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diff --git a/37064.txt b/37064.txt new file mode 100644 index 0000000..a6dd17c --- /dev/null +++ b/37064.txt @@ -0,0 +1,21115 @@ +The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition, +Volume 11, Slice 3, by Various + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 + "Frost" to "Fyzabad" + +Author: Various + +Release Date: August 13, 2011 [EBook #37064] + +Language: English + +Character set encoding: ASCII + +*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 11, SL 3 *** + + + + +Produced by Marius Masi, Don Kretz and the Online +Distributed Proofreading Team at https://www.pgdp.net + + + + + + + + +Transcriber's notes: + +(1) Numbers following letters (without space) like C2 were originally + printed in subscript. Letter subscripts are preceded by an + underscore, like C_n. + +(2) Characters following a carat (^) were printed in superscript. + +(3) Side-notes were relocated to function as titles of their respective + paragraphs. + +(4) Macrons and breves above letters and dots below letters were not + inserted. + +(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek + letters. + +(6) RN stands for Real Number symbol and (Pd) for Partial derivative + symbol + +(7) The following typographical errors have been corrected: + + ARTICLE FRUIT: "The covering of the seed is marked i. n is the + nucellus or perisperm, enclosing the embryo-sac es, in which the + endosperm is formed." 'in' amended from 'is'. + + ARTICLE FRUIT: "It occupies the whole cavity of the embryo-sac, or + is formed only at certain portions of it, at the apex, as in + Rhinanthus, ..." 'occupies' amended from 'occupied'. + + ARTICLE FUEL: "Owing to the siliceous nature of the ash of straw, + it is desirable to have a means of clearing the grate bars from + slags and clinkers at short intervals, and to use a steam jet to + clear the tubes from similar deposits." 'straw' amended from + 'sraw'. + + ARTICLE FUEL: "In a research upon the heating power and other + properties of coal for naval use, carried out by the German + admiralty, the results tabulated below were obtained with coals + from different localities." 'from' amended from 'form'. + + ARTICLE FUEL: "C + 2H2O = CO2 + 2H2" '+' amended from '='. + + ARTICLE FULHAM: "The principal thoroughfares are Fulham Palace Road + running S. from Hammersmith, Fulham Road and King's Road, W. from + Chelsea, converging and leading to Putney Bridge over the Thames; + ..." 'converging' amended from 'coverging'. + + ARTICLE FULLER, THOMAS: "Their eldest child, John, baptized at + Broadwindsor by his father, 6th June 1641, was afterwards rector of + Sidney Sussex College, edited the Worthies of England, 1662, and + became rector of Great Wakering, Essex, where he died in 1687." + added 'rector'. + + ARTICLE FUNCTION: "When Taylor's theorem leads to a representation + of the function by means of an infinite series, the function is + said to be "analytic" (cf. S 21)." 'Taylor's' amended from + 'Talyor's'. + + ARTICLE FUNCTION: "or a series which converges uniformly may be + integrated term by term." 'uniformly' amended from 'unformly'. + + ARTICLE FUNGI: "Parasitism.--Some fungi, though able to live as + saprophytes, occasionally enter the body of living plants, and are + thus termed facultative parasites." 'Parasitism' amended from + 'Parasatism'. + + ARTICLE FUNGI: "Recent experiments have shown that the difficulties + of getting orchid seeds to germinate are due to the absence of the + necessary fungus, which must be in readiness to infect the young + seedling immediately after it emerges from the seed." added + 'after'. + + + + + ENCYCLOPAEDIA BRITANNICA + + A DICTIONARY OF ARTS, SCIENCES, LITERATURE + AND GENERAL INFORMATION + + ELEVENTH EDITION + + + VOLUME XI, SLICE III + + FROST to FYZABAD + + + + +ARTICLES IN THIS SLICE: + + + FROST FULMAR + FROSTBITE FULMINIC ACID + FROSTBURG FULTON, ROBERT + FROTHINGHAM, OCTAVIUS BROOKS FULTON (Missouri, U.S.A.) + FROUDE, JAMES ANTHONY FULTON (New York, U.S.A.) + FRUCTOSE FUM + FRUGONI, CARLO INNOCENZIO MARIA FUMARIC AND MALEIC ACIDS + FRUIT FUMAROLE + FRUIT AND FLOWER FARMING FUMIGATION + FRUMENTIUS FUMITORY + FRUNDSBERG, GEORG VON FUNCHAL + FRUSTUM FUNCTION + FRUYTIERS, PHILIP FUNDY, BAY OF + FRY FUNERAL RITES + FRY, SIR EDWARD FUNGI + FRY, ELIZABETH FUNJ + FRYXELL, ANDERS FUNKIA + FUAD PASHA FUNNEL + FUCHOW FUR + FUCHS, JOHANN NEPOMUK VON FURAZANES + FUCHS, LEONHARD FURETIERE, ANTOINE + FUCHSIA FURFOOZ + FUCHSINE FURFURANE + FUCINO, LAGO DI FURIES + FUEL FURLONG + FUENTE OVEJUNA FURNACE + FUENTERRABIA FURNEAUX, TOBIAS + FUERO FURNES + FUERTEVENTURA FURNESS, HORACE HOWARD + FUGGER FURNESS + FUGITIVE SLAVE LAWS FURNISS, HARRY + FUGLEMAN FURNITURE + FUGUE FURNIVALL, FREDERICK JAMES + FUHRICH, JOSEPH VON FURSE, CHARLES WELLINGTON + FUJI FURST, JULIUS + FU-KIEN FURSTENBERG + FUKUI FURSTENWALDE + FUKUOKA FURTH + FULA FURTWANGLER, ADOLF + FULCHER OF CHARTRES FURZE + FULDA FUSARO, LAGO + FULGENTIUS, FABIUS PLANCIADES FUSELI, HENRY + FULGINIAE FUSEL OIL + FULGURITE FUSIBLE METAL + FULHAM FUSILIER + FULK (king of Jerusalem) FUSION + FULK (archbishop of Reims) FUSSEN + FULKE, WILLIAM FUST, JOHANN + FULK NERRA FUSTEL DE COULANGES, NUMA DENIS + FULLEBORN, GEORG GUSTAV FUSTIAN + FULLER, ANDREW FUSTIC + FULLER, GEORGE FUTURES + FULLER, MARGARET FUX, JOHANN JOSEPH + FULLER, MELVILLE WESTON FUZE + FULLER, THOMAS FYNE, LOCH + FULLER, WILLIAM FYRD + FULLER'S EARTH FYT, JOHANNES + FULLERTON, LADY GEORGIANA FYZABAD + + + + +FROST (a common Teutonic word, cf. Dutch, _vorst_, Ger. _Frost_, from +the common Teutonic verb meaning "to freeze," Dutch, _vriezen_, Ger. +_frieren_; the Indo-European root is seen in Lat. _pruina_, hoar-frost, +cf. _prurire_, to itch, burn, _pruna_, burning coal, Sansk. _plush_, to +burn), in meteorology, the act, or agent of the process, of freezing; +hence the terms "hoar-frost" and "white-frost" applied to visible frozen +vapour formed on exposed surfaces. A frost can only occur when the +surface temperature falls below 32 deg. F., the freezing-point of water; +if the temperature be between 28 deg. and 32 deg. it is a "light frost," +if below 28 deg. it is a "heavy," "killing" or "black frost"; the term +"black frost" is also used when no hoar-frost is present. The number of +degrees below freezing-point is termed "degrees of frost." As soon as a +mass of air is cooled to its dew-point, water begins to be precipitated +in the form of rain, dew, snow or hail. Hoarfrost is only formed at the +immediate surface of the land if the latter be at a temperature below 32 +deg., and this may occur even when the temperature of the air a few feet +above the ground is 12 deg.-16 deg. above the freezing-point. The +heaviest hoar-frosts are formed under weather conditions similar to +those under which the heaviest summer dews occur, namely, clear and calm +nights, when there is no cloud to impede the radiation of heat from the +surface of the land, which thereby becomes rapidly and completely +cooled. The danger of frost is minimized when the soil is very moist, as +for example after 10-12 mm. of rain; and it is a practice in America to +flood fields on the receipt of a frost warning, radiation being checked +by the light fog sheets which develop over moist soils, just as a +cloud-layer in the upper atmosphere impedes radiation on a grand scale. +A layer of smoke will also impede radiation locally, and to this end +smoky fires are sometimes lit in such positions that the smoke may drift +over planted ground which it is desirable to preserve from frost. +Similarly, frost may occur in open country when a town, protected by its +smoke-cloud above, is free of it. In a valley with fairly high and steep +flanks frost sometimes occurs locally at the bottom, because the layer +of air cooled by contact with the cold surface of the higher ground is +heavier than that not so cooled, and therefore tends to flow or settle +downwards along the slope of the land. When meteorological +considerations point to a frost, an estimate of the night temperature +may be obtained by multiplying the difference between the readings of +the wet and dry bulb thermometer by 2.5 and subtracting the result from +the dry bulb temperature. This rule applies when the evening air is at +about 50 deg. and 30.1 in. pressure, the sky being clear. An instrument +has been devised in France for the prediction of frost. It consists of a +wet bulb and a dry bulb thermometer, mounted on a board on which is also +a scale of lines corresponding to degrees of the dry bulb, and a pointer +traversing a scale graduated according to degrees of the wet bulb. +Observations for the night are taken about half an hour before sunset. +By means of the pointer and scale, the point may be found at which the +line of the dry-bulb reading meets the pointer set to the reading of the +wet bulb. The scale is further divided by colours so that the observed +point may fall within one of three zones, indicating certain frost, +probable frost or no probability of frost. + + + + +FROSTBITE, a form of mortification (q.v.), due to the action of extreme +cold in cutting off the blood-supply from the fingers, toes, nose, ears, +&c. In comparatively trifling forms it occurs as "chaps" and +"chilblains," but the term frostbite is usually applied only to more +severe cases, where the part affected becomes in danger of gangrene. An +immediate application of snow, or ice-water, will restore the +circulation; the application of heat would cause inflammation. But if +the mortification has gone too far for the circulation to be restored, +the part will be lost, and surgical treatment may be necessary. + + + + +FROSTBURG, a town of Allegany county, Maryland, U.S.A., 11 m. W. of +Cumberland. Pop. (1890) 3804; (1900) 5274 (578 foreign-born and 236 +negroes); (1910) 6028. It is served by the Cumberland & Pennsylvania +railway and the Cumberland & Westernport electric railway. The town is +about 2000 ft. above sea-level on a plateau between the Great Savage and +Dans mountains, and its delightful scenery and air have made it +attractive as a summer resort. It is the seat of the second state normal +school, opened in 1904. Frostburg is in the midst of the coal region of +the state, and is itself almost completely undermined; it has planing +mills and manufactures large quantities of fire-brick. The municipality +owns and operates its waterworks. Natural gas is piped to Frostburg from +the West Virginia fields, 120 m. away. Frostburg was first settled in +1812; was called Mount Pleasant until about 1830, when the present name +was substituted in honour of Meshech Frost, one of the town's founders; +and was incorporated in 1870. + + + + +FROTHINGHAM, OCTAVIUS BROOKS (1822-1895), American clergyman and author, +was born in Boston on the 26th of November 1822, son of Nathaniel +Langdon Frothingham (1793-1870), a prominent Unitarian preacher of +Boston, and through his mother's family related to Phillips Brooks. He +graduated from Harvard College in 1843 and from the Divinity School in +1846. He was pastor of the North Unitarian church of Salem, +Massachusetts, in 1847-1855. From 1855 to 1860 he was pastor of a new +Unitarian society in Jersey City, where he gave up the Lord's Supper, +thinking that it ministered to self-satisfaction; and it was as a +radical Unitarian that he became pastor of another young church in New +York City in 1860. Indeed in 1864 he was recognized as leader of the +radicals after his reply to Dr Hedge's address to the graduating +students of the Divinity School on _Anti-Supernaturalism in the Pulpit_. +In 1865, when he had practically given up "transcendentalism," his +church building was sold and his congregation began to worship in Lyric +Hall under the name of the Independent Liberal Church; in 1875 they +removed to the Masonic Temple, but four years later ill-health compelled +Frothingham's resignation, and the church dissolved. Paralysis +threatened him and he never fully recovered his health; in 1881 he +returned to Boston, where he died on the 27th of November 1895. To this +later period of his life belongs his best literary work. While he was in +New York he was for a time art critic of the _Tribune_. Always himself +on the unpopular side and an able but thoroughly fair critic of the +majority, he habitually under-estimated his own worth; he was not only +an anti-slavery leader when abolition was not popular even in New +England, and a radical and rationalist when it was impossible for him to +stay conveniently in the Unitarian Church, but he was the first +president of the National Free Religious Association (1867) and an early +and ardent disciple of Darwin and Spencer. To his radical views he was +always faithful. It is a mistake to say that he grew more conservative +in later years; but his judgment grew more generous and catholic. He was +a greater orator than man of letters, and his sermons in New York were +delivered to large audiences, averaging one thousand at the Masonic +Temple, and were printed each week; in eloquence and in the charm of his +spoken word he was probably surpassed in his day by none save George +William Curtis. Personally he seemed cold and distant, partly because of +his impressive appearance, and partly because of his own modesty, which +made him backward in seeking friendships. + + His principal published works are: _Stories from the Life of the + Teacher_ (1863), _A Child's Book of Religion_ (1866), and other works + of religious teaching for children; several volumes of sermons; + _Beliefs of Unbelievers_ (1876), _The Cradle of the Christ: a Study in + Primitive Christianity_ (1877), _The Spirit of New Faith_ (1877), _The + Rising and the Setting Faith_ (1878), and other expositions of the + "new faith" he preached; _Life of Theodore Parker_ (1874), + _Transcendentalism in New England_ (1876), which is largely + biographical, _Gerrit Smith, a Biography_ (1878), _George Ripley_ + (1882), in the "American Men of Letters" series, _Memoir of William + Henry Channing_ (1886), _Boston Unitarianism, 1820-1850_ (1890), + really a biography of his father; and _Recollections and Impressions, + 1822-1890_ (1891). + + + + +FROUDE, JAMES ANTHONY (1818-1894), English historian, son of R.H. +Froude, archdeacon of Totnes, was born at Dartington, Devon, on the 23rd +of April 1818. He was educated at Westminster and Oriel College, Oxford, +then the centre of the ecclesiastical revival. He obtained a second +class and the chancellor's English essay prize, and was elected a fellow +of Exeter College (1842). His elder brother, Richard Hurrell Froude +(1803-1836), had been one of the leaders of the High Church movement at +Oxford. Froude joined that party and helped J.H. Newman, afterwards +cardinal, in his _Lives of the English Saints_. He was ordained deacon +in 1845. By that time his religious opinions had begun to change, he +grew dissatisfied with the views of the High Church party, and came +under the influence of Carlyle's teaching. Signs of this change first +appeared publicly in his _Shadows of the Clouds_, a volume containing +two stories of a religious sort, which he published in 1847 under the +pseudonym of "Zeta," and his complete desertion of his party was +declared a year later in his _Nemesis of Faith_, an heretical and +unpleasant book, of which the earlier part seems to be autobiographical. + +On the demand of the college he resigned his fellowship at Oxford, and +mainly at least supported himself by writing, contributing largely to +_Fraser's Magazine_ and the _Westminster Review_. The excellence of his +style was soon generally recognized. The first two volumes of his +_History of England from the Fall of Wolsey to the Defeat of the Spanish +Armada_ appeared in 1856, and the work was completed in 1870. As an +historian he is chiefly remarkable for literary excellence, for the art +with which he represents his conception of the past. He condemns a +scientific treatment of history and disregards its philosophy. He held +that its office was simply to record human actions and that it should be +written as a drama. Accordingly he gives prominence to the personal +element in history. His presentations of character and motives, whether +truthful or not, are undeniably fine; but his doctrine that there should +be "no theorizing" about history tended to narrow his survey, and +consequently he sometimes, as in his remarks on the foreign policy of +Elizabeth, seems to misapprehend the tendencies of a period on which he +is writing. + +Froude's work is often marred by prejudice and incorrect statements. He +wrote with a purpose. The keynote of his _History_ is contained in his +assertion that the Reformation was "the root and source of the expansive +force which has spread the Anglo-Saxon race over the globe." Hence he +overpraises Henry VIII. and others who forwarded the movement, and +speaks too harshly of some of its opponents. So too, in his _English in +Ireland_ (1872-1874), which was written to show the futility of attempts +to conciliate the Irish, he aggravates all that can be said against the +Irish, touches too lightly on English atrocities, and writes unjustly of +the influence of Roman Catholicism. A strong anti-clerical prejudice is +manifest in his historical work generally, and is doubtless the result +of the change in his views on Church matters and his abandonment of the +clerical profession. Carlyle's influence on him may be traced both in +his admiration for strong rulers and strong government, which led him to +write as though tyranny and brutality were excusable, and in his +independent treatment of character. His rehabilitation of Henry VIII. +was a useful protest against the idea that the king was a mere +sanguinary profligate, but his representation of him as the self-denying +minister of his people's will is erroneous, and is founded on the false +theory that the preambles of the acts of Henry's parliaments represented +the opinions of the educated laymen of England. As an advocate he +occasionally forgets that sobriety of judgment and expression become an +historian. He was not a judge of evidence, and seems to have been +unwilling to admit the force of any argument or the authority of any +statement which militated against his case. In his _Divorce of Catherine +of Aragon_ (1891) he made an unfortunate attempt to show that certain +fresh evidence on the subject, brought forward by Dr Gairdner, Dr +Friedmann and others, was not inconsistent with the views which he has +expressed in his _History_ nearly forty years before. He worked +diligently at original manuscript authorities at Simancas, the Record +Office and Hatfield House; but he used his materials carelessly, and +evidently brought to his investigation of them a mind already made up as +to their significance. His _Life of Caesar_ (1879), a glorification of +imperialism, betrays an imperfect acquaintance with Roman politics and +the life of Cicero; and of his two pleasant books of travel, _The +English in the West Indies_ (1888) shows that he made little effort to +master his subject, and _Oceana_ (1886), the record of a tour in +Australia and New Zealand, among a multitude of other blunders, notes +the prosperity of the working-classes in Adelaide at the date of his +visit, when, in fact, owing to a failure in the wheat-crop, hundreds +were then living on charity. He was constitutionally inaccurate, and +seems to have been unable to represent the exact sense of a document +which lay before him, or even to copy from it correctly. Historical +scholars ridiculed his mistakes, and Freeman, the most violent of his +critics, never let slip a chance of hitting at him in the _Saturday +Review_. Froude's temperament was sensitive, and he suffered from these +attacks, which were often unjust and always too savage in tone. The +literary quarrel between him and Freeman excited general interest when +it blazed out in a series of articles which Freeman wrote in the +_Contemporary Review_ (1878-1879) on Froude's _Short Study_ of Thomas +Becket. + +Notwithstanding its defects, Froude's _History_ is a great achievement; +it presents an important and powerful account of the Reformation period +in England, and lays before us a picture of the past magnificently +conceived, and painted in colours which will never lose their freshness +and beauty. As with Froude's work generally, its literary merit is +remarkable; it is a well-balanced and orderly narrative, coherent in +design and symmetrical in execution. Though it is perhaps needlessly +long, the thread of the story is never lost amid a crowd of details; +every incident is made subordinate to the general idea, appears in its +appropriate place, and contributes its share to the perfection of the +whole. The excellence of its form is matched by the beauty of its style, +for Froude was a master of English prose. The most notable +characteristic of his style is its graceful simplicity; it is never +affected or laboured; his sentences are short and easy, and follow one +another naturally. He is always lucid. He was never in doubt as to his +own meaning, and never at a loss for the most appropriate words in which +to express it. Simple as his language is, it is dignified and worthy of +its subject. Nowhere perhaps does his style appear to more advantage +than in his four series of essays entitled _Short Studies on Great +Subjects_ (1867-1882), for it is seen there unfettered by the +obligations of narrative. Yet his narrative is admirably told. For the +most part flowing easily along, it rises on fit occasions to splendour, +picturesque beauty or pathos. Few more brilliant pieces of historical +writing exist than his description of the coronation procession of Anne +Boleyn through the streets of London, few more full of picturesque power +than that in which he relates how the spire of St Paul's was struck by +lightning; and to have once read is to remember for ever the touching +and stately words in which he compares the monks of the London +Charterhouse preparing for death with the Spartans at Thermopylae. +Proofs of his power in the sustained narration of stirring events are +abundant; his treatment of the Pilgrimage of Grace, of the sea fight at +St Helens and the repulse of the French invasion, and of the murder of +Rizzio, are among the most conspicuous examples of it. Nor is he less +successful when recording pathetic events, for his stories of certain +martyrdoms, and of the execution of Mary queen of Scots, are told with +exquisite feeling and in language of well-restrained emotion. And his +characters are alive. We may not always agree with his portraiture, but +the men and women whom he saw exist for us instinct with the life with +which he endows them and animated by the motives which he attributes to +them. His successes must be set against his failures. At the least he +wrote a great history, one which can never be disregarded by future +writers on his period, be their opinions what they may; which attracts +and delights a multitude of readers, and is a splendid example of +literary form and grace in historical composition. + +The merits of his work met with full recognition. Each instalment of his +_History_, in common with almost everything which he wrote, was widely +read, and in spite of some adverse criticisms was received with eager +applause. In 1868 he was elected rector of St Andrews University, +defeating Disraeli by a majority of fourteen. He was warmly welcomed in +the United States, which he visited in 1872, but the lectures on Ireland +which he delivered there caused much dissatisfaction. On the death of +his adversary Freeman in 1892, he was appointed, on the recommendation +of Lord Salisbury, to succeed him as regius professor of modern history +at Oxford. Except to a few Oxford men, who considered that historical +scholarship should have been held to be a necessary qualification for +the office, his appointment gave general satisfaction. His lectures on +Erasmus and other 16th-century subjects were largely attended. With some +allowance for the purpose for which they were originally written, they +present much the same characteristics as his earlier historical books. +His health gave way in the summer of 1894, and he died on the 20th of +October. + +His long life was full of literary work. Besides his labours as an +author, he was for fourteen years editor of _Fraser's Magazine_. He was +one of Carlyle's literary executors, and brought some sharp criticism +upon himself by publishing Carlyle's _Reminiscences_ and the _Memorials +of Jane Welsh Carlyle_, for they exhibited the domestic life and +character of his old friend in an unpleasant light. Carlyle had given +the manuscripts to him, telling him that he might publish them if he +thought it well to do so, and at the close of his life agreed to their +publication. Froude therefore declared that in giving them to the world +he was carrying out his friend's wish by enabling him to make a +posthumous confession of his faults. Besides publishing these +manuscripts he wrote a _Life of Carlyle_. His earlier study of Irish +history afforded him suggestions for a historical novel entitled _The +Two Chiefs of Dunboy_ (1889). In spite of one or two stirring scenes it +is a tedious book, and its personages are little more than machines for +the enunciation of the author's opinions and sentiments. Though Froude +had some intimate friends he was generally reserved. When he cared to +please, his manners and conversation were charming. Those who knew him +well formed a high estimate of his ability in practical affairs. In 1874 +Lord Carnarvon, then colonial secretary, sent Froude to South Africa to +report on the best means of promoting a confederation of its colonies +and states, and in 1875 he was again sent to the Cape as a member of a +proposed conference to further confederation. Froude's speeches in South +Africa were rather injudicious, and his mission was a failure (see SOUTH +AFRICA: _History_). He was twice married. His first wife, a daughter of +Pascoe Grenfell and sister of Mrs Charles Kingsley, died in 1860; his +second, a daughter of John Warre, M.P. for Taunton, died in 1874. + + Froude's _Life_, by Herbert Paul, was published in 1905. (W. Hu.) + + + + +FRUCTOSE, LAEVULOSE, or FRUIT-SUGAR, a carbohydrate of the formula +C6H12O6. It is closely related to ordinary d-glucose, with which it +occurs in many fruits, starches and also in honey. It is a hydrolytic +product of inulin, from which it may be prepared; but it is more usual +to obtain it from "invert sugar," the mixture obtained by hydrolysing +cane sugar with sulphuric acid. Cane sugar then yields a syrupy mixture +of glucose and fructose, which, having been freed from the acid and +concentrated, is mixed with water, cooled in ice and calcium hydroxide +added. The fructose is precipitated as a saccharate, which is filtered, +suspended in water and decomposed by carbon dioxide. The liquid is +filtered, the filtrate concentrated, and the syrup so obtained washed +with cold alcohol. On cooling the fructose separates. It may be obtained +as a syrup, as fine, silky needles, a white crystalline powder, or as a +granular crystalline, somewhat hygroscopic mass. When anhydrous it melts +at about 95 deg. C. It is readily soluble in water and in dilute alcohol, +but insoluble in absolute alcohol. It is sweeter than cane sugar and is +more easily assimilated. It has been employed under the name diabetin as +a sweetening agent for diabetics, since it does not increase the +sugar-content of the urine; other medicinal applications are in phthisis +(mixed with quassia or other bitter), and for children suffering from +tuberculosis or scrofula in place of cane sugar or milk-sugar. + +Chemically, fructose is an oxyketone or ketose, its structural formula +being CH2OH.(CH.OH)3.CO.CH2OH; this result followed from its conversion +by H. Kiliani into methylbutylacetic acid. The form described above is +_laevo_-rotatory, but it is termed d-fructose, since it is related to +d-glucose. Solutions exhibit mutarotation, fresh solutions having a +specific rotation of -104.0 deg., which gradually diminishes to -92 deg. +It was synthesized by Emil Fischer, who found the synthetic sugar which +he named [alpha]-acrose to be (d + l)-fructose, and by splitting this +mixture he obtained both the d and l forms. Fructose resembles d-glucose +in being fermentable by yeast (it is the one ketose which exhibits this +property), and also in its power of reducing alkaline copper and silver +solutions; this latter property is assigned to the readiness with which +hydroxyl and ketone groups in close proximity suffer oxidation. For the +structural (stereochemical) relations of fructose see SUGAR. + + + + +FRUGONI, CARLO INNOCENZIO MARIA (1692-1768), Italian poet, was born at +Genoa on the 21st of November 1692. He was originally destined for the +church and at the age of fifteen, in opposition to his strong wishes, +was shut up in a convent; but although in the following year he was +induced to pronounce monastic vows, he had no liking for this life. He +acquired considerable reputation as an elegant writer both of Latin and +Italian prose and verse; and from 1716 to 1724 he filled the chairs of +rhetoric at Brescia, Rome, Genoa, Bologna and Modena successively, +attracting by his brilliant fluency a large number of students at each +university. Through Cardinal Bentivoglio he was recommended to Antonio +Farnese, duke of Parma, who appointed him his poet laureate; and he +remained at the court of Parma until the death of Antonio, after which +he returned to Genoa. Shortly afterwards, through the intercession of +Bentivoglio, he obtained from the pope the remission of his monastic +vows, and ultimately succeeded in recovering a portion of his paternal +inheritance. After the peace of Aix-la-Chapelle he returned to the court +of Parma, and there devoted the later years of his life chiefly to +poetical composition. He died on the 20th of December 1768. As a poet +Frugoni was one of the best of the school of the Arcadian Academy, and +his lyrics and pastorals had great facility and elegance. + + His collected works were published at Parma in 10 vols. in 1799, and a + more complete edition appeared at Lucca in the same year in 15 vols. A + selection from his works was published at Brescia in 1782, in 4 vols. + + + + +FRUIT (through the French from the Lat. _fructus_; _frui_, to enjoy), in +its widest sense, any product of the soil that can be enjoyed by man or +animals; the word is so used constantly in the Bible, and extended, as a +Hebraism, to offspring or progeny of man and of animals, in such +expressions as "the fruit of the body," "of the womb," "fruit of thy +cattle" (Deut. xxviii. 4), &c., and generally to the product of any +action or effort. Between this wide and frequently figurative use of the +word and its application in the strict botanical sense treated below, +there is a popular meaning, regarding the objects denoted by the word +entirely from the standpoint of edibility, and differentiating them +roughly from those other products of the soil, which, regarded +similarly, are known as vegetables. In this sense "fruit" is applied to +such seed-envelopes of plants as are edible, either raw or cooked, and +are usually sweet, juicy or of a refreshing flavour. But applications of +the word in this sense are apt to be loose and shifting according to the +fashion of the time. + +Fruit, in the botanical sense, is developed from the flower as the +result of fertilization of the ovule. After fertilization various +changes take place in the parts of the flower. Those more immediately +concerned in the process, the anther and stigma, rapidly wither and +decay, while the filaments and style often remain for some time; the +floral envelopes become dry, the petals fall, and the sepals are either +deciduous, or remain persistent in an altered form; the ovary becomes +enlarged, forming the _pericarp_; and the ovules are developed as the +seeds, containing the embryo-plant. The term fruit is strictly applied +to the mature pistil or ovary, with the seeds in its interior; but it +often includes other parts of the flower, such as the bracts and floral +envelopes. Thus the fruit of the hazel and oak consists of the ovary +enveloped by the bracts; that of the apple and pear, of the ovary and +floral receptacle; and that of the pine-apple, of the whole +inflorescence. Such fruits are sometimes distinguished as _pseudocarps_. +In popular language, the fruit includes all those parts which exhibit a +striking change as the result of fertilization. In general, the fruit is +not ripened unless fertilization has been effected; but cases occur as +the result of cultivation in which the fruit swells and becomes to all +appearance perfect, while no seeds are produced. Thus, there are +seedless oranges, grapes and pineapples. When the ovules are +unfertilized, it is common to find that the ovary withers and does not +come to maturity; but in the case of bananas, plantains and bread-fruit, +the non-development of seeds seems to lead to a larger growth and a +greater succulence of fruit. + + The fruit, like the ovary, may be formed of a single carpel or of + several. It may have one cell or cavity, being _unilocular_; or many, + _multilocular_, &c. The number and nature of the divisions depend on + the number of carpels and the extent to which their edges are folded + inwards. The appearances presented by the ovary do not always remain + permanent in the fruit. Great changes are observed to take place, not + merely as regards the increased size of the ovary, its softening or + hardening, but also in its internal structure, owing to the + suppression, additional formation or enlargement of parts. Thus, in + the ash (fig. 1) an ovary with two cells, each containing an ovule + attached to a central placenta, is changed into a unilocular fruit + with one seed; one ovule becomes abortive, while the other, g, + gradually enlarging until the septum is pushed to one side, unites + with the walls of the cell, and the placenta appears to be parietal. + In the oak and hazel, an ovary with three and two cells respectively, + and two ovules in each, produces a one-celled fruit with one seed. In + the coco-nut, a trilocular and triovular ovary produces a one-celled, + one-seeded fruit. This abortion may depend on the pressure caused by + the development of certain ovules, or it may proceed from + non-fertilization of all the ovules and consequent non-enlargement of + the carpels. Again, by the growth of the placenta, or the folding + inwards of parts of the carpels, divisions occur in the fruit which + did not exist in the ovary. In _Cathartocarpus Fistula_ a one-celled + ovary is changed into a fruit having each of its seeds in a separate + cell, in consequence of spurious dissepiments being produced + horizontal from the inner wall of the ovary. In flax (_Linum_) by the + folding inwards of the back of the carpels a five-celled ovary becomes + a ten-celled fruit. In _Astragalus_ the folding inwards of the dorsal + suture converts a one-celled ovary into a two-celled fruit; and in + _Oxytropis_ the folding of the ventral suture gives rise to a similar + change. The development of cellular or pulpy matter, and the + enlargement of parts not forming whorls of the flower, frequently + alter the appearance of the fruit, and render it difficult to discover + its formation. In the gooseberry (fig. 29), grape, guava, tomato and + pomegranate, the seeds nestle in pulp formed by the placentas. In the + orange the pulpy matter surrounding the seeds is formed by succulent + cells, which are produced from the inner partitioned lining of the + pericarp. In the strawberry the receptacle becomes succulent, and + bears the mature carpels on its convex surface (fig. 2); in the rose + there is a fleshy hollow receptacle which bears the carpels on its + concave surface (fig. 3). In the juniper the scaly bracts grow up + round the seeds and become succulent, and in the fig (fig. 4) the + receptacle becomes succulent and encloses an inflorescence. + + [Illustration: + + Fig. 1.--Samara or winged fruit of Ash (_Fraxinus_). 1, Entire, with + its wing a; 2, lower portion cut transversely, to show that it + consists of two cells; one of which, l, is abortive, and is reduced + to a very small cavity, while the other is much enlarged and filled + with a seed g. + + Fig. 2.--Fruit of the Strawberry (_Fragaria vesca_), consisting of + an enlarged succulent receptacle, bearing on its surface the small + dry seed-like fruits (achenes). (After Duchartre.) + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + Fig. 3.--Fruit of the Rose cut vertically. s', Fleshy hollowed + receptacle; s, persistent sepals; _fr_, ripe carpels; e, stamens, + withered. + + Fig. 4.--Peduncle of Fig (_Ficus Carica_), ending in a hollow + receptacle enclosing numerous male and female flowers. + + Fig. 5.--Fruit of Cherry (_Prunus Cerasus_) in longitudinal section. + ep, Epicarp; m, mesocarp; en, endocarp. + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer.] + + The pericarp consists usually of three layers, the external, or + _epicarp_ (fig. 5, ep); the middle, or _mesocarp_, m; and the + internal, or _endocarp_, _en_. These layers are well seen in such a + fruit as the peach, plum or cherry, where they are separable one from + the other; in them the epicarp forms what is commonly called the skin; + the mesocarp, much developed, forms the flesh or pulp, and hence has + sometimes been called _sarcocarp_; while the endocarp, hardened by the + production of woody cells, forms the _stone_ or _putamen_ immediately + covering the kernel or seed. The pulpy matter found in the interior of + fruits, such as the gooseberry, grape and others, is formed from the + placentas, and must not be confounded with the sarcocarp. In some + fruits, as in the nut, the three layers become blended together and + are indistinguishable. In bladder senna (_Colutea arborescens_) the + pericarp retains its leaf-like appearance, but in most cases it + becomes altered both in consistence and in colour. Thus in the date + the epicarp is the outer brownish skin, the pulpy matter is the + mesocarp or sarcocarp, and the thin papery-like lining is the endocarp + covering the hard seed. In the medlar the endocarp becomes of a stony + hardness. In the melon the epicarp and endocarp are very thin, while + the mesocarp forms the bulk of the fruit, differing in texture and + taste in its external and internal parts. The rind of the orange + consists of epicarp and mesocarp, while the endocarp forms partitions + in the interior, filled with pulpy cells. The part of the pericarp + attached to the peduncle is the base, and the point where the style or + stigma existed is the apex. This latter is not always the apparent + apex, as in the case of the ovary; it may be lateral or even basilar. + The style sometimes remains in a hardened form, rendering the fruit + _apiculate_; at other times it falls off, leaving only traces of its + existence. The presence of the style or stigma serves to distinguish + certain single-seeded pericarps from seeds. + + [Illustration: + + FIG. 6.--Seed-vessel or capsule of Campion, opening by ten teeth at + the apex. The calyx c is seen surrounding the seed-vessel. + + FIG. 7.--Capsule of Poppy, opening by pores p, under the radiating + peltate stigma s.] + + + Dehiscence of fruits. + + When the fruit is mature and the seeds are ripe, the carpels usually + give way either at the ventral or dorsal suture or at both, and so + allow the seeds to escape. The fruit in this case is _dehiscent_. But + some fruits are _indehiscent_, falling to the ground entire, and the + seeds eventually reaching the soil by their decay. By dehiscence the + pericarp becomes divided into different pieces, or _valves_, the fruit + being univalvular, bivalvular or multivalvular, &c., according as + there are one, two or many valves. The splitting extends the whole + length of the fruit, or is partial, the valves forming teeth at the + apex, as in the order Caryophyllaceae (fig. 6). Sometimes the valves + are detached only at certain points, and thus dehiscence takes place + by pores at the apex, as in poppy (fig. 7), or at the base, as in + _Campanula_. Indehiscent fruits are either dry, as the nut, or fleshy, + as the cherry and apple. They are formed of one or several carpels. In + the former case they usually contain only a single seed, which may + become so incorporated with the pericarp as to appear to be naked, as + in the grain of wheat and generally in grasses. In such cases the + presence of the remains of style or stigma determines their true + nature. + + [Illustration: + + FIG. 8.