the first of these, and put x = c + u, it thus becomes _ /[epsilon] [phi](c + u) | ------------ cos n(c + u) du, which is of the form _/ 0 u^K _ /[epsilon] cos n(c + u) [phi](c + [theta][epsilon]) | ------------ du; _/ 0 u^K

now let nu = v, the integral becomes _ _ _ _ | cos nc /n[epsilon] cos v sin nc /n[epsilon] sin v | [phi](c + [theta][epsilon]) | ------- | ----- dv - ------- | ----- dv |; |_ n^(1-K) _/ 0 v^K n^(1-K) _/ 0 v^K _|

hence n^(1-K) [int]([pi] to -[pi]) [f](x) cos nxdx becomes, as n is definitely increased, of the form _ _ _ _ | /[oo] cos v /[oo] sin v | [phi](c) | cos nc | ----- dv - sin nc | ----- dv | |_ _/ 0 v^K _/ 0 v^K _|

which is finite, both the integrals being convergent and of known value. The other integral has a similar property, and we infer that n^(1-K) a_n, n^(1-K) b_n are less than fixed finite numbers.

_The Differentiation of Fourier's Series._--If we assume that the differential coefficient of a function [f](x) represented by a Fourier's Series exists, that function [f]'(x) is not necessarily representable by the series obtained by differentiating the terms of the Fourier's Series, such derived series being in fact not necessarily convergent. Stokes has obtained general formulae for finding the series which represent f'(x), [f]"(x)--the successive differential coefficients of a limited function [f](x). As an example of such formulae, consider the sine series (1); [f](x) is represented by _ 2 n[pi]x /l n[pi]x -- [Sigma] sin ------ | [f](x) sin ------ dx; l l _/0 l _ /l n[pi]x on integration by parts we have | [f](x)sin ------ dx _/0 l _ _ l | n[pi]a | = ---- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------ {[f]([alpha] + 0) - [f]([alpha] - 0)} | n[pi] |_ l _| _ l /l n[pi]x + ----- | [f]'(x) cos ------ dx n[pi] _/0 l

where [alpha] represent the points where [f](x) is discontinuous. Hence if f(x) is represented by the series [Sigma]a_n sin (n[pi]x/l), and [f]'(x) by the series [Sigma]b_n cos (n[pi]x/l), we have the relation _ _ n[pi] 2 | n[pi][alpha] | b_n = ----- [alpha]_n - -- | [f](+0) [+-] [f](l - 0) + [Sigma]cos ------------ {[f]([alpha] + 0) - [f](alpha - 0)} | l l |_ l _|

hence only when the function is everywhere continuous, and [f](+0) [f](l - 0) are both zero, is the series which represents [f]'(x) obtained at once by differentiating that which represents [f](x). The form of the coefficient [alpha]_n discloses the discontinuities of the function and of its differential coefficients, for on continuing the integration by parts we find _ _ 2 | n[pi][alpha] | [alpha]_n = ----- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------------ {[f]([alpha] + 0) - [f]([alpha] - 0)} | n[pi] |_ l _| _ _ 2l | n[pi][beta] | + --------- | [f]'(+0) [+-] [f]'(l - 0) + [Sigma] sin ----------- {[f]'([beta] + 0) - [f]'([beta] - 0)} | + &c. n^2[pi]^2 |_ l _|

where [beta] are the points at which [f]'(x) is discontinuous.

HISTORY AND LITERATURE OF THE THEORY

The history of the theory of the representation of functions by series of sines and cosines is of great interest in connexion with the progressive development of the notion of an arbitrary function of a real variable, and of the peculiarities which such a function may possess; the modern views on the foundations of the infinitesimal calculus have been to a very considerable extent formed in this connexion (see FUNCTION). The representation of functions by these series was first considered in the 18th century, in connexion with the problem of a vibrating cord, and led to a controversy as to the possibility of such expansions. In a memoir published in 1747 (_Memoirs of the Academy of Berlin_, vol. iii.) D'Alembert showed that the ordinate y at any time t of a vibrating cord satisfies a differential equation of the form [delta]^y/[delta]t^2 = a^2 [delta]^y/[delta]x^2, where x is measured along the undisturbed length of the cord, and that with the ends of the cord of length l fixed, the appropriate solution is y = [f](at + x) - [f](at - x), where [f] is a function such that [f](x) = [f](x + 2l); in another memoir in the same volume he seeks for functions which satisfy this condition. In the year 1748 (_Berlin Memoirs_, vol. iv.) Euler, in discussing the problem, gave [f](x) = [alpha] sin [pi]x/l + [beta] sin 2[pi]x/l + ... as a particular solution, and maintained that every curve, whether regular or irregular, must be representable in this form. This was objected to by D'Alembert (1750) and also by Lagrange on the ground that irregular curves are inadmissible. D. Bernoulli (_Berlin Memoirs_, vol. ix., 1753) based a similar result to that of Euler on physical intuition; his method was criticized by Euler (1753). The question was then considered from a new point of view by Lagrange, in a memoir on the nature and propagation of sound (_Miscellanea Taurensia_, 1759; [_OE]uvres_, vol. i.), who, while criticizing Euler's method, considers a finite number of vibrating particles, and then makes the number of them infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli's Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous; a discontinuous function was considered only as a succession of several different functions. Thus the possibility of the expansion of a broken function was not generally admitted. The first cases in which rational functions are expressed in sines and cosines were given by Euler (_Subsidium calculi sinuum_, Novi Comm. Petrop., vol. v., 1754-1755), who obtained the formulae

1/2 [phi] = sin [phi] - 1/2 sin 2[phi] + 1/3 sin 3[phi] ...

[pi]^2 [phi]^2 ------ - ------- = cos [phi] - 1/4 cos 2[phi] + 1/9 cos 3[phi] ... 12 4

In a memoir presented to the Academy of St Petersburg in 1777, but not published until 1798, Euler gave the method afterwards used by Fourier, of determining the coefficients in the expansions; he remarked that if [Phi] is expansible in the form _ _ 1 /[pi] 2 /[pi] A + B cos[phi] + C cos 2[phi] + ..., then A = ---- | [Phi]d[phi], B = ---- | cos [phi]d[phi], &c. [pi] _/ 0 [pi] _/ 0

The second period in the development of the theory commenced in 1807, when Fourier communicated his first memoir on the Theory of Heat to the French Academy. His exposition of the present theory is contained in a memoir sent to the Academy in 1811, of which his great treatise the _Theorie analytique de la chaleur_, published in 1822, is, in the main, a reproduction. Fourier set himself to consider the representation of a function given graphically, and was the first fully to grasp the idea that a single function may consist of detached portions given arbitrarily by a graph. He had an accurate conception of the convergence of a series, and although he did not give a formally complete proof that a function with discontinuities is representable by the series, he indicated in particular cases the method of procedure afterwards carried out by Dirichlet. As an exposition of principles, Fourier's work is still worthy of careful perusal by all students of the subject. Poisson's treatment of the subject, which has been adopted in English works (see the _Journal de l'ecole polytechnique_, vol. xi., 1820, and vol. xii., 1823, and also his treatise, _Theorie de la chaleur_, 1835), depends upon the equality _ /[pi] 1 - h^2 | [f]([alpha]) ------------------------------ d[alpha] _/-[pi] 1 - 2h cos (x - [alpha]) + h^2