--Dry dehiscent fruit. The pod (legume) of the Pea; r, the + dorsal suture; b, the ventral; c, calyx; s, seeds. + + From Vines' _Students' Text-Book of Botany_, by permission of Swan + Sonnenschein & Co. + + FIG. 9.--(1) Fruit or capsule of Meadow Saffron (_Colchicum + autumnale_), dehiscing along the septa (septicidally); (2) same cut + across, showing the three chambers with the seeds attached along the + middle line (axile placentation). + + FIG. 10.--Diagram to illustrate the septicidal dehiscence in a + pentalocular capsule. The loculaments l correspond to the number of + the carpels, which separate by splitting through the septa, s. + + FIG. 11.--The seed vessel (capsule) of the Flower-de-Luce (_Iris_), + opening in a loculicidal manner. The three valves bear the septa in + the centre, and the opening takes place through the back of the + loculaments. Each valve is formed by the halves of contiguous + carpels. + + FIG. 12.--Diagram to illustrate loculicidal dehiscence. The + loculaments l, split at the back, and the valves separate, bearing + the septa s on their centres. + + FIG. 13.--Diagram to illustrate septifragal dehiscence, in which the + dehiscence takes place through the back of the loculaments l, and + the valves separate from the septa s, which are left attached to the + placentas in the centre.] + + Dehiscent fruits, when composed of single carpels, may open by the + ventral suture only, as in the paeony, hellebore, _Aquilegia_ (fig. + 28) and _Caltha_; by the dorsal suture only, as in magnolias and some + _Proteaceae_, or by both together, as in the pea (fig. 8) and bean; in + these cases the dehiscence is _sutural_. When composed of several + united carpels, two types of dehiscence occur--a longitudinal and a + transverse. In the longitudinal the separation may take place by the + dissepiments throughout their length, so that the fruit is resolved + into its original carpels, and each valve represents a carpel, as in + rhododendron, _Colchicum_, &c.; this dehiscence, in consequence of + taking place through the septum, is called _septicidal_ (figs. 9, 10). + The valves separate from their commissure, or central line of union, + carrying the placentas with them, or they leave the latter in the + centre, so as to form with the axis a column of a cylindrical, conical + or prismatic shape. Dehiscence is _loculicidal_ when the union between + the edges of the carpels is persistent, and they dehisce by the dorsal + suture, or through the back of the loculaments, as in the lily and + iris (figs. 11, 12). In these cases each valve consists of a half of + each of two contiguous carpels. The placentas either remain united to + the axis, or they separate from it, being attached to the septa on the + valves. When the outer walls of the carpels break off from the septa, + leaving them attached to the central column, the dehiscence is said to + be _septifragal_ (fig. 13), and where, as in _Linum catharticum_ and + _Calluna_, the splitting takes place first of all in a septicidal + manner, the fruit is described as _septicidally septifragal_; while in + other cases, as in thorn apple (_Datura Stramonium_), where the + splitting is at first loculicidal, the dehiscence is _loculicidally + septifragal_. In all those forms the separation of the valves takes + place either from above downwards or from below upwards. In + _Saxifraga_ a splitting for a short distance of the ventral sutures of + the carpels takes place, so that a large apical pore is formed. In the + fruit of Cruciferae, as wallflower (fig. 14), the valves separate from + the base of the fruit, leaving a central _replum_, or frame, which + supports the false septum formed by a prolongation from the parietal + placentas on opposite sides of the fruit, extending between the + ventral sutures of the carpels. In Orchidaceae (fig. 15) the pericarp, + when ripe, separates into three valves in a loculicidal manner, but + the midribs of the carpels, to which the placentas are attached, often + remain adherent to the axis both at the apex and base after the valves + bearing the seeds have fallen. The other type of dehiscence is + transverse, or _circumscissile_, when the upper part of the united + carpels falls off in the form of a lid or operculum, as in _Anagallis_ + and in henbane (_Hyoscyamus_) (fig. 16). + + [Illustration: + + FIG. 14.--Siliqua or seed-vessel of Wallflower (_Cheiranthus + Cheiri_), opening by two valves, which separate from the base + upwards, leaving the seeds attached to the dissepiment which is + supported by the replum. + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 15.--Capsule of an Orchid (_Xylobium_). v, valve. + + FIG. 16.--Seed-vessel of _Anagallisarvensis_, opening by + circumscissile dehiscence. + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 17.--Lomentum of _Hedysarum_ which, when ripe, separates + transversely into single-seeded portions or mericarps. + + FIG. 18.--Fruit of _Geranium pratense_, after splitting.] + + Sometimes the axis is prolonged beyond the base of the carpels, as in + the mallow and castor-oil plant, the carpels being united to it + throughout their length by their faces, and separating from it without + opening. In the Umbelliferae the two carpels separate from the lower + part of the axis, and remain attached by their apices to a + prolongation of it, called a _carpophore_ or _podocarp_, which splits + into two (fig. 25) and suspends them; hence the fruit is termed a + _cremocarp_, which divides into two _mericarps_. The general term + _schizocarp_ is applied to all dry fruits, which break up into two or + more one-seeded indehiscent mericarps, as in _Hedysarum_ (fig. 17). In + the order Geraniaceae the styles remain attached to a central column, + and the mericarps separate from below upwards, before dehiscing by + their ventral suture (fig. 18). Carpels which separate one from + another in this manner are called _cocci_. They are well seen in the + order Euphorbiaceae, where there are usually three such carpels, and + the fruit is termed tricoccus. In many of them, as _Hura crepitans_, + the cocci separate with great force and elasticity. In many leguminous + plants, such as _Ornithopus_, _Hedysarum_ (fig. 17), _Entada_, + _Coronilla_ and the gum-arabic plant (_Acacia arabica_), the fruit + becomes a schizocarp by the formation of transverse partitions from + the folding in of the sides of the pericarp, and distinct separations + taking place at these partitions. + + Fruits are formed by one flower, or are the product of several flowers + combined. In the former case they are either _apocarpous_, of one + mature carpel or of several separate free carpels; or _syncarpous_, of + several carpels, more or less completely united. When the fruit is + composed of the ovaries of several flowers united, it is usual to find + the bracts and floral envelopes also joined with them, so as to form + one mass; hence such fruits are known as multiple, confluent or + _anthocarpous_. The term simple is applied to fruits which are formed + by the ovary of a single flower, whether they are composed of one or + several carpels, and whether these carpels are separate or combined. + + [Illustration: From Vines' _Students' Text-Book of Botany_, by + permission of Swan Sonnenschein & Co. + + FIG. 19.--Dry one-seeded fruit of dock (_Rumex_) cut vertically. ov, + Pericarp formed from ovary wall; s, seed; e, endosperm; pl, embryo + with radicle pointing upwards and cotyledons downwards--enlarged. + + FIG. 20.--Achene of _Ranunculus arvensis_ in longitudinal section; + e, endosperm; pl, embryo. (After Baillon, enlarged.) + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 21.--Fruit of Common Sycamore (_Acer Pseudoplatanus_), dividing + into two mericarps m; s, pedicel; fl, wings (nat. size).] + + + Dispersal of fruit or seed. + + The object of the fruit in the economy of the plant is the protection + and nursing of the developing seed and the dispersion of the ripe + seeds. Hence, generally, one-seeded fruits are indehiscent, while + fruits containing more than one seed open to allow of the dispersal of + the seeds over as wide an area as possible. The form, colour, + structure and method of dehiscence of fruits and the form of the + contained seeds are intimately associated with the means of dispersal, + which fall into several categories. (1) By a mechanism residing in the + fruit. Thus many fruits open suddenly when they are dry, and the seeds + are ejected by the twisting or curving of the valves, or in some other + way; e.g. in gorse, by the spiral curving of the valves; in + _Impatiens_, by the twisting of the cocci; in squirting cucumber, by + the pressure exerted on the pulpy contents by the walls of the + pericarp. (2) By aid of various external agencies such as water. + Fruits or seeds are sometimes sufficiently buoyant to float for a long + time on sea- or fresh-water; e.g. coco-nut, by means of its thick, + fibrous coat (mesocarp), is carried hundreds of miles in the sea, the + tough, leathery outer coat (epicarp) preventing it from becoming + water-soaked. Fruits and seeds of West Indian plants are thrown up on + the coasts of north-west Europe, having been carried by the Gulf + Stream, and will often germinate; many are rendered buoyant by + air-containing cavities, and the embryo is protected from the seawater + by the tough coat of fruit or seed. Water-lily seeds are surrounded + with a spongy tissue when set free from the fruit, and float for some + distance before dropping to the bottom. (3) The most general agent in + the dispersal of seeds is the wind or currents of air--the fruit or + seed being rendered buoyant by wing-developments as in fruits of ash + (fig. 1) or maple (fig. 21), seeds of pines and firs, or many members + of the order Bignoniaceae; or hair-developments as in fruits of + clematis, where the style forms a feathery appendage, fruits of many + Compositae (dandelion, thistle, &c.), which are crowned by a plumose + pappus, or seeds of willow and poplar, or _Asclepias_ (fig. 36), which + bear tufts of silky hairs; to this category belong bladder-like + fruits, such as bladder-senna, which are easily rolled by the wind, or + cases like the so-called rose of Jericho, a small cruciferous plant + (_Anastatica hierocuntica_), where the plant dries up after developing + its fruits and becomes detached from the ground; the branches curl + inwards, and the whole plant is rolled over the dry ground by the + wind. The wind also aids the dispersal of the seeds in the case of + fruits which open by small teeth (many Caryophyllaceae [fig. 6]) or + pores (poppy [fig. 7], _Campanula_, &c.); the seeds are in these cases + small and numerous, and are jerked through the pores when the + capsules, which are generally borne on long, dry stems or stalks, are + shaken by the wind. (4) In other cases members of the animal world aid + in seed-dispersal. Fruits often bear stiff hairs or small hooks, which + cling to the coat of an animal or the feathers of a bird; such are + fruits of cleavers (_Galium Aparine_), a common hedge-row plant, + _Ranunculus arvensis_ (fig. 20), carrot, _Geum_, &c.; or the fruit or + seed has an often bright-coloured, fleshy covering, which is sought by + birds as food, as in stone-fruits such as plum, cherry (fig. 5), &c., + where the seed is protected from injury in the mouth or stomach of the + animal by the hard endocarp; or the hips of the rose (fig. 3), where + the succulent scarlet "fruit" (the swollen receptacle) envelops a + number of small dry true fruits (achenes), which cling by means of + stiff hairs to the beak of the bird. + + [Illustration: + + FIG. 22.--Vertical section of a grain of wheat, showing embryo below + at the base of the endosperm e; s, scutellum separating embryo from + endosperm; f.l, foliage leaf; p.s, sheath of plumule; p.r, primary + root; s.p.r, sheath of primary root. + + FIG. 23.--Fruit of Comfrey (_Symphytum_) surrounded by persistent + calyx, c. The style s appears to arise from the base of the carpels, + enlarged. + + FIG. 24.--Ovary of _Foeniculum officinale_ with pendulous ovules, in + longitudinal section. (After Berg and Schmidt, magnified.) + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 25.--Fruit of _Carum Carui_. A, Ovary of the flower; B, ripe + fruit. The two carpels have separated so as to form two mericarps + (m). Part of the septum constitutes the carpophore (a). p, Top of + flower-stalk; d, disk on top of ovary; n, stigma. + + From Vines' _Students' Text-Book of Botany_, by permission of Swan + Sonnenschein & Co.] + + + Forms of fruit. + + Simple fruits have either a _dry_ or _succulent_ pericarp. The + _achene_ is a dry, one-seeded, indehiscent fruit, the pericarp of + which is closely applied to the seed, but separable from it. It is + solitary, forming a single fruit, as in the dock (fig. 19) and in the + cashew, where it is supported on a fleshy peduncle; or _aggregate_, as + in _Ranunculus_ (fig. 20), where several achenes are placed on a + common elevated receptacle. In the strawberry the achenes (fig. 2) are + aggregated on a convex succulent receptacle. In the rose they are + supported on a concave receptacle (fig. 3), and in the fig the + succulent receptacle completely encloses the achenes (fig. 4). In + _Dorstenia_ the achenes are situated on a flat or slightly concave + receptacle. Hence what in common language are called the seeds of the + strawberry, rose and fig, are in reality ripe carpels. The styles + occasionally remain attached to the achenes in the form of feathery + appendages, as in _Clematis_. In Compositae, the fruit is an inferior + achene (_cypsela_), to which the pappus (modified calyx) remains + adherent. Such is also the nature of the fruit in Dipsacaceae (e.g. + scabious). When the pericarp is thin, and appears like a bladder + surrounding the seed, the achene is termed a _utricle_, as in + Amarantaceae. When the pericarp is extended in the form of a winged + appendage, a _samara_ or _samaroid achene_ is produced, as in the ash + (fig. 1) and common sycamore (fig. 21). In these cases there are + usually two achenes united, one of which, however, as in _Fraxinus_ + (fig. 1), may be abortive. The wing surrounds the fruit longitudinally + in the elm. When the pericarp becomes so incorporated with the seed as + to be inseparable from it, as in grains of wheat (fig. 22), maize, + oats and other grasses, then the name _caryopsis_ is given. The + one-seeded portions (mericarps) of schizocarps often take the form of + achenes, e.g. the mericarps of the mallows or of umbellifers (figs. + 24, 25). In Labiatae and Boraginaceae (e.g. comfrey, fig. 23), where + the bicarpellary ovary becomes our one-seeded portions in the fruit, + the partial fruits are of the nature of achenes or nutlets according + to the texture (leathery or hard) of the pericarp. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 26.--Cupule of _Quercus Aegilops_. cp, cupule; gl, fruit. + (After Duchartre.)] + + The _nut_ or _glans_ is a dry one-celled indehiscent fruit with a + hardened pericarp, often surrounded by bracts at the base, and, when + mature, containing only one seed. In the young state the ovary often + contains two or more ovules, but only one comes to maturity. It is + illustrated by the fruits of the hazel and chestnut, which are covered + by leafy bracts, in the form of a _husk_, and by the acorn, in which + the bracts and receptacle form a _cupula_ or _cup_ (fig. 26). The + parts of the pericarp of the nut are united so as to appear one. In + common language the term nut is very vaguely applied both to fruit and + seeds. + + The _drupe_ is a succulent usually one-seeded indehiscent fruit, with + a pericarp easily distinguishable into epicarp, mesocarp and endocarp. + This term is applied to such fruits as the cherry (fig. 5), peach, + plum, apricot or mango. The endocarp is usually hard, forming the + stone (putamen) of the fruit, which encloses the kernel or seed. The + mesocarp is generally pulpy and succulent, so as to be truly a + sarcocarp, as in the peach, but it is sometimes of a tough texture, as + in the almond, and at other times is more or less fibrous, as in the + coco-nut. In the almond there are often two ovules formed, only one of + which comes to perfection. In the raspberry and bramble several small + drupes or _drupels_ are aggregated so as to constitute an _etaerio_. + + The _follicle_ is a dry unilocular many-seeded fruit, formed from one + carpel and dehiscing by the ventral suture. It is rare to meet with a + solitary follicle forming the fruit. There are usually several + aggregated together, either in a whorl on a shortened receptacle, as + in hellebore, aconite, larkspur, columbine (figs. 27, 28) or the order + Crassulaceae, or in a spiral manner on an elongated receptacle, as in + _Magnolia_ and _Banksia_. Occasionally, follicles dehisce by the + dorsal suture, as in _Magnolia grandiflora_ and _Banksia_. + + [Illustration: + + FIG. 27.--Fruit of Columbine (_Aquilegia_), formed of five + follicles. + + FIG. 28.--Single follicle, showing dehiscence by the ventral suture. + + FIG. 29.--Transverse section of berry of Gooseberry, showing the + seeds attached to the parietal placentas and immersed in pulp, which + is formed partly from the endocarp, partly from the seed-coat. + + FIG. 30.--Section of the fruit of the Apple (_Pyrus Malus_), or + pome, consisting of a fleshy covering formed by the floral + receptacle and the true fruit or core with five cavities with + seeds.] + + The _legume_ or _pod_ is a dry monocarpellary unilocular many-seeded + fruit, formed from one carpel, dehiscing both by the ventral and the + dorsal suture. It characterizes leguminous plants, as the bean and pea + (fig. 8). In the bladder-senna it forms an inflated legume. In some + Leguminosae, as _Arachis_, _Cathartocarpus Fistula_ and the tamarind, + the fruit must be considered a legume, although it does not dehisce. + The first of these plants produces its fruit underground, and is + called earth-nut; the second has a partitioned legume and is + schizocarpic; and both the second and third have pulpy matter + surrounding the seeds. Some legumes are schizocarpic by the formation + of constrictions externally. Such a form is the _lomentum_ or + _lomentaceous legume_ of _Hedysarum_ (fig. 17), _Coronilla_, + _Ornithopus_, _Entada_ and of some Acacias. In _Medicago_ the legume + ~~ is twisted like a snail, and in _Caesalpinia coriaria_, or + Divi-divi, it is vermiform or curved like a worm. Sometimes the number + of seeds is reduced, as in _Erythrina monosperma_ and _Geoffroya + superba_, which are one-seeded, and in _Pterocarpus_ and _Dalbergia_, + which are two-seeded. + + [Illustration: FIG. 31.--Transverse section of the fruit of the Melon + (_Cucumis Melo_), showing the placentas with the seeds attached to + them. The three carpels forming the pepo are separated by partitions. + From the centre processes pass outwards, ending in the curved + placenta. + + The _berry_ (_bacca_) is a term applied generally to all fruits with + seeds immersed in pulp, and includes fruits of very various origin. In + _Actaea_ (baneberry) or _Berberis_ (barberry) it is derived from a + single free carpel; generally, however, it is the product of a + syncarpous ovary, which is superior, as in grape or potato, or + inferior, as in gooseberry (fig. 29) or currant. In the pomegranate + there is a peculiar baccate many-celled inferior fruit, having a tough + rind, enclosing two rows of carpels placed one above the other. The + seeds are immersed in pulp, and are attached irregularly to the wall, + base and centre of the loculi. In the baobab there is a multilocular + syncarpous fruit, in which the seeds are immersed in pulp. + + The _pepo_, another indehiscent syncarpous fruit, is illustrated by + the fruit of the gourd, melon (fig. 31) and other Cucurbitaceae. It is + formed of three carpels, surmounted by the calyx; the rind is thick + and fleshy, and there are three or more seed-bearing parietal + placentas, either surrounding a central cavity or prolonged inwards + into it. The fruit of the papaw resembles the pepo, but the calyx is + not superior. + + The _hesperidium_ is the name given to such indehiscent fleshy + syncarpous fruits as the orange, lemon and shaddock, in which the + epicarp and mesocarp form a separable rind, and the endocarp sends + prolongations inwards, forming triangular divisions, to the inner + angle of which the seeds are attached, pulpy cells being developed + around them from the wall. Both pepo and hesperidium may be considered + as modifications of the berry. + + The _pome_ (fig. 30), seen in the apple, pear, quince, medlar and + hawthorn, is a fleshy indehiscent syncarpous fruit, in the formation + of which the receptacle takes part. The outer succulent part is the + swollen receptacle, the horny core being the true fruit developed from + the usually five carpels and enclosing the seeds. In the medlar the + core (or true pericarp) is of a stony hardness, while the outer + succulent covering is open at the summit. The pome somewhat resembles + the fruit of the rose (fig. 3), where the succulent receptacle + surrounds a number of separate achenes. + + The name _capsule_ is applied generally to all dry syncarpous fruits, + which dehisce by valves. It may thus be unilocular or multilocular, + one- or many-seeded. The true valvular capsule is observed in + _Colchicum_ (fig. 9), lily and iris (fig. 11). The _porose capsule_ is + seen in the poppy (fig. 7), _Antirrhinum_ and _Campanula_. In + _Campanula_ the pores occur at the base of the capsule, which becomes + inverted when ripe. When the capsule opens by a lid, or by + circumscissile dehiscence, it is called a _pyxidium_, as in pimpernel + (_Anagallis arvensis_) (fig. 16), henbane and monkey-pot (_Lecythis_). + The capsule assumes a screw-like form in _Helicteres_, and a star-like + form in star-anise (_Illicium anisatum_). In certain instances the + cells of the capsule separate from each other, and open with + elasticity to scatter the seeds. This kind of capsule is met with in + the sandbox tree (_Hura crepitans_) and other Euphorbiaceae, where the + cocci, containing each a single seed, burst asunder with force; and in + Geraniaceae, where the cocci, each containing, when mature, usually + one seed, separate from the carpophore, become curved upwards by their + adherent styles, and open by the ventral suture (fig. 18). + + The _siliqua_ is a dry syncarpous bilocular many-seeded fruit, formed + from two carpels, with a false septum, dehiscing by two valves from + below upwards, the valves separating from the placentas and leaving + them united by the septum (fig. 32). The seeds are attached on both + sides of the septum, either in one row or in two. When the fruit is + long and narrow it is a _siliqua_ (fig. 14); when broad and short, + _silicula_ (fig. 33). It occurs in cruciferous plants, as wallflower, + cabbage and cress. In _Glaucium_ and _Eschscholtzia_ (Papaveraceae) + the dissepiment is of a spongy nature. It may become transversely + constricted (_lomentaceous_), as in radish (_Raphanus_) and sea-kale, + and it may be reduced, as in woad (_Isatis_), to a one-seeded + condition. + + It sometimes happens that the ovaries of two flowers unite so as to + form a double fruit (_syncarp_). This may be seen in many species of + honeysuckle. But the fruits which are now to be considered consist + usually of the floral envelopes, as well as the ovaries of several + flowers united into one, and are called _multiple_ or _confluent_. The + term _anthocarpous_ has also been applied as indicating that the + floral envelopes as well as the carpels are concerned in the formation + of the fruit. + + The _sorosis_ is a succulent multiple fruit formed by the confluence + of a spike of flowers, as in the fruit of the pine-apple (fig. 34), + the bread-fruit and jack-fruit. Similarly the fruit of the mulberry + represents a catkin-like inflorescence. + + The _syconus_ is an anthocarpous fruit, in which the receptacle + completely encloses numerous flowers and becomes succulent. The fig + (fig. 4) is of this nature, and what are called its seeds are the + achenes of the numerous flowers scattered over the succulent hollowed + receptacle. In _Dorstenia_ the axis is less deeply hollowed, and of a + harder texture, the fruit exhibiting often very anomalous forms. + + The _strobilus_, or _cone_, is a seed-bearing spike, more or less + elongated, covered with scales, each of which may be regarded as + representing a separate flower, and has often two seeds at its base; + the seeds are naked, no ovary being present. This fruit is seen in the + cones of firs, spruces, larches and cedars, which have received the + name of Coniferae, or cone-bearers, on this account. Cone-like fruit + is also seen in most Cycadaceae. The scales of the strobilus are + sometimes thick and closely united, so as to form a more or less + angular and rounded mass, as in the cypress; while in the juniper they + become fleshy, and are so incorporated as to form a globular fruit + like a berry. The dry fruit of the cypress and the succulent fruit of + the juniper have received the name of _galbulus_. In the hop the fruit + is called also a strobilus, but in it the scales are thin and + membranous, and the seeds are not naked but are contained in + pericarps. + + [Illustration: + + FIG. 32.--Honesty (_Lunaria biennis_), showing the septum after the + carpels have fallen away. + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 33.--Silicula or pouch of shepherd's purse (_Capsella_), + opening by two folded valves, which separate from above downwards. + The partition is narrow, hence the silicula is angustiseptal. + + From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav + Fischer. + + FIG. 34.--Fruit of the pine-apple (_Ananassa sativa_), developed + from a spike of numerous flowers with bracts, united so as to form a + collective or anthocarpous fruit. The crown of the pine-apple, c, + consists of a series of empty bracts prolonged beyond the fruit.] + + The same causes which produce alterations in the other parts of the + flower give rise to anomalous appearances in the fruit. The carpels, + in place of bearing seeds, are sometimes changed into leaves, with + lobes at their margins. Leaves are sometimes produced from the upper + part of the fruit. In the genus _Citrus_, to which the orange and + lemon belong, it is very common to meet with a separation of the + carpels, so as to produce what are called horned oranges and fingered + citrons. In this case a syncarpous fruit has a tendency to become + apocarpous. In the orange we occasionally find a supernumerary row of + carpels produced, giving rise to the appearance of small and imperfect + oranges enclosed within the original one; the navel orange is of this + nature. It sometimes happens that, by the union of flowers, double + fruits are produced. Occasionally a double fruit is produced, not by + the incorporation of two flowers, but by the abnormal development of a + second carpel in the flower. + + _Arrangement of Fruits._ + + A. True fruits--developed from the ovary alone. + 1. Pericarp not fleshy or fibrous. + i. Indehiscent--not opening to allow the escape of the + seeds--generally one-seeded. Achene; caryopsis; cypsela; + nut; schizocarp. + ii. Dehiscent--the pericarp splits to allow the escape of the + seeds--generally many-seeded. Follicle; legume; siliqua; + capsule. + 2. Pericarp generally differentiated into distinct layers, one + of which is succulent or fibrous. Drupe; berry. + B. Pseudocarps--the development extends beyond the ovary. Pome; + syconus; sorosis. + + _The Seed._--The _seed_ is formed from the ovule as the result of + fertilization. It is contained in a seed-vessel formed from the ovary + in the plants called _angiospermous_; while in _gymnospermous_ + plants, such as Coniferae and Cycadaceae, it is naked, or, in other + words, has no true pericarp. It sometimes happens in Angiosperms, that + the seed-vessel is ruptured at an early period of growth, so that the + seeds become more or less exposed during their development; this + occurs in mignonette, where the capsule opens at the apex, and in + _Cuphea_, where the placenta bursts through the ovary and floral + envelopes, and appears as an erect process bearing the young seeds. + After fertilization the ovule is greatly changed, in connexion with + the formation of the embryo. In the embryo-sac of most Angiosperms + (q.v.) there is a development of cellular tissue, the endosperm, more + or less filling the embryo-sac. In Gymnosperms (q.v.) the endosperm is + formed preparatory to fertilization. The fertilized egg enlarges and + becomes multicellular, forming the embryo. The embryo-sac enlarges + greatly, displacing gradually the surrounding nucellus, which + eventually forms merely a thin layer around the sac, or completely + disappears. The remainder of the nucellus and the integuments of the + ovules form the seed-coats. In some cases (fig. 35) a delicate inner + coat or _tegmen_ can be distinguished from a tougher outer coat or + _testa_; often, however, the layers are not thus separable. The + consistency of the seed-coat, its thickness, the character of its + surface, &c., vary widely, the variations being often closely + associated with the environment or with the means of seed-dispersal. + An account of the development of the seed from the ovule will be found + in the article ANGIOSPERMS. When the pericarp is dehiscent the + seed-covering is of a strong and often rough character; but when the + pericarp is indehiscent and encloses the seed for a long period, the + outer seed-coat is thin and soft. The cells of the testa are often + coloured, and have projections and appendages of various kinds. Thus + in _Abrus precatorius_ and _Adenanthera pavonina_ it is of a bright + red colour; in French beans it is beautifully mottled; in the almond + it is veined; in the tulip and primrose it is rough; in the snapdragon + it is marked with depressions; in cotton and _Asclepias_ (fig. 36) it + has hairs attached to it; and in mahogany, _Bignonia_, and the pines + and firs it is expanded in the form of wing-like appendages (fig. 37). + In _Collomia_, _Acanthodium_, _Cobaea scandens_ and other seeds, it + contains spiral cells, from which, when moistened with water, the + fibres uncoil in a beautiful manner; and in flax (_Linum_) and others + the cells are converted into mucilage. These structural peculiarities + of the testa in different plants have relation to the scattering of + the seed and its germination upon a suitable nidus. But in some plants + the pericarps assume structures which subserve the same purpose; this + especially occurs in small pericarps enclosing single seeds, as + achenes, caryopsides, &c. Thus in Compositae and valerian, the pappose + limb of the calyx forms a parachute to the pericarp; in Labiatae and + some Compositae spiral cells are formed in the epicarp; and the + epicarp is prolonged as a wing in _Fraxinus_ (fig. 1) and _Acer_ (fig. + 21). + + [Illustration: + + FIG. 35.--Seed of Pea (_Pisum_) with one cotyledon removed. c, + Remaining cotyledon; ch, chalaza-point at which the nourishing + vessels enter; e, tegmen or inner coat; f, funicle or stalk; g, + plumule of embryo; m, micropyle; pl, placenta; r, radicle of embryo; + t, tigellum or stalk between root and plumule; te, testa. + + FIG. 36.--Seed of _Asclepias_, with a cluster of hairs arising from + the edges of the micropyle.] + + Sometimes there is an additional covering to the seed, formed after + fertilization, to which the name _arillus_ has been given (fig. 38). + This is seen in the passion-flower, where the covering arises from the + placenta or extremity of the funicle at the base of the ovule and + passes upwards towards the apex, leaving the micropyle uncovered. In + the nutmeg and spindle tree this additional coat is formed from above + downwards, constituting in the former case a laciniated scarlet + covering called _mace_. In such instances it has been called an + _arillode_ (fig. 39). This arillode, after growing downwards, may be + reflected upwards so as to cover the micropyle. The fleshy scarlet + covering formed around the naked seed in the yew is by some considered + of the nature of an aril. On the testa, at various points, there are + produced at times other cellular bodies, to which the name of + _strophioles_, or _caruncles_, has been given, the seeds being + strophiolate or carunculate. These tumours may occur near the base of + the seed, as in _Polygala_, or at the apex, as in Castor-oil plant + (_Ricinus_); or they may occur in the course of the raphe, as in + blood-root (_Sanguinaria_) and _Asarabacca_. The funicles of the + ovules frequently attain a great length in the seed, and in some + magnolias, when the fruit dehisces, they appear as long scarlet cords + suspending the seeds outside. The hilum or umbilicus of the seed is + usually well marked, as a scar of varying size; in the calabar bean + and in some species of Mucuna and Dolichos it extends along a large + portion of the edge of the seed; it frequently exhibits marked + colours, being black in the bean, white in many species of Phaseolus, + &c. The micropyle (fig. 35, m) of the seed may be recognizable by the + naked eye, as in the pea and bean tribe, _Iris_, &c., or it may be + very minute or microscopic. It indicates the true apex of the seed, + and is important as marking the point to which the root of the embryo + is directed. At the micropyle in the bean is observed a small process + of integument, which, when the young plant sprouts, is pushed up like + a lid; it is called the _embryotega_. The chalaza (fig. 38, ch) is + often of a different colour from the rest of the seed. In the orange + (fig. 40) it is of a reddish-brown colour, and is easily recognized at + one end of the seed when the integuments are carefully removed. In + anatropal seeds the raphe forms a distinct ridge along one side of the + seed (fig. 41). + + The position of the seed as regards the pericarp resembles that of the + ovule in the ovary, and the same terms are applied--erect, ascending, + pendulous, suspended, curved, &c. These terms have no reference to the + mode in which the fruit is attached to the axis. Thus the seed may be + erect while the fruit itself is pendent, in the ordinary meaning of + that term. The part of the seed next the axis or the ventral suture is + its face, the opposite side being the back. Seeds exhibit great + varieties of form. They may be flattened laterally (_compressed_), or + from above downwards (_depressed_). They may be round, oval, + triangular, polygonal, rolled up like a snail, as in _Physostemon_, or + coiled up like a snake, as in _Ophiocaryon paradoxum_. + + [Illustration: + + FIG. 37.--Seed of Pine (_Pinus_), with a membranous appendage w to + the testa, called a wing. + + FIG. 38.--Young anatropal seed of the white Water-lily (_Nymphaea + alba_), cut vertically. It is attached to the placenta by the + funicle f, cellular prolongations from which form an aril a a. The + vessels of the cord are prolonged to the base of the nucellus n by + means of the raphe r. The base of the nucellus is indicated by the + chalaza ch, while the apex is at the micropyle m. The covering of + the seed is marked i. n is the nucellus or perisperm, enclosing the + embryo-sac es, in which the endosperm is formed. The embryo e, with + its suspensor, is contained in the sac, the radicle pointing to the + micropyle m. + + FIG. 39.--Arillode a, or false aril, of the Spindle-tree + (_Euonymus_), arising from the micropyle f. + + FIG. 40.--Anatropal seed of the Orange (_Citrus Aurantium_) opened + to show the chalaza c, which forms a brown spot at one end. + + FIG. 41.--Entire anatropal seed of the Orange (_Citrus Aurantium_), + with its rugose or wrinkled testa, and the raphe r ramifying in the + thickness of the testa on one side.] + + The endosperm formed in the embryo-sac of angiosperms after + fertilization, and found previous to it in gymnosperms, consists of + cells containing nitrogenous and starchy or fatty matter, destined for + the nutriment of the embryo. It occupies the whole cavity of the + embryo-sac, or is formed only at certain portions of it, at the apex, + as in _Rhinanthus_, at the base, as in _Vaccinium_, or in the middle, + as in _Veronica_. As the endosperm increases in size along with the + embryo-sac and the embryo, the substance of the original nucellus of + the ovule is gradually absorbed. Sometimes, however, as in Musaceae, + Cannaceae, Zingiberaceae, no endosperm is formed; the cells of the + original nucellus, becoming filled with food-materials for the embryo, + are not absorbed, but remain surrounding the embryo-sac with the + embryo, and constitute the _perisperm_. Again, in other plants, as + Nymphaeaceae (fig. 38) and Piperaceae, both endosperm and perisperm + are present. It was from observations on cases such as these that old + authors, imagining a resemblance betwixt the plant-ovule and the + animal ovum, applied the name _albumen_ to the outer nutrient mass or + perisperm, and designated the endosperm as _vitellus_. The term + albumen is very generally used as including all the nutrient matter + stored up in the seed, but it would be advisable to discard the name + as implying a definite chemical substance. There is a large class of + plants in which although at first after fertilization a mass of + endosperm is formed, yet, as the embryo increases in size, the + nutrient matter from the endospermic cells passes out from them, and + is absorbed by the cells of the embryo plant. In the mature seed, in + such cases, there is no separate mass of tissue containing nutrient + food-material apart from the embryo itself. Such a seed is said to be + _exalbuminous_, as in Compositae, Cruciferae and most Leguminosae + (e.g. pea, fig. 35). When either endosperm or perisperm or both are + present the seed is said to be _albuminous_. + + [Illustration: FIG. 42.--The dicotyledonous embryo of the Pea laid + open. c, c, The two fleshy cotyledons, or seed-lobes, which remain + under ground when the plant sprouts; r, the radicular extremity of the + axis whence the root arises; t, the axis (hypocotyl) bearing the young + stalk and leaves g (plumule), which lie in a depression of the + cotyledons f.] + + The albumen varies much in its nature and consistence, and furnishes + important characters. It may be farinaceous or mealy, consisting + chiefly of cells filled with starch, as in cereal grains, where it is + abundant; fleshy or cartilaginous, consisting of thicker cells which + are still soft, as in the coco-nut, and which sometimes contain oil, + as in the oily albumen of _Croton_, _Ricinus_ and poppy; horny, when + the cell-walls are slightly thickened and capable of distension, as in + date and coffee; the cell-walls sometimes become greatly thickened, + filling up the testa as a hard mass, as in vegetable ivory + (_Phytelephas_). The albumen may be uniform throughout, or it may + present a mottled appearance, as in the nutmeg, the seeds of Anonaceae + and some Palms, where it is called _ruminated_. This mottled + appearance is due to a protrusion of a dark lamella of the integument + between folded protuberances of albumen. A cavity is sometimes left in + the centre which is usually filled with fluid, as in the coco-nut. The + relative size of the embryo and of the endosperm varies much. In + Monocotyledons the embryo is usually small, and the endosperm large, + and the same is true in the case of coffee and many other plants + amongst Dicotyledons. The opposite is the case in other plants, as in + the Labiatae, Plumbaginaceae, &c. + + The embryo consists of an axis bearing the _cotyledons_ (fig. 42, c), + or the first leaves of the plant. To that part of this axis + immediately beneath the cotyledons the terms _hypocotyl_, _caulicle_ + or _tigellum_ (t) have been applied, and continuous backwards with it + is the young root or _radicle_ (r), the descending axis, their point + of union being the collar or neck. The terminal growing bud of the + axis is called the _plumule_ or _gemmule_ (g), and represents the + ascending axis. The radicular extremity points towards the micropyle, + while the cotyledonary extremity is pointed towards the base of the + ovule or the chalaza. Hence, by ascertaining the position of the + micropyle and chalaza, the two extremities of the embryo can in + general be discovered. It is in many cases difficult to recognize the + parts in an embryo; thus in _Cuscuta_, the embryo appears as an + elongated axis without divisions; and in _Caryocar_ the mass of the + embryo is made up by the radicular extremity and hypocotyl, in a + groove of which the cotyledonary extremity lies embedded (fig. 52). In + some monocotyledonous embryos, as in Orchidaceae, the embryo is a + cellular mass showing no parts. In parasitic plants also which form no + chlorophyll, as _Orobanche_, _Monotropa_, &c., the embryo remains + without differentiation, consisting merely of a mass of cells until + the ripening of the seed. When the embryo is surrounded by the + endosperm on all sides except its radicular extremity it is internal + (see figs. 19, 20); when lying outside the endosperm, and only coming + into contact with it at certain points, it is external, as in grasses + (e.g. wheat, fig. 22). When the embryo follows the direction of the + axis of the seed, it is axile or axial (fig. 43); when it is not in + the direction of the axis, it becomes abaxile or abaxial. In + campylotropal seeds the embryo is curved, and in place of being + embedded in endosperm, is frequently external to it, following the + concavity of the seed (fig. 44), and becoming peripherical, with the + chalaza situated in the curvature of the embryo, as in + Caryophyllaceae. + + It has been already stated that the radicle of the embryo is directed + to the micropyle, and the cotyledons to the chalaza. In some cases, by + the growth of the integuments, the former is turned round so as not to + correspond with the apex of the nucellus, and then the embryo has the + radicle directed to one side, and is called excentric, as is seen in + Primulaceae, Plantaginaceae and many palms, especially the date. The + position of the embryo in different kinds of seeds varies. In an + orthotropal seed the embryo is inverted or _antitropal_, the radicle + pointing to the apex of the seed, or to the part opposite the hilum. + Again, in an anatropal seed the embryo is erect or _homotropal_ (fig. + 43), the radicle being directed to the base of the seed. In curved or + campylotropal seeds the embryo is folded so that its radicular and + cotyledonary extremities are approximated, and it becomes + _amphitropal_ (fig. 44). In this instance the seed may be + exalbuminous, and the embryo may be folded on itself; or albuminous, + the embryo surrounding more or less completely the endosperm and being + peripherical. According to the mode in which the seed is attached to + the pericarp, the radicle may be directed upwards or downwards, or + laterally, as regards the ovary. In an orthotropal seed attached to + the base of the pericarp it is superior, as also in a suspended + anatropal seed. In other anatropal seeds the radicle is inferior. When + the seed is horizontal as regards the pericarp, the radicle is either + centrifugal, when it points to the outer wall of the ovary; or + centripetal, when it points to the axis or inner wall of the ovary. + These characters are of value for purposes of classification, as they + are often constant in large groups of genera. + + Plants in which there are two cotyledons produced in the embryo are + _dicotyledonous_. The two cotyledons thus formed are opposite to each + other (figs. 42 and 45), but are not always of the same size. Thus, in + Abronia and other members of the order Nyctaginaceae, one of them is + smaller than the other (often very small), and in _Carapa guianensis_ + there appears to be only one, in consequence of the intimate union + which takes place between the two. The union between the cotyledonary + leaves may continue after the young plant begins to germinate. Such + embryos have been called _pseudomonocotyledonous_. The texture of the + cotyledons varies. They may be thick, as in the pea (fig. 42), + exhibiting no traces of venation, with their flat internal surfaces in + contact, and their backs more or less convex; or they may be in the + form of thin and delicate laminae, flattened on both sides, and having + distinct venation, as in _Ricinus_, _Jatropha_, _Euonymus_, &c. The + cotyledons usually form the greater part of the mature embryo, and + this is remarkably well seen in such exalbuminous seeds as the bean + and pea. + + [Illustration: + + FIG. 43.--Seed of Pansy (_Viola tricolor_) cut vertically. The + embryo pl is axial, in the midst of fleshy endosperm al. The seed is + anatropal, and the embryo is homotropal; the cotyledons co point to + the base of the nucellus or chalaza ch, while the radicle, or the + other extremity of the embryo, points to the micropyle, close to the + hilum h. The hilum or base of the seed, and the chalaza or base of + the nucellus are united by means of the raphe r. + + FIG. 44.--Seed of the Red Campion (_Lychnis_), cut vertically, + showing the peripheral embryo, with its two cotyledons and its + radicle. The embryo is curved round the albumen, so that its + cotyledons and radicle both come near the hilum (_amphitropal_). + + FIG. 45.--Mature dicotyledonous embryo of the Almond, with one of + the cotyledons removed. r, Radicle; t, young stem or caulicle; c, + one of the cotyledons left; i, line of insertion of the cotyledon + which has been removed; g, plumule. + + FIG. 46.--Exalbuminous seed of Wallflower (Cheiranthus) cut + vertically. The radicle r is folded on the edges of the cotyledons c + which are accumbent. + + FIG. 47.--Transverse section of the seed of the Wallflower + (_Cheiranthus_), showing the radicle r folded on the edges of the + accumbent cotyledons c. + + FIG. 48.--Transverse section of the seed of the Dame's Violet + (_Hesperis_). The radicle r is folded on the back of the cotyledons + c, which are said to be incumbent.] + + Cotyledons are usually entire and sessile. But they occasionally + become lobed, as in the walnut and the lime; or petiolate, as in + _Geranium molle_; or auriculate, as in the ash. Like leaves in the + bud, cotyledons may be either applied directly to each other, or may + be folded in various ways. In geranium the cotyledons are twisted and + doubled; in convolvulus they are corrugated; and in the potato and in + _Bunias_, they are spiral,--the same terms being applied as to the + foliage leaves. The radicle and cotyledons are either straight or + variously curved. Thus, in some cruciferous plants, as the wallflower, + the cotyledons are applied by their faces, and the radicle (figs. 46, + 47) is folded on their edges, so as to be lateral; the cotyledons are + here _accumbent_. In others, as _Hesperis_, the cotyledons (fig. 48) + are applied to each other by their faces, and the radicle, r, is + folded on their back, so as to be dorsal, and the cotyledons are + _incumbent_. Again, the cotyledons are _conduplicate_ when the radicle + is dorsal, and enclosed between their folds. In other divisions the + radicle is folded in a spiral manner, and the cotyledons follow the + same course. + + In many gymnosperms more than two cotyledons are present, and they are + arranged in a whorl. This occurs in Coniferae, especially in the pine, + fir (fig. 49), spruce and larch, in which six, nine, twelve and even + fifteen have been observed. They are linear, and resemble in their + form and mode of development the clustered or fasciculated leaves of + the larch. Plants having numerous cotyledons are termed + _polycotyledonous_. In species of _Streptocarpus_ the cotyledons are + permanent, and act the part of leaves. One of them is frequently + largely developed, while the other is small or abortive. + + [Illustration: + + FIG. 49.--Polycotylodonous embryo of the Pine (_Pinus_) beginning to + sprout. t, Hypocotyl; r, radicle. The cotyledons c are numerous. + Within the cotyledons the primordial leaves are seen, constituting + the plumule or first bud of the plant. + + FIG. 50.--Embryo of a species of Arrow-grass (_Triglochin_), showing + a uniform conical mass, with a slit s near the lower part. The + cotyledon c envelops the young bud, which protrudes at the slit + during germination. The radicle is developed from the lower part of + the axis r. + + FIG. 51.--Grain of wheat (_Triticum_) germinating, showing (b) the + cotyledon and (c) the rootlets surrounded by their sheaths + (_coleorrhizae_). + + FIG. 52.--Embryo of _Caryocar_. t, Thick hypocotyl, forming nearly + the whole mass, becoming narrowed and curved at its extremity, and + applied to the groove s. In the figure this narrowed portion is + slightly separated from the groove; c, two rudimentary cotyledons.] + + In those plants in which there is only a single cotyledon in the + embryo, hence called _monocotyledonous_, the embryo usually has a + cylindrical form more or less rounded at the extremities, or elongated + and fusiform, often oblique. The axis is usually very short compared + with the cotyledon, which in general encloses the plumule by its lower + portion, and exhibits on one side a small slit which indicates the + union of the edges of the vaginal or sheathing portion of the leaf + (fig. 50). In grasses, by the enlargement of the embryo in a + particular direction, the endosperm is pushed on one side, and thus + the embryo comes to lie outside at the base of the endosperm (figs. + 22, 51). The lamina of the cotyledon is not developed. Upon the side + of the embryo next the endosperm and enveloping it is a large + shield-shaped body, termed the _scutellum_. This is an outgrowth from + the base of the cotyledon, enveloping more or less the cotyledon and + plumule, in some cases, as in maize, completely investing it; in other + cases, as in rice, merely sending small prolongations over its + anterior face at the apex. By others this scutellum is considered as + the true cotyledon, and the sheathing structure covering the plumule + is regarded as a ligule or axillary stipule (see GRASSES). In many + aquatic monocotyledons (e.g. _Potamogeton_, _Ruppia_ and others) there + is a much-developed hypocotyl, which forms the greater part of the + embryo and acts as a store of nutriment in germination; these are + known as _macropodous_ embryos. A similar case is that of _Caryocar_ + among Dicotyledons, where the swollen hypocotyl occupies most of the + embryo (fig. 52). In some grasses, as oats and rice, a projection of + cellular tissue is seen upon the side of the embryo opposite to the + scutellum, that is, on the anterior side. This has been termed the + _epiblast_. It is very large in rice. This by some was considered the + rudimentary second cotyledon; but is now generally regarded as an + outgrowth of the sheath of the true cotyledon. (A. B. R.) + + + + +FRUIT AND FLOWER FARMING. The different sorts of fruits and flowers are +dealt with in articles under their own headings, to which reference may +be made; and these give the substantial facts as to their cultivation. +See also the article HORTICULTURE. + + +TABLE I.--_Extent of Orchards in Great Britain in each Year, 1887 to +1901._ + + +------+---------++------+---------++------+---------+ + | Year.| Acres. || Year.| Acres. || Year.| Acres. | + +------+---------++------+---------++------+---------+ + | 1887 | 202,234 || 1892 | 208,950 || 1897 | 224,116 | + | 1888 | 199,178 || 1893 | 211,664 || 1898 | 226,059 | + | 1889 | 199,897 || 1894 | 214,187 || 1899 | 228,603 | + | 1890 | 202,305 || 1895 | 218,428 || 1900 | 232,129 | + | 1891 | 209,996 || 1896 | 221,254 || 1901 | 234,660 | + +------+---------++------+---------++------+---------+ + + +TABLE II.--_Areas under Orchards in England, Wales and Scotland--Acres._ + + +------+----------+--------+---------+--------------+ + | Year.| England. | Wales. |Scotland.|Great Britain.| + +------+----------+--------+---------+--------------+ + | 1896 | 215,642 | 3677 | 1935 | 221,254 | + | 1897 | 218,261 | 3707 | 2148 | 224,116 | + | 1898 | 220,220 | 3690 | 2149 | 226,059 | + | 1899 | 222,712 | 3666 | 2225 | 228,603 | + | 1900 | 226,164 | 3695 | 2270 | 232,129 | + | 1901 | 228,580 | 3767 | 2313 | 234,660 | + | 1908 | 244,430 | 3577 | 2290 | 250,297 | + +------+----------+--------+---------+--------------+ + + +GREAT BRITAIN + +The extent of the fruit industry may be gathered from the figures for +the acreage of land under cultivation in orchards and small fruit +plantations. The Board of Agriculture returns concerning the orchard +areas of Great Britain showed a continuous expansion year by year from +199,178 acres in 1888 to 234,660 acres in 1901, as will be learnt from +Table I. There was, it is true, an exception in 1892, but the decline in +that year is explained by the circumstance that since 1891 the +agricultural returns have been collected only from holdings of more than +one acre, whereas they were previously obtained from all holdings of a +quarter of an acre or more. As there are many holdings of less than an +acre in extent upon which fruit is grown, and as fruit is largely raised +also in suburban and other gardens which do not come into the returns, +it may be taken for granted that the actual extent of land devoted to +fruit culture exceeds that which is indicated by the official figures. +In the Board of Agriculture returns up to June 1908, 308,000 acres are +stated to be devoted to fruit cultivation of all kinds in Great Britain. +Table II. shows that the expansion of the orchard area of Great Britain +is mainly confined to England, for it has slightly decreased in Wales +and Scotland. The acreage officially returned as under orchards is that +of arable or grass land which is also used for fruit trees of any kind. +Conditions of soil and climate determine the irregular distribution of +orchards in Great Britain. The dozen counties which possess the largest +extent of orchard land all lie in the south or west of the island. +According to the returns for 1908 (excluding small fruit areas) they +were the following:-- + + +----------+--------++-----------+--------++----------+------+ + | County. | Acres. || County. | Acres. || County. |Acres.| + +----------+--------++-----------+--------++----------+------+ + | Kent | 32,751 || Worcester | 23,653 || Salop | 4685 | + | Devon | 27,200 || Gloucester| 20,424 || Dorset | 4464 | + | Hereford | 28,316 || Cornwall | 5,415 || Monmouth | 3914 | + | Somerset | 25,279 || Middlesex | 5,300 || Wilts | 3630 | + +----------+--------++-----------+--------++----------+------+ + +Leaving out of consideration the county of Kent, which grows a greater +variety of fruit than any of the others, the counties of Devon, +Hereford, Somerset, Worcester and Gloucester have an aggregate orchard +area of 124,872 acres. These five counties of the west and south-west of +England--constituting in one continuous area what is essentially the +cider country of Great Britain--embrace therefore rather less than half +of the entire orchard area of the island, while Salop, Monmouth and +Wilts have about 300 less than they had a few years ago. Five English +counties have less than 1000 acres each of orchards, namely, the county +of London, and the northern counties of Cumberland, Westmorland, +Northumberland and Durham. Rutland has just over 100 acres. The largest +orchard areas in Wales are in the two counties adjoining +Hereford--Brecon with 1136 acres and Radnor with 727 acres; at the other +extreme is Anglesey, with a decreasing orchard area of only 22 acres. Of +the Scottish counties, Lanark takes the lead with 1285 acres, Perth, +Stirling and Haddington following with 684 and 129 acres respectively. +Ayr and Midlothian are the only other counties possessing 100 acres or +more of orchards, whilst Kincardine, Orkney and Shetland return no +orchard area, and Banff, Bute, Kinross, Nairn, Peebles, Sutherland and +Wigtown return less than 10 acres each. It may be added that in 1908 +Jersey returned 1090 acres of orchards, Guernsey, &c., 144 acres, and +the Isle of Man, 121 acres; the two last-named places showing a decline +as compared with eight years previously. + +Outside the cider counties proper of England, the counties in which +orchards for commercial fruit-growing have increased considerably in +recent years include Berks, Buckingham, Cambridge, Essex, Lincoln, +Middlesex, Monmouth, Norfolk, Oxford, Salop, Sussex, Warwick and Wilts. +Apples are the principal fruit grown in the western and south-western +counties, pears also being fairly common. In parts of Gloucestershire, +however, and in the Evesham and Pershore districts of Worcestershire, +plum orchards exist. Plums are almost as largely grown as apples in +Cambridgeshire. Large quantities of apples, plums, damsons, cherries, +and a fair quantity of pears are grown for the market in Kent, whilst +apples, plums and pears predominate in Middlesex. In many counties +damsons are cultivated around fruit plantations to shelter the latter +from the wind. + +Of small fruit (currants, gooseberries, strawberries, raspberries, &c.) +no return was made of the acreage previous to 1888, in which year it was +given as 36,724 acres for Great Britain. In 1889 it rose to 41,933 +acres. + +Later figures are shown in Table III. It will be observed that, owing to +corrections made in the enumeration in 1897, a considerable reduction in +the area is recorded for that year, and presumably the error then +discovered existed in all the preceding returns. The returns for 1907 +gave the acreage of small fruit as 82,175 acres, and in 1908 at 84,880 +acres--an area more than double that of 1889. + + +TABLE III.--_Areas of Small Fruit in Great Britain._ + + +-------+--------+-------+--------+-------+--------+ + | Year. | Acres. | Year. | Acres. | Year. | Acres. | + +-------+--------+-------+--------+-------+--------+ + | 1890 | 46,234 | 1894 | 68,415 | 1898 | 69,753 | + | 1891 | 58,704 | 1895 | 74,547 | 1899 | 71,526 | + | 1892 | 62,148 | 1896 | 76,245 | 1900 | 73,780 | + | 1893 | 65,487 | 1897 | 69,792 | 1901 | 74,999 | + +-------+--------+-------+--------+-------+--------+ + + +TABLE IV.--_Areas under Small Fruit in England, Wales and +Scotland--Acres._ + + +-------+----------+--------+-----------+----------------+ + | Year. | England. | Wales. | Scotland. | Great Britain. | + +-------+----------+--------+-----------+----------------+ + | 1898 | 63,438 | 1044 | 5271 | 69,753 | + | 1899 | 64,867 | 1106 | 5553 | 71,526 | + | 1900 | 66,749 | 1109 | 5922 | 73,780 | + | 1901 | 67,828 | 1092 | 6079 | 74,999 | + | 1908 | 75,750 | 1200 | 7930 | 84,880 | + +-------+----------+--------+-----------+----------------+ + +There has undoubtedly been a considerable expansion, rather than a +contraction, of small fruit plantations since 1896. The acreage of small +fruit in Great Britain is about one-third that of the orchards. As may +be seen in Table IV., it is mainly confined to England, though Scotland +has over 4000 more acres of small fruit than of orchards. About +one-third of the area of small fruit in England belongs to Kent alone, +that county having returned 24,137 acres in 1908. Cambridge now ranks +next with 6878 acres, followed by Norfolk with 5876 acres, +Worcestershire with 4852 acres, Middlesex with 4163 acres, Hants with +3320 acres and Essex with 2150 acres. It should be remarked that between +1900 and 1908 Cambridgeshire had almost doubled its area of small +fruits, from 3740 to 6878 acres; whilst both Norfolk and Worcestershire +in 1908 had larger areas devoted to small fruits than Middlesex--in +which county there had been a decrease of about 400 acres during the +same period. The largest county area of small fruit in Wales is 806 +acres in Denbighshire, and in Scotland 2791 acres in Perthshire, 2259 +acres in Lanarkshire, followed by 412 acres in Forfarshire. The only +counties in Great Britain which make no return under the head of small +fruit are Orkney and Shetland; and Sutherland only gives 2-1/2 acres. It +is hardly necessary to say that considerable areas of small fruit, in +kitchen gardens and elsewhere, find no place in the official returns, +which, however, include small fruit grown between and under orchard +trees. + +Gooseberries are largely grown in most small fruit districts. Currants +are less widely cultivated, but the red currant is more extensively +grown than the black, the latter having suffered seriously from the +ravages of the black currant mite. Kent is the great centre for +raspberries and for strawberries, though, in addition, the latter fruit +is largely grown in Cambridgeshire (2411 acres), Hampshire (2327 acres), +Norfolk (2067 acres) and Worcestershire (1273 acres). Essex, +Lincolnshire, Cheshire, Cornwall and Middlesex each has more than 500 +acres devoted to strawberry cultivation. + +The following statement from returns for 1908 shows the area under +different kinds of fruit in 1907 and 1908 in Great Britain, and also +whether there had been an increase or decrease: + + +-----------------+---------+---------+-------------+ + | | 1907. | 1908. | Increase or | + | | | | Decrease. | + +-----------------+---------+---------+-------------+ + | | Acres. | Acres. | Acres. | + | Small Fruit-- | | | | + | Strawberries | 27,827 | 28,815 | + 988 | + | Raspberries | 8,878 | 9,323 | + 445 | + | Currants and | | | | + | Gooseberries | 25,590 | 26,241 | + 651 | + | Other kinds | 19,880 | 20,501 | + 621 | + | +---------+---------+-------------+ + | | 82,175 | 84,880 | +2705 | + | +---------+---------+-------------+ + | Orchards-- | | | | + | Apples | 172,643 | 172,751 | + 108 | + | Pears | 8,911 | 9,604 | + 693 | + | Cherries | 12,027 | 11,868 | - 159 | + | Plums | 14,901 | 15,683 | + 782 | + | Other kinds | 41,694 | 40,391 | -1303 | + | +---------+---------+-------------+ + | | 250,176 | 250,297 | + 121 | + +-----------------+---------+---------+-------------+ + +It appears from the Board of Agriculture returns that 27,433 acres of +small fruit was grown in orchards, so that the total extent of land +under fruit cultivation in Great Britain at the end of 1908 was about +308,000 acres. + +There are no official returns as to the acreage devoted to orchard +cultivation in Ireland. The figures relating to small fruit, moreover, +extend back only to 1899, when the area under this head was returned as +4809 acres, which became 4359 acres in 1900 and 4877 acres in 1901. In +most parts of the country there are districts favourable to the culture +of small fruits, such as strawberries, raspberries, gooseberries and +currants, and of top fruits, such as apples, pears, plums and damsons. +The only localities largely identified with fruit culture as an industry +are the Drogheda district and the Armagh district. In the former all the +kinds named are grown except strawberries, the speciality being +raspberries, which are marketed in Dublin, Belfast and Liverpool. In the +Armagh district, again, all the kinds named are grown, but in this case +strawberries are the speciality, the markets utilized being Richhill, +Belfast, and those in Scotland. In the Drogheda district the grower +bears the cost of picking, packing and shipping, but he cannot estimate +his net returns until his fruit is on the market. Around Armagh the +Scottish system prevails--that is, the fruit is sold while growing, the +buyer being responsible for the picking and marketing. + +The amount of fruit imported into the United Kingdom has such an +important bearing on the possibilities of the industry that the +following figures also may be useful: + + The quantities of apples, pears, plums, cherries and grapes imported + in the raw condition into the United Kingdom in each year, 1892 to + 1901, are shown in Table V. Previous to 1892 apples only were + separately enumerated. Up to 1899 inclusive the quantities were given + in bushels, but in 1900 a change was made to hundred-weights. This + renders the quantities in that and subsequent years not directly + comparable with those in earlier years, but the comparison of the + values, which are also given in the table, continues to hold good. The + figures for 1908 have been added to show the increase that had taken + place. In some years the value of imported apples exceeds the + aggregate value of the pears, plums, cherries and grapes imported. The + extreme values for apples shown in the table are L844,000 in 1893 and + L2,079,000 in 1908. Grapes rank next to apples in point of value, and + over the seventeen years the amount ranged between L394,000 in 1892 + and L728,000 in 1908. On the average, the annual outlay on imported + pears is slightly in excess of that on plums. The extremes shown are + L167,000 in 1895 and L515,000 in 1908. In the case of plums, the + smallest outlay tabulated is L166,000 in 1895, whilst the largest is + L498,000 in 1897. The amounts expended upon imported cherries varied + between L96,000 in 1895 and L308,000 in 1900. In 1900 apricots and + peaches, imported raw, previously included with raw plums, were for + the first time separately enumerated, the import into the United + Kingdom for that year amounting to 13,689 cwt., valued at L25,846; in + 1901 the quantity was 13,463 cwt. and the value L32,350. The latter + rose in 1908 to L60,000. In 1900, also, currants, gooseberries and + strawberries, hitherto included in unenumerated raw fruit, were + likewise for the first time separately returned. Of raw currants the + import was 64,462 cwt., valued at L87,170 (1908, L121,850); of raw + gooseberries 26,045 cwt., valued at L14,626 (1908, L25,520); and of + raw strawberries, 52,225 cwt., valued at L85,949. In 1907 only 44,000 + cwt. of strawberries were imported. In 1901 the quantities and values + were respectively--currants, 70,402 cwt., L75,308; gooseberries, + 21,735 cwt., L11,420; strawberries, 38,604 cwt., L51,290. Up to 1899 + the imports of tomatoes were included amongst unenumerated raw + vegetables, so that the quantity was not separately ascertainable. For + 1900 the import of tomatoes was 833,032 cwt., valued at L792,339, + which is equivalent to a fraction under 2-1/2d. per lb. For 1901 the + quantity was 793,991 cwt., and the value L734,051; for 1906, there + were 1,124,700 cwt., valued at L953,475; for 1907, 1,135,499 cwt., + valued at L1,020,805; and for 1908, 1,160,283 cwt., valued at + L955,983. + + + TABLE V.--_Imports of Raw Apples, Pears, Plums, Cherries and Grapes + into the United Kingdom, 1892 to 1901. Quantities in Thousands of + Bushels (thousands of cwt. in 1900 and 1901). Values in Thousands of + Pounds Sterling._ + + +------+-------------------------------------------------+ + | | Quantities. | + | Year.+---------+--------+--------+-----------+---------+ + | | Apples. | Pears. | Plums. | Cherries. | Grapes. | + +------+---------+--------+--------+-----------+---------+ + | 1892 | 4515 | 637 | 413 | 217 | 762 | + | 1893 | 3460 | 915 | 777 | 346 | 979 | + | 1894 | 4969 | 1310 | 777 | 311 | 833 | + | 1895 | 3292 | 407 | 401 | 196 | 865 | + | 1896 | 6177 | 483 | 560 | 219 | 883 | + | 1897 | 4200 | 1052 | 1044 | 312 | 994 | + | 1898 | 3459 | 492 | 922 | 402 | 1136 | + | 1899 | 3861 | 572 | 558 | 281 | 1158 | + | 1900 | 2129* | 477* | 423* | 243* | 593* | + | 1901 | 1830* | 349* | 264* | 213* | 680* | + +------+---------+--------+--------+-----------+---------+ + | Values. | + +------+---------+--------+--------+-----------+---------+ + | 1892 | 1354 | 297 | 200 | 135 | 394 | + | 1893 | 844 | 347 | 332 | 195 | 530 | + | 1894 | 1389 | 411 | 302 | 167 | 470 | + | 1895 | 960 | 167 | 166 | 96 | 487 | + | 1896 | 1582 | 207 | 242 | 106 | 443 | + | 1897 | 1187 | 378 | 498 | 178 | 495 | + | 1898 | 1108 | 222 | 435 | 231 | 550 | + | 1899 | 1186 | 266 | 294 | 154 | 588 | + | 1900 | 1225 | 367 | 393 | 308 | 595 | + | 1901 | 1183 | 296 | 244 | 214 | 695 | + | 1908 | 2079 | 515 | 428 | 235 | 728 | + +------+---------+--------+--------+-----------+---------+ + * Thousands of cwts. + + In 1908 the outlay of the United Kingdom upon imported raw fruits, + such as can easily be produced at home, was L4,195,654, made up as + follows: + + Apples L2,079,703 | Plums L428,966 + Grapes 728,026 | Currants 121,852 + Pears 515,914 | Apricots and peaches 60,141 + Cherries 235,523 | Gooseberries 25,529 + + In addition about L280,000 was spent upon "unenumerated" raw fruit, + and L560,000 on nuts other than almonds "used as fruit," which would + include walnuts and filberts, both produced at home. It is certain, + therefore, that the expenditure on imported fruits, such as are grown + within the limits of the United Kingdom, exceeds four millions + sterling per annum. The remainder of the outlay on imported fruit in + 1908, amounting to over L5,000,000, was made up of L2,269,651 for + oranges, L471,713 for lemons, L1,769,249 for bananas, and L560,301 for + almond-nuts; these cannot be grown on an industrial scale in the + British Isles. + + It may be interesting to note the source of some of these imported + fruits. The United States and Canada send most of the apples, the + quantity for 1907 being 1,413,000 cwt. and 1,588,000 cwt. + respectively, while Australia contributes 280,000 cwt. Plums come + chiefly from France (200,000 cwt.), followed with 38,000 cwt. from + Germany and 28,000 cwt. from the Netherlands. Pears are imported + chiefly from France (204,000 cwt.) and Belgium (176,000); but the + Netherlands send 52,000 cwt., and the United States 24,000 cwt. The + great bulk of imported tomatoes comes from the Canary Islands, the + quantity in 1907 being 604,692 cwt. The Channel Islands also sent + 223,800 cwt., France 115,500 cwt., Spain 169,000 cwt., and Portugal a + long way behind with 11,700 cwt. Most of the strawberries imported + come from France (33,800 cwt.) and the Netherlands (10,300 cwt.). + +_Fruit-growing in Kent._--Kent is by far the largest fruit-growing +county in England. For centuries that county has been famous for its +fruit, and appears to have been the centre for the distribution of trees +and grafts throughout the country. The cultivation of fruit land upon +farms in many parts of Kent has always been an important feature in its +agriculture. An excellent description of this noteworthy characteristic +of Kentish farming is contained in a comprehensive paper on the +agriculture of Kent by Mr Charles Whitehead,[1] whose remarks, with +various additions and modifications, are here reproduced. + + Where the conditions are favourable, especially in East and Mid Kent, + there is a considerable acreage of fruit land attached to each farm, + planted with cherry, apple, pear, plum and damson trees, and with bush + fruits, or soft fruits as they are sometimes called, including + gooseberries, currants, raspberries, either with or without standard + trees, and strawberries, and filberts and cob-nuts in Mid Kent. This + acreage has largely increased, and will no doubt continue to increase, + as, on the whole, fruit-growing has been profitable and has materially + benefited those fortunate enough to have fruit land on their farms. + There are also cultivators who grow nothing but fruit. These are + principally in the district of East Kent, between Rochester and + Canterbury, and in the district of Mid Kent near London, and they + manage their fruit land, as a rule, better than farmers, as they give + their undivided attention to it and have more technical knowledge. But + there has been great improvement of late in the management of fruit + land, especially of cherry and apple orchards, the grass of which is + fed off by animals having corn or cake, or the land is well manured. + Apple trees are grease-banded and sprayed systematically by advanced + fruit-growers to prevent or check the attacks of destructive insects. + Far more attention is being paid to the selection of varieties of + apples and pears having colour, size, flavour, keeping qualities, and + other attributes to meet the tastes of the public, and to compete with + the beautiful fruit that comes from the United States and Canada. + + Of the various kinds of apples at present grown in Kent mention should + be made of Mr Gladstone, Beauty of Bath, Devonshire Quarrenden, Lady + Sudely, Yellow Ingestre and Worcester Pearmain. These are dessert + apples ready to pick in August and September, and are not stored. For + storing, King of the Pippins, Cox's Orange Pippin (the best dessert + apple in existence), Cox's Pomona, Duchess, Favourite, Gascoyne's + Scarlet Seedling, Court Pendu Plat, Baumann's Red Reinette, Allington + Pippin, Duke of Devonshire and Blenheim Orange. Among kitchen apples + for selling straight from the trees the most usually planted are Lord + Grosvenor, Lord Suffield, Keswick Codlin, Early Julian, Eclinville + Seedling, Pott's Seedling, Early Rivers, Grenadier, Golden Spire, + Stirling Castle and Domino. For storing, the cooking sorts favoured + now are Stone's or Loddington, Warner's King, Wellington, Lord Derby, + Queen Caroline, Tower of Glamis, Winter Queening, Lucombe's Seedling, + Bismarck, Bramley's Seedling, Golden Noble and Lane's Prince Albert. + Almost all these will flourish equally as standards, pyramids and + bushes. Among pears are Hessle, Clapp's Favourite, William's Bon + Chretien, Beurre de Capiaumont, Fertility, Beurre Riche, Chissel, + Beurre Clairgeau, Louise Bonne of Jersey, Doyenne du Comice and Vicar + of Winkfield. Among plums, Rivers's Early Prolific, Tsar, Belgian + Purple, Black Diamond, Kentish Bush Plum, Pond's Seedling, Magnum + Bonum and Victoria are mainly cultivated. The damson known as Farleigh + Prolific, or Crittenden's, is most extensively grown throughout the + county, and usually yields large crops, which make good prices. As a + case in point, purchasers were offering to contract for quantities of + this damson at L20 per ton in May of 1899, as the prospects of the + yield were unsatisfactory. On the other hand, in one year recently + when the crop was abnormally abundant, some of the fruit barely paid + the expenses of sending to market. The varieties of cherries most + frequently grown are Governor Wood, Knight's Early Black, Frogmore + Blackheart, Black Eagle, Waterloo, Amberheart, Bigarreau, Napoleon + Bigarreau and Turk. A variety of cherry known as the Kentish cherry, + of a light red colour and fine subacid flavour, is much grown in Kent + for drying and cooking purposes. Another cherry, similar in colour and + quality, which comes rather late, known as the Flemish, is also + extensively cultivated, as well as the very dark red large Morello, + used for making cherry brandy. These three varieties are grown + extensively as pyramids, and the last-named also on walls and sides of + buildings. Sometimes the cherry crop is sold by auction to dealers, + who pick, pack and consign the fruit to market. Large prices are often + made, as much as L80 per acre being not uncommon. The crop on a large + cherry orchard in Mid Kent has been sold for more than L100 per acre. + + Where old standard trees have been long neglected and have become + overgrown by mosses and lichens, the attempts made to improve them + seldom succeed. The introduction of bush fruit trees dwarfed by + grafting on the Paradise stock has been of much advantage to fruit + cultivators, as they come into bearing in two or three years, and are + more easily cultivated, pruned, sprayed and picked than standards. + Many plantations of these bush trees have been formed in Kent of + apples, pears and plums. Half standards and pyramids have also been + planted of these fruits, as well as of cherries. Bushes of + gooseberries and currants, and clumps or stools of raspberry canes, + have been planted to a great extent in many parts of the East and Mid + divisions of Kent, but not much in the Weald, where apples are + principally grown. Sometimes fruit bushes are put in alternate rows + with bush of standard trees of apple, pear, plum or damson, or they + are planted by themselves. The distances apart for planting are + generally for cherry and apple trees on grass 30 ft. by 30 ft.; for + standard apples and pear trees from 20 ft. to 24 ft. upon arable land, + with bush fruit, as gooseberries and currants, under them. These are + set 6 ft. by 6 ft. apart, and 5 ft. by 2 ft. for raspberries, and + strawberries 2 ft. 6 in. to 3 ft. by 1 ft. 6 in. to 1 ft. 3 in. apart. + On some fruit farms bush or dwarf trees--apples, pears, plums--are + planted alone, at distances varying from 8 ft. to 10 ft. apart, giving + from 485 to 680 bush trees per acre, nothing being grown between them + except perhaps strawberries or vegetables during the first two or + three years. It is believed that this is the best way of ensuring + fruit of high quality and colour. Another arrangement consists in + putting standard apple or pear trees 30 ft. apart (48 trees per acre), + and setting bush trees of apples or pears 15 ft. apart between them; + these latter come quickly into bearing, and are removed when the + standards are fully grown. Occasionally gooseberry or currant bushes, + or raspberry canes or strawberry plants, are set between the bush + trees, and taken away directly they interfere with the growth of + these. Half standard apple or plum trees are set triangularly 15 ft. + apart, and strawberry plants at a distance of 1-1/2 ft. from plant to + plant and 2-1/2 ft. from row to row. Or currant or gooseberry bushes + are set between the half standards, and strawberry plants between + these. + + These systems involve high farming. The manures used are London + manure, where hops are not grown, and bone meal, super-phosphate, + rags, shoddy, wool-waste, fish refuse, nitrate of soda, kainit and + sulphate of ammonia. Where hops are grown the London manure is wanted + for them. Fruit plantations are always dug by hand with the Kent spud. + Fruit land is never ploughed, as in the United States and Canada. The + soil is levelled down with the "Canterbury" hoe, and then the + plantations are kept free from weeds with the ordinary draw or "plate" + hoe. The best fruit farmers spray fruit trees regularly in the early + spring, and continue until the blossoms come out, with quassia and + soft soap and paraffin emulsions, and a very few with Paris green + only, where there is no under fruit, in order to prevent and check the + constant attacks of the various caterpillars and other insect pests. + This is a costly and laborious process, but it pays well, as a rule. + The fallacy that fruit trees on grass land require no manure, and that + the grass may be allowed to grow up to their trunks without any harm, + is exploding, and many fruit farmers are well manuring their grass + orchards and removing the grass for some distance round the stems, + particularly where the trees are young. + + Strawberries are produced in enormous quantities in the northern part + of the Mid Kent district round the Crays, and from thence to + Orpington; also near Sandwich, and to some extent near Maidstone. + Raspberry canes have been extensively put in during the last few + years, and in some seasons yield good profits. There is a very great + and growing demand for all soft fruits for jam-making, and prices are + fairly good, taking an average of years, notwithstanding the heavy + importations from France, Belgium, Holland, Spain and Italy. The + extraordinary increase in the national demand for jam and other fruit + preserves has been of great benefit to Kent fruit producers. The + cheapness of duty-free sugar, as compared with sugar paying duty in + the United States and other large fruit-producing countries, afforded + one of the very few advantages possessed by British cultivators, but + the reimposition of the sugar duty in the United Kingdom in 1901 has + modified the position in this respect. Jam factories were established + in several parts of Kent about 1889 or 1890, but most of them + collapsed either from want of capital or from bad management. There + are still a few remaining, principally in connexion with large fruit + farms. One of these is at Swanley, whose energetic owners farm nearly + 2000 acres of fruit land in Kent. The fruit grown by them that will + not make satisfactory prices in a fresh raw state is made into jam, or + if time presses it is first made into pulp, and kept until the + opportunity comes for making it into jam. In this factory there are + fifteen steam-jacketed vats in one row, and six others for candied + peel. A season's output on a recent occasion comprised about 3500 tons + of jam, 850 tons of candied peel and 750 gross (108,000 bottles) of + bottled fruit. A great deal of the fruit preserved is purchased, + whilst much of that grown on the farms is sold. A strigging machine is + employed, which does as much work as fifty women in taking currants + off their strigs or stalks. Black currant pulp is stored in casks till + winter, when there is time to convert it into jam. Strawberries cannot + be pulped to advantage, but it is otherwise with raspberries, the pulp + of which is largely made. Apricots for jam are obtained chiefly from + France and Spain. There is another flourishing factory near + Sittingbourne worked on the same lines. It is very advantageous to + fruit farmers to have jam factories in connexion with their farms or + to have them near, as they can thoroughly grade their fruit, and send + only the best to market, thus ensuring a high reputation for