_ _ 1 /[pi] 1 /[pi] = ----- | [f]([alpha]) d[alpha] + ---- [Sigma]h^n | [f]([alpha]) cos n(x - [alpha]) d[alpha] 2[pi] _/-[pi] [pi] _/-[pi]

where 0 < h < 1; the limit of the integral on the left-hand side is evaluated when h=1, and found to be 1/2 {[f](x + 0) + [f](x - 0)}, the series on the right-hand side becoming Fourier's Series. The equality of the two limits is then inferred. If the series is assumed to be convergent when h = 1, by a theorem of Abel's its sum is continuous with the sum for values of h less than unity, but a proof of the convergency for h = 1 is requisite for the validity of Poisson's proof; as Poisson gave no such proof of convergency, his proof of the general theorem cannot be accepted. The deficiency cannot be removed except by a process of the same nature as that afterwards applied by Dirichlet. The definite integral has been carefully studied by Schwarz (see two memoirs in his collected works on the integration of the equation [delta]^2u/[delta]x^2 + [delta]^2u/[delta]y^2 = 0), who showed that the limiting value of the integral depends upon the manner in which the limit is approached. Investigations of Fourier's Series were also given by Cauchy (see his "Memoire sur les developpements des fonctions en series periodiques," _Mem. de l'Inst_., vol. vi., also _Oeuvres completes_, vol. vii.); his method, which depends upon a use of complex variables, was accepted, with some modification, as valid by Riemann, but one at least of his proofs is no longer regarded as satisfactory. The first completely satisfactory investigation is due to Dirichlet; his first memoir appeared in _Crelle's Journal_ for 1829, and the second, which is a model of clearness, in Dove's _Repertorium der Physik_. Dirichlet laid down certain definite sufficient conditions in regard to the nature of a function which is expansible, and found under these conditions the limiting value of the sum of n terms of the series. Dirichlet's determination of the sum of the series at a point of discontinuity has been criticized by Schlafli (see _Crelle's Journal_, vol. lxxii.) and by Du Bois-Reymond (_Mathem. Annalen_, vol. vii.), who maintained that the sum is really indeterminate. Their objection appears, however, to rest upon a misapprehension as to the meaning of the sum of the series; if x1 be the point of discontinuity, it is possible to make x approach x1, and n become indefinitely great, so that the sum of the series takes any assigned value in a certain interval, whereas we ought to make x = x1 first and afterwards n = [oo], and no other way of going to the double limit is really admissible. Other papers by Dircksen (_Crelle_, vol. iv.) and Bessel (_Astronomische Nachrichten_, vol. xvi.), on similar lines to those by Dirichlet, are of inferior importance. Many of the investigations subsequent to Dirichlet's have the object of freeing a function from some of the restrictions which were imposed upon it in Dirichlet's proof, but no complete set of necessary and sufficient conditions as to the nature of the function has been obtained. Lipschitz ("De explicatione per series trigonometricas," _Crelle's Journal_, vol. lxiii., 1864) showed that, under a certain condition, a function which has an infinite number of maxima and minima in the neighbourhood of a point is still expansible; his condition is that at the point of discontinuity [beta], |[f]([beta] + [delta]) -f([beta])| < B[delta]^[alpha] as [delta] converges to zero, B being a constant, and a a positive exponent. A somewhat wider condition is

{[f]([beta] + [delta]) - [f]([beta])} log [delta]) = 0, [delta] = 0

for which Lipschitz's results would hold. This last condition is adopted by Dini in his treatise (_Sopra la serie di Fourier_, &c., Pisa, 1880).

The modern period in the theory was inaugurated by the publication by Riemann in 1867 of his very important memoir, written in 1854, _Uber die Darstellbarkeit einer Function durch eine trigonometrische Reihe_. The first part of his memoir contains a historical account of the work of previous investigators; in the second part there is a discussion of the foundations of the Integral Calculus, and the third part is mainly devoted to a discussion of what can be inferred as to the nature of a function respecting the changes in its value for a continuous change in the variable, if the function is capable of representation by a trigonometrical series. Dirichlet and probably Riemann thought that all continuous functions were everywhere representable by the series; this view was refuted by Du Bois-Reymond (_Abh. der Bayer. Akad._ vol. xii. 2). It was shown by Riemann that the convergence or non-convergence of the series at a particular point x depends only upon the nature of the function in an arbitrarily small neighbourhood of the point x. The first to call attention to the importance of the theory of uniform convergence of series in connexion with Fourier's Series was Stokes, in his memoir "On the Critical Values of the Sums of Periodic Series" (_Camb. Phil. Trans._, 1847; _Collected Papers_, vol. i.). As the method of determining the coefficients in a trigonometrical series is invalid unless the series converges in general uniformly, the question arose whether series with coefficients other than those of Fourier exist which represent arbitrary functions. Heine showed (_Crelle's Journal_, vol. lxxi., 1870, and in his treatise _Kugelfunctionen_, vol. i.) that Fourier's Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (_Crelle's Journal_, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the _Math. Ann._ vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. Important contributions to the theory of the series have been published by Du Bois-Reymond (_Abh. der Bayer. Akademie_, vol. xii., 1875, two memoirs, also in Crelle's Journal, vols. lxxiv. lxxvi. lxxix.), by Kronecker (_Berliner Berichte_, 1885), by O. Holder (_Berliner Berichte_, 1885), by Jordan (_Comptes rendus_, 1881, vol. xcii.), by Ascoli (_Math. Annal._, 1873, and _Annali di matematica_, vol. vi.), and by Genocchi (_Atti della R. Acc. di Torino_, vol. x., 1875). Hamilton's memoir on "Fluctuating Functions" (_Trans. R.I.A._, vol. xix., 1842) may also be studied with profit in this connexion. A memoir by Broden (_Math. Annalen_, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier's Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue's work on Fourier's Series will be found in his treatise, _Lecons sur les series trigonometriques_ (1906); also in a memoir, "Sur les series trigonometriques," _Annales sc. de l'ecole normale superieure_, series ii. vol. xx. (1903), and in a paper "Sur la convergence des series de Fourier," _Math. Annalen_, vol. lxiv. (1905).

AUTHORITIES.--The foregoing historical account has been mainly drawn from A. Sachse's work, "Versuch einer Geschichte der Darstellung willkurlicher Functionen einer Variabeln durch trigonometrische Reihen," published in _Schlomilch's Zeitschrift fur Mathematik_, Supp., vol. xxv. 1880, and from a paper by G.A. Gibson "On the History of the Fourier Series" (_Proc. Ed. Math. Soc._ vol. xi.). Reiff's _Geschichte der unendlichen Reihen_ may also be consulted, and also the first part of Riemann's memoir referred to above. Besides Dini's treatise already referred to, there is a lucid treatment of the subject from an elementary point of view in C. Neumann's treatise, _Uber die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen_. Jordan's discussion of the subject in his _Cours d'analyse_ is worthy of attention: an account of functions with limited variation is given in vol. i.; see also a paper by Study in the _Math. Annalen_, vol. xlvii. On the second mean-value theorem papers by Bonnet (Brux. Memoires, vol. xxiii., 1849, _Lionville's Journal_, vol. xiv., 1849), by Du Bois-Reymond (_Crelle's Journal_, vol. lxxix., 1875), by Hankel (_Zeitschrift fur Math. und Physik_, vol. xiv., 1869), by Meyer (_Math. Ann._, vol. vi., 1872) and by Holder (_Gottinger Anzeigen_, 1894) may be consulted; the most general form of the theorem has been given by Hobson (_Proc. London Math. Soc._, Series II. vol. vii., 1909). On the theory of uniform convergence of series, a memoir by W.F. Osgood (_Amer. Journal of Math._ xix.) may be with advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (_Annali di matematica_, Series III. vol. iii.) is of great importance. Bromwich's _Theory of Infinite Series_ (1908) contains much information on the general theory of series. Bocher's "Introduction to the Theory of Fourier's Series," _Annals of Math._, Series II. vol. vii., 1906, will be found useful. See also Carslaw's _Introduction to the Theory of Fourier's Series and Integrals, and the Mathematical Theory of the Conduction of Heat_ (1906). A full account of the theory will be found in Hobson's treatise _On the Theory of Functions of a Real Variable and on the Theory of Fourier's Series_ (1907). (E. W. H.)

FOURMIES, a town of northern France, in the department of Nord, on an affluent of the Sambre, 39 m. S.E. of Valenciennes by rail. Pop. (1906) 13,308. It is one of the chief centres in France for wool combing and spinning, and produces a great variety of cloths. The glass-works of Fourmies date from 1599, and were the first established in the north of France. Iron is worked in the vicinity, and there are important forges and foundries. Enamel-ware is also manufactured. In 1891 labour troubles brought about military intervention and consequent bloodshed. A board of trade arbitration and a school of commerce and industry are among the public institutions.