its + quality. Carriage is saved, which is a serious charge, though railway + rates from Kent to the great manufacturing towns and to Scotland are + very much less proportionally than those to London, and consequently + Kent growers send increasing quantities to these distant markets, + where prices are better, not being so directly interfered with by + imported fruit, which generally finds its way to London. + + Kentish fruit-growers are becoming more particular in picking, + grading, packing and storing fruit, as well as in marketing it. A + larger quantity of fruit is now carefully stored, and sent to selected + markets as it ripens, or when there is an ascertained demand, as it is + found that if it is consigned to market direct from the trees there + must frequently be forced sales and competition with foreign fruit + that is fully matured and in good order. It was customary formerly for + Kentish growers to consign all their fruit to the London markets; now + a good deal of it is sent to Manchester, Birmingham, Liverpool, + Sheffield, Newcastle and other large cities. Some is sent even to + Edinburgh and Glasgow. Many large growers send no fruit to London now. + It is by no means uncommon for growers to sell their fruit crops on + the trees or bushes by auction or private treaty, or to contract to + supply a stipulated quantity of specified fruit, say of currants, + raspberries or strawberries, to jam manufacturers. There is a + considerable quantity of fruit, such as grapes, peaches, nectarines, + grown under glass, and this kind of culture tends to increase. + + Filberts and cob-nuts are a special product of Kent, in the + neighbourhood of Maidstone principally, and upon the Ragstone soils, + certain conditions of soil and situation being essential for their + profitable production. A part of the filbert and cob-nut crop is + picked green in September, as they do well for dessert, though their + kernels are not large or firm, and it pays to sell them green, as they + weigh more heavily. One grower in Mid Kent has 100 acres of nuts, and + has grown 100 tons in a good year. The average price of late years has + been about 5d. per lb., which would make the gross return of the 100 + acres amount to L4660. Kentish filberts have long been proverbial for + their excellence. Cobs are larger and look better for dessert, though + their flavour is not so fine. They are better croppers, and are now + usually planted. This cultivation is not much extending, as it is very + long before the trees come into full bearing. The London market is + supplied entirely with these nuts from Kent, and there is some demand + in America for them. Filbert and cob trees are most closely pruned. + All the year's growth is cut away except the very finest young wood, + which the trained eye of the tree-cutter sees at a glance is + blossom-bearing. The trees are kept from 5-1/2 to 7 ft. high upon + stems from 1-1/2 to 2 ft. high, and are trained so as to form a cup of + from 7 to 8 ft. in diameter. + + There seems no reason to expect any decrease in the acreage of fruit + land in Kent, and if the improvement in the selection of varieties and + in the general management continues it will yet pay. A hundred years + ago every one was grubbing fruit land in order that hops might be + planted, and for this many acres of splendid cherry orchards were + sacrificed. Now the disposition is to grub hop plants and substitute + apples, plums, or small fruit or cherry trees. + + _Fruit-growing in other Districts._--The large fruit plantations in + the vicinity of London are to be found mostly in the valley of the + Thames, around such centres as Brentford, Isleworth, Twickenham, + Heston, Hounslow, Cranford and Southall. All varieties of orchard + trees, but mostly apples, pears, and plums and small fruit, are grown + in these districts, the nearness of which to the metropolitan fruit + market at Covent Garden is of course an advantage. Some of the + orchards are old, and are not managed on modern principles. They + contain, moreover, varieties of fruit many of which are out of date + and would not be employed in establishing new plantations. In the + better-managed grounds the antiquated varieties have been removed, and + their places taken by newer and more approved types. In addition to + apples, pears, plums, damsons, cherries and quinces as top fruit, + currants, gooseberries and raspberries are grown as bottom fruit. + Strawberries are extensively grown in some of the localities, and in + favourable seasons outdoor tomatoes are ripened and marketed. + + Fruit is extensively grown in Cambridgeshire and adjacent counties in + the east of England. A leading centre is Cottenham, where the Lower + Greensand crops out and furnishes one of the best of soils for + fruit-culture. In Cottenham about a thousand acres are devoted to + fruit, and nearly the same acreage to asparagus, which is, however, + giving place to fruit. Currants, gooseberries and strawberries are the + most largely grown, apples, plums and raspberries following. Of + varieties of plums the Victoria is first in favour, and then Rivers's + Early Prolific, Tsar and Gisborne. London is the chief market, as it + receives about half the fruit sent away, whilst a considerable + quantity goes to Manchester, and some is sent to a neighbouring jam + factory at Histon, where also a moderate acreage of fruit is grown. + Another fruit-growing centre in Cambridgeshire is at Willingham, + where--besides plums, gooseberries and raspberries--outdoor tomatoes + are a feature. Greengages are largely grown near Cambridge. Wisbech is + the centre of an extensive fruit district, situated partly in + Cambridgeshire and partly in Norfolk. Gooseberries, strawberries and + raspberries are largely grown, and as many as 80 tons of the + first-named fruit have been sent away from Wisbech station in a single + day. In the fruit-growing localities of Huntingdonshire apples, plums + and gooseberries are the most extensively grown, but pears, + greengages, cherries, currants, strawberries and raspberries are also + cultivated. As illustrating variations in price, it may be mentioned + that about the year 1880 the lowest price for gooseberries was L10 per + ton, whereas it has since been down to L4. Huntingdonshire fruit is + sent chiefly to Yorkshire, Scotland and South Wales, but railway + freights are high. + + Essex affords a good example of successful fruit-farming at Tiptree + Heath, near Kelvedon, where under one management about 260 acres out + of a total of 360 are under fruit. The soil, a stiff loam, grows + strawberries to perfection, and 165 acres are allotted to this fruit. + The other principal crops are 43 acres of raspberries and 30 acres of + black currants, besides which there are small areas of red currants, + gooseberries, plums, damsons, greengages, cherries, apples, quinces + and blackberries. The variety of strawberry known as the Small Scarlet + is a speciality here, and it occupies 55 acres, as it makes the best + of jam. The Paxton, Royal Sovereign and Noble varieties are also + grown. Strawberries stand for six or seven years on this farm, and + begin to yield well when two years old. A jam factory is worked in + conjunction with the fruit farm. Pulp is not made except when there is + a glut of fruit. Perishable fruit intended for whole-fruit preserves + is never held over after it is gathered. The picking of strawberries + begins at 4 A.M., and the first lot is made into jam by 6 A.M. + + Hampshire, like Cambridgeshire and Norfolk, are the only counties in + which the area of small fruit exceeds that of orchards. The returns + for 1908 show that Hampshire had 3320 acres of small fruit to 2236 + acres of orchards; Cambridge had 6878 acres of small fruit to 5221 of + orchards; and Norfolk had 5876 acres of small fruit against 5188 acres + of orchards. Compared with twenty years previously, the acreage of + small fruit had trebled. This is largely due in Hampshire to the + extension of strawberry culture in the Southampton district, where the + industry is in the hands of many small growers, few of whom cultivate + more than 20 acres each. Sarisbury and Botley are the leading parishes + in which the business is carried on. Most of the strawberry holdings + are from half an acre to 5 acres in extent, a few are from 5 to 10 + acres, fewer still from 10 to 20 acres and only half-a-dozen over that + limit. Runners from one-year plants are used for planting, being found + more fruitful than those from older plants. Peat-moss manure from + London stables is much used, but artificial manures are also employed + with good results. Shortly after flowering the plants are bedded down + with straw at the rate of about 25 cwt. per acre. Picking begins some + ten days earlier than in Kent, at a date between 1st June and 15th + June. The first week's gathering is sent mostly to London, but + subsequently the greater part of the fruit goes to the Midlands and to + Scotland and Ireland. + + In recent years fruit-growing has much increased in South + Worcestershire, in the vicinity of Evesham and Pershore. Hand-lights + are freely used in the market gardens of this district for the + protection of cucumbers and vegetable marrows, besides which tomatoes + are extensively grown out of doors. At one time the egg plum and the + Worcester damson were the chief fruit crops, apples and cherries + ranking next, pears being grown to only a moderate extent. According + to the 1908 returns, however, apples come first, plums second, pears + third and cherries fourth. In a prolific season a single tree of the + Damascene or Worcester damson will yield from 400 to 500 lb. of + fruit. There is a tendency to grow plum trees in the bush shape, as + they are less liable than standards to injury from wind. The manures + used include soot, fish guano, blood manure and phosphates--basic slag + amongst the last-named. In the Pershore district, where there is a jam + factory, plums are the chief tree fruit, whilst most of the orchard + apples and pears are grown for cider and perry. Gooseberries are a + feature, as are also strawberries, red and black currants and a few + white, but raspberries are little grown. The soil, a strong or medium + loam of fair depth, resting on clay, is so well adapted to plums that + trees live for fifty years. In order to check the ravages of the + winter moth, plum and apple trees are grease-banded at the beginning + of October and again at the end of March. The trees are also sprayed + when necessary with insecticidal solutions. Pruning is done in the + autumn. An approved distance apart at which to grow plum trees is 12 + ft. by 12 ft. In the Earl of Coventry's fruit plantation, 40 acres in + extent, at Croome Court, plums and apples are planted alternately, the + bottom fruit being black currants, which are less liable to injury + from birds than are red currants or gooseberries. Details concerning + the methods of cultivation of fruit and flowers in various parts of + England, the varieties commonly grown, the expenditure involved, and + allied matters, will be found in Mr W.E. Bear's papers in the _Journal + of the Royal Agricultural Society_ in 1898 and 1899. + + Apart altogether from market gardening and commercial fruit-growing, + it must be borne in mind that an enormous business is done in the + raising of young fruit-trees every year. Hundreds of thousands of + apples, pears, plums, cherries, peaches, nectarines and apricots are + budded or grafted each year on suitable stocks. They are trained in + various ways, and are usually fit for sale the third year. These young + trees replace old ones in private and commercial gardens, and are also + used to establish new plantations in different parts of the kingdom. + + _The Woburn Experimental Fruit Farm._--The establishment in 1894 of + the experimental fruit farm at Ridgmont, near Woburn, Beds, has + exercised a healthy influence upon the progress and development of + fruit-farming in England. The farm was founded and carried on by the + public-spirited enterprise of the Duke of Bedford and Mr Spencer U. + Pickering, the latter acting as director. The main object of the + experimental station was "to ascertain facts relative to the culture + of fruit, and to increase our knowledge of, and to improve our + practice in, this industry." The farm is 20 acres in extent, and + occupies a field which up to June 1894 had been used as arable land + for the ordinary rotation of farm crops. The soil is a sandy loam 9 or + 10 in. deep, resting on a bed of Oxford Clay. Although it contains a + large proportion of sand, the land would generally be termed very + heavy, and the water often used to stand on it in places for weeks + together in a wet season. The tillage to which the ground was + subjected for the purposes of the fruit farm much improved its + character, and in dry weather it presents as good a tilth as could be + desired. Chemical analyses of the soil from different parts of the + field show such wide differences that it is admitted to be by no means + an ideal one for experimental purposes. Without entering upon further + details, it may be useful to give a summary of the chief results + obtained. + + Apples have been grown and treated in a variety of ways, but of the + different methods of treatment careless planting, coupled with + subsequent neglect, has given the most adverse results, the crop of + fruit being not 5% of that from trees grown normally. Of the separate + deleterious items constituting total neglect, by far the most + effective was the growth of weeds on the surface; careless planting, + absence of manure, and the omission of trenching all had comparatively + little influence on the results. A set of trees that had been + carelessly planted and neglected, but subsequently tended in the early + part of 1896, were in the autumn of that year only 10% behind their + normally-treated neighbours, thus demonstrating that the response to + proper attention is prompt. The growth of grass around young apple + trees produced a very striking effect, the injury being much greater + than that due to weeds. It is possible, however, that in wet years the + ill-effects of both grass and weeds would be less than in dry seasons. + Nevertheless, the grass-grown trees, after five years, were scarcely + bigger than when planted, and the actual increase in weight which they + showed during that time was about eighteen times smaller than in the + case of similar trees in tilled ground. It is believed that one of the + main causes of the ill-effects is the large increase in the + evaporation of water from the soil which is known to be produced by + grass, the trees being thereby made to suffer from drought, with + constant deprivation of other nourishment as well. That grass growing + round young apple trees is deleterious was a circumstance known to + many horticulturists, but the extent to which it interferes with the + development of the trees had never before been realized. Thousands of + pounds are annually thrown away in England through want of knowledge + of this fact. Yet trees will flourish in grass under certain + conditions. Whether the dominant factor is the age (or size) of the + tree has been investigated by grassing over trees which have hitherto + been in the open ground, and the results appear to indicate that the + grass is as deleterious to the older trees as it was to the younger + ones. Again, it appears to have been demonstrated that young apple + trees, at all events in certain soils, require but little or no manure + in the early stages of their existence, so that in this case also + large sums must be annually wasted upon manurial dressings which + produce no effects. The experiments have dealt with dwarf trees of + Bramley, Cox and Potts, six trees of each variety constituting one + investigation. Some of the experiments were repeated with Stirling + Castle, and others with standard trees of Bramley, Cox and Lane's + Prince Albert. All were planted in 1894-1895, the dwarfs being then + three years old and the standards four. In each experiment the + "normal" treatment is altered in some one particular, this normal + treatment consisting of planting the trees carefully in trenched + ground, and subsequently keeping the surface clean; cutting back after + planting, pruning moderately in autumn, and shortening the growths + when it appeared necessary in summer; giving in autumn a dressing of + mixed mineral manures, and in February one of nitrate of soda, this + dressing being probably equivalent to one of 12 tons of dung per acre. + In the experiments on branch treatment, the bad effects of omitting to + cut the trees back on planting, or to prune them subsequently, is + evident chiefly in the straggling and bad shape of the resulting + trees, but such trees also are not so vigorous as they should be. The + quantity of fruit borne, however, is in excess of the average. The + check on the vigour and growth of a tree by cutting or injuring its + roots is in marked contrast with the effects of a similar interference + with the branches. Trees which had been root-pruned each year were in + 1898 little more than half as big as the normal trees, whilst those + root-pruned every second year were about two-thirds as big as the + normal. The crops borne by these trees were nevertheless heavy in + proportion to the size of the trees. Such frequent root-pruning is + not, of course, a practice which should be adopted. It was found that + trees which had been carefully lifted every other year and replanted + at once experienced no ill-effects from the operation; but in a case + where the trees after being lifted had been left in a shed for three + days before replanting--which would reproduce to a certain extent the + conditions experienced when trees are sent out from a + nursery--material injury was suffered, these trees after four years + being 28% smaller than similar ones which had not been replanted. Sets + of trees planted respectively in November, January and March have, on + the whole, shown nothing in favour of any of these different times for + planting purposes. Some doubt is thrown on the accepted view that + there is a tendency, at any rate with young apple and pear trees, to + fruit in alternate seasons. + + Strawberries of eighty-five different varieties have been experimented + with, each variety being represented in 1900 by plants of five + different ages, from one to five years. In 1896 and 1898 the crops of + fruit were about twice as heavy as in 1897 and 1899, but it has not + been found possible to correlate these variations with the + meteorological records of the several seasons. Taking the average of + all the varieties, the relative weights of crop per plant, when these + are compared with the two-year-old plants in the same season, are, for + the five ages of one to five years, 31, 100, 122, 121 and 134, + apparently showing that the bearing power increases rapidly up to two + years, less rapidly up to three years, after which age it remains + practically constant. The relative average size of the berries shows a + deterioration with the age of the plant. The comparative sizes from + plants of one to five years old were 115, 100, 96, 91 and 82 + respectively. If the money value of the crop is taken to be directly + dependent on its total weight, and also on the size of the fruits, the + relative values of the crop for the different ages would be 34, 100, + 117, 111 and 110, so that, on the Ridgmont ground, strawberry plants + could be profitably retained up to five years and probably longer. As + regards what may be termed the order of merit of different varieties + of strawberries, it appears that even small differences in position + and treatment cause large variations, not only in the features of the + crop generally, but also in the relative behaviour of the different + varieties. The relative cropping power of the varieties under + apparently similar conditions may often be expressed by a number five + or tenfold as great in one case as in the other. A comparison of the + relative behaviour of the same varieties in different seasons is + attended by similar variations. The varying sensitiveness of different + varieties of strawberry plants to small and undefinable differences in + circumstances is indeed one of the most important facts brought to + light in the experiments. + + _Fruit Culture in Ireland._--The following figures have been kindly + supplied by the Irish Board of Agriculture, and deal with the acreage + under fruit culture in Ireland up to the end of the year 1907. + + 1. _Orchard Fruit_-- Statute Acres. + Apples 5829 + Pears 224 + Plums 223 + Damsons 138 + Other kinds 129 + ---- + Total 6543 + + 2. _Small Fruit_-- + Currants, black 234 + Currants, red and white 159 + Gooseberries 675 + Raspberries 374 + Strawberries 994 + Mixed fruit 2470 + ---- + Total 4906 + + It therefore appears that while Ireland grows only about + one-thirty-third the quantity of apples that England does, it is + nevertheless nearly 5000 acres ahead of Scotland and about 2000 acres + ahead of Wales. It grows 41 times fewer pears than England, but still + is ahead of Scotland and a long way ahead of Wales in this fruit. + There are 70 times fewer plums grown in Ireland than in England, and + about the same in Scotland, while Wales does very little indeed. In + small fruit Ireland is a long way behind Scotland in the culture of + strawberries and raspberries, although with currants and gooseberries + it is very close. Considering the climate, and the fact that there + are, according to the latest available returns, over 62,000 holdings + above 1 acre but not exceeding 5 acres (having a total of 224,000 + acres), it is possible fruit culture may become more prevalent than it + has been in the past. + +_The Flower-growing Industry._--During the last two or three decades of +the 19th century a very marked increase in flower production occurred in +England. Notably was this the case in the neighbourhood of London, +where, within a radius of 15 or 20 m., the fruit crops, which had +largely taken the place of garden vegetables, were themselves ousted in +turn to satisfy the increasing demand for land for flower cultivation. +No flower has entered more largely into the development of the industry +than the narcissus or daffodil, of which there are now some 600 +varieties. Comparatively few of these, however, are grown for market +purposes, although all are charming from the amateur point of view. On +some flower farms a dozen or more acres are devoted to narcissi alone, +the production of bulbs for sale as well as of flowers for market being +the object of the growers. + +In the London district the country in the Thames valley west of the +metropolis is as largely occupied by flower farms as it is by fruit +farms--in fact, the cultivation of flowers is commonly associated with +that of fruit. In the vicinity of Richmond narcissi are extensively +grown, as they also are more to the west in the Long Ditton district, +and likewise around Twickenham, Isleworth, Hounslow, Feltham and +Hampton. Roses come more into evidence in the neighbourhood of Hounslow, +Cranford, Hillingdon and Uxbridge, and in some gardens daffodils and +roses occupy alternate rows. In this district also such flowers as +herbaceous paeonies, Spanish irises, German irises, Christmas roses, +lilies of the valley, chrysanthemums, foxgloves, hollyhocks, +wallflowers, carnations, &c., are extensively grown in many market +gardens. South of London is the Mitcham country, long noted for its +production of lavender. The incessant growth of the lavender plant upon +the same land, however, has led to the decline of this industry, which +has been largely transferred to districts in the counties of Bedford, +Essex and Hertford. At Mitcham, nevertheless, mixed flowers are very +largely grown for the supply of the metropolis, and one farm alone has +nearly 100 acres under flowers and glass-houses. Chrysanthemums, asters, +Iceland poppies, gaillardias, pansies, bedding calceolarias, zonal +pelargoniums and other plants are cultivated in immense quantities. At +Swanley and Eynsford, in Kent, flowers are extensively cultivated in +association with fruit and vegetables. Narcissi, chrysanthemums, +violets, carnations, campanulas, roses, pansies, irises, sweet peas, and +many other flowers are here raised, and disposed of in the form both of +cut flowers and of plants. + +The Scilly Isles are important as providing the main source of supply of +narcissi to the English markets in the early months of the year. This +trade arose almost by accident, for it was about the year 1865 that a +box of narcissi sent to Covent Garden Market, London, realized L1; and +the knowledge of this fact getting abroad, the farmers of the isles +began collecting wild bulbs from the fields in order to cultivate them +and increase their stocks. Some ten years, however, elapsed before the +industry promised to become remunerative. In 1885 a Bulb and Flower +Association was established to promote the industrial growth of flowers. +The exports of flowers in that year reached 65 tons, and they steadily +increased until 1893, when they amounted to 450 tons. A slight decline +followed, but in 1896 the quantity exported was no less than 514 tons. +This would represent upwards of 3-1/2 million bunches of flowers, +chiefly narcissi and anemones. Rather more than 500 acres are devoted to +flower-growing in the isles, by far the greater part of this area being +assigned to narcissi, whilst anemones, gladioli, marguerites, arum +lilies, Spanish irises, pinks and wallflowers are cultivated on a much +smaller scale. The great advantage enjoyed by the Scilly flower-growers +is earliness of production, due to climatic causes; the soil, moreover, +is well suited to flower culture and there is an abundance of sunshine. +The long journey to London is somewhat of a drawback, in regard to both +time and freight, but the earliness of the flowers more than compensates +for this. Open-air narcissi are usually ready at the beginning of +January, and the supply is maintained in different varieties up to the +middle or end of May. The narcissus bulbs are usually planted in +October, 4 in. by 3 in. apart for the smaller sorts and 6 in. by 4 to 6 +in. for the larger. A compost of farmyard manure, seaweed, earth and +road scrapings is the usual dressing, but nitrate of soda, guano and +bones are also occasionally employed. A better plan, perhaps, is to +manure heavily the previous crop, frequently potatoes, no direct +manuring then being needed for the bulbs, these not being left in the +ground more than two or three years. The expenses of cultivation are +heavy, the cost of bulbs alone--of which it requires nearly a quarter of +a million of the smaller varieties, or half as many of the largest, to +plant an acre--being considerable. The polyanthus varieties of narcissus +are likely to continue the most remunerative to the flower-growers of +Scilly, as they flourish better in these isles than on the mainland. + +In the district around the Wash, in the vicinity of such towns as +Wisbech, Spalding and Boston, the industrial culture of bulbs and +flowers underwent great expansion in the period between 1880 and 1909. +At Wisbech one concern alone has a farm of some 900 acres, devoted +chiefly to flowers and fruit, the soil being a deep fine alluvium. Roses +are grown here, one field containing upwards of 100,000 trees. Nearly 20 +acres are devoted to narcissi, which are grown for the bulbs and also, +together with tulips, for cut flowers. Carnations are cultivated both +in the field and in pots. Cut flowers are sent out in large quantities, +neatly and effectively packed, the parcel post being mainly employed as +a means of distribution. In the neighbourhood of Spalding crocuses and +snowdrops are less extensively grown than used to be the case. On one +farm, however, upwards of 20 acres are devoted to narcissi alone, whilst +gladioli, lilies and irises are grown on a smaller scale. Around Boston +narcissi are also extensively grown for the market, both bulbs and cut +blooms being sold. The bulbs are planted 3 in. apart in rows, the latter +being 9 in. apart, and are allowed to stand from two to four years. + + The imports of fresh flowers into the United Kingdom were not + separately shown prior to 1900. In that year, however, their value + amounted to L200,585, in 1901 to L225,011, in 1906 to L233,884, in + 1907 to L233,641, and in 1908 to L229,802, so that the trade showed a + fairly steady condition. From the monthly totals quoted in Table VI. + it would appear that the trade sinks to its minimum dimensions in the + four months July to October inclusive, and that after September the + business continually expands up to April, subsequent to which + contraction again sets in. About one-half of the trade belongs + practically to the three months of February, March and April. + + + TABLE VI.--_Values of Fresh Flowers imported into the United Kingdom._ + + +-----------+----------+----------+----------+ + | Month. | 1906. | 1907. | 1908. | + +-----------+----------+----------+----------+ + | January | L31,035 | L18,545 | L29,180 | + | February | 34,647 | 25,541 | 30,541 | + | March | 50,232 | 42,611 | 35,185 | + | April | 30,809 | 50,418 | 42,681 | + | May | 22,980 | 21,767 | 23,129 | + | June | 17,641 | 18,358 | 16,904 | + | July | 3,386 | 4,509 | 3,467 | + | August | 1,646 | 1,539 | 1,081 | + | September | 852 | 736 | 953 | + | October | 4,481 | 3,180 | 4,504 | + | November | 17,506 | 15,763 | 15,097 | + | December | 18,669 | 30,674 | 27,080 | + | +----------+----------+----------+ + | Total | L233,884 | L233,641 | L229,802 | + +-----------+----------+----------+----------+ + +_Hothouse Culture of Fruit and Flowers._--The cultivation of fruit and +flowers under glass has increased enormously since about the year 1880, +especially in the neighbourhood of London, where large sums of money +have been sunk in the erection and equipment of hothouses. In the parish +of Cheshunt, Herts, alone there are upwards of 130 acres covered with +glass, and between that place on the north and London on the south +extensive areas of land are similarly utilized. In Middlesex, in the +north, in the districts of Edmonton, Enfield, Ponders End and Finchley, +and in the west from Isleworth to Hampton, Feltham, Hillingdon, Sipson +and Uxbridge, many crops are now cultivated under glass. At Erith, +Swanley, and other places in Kent, as also at Worthing, in Sussex, +glass-house culture has much extended. A careful estimate puts the area +of industrial hothouses in England at about 1200 acres, but it is +probably much more than this. Most of the greenhouses are fixtures, but +in some parts of the kingdom structures that move on rails and wheels +are used, to enable the ground to be prepared in the open for one crop +while another is maturing under glass. The leading products are grapes, +tomatoes and cucumbers, the last-named two being true fruits from the +botanist's point of view, though commercially included with vegetables. +To these may be added on the same ground dwarf or French beans, and +runner or climbing beans. Peaches, nectarines and strawberries are +largely grown under glass, and, in private hothouses--from which the +produce is used mainly for household consumption, and which are not +taken into consideration here--pineapples, figs and other fruit. +Conservative estimates indicate the average annual yield of hothouse +grapes to be about 12 tons per acre and of tomatoes 20 tons. The greater +part of the space in the hothouses is assigned to fruit, but whilst some +houses are devoted exclusively to flowers, in others, where fruit is the +main object, flowers are forced in considerable quantities in winter and +early spring. The flowers grown under glass include tulips, hyacinths, +primulas, cyclamens, spiraeas, mignonettes, fuchsias, calceolarias, +roses, chrysanthemums, daffodils, arum lilies or callas, liliums, +azaleas, eucharises, camellias, stephanotis, tuberoses, bouvardias, +gardenias, heaths or ericas, poinsettias, lilies of the valley, zonal +pelargoniums, tuberous and fibrous rooted begonias, and many others. +There is an increasing demand for foliage hothouse plants, such as +ferns, palms, crotons, aspidistras, araucarias, dracaenas, India-rubber +plants, aralias, grevilleas, &c. Berried plants like solanums and +aucubas also find a ready sale, while the ornamental kinds of asparagus +such as _sprengeri_ and _plumosus_ nanus, are ever in demand for +trailing decorations, as well as myrsiphyilum. Special mention must be +made of the winter or perpetual flowering carnations which are now grown +by hundreds of thousands in all parts of the kingdom for decorative work +during the winter season. The converse of forcing plants into early +blossom is adopted with such an important crop as lily of the valley. +During the summer season the crowns are placed in refrigerators with +about 2 degrees of frost, and quantities are taken out as required every +week and transferred to the greenhouse to develop. Tomatoes are grown +largely in houses exclusively occupied by them, in which case two and +sometimes three crops can be gathered in the year. In the Channel +Islands, where potatoes grown under glass are lifted in April and May, +in order to secure the high prices of the early markets, tomato +seedlings are planted out from boxes into the ground as quickly as the +potatoes are removed, the tomato planter working only a few rows behind +the potato digger. The trade in imported tomatoes is so considerable +that home growers are well justified in their endeavours to meet the +demand more fully with native produce, whether raised under glass or in +the open. Tomatoes were not separately enumerated in the imports +previous to 1900. It has already been stated that in 1900 the raw +tomatoes imported amounted to 833,032 cwt., valued at L792,339, and in +1901 to 793,991 cwt., valued at L734,051. From the monthly quantities +given in Table VII., it would appear that the imports are largest in +June, July and August, about one-half of the year's total arriving +during those three months. It is too early in June and July for +home-grown outdoor tomatoes to enter into competition with the imported +product, but home-grown hothouse tomatoes should be qualified to +challenge this trade. + + +TABLE VII.--_Quantities of Tomatoes imported into the United Kingdom._ + + +-----------+-----------+------------+------------+ + | Month. | 1906. | 1907. | 1908. | + +-----------+-----------+------------+------------+ + | January | 61,940 | 56,022 | 73,409 | + | February | 58,187 | 58,289 | 69,350 | + | March | 106,458 | 98,028 | 86,928 | + | April | 103,273 | 109,057 | 74,917 | + | May | 67,933 | 114,041 | 88,901 | + | June | 62,906 | 144,379 | 127,793 | + | July | 238,362 | 150,907 | 171,978 | + | August | 180,046 | 102,600 | 124,757 | + | September | 114,860 | 101,198 | 119,224 | + | October | 52,678 | 67,860 | 75,722 | + | November | 41,513 | 66,522 | 74,292 | + | December | 36,316 | 66,591 | 73,012 | + | +-----------+------------+------------+ + | Total | 1,124,472 | 1,135,494 | 1,160,283 | + | +-----------+------------+------------+ + | Value | L953,475 | L1,135,499 | L1,160,283 | + +-----------+-----------+------------+------------+ + +An important feature of modern flower growing is the production and +cultivation of what are known as "hardy herbaceous perennials." Some +2000 or 3000 different species and varieties of these are now raised in +special nurseries; and during the spring, summer and autumn seasons +magnificent displays are to be seen not only in the markets but at the +exhibitions in London and at the great provincial shows held throughout +the kingdom. The production of many of these perennials is so easy that +amateurs in several instances have taken it up as a business hobby; and +in some cases, chiefly through advertising in the horticultural press, +very lucrative concerns have been established. + +Ornamental flowering trees and shrubs constitute another feature of +modern gardening. These are grown and imported by thousands chiefly for +their sprays of blossom or foliage, and for planting in large or small +gardens, public parks, &c., for landscape effect. Indeed there is +scarcely an easily grown plant from the northern or southern temperate +zones that does not now find a place in the nursery or garden, provided +it is sufficiently attractive to sell for its flowers, foliage or +appearance. + +_Conditions of the Fruit and Flower growing Industries._--As regards +open-air fruit-growing, the outlook for new ventures is perhaps brighter +than in the hothouse industry, not--as Mr Bear has pointed out--because +the area of fruit land in England is too small, but because the level of +efficiency, from the selection of varieties to the packing and marketing +of the produce, is very much lower in the former than in the latter +branch of enterprise. In other words, whereas the practice of the +majority of hothouse nurserymen is so skilled, so up-to-date, and so +entirely under high pressure that a new competitor, however well +trained, will find it difficult to rise above mediocrity, the converse +is true of open-air fruit-growers. Many, and an increasing proportion, +of the latter are thoroughly efficient in all branches of their +business, and are in possession of plantations of the best market +varieties of fruit, well cultivated, pruned and otherwise managed. But +the extent of fruit plantations completely up to the mark in relation to +varieties and treatment of trees and bushes, and in connexion with which +the packing and marketing of the produce are equally satisfactory, is +small in proportion to the total fruit area of the country. Information +concerning the best treatment of fruit trees has spread widely in recent +years, and old plantations, as a rule, suffer from the neglect or errors +of the past, however skilful their present holders may be. Although the +majority of professional market fruit-growers may be well up to the +standard in skill, there are numerous contributors to the fruit supply +who are either ignorant of the best methods of cultivation and marketing +or careless in their application. The bad condition of the great +majority of farm orchards is notorious, and many landowners, farmers and +amateur gardeners who have planted fruit on a more or less extensive +scale have mismanaged their undertakings. For these reasons new growers +of open-air fruit for market have opportunities of succeeding by means +of superiority to the majority of those with whom they will compete, +provided that they possess the requisite knowledge, energy and capital. +It has been asserted on sound authority that there is no chance of +success for fruit-growers except in districts favourable as regards +soil, climate and nearness to a railway or a good market; and, even +under these conditions, only for men who have had experience in the +industry and are prepared to devote their unremitting attention to it. +Most important is it to a beginner that he should ascertain the +varieties of fruit that flourish best in his particular district. +Certain kinds seem to do well or fairly well in all parts of the +country; others, whilst heavy croppers in some localities, are often +unsatisfactory in others. + +As has been intimated, there is probably in England less room for +expansion of fruit culture under glass than in the open. The large +increase of glass-houses in modern times appears to have brought the +supply of hothouse produce, even at greatly reduced prices, at least up +to the level of the demand; and as most nurserymen continue to extend +their expanse of glass, the prospect for new competitors is not a bright +one. Moreover, the vast scale upon which some of the growers conduct the +hothouse industry puts small producers at a great disadvantage, not only +because the extensive producers can grow grapes and other fruit more +economically than small growers--with the possible exception of those +who do all or nearly all their own work--but also, and still more, +because the former have greater advantages in transporting and marketing +their fruit. There has, in recent years, been a much greater fall in the +prices of hothouse than of open-air fruit, especially under the existing +system of distribution, which involves the payment by consumers of 50 to +100% more in prices than growers receive. The best openings for new +nurseries are probably not where they are now to be found in large +groups, and especially not in the neighbourhood of London, but in +suitable spots near the great centres of population in the Midlands and +the North, or big towns elsewhere not already well supplied with +nurseries. By such a selection of a locality the beginner may build up a +retail trade in hothouse fruit, or at least a trade with local +fruiterers and grocers, thus avoiding railway charges and salesmen's +commissions to a great extent, though it may often be advantageous to +send certain kinds of produce to a distant market. Above all, a man who +has no knowledge of the hothouse industry should avoid embarking his +capital in it, trusting himself in the hands of a foreman, as experience +shows that such a venture usually leads to disaster. Some years of +training in different nurseries are desirable for any young man who is +desirous of becoming a grower of hothouse fruits or flowers. + +There can be no doubt that flower-growing is greatly extending in +England, and that competition among home growers is becoming more +severe. Foreign supplies of flowers have increased, but not nearly as +greatly in proportion as home supplies, and it seems clear that home +growers have gained ground in relation to their foreign rivals, except +with respect to flowers for the growth of which foreigners have +extraordinary natural advantages. There seems some danger of the home +culture of the narcissus being over-done, and the florists' +chrysanthemum appears to be produced in excess of the demand. Again, in +the production of violets the warm and sunny South of France has an +advantage not possessed by England, whilst Holland, likewise for +climatic reasons, maintains her hold upon the hyacinth and tulip trade. +Whether the production of flowers as a whole is gaining ground upon the +demand or not is a difficult question to answer. It is true that the +prices of flowers have fallen generally; but production, at any rate +under glass, has been cheapened, and if a fair profit can be obtained, +the fall in prices, without which the existing consumption of flowers +would be impossible, does not necessarily imply over-production. There +is some difference of opinion among growers upon this point; but nearly +all agree that profits are now so small that production on a large scale +is necessary to provide a fair income. Industrial flower-growing affords +such a wide scope for the exercise of superior skill, industry and +alertness, that it is not surprising to find some who are engaged in it +doing remarkably well to all appearance, while others are struggling on +and hardly paying their way. That a man with only a little capital, +starting in a small way, has many disadvantages is certain; also, that +his chance of saving money and extending his business quickly is much +smaller than it was. To the casual looker-on, who knows nothing of the +drudgery of the industry, flower-growing seems a delightful method of +getting a living. That it is an entrancing pursuit there is no doubt; +but it is equally true that it is a very arduous one, requiring careful +forethought, ceaseless attention and abundant energy. Fortunately for +those who might be tempted, without any knowledge of the industry, to +embark capital in it, flower-growing, if at all comprehensive in scope, +so obviously requires a varied and extensive technical knowledge, +combined with good commercial ability, that any one can see that a +thorough training is necessary to a man who intends to adopt it as a +business, especially if hothouse flowers are to be produced. + + The market for fruit, and more especially for flowers, is a fickle + one, and there is nearly always some uncertainty as to the course of + prices. The perishable nature of soft fruit and cut flowers renders + the markets very sensitive to anything in the nature of a glut, the + occurrence of which is usually attended with disastrous results to + producers. Foreign competition, moreover, has constantly to be faced, + and it is likely to increase rather than diminish. French growers have + a great advantage over the open-air cultivators of England, for the + climate enables them to get their produce into the markets early in + the season, when the highest prices are obtainable. The geographical + advantage which France enjoys in being so near to England is, however, + considerably discounted by the increasing facilities for cold storage + in transit, both by rail and sea. The development of such facilities + permits of the retail sale in England of luscious fruit as fresh and + attractive as when it was gathered beneath the sunny skies of + California. In the case of flowers, fashion is an element not to be + ignored. Flowers much in request in one season may meet with very + little demand in another, and it is difficult for the producer to + anticipate the changes which caprice may dictate. Even for the same + kind of flower the requirements are very uncertain, and the white + blossom which is all the rage in one season may be discarded in favour + of one of another colour in the next. The sale of fresh flowers for + church decoration at Christmas and Easter has reached enormous + dimensions. The irregularity in the date of the festival, however, + causes some inconvenience to growers. If it falls very early the great + bulk of suitable flowers may not be sufficiently forward for sale, + whilst a late Easter may find the season too far advanced. The trade + in cut flowers, therefore, is generally attended by uncertainty, and + often by anxiety. (W. FR.; J. WS.) + + +UNITED STATES + +In the United States horticulture and market gardening have now assumed +immense proportions. In a country of over 3,000,000 sq. m., stretching +from the Atlantic to the Pacific on the one hand, and from the Gulf of +Mexico to the great northern lakes and the Dominion of Canada on the +other, a great variation of climatic conditions is not unnatural. From a +horticultural point of view there are practically two well-defined +regions: (1) that to the east of the Rocky Mountains across to the +Atlantic, where the climate is more like that of eastern Asia than of +western Europe so far as rainfall, temperature and seasonable conditions +are concerned; (2) that to the west of the Rockies, known as the Pacific +coast region, where the climate is somewhat similar to that of western +Europe. It may be added that in the northern states--in Washington, +Montana, North Dakota, Minnesota, Wisconsin, &c.--the winters are often +very severe, while the southern states practically enjoy a temperature +somewhat similar to that of the Riviera. Indeed the range of temperature +between the extreme northern states and the extreme southern may vary as +much as 120 deg. F. The great aim of American gardeners, therefore, has +been to find out or to produce the kinds of fruits, flowers and +vegetables that are likely to flourish in different parts of this +immense country. + +_Fruit Culture._--There is probably no country in the world where so +many different kinds of fruit can be grown with advantage to the nation +as in the United States. In the temperate regions apples, pears and +plums are largely grown, and orchards of these are chiefly to be found +in the states of New York, Massachusetts, Pennsylvania, Michigan, +Missouri, Colorado, and also in northern Texas, Arkansas and N. +California. To these may be added cranberries and quinces, which are +chiefly grown in the New England states. The quinces are not a crop of +first-rate importance, but as much as 800,000 bushels of cranberries are +grown each year. The peach orchards are assuming great proportions, and +are chiefly to be found in Georgia and Texas, while grapes are grown +throughout the Republic from east to west in all favourable localities. +Oranges, lemons and citrons are more or less extensively grown in +Florida and California, and in these regions what are known as Japanese +or "Kelsey" plums (forms of _Prunus triflora_) are also grown as +marketable crops. Pomegranates are not yet largely grown, but it is +possible their culture will develop in southern Texas and Louisiana, +where the climate is tempered by the waters of the Gulf of Mexico. +Tomatoes are grown in most parts of the country so easily that there is +frequently a glut; while the strawberry region extends from Florida to +Virginia, Pennsylvania and other states--thus securing a natural +succession from south to north for the various great market centres. + +Of the fruits mentioned apples are undoubtedly the most important. Not +only are the American people themselves supplied with fresh fruit, but +immense quantities are exported to Europe--Great Britain alone absorbing +as much as 1,430,000 cwt. in 1908. The varieties originally grown were +of course those taken or introduced from Europe by the early settlers. +Since the middle of the 19th century great changes have been brought +about, and the varieties mostly cultivated now are distinctly American. +They have been raised by crossing and intercrossing the most suitable +European forms with others since imported from Russia. In the extreme +northern states indeed, where it is essential to have apple trees that +will stand the severest winters, the Russian varieties crossed with the +berry crab of eastern Europe (_Pyrus baccata_) have produced a race +eminently suited to that particular region. The individual fruits are +not very large, but the trees are remarkably hardy. Farther south larger +fruited varieties are grown, and among these may be noted Baldwins, +Newton pippins, Spitzenbergs and Rhode Island greening. Apple orchards +are numerous in the State of New York, where it is estimated that over +100,000 acres are devoted to them. In the hilly regions of Missouri, +Arkansas and Colorado there are also great plantations of apples. The +trees, however, are grown on different principles from those in New York +State. In the latter state apple trees with ordinary care live to more +than 100 years of age and produce great crops; in the other states, +however, an apple tree is said to be middle-aged at 20, decrepit at 30 +and practically useless at 40 years of age. They possess the advantage, +however, of bearing early and heavily. + +Until the introduction of the cold-storage system, about the year 1880, +America could hardly be regarded as a commercial fruit-growing country. +Since then, however, owing to the great improvements made in railway +refrigerating vans and storage houses, immense quantities of fruit can +be despatched in good condition to any part of the world; or they can be +kept at home in safety until such time as the markets of Chicago, New +York, Boston, Baltimore, Philadelphia, &c., are considered favourable +for their reception. + +Apple trees are planted at distances varying from 25 ft. to 30 ft. apart +in the middle western states, to 40 ft. to 50 ft. apart in New York +State. Here and there, however, in some of the very best orchards the +trees are planted 60 ft. apart every way. Each tree thus has a chance to +develop to its utmost limits, and as air and light reach it better, a +far larger fruit-bearing surface is secured. Actual experience has shown +that trees planted at 60 ft. apart--about 28 to the acre--produce more +fruit by 43 bushels than trees at 30 ft. apart--i.e. about 48 to the +acre. + +Until recent years pruning as known to English and French gardeners was +practically unknown. There was indeed no great necessity for it, as the +trees, not being cramped for space, threw their branches outwards and +upwards, and thus rarely become overcrowded. When practised, however, +the operation could scarcely be called pruning; lopping or trimming +would be more accurate descriptions. + +Apple orchards are not immune from insect pests and fungoid diseases, +and an enormous business is now done in spraying machines and various +insecticides. It pays to spray the trees, and figures have been given to +show that orchards that have been sprayed four times have produced an +average income of L211 per acre against L103 per acre from unsprayed +orchards. + +The spring frosts are also troublesome, and in the Colorado and other +orchards the process known as "smudging" is now adopted to save the +crops. This consists in placing 20 or 30, or even more, iron or tin pots +to an acre, each pot containing wooden chips soaked in tar (or pitch) +mixed with kerosene. Whenever the thermometer shows 3 or 4 degrees of +frost the smudge-pots are lighted. A dense white smoke then arises and +is diffused throughout the orchards, enveloping the blossoming heads of +the trees in a dense cloud. This prevents the frost from killing the +tender pistils in the blossoms, and when several smudge-pots are alight +at the same time the temperature of the orchard is raised two or three +degrees. This work has generally to be done between 3 and 5 A.M., and +the growers naturally have an anxious time until all danger is over. The +failure to attend to smudging, even on one occasion, may result in the +loss of the entire crop of plums, apples or pears. + +Next to apples perhaps peaches are the most important fruit crop. The +industry is chiefly carried on in Georgia, Texas and S. Carolina, and on +a smaller scale in some of the adjoining states. Peaches thus flourish +in regions that are quite unsuitable for apples or pears. In many +orchards in Georgia, where over 3,000,000 acres have been planted, there +are as many as 100,000 peach trees; while some of the large fruit +companies grow as many as 365,000. In one place in West Virginia there +is, however, a peach orchard containing 175,000 trees, and in Missouri +another company has 3 sq. m. devoted to peach culture. As a rule the +crops do well. Sometimes, however, a disease known as the "yellows" +makes sad havoc amongst them, and scarcely a fruit is picked in an +orchard which early in the season gave promise of a magnificent crop. + +Plums are an important crop in many states. Besides the European +varieties and those that have been raised by crossing with American +forms, there is now a growing trade done in Japanese plums. The largest +of these is popularly known as "Kelseys," named after John Kelsey, who +raised the first fruit in 1876 from trees brought to California in 1870. +Sometimes the fruits are 3 in. in diameter, and like most of the +Japanese varieties are more heart-shaped and pointed than plums of +European origin. One apparent drawback to the Kelsey plum is its +irregularity in ripening. It has been known in some years to be quite +ripe in June, while in others the fruits are still green in October. + +Pears are much grown in such states as Massachusetts, New York, +Pennsylvania, Missouri and California; while bush fruits like currants, +gooseberries and raspberries find large spaces devoted in most of the +middle and northern states. Naturally a good deal of crossing and +intercrossing has taken place amongst the European and American forms of +these fruits, but so far as gooseberries are concerned no great advance +seems to have been made in securing varieties capable of resisting the +devastating gooseberry mildew. + +Other fruits of more or less commercial value are oranges, lemons and +citrons, chiefly in Florida. Lemons are practically a necessity to the +American people, owing to the heat of the summers, when cool and +refreshing drinks with an agreeable acidulous taste are in great demand. +The pomelo (grape-fruit) is a kind of lemon with a thicker rind and a +more acid flavour. At one time its culture was confined to Florida, but +of recent years it has found its way into Californian orchards. +Notwithstanding the prevailing mildness of the climate in both +California and Florida, the crops of oranges, lemons, citrons, &c., are +sometimes severely injured by frosts when in blossom. + +Other fruits likely to be heard of in the future are the kaki or +persimmon, the loquat, which is already grown in Louisiana, as well as +the pomegranate. + +Great aid and encouragement are given by the government to the progress +of American fruit-growing, and by the experiments that are being +constantly carried out and tabulated at Cornell University and by the +U.S.A. department of agriculture. + +_Flower Culture._--So far as flowers are concerned there appears to be +little difference between the kinds of plants grown in the United States +and in England, France, Belgium, Germany, Holland, &c. Indeed there is a +great interchange of new varieties of plants between Europe and America, +and modifications in systems of culture are being gradually introduced +from one side of the Atlantic to the other. The building of greenhouses +for commercial purposes is perhaps on a somewhat different scale from +that in England, but there are probably no extensive areas of glass such +as are to be seen north of London from Enfield Highway to Broxburne. Hot +water apparatus differs merely in detail, although most of the boilers +used resemble those on the continent of Europe rather than in England. +Great business is done in bulbs--mostly imported from Holland--stove and +greenhouse plants, hardy perennials, orchids, ferns of the "fancy" and +"dagger" types of Nephrolepis, and in carnations and roses. Amongst the +latter thousands of such varieties as Beauty, Liberty, Killarney, +Richmond and Bride are grown, and realize good prices as a rule in the +markets. Carnations of the winter-flowering or "perpetual" type have +long been grown in America, and enormous prices have been given for +individual plants on certain occasions, rivalling the fancy prices paid +in England for certain orchids. The American system of carnation-growing +has quite captivated English cultivators, and new varieties are being +constantly raised in both countries. Chrysanthemums are another great +feature of American florists, and sometimes during the winter season a +speculative grower will send a living specimen to one of the London +exhibitions in the hope of booking large orders for cuttings of it later +on. Sweet peas, dahlias, lilies of the valley, arum lilies and indeed +every flower that is popular in England is equally popular in America, +and consequently is largely grown. + + _Vegetables._--So far as these are concerned, potatoes, cabbages, + cauliflowers, beans of all kinds, cucumbers, tomatoes (already + referred to under fruits), musk-melons, lettuces, radishes, endives, + carrots, &c.; are naturally grown in great quantities, not only in the + open air, but also under glass. The French system of intensive + cultivation as practised on hot beds of manure round Paris is + practically unknown at present. In the southern states there would be + no necessity to practise it, but in the northern ones it is likely to + attract attention. (J. Ws.) + + +FOOTNOTE: + + [1] _Jour. Roy. Agric. Soc._, 1899. + + + + +FRUMENTIUS (c. 300-c. 360), the founder of the Abyssinian church, +traditionally identified in Abyssinian literature with Abba Salama or +Father of Peace (but see ETHIOPIA), was a native of Phoenicia. According +to the 4th-century historian Rufinus (x. 9), who gives Aedesius himself +as his authority, a certain Tyrian, Meropius, accompanied by his kinsmen +Frumentius and Aedesius, set out on an expedition to "India," but fell +into the hands of Ethiopians on the shore of the Red Sea and, with his +ship's crew, was put to death. The two young men were taken to the king +at Axum, where they were well treated and in time obtained great +influence. With the help of Christian merchants who visited the country +Frumentius gave Christianity a firm footing, which was strengthened when +in 326 he was consecrated bishop by Athanasius of Alexandria, who in his +_Epistola ad Constantinum_ mentions the consecration, and gives some +details of the history of Frumentius's mission. Later witnesses speak of +his fidelity to the homoousian during the Arian controversies. Aedesius +returned to Tyre, where he was ordained presbyter. + + + + +FRUNDSBERG, GEORG VON (1473-1528), German soldier, was born at +Mindelheim on the 24th of September 1473. He fought for the German king +Maximilian I. against the Swiss in 1499, and in the same year was among +the imperial troops sent to assist Ludovico Sforza, duke of Milan, +against the French. Still serving Maximilian, he took part in 1504 in +the war over the succession to the duchy of Bavaria-Landshut, and +afterwards fought in the Netherlands. Convinced of the necessity of a +native body of trained infantry Frundsberg assisted Maximilian to +organize the _Landsknechte_ (q.v.), and subsequently at the head of +bands of these formidable troops he was of great service to the Empire +and the Habsburgs. In 1509 he shared in the war against Venice, winning +fame for himself and his men; and after a short visit to Germany +returned to Italy, where in 1513 and 1514 he gained fresh laurels by his +enterprises against the Venetians and the French. Peace being made, he +returned to Germany, and at the head of the infantry of the Swabian +league assisted to drive Ulrich of Wurttemberg from his duchy in 1519. +At the diet of Worms in 1521 he spoke words of encouragement to Luther, +and when the struggle between France and the Empire was renewed he took +part in the invasion of Picardy, and then proceeding to Italy brought +the greater part of Lombardy under the influence of Charles V. through +his victory at Bicocca in April 1522. He was partly responsible for the +great victory over the French at Pavia in February 1525, and, returning +to Germany, he assisted to suppress the Peasant revolt, using on this +occasion, however, diplomacy as well as force. When the war in Italy was +renewed Frundsberg raised an army at his own expense, and skilfully +surmounting many difficulties, joined the constable de Bourbon near +Piacenza and marched towards Rome. Before he reached the city, however, +his unpaid troops showed signs of mutiny, and their leader, stricken +with illness and unable to pacify them, gave up his command. Returning +to Germany, he died at Mindelheim on the 20th of August 1528. He was a +capable and chivalrous soldier, and a devoted servant of the Habsburgs. +His son Caspar (1500-1536) and his grandson Georg (d. 1586) were both +soldiers of some distinction. With the latter's death the family became +extinct. + + See Adam Reissner, _Historia Herrn Georgs und Herrn Kaspars von + Frundsberg_ (Frankfort, 1568). A German translation of this work was + published at Frankfort in 1572. F.W. Barthold, _Georg von Frundsberg_ + (Hamburg, 1833); J. Heilmann, _Kriegsgeschichte von Bayern, Franken, + Pfalz und Schwaben_ (Munich, 1868). + + + + + +FRUSTUM (Latin for a "piece broken off"), a term in geometry for the +part of a solid figure, such as a cone or pyramid, cut off by a plane +parallel to the base, or lying between two parallel planes; and hence in +architecture a name given to the drum of a column. + + + + +FRUYTIERS, PHILIP (1627-1666), Flemish painter and engraver, was a pupil +of the Jesuits' college at Antwerp in 1627, and entered the Antwerp gild +of painters without a fee in 1631. He is described in the register of +that institution as "illuminator, painter and engraver." The current +account of his life is "that he worked exclusively in water colours, yet +was so remarkable in this branch of his art for arrangement, drawing, +and especially for force and clearness of colour, as to excite the +admiration of Rubens, whom he portrayed with all his family." The truth +is that he was an artist of the most versatile talents, as may be judged +from the fact that in 1646 he executed an Assumption with figures of +life size, and four smaller pictures in oil, for the church of St +Jacques at Antwerp, for which he received the considerable sum of 1150 +florins. Unhappily no undoubted production of his hand has been +preserved. All that we can point to with certainty is a series of etched +plates, chiefly portraits, which are acknowledged to have been +powerfully and skilfully handled. If, however, we search the portfolios +of art collections on the European continent, we sometimes stumble upon +miniatures on vellum, drawn with great talent and coloured with +extraordinary brilliancy. In form they quite recall the works of Rubens, +and these, it may be, are the work of Philip Fruytiers. + + + + +FRY, the name of a well-known English Quaker family, originally living +in Wiltshire. About the middle of the 18th century JOSEPH FRY +(1728-1787), a doctor, settled in Bristol, where he acquired a large +practice, but eventually abandoned medicine for commerce. He became +interested in china-making, soap-boiling and type-founding businesses in +Bristol, and in a chemical works at Battersea, all of which ventures +proved very profitable. The type-founding business was subsequently +removed to London and conducted by his son Edmund. Joseph Fry, however, +is best remembered as the founder of the great Bristol firm of J.S. Fry +& Sons, chocolate manufacturers. He purchased the chocolate-making +patent of William Churchman and on it laid the foundations of the +present large business. After his death the Bristol chocolate factory +was carried on with increasing success by his widow and by his son, +JOSEPH STORRS FRY (1767-1835). + +In 1795 a new and larger factory was built in Union Street, Bristol, +which still forms the centre of the firm's premises, and in 1798 a +Watt's steam-engine was purchased and the cocoa-beans ground by steam. +On the death of Joseph Storrs Fry his three sons, Joseph (1795-1879), +Francis, and Richard (1807-1878) became partners in the firm, the +control being mainly in the hands of FRANCIS FRY (1803-1886). Francis +Fry was in every way a remarkable character. The development of the +business to its modern enormous proportion was chiefly his work, but +this did not exhaust his activities. He took a principal part in the +introduction of railways to the west of England, and in 1852 drew up a +scheme for a general English railway parcel service. He was an ardent +bibliographer, taking a special interest in early English Bibles, of +which he made in the course of a long life a large and striking +collection, and of the most celebrated of which he published facsimiles +with bibliographical notes. Francis Fry died in 1886, and his son +Francis J. Fry and nephew Joseph Storrs Fry carried on the business, +which in 1896 was for family reasons converted into a private limited +company, Joseph Storrs Fry being chairman and all the directors members +of the Fry family. + + + + +FRY, SIR EDWARD (1827- ), English judge, second son of Joseph Fry +(1795-1879), was born at Bristol on the 4th of November 1827, and +educated at University College, London, and London University. He was +called to the bar in 1854 and was made a Q.C. in 1869, practising in the +rolls court and becoming recognized as a leading equity lawyer. In 1877 +he was raised to the bench and knighted. As chancery judge he will be +remembered for his careful interpretations and elucidations of the +Judicature Acts, then first coming into operation. In 1883 he was made a +lord justice of appeal, but resigned in 1892; and subsequently his +knowledge of equity and talents for arbitration were utilized by the +British government from time to time in various special directions, +particularly as chairman of many commissions. He was also one of the +British representatives at the Paris North Sea Inquiry Commission +(1905), and was appointed a member of the Hague Permanent Arbitration +Court. He wrote _A Treatise on the Specific Performance of Public +Contracts_ (London, 1858, and many subsequent editions). + + + + +FRY, ELIZABETH (1780-1845), English philanthropist, and, after Howard, +the chief promoter of prison reform in Europe, was born in Norwich on +the 21st of May 1780. Her father, John Gurney, afterwards of Earlham +Hall, a wealthy merchant and banker, represented an old family which for +some generations had belonged to the Society of Friends. While still a +girl she gave many indications of the benevolence of disposition, +clearness and independence of judgment, and strength of purpose, for +which she was afterwards so distinguished; but it was not until after +she had entered her eighteenth year that her religion assumed a decided +character, and that she was induced, under the preaching of the American +Quaker, William Savery, to become an earnest and enthusiastic though +never fanatical "Friend." In August 1800 she became the wife of Joseph +Fry, a London merchant. + +Amid increasing family cares she was unwearied in her attention to the +poor and the neglected of her neighbourhood; and in 1811 she was +acknowledged by her co-religionists as a "minister," an honour and +responsibility for which she was undoubtedly qualified, not only by +vigour of intelligence and warmth of heart, but also by an altogether +unusual faculty of clear, fluent and persuasive speech. Although she had +made several visits to Newgate prison as early as February 1813, it was +not until nearly four years afterwards that the great public work of her +life may be said to have begun. The association for the Improvement of +the Female Prisoners in Newgate was formed in April 1817. Its aim was +the much-needed establishment of some of what are now regarded as the +first principles of prison discipline, such as entire separation of the +sexes, classification of criminals, female supervision for the women, +and adequate provision for their religious and secular instruction, as +also for their useful employment. The ameliorations effected by this +association, and largely by the personal exertions of Mrs Fry, soon +became obvious, and led to a rapid extension of similar methods to other +places. In 1818 she, along with her brother, visited the prisons of +Scotland and the north of England; and the publication (1819) of the +notes of this tour, as also the cordial recognition of the value of her +work by the House of Commons committee on the prisons of the metropolis, +led to a great increase of her correspondence, which now extended to +Italy, Denmark and Russia, as well as to all parts of the United +Kingdom. Through a visit to Ireland, which she made in 1827, she was led +to direct her attention to other houses of detention besides prisons; +and her observations resulted in many important improvements in the +British hospital system, and in the treatment of the insane. In 1838 she +visited France, and besides conferring with many of the leading prison +officials, she personally visited most of the houses of detention in +Paris, as well as in Rouen, Caen and some other places. In the following +year she obtained an official permission to visit all the prisons in +that country; and her tour, which extended from Boulogne and Abbeville +to Toulouse and Marseilles, resulted in a report which was presented to +the minister of the interior and the prefect of police. Before returning +to England she had included Geneva, Zurich, Stuttgart and +Frankfort-on-Main in her inspection. The summer of 1840 found her +travelling through Belgium, Holland and Prussia on the same mission; and +in 1841 she also visited Copenhagen. In 1842, through failing health, +Mrs Fry was compelled to forgo her plans for a still more widely +extended activity, but had the satisfaction of hearing from almost every +quarter of Europe that the authorities were giving increased practical +effect to her suggestions. In 1844 she was seized with a lingering +illness, of which she died on the 12th of October 1845. She was +survived by a numerous family, the youngest of whom was born in 1822. + +Two interesting volumes of _Memoirs, with Extracts from her Journals and +Letters_, edited by two of her daughters, were published in 1847. See +also _Elizabeth Fry_, by G. King Lewis (1910). + + + + +FRYXELL, ANDERS (1795-1881), Swedish historian, was born at Hesselskog, +Dalsland, Sweden, on the 7th of February 1795. He was educated at +Upsala, took holy orders in 1820, was made a doctor of philosophy in +1821, and in 1823 began to publish the great work of his life, the +_Stories from Swedish History_. He did not bring this labour to a close +until, fifty-six years later, he published the forty-sixth and crowning +volume of his vast enterprise. Fryxell, as a historian, appealed to +every class by the picturesqueness of his style and the breadth of his +research; he had the gift of awakening to an extraordinary degree the +national sense in his readers. In 1824 he published his _Swedish +Grammar_, which was long without a rival. In 1833 he received the title +of professor, and in 1835 he was appointed to the incumbency of Sunne, +in the diocese of Karlstad, where he resided for the remainder of his +life. In 1840 he was elected to the Swedish Academy in succession to the +poet Wallin (1779-1839). In 1847 Fryxell received from his bishop +permission to withdraw from all the services of the Church, that he +might devote himself without interruption to historical investigation. +Among his numerous minor writings are prominent his _Characteristics of +Sweden between 1592 and 1600_ (1830), his _Origins of the Inaccuracy +with which the History of Sweden in Catholic Times has been Treated_ +(1847), and his _Contributions to the Literary History of Sweden_. It is +now beginning to be seen that the abundant labours of Fryxell were +rather of a popular than of a scientific order, and although their +influence during his lifetime was unbounded, it is only fair to later +and exacter historians to admit that they threaten to become obsolete in +more than one direction. On the 21st of March 1881 Anders Fryxell died +at Stockholm, and in 1884 his daughter Eva Fryxell (born 1829) published +from his MS. an interesting _History of My History_, which was really a +literary autobiography and displays the persistency and tirelessness of +his industry. (E. G.) + + + + +FUAD PASHA (1815-1869), Turkish statesman, was the son of the +distinguished poet Kecheji-zade Izzet Molla. He was educated at the +medical school and was at first an army surgeon. About 1836 he entered +the civil service as an official of the foreign ministry. He became +secretary of the embassy in London; was employed on special missions in +the principalities and at St Petersburg (1848), and was sent to Egypt as +special commissioner in 1851. In that year he became minister for +foreign affairs, a post to which he was appointed also on four +subsequent occasions and which he held at the time of his death. During +the Crimean War he commanded the troops on the Greek frontier and +distinguished himself by his bravery. He was Turkish delegate at the +Paris conference of 1856; was charged with a mission to Syria in 1860; +grand vizier in 1860 and 1861, and also minister of war. He accompanied +the sultan Abd-ul-Aziz on his journey to Egypt and Europe, when the +freedom of the city of London was conferred on him. He died at Nice +(whither he had been ordered for his health) in 1869. Fuad was renowned +for his boldness and promptness of decision, as well as for his ready +wit and his many bons mots. Generally regarded as the partisan of a +pro-English policy, he rendered most valuable service to his country by +his able management of the foreign relations of Turkey, and not least by +his efficacious settlement of affairs in Syria after the massacres of +1860. + + + + +FUCHOW, FU-CHAU, FOOCHOW, a city of China, capital of the province of +Fu-kien, and one of the principal ports open to foreign commerce. In the +local dialect it is called Hokchiu. It is situated on the river Min, +about 35 m. from the sea, in 26 deg. 5' N. and 119 deg. 20' E., 140 m. +N. of Amoy and 280 S. of Hang-chow. The city proper, lying nearly 3 m. +from the north bank of the river, is surrounded by a wall about 30 ft. +high and 12 ft. thick, which makes a circuit of upwards of 5 m. and is +pierced by seven gateways surrounded by tall fantastic watch-towers. +The whole district between the city and the river, the island of +Nantai, and the southern banks of the Min are occupied by extensive +suburbs; and the river itself bears a large floating population. +Communication from bank to bank is afforded by a long stone bridge +supported by forty solid stone piers in its northern section and by nine +in its southern. The most remarkable establishment of Fuchow is the +arsenal situated about 3 m. down the stream at Pagoda Island, where the +sea-going vessels usually anchor. It was founded in 1867, and is +conducted under the direction of French engineers according to European +methods. In 1870 it employed about 1000 workmen besides fifty European +superintendents, and between that date and 1880 it turned out about 20 +or 30 small gunboats. In 1884 it was partially destroyed by the French +fleet, and for a number of years the workshops and machinery were +allowed to stand idle and go to decay. On the 1st of August 1895 an +attack was made on the English mission near the city of Ku-chang, 120 m. +west of Fuchow, on which occasion nine missionaries, of whom eight were +ladies, were massacred. The port was opened to European commerce in +1842; and in 1853 the firm of Russell and Co. shipped the first cargoes +of tea from Fuchow to Europe and America. The total trade in foreign +vessels in 1876 was imports to the value of L1,531,617, and exports to +the value of L3,330,489. In 1904 the imports amounted to L1,440,351, and +the exports to L1,034,436. The number of vessels that entered in 1876 +was 275, and of these 211 were British, 27 German, 11 Danish and 9 +American. While in 1904, 480 vessels entered the port, 216 of which were +British. A large trade is carried on by the native merchants in timber, +paper, woollen and cotton goods, oranges and olives; but the foreign +houses mainly confine themselves to opium and tea. Commercial +intercourse with Australia and New Zealand is on the increase. The +principal imports, besides opium, are shirtings, T-cloths, lead and tin, +medicines, rice, tobacco, and beans and peas. Two steamboat lines afford +regular communication with Hong-Kong twice a month. The town is the seat +of several important missions, of which the first was founded in 1846. +That supported by the American board had in 1876 issued 1,3000,000 +copies of Chinese books and tracts. + + + + +FUCHS, JOHANN NEPOMUK VON (1774-1856), German chemist and mineralogist, +was born at Mattenzell, near Brennberg in the Bavarian Forest, on the +15th of May 1774. In 1807 he became professor of chemistry and +mineralogy at the university of Landshut, and in 1823 conservator of the +mineralogical collections at Munich, where he was appointed professor of +mineralogy three years later, on the removal thither of the university +of Landshut. He retired in 1852, was ennobled by the king of Bavaria in +1854, and died at Munich on the 5th of March 1856. His name is chiefly +known for his mineralogical observations and for his work on soluble +glass. + + His collected works, including _Uber den Einfluss der Chemie und + Mineralogie_ (1824), _Die Naturgeschichte des Mineralreichs_ (1842), + _Uber die Theorien der Erde_ (1844), were published at Munich in 1856. + + + + +FUCHS, LEONHARD (1501-1566), German physician and botanist, was born at +Wembdingen in Bavaria on the 17th of January 1501. He attended school at +Heilbronn and Erfurt, and in 1521 graduated at the university of +Ingolstadt. About the same time he espoused the doctrines of the +Reformation. Having in 1524 received his diploma as doctor of medicine, +he practised for two years in Munich. He became in 1526 professor of +medicine at Ingolstadt, and in 1528 physician to the margrave of +Anspach. In Anspach he was the means of saving the lives of many during +the epidemic locally known as the "English sweating-sickness." By the +duke of Wurttemberg he was, in 1535, appointed to the professorship of +medicine at the university of Tubingen, a post held by him till his +death on the 10th of May 1566. Fuchs was an advocate of the Galenic +school of medicine, and published several Latin translations of +treatises by its founder and by Hippocrates. But his most important +publication was _De historia stirpium commentarii insignes_ (Basel, +1542), a work illustrated with more than five hundred excellent outline +illustrations, including figures of the common foxglove and of another +species of the genus _Digitalis_, which was so named by him. + + + + +FUCHSIA, so named by Plumier in honour of the botanist Leonhard Fuchs, a +genus of plants of the natural order Onagraceae, characterized by +entire, usually opposite leaves, pendent flowers, a funnel-shaped, +brightly coloured, quadripartite, deciduous calyx, 4 petals, alternating +with the calycine segments, 8, rarely 10, exserted stamens, a long +filiform style, an inferior ovary, and fruit, a fleshy ovoid many-seeded +berry. All the members of the genus, with the exception of the New +Zealand species, _F. excorticata, F. Colensoi_ and _F. procumbens_, are +natives of Central and South America--occurring in the interior of +forests or in damp and shady mountainous situations. The various species +differ not a little in size as well as in other characters; some, as _F. +verrucosa_, being dwarf shrubs; others, as _F. arborescens_ and _F. +apetala_, attaining a height of 12 to 16 ft., and having stems several +inches in diameter. Plumier, in his _Nova plantarum Americanarum genera_ +(p. 14, tab. 14, Paris, 1703), gave a description of a species of +fuchsia, the first known, under the name of _Fuchsia triphylla, flore +coccineo_, and a somewhat conventional outline figure of the same plant +was published at Amsterdam in 1757 by Burmann. In the _Histoire des +plantes medicinales_ of the South American traveller Feuillee (p. 64, +pl. XLVII.), written in 1709-1711, and published by him with his +_Journal_, Paris, 1725, the name _Thilco_ is applied to a species of +fuchsia from Chile, which is described, though not evidently so figured, +as having a pentamerous calyx. The _F. coccinea_ of Alton (fig.) (see +J.D. Hooker, in _Journal Linnean Soc_., Botany, vol. x. p. 458, 1867), +the first species of fuchsia cultivated in England, where it was long +confined to the greenhouse, was brought from South America by Captain +Firth in 1788 and placed in Kew Gardens. Of this species Mr Lee, a +nurseryman at Hammersmith, soon afterwards obtained an example, and +procured from it by means of cuttings several hundred plants, which he +sold at a guinea each. In 1823 _F. macrostemma_ and _F. gracilis_, and +during the next two or three years several other species, were +introduced into England; but it was not until about 1837, or soon after +florists had acquired _F. fulgens_, that varieties of interest began to +make their appearance. The numerous hybrid forms now existing are the +result chiefly of the intercrossing of that or other long-flowered with +globose-flowered plants. _F. Venus-victrix_, raised by Mr Gulliver, +gardener to the Rev. S. Marriott of Horsemonden, Kent, and sold in 1822 +to Messrs Cripps, was the earliest white-sepalled fuchsia. The first +fuchsia with a white corolla was produced about 1853 by Mr Storey. In +some varieties the blossoms are variegated, and in others they are +double. There appears to be very little limit to the number of forms to +be obtained by careful cultivation and selection. To hybridize, the +flower as soon as it opens is emasculated, and it is then fertilized +with pollen from some different flower. + +[Illustration: _Fuchsia coccinea_. 