FOURMONT, ETIENNE (1683-1745), French orientalist, was born at Herbelai, near Saint Denis, on the 23rd of June 1683. He studied at the College Mazarin, Paris, and afterwards in the College Montaigu, where his attention was attracted to Oriental languages. Shortly after leaving the college he published a _Traduction du commentaire du Rabbin Abraham Aben Esra sur l'ecclesiast_e. In 1711 Louis XIV. appointed Fourmont to assist a young Chinese, Hoan-ji, in compiling a Chinese grammar. Hoan-ji died in 1716, and it was not until 1737 that Fourmont published _Meditationes Sinicae_ and in 1742 _Grammatica Sinica_. He also wrote _Reflexions critiques sur les histoires des anciens peuples_ (1735), and several dissertations printed in the _Memoires_ of the Academy of Inscriptions. He became professor of Arabic in the College de France in 1715. In 1713 he was elected a member of the Academy of Inscriptions, in 1738 a member of the Royal Society of London, and in 1742 a member of that of Berlin. He died at Paris on the 19th of December 1745.

His brother, Michel Fourmont (1690-1746), was also a member of the Academy of Inscriptions, and professor of the Syriac language in the Royal College, and was sent by the government to copy inscriptions in Greece.

An account of Etienne Fourmont's life and a catalogue of his works will be found in the second edition (1747) of his _Reflexions critiques_.

FOURNET, JOSEPH JEAN BAPTISTE XAVIER (1801-1869), French geologist and metallurgist, was born at Strassburg on the 15th of May 1801. He was educated at the Ecole des Mines at Paris, and after considerable experience as a mining engineer he was in 1834 appointed professor of geology at Lyons. He was a man of wide knowledge and extensive research, and wrote memoirs on chemical and mineralogical subjects, on eruptive rocks, on the structure of the Jura, the metamorphism of the Western Alps, on the formation of oolitic limestones, on kaolinization and on metalliferous veins. On metallurgical subjects also he was an acknowledged authority; and he published observations on the order of sulphurability of metals (_loi de Fournet_). He died at Lyons on the 8th of January 1869. His chief publications were: E_tudes sur les depots metalliferes_ (Paris, 1834); _Histoire de la dolomie_ (Lyons, 1847); _De l'extension des terrains houillers_ (1855); _Geologie lyonnaise_ (Lyons, 1861).

FOURNIER, PIERRE SIMON (1712-1768), French engraver and typefounder, was born at Paris on the 15th of September 1712. He was the son of a printer, and was brought up to his father's business. After studying drawing under the painter Colson, he practised for some time the art of wood-engraving, and ultimately turned his attention to the engraving and casting of types. He designed many new characters, and his foundry became celebrated not only in France, but in foreign countries. Not content with his practical achievements, he sought to stimulate public interest in his art by the production of various works on the subject. In 1737 he published his _Table des proportions qu'il faut observer entre les caracteres_, which was followed by several other technical treatises. In 1758 he assailed the title of Gutenberg to the honour awarded him as inventor of printing, claiming it for Schoffer, in his _Dissertation sur l'origine et les progres de l'art de graver en bois_. This gave rise to a controversy in which Schopflin and Baer were his opponents. Fournier's contributions to this debate were collected and reprinted under the title of _Traites historiques et critiques sur l'origine de l'imprimerie_. His principal work, however, was the _Manuel typographique_, which appeared in 2 vols. 8vo in 1764, the first volume treating of engraving and type-founding, the second of printing, with examples of different alphabets. It was the author's design to complete the work in four volumes, but he did not live to execute it. He died at Paris on the 8th of October 1768.

FOURNIER L'HERITIER, CLAUDE (1745-1825), French revolutionist, called "l'Americain," was born at Auzon (Haute-Loire) on the 21st of December 1745, the son of a poor weaver. He went to America to seek his fortune, and started at San Domingo an establishment for making _tafia_ (an inferior quality of rum), but lost his money in a fire. Returning to France he threw himself into the Revolution with enthusiasm, and specially distinguished himself by the active part he took in the organization of the popular armed force by means of which the most famous of the revolutionary _coups_ were effected. His influence was principally manifested in the insurrections of the 5th and 6th of October 1789, the 17th of July 1791, and the 20th of June and the 10th of August 1792. He was on bad terms with the majority of the politicians, and particularly with Marat, and spent a great part of his time in prison, all the governments regarding him as an agitator and accusing him of inciting to insurrection. Arrested for the first time for trying to force an entrance into the club of the Cordeliers, from which he had been expelled, he was released, but was in prison from the 12th of December 1793 to the 21st of September 1794, and again from the 9th of March 1795 to the 26th of October 1795. After the attempt on the First Consul in the rue Sainte-Nicaise he was deported to Guiana, but was allowed to return to France in 1809. In 1811, while under surveillance at Auxerre, he was accused of having provoked an _emeute_ against taxes known as the _droits reunis_ (afterwards called c_ontributions indirectes_), and was imprisoned in the Chateau d'If, where he remained till 1814. On the second restoration of the Bourbons Fournier was confined for about nine months in the prison of La Force. After 1816 he was left unmolested, turned royalist, and passed his last years in importuning the Restoration government for compensation for his lost property in San Domingo. He died in obscurity.

For further details see preface to F.A. Aulard's edition of Fournier's _Memoires secrets_ (Paris, 1890), published by the Societe de l'histoire de la Revolution.

FOURTOU, MARIE FRANCOIS OSCAR BARDY DE (1836-1897), French politician, was born at Riberac (Dordogne) on the 3rd of January 1836, and represented his native department in the National Assembly after the Franco-German War. There he proved a useful adherent to Thiers, who made him minister of public works in December 1872. He was minister of religion in the cabinet of May 18-24, 1873, being the only member of the Right included by Thiers in that short-lived ministry. As minister of education, religion and the fine arts in the reconstructed cabinet of the duc de Broglie he had used his administrative powers to further clerical ends, and as minister of the interior in Broglie's cabinet in 1877 he resumed the administrative methods of the Second Empire. With a well-known Bonapartist, Baron R.C.F. Reille, as his secretary, he replaced republican functionaries by Bonapartist partisans, reserving a few places for the Legitimists. In the general elections of that year he used the whole weight of officialdom to secure a majority for the Right, to support a clerical and reactionary programme. He accompanied Marshal MacMahon in his tour through southern France, and the presidential manifesto of September, stating that the president would rely solely on the Senate should the elections prove unfavourable, was generally attributed to Fourtou. In spite of these efforts the cabinet fell, and a commission was appointed to inquire into their unconstitutional abuse of power. Fourtou was unseated in consequence of the revelations made in the report of the commission. In the Chamber of Deputies Gambetta gave the lie direct to Fourtou's allegation that the republican party opposed every republican principle that was not antiquated. A duel was fought in consequence, but neither party was injured. He was re-elected to the chamber in 1879 and entered the Senate the next year. Failing to secure re-election to the Senate in 1885 he again entered the popular chamber as Legitimist candidate in 1889, but he took no further active part in politics. He died in Paris in 1897.

His works include _Histoire de Louis XVI_ (1840); _Histoire de Saint Pie V_ (1845); _Mme Swetchine, sa vie et ses oeuvres_ (2 vols., 1859); _La Question italienne_ (1860); _De la contre-revolution_ (1876); and _Memoires d'un royaliste_ (2 vols., 1888).

FOUSSA, or FOSSA, the native name of _Cryptoprocta ferox_, a somewhat cat-like or civet-like mammal peculiar to Madagascar, where it is the largest carnivorous animal. It is about twice the size of a cat (5 ft. from nose to end of tail), with short close fur of nearly uniform pale brown. Little is known of its habits, except that it is nocturnal, frequently attacks and carries off goats, and especially kids, and shows great ferocity when wounded, on which account it is much dreaded by the natives. An example lived in the London zoological gardens for nearly fourteen years. See CARNIVORA.

FOWEY (usually pronounced _Foy_), a seaport and market-town in the Bodmin parliamentary division of Cornwall, England, on the Great Western railway, 25 m. by sea W. of Plymouth. Pop. (1901) 2258. It lies on the west shore of the picturesque estuary of the river Fowey, close to the water's edge, and sheltered by a screen of hills. Its church of St Nicholas is said to have been built in the 14th century, on the site of a still older edifice dedicated to St Finbar of Cork. It has a fine tower and late Norman doorway. Within are a priest's chamber over the porch, a handsome oak ceiling, a 15th-century pulpit, and some curious monuments and brasses. Place House, adjacent to the church, is a highly ornate Tudor building. A few ancient houses remain in the town. Deep-sea fishing is carried on; but the staple trade consists in the export of china clay and minerals, coal being imported. Fowey harbour, which is easy of access in clear weather, will admit large vessels at any state of the tide. St Catherine's Fort, dating from the days of Henry VIII. and now ruined, stands at the harbour's mouth, and once formed the main defence of the town. Opposite the town, and connected with it by Bodeneck Ferry, is the village of Polruan. Its main features are St Saviour's Chapel, with an ancient rood-stone, and the remains of Hall House, which was garrisoned during the civil wars of the 17th century.