1, Flower cut open after removal of +sepals; 2, fruit; 3, floral diagram.] + +Ripe seed is sown either in autumn or about February or March in light, +rich, well-drained mould, and is thinly covered with sandy soil and +watered. A temperature of 70 deg. to 75 deg. Fahr. has been found +suitable for raising. The seedlings are pricked off into shallow pots or +pans, and when 3 in. in height are transferred to 3-in. pots, and are +then treated the same as plants from cuttings. Fuchsias may be grafted +as readily as camellias, preferably by the splice or whip method, the +apex of a young shoot being employed as a scion; but the easiest and +most usual method of propagation is by cuttings. The most expeditious +way to procure these is to put plants in heat in January, and to take +their shoots when 3 in. in length. For summer flowering in England they +are best made about the end of August, and should be selected from the +shortest-jointed young wood. They root readily in a compost of loam and +silver-sand if kept close and sprinkled for a short time. In from two to +three weeks they may be put into 3-in. pots containing a compost of +equal parts of rich loam, silver-sand and leaf-mould. They are +subsequently moved from the frame or bed, first to a warm and shady, and +then to a more airy part of the greenhouse. In January a little +artificial heat may be given, to be gradually increased as the days +lengthen. The side-shoots are generally pruned when they have made three +or four joints, and for bushy plants the leader is stopped soon after +the first potting. Care is taken to keep the plants as near the glass as +possible, and shaded from bright sunshine, also to provide them +plentifully with water, except at the time of shifting, when the roots +should be tolerably dry. For the second potting a suitable soil is a +mixture of well-rotted cow-dung or old hotbed mould with leaf-mould and +sandy peat, and to promote drainage a little peat-moss may be placed +immediately over the crocks in the lower part of the pot. Weak liquid +manure greatly promotes the advance of the plants, and should be +regularly supplied twice or thrice a week during the flowering season. +After this, water is gradually withheld from them, and they may be +placed in the open air to ripen their wood. + +Among the more hardy or half-hardy plants for inside borders are +varieties of the Chilean species, _F. macrostemma_ (or _F. +magellanica_), a shrub 6 to 12 ft. high with a scarlet calyx, such as +_F. m. globosa, F. m. gracilis_; one of the most graceful and hardy of +these, a hybrid _F. riccartoni_, was raised at Riccarton, near +Edinburgh, in 1830. For inside culture may be mentioned _F. boliviana_ +(Bolivia), 2 to 4 ft. high, with rich crimson flowers with a +trumpet-shaped tube; _F. corymbiflora_ (Peru), 4 to 6 ft. high, with +scarlet flowers nearly 2 in. long in long terminal clusters; F. fulgens +(Mexico), 4 to 6 ft., with drooping apical clusters of scarlet flowers; +_F. microphylla_ (Central America), with small leaves and small scarlet +funnel-shaped flowers, the petals deep red; _F. procumbens_ (New +Zealand), a pretty little creeper, the small flowers of which are +succeeded by oval magenta-crimson berries which remain on for months; +and _F. splendens_ (Mexico), 6 ft. high, with very showy scarlet and +green flowers. But these cannot compare in beauty or freedom of blossom +with the numerous varieties raised by gardeners. The nectar of fuchsia +flowers has been shown to contain nearly 78% of cane sugar, the +remainder being fruit sugar. The berries of some fuchsias are subacid or +sweet and edible. From certain species a dye is obtainable. The +so-called "native fuchsias" of southern and eastern Australia are plants +of the genus _Correa_, natural order Rutaceae. + + + + +FUCHSINE, or MAGENTA, a red dye-stuff consisting of a mixture of the +hydrochlorides or acetates of pararosaniline and rosaniline. It was +obtained in 1856 by J. Natanson (_Ann_., 1856, 98, p. 297) by the action +of ethylene chloride on aniline, and by A.W. Hofmann in 1858 from +aniline and carbon tetrachloride. It is prepared by oxidizing "aniline +for red" (a mixture of aniline and ortho- and para-toluidine) with +arsenic acid (H. Medlock, _Dingler's Poly. Jour_., 1860, 158, p. 146); +by heating aniline for red with nitrobenzene, concentrated hydrochloric +acid and iron (Coupier, _Ber_., 1873, 6, p. 423); or by condensing +formaldehyde with aniline and ortho-toluidine and oxidizing the mixture. +It forms small crystals, showing a brilliant green reflex, and is +soluble in water and alcohol with formation of a deep red solution. It +dyes silk, wool and leather direct, and cotton after mordanting with +tannin and tartar emetic (see DYEING). An aqueous solution of fuchsine +is decolorized on the addition of sulphurous acid, the easily soluble +fuchsine sulphurous acid being formed. This solution is frequently used +as a test reagent for the detection of aldehydes, giving, in most cases, +a red coloration on the addition of a small quantity of the aldehyde. + + The constitution of the fuchsine bases (pararosaniline and rosaniline) + was determined by E. and O. Fischer in 1878 (_Ann_., 1878, 194, p. + 242); A.W. Hofmann having previously shown that oxidation of pure + aniline alone or of pure toluidine yielded no fuchsine, whilst + oxidation of a mixture of aniline and para-toluidine gave rise to the + fine red dye-stuff para-fuchsine (pararosaniline hydrochloride) + + CH3.C6H4NH2 + 2C6H5NH2 + 3O = HO.C(C6H4NH2)3 + 2H2O. + Colour base (pararosaniline). + + HO.C(C6H4NH2)3.HCl = H2O + (H2N.C6H4)2C : C6H4 : NH2Cl. + Pararosaniline hydrochloride. + + A. Rosenstiehl (_Jahres_., 1869, p. 693) found also that different + rosanilines were obtained according to whether ortho- or + para-toluidine was oxidized with aniline; and he gave the name + rosaniline to the one obtained from aniline and ortho-toluidine, + reserving the term pararosaniline for the other. E. and O. Fischer + showed that these compounds were derivatives of triphenylmethane and + tolyldiphenylmethane respectively. Pararosaniline was reduced to the + corresponding leuco compound (paraleucaniline), from which by + diazotization and boiling with alcohol, the parent hydrocarbon was + obtained + + (H2N.C5H4)2C : C6H4:NH2Cl --> HC(C6H4NH2.HCl)3 --> HC(C6H4N2Cl3) + Pararosaniline hydrochloride. Paraleucaniline. + + --> HC(C6H5)3. + Triphenylmethane. + + The reverse series of operations was also carried out by the Fischers, + triphenylmethane being nitrated, and the nitro compound then reduced + to triaminotriphenylmethane or paraleucaniline, which on careful + oxidation is converted into the dye-stuff. A similar series of + reactions was carried out with rosaniline, which was shown to be the + corresponding derivative of tolyldiphenylmethane. + + The free pararosaniline, C19H19N3O, and rosaniline, C20H21N3O, may be + obtained by precipitating solutions of their salts with a caustic + alkali, colourless precipitates being obtained, which crystallize from + hot water in the form of needles or plates. The position of the amino + groups in pararosaniline was determined by the work of H. Caro and C. + Graebe (_Ber_., 1878, II, p. 1348) and of E. and O. Fischer (_Ber._, + 1880, 13, p. 2204) as follows: Nitrous acid converts pararosaniline + into aurin, which when superheated with water yields + para-dioxybenzophenone. As the hydroxyl groups in aurin correspond to + the amino groups in pararosaniline, two of these in the latter + compound must be in the para position. The third is also in the para + position; for if benzaldehyde be condensed with aniline, condensation + occurs in the para position, for the compound formed may be converted + into para-dioxybenzophenone, + + C6H5CHO --> C6H5CH(C6H4NH2)2 --> C6H5CH(C6H4OH)2 --> CO(C6H4OH)2; + + but if para-nitrobenzaldehyde be used in the above reaction and the + resulting nitro compound NO2.C6H4.CH(C6H4NH2)2 be reduced, then + pararosaniline is the final product, and consequently the third amino + group occupies the para position. Many derivatives of pararosaniline + and rosaniline are known, in which the hydrogen atoms of the amino + groups are replaced by alkyl groups; this has the effect of producing + a blue or violet shade, which becomes deeper as the number of groups + increases (see DYEING). + + + + +FUCINO, LAGO DI [Lat. _Lacus Fucinus_], a lake bed of the Abruzzi, +Italy, in the province of Aquila, 2 m. E. of the town of Avezzano. The +lake was 37 m. in circumference and 65 ft. deep. From the lack of an +outlet, the level of the lake was subject to great variations, often +fraught with disastrous consequences. As early as A.D. 52 the emperor +Claudius, realizing a project of Julius Caesar, constructed a tunnel +3-1/2 m. long, with 40 shafts at intervals, by which the surplus waters +found an outlet to the Liris (or Garigliano). No less than 30,000 +workmen were employed for eleven years in driving this tunnel. In the +following reign the tunnel was allowed to fall into disrepair, but was +repaired by Trajan. When, however, it finally went out of use is +uncertain. The various attempts made to reopen it from 1240 onwards were +unsuccessful. By 1852 the lake had gradually risen until it was 30 ft. +above its original level, and had become a source of danger to the +surrounding countryside. A company undertook to drain it on condition of +becoming proprietors of the site when dry; in 1854, however, the rights +and privileges were purchased by Prince Giulio Torlonia (d. 1886), the +great Roman banker, who carried on the work at his own expense until, in +1876, the lake was finally drained at the cost of some L1,700,000. The +reclaimed area is 12-1/2 m. long, 7 m. broad, and is cultivated by +families from the Torlonia estates. The outlet by which it was drained +is 4 m. long and 24 sq. yds. in section. + + See A. Brisse and L. de Rotron, _Le Dessechement du lac Fucin, execute + par S.E. le Prince A. Torlonia_ (Rome, 1876). (T. As.) + + + + +FUEL (O. Fr. _feuaile_, popular Lat. _focalia_, from _focus_, hearth, +fire), a term applicable to all substances that can be usefully employed +for the production of heat by combustion. Any element or combination of +elements susceptible of oxidation may under appropriate conditions be +made to burn; but only those that ignite at a moderate initial +temperature and burn with comparative rapidity, and, what is practically +of more importance, are obtainable in quantity at moderate prices, can +fairly be regarded as fuels. The elementary substances that can be so +classed are primarily hydrogen, carbon and sulphur, while others finding +more special applications are silicon, phosphorus, and the more readily +oxidizable metals, such as iron, manganese, aluminium and magnesium. +More important, however, than the elements are the carbohydrates or +compounds of carbon, oxygen and hydrogen, which form the bulk of the +natural fuels, wood, peat and coal, as well as of their liquid and +gaseous derivatives--coal-gas, coal-tar, pitch, oil, &c., which have +high values as fuel. Carbon in the elementary form has its nearest +representative in the carbonized fuels, charcoal from wood and coke from +coal. + + +_Solid Fuels._ + + Wood. + +Wood may be considered as having the following average composition when +in the air-dried state: Carbon, 39.6; hydrogen, 4.8; oxygen, 34.8; ash, +1.0; water, 20%. When it is freshly felled, the water may be from 18 to +50%. Air-dried or even green wood ignites readily when a considerable +surface is exposed to the kindling flame, but in large masses with +regular or smooth surfaces it is often difficult to get it to burn. When +previously torrefied or scorched by heating to a temperature of about +200 deg., at which incipient charring is set up, it is exceedingly +inflammable. The ends of imperfectly charred boughs from the charcoal +heaps in this condition are used in Paris and other large towns in +France for kindling purposes, under the name of _fumerons_. The +inflammability, however, varies with the density,--the so-called hard +woods, oak, beech and maple, taking fire less readily than the softer, +and, more especially, the coniferous varieties rich in resin. The +calorific power of absolutely dry woods may as an average be taken at +about 4000 units, and when air-dried, i.e. containing 25% of water, at +2800 to 3000 units. Their evaporative values, i.e. the quantities of +water evaporated by unit weight, are 3.68 and 4.44. + +Wood being essentially a flaming fuel is admirably adapted for use with +heat-receiving surfaces of large extent, such as locomotive and marine +boilers, and is also very clean in use. The absence of all cohesion in +the cinders or unburnt carbonized residue causes a large amount of +ignited particles to be projected from the chimney, when a rapid draught +is used, unless special spark-catchers of wire gauze or some analogous +contrivance are used. When burnt in open fireplaces the volatile +products given off in the apartment on the first heating have an acrid +penetrating odour, which is, however, very generally considered to be +agreeable. Owing to the large amount of water present, no very high +temperatures can be obtained by the direct combustion of wood, and to +produce these for metallurgical purposes it is necessary to convert it +previously either into charcoal or into inflammable gas. + + + Peat. + +Peat includes a great number of substances of very unequal fuel value, +the most recently formed spongy light brown kind approximating in +composition to wood, while the dense pitchy brown compact substance, +obtained from the bottom of bogs of ancient formation, may be compared +with lignite or even in some instances with coal. Unlike wood, however, +it contains incombustible matter in variable but large quantity, from 5 +to 15% or even more. Much of this, when the amount is large, is often +due to sand mechanically intermixed; when air-dried the proportion of +water is from 8 to 20%. When these constituents are deducted the average +composition may be stated to be--carbon, 52 to 66; hydrogen, 4.7 to +7.4; oxygen, 28 to 39; and nitrogen, 1.5 to 3%. Average air-dried peat +may be taken as having a calorific value of 3000 to 3500 units, and when +dried at 100 deg. C., and with a minimum of ash (4 to 5%), at about 5200 +units, or from a quarter to one-third more than that of an equal weight +of wood. The lighter and more spongy varieties of peat when air-dried +are exceedingly inflammable, firing at a temperature of 200 deg. C.; the +denser pulpy kinds ignite less readily when in the natural state, and +often require a still higher temperature when prepared by pulping and +compression or partial carbonization. Most kinds burn with a red smoky +flame, developing a very strong odour, which, however, has its admirers +in the same way that wood smoke has. This arises from the destructive +distillation of imperfectly carbonized organic matter. The ash, like +that of wood, is light and powdery, except when much sand is present, +when it is of a denser character. + +Peat is principally found in high latitudes, on exposed high tablelands +and treeless areas in more temperate climates, and in the valleys of +slow-flowing rivers,--as in Ireland, the west of Scotland, the tableland +of Bavaria, the North German plain, and parts of the valleys of the +Somme, Oise and a few other rivers in northern France. A principal +objection to its use is its extreme bulk, which for equal evaporative +effect is from 8 to 18 times that of coal. Various methods have been +proposed, and adopted more or less successfully, for the purpose of +increasing the density of raw peat by compression, either with or +without pulping; the latter process gives the heaviest products, but the +improvement is scarcely sufficient to compensate for the cost. + + + Lignite. + +Lignite or brown coal is of intermediate character between peat and coal +proper. The best kinds are undistinguishable in quality from +free-burning coals, and the lowest earthy kinds are not equal to average +peat. When freshly raised, the proportion of water may be from 45 to 50% +and even more, which is reduced from 28 to 20% by exposure to dry air. +Most varieties, however, when fully dried, break up into powder, which +considerably diminishes their utility as fuel, as they cannot be +consolidated by coking. Lignite dust may, however, be compacted into +serviceable blocks for burning, by pressure in machines similar to those +used for brickmaking, either in the wet state as raised from the mines +or when kiln-dried at 200 deg. C. This method was adopted to a very large +extent in Prussian Saxony. The calorific value varies between 3500 and +5000 units, and the evaporative factor from 2.16 when freshly raised to +5.84 for the best kinds of lignite when perfectly dried. + + + Other natural fuels. + +Of the other natural fuels, apart from coal (q.v.), the most important +is so-called vegetable refuse, such as cotton stalks, brushwood, straw, +and the woody residue of sugar-cane after the extraction of the +saccharine juice known as megasse or cane trash. These are extensively +used in countries where wood and coal are scarce, usually for providing +steam in the manufactures where they arise, e.g. straw for thrashing, +cotton stalks for ploughing, irrigating, or working presses, and cane +trash for boiling down sugar or driving the cane mill. According to J. +Head (_Proc. Inst. of Civil Engineers_, vol. xlviii. p. 75), the +evaporative values of 1 lb. of these different articles when burnt in a +tubular boiler are--coal, 8 lb.; dry peat, 4 lb.; dry wood, 3.58-3.52 +lb.; cotton stalks or megasse, 3.2-2.7 lb.; straw, 2.46-2.30 lb. +Owing to the siliceous nature of the ash of straw, it is desirable to +have a means of clearing the grate bars from slags and clinkers at short +intervals, and to use a steam jet to clear the tubes from similar +deposits. + +The common fuel of India and Egypt is derived from the dung of camels +and oxen, moulded into thin cakes, and dried in the sun. It has a very +low heating power, and in burning gives off acrid ammoniacal smoke and +vapour. + +Somewhat similar are the tan cakes made from spent tanners' bark, which +are used to some extent in eastern France and in Germany. They are made +by moulding the spent bark into cakes, which are then slowly dried by +exposure to the air. Their effect is about equivalent to 80 and 30% of +equal weights of wood and coal respectively. + +Sulphur, phosphorus and silicon, the other principal combustible +elements, are only of limited application as fuels. The first is used in +the liquidation of sulphur-bearing rocks. The ore is piled into large +heaps, which are ignited at the bottom, a certain proportion, from +one-fourth to one-third, of the sulphur content being sacrificed, in +order to raise the mass to a sufficient temperature to allow the +remainder to melt and run down to the collecting basin. Another +application is in the so-called "pyritic smelting," where ores of copper +(q.v.) containing iron pyrites, FeS2, are smelted with appropriate +fluxes in a hot blast, without preliminary roasting, the sulphur and +iron of the pyrites giving sufficient heat by oxidation to liquefy both +slag and metal. Phosphorus, which is of value from its low igniting +point, receives its only application in the manufacture of lucifer +matches. The high temperature produced by burning phosphorus is in part +due to the product of combustion (phosphoric acid) being solid, and +therefore there is less heat absorbed than would be the case with a +gaseous product. The same effect is observed in a still more striking +manner with silicon, which in the only special case of its application +to the production of heat, namely, in the Bessemer process of +steel-making, gives rise to an enormous increase of temperature in the +metal, sufficient indeed to keep the iron melted. The absolute calorific +value of silicon is lower than that of carbon, but the product of +combustion (silica) being non-volatile at all furnace temperatures, the +whole of the heat developed is available for heating the molten iron, +instead of a considerable part being consumed in the work of +volatilization, as is the case with carbonic oxide, which burns to waste +in the air. + + + Calorific power. + + _Assay and Valuation of Carbonaceous Fuels._--The utility or value of + a fuel depends upon two principal factors, namely, its calorific power + and its calorific intensity or pyrometric effect, that is, the + sensible temperature of the products of combustion. The first of these + is constant for any particular product of combustion independently of + the method by which the burning is effected, whether by oxygen, air or + a reducible metallic oxide. It is most conveniently determined in the + laboratory by measuring the heat evolved during the combustion of a + given weight of the fuel. The method of Lewis Thompson is one of the + most useful. The calorimeter consists of a copper cylinder in which a + weighed quantity of coal intimately mixed with 10-12 parts of a + mixture of 3 parts of potassium chlorate and 1 of potassium nitrate is + deflagrated under a copper case like a diving-bell, placed at the + bottom of a deep glass jar filled with a known weight of water. The + mixture is fired by a fuse of lamp-cotton previously soaked in a nitre + solution and dried. The gases produced by the combustion rising + through the water are cooled, with a corresponding increase of + temperature in the latter, so that the difference between the + temperature observed before and after the experiment measures the heat + evolved. The instrument is so constructed that 30 grains (2 grammes) + of coal are burnt in 29,010 grains of water, or in the proportion of 1 + to 937, these numbers being selected that the observed rise of + temperature in Fahrenheit degrees corresponds to the required + evaporative value in pounds, subject only to a correction for the + amount of heat absorbed by the mass of the instrument, for which a + special coefficient is required and must be experimentally determined. + The ordinary bomb calorimeter is also used. An approximate method is + based upon the reduction of lead oxide by the carbon and hydrogen of + the coal, the amount of lead reduced affording a measure of the oxygen + expended, whence the heating power may be calculated, 1 part of pure + carbon being capable of producing 34-1/2 times its weight of lead. The + operation is performed by mixing the weighed sample with a large + excess of litharge in a crucible, and exposing it to a bright red heat + for a short time. After cooling, the crucible is broken and the + reduced button of lead is cleaned and weighed. The results obtained by + this method are less accurate with coals containing much disposable + hydrogen and iron pyrites than with those approximating to anthracite, + as the heat equivalent of the hydrogen in excess of that required to + form water with the oxygen of the coal is calculated as carbon, while + it is really about four times as great. Sulphur in iron pyrites also + acts as a reducing agent upon litharge, and increases the apparent + effect in a similar manner. + + The evaporative power of a coal found by the above methods, and also + by calculating the separate calorific factors of the components as + determined by the chemical analysis, is always considerably above that + obtained by actual combustion under a steam boiler, as in the latter + case numerous sources of loss, such as imperfect combustion of gases, + loss of unburnt coal in cinders, &c., come into play, which cannot be + allowed for in laboratory experiments. It is usual, therefore, to + determine the value of a coal by the combustion of a weighed quantity + in the furnace of a boiler, and measuring the amount of water + evaporated by the heat developed. + + In a research upon the heating power and other properties of coal for + naval use, carried out by the German admiralty, the results tabulated + below were obtained with coals from different localities. + + +-----------------------+-----------+------------+-----------+------------------+ + | | Slag left | Ashes in | Soot in | Water evaporated | + | | in Grate. | Ashpit. | Flues. | by 1 lb. of Coal.| + +-----------------------+-----------+------------+-----------+------------------+ + | Westphalian gas coals | 0.33-6.42 | 2.83- 6.53 | 0.32-0.46 | 6.60-7.45 lb. | + | Do. bituminous coals | 0.98-9.10 | 1.97- 9.63 | 0.24-0.88 | 7.30-8.66 | + | Do. dry coals | 1.93-5.70 | 4.37-10.63 | 0.24-0.48 | 7.03-8.51 | + | Silesian coals | 0.92-1.30 | 3.15- 3.50 | 0.24-0.30 | 6.73-7.10 | + | Welsh steam coals | 1.20-4.07 | 4.07 | 0.32 | 8.41 | + | Newcastle coals | 1.92 | 2.57 | 0.35 | 7.28 | + +-----------------------+-----------+------------+-----------+------------------+ + + The heats of combustion of elements and compounds will be found in + most of the larger works on physical and chemical constants; a + convenient series is given in the _Annuaire du Bureau des Longitudes_, + appearing in alternate years. The following figures for the principal + fuel elements are taken from the issue for 1908; they are expressed in + gramme "calories" or heat units, signifying the weight of water in + grammes that can be raised 1 deg. C. in temperature by the combustion + of 1 gramme of the substance, when it is oxidized to the condition + shown in the second column: + + +----------------+---------------------------------+-----------+ + | Element. | Product of Combustion. | Calories. | + +----------------+---------------------------------+-----------+ + | Hydrogen \ | Water, H2O, condensed to liquid | 34,500 | + | / | " as vapour | 29,650 | + | Carbon-- | | | + | Diamond | Carbon Dioxide, CO2 | 7,868 | + | Graphite | " " | 7,900 | + | Amorphous | " " | 8,133 | + | Silicon-- | | | + | Amorphous | Silicon Dioxide, SiO2 | 6,414 | + | Crystallized | " " | 6,570 | + | Phosphorus | Phosphoric pentoxide, P2O5 | 5,958 | + | Sulphur | Sulphur dioxide, SO2, gaseous | 2,165 | + +----------------+---------------------------------+-----------+ + + The results may also be expressed in terms of the atomic equivalent of + the combustible by multiplying the above values by the atomic weight + of the substance, 12 for carbon, 28 for silicon, &c. + + In all fuels containing hydrogen the calorific value as found by the + calorimeter is higher than that obtainable under working conditions by + an amount equal to the latent heat of volatilization of water which + reappears as heat when the vapour is condensed, though under ordinary + conditions of use the vapour passes away uncondensed. This gives rise + to the distinction of higher and lower calorific values for such + substances, the latter being those generally used in practice. The + differences for the more important compound gaseous fuels are as + follows:-- + + Calorific Value. + Higher. Lower. + Acetylene, C2H2 11,920 11,500 + Ethylene, C2H4 11,880 11,120 + Methane, CH4 13,240 11,910 + Carbon monoxide, CO 2,440 2,440 + + + Caloric intensity. + + The calorific intensity or pyrometric effect of any particular fuel + depends upon so many variable elements that it cannot be determined + except by actual experiment. The older method was to multiply the + weight of the products of combustion by their specific heats, but this + gave untrustworthy results as a rule, on account of two + circumstances--the great increase in specific heat at high + temperatures in compound gases such as water and carbon dioxide, and + their instability when heated to 1800 deg. or 2000 deg. At such + temperatures dissociation to a notable extent takes place, especially + with the latter substance, which is also readily reduced to carbon + monoxide when brought in contact with carbon at a red heat--a change + which is attended with a large heat absorption. This effect is higher + with soft kinds of carbon, such as charcoal or soft coke, than with + dense coke, gas retort carbon or graphite. These latter substances, + therefore, are used when an intense local heat is required, as for + example, in the Deville furnace, to which air is supplied under + pressure. Such a method is, however, only of very special application, + the ordinary method being to supply air to the fire in excess of that + required to burn the fuel to prevent the reduction of the carbon + dioxide. The volume of flame, however, is increased by inert gas, and + there is a proportionate diminution of the heating effect. Under the + most favourable conditions, when the air employed has been previously + raised to a high temperature and pressure, the highest attainable + flame temperature from carbonaceous fuel seems to be about 2100 + deg.-2300 deg. C.; this is realized in the bright spots or "eyes" of + the tuyeres of blast furnaces. + + Very much higher temperatures may be reached when the products of + combustion are not volatile, and the operation can be effected by + using the fuel and oxidizing agent in the proportions exactly + required for perfect combustion and intimately mixed. These + conditions are met in the "Thermit" process of Goldschmidt, where + finely divided aluminium is oxidized by the oxide of some similar + metal, such as iron, manganese or chromium, the reaction being started + by a primer of magnesium and barium peroxide. The reaction is so + rapidly effected that there is an enormous rise in temperature, + estimated to be 5400 deg. F. (3000 deg. C.), which is sufficient to + melt the most refractory metals, such as chromium. The slag consists + of alumina which crystallizes in the forms of corundum and ruby, and + is utilized as an abrasive under the name of corubin. + + The chemical examination includes the determination of (1) moisture, + (2) ash, (3) coke, (4) volatile matter, (5) fixed carbon in coke, (6) + sulphur, (7) chlorine, (8) phosphorus. Moisture is determined by + noting the loss in weight when a sample is heated at 100 deg. for + about one hour. The ash is determined by heating a sample in a muffle + furnace until all the combustible matter has been burnt off. The ash, + which generally contains silica, oxides of the alkaline earths, ferric + oxide (which gives the ash a red colour), sulphur, &c., is analysed by + the ordinary gravimetric methods. The determination of coke is very + important on account of the conclusions concerning the nature of the + coal which it permits to be drawn. A sample is finely powdered and + placed in a covered porcelain crucible, which is surrounded by an + outer one, the space between them being packed with small coke. The + crucibles are heated in a wind furnace for 1 to 1-1/2 hours, then + allowed to cool, the inner crucible removed, and the coke weighed. The + coke may be (1) pulverulent, (2) slightly fritted, (3) spongy and + swelled, (4) compact. Pulverulent cokes indicate a non-caking + bituminous coal, rich in oxygen if the amount be below 60%, but if the + amount be very much less it generally indicates a lignite; if the + amount be above 80% it indicates an anthracite containing little + oxygen or hydrogen. A fritted coke indicates a slightly coking coal, + while the spongy appearance points to a highly coking coal which has + been partly fused in the furnace. A compact coke is yielded by good + coking coals, and is usually large in amount. The volatile matters are + determined as the loss of weight on coking less the amount of + moisture. The "fixed carbon" is the carbon retained in the coke, which + contains in addition the ash already determined. The fixed carbon is + therefore the difference between the coke and the ash, and may be + determined from these figures; or it may be determined directly by + burning off the coke in a muffle and noting the loss in weight. + Sulphur may be present as (1) organic sulphur, (2) as iron pyrites or + other sulphides, (3) as the sulphates of calcium, aluminium and other + metals; but the amount is generally so small that only the total + sulphur is determined. This is effected by heating a mixture of the + fuel with lime and sodium carbonate in a porcelain dish to redness in + a muffle until all the carbonaceous matter has been burnt off. The + residue, which contains the sulphur as calcium sulphate, is + transferred to a beaker containing water to which a little bromine has + been added. Hydrochloric acid is carefully added, the liquid filtered + and the residue washed. To the filtrate ammonia is added, and then + barium chloride, which precipitates the sulphur as barium sulphate. + Sulphur existing in the form of sulphates may be removed by washing a + sample with boiling water and determining the sulphuric acid in the + solution. The washed sample is then fused in the usual way to + determine the proportion of sulphur existing as iron pyrites. The + distinction between sulphur present as sulphate and sulphide is of + importance in the examination of coals intended for iron smelting, as + the sulphates of the earthy metals are reduced by the gases of the + furnace to sulphides, which pass into the slag without affecting the + quality of the iron produced, while the sulphur of the metallic + sulphides in the ash acts prejudicially upon the metal. Coals for + gas-making should contain little sulphur, as the gases produced in the + combustion are noxious and have very corrosive properties. Chlorine is + rarely determined, but when present in quantity it corrodes copper and + brass boiler tubes, with which consequently chlorine-bearing coals + cannot be used. The element is determined by fusing with soda lime in + a muffle, dissolving the residue in water and precipitating with + silver nitrate. Phosphorus is determined in the ash by fusing it with + a mixture of sodium and potassium carbonates, extracting the residue + with hydrochloric acid, and twice evaporating to dryness with the same + acid. The residue is dissolved in hydrochloric acid, a few drops of + ferric chloride added, and then ammonia in excess. The precipitate of + ferric phosphate is then treated as in the ordinary estimation of + phosphates. If it be necessary to determine the absolute amount of + carbon and hydrogen in a fuel, the dried sample is treated with copper + oxide as in the ordinary estimation of these elements in organic + compounds. (H. B.) + + +_Liquid Fuel._ + +Vegetable oil is not used for fuel except for laboratory purposes, +partly because its constituent parts are less adaptable for combustion +under the conditions necessary for steam-raising, but chiefly because of +the commercial difficulty of producing it with sufficient economy to +compete with mineral fuel either solid or liquid. + +The use of petroleum as fuel had long been recognized as a scientific +possibility, and some attempts had been made to adopt it in practice +upon a commercial scale, but the insufficiency, and still more the +irregularity, of the supplies prevented it from coming into practical +use to any important extent until about 1898, when discoveries of oil +specially adapted by chemical composition for fuel purposes changed the +aspect of the situation. These discoveries of special oil were made +first in Borneo and later in Texas, and experience in treating the oils +from both localities has shown that while not less adapted to produce +kerosene or illuminating oil, they are better adapted to produce fuel +oil than either the Russian or the Pennsylvanian products. Texas oil did +not hold its place in the market for long, because the influx of water +into the wells lowered their yield, but discoveries of fuel oil in +Mexico have come later and will help to maintain the balance of the +world's supply, although this is still a mere fraction of the assured +supply of coal. + +With regard to the chemical properties of petroleum, it is not necessary +to say more in the present place than that the lighter and more volatile +constituents, known commercially as naphtha and benzene, must be removed +by distillation in order to leave a residue composed principally of +hydrocarbons which, while containing the necessary carbon for +combustion, shall be sufficiently free from volatile qualities to avoid +premature ignition and consequent danger of explosion. Attempts have +been made to use crude oil for fuel purposes, and these have had some +success in the neighbourhood of the oil wells and under boilers of +unusually good ventilation both as regards their chimneys and the +surroundings of their stokeholds; but for reasons both of commerce and +of safety it is not desirable to use crude oil where some distillation +is possible. The more complete the process of distillation, and the +consequent removal of the volatile constituents, the higher the +flash-point, and the more turgid and viscous is the fuel resulting; and +if the process is carried to an extreme, the residue or fuel becomes +difficult to ignite by the ordinary process of spraying or atomizing +mechanically at the moment immediately preceding combustion. The +proportions which have been found to work efficiently in practice are as +follows:-- + + Carbon 88.00 % + Hydrogen 10.75 % + Oxygen 1.25 % + --------- + Total 100 + +The standards of safety for liquid fuel as determined by flash-point are +not yet finally settled, and are changing from time to time. The British +admiralty require a flash-point of 270 deg. F., and to this high +standard, and the consequent viscosity of the fuel used by vessels in +the British fleet, may partly be attributed the low rate of combustion +that was at first found possible in them. The German admiralty have +fixed a flash-point of 187 deg. F., and have used oil of this standard +with perfect safety, and at the same time with much higher measure of +evaporative duty than has been attained in British war-vessels. In the +British mercantile marine Lloyd's Register has permitted fuel with a +flash-point as low as 150 deg. F. as a minimum, and no harm has +resulted. The British Board of Trade, the department of the government +which controls the safety of passenger vessels, has fixed a higher +standard upon the basis of a minimum of 185 deg. In the case of +locomotives the flash-point as a standard of safety is of less +importance than in the case of stationary or marine boilers, because