Fowey (Fawy, Vawy, Fowyk) held a leading position amongst Cornish ports from the reign of Edward I. to the days of the Tudors. The numerous references to the privateering exploits of its ships in the Patent and Close Rolls and the extraordinary number of them at the siege of Calais in 1346 alike testify to its importance. During this period the king's mandates were addressed to the bailiffs or to the mayor and bailiffs, and no charter of incorporation appears to have been granted until the reign of James II. Under the second charter of 1690 the common council consisted of a mayor and eight aldermen and these with a recorder elected the free burgesses. A member for Fowey and Looe was summoned to a council at Westminster in 1340, but from that date until 1571, when it was entrusted with the privilege of returning two members, it had no parliamentary representation. By the Reform Act of 1832 it lost both its members. It had ceased to exercise its municipal functions a few years previously. In 1316 the prior of Tywardreath, as lord of the manor, obtained the right to hold a Monday market and two fairs on the feasts of St Finbar and St Lucy, but by the charter of 1690 provision was made for a Saturday market and three fairs, on the 1st of May, 10th of September and Shrove Tuesday, and only these three continue to be held.

FOWL (Dan. _Fugl_, Ger. _Vogel_), a term originally used in the sense that bird[1] now is, but, except in composition,--as sea-fowl, wild-fowl and the like,--practically almost confined[2] at present to designate the otherwise nameless species which struts on our dunghills, gathers round our barn-doors, or stocks our poultry yards--the type of the genus _Gallus_ of ornithologists, of which four well-marked species are known. The _first_ of these is the red jungle-fowl of the greater part of India, _G. ferrugineus_,--called by many writers _G. bankiva_,--which is undoubtedly the parent stock of all the domestic races (cf. Darwin, _Animals and Plants under Domestication_, i. pp. 233-246). It inhabits northern India from Sind to Burma and Cochin China, as well as the Malay Peninsula and many of the islands as far as Timor, besides the Philippines. It occurs on the Himalayas up to the height of 4000 ft., and its southern limits in the west of India proper are, according to Jerdon, found on the Raj-peepla hills to the south of the Nerbudda, and in the east near the left bank of the Godavery, or perhaps even farther, as he had heard of its being killed at Cummum. This species resembles in plumage what is commonly known among poultry-fanciers as the "Black-breasted game" breed, and this is said to be especially the case with examples from the Malay countries, between which and examples from India some differences are observable--the latter having the plumage less red, the ear-lappets almost invariably white, and slate-coloured legs, while in the former the ear-lappets are crimson, like the comb and wattles, and the legs yellowish. If the Malayan birds be considered distinct, it is to them that the name _G. bankiva_ properly applies. This species is said to be found in lofty forests and in dense thickets, as well as in ordinary bamboo-jungles, and when cultivated land is near its haunts, it may be seen in the fields after the crops are cut in straggling parties of from 10 to 20. The crow to which the cock gives utterance morning and evening is just like that of a bantam, never prolonged as in most domestic birds. The hen breeds from January to July, according to the locality; and lays from 8 to 12 creamy-white eggs, occasionally scraping together a few leaves or a little dry grass by way of a nest. The so-called _G. giganteus_, formerly taken by some ornithologists for a distinct species, is now regarded as a tame breed of _G. ferrugineus_ or _bankiva_. The _second_ good species is the grey jungle-fowl, _G. sonnerati_, whose range begins a little to the northward of the limits of the preceding, and it occupies the southern part of the Indian peninsula, without being found elsewhere. The cock has the end of the shaft of the neck-hackles dilated, forming a horny plate, like a drop of yellow sealing-wax. His call is very peculiar, being a broken and imperfect kind of crow, quite unlike that of _G. ferrugineus_ and more like a cackle. The two species where their respective ranges overlap, occasionally interbreed in a wild state, and the present readily crosses in confinement with domestic poultry, but the hybrids are nearly always sterile. The _third_ species is the Sinhalese jungle-fowl, _G. stanleyi_ (the _G. lafayettii_ of some authors), peculiar to Ceylon. This also greatly resembles in plumage some domestic birds, but the cock is red beneath, and has a yellow comb with a red edge and purplish-red cheeks and wattles. He has also a singularly different voice, his crow being dissyllabic. This bird crosses readily with tame hens, but the hybrids are believed to be infertile. The _fourth_ species, _G. varius_ (the _G. furcatus_ of some authors), inhabits Java and the islands eastwards as far as Flores. This differs remarkably from the others in not possessing hackles, and in having a large unserrated comb of red and blue and only a single chin wattle. The predominance of green in its plumage is another easy mark of distinction. Hybrids between this species and domestic birds are often produced, but they are most commonly sterile. Some of them have been mistaken for distinct species, as those which have received the names of _G. aeneus_ and _G. temmincki_.

Several circumstances seem to render it likely that fowls were first domesticated in Burma or the countries adjacent thereto, and it is the tradition of the Chinese that they received their poultry from the West about the year 1400 B.C. By the Institutes of Manu, the tame fowl is forbidden, though the wild is allowed to be eaten--showing that its domestication was accomplished when they were written. The bird is not mentioned in the Old Testament nor by Homer, though he has [Greek: 'Alektor] (cock) as the name of a man, nor is it figured on ancient Egyptian monuments. Pindar mentions it, and Aristophanes calls it the Persian bird, thus indicating it to have been introduced to Greece through Persia, and it is figured on Babylonian cylinders between the 6th and 7th centuries B.C. It is sculptured on the Lycian marbles in the British Museum (c. 600 B.C.), and E. Blyth remarks (Ibis, 1867, p. 157) that it is there represented with the appearance of a true jungle-fowl, for none of the wild _Galli_ have the upright bearing of the tame breed, but carry their tail in a drooping position. For further particulars of these breeds see POULTRY. (A. N.)

FOOTNOTES:

[1] _Bird_ (cognate with _breed_ and _brood_) was originally the young of any animal, and an early Act of the Scottish parliament speaks of "Wolf-birdis," i.e. Wolf-cubs.

[2] Like _Deer_ (Dan. _Dyr_, Ger. _Tier_). _Beast_, too, with some men has almost attained as much specialization.

FOWLER, CHARLES (1792-1867), English architect, was born at Cullompton, Devon, on the 17th of May 1792. After serving an apprenticeship of five years at Exeter, he went to London in 1814, and entered the office of David Laing, where he remained till he commenced practice for himself. His first work of importance was the court of bankruptcy in Basinghall Street, finished in 1821. In the following year he gained the first premium for a design for the new London bridge, which, however, was ultimately built according to the design of another architect. Fowler's other designs for bridges include one constructed across the Dart at Totnes. He was also the architect for the markets of Covent Garden and Hungerford, the new market at Gravesend, and Exeter lower market, and besides several churches he designed Devon lunatic asylum (1845), the London fever hospital (1849), and the hall of the Wax Chandlers' Company, Gresham Street (1853). For some years he was honorary secretary of the institute of British architects, and he was afterwards created vice-president. He retired from his profession in 1853, and died at Great Marlow, Bucks, on the 26th of September 1867.

FOWLER, EDWARD (1632-1714), English divine, was born in 1632 at Westerleigh, Gloucestershire, and was educated at Corpus Christi College, Oxford, afterwards migrating to Trinity College, Cambridge. He was successively rector of Norhill, Bedfordshire (1656) and of All Hallows, Bread Street, London (1673), and in 1676 was elected a canon of Gloucester, his friend Henry More, the Cambridge Platonist, resigning in his favour. In 1681 he became vicar of St Giles, Cripplegate, but after four years was suspended for Whiggism. When the Declaration of Indulgence was published in 1687 he successfully influenced the London clergy against reading it. In 1691 he was consecrated bishop of Gloucester and held the see until his death on the 26th of August 1714. Fowler was suspected of Pelagian tendencies, and his earliest book was a _Free Discourse_ in defence of _The Practices of Certain Moderate Divines called Latitudinarians_ (1670). _The Design of Christianity_, published by him in the following year, in which he laid stress on the moral design of revelation, was criticized by Baxter in his _How far Holiness is the Design of Christianity_ (1671) and by Bunyan in his _Defence of the Doctrine of Justification by Faith_ (1672), the latter describing the _Design_ as "a mixture of Popery, Socinianism and Quakerism," a horrid accusation to which Fowler replied in a scurrilous pamphlet entitled _Dirt Wip'd Off_. He also published, in 1693, _Twenty-Eight Propositions, by which the Doctrine of the Trinity is endeavoured to be explained_, challenging with some success the Socinian position.

FOWLER, JOHN (1826-1864), English inventor, was born at Melksham, Wilts, on the 11th of July 1826. He learned practical engineering at Middlesborough-on-Tees, and about 1850 invented a mechanical system for the drainage of land. In 1852 he began experiments in steam cultivation, and in 1858 the Royal Agricultural Society awarded him the prize of L500 which it had offered for a steam-cultivator that should be an economic substitute for the plough or the spade. In 1860 he founded at Hunslet, Leeds, the firm of Fowler & Co., manufacturers of agricultural machinery, traction engines, &c. He died at Ackworth, Yorkshire, on the 4th of December 1864.

FOWLER, SIR JOHN (1817-1898), English civil engineer, was born on the 15th of July 1817 at Wadsley Hall, near Sheffield, where his father was a land-surveyor. At the age of sixteen he became a pupil of John Towlerton Leather, the engineer of the Sheffield water-works. The latter's uncle, George Leather, was engineer of the Great Aire and Calder Navigation Company, of the Goole Docks, and other similar works, and Fowler passed occasionally into his employment, in which he acquired a thorough knowledge of hydraulic engineering. The era of railway construction soon swept both Fowler and his employers into its service, and one of his first employments was to oppose the route of the Midland railway, chosen by the Stephensons, which left Sheffield on a branch line, and was therefore strongly resented by the inhabitants. The prestige of the Stephensons carried all before it, but in later life Sir John Fowler had the satisfaction of seeing the opposition of his clients justified, and Sheffield placed on the main line. In 1838 he went into the office of John Urpeth Rastrick, one of the leading railway engineers of the day, where he was employed in designing bridges for the line from London to Brighton, and also in surveying for railways in Lancashire. In 1839 he went as representative of Mr Leather to take charge of the construction of the Stockton & Hartlepool railway and remained as manager of the line after it was finished. In 1844 he began his independent career as an engineer, and from the first was largely employed, more particularly in laying out the small railway systems which eventually were amalgamated under the title of the Manchester, Sheffield & Lincolnshire. In the course of this work he designed a bridge known as Torksey Bridge, which was disallowed by the Board of Trade inspector, Captain (afterwards Field-Marshal Sir) Lintorn Simmons. The engineering profession espoused Fowler's side in the controversy which followed, and as a result the verdict of the Board of Trade was modified. The episode was the beginning of a warm friendship between these distinguished representatives of civil and military engineering. Fowler was engineer of the London Metropolitan railway, the pioneer of underground railways, and noteworthy in that it was mostly made not by tunnelling, but by excavating from the surface and then covering in the permanent way; and he lived to be one of the engineers officially connected with the deep tunnelling "tube" system extensively adopted for electric railways in London. He was also engaged in the making of railways in Ireland, and in 1867 he was selected by Disraeli to serve on a commission to advise the government in respect of a proposal for a state-purchase of the Irish railway system. He also carried out considerable works in relation to the Nene Valley drainage and the reclamation of land at the Norfolk estuary.

In 1865 he was elected president of the Institution of Civil Engineers, the youngest president who had ever sat in the chair. He was strongly opposed to the project of a Channel tunnel to France, and in 1872 he endeavoured to obtain the consent of parliament to a Channel ferry scheme, whereby trains were to be transported across the strait in large ferry steamers. The proposal involved the making of enlarged harbours at Dover and Audresselles on the French coast, and the bill, after passing the Commons, was thrown out by the casting vote of the chairman of a committee of the House of Lords. In 1875 he was enabled to render, in his private capacity, a signal service to the Italian government, which was much embarrassed by impracticable proposals pressed on it by Garibaldi for a rectification of the course of the Tiber and other engineering works. He had several interviews with the Italian patriot, and persuaded him of the impracticable nature of his plan, thereby obtaining for the government leisure to devise a more reasonable scheme. For eight years from 1871 he acted as general engineering adviser in Egypt to the Khedive Ismail. He projected a railway to the Sudan, and also the reparation of the barrage. These and many other plans came to an end owing to financial reasons. But the maps and surveys for the railway were given to the war office, and proved most useful to Lord Wolseley in his Nile expedition. For his service Fowler was made K.C.M.G. (1885). He was created a baronet in 1890 on the completion of the Forth bridge, of which with his partner Sir Benjamin Baker he was joint engineer. He died at Bournemouth on the 20th of November 1898.

FOWLER, WILLIAM (c. 1560-1614), Scottish poet, was born about the year 1560. He attended St Leonard's college, St Andrews, between 1574 and 1578, and in 1581 he was in Paris studying civil law. In 1581 he issued a pamphlet against John Hamilton and other Catholics, who had, he said, driven him from his country. He subsequently (about ?1590) became private secretary and Master of Requests to Anne of Denmark, wife of James VI., and was renominated to these offices when the queen went to England. In 1609 his services were rewarded by a grant of 2000 acres in Ulster. His sister Susannah Fowler married Sir John Drummond, and was mother of the poet William Drummond of Hawthornden. On the title-page of _The Triumphs of Petrarke_, Fowler styles himself "P. of Hawick," which has been held to mean that he was parson of Hawick, but this is doubtful. A MS. collection of seventy-two sonnets, entitled _The Tarantula of Love_, and a translation (1587) from the Italian of the _Triumphs of Petrarke_ are preserved in the library of the university of Edinburgh, in the collection bequeathed by his nephew, William Drummond. Two other volumes of his manuscript notes, scrolls of poems, &c., are preserved among the Drummond MSS., now in the library of the Society of Antiquaries of Scotland. Specimens of Fowler's verses were published in 1803 by John Leyden in his _Scottish Descriptive Poems_. Fowler contributed a prefatory sonnet to James VI.'s _Furies_; and James, in return, commended, in verse, Fowler's _Triumphs_.

FOX, CHARLES JAMES (1749-1806), British statesman and orator, was the third son of Henry Fox, 1st Lord Holland, and his wife. Lady Caroline Lennox, eldest daughter of Charles Lennox, 2nd duke of Richmond. He was born at 9 Conduit Street, Westminster, on the 24th of January 1749. The father, who treated his children with extreme indulgence, allowed him to choose his school, and he elected to go to one kept at Wandsworth by a French refugee, named Pampelonne. In a very short time he asked to be sent to Eton, where he went in 1757. At Eton he did no more work than was acceptable to him, but he had an inborn love of literature, and he laid the foundation of that knowledge of the classic languages which in after years was the delight of his life. The vehemence of his temper was controlled by an affectionate disposition. When quite a boy he checked his own tendency to fits of passion on learning that his father trusted him to cure his defects.

That he learnt anything, and that he grew up an amiable and magnanimous man, were solely due to his natural worth, for no one ever owed less to education or to family example. The relations of Lord Holland to his sons would be difficult to parallel. He not only treated them, and in

## particular Charles, as friends and companions in pleasure from the

first, but he did his best to encourage them in dissipation. In 1763 he took Charles for a tour on the continent, introduced him to the most immoral society of the time and gave him money with which to gamble. The boy came back to Eton a precocious rake. It was his good fortune that he did go back, for he was subjected to a wholesome course of ridicule by the other boys, and was flogged by Dr Barnard, the headmaster. In 1764 Charles proceeded to Hertford College, Oxford. At Oxford, as at Eton, he read literature from natural liking, and he paid some attention to mathematics. His often quoted saying that he found mathematics entertaining was probably meant as a jest at the expense of Sir G. Macartney, to whom he was writing, and who was known to maintain that it was useless. His own account of his school and college training, given in a letter to the same correspondent (6th August 1767), is: "I employed almost my whole time at Oxford in the mathematical and classical knowledge, but more particularly in the latter, so that I understand Latin and Greek tolerably well. I am totally ignorant in every part of useful knowledge. I am more convinced every day how little advantage there is in being what at school and the university is called a good scholar: one receives a good deal of amusement from it, but that is all. At present I read nothing but Italian, which I am immoderately fond of,

## particularly of the poetry.... As for French, I am far from being so

thorough a master of it as I could wish, but I know so much of it that I could perfect myself in it at any time with very little trouble, especially if I pass three or four months in France." The passage is characteristic. It shows at once his love of good literature and his thoroughness. Fox's youth was disorderly, but it was never indolent. He was incapable of half doing anything which he did at all. He did perfect himself in French, and he showed no less determination to master mere sports. At a later period when he had grown fat he accounted for his skill in taking "cut balls" at tennis by saying that he was a very "painstaking man." He was all his life a great and steady walker.