the +storage is more open, and the ventilation, both of the storage tanks and +the boilers during combustion, much more perfect than in any other class +of steam-boilers. + +The process of refining by distillation is also necessary to reduce two +impurities which greatly retard storage and combustion, i.e. water and +sulphur. Water is found in all crude petroleum as it issues from the +wells, and sulphur exists in important quantities in oil from the Texas +wells. Its removal was at first found very expensive, but there no +longer exists difficulty in this respect, and large quantities of +petroleum fuel practically free from sulphur are now regularly exported +from Texas to New York and to Europe. + +Water mixed with fuel is in intimate mechanical relation, and frequently +so remains in considerable quantities even after the process of +distillation. It is in fact so thoroughly mixed as to form an emulsion. +The effect of feeding such a mixture into a furnace is extremely +injurious, because the water must be decomposed chemically into its +constituents, hydrogen and oxygen, thus absorbing a large quantity of +heat which would otherwise be utilized for evaporation. Water also +directly delays combustion by producing from the jet a long, dull, red +flame instead of a short bright, white flame, and the process of +combustion, which should take place by vaporization of the oil near the +furnace mouth, is postponed and transferred to the upper part of the +combustion-box, the tubes, and even the base of the chimney, producing +loss of heat and injury to the boiler structure. The most effective +means of ridding the fuel of this dangerous impurity is by heat and +settlement. The coefficients of expansion of water and oil by heat are +substantially different, and a moderate rise of temperature therefore +separates the particles and precipitates the water, which is easily +drawn off--leaving the oil available for use. The heating and +precipitation are usually performed upon a patented system of settling +tanks and heating apparatus known as the Flannery-Boyd system, which has +proved itself indispensable for the successful use at sea of petroleum +fuel containing any large proportion of water. + + + Progress of liquid fuel. + +The laboratory and mechanical use of petroleum for fuel has already been +referred to, but it was not until the year 1870 that petroleum was +applied upon a wider and commercial scale. In the course of distillation +of Russian crude petroleum for the production of kerosene or lamp oil, +large quantities of refuse were produced--known by the Russian name of +_astatki_--and these were found an incumbrance and useless for any +commercial purpose. To a Russian oil-refiner gifted with mechanical +instinct and the genius for invention occurred the idea of utilizing the +waste product as fuel by spraying or atomizing it with steam, so that, +the thick and sluggish fluid being broken up into particles, the air +necessary for combustion could have free access to it. The earliest +apparatus for this purpose was a simple piece of gas-tube, into which +the thick oil was fed; by another connexion steam at high pressure was +admitted to an inner and smaller tube, and, the end of the tube nearest +to the furnace being open, the pressure of the steam blew the oil into +the furnace, and by its velocity broke it up into spray. The apparatus +worked with success from the first. Experience pointed out the proper +proportionate sizes for the inlets of steam and oil, the proper pressure +for the steam, and the proportionate sizes for the orifices of admission +to the furnaces, as well as the sizes of air-openings and best +arrangements of fire-bricks in the furnaces themselves; and what had +been a waste product now became a by-product of great value. Practically +all the steam power in South Russia, both for factories and navigation +of the inland seas and rivers, is now raised from _astatki_ fuel. + +In the Far East, including Burma and parts of China and Japan, the use +of liquid fuel spread rapidly during the years 1899, 1900 and 1901, +owing entirely to the development of the Borneo oil-fields by the +enterprise of Sir Marcus Samuel and the large British corporation known +as the Shell Transport and Trading Company, of which he is the head. +This corporation has since amalgamated with the Royal Dutch Petroleum +Company controlling the extensive wells in Dutch Borneo, and together +they supply large quantities of liquid fuel for use in the Far East. In +the United States of America liquid fuel is not only used for +practically the whole of the manufacturing and locomotive purposes of +the state of Texas, but factories in New York, and a still larger number +in California, are now discarding the use of coal and adopting +petroleum, because it is more economical in its consumption and also +more easily handled in transit, and saves nearly all the labour of +stoking. So far the supplies for China and Japan have been exported from +Borneo, but the discoveries of new oil-fields in California, of a +character specially adapted for fuel, have encouraged the belief that it +may be possible to supply Chile and Peru and other South American +countries, where coal is extremely expensive, with Californian fuel; and +it has also found its way across the Pacific to Japan. There are +believed to be large deposits in West Africa, but in the meantime the +only sources of supply to those parts of Africa where manufacture is +progressing, i.e. South Africa and Egypt, are the oil-fields of Borneo +and Texas, from which the import has well begun, from Texas to +Alexandria via the Mediterranean, and from Borneo to Cape Town via +Singapore. + +In England, notwithstanding the fact that there exist the finest +coal-fields in the world, there has been a surprising development of the +use of petroleum as fuel. The Great Eastern railway adapted 120 +locomotive engines to its use, and these ran with regularity and success +both on express passenger and goods trains until the increase in price +due to short supply compelled a return to coal fuel. The London, +Brighton & South Coast railway also began the adaptation of some of +their locomotive engines, but discontinued the use of liquid fuel from +the same cause. Several large firms of contractors and cement +manufacturers, chiefly on the banks of the Thames, made the same +adaptations which proved mechanically successful, but were not continued +when the price of liquid fuel increased with the increased demand. + +[Illustration: FIG. 1.--Holden Burner.] + + + Economy of liquid fuel. + +The chief factors of economy are the greater calorific value of oil than +coal (about 16 lb. of water per lb. of oil fuel evaporated from a +temperature of 212 deg. F.), not only in laboratory practice, but in +actual use on a large scale, and the saving of labour both in transit +from the source of supply to the place of use and in the act of stoking +the furnaces. The use of cranes, hand labour with shovels, wagons and +locomotives, horses and carts, is unavoidable for the transit of coal; +and labour to trim the coal, to stoke it when under combustion, and to +handle the residual ashes, are all indispensable to steam-raising by +coal. On the other hand, a system of pipes and pumps, and a limited +quantity of skilled labour to manage them, is all that is necessary for +the transit and combustion of petroleum fuel; and it is certain that +even in England will be found places which, from topographical and other +circumstances, will use petroleum more economically than coal as fuel +for manufacturing purposes under reasonable conditions of price for the +fuel. + +[Illustration: FIG. 2.--Rusden and Eeles Burner.] + +The theoretical calorific value of oil fuel is more nearly realized in +practice than the theoretical calorific value of coal, because the +facilities for complete combustion, due to the artificial admixture of +the air by the atomizing process, are greater in the case of oil than +coal, and for this reason, among others, the practical evaporative +results are proportionately higher with liquid fuel. In some cases the +work done in a steam-engine by 2 tons of coal has been performed by 1 +ton of oil fuel, but in others the proportions have been as 3 to 2, and +these latter can be safely relied on in practice as a minimum. This +saving, combined with the savings of labour and transit already +explained, will in the near future make the use of liquid fuel +compulsory, except in places so near to coal-fields that the cost of +coal becomes sufficiently low to counterbalance the savings in weight of +fuel consumed and in labour in handling it. In some locomotives on the +Great Eastern railway the consumption of oil and coal for the same +development of horse-power was as 17 lb. oil is to 35 lb. coal; all, +however, did not realize so high a result. + + + Liquid fuel in locomotives. + +The mechanical apparatus for applying petroleum to steam-raising in +locomotives is very simple. The space in the tender usually occupied by +coal is closed up by steel-plating closely riveted and tested, so as to +form a storage tank. From this tank a feed-pipe is led to a burner of +the combined steam-and-oil type already indicated, and this burner is so +arranged as to enter a short distance inside the furnace mouth. The +ordinary fire-bars are covered with a thin layer of coal, which starts +the ignition in the first place, and the whole apparatus is ready for +work. The burner best adapted for locomotive practice is the Holden +Burner (fig. 1), which was used on the Great Eastern railway. The +steam-pipe is connected at A, the oil-pipe at B, and the hand-wheels C +and D are for the adjustment of the internal orifices according to the +rate of combustion required. The nozzle E is directed towards the +furnace, and the external ring FF, supplied by the small pipe G and the +by-pass valve H, projects a series of steam jets into the furnace, +independent of the injections of atomized fuel, and so induces an +artificial inrush of air for the promotion of combustion. This type of +burner has also been tried on stationary boilers and on board ship. It +works well, although the great consumption of steam by the supplementary +ring is a difficulty at sea, where the water lost by the consumption of +steam cannot easily be made up. + + + Liquid fuel at sea. + +Although the application of the new fuel for land and locomotive boilers +has already been large, the practice at sea has been far more extensive. +The reason is chiefly to be found in the fact that although the sources +of supply are at a distance from Great Britain, yet they are in +countries to whose neighbourhood British steamships regularly trade, and +in which British naval squadrons are regularly stationed, so that the +advantages of adopting liquid fuel have been more immediate and the +economy more direct. The certainty of continuous supply of the fuel and +the wide distribution of storage stations have so altered the conditions +that the general adoption of the new fuel for marine purposes becomes a +matter of urgency for the statesman, the merchant and the engineer. None +of these can afford to neglect the new conditions, lest they be noted +and acted upon by their competitors. Storage for supply now exists at a +number of sea ports: London, Barrow, Southampton, Amsterdam, Copenhagen, +New Orleans, Savannah, New York, Philadelphia, Singapore, Hong Kong, +Madras, Colombo, Suez, Hamburg, Port Arthur, Rangoon, Calcutta, Bombay, +Alexandria, Bangkok, Saigon, Penang, Batavia, Surabaya, Amoy, Swatow, +Fuchow, Shanghai, Hankow, Sydney, Melbourne, Adelaide, Zanzibar, +Mombasa, Yokohama, Kobe and Nagasaki; also in South African and South +American ports. + +[Illustration: FIG. 3.--Storage of Liquid Fuel on Oil-carrying Steamers +(Flannery-Boyd System).] + +The British admiralty have undertaken experiments with liquid fuel at +sea, and at the same time investigations of the possibility of supply +from sources within the regions of the British empire. There is an +enormous supply of shale under the north-eastern counties of England, +but no oil that can be pumped--still less oil with a pressure above it +so as to "gush" like the wells in America--and the only sources of +liquid supply under the British flag appear to be in Burma and Trinidad. +The Borneo fields are not under British control, although developed +entirely by British capital. The Italian admiralty have fitted several +large warships with boiler apparatus to burn petroleum. The German +admiralty are regularly using liquid fuel on the China station. The +Dutch navy have fitted coal fuel and liquid fuel furnaces in +combination, so that the smaller powers required may be developed by +coal alone, and the larger powers by supplementing coal fuel with oil +fuel. The speeds of some vessels of the destroyer type have by this +means been accelerated nearly two knots. + +[Illustration: FIG. 4.--Installation on ss. "Trochas."] + +[Illustration: FIG. 5.--Details of Furnace, Meyer System.] + +[Illustration: FIG. 6.--Details of Exterior Elongation of Furnace, Meyer +System.] + + + Advantages in warships. + +The questions which govern the use of fuel in warships are more largely +those of strategy and fighting efficiency than economy of evaporation. +Indeed, the cost of constructing and maintaining in fighting efficiency +a modern warship is so great that the utmost use strategically must be +obtained from the vessel, and in this comparison the cost of fuel is +relatively so small an item that its increase or decrease may be +considered almost a negligible quantity. The desideratum in a warship is +to obtain the greatest fighting efficiency based on the thickest +armour, the heaviest and most numerous guns, the highest maximum speed, +and, last and not least, the greatest range of effective action based +upon the maximum supplies of fuel, provisions and other consumable +stores that the ship can carry. Now, if by changing the type of fuel it +be possible to reduce its weight by 30%, and to abolish the stokers, who +are usually more than half the ship's company, the weight saved will be +represented not merely by the fuel, but by the consumable stores +otherwise necessary for the stokers. Conversely, the radius of effective +action of the ship will be doubled as regards consumable stores if the +crew be halved, and will be increased by 50% if the same weight of fuel +be carried in the form of liquid instead of coal. In space the gain by +using oil fuel is still greater, and 36 cubic feet of oil as stored are +equal in practical calorific value to 67 cubic feet of coal according to +the allowance usual for ship's bunkering. On the other hand, coal has +been relied upon, when placed in the side bunkers of unarmoured ships, +as a protection against shot and shell, and this advantage, if it really +exists, could not be claimed in regard to liquid fuel. + +Recent experiments in coaling warships at sea have not been very +successful, as the least bad weather has prevented the safe transmission +of coal bags from the collier to the ship. The same difficulty does not +exist for oil fuel, which has been pumped through flexible tubing from +one ship to the other even in comparatively rough weather. +Smokelessness, so important a feature of sea strategy, has not always +been attained by liquid fuel, but where the combustion is complete, by +reason of suitable furnace arrangements and careful management, there is +no smoke. The great drawback, however, to the use of liquid fuel in fast +small vessels is the confined space allotted to the boilers, such +confinement being unavoidable in view of the high power concentrated in +a small hull. The British admiralty's experiments, however, have gone +far to solve the problem, and the quantity of oil which can be consumed +by forced draught in confined boilers now more nearly equals the +quantity of coal consumed under similar conditions. All recent vessels +built for the British navy are so constructed that the spaces between +their double bottoms are oil-tight and capable of storing liquid fuel in +the tanks so formed. Most recent battleships and cruisers have also +liquid fuel furnace fittings, and in 1910 it already appeared probable +that the use of oil fuel in warships would rapidly develop. + +In view of recent accusations of insufficiency of coal storage in +foreign naval depots, by reason of the allegation that coal so stored +quickly perishes, it is interesting to note that liquid fuel may be +stored in tanks for an indefinite time without any deterioration +whatever. + + + Advantages in merchant ships. + +In the case of merchant steamers large progress has also been made. The +Shell Transport and Trading Company have twenty-one vessels successfully +navigating in all parts of the world and using liquid fuel. The +Hamburg-American Steamship Company have four large vessels similarly +fitted for oil fuel, which, however, differ in furnace arrangements, as +will be hereafter described, although using coal when the fluctuation of +the market renders that the more economical fuel. One of the large +American transatlantic lines is adopting liquid fuel, and French, +German, Danish and American mercantile vessels are also beginning to use +it in considerable amounts. + +In the case of very large passenger steamers, such as those of 20 knots +and upwards in the Atlantic trade, the saving in cost of fuel is +trifling compared with the advantage arising from the greater weight and +space available for freight. Adopting a basis of 3 to 2 as between coal +consumption and oil consumption, there is an increase of 1000 tons of +dead weight cargo in even a medium-sized Atlantic steamer, and a +collateral gain of about 100,000 cub. ft. of measurement cargo, by +reason of the ordinary bunkers being left quite free, and the oil being +stored in the double bottom spaces hitherto unutilized except for the +purpose of water ballast. The cleanliness and saving of time from +bunkering by the use of oil fuel is also an important factor in +passenger ships, whilst considerable additional speed is obtainable. The +cost of the installation, however, is very considerable, as it includes +not only burners and pipes for the furnaces, but also the construction +of oil-tight tanks, with pumps and numerous valves and pipe connexions. + +[Illustration: FIG. 7.--Furnace on ss. "Ferdinand Laeisz." A, it is +proposed to do away with this ring of brickwork as being useless; B, it +is proposed to fill this space up, thus continuing lining of furnace to +combustion chamber, and also to fit protection bricks in way of saddle +plate.] + +[Illustration: FIG. 8.--Fuel Tanks, &c., of ss. "Murex."] + +[Illustration: FIG. 9.--Furnace Gear of ss. "Murex."] + + Fig. 2 shows a burner of Rusden and Eeles' patent as generally used on + board ships for the purpose of injecting the oil. A is a movable cap + holding the packing B, which renders the annular spindle M oil and + steam tight. E is the outer casing containing the steam jacket from + which the steam, after being fed through the steam-supply pipe G, + passes into the annular space surrounding the spindle P. It will be + seen that if the spindle P be travelled inwards by turning the handle + N, the orifice at the nozzle RR will be opened so as to allow the + steam to flow out radially. If at the same time the annular spindle M + be drawn inwards by revolving the handle L, the oil which passes + through the supply pipe F will also have emission at RR, and, coming + in contact with the outflowing steam, will be pulverized and sprayed + into the furnace. Fig. 3 is a profile and plan of a steamer adapted + for carrying oil in bulk, and showing all the storage arrangements for + handling liquid fuel. Fig. 4 shows the interior arrangement of the + boiler furnace of the steamship "Trocas." A is broken fire-brick + resting on the ordinary fire-bars, B is a brick bridge, C a casing of + fire-brick intended to protect the riveted seam immediately above it + from the direct impact of the flame, and D is a lining of fire-brick + at the back of the combustion-box, also intended to protect the + plating from the direct impact of the petroleum flame. The arrangement + of the furnace on the Meyer system is shown in fig. 5, where E is an + annular projection built at the mouth of the furnace, and BB are + spiral passages for heating the air before it passes into the furnace. + Fig. 6 shows the rings CC and details of the casting which forms the + projection or exterior elongation of the furnace. The brickwork + arrangement adopted for the double-ended boilers on the + Hamburg-American Steamship Company's "Ferdinand Laeisz" is represented + in fig. 7. The whole furnace is lined with fire-brick, and the burner + is mounted upon a circular disk plate which covers the mouth of the + furnace. The oil is injected not by steam pulverization, but by + pressure due to a steam-pump. The oil is heated to about 60 deg. C. + before entering the pump, and further heated to 90 deg. C. after + leaving the pump. It is then filtered, and passes to the furnace + injector C at about 30-lb. pressure; and its passage through this + injector and the spiral passages of which it consists pulverizes the + oil into spray, in which form it readily ignites on reaching the + interior of the furnace. The injector is on the Korting principle, + that is, it atomizes by fracture of the liquid oil arising from its + own momentum under pressure. The advantage of this system as compared + with the steam-jet system is the saving of fresh water, the + abstraction of which is so injurious to the boiler by the formation of + scale. + + [Illustration: FIG. 10.--Section through Furnace of ss. "Murex."] + + The general arrangement of the fuel tanks and filling pipes on the ss. + "Murex" is shown in fig. 8; and fig. 9 represents the furnace gear of + the same vessel, A being the steam-pipe, B the oil-pipe, C the + injector, D the swivel upon which the injector is hung so that it may + be swung clear of the furnace, E the fire-door, and F the handle for + adjusting the injector. In fig. 10, which represents a section of the + furnace, H is a fire-brick pier and K a fire-brick baffling bridge. + + It is found in practice that to leave out the fire-bars ordinarily + used for coal produces a better result with liquid fuel than the + alternative system of keeping them in place and protecting them by a + layer of broken fire-brick. + + Boilers fitted upon all the above systems have been run for thousands + of miles without trouble. In new construction it is desirable to give + larger combustion chambers and longer and narrower boiler tubes than + in the case of boilers intended for the combustion of coal alone. + (F. F.*) + + +_Gaseous Fuel._ + +Strictly speaking, much, and sometimes even most, of the heating +effected by solid or liquid fuel is actually performed by the gases +given off during the combustion. We speak, however, of gaseous fuel only +in those cases where we supply a combustible gas from the outset, or +where we produce from ordinary solid (or liquid) fuel in one place a +stream of combustible gas which is burned in another place, more or less +distant from that where it has been generated. + + The various descriptions of gaseous fuel employed in practice may be + classified under the following heads: + + I. Natural Gas. + + II. Combustible Gases obtained as by-products in various technical + operations. + + III. Coal Gas (Illuminating Gas). + + IV. Combustible Gases obtained by the partial combustion of coal, &c. + +I. _Natural Gas._--From time immemorial it has been known that in some +parts of the Caucasus and of China large quantities of gases issue from +the soil, sometimes under water, which can be lighted and burn with a +luminous flame. The "eternal fires" of Baku belong to this class. In +coal-mines frequently similar streams of gas issue from the coal; these +are called "blowers," and when they are of somewhat regular occurrence +are sometimes conducted away in pipes and used for underground lighting. +As a regular source of heating power, however, natural gas is employed +only in some parts of the United States, especially in Pennsylvania, +Kansas, Ohio and West Virginia, where it always occurs in the +neighbourhood of coal and petroleum fields. The first public mention of +it was made in 1775, but it was not till 1821 that it was turned to use +at Fredonia, N.Y. In Pennsylvania natural gas was discovered in 1859, +but at first very little use was made of it. Its industrial employment +dates only from 1874, and became of great importance about ten years +later. Nobody ever doubted that the gas found in these localities was an +accumulation of many ages and that, being tapped by thousands of +bore-holes, it must rapidly come to an end. This assumption was +strengthened by the fact that the "gas-wells," which at first gave out +the gas at a pressure of 700 or 800, sometimes even of 1400 lb. per sq. +in., gradually showed a more and more diminishing pressure and many of +them ceased to work altogether. About the year 1890 the belief was +fairly general that the stock of natural gas would soon be entirely +exhausted. Indeed, the value of the annual production of natural gas in +the United States, computed as its equivalent of coal, was then +estimated at twenty-one million dollars, in 1895 at twelve millions, in +1899 at eleven and a half millions. But the output rose again to a value +of twenty-seven millions in 1901, and to fifty million dollars in 1907. +Mostly the gas, derived from upwards of 10,000 gas-wells, is now +artificially compressed to a pressure of 300 or 400 lb. per sq. in. by +means of steam-power or gas motors, fed by the gas itself, and is +conveyed over great distances in iron pipes, from 9 or 10 to 36 in. in +diameter. In 1904 nearly 30,000 m. of pipe lines were in operation. In +1907 the quantity of natural gas consumed in the United States (nearly +half of which was in Pennsylvania) was 400,000 million cub. ft., or +nearly 3 cub. m. Canada (Ontario) also produces some natural gas, +reaching a maximum of about $746,000 in 1907. + +The principal constituent of natural gas is always methane, CH4, of +which it contains from 68.4 to 94.0% by volume. Those gases which +contain less methane contain all the more hydrogen, viz. 2.9 to 29.8%. +There is also some ethylene, ethane and carbon monoxide, rarely +exceeding 2 or 3%. The quantity of incombustible gases--oxygen, carbon +dioxide, nitrogen--ranges from mere traces to about 5%. The density is +from 0.45 to 0.55. The heating power of 1000 cub. ft. of natural gas is +equal to from 80 to 120 lb., on the average 100 lb., of good coal, but +it is really worth much more than this proportion would indicate, as it +burns completely, without smoke or ashes, and without requiring any +manual labour. It is employed for all domestic and for most industrial +purposes. + +The origin of natural gas is not properly understood, even now. The most +natural assumption is, of course, that its formation is connected with +that of the petroleum always found in the same neighbourhood, the latter +principally consisting of the higher-boiling aliphatic hydrocarbons of +the methane series. But whence do they both come? Some bring them into +connexion with the formation of coal, others with the decomposition of +animal remains, others with that of _diatomaceae_, &c., and even an +inorganic origin of both petroleum and natural gas has been assumed by +chemists of the rank of D.I. Mendeleeff and H. Moissan. + +II. _Gases obtained as By-products._--There are two important cases in +which gaseous by-products are utilized as fuel; both are intimately +connected with the manufacture of iron, but in a very different way, and +the gases are of very different composition. + +(a) _Blast-furnace Gases._--The gases issuing from the mouths of +blast-furnaces (see IRON AND STEEL) were first utilized in 1837 by Faber +du Faur, at Wasseralfingen. Their use became more extensive after 1860, +and practically universal after 1870. The volume of gas given off per +ton of iron made is about 158,000 cub. ft. Its percentage composition by +volume is: + + Carbon monoxide 21.6 to 29.0, mostly about 26% + Hydrogen 1.8 " 6.3, " " 3% + Methane 0.1 " 0.8, " " 0.5% + Carbon dioxide 6 " 12, " " 9.5% + Nitrogen 51 " 60, " " 56 % + Steam 5 " 12, " " 5 % + ----- + 100 % + +There is always a large amount of mechanically suspended flue-dust in +this gas. It is practically equal to a poor producer-gas (see below), +and is everywhere used, first for heating the blast in Cowper stoves or +similar apparatus, and secondly for raising all the steam required for +the operation of the blast-furnace, that is, for driving the +blowing-engines, hoisting the materials, &c. Where the iron ore is +roasted previously to being fed into the furnace, this can also be done +by this gas, but in some cases the waste in using it is so great that +there is not enough left for the last purpose. The calorific power of +this gas per cubic foot is from 80 to 120 B.Th.U. + +Since about 1900 a great advance has been made in this field. Instead of +burning the blast-furnace gas under steam boilers and employing the +steam for producing mechanical energy, the gas is directly burned in +gas-motors on the explosion principle. Thus upwards of three times the +mechanical energy is obtained in comparison with the indirect way +through the steam boiler. After all the power required for the +operations of the blast-furnace has been supplied, there is a surplus of +from 10 to 20 h.p. for each ton of pig-iron made, which may be applied +to any other purpose. + +(b) _Coke-oven Gases._--Where the coking of coal is performed in the old +beehive ovens or similar apparatus the gas issuing at the mouth of the +ovens is lost. The attempts at utilizing the gases in such cases have +not been very successful. It is quite different where coke is +manufactured in the same way as illuminating gas, viz. by the +destructive distillation of coal in closed apparatus (retorts), heated +from the outside. This industry, which is described in detail in G. +Lunge's _Coal-Tar and Ammonia_ (4th ed., 1909), originated in France, +but has spread far more in Germany, where more than half of the coke +produced is made by it; in the United Kingdom and the United States its +progress has been much slower, but there also it has long been +recognized as the only proper method. The output of coke is increased by +about 15% in comparison with the beehive ovens, as the heat required for +the process of distillation is not produced by burning part of the coal +itself (as in the beehive ovens), but by burning part of the gas. The +quality of the coke for iron-making is quite as good as that of beehive +coke, although it differs from it in appearance. Moreover, the gases can +be made to yield their ammonia, their tar, and even their benzene +vapours, the value of which products sometimes exceeds that of the coke +itself. And after all this there is still an excess of gas available for +any other purpose. + +As the principle of distilling the coal is just the same, whether the +object is the manufacture of coal gas proper or of coke as the main +product, although there is much difference in the details of the +manufacture, it follows that the quality of the gas is very similar in +both cases, so far as its heating value is concerned. Of course this +heating value is less where the benzene has been extracted from +coke-oven gas, since this compound is the richest heat-producer in the +gas. This is, however, of minor importance in the present case, as there +is only about 1% benzene in these gases. + +The composition of coke-oven gases, after the extraction of the ammonia +and tar, is about 53% hydrogen, 36% methane, 6% carbon monoxide, 2% +ethylene and benzene, 0.5% sulphuretted hydrogen, 1.5% carbon dioxide, +1% nitrogen. + +III. _Coal Gas (Illuminating Gas)._--Although ordinary coal gas is +primarily manufactured for illuminating purposes, it is also extensively +used for cooking, frequently also for heating domestic rooms, baths, +&c., and to some extent also for industrial operations on a small scale, +where cleanliness and exact regulation of the work are of particular +importance. In chemical laboratories it is preferred to every other kind +of fuel wherever it is available. The manufacture of coal gas being +described elsewhere in this work (see GAS, S _Manufacture_), we need +here only point out that it is obtained by heating bituminous coal in +fireclay retorts and purifying the products of this destructive +distillation by cooling, washing and other operations. The residual gas, +the ordinary composition of which is given in the table below, amounts +to about 10,000 cub. ft. for a ton of coal, and represents about 21% of +its original heating value, 56.5% being left in the coke, 5.5% in the +tar and 17% being lost. As we must deduct from the coke that quantity +which is required for the heating of the retorts, and which, even when +good gas producers are employed, amounts to 12% of the weight of the +coal, or 10% of its heat value, the total loss of heat rises to 27%. +Taking, further, into account the cost of labour, the wear and tear, and +the capital interest on the plant, coal gas must always be an expensive +fuel in comparison with coal itself, and cannot be thought of as a +general substitute for the latter. But in many cases the greater expense +of the coal gas is more than compensated by its easy distribution, the +facility and cleanliness of its application, the general freedom from +the mechanical loss, unavoidable in the case of coal fires, the +prevention of black smoke and so forth. The following table shows the +average composition of coal gas by volume and weight, together with the +heat developed by its single constituents, the latter being expressed in +kilogram-calories per cub. metre (0.252 kilogram-calories = 1 British +heat unit; 1 cub. metre = 35.3 cub. ft.; therefore 0.1123 calories per +cub. metre = 1 British heat unit per cub. foot). + + +----------------------+----------+----------+------------+--------------+------------+ + | | | | Heat-value | Heat-value | Heat-value | + | Constituents. | Volume | Weight | per Cubic | per Quantity | per cent. | + | | per cent.| per cent.| Metre | contained in | of Total. | + | | | | Calories. | 1 Cub. Met. | | + +----------------------+----------+----------+-----------+---------------+------------+ + | Hydrogen, H2 | 47 | 7.4 | 2,582 | 1213 | 22.8 | + | Methane, CH4 | 34 | 42.8 | 8,524 | 2898 | 54.5 | + | Carbon monoxide, CO | 9 | 19.9 | 3,043 | 273 | 5.1 | + | Benzene vapour, C6H6 | 1.2 | 7.4 | 33,815 | 405 | 7.7 | + | Ethylene, C2H4 | 3.8 | 8.4 | 13,960 | 530 | 9.9 | + | Carbon dioxide, CO2 | 2.5 | 8.6 | .. | .. | .. | + | Nitrogen, N2 | 2.5 | 5.5 | .. | .. | .. | + | +----------+----------+------------+--------------+------------+ + | Total | 100.0 | 100.0 | .. | 5319 | 100.0 | + +----------------------+----------+----------+------------+--------------+------------+ + +One cubic metre of such gas weighs 568 grammes. _Rich gas_, or gas made +by the destructive distillation of certain bituminous schists, of oil, +&c., contains much more of the heavy hydrocarbons, and its heat-value is +therefore much higher than the above. The carburetted water gas, very +generally made in America, and sometimes employed in England for mixing +with coal gas, is of varying composition; its heat-value is generally +rather less than that of coal gas (see below). + +IV. _Combustible Gases produced by the Partial Combustion of Coal, +&c._--These form by far the most important kind of gaseous fuel. When +coal is submitted to destructive distillation to produce the +illuminating gas described in the preceding paragraph, only a +comparatively small proportion of the heating value of the coal (say, a +sixth or at most a fifth part) is obtained in the shape of gaseous fuel, +by far the greater proportion remaining behind in the shape of coke. + +An entirely different class of gaseous fuels comprises those produced by +the incomplete combustion of the total carbon contained in the raw +material, where the result is a mixture of gases which, being capable of +combining with more oxygen, can be burnt and employed for heating +purposes. Apart from some descriptions of waste gases belonging to this +class (of which the most notable are those from blast-furnaces), we must +distinguish two ways of producing such gaseous fuels entirely different +in principle, though sometimes combined in one operation. The incomplete +combustion of carbon may be brought about by means of atmospheric +oxygen, by means of water, or by a simultaneous combination of these two +actions. In the first case the chemical reaction is + + C + O= CO (a); + +the nitrogen accompanying the oxygen in the atmospheric air necessarily +remains mixed with carbon monoxide, and the resulting gases, which +always contain some carbon dioxide, some products of the destructive +distillation of the coal, &c., are known as _producer gas_ or _Siemens +gas_. In the second case the chemical reaction is mainly + + C + H2O = CO + H2 (b); + +that is to say, the carbon is converted into monoxide and the hydrogen +is set free. As both of these substances can combine with oxygen, and as +there is no atmospheric nitrogen to deal with, the resulting gas (_water +gas_) is, apart from a few impurities, entirely combustible. Another +kind of water gas is formed by the reaction + + C + 2H2O = CO2 + 2H2 (c), + +but this reaction, which converts all the carbon into the incombustible +form of CO2, is considered as an unwelcome, although never entirely +avoidable, concomitant of (b). + +The reaction by which water gas is produced being endothermic (as we +shall see), this gas cannot be obtained except by introducing the +balance of energy in another manner. This might be done by heating the +apparatus from without, but as this method would be uneconomical, the +process is carried out by alternating the endothermic production of +water gas with the exothermic combustion of carbon by atmospheric air. +Pure water gas is not, therefore, made by a continuous process, but +alternates with the production of other gases, combustible or not. But +instead of constantly interrupting the process in this way, a continuous +operation may be secured by simultaneously carrying on both the +reactions (a) and (b) in such proportions that the heat generated by (a) +at least equals the heat absorbed by (b). For this purpose the apparatus +is fed at the same time with atmospheric air and with a certain quantity +of steam, preferably in a superheated state. Gaseous mixtures of this +kind have been made, more or less intentionally, for a long time past. +One of the best known of them, intended less for the purpose of serving +as ordinary fuel than for that of driving machinery, is the Dowson gas. + +An advantage common to all kinds of gaseous fuel, which indeed forms the +principal reason why it is intentionally produced from solid fuel, in +spite of inevitable losses in the course of the operation, is the +following. The combustion of solid fuel (coal, &c.) cannot be carried on +with the theoretically necessary quantity of atmospheric air, but +requires a considerable excess of the latter, at least 50%, sometimes +100% and more. This is best seen from the analyses of smoke gases. If +all the oxygen of the air were converted into CO2 and H2O, the amount of +CO2 in the smoke gases should be in the case of pure carbon nearly 21 +volumes %, as carbon dioxide occupies the same volume as oxygen; while +ordinary coal, where the hydrogen takes up a certain quantity of oxygen +as well, should show about 18.5% CO2. But the best smoke gases of steam +boilers show only 12 or 13%, much more frequently only 10% CO2, and +gases from reverberatory furnaces often show less than 5%. This means +that the volume of the smoke gases escaping into the air is from 1-1/2 +to 2 times (in the case of high-temperature operations often 4 times) +greater than the theoretical minimum; and as these gases always carry +off a considerable quantity of heat, the loss of heat is all the greater +the less complete is the utilization of the oxygen and the higher the +temperature of the operation. This explains why, in the case of the +best-constructed steam-boiler fires provided with heat economizers, +where the smoke gases are deprived of most of their heat, the proportion +of the heat value of the fuel actually utilized may rise to 70 or even +75%, while in some metallurgical operations, in glass-making and similar +cases, it may be below 5%. + +One way of overcoming this difficulty to a certain extent is to reduce +the solid fuel to a very