The disorders of his early years were notorious, and were a common subject of gossip. In the spring of 1767 he left Oxford and joined his father on the continent during a tour in France and Italy. In 1768 Lord Holland bought the pocket borough of Midhurst for him, and he entered on his parliamentary career, and on London society, in 1769. Within the next few years Lord Holland reaped to the full the reward for all that was good, and whatever was evil, in the training he had given his son. The affection of Charles Fox for his father was unbounded, but the passion for gambling which had been instilled in him as a boy proved the ruin of the family fortune. He kept racehorses, and bet on them largely. On the racecourse he was successful, and it is another proof of his native thoroughness that he gained a reputation as a handicapper. It is said that he won more than he lost on the course. At the gambling table he was unfortunate, and there can be little question that he was fleeced both in London and in Paris by unscrupulous players of his own social rank, who took advantage of his generosity and whose worthlessness he knew. In the ardour of his passion Fox took his losses and their consequences with an attractive gaiety. He called the room in which he did business with the Jew moneylenders his "Jerusalem chamber." When his elder brother had a son, and his prospects were injured, he said that the boy was a second Messiah, who had appeared for the destruction of the Jews. "He had his jest, and they had his estate." In 1774 Lord Holland had to find L140,000 to pay the gambling debts of his sons. For years Charles lived in pecuniary embarrassment, and during his later years, when he had given up gambling, he was supported by the contributions of wealthy friends, who in 1793 formed a fund of L70,000 for his benefit.

His public career did not supply him with a check on habits of dissipation in the shape of the responsibilities of office. He began, as was to be expected in his father's son, by supporting the court; and in 1770, when only twenty-one, he was appointed a junior lord of the admiralty with Lord North. During the violent conflict over the Middlesex election (see WILKES, JOHN) he took the unpopular side, and vehemently asserted the right of the House of Commons to exclude Wilkes. In 1772 during the proceedings against Crosby and Oliver--a part of the "Wilkes and liberty" agitation--he and Lord North were attacked by a mob and rolled in the mud. But Fox's character was incompatible with ministerial service under King George III. The king, himself a man of orderly life, detested him as a gambler and a rake. And Fox was too independent to please a master who expected obedience. In February 1772 he threw up his place to be free to oppose the Royal Marriage Act, on which the king's heart was set. He returned to office as junior lord of the treasury in December. But he was insubordinate; his sympathy with the American colonies, which were now beginning to resist the claims of the mother country to tax them, made him intolerable to the king and he was dismissed in February 1774. The death of his father on the 1st of July of that year removed an influence which tended to keep him subordinate to the court, and his friendship for Burke drew him into close alliance with the Rockingham Whigs. From the first his ability had won him admiration in the House of Commons. He had prepared to distinguish himself as an orator by the elaborate cultivation of his voice, which was naturally harsh and shrill. His argumentative force was recognized at once, but the full scope of his powers was first shown on the 2nd of February 1775, when he spoke on the disputes with the colonies. The speech is unfortunately lost, but Gibbon, who heard it, told his friend Holroyd (afterwards Earl of Sheffield) that Fox, "taking the vast compass of the question before us, discovered powers for regular debate which neither his friends hoped nor his enemies dreaded."

His great political career dates from that day. It is unique among the careers of British statesmen of the first rank, for it was passed almost wholly in opposition. Except for a few months in 1782 and 1783, and again for a few months before his death in 1806, he was out of office. If he was absolutely sincere in the statement he made to his friend Fitzpatrick, in a letter of the 3rd of February 1778, his life was all he could have wished. "I am," he wrote, "certainly ambitious by nature, but I really have, or think I have, totally subdued that passion. I have still as much vanity as ever, which is a happier passion by far, because great reputation I think I may acquire and keep, great situation I never can acquire, nor if acquired keep, without making sacrifices that I never will make." His words show that he judged himself and read the future accurately. Yet it was certainly a cause of bitter disappointment to him that he had to stand by while the country was in his opinion not only misgoverned, but led to ruin. His reputation as an orator and a political critic, which was great from the first and grew as he lived, most assuredly did not console him for his impotence as a statesman. Of the causes which rendered his brilliant capacity useless for the purpose of obtaining practical success the most important, perhaps the only one of real importance, was his personal character. Lord John Russell (afterwards Earl Russell), his friendly biographer, has to confess that Fox might have joined in the confession of Mirabeau: "The public cause suffers for the immoralities of my youth." His reputation as a rake and gambler was so well established at the very beginning of his career that when he was dismissed from office in 1774 there was a general belief among the vulgar that he had been detected in actual theft. His perfect openness, the notoriety of his bankruptcies and of the seizure of his books and furniture in execution, kept him before the world as a model of dissipation. In 1776, when he was leading the resistance to Lord North's colonial policy, he "neither abandoned gaming nor his rakish life. He was seldom in bed before five in the morning nor out of it before two at noon." At the most important crisis of his life in 1783, he almost made an ostentation of disorder and of indifference not only to appearances, but even to decency. Horace Walpole has drawn a picture of him at that time which Lord Holland, Fox's beloved and admiring nephew, speaking from his early recollections of his uncle, confesses has "some justification." Coming from such an authority the certificate may be held to confirm the substantial accuracy of Walpole. "Fox lodged in St James's Street, and as soon as he rose, which was very late, had a levee of his followers and of the gaming club at Brooks's--all his disciples. His bristly black person, and shagged breast quite open and rarely purified by any ablutions, was wrapped in a foul linen nightgown and his bushy hair dishevelled. In these cynic weeds and with Epicurean good humour did he dictate his politics, and in this school did the heir of the empire attend his lessons and imbibe them." That this cynic manner, and Epicurean speech, were only the outside of a manly and generous nature was well known to the personal friends of Fox, and is now universally allowed. But by the bulk of his contemporaries, who could not fail to see the weaknesses he ostentatiously displayed, Fox was, not unnaturally, suspected as being immoral and untrustworthy. Therefore when he came into collision with the will of the king he failed to secure the confidence of the nation which was his only support. Nor ought any critical admirer of Fox to deny that George III. was not wholly wrong when he said that the great orator "was totally destitute of discretion and sound judgment." Fox made many mistakes, due in some cases to vehemence of temperament, and in others only to be ascribed to want of sagacity. That he fought unpopular causes is a very insufficient explanation of his failure as a practical statesman. He could have profited by the reaction which followed popular excitement but for his bad reputation and his want of discretion.

During the eight years between his expulsion from office in 1774 and the fall of Lord North's ministry in March 1782 he may indeed be said to have done one very great thing in politics. He planted the seed of the modern Liberal party as opposed to the pure Whigs. In political allegiance he became a member of the Rockingham party and worked in alliance with the marquis and with Burke, whose influence on him was great. In opposing the attempt to coerce the American colonists, and in assailing the waste and corruption of Lord North's administration, as well as the undue influence of the crown, he was at one with the Rockingham Whigs. During the agitation against corruption, and in favour of honest management of the public money, which was very strong between 1779 and 1782, he and they worked heartily together. It had a considerable effect, and prepared the way for the reforms begun by Burke and continued by Pitt. But if Fox learnt much from Burke he learnt with originality. He declined to accept the revolution settlement as final, or to think with Burke that the constitution of the House of Commons could not be bettered. Fox acquired the conviction that, if the House was to be made an efficient instrument for restraining the interference of the king and for securing good government, it must cease to be filled to a very large extent by the nominees of boroughmongers and the treasury. He became a strong advocate for parliamentary reform. In all ways he was the ardent advocate of what have in later times been known as "Liberal causes," the removal of all religious disabilities and tests, the suppression of private interests which hampered the public good, the abolition of the slave trade, and the emancipation of all classes and races of men from the strict control of authority.

A detailed account of his activity from 1774 to 1782 would entail the mention of every crisis of the American War of Independence and of every serious debate in parliament. Throughout the struggle Fox was uniformly opposed to the coercion of the colonies and was the untiring critic of Lord North. While the result must be held to prove that he was right, he prepared future difficulties for himself by the fury of his language. He was the last man in the world to act on the worldly-wise maxim that an enemy should always be treated as if he may one day be a friend, and a friend as if he might become an enemy. On the 29th of November 1779 Fox was wounded in a duel with Mr William Adam, a supporter of Lord North's whom he had savagely denounced. He assailed Lord North with unmeasured invective, directed not only at his policy but at his personal character, though he well knew that the prime minister was an amiable though pliable man, who remained in office against his own wish, in deference to the king who appealed to his loyalty. When the disasters of the American war had at last made a change of ministry necessary, and the king applied to the Whigs, through the intermediary of Lord Shelburne, Fox made a very serious mistake in persuading the marquess of Rockingham not to insist on dealing directly with the sovereign. The result was the formation of a cabinet belonging, in Fox's own words,

## partly to the king and partly to the country--that is to say, partly of

Whigs who wished to restrain the king, and partly of the king's friends, represented by Lord Shelburne, whose real function was to baffle the Whigs. Dissensions began from the first, and were peculiarly acute between Shelburne and Fox, the two secretaries of state. The old division of duties by which the southern secretary had the correspondence with the colonies and the western powers of Europe, and the northern secretary with the others, had been abolished on the formation of the Rockingham cabinet. All foreign affairs were entrusted to Fox. Lord Shelburne meddled in the negotiations for the peace at Paris. He also persuaded his colleagues to grant some rather scandalous pensions, and Fox's acquiescence in this abuse after his recent agitation against Lord North's waste did him injury. When the marquess of Rockingham died on the 1st of July 1782, and the king offered the premiership to Shelburne, Fox resigned, and was followed by a part of the Rockingham Whigs.

In refusing to serve under Shelburne he was undoubtedly consistent, but his next step was ruinous to himself and his party. On the 14th of February 1783 he formed a coalition with Lord North, based as they declared on "mutual goodwill and confidence." Plausible excuses were made for the alliance, but to the country at large this union, formed with a man whom he had denounced for years, had the appearance of an unscrupulous conspiracy to obtain office on any terms. In the House of Commons the coalition was strong enough to drive Shelburne from office on the 24th of February. The king made a prolonged resistance to the pressure put on him to accept Fox and North as his ministers (see PITT, WILLIAM). On the 2nd of April he was constrained to submit to the formation of a new ministry, in which the duke of Portland was prime minister and Fox and North were secretaries of state. The new administration was ill liked by some of the followers of both. Fox increased its unpopularity both in the House and in the country by consenting against the wish of most of his colleagues to ask for the grant of a sum of L100,000 a year to the prince of Wales. The act had the appearance of a deliberate offence to the king, who was on bad terms with his son. The magnitude of the sum, and his acquiescence in the grant of pensions by the Shelburne ministry, convinced the country that his zeal for economy was hypocritical. The introduction of the India Bill in November 1783 alarmed many vested interests, and offended the king by the provision which gave the patronage of India to a commission to be named by the ministry and removable only by parliament. The coalition, and Fox in particular, were assailed in a torrent of most telling invective and caricature. Encouraged by the growing unpopularity of his ministers, George III. gave it to be understood that he would not look upon any member of the House of Lords who voted for the India Bill as his friend. The bill was thrown out in the upper House on the 17th of December, and next day the king dismissed his ministers.

Fox now went into opposition again. The remainder of his life may be divided into four portions--his opposition to Pitt during the session of 1784; his parliamentary activity till his secession in 1797; his retirement till 1800; his return to activity and his short tenure of office before his death in 1806. During the first of these periods he deepened his unpopularity by assailing the undoubted prerogatives of the crown, by claiming for the House of Commons the right to override not only the king and the Lords but the opinion of the country, and by resisting a dissolution. This last pretension came very ill from a statesman who in 1780 had advocated yearly elections. He lost ground daily before the steady good judgment and unblemished character of Pitt. When parliament was dissolved at the end of the session of 1784, the country showed its sentiments by unseating 180 of the followers of Fox and North. Immense harm was done to both by the publication of a book called _The Beauties of Fox, North and Burke_, a compilation of their abuse of one another in recent years.

Fox himself was elected for Westminster with fewer votes than Admiral Lord Hood, but with a majority over the ministerial candidate, Sir Cecil Wray. The election was marked by an amazing outflow of caricatures and squibs, by weeks of rioting in which Lord Hood's sailors fought pitched battles in St James's Street with Fox's hackney coachmen, and by the intrepid canvassing of Whig ladies. The beautiful duchess of Devonshire (Georgiana Spencer) is said to have won at least one vote for Fox by kissing a shoemaker who had a romantic idea of what constituted a desirable bribe. The high bailiff refused to make a return, and the confirmation of Fox's election was delayed by the somewhat mean action of the ministry. He had, however, been chosen for Kirkwall, and could fight his cause in the House. In the end he recovered damages from the high bailiff. In his place in parliament he sometimes supported Pitt and sometimes opposed him with effect. His criticism on the ministers' bill for the government of India was sound in principle, though the evils he foresaw did not arise. Little excuse can be made for his opposition to Pitt's commercial policy towards Ireland. But as Fox on this occasion aided the vested interests of some English manufacturers he secured a certain revival of popularity. His support of Pitt's Reform Bill was qualified by a just dislike of the ministers' proposal to treat the possession of the franchise by a constituency as a property and not as a trust. His unsuccessful opposition to the commercial treaty with France in 1787 was unwise and most injurious to himself. He committed himself to the proposition that France was the natural enemy of Great Britain, a saying often quoted against him in coming years. It has been excused on the ground that when he said France he meant the aggressive house of Bourbon. A statesman whose words have to be interpreted by an esoteric meaning cannot fairly complain if he is often misunderstood. In 1788 he travelled in Italy, but returned in haste on hearing of the illness of the king. Fox supported the claim of the prince of Wales to the regency as a right, a doctrine which provoked Pitt into declaring that he would "unwhig the gentleman for the rest of his life." The friendship between him and the prince of Wales (see GEORGE IV.) was always injurious to Fox. In 1787 he was misled by the prince's ambiguous assurances into denying the marriage with Mrs Fitzherbert. On discovering that he had been deceived he broke off all relations with the prince for a year, but their alliance was renewed. During these years he was always in favour of whatever measures could be described as favourable to emancipation and to humanity. He actively promoted the impeachment of Warren Hastings, which had the support of Pitt. He was always in favour of the abolition of the slave trade (which he actually effected during his short tenure of office in 1806), of the repeal of the Test Acts, and of concessions to the Roman Catholics, both in Great Britain and in Ireland.

The French Revolution affected Fox profoundly. Together with almost all his countrymen he welcomed the meeting of the states-general in 1789 as the downfall of a despotism hostile to Great Britain. But when the development of the Revolution caused a general reaction, he adhered stoutly to his opinion that the Revolution was essentially just and ought not to be condemned for its errors or even for its crimes. As a natural consequence he was the steady opponent of Pitt's foreign policy, which he condemned as a species of crusade against freedom in the interest of despotism. Between 1790 and 1800 his unpopularity reached its height. He was left almost alone in parliament, and was denounced as the enemy of his country. On the 6th of May 1791 occurred the painful

## scene in the House of Commons, in which Burke renounced his friendship.

In 1792 there was some vague talk of a coalition between him and Pitt, which came to nothing. It should be noted that the scene with Burke took place in the course of the debate on the Quebec Bill, in which Fox displayed real statesmanship by criticizing the division of Upper from Lower Canada, and other provisions of the bill, which in the end proved so injurious as to be unworkable. In this year he carried the Libel Bill. In 1792 his ally, the duke of Portland, and most of his party left him. In 1797 he withdrew from parliament, and only came forward in 1798 to reaffirm the doctrine of the sovereignty of the people at a great Whig dinner. On the 9th of May he was dismissed from the privy council.

The interval of secession was perhaps the happiest in his life. In 1783 he formed a connexion with Elizabeth Bridget Cane, commonly known as Mrs Armstead or Armistead, an amiable and well-mannered woman to whom he was passionately attached. In company with her he established himself at St Anne's Hill near Chertsey in Surrey. In 1795 he married her privately, but did not avow his marriage till 1802. In his letters he spoke of her always as Mrs Armistead, and some of his friends--Mr Coke of Holkham, afterwards Lord Leicester, with whom he stayed every year, being one of them--would not invite her to their houses. It is hard to explain this solitary instance of shabby conduct in a thoroughly generous man towards a person to whom he was unalterably attached and who fully deserved his affection. Fox's time at St Anne's was largely spent in gardening, in the enjoyment of the country, and in correspondence on literary subjects with his nephew, the 3rd Lord Holland, and with Gilbert Wakefield, the editor of Euripides. His letters show that he had a very sincere love for, and an enlightened appreciation of, good literature. Greek and Italian were his first favourites, but he was well read in English literature and in French, and acquired some knowledge of Spanish. His favourite authors were Euripides, Virgil and Racine, whom he defends against the stock criticisms of the admirers of Corneille with equal zeal and insight.

Fox reappeared in parliament to take part in the vote of censure on ministers for declining Napoleon's overtures for a peace. The fall of Pitt's first ministry and the formation of the Addington cabinet, the peace of Amiens, and the establishment of Napoleon as first consul with all the powers of a military despot, seemed to offer Fox a chance of resuming power in public life. The struggle with Jacobinism was over, and he could have no hesitation in supporting resistance to a successful general who ruled by the sword, and who pursued a policy of perpetual aggression. During 1802 he visited Paris in company with his wife. An account of his journey was published in 1811 by his secretary, Mr Trotter, in an otherwise poor book of reminiscence. It gives an attractive picture of Fox's good-humour, and of his enjoyment of the "species of minor comedy which is constantly exhibited in common life." His main purpose in visiting Paris was to superintend the transcription of the correspondence of Barillon, which he needed for his proposed life of James II. The book was never finished, but the fragment he completed was published in 1808, and was translated into French by Armand Carrel in 1846. Fox was not favourably impressed by Napoleon. He saw a good deal of French society, and was himself much admired for his hearty defence of his rival Pitt against a foolish charge of encouraging plots for Napoleon's assassination. On his return he resumed his regular attendance in the House of Commons. The history of the renewal of the war, of the fall of Addington's ministry, and of the formation of Pitt's second administration is so fully dealt with in the article on Pitt (q.v.) that it need not be repeated here.

The death of Pitt left Fox so manifestly the foremost man in public life that the king could no longer hope to exclude him from office. The formation of a ministry was entrusted by the king to Lord Grenville, but when he named Fox as his proposed secretary of state for foreign affairs George III. accepted him without demur. Indeed his hostility seems to a large extent to have died out. A long period of office might now have appeared to lie before Fox, but his health was undermined. Had he lived it may be considered as certain that the war with Napoleon would have been conducted with a vigour which was much wanting during the next few years. In domestic politics Fox had no time to do more than insist on the abolition of the slave trade. He, like Pitt, was compelled to bow to the king's invincible determination not to allow the emancipation of the Roman Catholics. When a French adventurer calling himself Guillet de la Gevrilliere, whom Fox at first "did the honour to take for a spy," came to him with a scheme for the murder of Napoleon, he sent a warning on the 20th of February to Talleyrand. The incident gave him an opportunity for reopening negotiations for peace. A correspondence ensued, and British envoys were sent to Paris. But Fox was soon convinced that the French ministers were playing a false game. He was resolved not to treat apart from Russia, then the ally of Great Britain, nor to consent to the surrender of Sicily, which Napoleon insisted upon, unless full compensation could be obtained for King Ferdinand. The later stages of the negotiation were not directed by Fox, but by colleagues who took over his work at the foreign office when his health began to fail in the summer of 1806. He showed symptoms of dropsy, and operations only procured him temporary relief. After carrying his motion for the abolition of the slave trade on the 10th of June, he was forced to give up attendance in parliament, and he died in the house of the duke of Devonshire, at Chiswick, on the 13th of September 1806. His wife survived him till the 8th of July 1842. No children were born of the marriage. Fox is buried in Westminster Abbey by the side of Pitt.

The striking personal appearance of Fox has been rendered very familiar by portraits and by innumerable caricatures. The latter were no doubt deliberately exaggerated, and yet a comparison between the head of Fox in Sayer's plate "Carlo Khan's triumphal entry into Leadenhall," and in Abbot's portrait, shows that the caricaturist did not depart from the original. Fox was twice painted by Sir Joshua Reynolds, once when young in a group with Lady Sarah Bunbury and Lady Susan Strangeways, and once at full length. A half-length portrait by the German painter, Karl Anton Hickel, is in the National Portrait Gallery, where there is also a terra-cotta bust by Nollekens.

AUTHORITIES.--The materials for a life of Fox were first collected by his nephew, Lord Holland, and were then revised and rearranged by Mr Allen and Lord John Russell. These materials appear as _Memoirs and Correspondence of C.J. Fox_ (London, 1853-1857). On them Lord John Russell based his _Life and Times of C.J. Fox_ (London, 1859-1866); Sir G.O. Trevelyan's _Early History of C.J. Fox_ (London. 1880) brings new evidence; _Charles James Fox, a Political Study_, by J.L. Le B. Hammond (London, 1903), is a series of studies written by an extreme admirer. His _Speeches_ were collected and published in 1815. The newspaper articles (e.g. in _The Times_) published on the occasion of the centenary of his death contain interesting appreciations. See also Lloyd Sanders, _The Holland House Circle_ (1908). (D. H.)

FOX, EDWARD (c. 1496-1538), bishop of Hereford, was born about 1496 at Dursley in Gloucestershire; he is said on very doubtful authority to have been related to Richard Fox (q.v.). From Eton he proceeded to King's College, Cambridge, and after graduating was made secretary to Wolsey. In 1528 he was sent with Gardiner to Rome to obtain from Clement VII. a decretal commission for the trial and decision of the case between Henry VIII. and Catherine of Aragon. On his return he was elected provost of King's College, and in August 1529 was the means of conveying to the king Cranmer's historic advice that he should apply to the universities of Europe rather than to the pope. This introduction led eventually to Cranmer's promotion over Fox's head to the archbishopric of Canterbury. After a brief mission to Paris in October 1529, Fox in January 1530 befriended Latimer at Cambridge and took an

## active part in persuading that university and Oxford to decide in the

king's favour. He was sent to employ similar methods of persuasion at the French universities in 1530-1531, and was also engaged in negotiating a closer league between England and France. In April 1533 he was prolocutor of convocation when it decided against the validity of Henry's marriage with Catherine, and in 1534 published his treatise _De vera differentia regiae potestatis et ecclesiae_ (second ed. 1538, English transl. 1548). Various ecclesiastical preferments were now granted him, including the archdeaconry of Leicester (1531) and the bishopric of Hereford (1535). In 1535-1536 he was sent to Germany to discuss the basis of a political and theological understanding with the Lutheran princes and divines, and had several interviews with Luther, who could not be persuaded of the justice of Henry VIII.'s divorce. The principal result of the mission was the Wittenberg articles of 1536, which had no slight influence on the English Ten Articles of the same year. Bucer dedicated to him in 1536 his _Commentaries on the Gospels_, and Fox's Protestantism was also illustrated by his patronage of Alexander Aless, whom he defended before Convocation. Fox is credited with the authorship of several proverbial sayings, such as "the surest way to peace is a constant preparedness for war" and "time and I will challenge any two in the world." The former at any rate is only a variation of the Latin _si vis pacem, para bellum_, and probably the latter is not more original in Fox than in Philip II., to whom it is usually ascribed. Fox died on the 8th of May 1538 and was buried in the church of St Mary Mounthaw, London. His chief distinction is perhaps that he was the most Lutheran of Henry VIII.'s bishops, and was largely responsible for the Ten Articles of 1536.

See _Letters and Papers of Henry VIII._, vols. iv.-xiv.; Cooper's _Athenae Cantabrigienses_; _Dict. Nat. Biogr._; R.W. Dixon's _Church History_; G. Mentz, _Die Wittenberger Artikel von 1536_ (1905). (A. F. P.)