fine powder, which can be intimately mixed with +the air so that the consumption of the latter is only very slightly in +excess of the theoretical quantity; but this process, which has been +only recently introduced on a somewhat extended scale, involves much +additional expense and trouble, and cannot as yet be considered a real +success. Generally, too, it is far less easily applied than gaseous +fuel. The latter can be readily and intimately mixed with the exact +quantity of air that is required and distributed in any suitable way, +and much of the waste heat can be utilized for a preliminary heating of +the air and the gas to be burned by means of "recuperators." + +We shall now describe the principal classes of gaseous fuel, produced by +the partial combustion of coal. + +A. _Producer Gas, Siemens Gas._--As we have seen above, this gas is made +by the incomplete combustion of fuel. The materials generally employed +for its production are anthracite, coke or other fuels which are not +liable to cake during the operation, and thus stop the draught or +otherwise disturb the process, but by special measures also bituminous +coal, lignite, peat and other fuel may be utilized for gas producers. +The fuel is arranged in a deep layer, generally from 4 ft. up to 10 ft., +and the air is introduced from below, either by natural draught or by +means of a blast, and either by a grate or only by a slit in the wall of +the "gas producer." Even if the primary action taking place at the +entrance of the air consisted in the complete combustion of the carbon +to dioxide, CO2, the latter, in rising through the high column of +incandescent fuel, must be reduced to monoxide: CO2 + C = 2CO. But as +the temperature in the producer rises rather high, and as in ordinary +circumstances the action of oxygen on carbon above 1000 deg. C. consists +almost entirely in the direct formation of CO, we may regard this +compound as primarily formed in the hotter parts of the gas-producer. It +is true that ordinary producer gas always contains more or less CO2, but +this may be formed higher up by air entering through leakages in the +apparatus. If we ignore the hydrogen contained in the fuel, the +theoretical composition of producer gas would be 33.3% CO and 66.7% N, +both by volume and weight. Its weight per cubic metre is 1.251 grammes, +and its heat value 1013 calories per cubic metre, or less than one-fifth +of the heat-value of coal gas. Practically, however, producer gas +contains a small percentage of gases, increasing its heat-value, like +hydrogen, methane, &c., but on the other hand it is never free from +carbon dioxide to the extent of from 2 to 8%. Its heat-value may +therefore range between 800 and 1100 calories per cubic metre. Even when +taking as the basis of our calculation a theoretical gas of 33.3% CO, we +find that there is a great loss of heat-value in the manufacture of this +gas. Thermochemistry teaches us that the reaction C + O develops 29.5% +of the heat produced by the complete oxidation of C to CO2, thus leaving +only 70.5% for the stage CO + O = CO2. If, therefore, the gas given off +in the producer is allowed to cool down to ordinary temperature, nearly +30% of the heat-value of the coal is lost by radiation. If, however, the +gas producer is built in close proximity to the place where the +combustion takes place, so that the gas does not lose very much of its +heat, the loss is correspondingly less. Even then there is no reason why +this mode of burning the fuel, i.e. first with "primary air" in the +producer (C + O = CO), then with "secondary air" in the furnace (CO + O += CO2), should be preferred to the direct complete burning of the fuel +on a grate, unless the above-mentioned advantage is secured, viz. +reduction of the smoke gases to a minimum by confining the supply of air +as nearly as possible to that required for the formation of CO2, which +is only possible by producing an intimate mixture of the producer gas +with the secondary air. The advantage in question is not very great +where the heat of the smoke gases can be very fully utilized, e.g. in +well-constructed steam boilers, salt-pans and the like, and as a matter +of fact gas producers have not found much use in such cases. But a very +great advantage is attained in high-temperature operations, where the +smoke gases escape very hot, and where it is on that account +all-important to confine their quantity to a minimum. + +It is precisely in these cases that another requirement frequently comes +in, viz. the production at a given point of a higher temperature than is +easily attained by ordinary fires. Gas-firing lends itself very well to +this end, as it is easily combined with a preliminary heating up of the +air, and even of the gas itself, by means of "recuperators." The +original and best-known form of these, due to Siemens Brothers, consists +of two brick chambers filled with loosely stacked fire-bricks in such +manner that any gases passed through the chambers must seek their way +through the interstices left between the bricks, by which means a +thorough interchange of temperature takes place. The smoke gases, +instead of escaping directly into the atmosphere, are made to pass +through one of these chambers, giving up part of their heat to the +brickwork. After a certain time the draught is changed by means of +valves, the smoke gases are passed through another chamber, and the cold +air intended to feed the combustion is made to pass through the first +chamber, where it takes up heat from the white-hot bricks, and is thus +heated up to a bright red heat until the chamber is cooled down too far, +when the draughts are again reversed. Sometimes the producer gas itself +is heated up in this manner (especially when it has been cooled down by +travelling a long distance); in that case four recuperator chambers must +be provided instead of two. Another class of recuperators is not founded +on the alternating system, but acts continuously; the smoke gases travel +always in the same direction in flues contiguous to other flues or pipes +in which the air flows in the opposite direction, an interchange of heat +taking place through the walls of the flues or pipes. Here the surface +of contact must be made very large if a good effect is to be produced. +In both cases not merely is a saving effected of all the calories which +are abstracted by the cold air from the recuperator, but as less fuel +has to be burned to get a given effect, the quantity of smoke gas is +reduced. For details and other producer gases, see GAS, II. _For Fuel +and Power._ + +Gas-firing in the manner just described can be brought about by very +simple means, viz. by lowering the fire-grate of an ordinary fire-place +to at least 4 ft. below the fire-bridge, and by introducing the air +partly below the grate and partly behind the fire-place, at or near the +point where the greatest heat is required. Usually, however, more +elaborate apparatus is employed, some of which we shall describe below. +Gas-firing has now become universal in some of the most important +industries and nearly so in others. The present extension of +steel-making and other branches of metallurgy is intimately connected +with this system, as is the modern method of glass-making, of heating +coal gas retorts and so forth. + +The composition of producer gas differs considerably, principally +according to the material from which it is made. Analyses of ordinary +producer gas (not such as falls under the heading of "semi-water gas," +see _sub_ C) by volume show 22 to 33% CO, 1 to 7% CO2, 0.5 to 2% H2, 0.5 +to 3% hydrocarbons, and 64 to 68% N2. + +B. _Water Gas._--The reaction of steam on highly heated carbonaceous +matter was first observed by Felice Fontana in 1780. This was four years +before Henry Cavendish isolated hydrogen from water, and thirteen years +before William Murdoch made illuminating gas by the distillation of +coal, so that it was no wonder that Fontana's laboratory work was soon +forgotten. Nor had the use of carburetted water gas, as introduced by +Donovan in 1830 for illuminating purposes, more than a very short life. +More important is the fact that during nine years the illumination of +the town of Narbonne was carried on by incandescent platinum wire, +heated by water gas, where also internally heated generators were for +the first time regularly employed. The Narbonne process was abandoned in +1865, and for some time no real progress was made in this field in +Europe. But in America, T.S.C. Lowe, Strong, Tessie du Motay and others +took up the matter, the first permanent success being obtained by the +introduction (1873) of Lowe's system at Phoenixville, Pa. In the United +States the abundance of anthracite, as well as of petroleum naphtha, +adapted for carburetting the gas, secures a great commercial advantage +to this kind of illuminant over coal gas, so that now three-fourths of +all American gas-works employ carburetted water gas. In Europe the +progress of this industry was naturally much less rapid, but here also +since 1882, when the apparatus of Lowe and Dwight was introduced in the +town of Essen, great improvements have been worked out, principally by +E. Blass, and by these improvements water gas obtained a firm footing +also for certain heating purposes. The American process for making +carburetted water gas, as an auxiliary to ordinary coal gas, was first +introduced by the London Gas Light and Coke Company on a large scale in +1890. + +Water gas in its original state is called "blue gas," because it burns +with a blue, non-luminous flame, which produces a very high temperature. +According to the equation C + H2O = CO + H2, this gas consists +theoretically of equal volumes of carbon monoxide and hydrogen. We shall +presently see why it is impossible to avoid the presence of a little +carbon dioxide and other gases, but we shall for the moment treat of +water gas as if it were composed according to the above equation. The +reaction C + H2O = CO + H2 is endothermic, that is, its thermal value is +negative. One gram-molecule of carbon produces 97 great calories (1 +great calorie or kilogram-calorie = 1000 gram-calories) when burning to +CO2, and this is of course the maximum effect obtainable from this +source. If the same gram-molecule of carbon is used for making water +gas, that is, CO + H2, the heat produced by the combustion of the +product is 68.4 + 57.6 = 126 great calories, an apparent surplus of 29 +calories, which cannot be got out of nothing. This is made evident by +another consideration. In the above reaction C is not burned to CO2, but +to CO, a reaction which produces 28.6 calories per gram-molecule. But as +the oxygen is furnished from water, which must first be decomposed by +the expenditure of energy, we must introduce this amount, 68.5 calories +in the case of liquid water, or 57.6 calories in the case of steam, as a +negative quantity, and the difference, viz. + 28.6 - 57.6 = 29 great +calories, represents the amount of heat to be expended from another +source in order to bring about the reaction of one gram-molecule of +carbon on one gram-molecule of H2O in the shape of steam. This explains +why steam directed upon incandescent coal will produce water gas only +for a very short time: even a large mass of coal will quickly be cooled +down so much that at first a gas of different composition is formed and +soon the process will cease altogether. We can avoid this result by +carrying on the process in a retort heated from without by an ordinary +coal fire, and all the early water gas apparatus was constructed in this +way; but such a method is very uneconomical, and was long ago replaced +by a process first patented by J. and T.N. Kirkham in 1854, and very +much improved by successive inventors. This process consists in +conducting the operation in an upright brick shaft, charged with +anthracite, coke or other suitable fuel. This shaft resembles an +ordinary gas producer, but it differs in being worked, not in a +continuous manner, which, as shown above, would be impossible, but by +alternately blowing air and steam through the coal for periods of a few +minutes each. During the first phase, when carbon is burned by +atmospheric oxygen, and thereby heat is produced, this heat, or rather +that part of it which is not carried away by radiation and by the +products of combustion on leaving the apparatus, is employed in raising +the temperature of the remaining mass of fuel, and is thus available for +the second phase, in which the reaction (b) C + H2O = CO + H2 goes on +with the abstraction of a corresponding amount of heat from the +incandescent fuel, so that the latter rapidly cools down, and the +process must be reversed by blowing in air and so forth. The formation +of exactly equal volumes of carbon monoxide and hydrogen goes on only at +temperatures over 1200 deg. C., that is, for a very few minutes. Even at +1100 deg. C. a little CO2 can be proved to exist in the gas, and at 900 +deg. its proportion becomes too high to allow the process to go on. +About 650 deg. C. the CO has fallen to a minimum, and the reaction is +now essentially (c) C + 2H2O = CO2 + 2H2; soon after the temperature of +the mass will have fallen to such a low point that the steam passes +through it without any perceptible action. The gas produced by reaction +(c) contains only two-thirds of combustible matter, and is on that +account less valuable than proper water gas formed by reaction (b); +moreover, it requires the generation of twice the amount of steam, and +its presence is all the less desirable since it must soon lead to a +total cessation of the process. In ordinary circumstances it is evident +that the more steam is blown in during a unit of time, the sooner +reaction (c) will set in; on the other hand, the more heat has been +accumulated in the producer the longer can the blowing-in of steam be +continued. + +The process of making water gas consequently comprises two alternating +operations, viz. first "blowing-up" by means of a current of air, by +which the heat of the mass of fuel is raised to about 1200 deg. C.; and, +secondly "steaming," by injecting a current of (preferably superheated) +steam until the temperature of the fuel had fallen to about 900 deg. C., +and too much carbon dioxide appears in the product. During the steaming +the gas is carried off by a special conduit into a scrubber, where the +dust mechanically carried away in the current is washed out, and the gas +is at the same time cooled down nearly to the ordinary temperature. It +is generally stored in a gas-holder, from which it is conducted away as +required. It is never quite free from nitrogen, as the producer at the +beginning of steaming contains much of this gas, together with CO or +CO2. The proportion of hydrogen may exceed 50%, in consequence of +reaction (c) setting in at the close of the steaming. Ordinary "blue" +water gas, if, as usual, made from coke or anthracite, contains 48-52% +H2, 40-41% CO, 1-5% CO2, 4-5% N2, and traces of hydrocarbons, especially +methane. If made from bituminous coal, it contains more of the latter. +If "carburetted" (a process which increases its volume 50% and more) by +the vapours from superheated petroleum naphtha, the proportion of CO +ranges about 25%, with about as much methane, and from 10 to 15% of +"illuminants" (heavy hydrocarbons). The latter, of course, greatly +enhance the fuel-value of the gas. Pure water gas would possess the +following fuel-value per cubic metre: + + 0.5 cub. met. H2 = 1291 calories + 0.5 " " CO = 1522 " + ---- + 2813 " + +Ordinary "blue" water gas has a fuel-value of at least 2500 calories. +Carburetted water gas, which varies very much in its percentage of +hydrocarbons, sometimes reaches nearly the heat-value of coal gas, but +such gas is only in exceptional cases used for heating purposes. + +We must now turn to the "blowing-up" stage of the process. Until +recently it was assumed that during this stage the combustion of carbon +cannot be carried on beyond the formation of carbon monoxide, for as the +gas-producer must necessarily contain a deep layer of fuel (generally +about 6 to 10 ft.), any CO2 formed at first would be reduced to CO; and +it was further assumed that hardly any CO2 would be formed from the +outset, as the temperature of the apparatus is too high for this +reaction to take place. But as the combustion of C to CO produces only +about 30% of the heat produced when C is burned into CO2, the quantity +of fuel consumed for "blowing-up" is very large, and in fact +considerably exceeds that consumed in "steaming." There is, of course, a +further loss by radiation and minor sources, and the result is that 1 +kilogram of carbon yields only about 1.2 cub. met. of water gas. Each +period of blowing-up generally occupies from 8 to 12 minutes, that of +steaming only 4 or 5 minutes. This low yield of water gas until quite +recently appeared to be unavoidable, and the only question seemed to be +whether and to what extent the gas formed during blowing-up, which is in +fact identical with ordinary producer gas (Siemens gas), could be +utilized. In America, where the water gas is mostly employed for +illuminating purposes, at least part of the blowing-up gas is utilized +for heating the apparatus in which the naphtha is volatilized and the +vapours are "fixed" by superheating. This process, however, never +utilizes anything like the whole of the blowing-up gas, nor can this be +effected by raising and superheating the steam necessary for the second +operation; indeed, the employment of this gas for raising steam is not +very easy, owing to the irregularities of and constant interruptions in +the supply. In some systems the gas made during the blowing-up stage is +passed through chambers, loosely filled with bricks, like Siemens +recuperators, where it is burned by "secondary" air: the heat thus +imparted to the brickwork is utilized by passing through the +recuperator, and thus superheating, the steam required for the next +steaming operation. In many cases, principally where no carburetting is +practised, the blowing-up gas is simply burned at the mouth of the +producer, and is thus altogether lost; and in no case can it be utilized +without great waste. A very important improvement in this respect was +effected by C. Dellwik and E. Fleischer. They found that the view that +it is unavoidable to burn the carbon to monoxide during the blowing-up +holds good only for the pressure of blast formerly applied. This did not +much exceed that which is required for overcoming the frictional +resistance within the producer. If, however, the pressure is +considerably increased, and the height of the column of fuel reduced, +both of these conditions being strictly regulated in accordance with the +result desired, it is easy to attain a combustion of the carbon to +dioxide, with only traces of monoxide, in spite of the high temperature. +Evidently the excess of oxygen coming into contact with each particle of +carbon in a given unit of time produces other conditions of chemical +equilibrium than those existing at lower pressures. At any rate, +experience has shown that by this process, in which the full heat-value +of carbon is utilized during the blowing-up stage, the time of +heating-up can be reduced from 10 to 1-1/2 or 2 minutes, and the +steaming can be prolonged from 4 or 5 to 8 or 10 minutes, with the +result that twice the quantity of water gas is obtained, viz. upwards of +2 cub. metres from 1 kilogram of carbon. + +The application of water gas as a fuel mainly depends upon the high +temperatures which it is possible to attain by its aid, and these are +principally due to the circumstance that it forms a much smaller flame +than coal gas, not to speak of Siemens gas, which contains at most 33% +of combustible matter against 90% or more in water gas. The latter +circumstance also allows the gas to be conducted and distributed in +pipes of moderate dimensions. Its application, apart from its use as an +illuminant (with which we are not concerned here), was formerly retarded +by its high cost in comparison with Siemens gas and other sources of +heat, but as this state of affairs has been changed by the modern +improvements, its use is rapidly extending, especially for metallurgical +purposes. + +C. _Mixed Gas (Semi-Water Gas)._--This class is sometimes called Dowson +gas, irrespective of its method of production, although it was made and +extensively used a long time before J.E. Dowson constructed his +apparatus for generating such a gas principally for driving gas-engines. +By a combination of the processes for generating Siemens gas and water +gas, it is produced by injecting into a gas-producer at the same time a +certain quantity of air and a corresponding quantity of steam, the +latter never exceeding the amount which can be decomposed by the +heat-absorbing reaction, C + H2O = CO + H2, at the expense of the heat +generated by the action of the air in the reaction C + O = CO. Such gas +used to be frequently obtained in an accidental way by introducing +liquid water or steam into an ordinary gas-producer for the purpose of +facilitating its working by avoiding an excessive temperature, such as +might cause the rapid destruction of the brickwork and the fusion of the +ashes of the fuel into troublesome cakes. It was soon found that by +proceeding in this way a certain advantage could be gained in regard to +the consumption of fuel, as the heat abstracted by the steam from the +brickwork and the fuel itself was usefully employed for decomposing +water, its energy thus reappearing in the shape of a combustible gas. It +is hardly necessary to mention explicitly that the total heat obtained +by any such process from a given quantity of carbon (or hydrogen) can in +no case exceed that which is generated by direct combustion; some +inventors, however, whether inadvertently or intentionally, have +actually represented this to be possible, in manifest violation of the +law of the conservation of energy. + +Roughly speaking, this gas may be said to be produced by the combination +of the reactions, described _sub_ A and B, to the joint reaction: 2C + O ++ H2O = 2CO + H2. The decomposition of H2O (applied in the shape of +steam) absorbs 57.6 gram calories, the formation of 2CO produces 59 gram +calories; hence there is a small positive excess of 1.4 calories at +disposal. This in reality would not be sufficient to cover the loss by +radiation, &c.; hence rather more free oxygen (i.e. atmospheric air) +must be employed than is represented by the above equation. All this +free oxygen is, of course, accompanied by nearly four times its volume +of nitrogen. + +The mixed gas thus obtained differs very much in composition, but is +always much richer in hydrogen (of which it contains sometimes as much +as 20%) and poorer in carbon monoxide (sometimes down to 20%) than +Siemens gas; generally it contains more of CO2 than the latter. The +proportion of nitrogen is always less, about 50%. It is therefore a more +concentrated fuel than Siemens gas, and better adapted to the driving of +gas-engines. It scarcely costs more to make than ordinary Siemens gas, +except where the steam is generated and superheated in special +apparatus, as is done in the Dowson producer, which, on the other hand, +yields a correspondingly better gas. As is natural, its properties are +some way between those of Siemens gas and of water gas; but they +approach more nearly the former, both as to costs and as to fuel-value, +and also as to the temperatures reached in combustion. This is easily +understood if we consider that gas of just the same description can be +obtained by mixing one volume of real water gas with the four volumes of +Siemens gas made during the blowing-up stage--an operation which is +certainly too expensive for practical use. + +A modification of this gas is the _Mond gas_, which is made, according +to Mond's patent, by means of such an excess of steam that most of the +nitrogen of the coke is converted into ammonia (Grouven's reaction). Of +course much of this steam passes on undecomposed, and the quantity of +the gas is greatly increased by the reaction C + 2H2O = CO2 + 2H2; hence +the fuel-value of this gas is less than that of semi-water gas made in +other ways. Against this loss must be set the gain of ammonia which is +recovered by means of an arrangement of coolers and scrubbers, and, +except at very low prices of ammonia, the profit thus made is probably +more than sufficient to cover the extra cost. But as the process +requires very large and expensive plant, and its profits would vanish in +the case of the value of ammonia becoming much lower (a result which +would very probably follow if it were somewhat generally introduced), it +cannot be expected to supplant the other descriptions of gaseous fuel to +more than a limited extent. + +Semi-water gas is especially adapted for the purpose of driving +gas-engines on the explosive principle (gas-motors). Ordinary +producer-gas is too poor for this purpose in respect of heating power; +moreover, owing to the prevalence of carbon monoxide, it does not light +quickly enough. These defects are sufficiently overcome in semi-water +gas by the larger proportion of hydrogen contained in it. For the +purpose in question the gas should be purified from tar and ashes, and +should also be cooled down before entering the gas-engine. The Dowson +apparatus and others are constructed on this principle. + +_Air Gas._--By forcing air over or through volatile inflammable liquids +a gaseous mixture can be obtained which burns with a bright flame and +which can be used for illumination. Its employment for heating purposes +is quite exceptional, e.g. in chemical laboratories, and we abstain, +therefore, from describing any of the numerous appliances, some of them +bearing very fanciful names, which have been devised for its +manufacture. (G. L.) + + + + +FUENTE OVEJUNA [_Fuenteovejuna_], a town of Spain, in the province of +Cordova; near the sources of the river Guadiato, and on the Fuente del +Arco-Belmez-Cordova railway. Pop. (1900) 11,777. Fuente Ovejuna is built +on a hill, in a well-irrigated district, which, besides producing an +abundance of wheat, wine, fruit and honey, also contains argentiferous +lead mines and stone quarries. Cattle-breeding is an important local +industry, and leather, preserved meat, soap and flour are manufactured. +The parish church formerly belonged to the knights of Calatrava (c. +1163-1486). + + + + +FUENTERRABIA (formerly sometimes written _Fontarabia_; Lat. _Fons +Rapidus_), a town of northern Spain, in the province of Guipuzcoa; on +the San Sebastian-Bayonne railway; near the Bay of Biscay and on the +French frontier. Pop. (1870) about 750; (1900) 4345. Fuenterrabia stands +on the slope of a hill on the left bank of the river Bidassoa, and near +the point where its estuary begins. Towards the close of the 19th +century the town became popular as a summer resort for visitors from the +interior of Spain, and, in consequence, its appearance underwent many +changes and much of its early prosperity returned. Hotels and villas +were built in the new part of the town that sprang up outside the +picturesque walled fortress, and there is quite a contrast between the +part inside the heavy, half-ruined ramparts, with its narrow, steep +streets and curious gable-roofed houses, its fine old church and castle +and its massive town hall, and the new suburbs and fishermen's quarter +facing the estuary of the Bidassoa. Many industries flourish on the +outskirts of the town, including rope and net manufactures, flour mills, +saw mills, mining railways, paper mills. + +Fuenterrabia formerly possessed considerable strategic importance, and +it has frequently been taken and retaken in wars between France and +Spain. The rout of Charlemagne in 778, which has been associated with +Fontarabia, by Milton (_Paradise Lost_, i. 587), is generally understood +to have taken place not here but at Roncesvalles (q.v.), which is nearly +40 m. E.S.E. Unsuccessful attempts to seize Fuenterrabia were made by +the French troops in 1476 and again in 1503. In a subsequent campaign +(1521) these were more successful, but the fortress was retaken in 1524. +The prince of Conde sustained a severe repulse under its walls in 1638, +and it was on this occasion that the town received from Philip IV. the +rank of city (_muy noble, muy leal, y muy valerosa ciudad_, "most noble, +most loyal, and most valiant city"), a privilege which involved some +measure of autonomy. After a severe siege, Fuenterrabia surrendered to +the duke of Berwick and his French troops in 1719; and in 1794 it again +fell into the hands of the French, who so dismantled it that it has +never since been reckoned by the Spaniards among their fortified places. +It was by the ford opposite Fuenterrabia that the duke of Wellington, on +the 8th of October 1813, successfully forced a passage into France in +the face of an opposing army commanded by Marshal Soult. Severe fighting +also took place here during the Carlist War in 1837. + + + + +FUERO, a Spanish term, derived from the Latin _forum_. The Castillan use +of the word in the sense of a right, privilege or charter is most +probably to be traced to the Roman _conventus juridici_, otherwise known +as _jurisdictiones_ or _fora_, which in Pliny's time were already +numerous in the Iberian peninsula. In each of these provincial _fora_ +the Roman magistrate, as is well known, was accustomed to pay all +possible deference to the previously established common law of the +district; and it was the privilege of every free subject to demand that +he should be judged in accordance with the customs and usages of his +proper forum. This was especially true in the case of the inhabitants of +those towns which were in possession of the _jus italicum_. It is not, +indeed, demonstrable, but there are many presumptions, besides some +fragments of direct evidence, which make it more than probable that the +old administrative arrangements both of the provinces and of the towns, +but especially of the latter, remained practically undisturbed at the +period of the Gothic occupation of Spain.[1] The Theodosian Code and the +Breviary of Alaric alike seem to imply a continuance of the municipal +system which had been established by the Romans; nor does the later Lex +Visigothorum, though avowedly designed in some points to supersede the +Roman law, appear to have contemplated any marked interference with the +former _fora_, which were still to a large extent left to be regulated +in the administration of justice by unwritten, immemorial, local custom. +Little is known of the condition of the subject populations of the +peninsula during the Arab occupation; but we are informed that the +Christians were, sometimes at least, judged according to their own laws +in separate tribunals presided over by Christian judges;[2] and the mere +fact of the preservation of the name _alcalde_, an official whose +functions corresponded so closely to those of the _judex_ or _defensor +civitatis_, is fitted to suggest that the old municipal _fora_, if much +impaired, were not even then in all cases wholly destroyed. At all +events when the word _forum_[3] begins to appear for the first time in +documents of the 10th century in the sense of a liberty or privilege, +it is generally implied that the thing so named is nothing new. The +earliest extant written fuero is probably that which was granted to the +province and town of Leon by Alphonso V. in 1020. It emanated from the +king in a general council of the kingdom of Leon and Castile, and +consisted of two separate parts; in the first 19 chapters were contained +a series of statutes which were to be valid for the kingdom at large, +while the rest of the document was simply a municipal charter.[4] But in +neither portion does it in any sense mark a new legislative departure, +unless in so far as it marks the beginning of the era of written +charters for towns. The "fuero general" does not profess to supersede +the _consuetudines antiquorum jurium_ or Chindaswint's codification of +these in the Lex Visigothorum; the "fuero municipal" is really for the +most part but a resuscitation of usages formerly established, a +recognition and definition of liberties and privileges that had long +before been conceded or taken for granted. The right of the burgesses to +self-government and self-taxation is acknowledged and confirmed, they, +on the other hand, being held bound to a constitutional obedience and +subjection to the sovereign, particularly to the payment of definite +imperial taxes, and the rendering of a certain amount of military +service (as the ancient municipia had been). Almost contemporaneous with +this fuero of Leon was that granted to Najera (Naxera) by Sancho el +Mayor of Navarre (_ob._ 1035), and confirmed, in 1076, by Alphonso +VI.[5] Traces of others of perhaps even an earlier date are occasionally +to be met with. In the fuero of Cardena, for example, granted by +Ferdinand I. in 1039, reference is made to a previous forum Burgense +(Burgos), which, however, has not been preserved, if, indeed, it ever +had been reduced to writing at all. The phraseology of that of Sepulveda +(1076) in like manner points back to an indefinitely remote +antiquity.[6] Among the later fueros of the 11th century, the most +important are those of Jaca (1064) and of Logrono (1095). The former of +these, which was distinguished by the unusual largeness of its +concessions, and by the careful minuteness of its details, rapidly +extended to many places in the neighbourhood, while the latter charter +was given also to Miranda by Alphonso VI., and was further extended in +1181 by Sancho el Sabio of Navarre to Vitoria, thus constituting one of +the earliest written _fora_ of the "Provincias Vascongadas." In the +course of the 12th and 13th centuries the number of such documents +increased very rapidly; that of Toledo especially, granted to the +Mozarabic population in 1101, but greatly enlarged and extended by +Alphonso VII. (1118) and succeeding sovereigns, was used as a basis for +many other Castilian fueros. Latterly the word fuero came to be used in +Castile in a wider sense than before, as meaning a general code of laws; +thus about the time of Saint Ferdinand the old Lex Visigothorum, then +translated for the first time into the vernacular, was called the Fuero +Juzgo, a name which was soon retranslated into the barbarous Latin of +the period as Forum Judicum;[7] and among the compilations of Alphonso +the Learned in like manner were an _Espejo de Fueros_ and also the +_Fuero de las leyes_, better known perhaps as the _Fuero Real_. The +famous code known as the _Ordenamiento Real de Alcala_, or _Fuero Viejo +de Castilla_, dates from a still later period. As the power of the +Spanish crown was gradually concentrated and consolidated, royal +pragmaticas began to take the place of constitutional laws; the local +fueros of the various districts slowly yielded before the superior force +of imperialism; and only those of Navarre and the Basque provinces (see +BASQUES) have had sufficient vitality to enable them to survive to +comparatively modern times. While actually owning the lordship of the +Castilian crown since about the middle of the 14th century, these +provinces rigidly insisted upon compliance with their consuetudinary +law, and especially with that which provided that the _senor_, before +assuming the government, should personally appear before the assembly +and swear to maintain the ancient constitutions. Each of the provinces +mentioned had distinct sets of fueros, codified at different periods, +and varying considerably as to details; the main features, however, were +the same in all. Their rights, after having been recognized by +successive Spanish sovereigns from Ferdinand the Catholic to Ferdinand +VII., were, at the death of the latter in 1833, set aside by the +government of Castanos. The result was a civil war, which terminated in +a renewed acknowledgment of the fueros by Isabel II. (1839). The +provisional government of 1868 also promised to respect them, and +similar pledges were given by the governments which succeeded. In +consequence, however, of the Carlist rising of 1873-1876, the Basque +fueros were finally extinguished in 1876. The history of the _Foraes_ of +the Portuguese towns, and of the _Fors du Bearn_, is precisely analogous +to that of the fueros of Castile. + + Among the numerous works that more or less expressly deal with this + subject, that of Marina (_Ensayo historico-critico sobre la antigua + legislacion y principales cuerpos legales de los reynos de Leon y + Castilla_) still continues to hold a high place. Reference may also be + made to Colmeiro's _Curso de derecho politico segun la historia de + Leon y de Castilla_ (Madrid, 1873); to Schafer's _Geschichte von + Spanien_, ii. 418-428, iii. 293 seq.; and to Hallam's _Middle Ages_, + c. iv. + + +FOOTNOTES: + + [1] The nature of the evidence may be gathered from Savigny, _Gesch. + d. rom. Rechts_. See especially i. pp. 154, 259 seq. + + [2] Compare Lembke u. Schafer, _Geschichte von Spanien_, i. 314; ii. + 117. + + [3] Or rather _forus_. See Ducange, s.v. + + [4] Cap. xx. begins: "Constituimus etiam ut Legionensis civitas, quae + depopulata fuit a Sarracenis in diebus patris mei Veremundi regis, + repopulatur _per hos foros subscriptos_." + + [5] "Mando et concedo et confirmo ut ista civitas cum sua plebe et + cum omnibus suis pertinentiis sub tali lege et sub tali foro maneat + per saecula cuncta. Amen. Isti sunt fueros quae habuerunt in Naxera + in diebus Sanctii regis et Gartiani regis." + + [6] "Ego Aldefonsus rex et uxor mea Agnes confirmamus ad + Septempublica suo foro quod habuit in tempore antiquo de avolo meo et + in tempore comitum Ferrando Gonzalez et comite Garcia Ferdinandez et + comite Domno Santio." + + [7] This Latin is later even than that of Ferdinand, whose words are: + "Statuo et mando quod Liber Judicum, quo ego misi Cordubam, + translatetur in vulgarem et vocetur forum de Corduba ... et quod per + saecula cuncta sit pro foro et nullus sit ausus istud forum aliter + appellare nisi forum de Corduba, et jubeo et mando quod omnis morator + et populator ... veniet ad judicium et ad forum de Corduba." + + + + +FUERTEVENTURA, an island in the Atlantic Ocean, forming part of the +Spanish archipelago of the Canary Islands (q.v.). Pop. (1900) 11,669; +area 665 sq. m. Fuerteventura lies between Lanzarote and Grand Canary. +It has a length of 52 m., and an average width of 12 m. Though less +mountainous than the other islands, its aspect is barren. There are only +two springs of fresh water, and these are confined to one valley. Lava +streams and other signs of volcanic action abound, but there has been no +igneous activity since the Spaniards took possession. At each extremity +of the island are high mountains, which send off branches along the +coast so as to enclose a large arid plain. The highest peak reaches 2500 +ft. In external appearance, climate and productions, Fuerteventura +greatly resembles Lanzarote. An interval of three years without rain has +been known. Oliva (pop. 1900, 2464) is the largest town. A smaller place +in the centre of the island named Betancuria (586) is the administrative +capital. Cabras (1000) on the eastern coast is the chief port. +Dromedaries are bred here. + + + + +FUGGER, the name of a famous German family of merchants and bankers. The +founder of the family was Johann Fugger, a weaver at Graben, near +Augsburg, whose son, Johann, settled in Augsburg probably in 1367. The +younger Johann added the business of a merchant to that of a weaver, and +through his marriage with Clara Widolph became a citizen of Augsburg. +After a successful career he died in 1408, leaving two sons, Andreas and +Jakob, who greatly extended the business which they inherited from their +father. Andreas, called the "rich Fugger," had several sons, among them +being Lukas, who was very prominent in the municipal politics of +Augsburg and who was very wealthy until he was ruined by the repudiation +by the town of Louvain of a great debt owing to him, and Jakob, who was +granted the right to bear arms in 1452, and who founded the family of +Fugger vom Reh--so called from the first arms of the Fuggers, a roe +(_Reh_) or on a field azure--which became extinct on the death of his +great-grandson, Ulrich, in 1583. Johann Fugger's son, Jakob, died in +1469, and three of his seven sons, Ulrich (1441-1510), Georg (1453-1506) +and Jakob (1459-1525), men of great resource and industry, inherited the +family business and added enormously to the family wealth. In 1473 +Ulrich obtained from the emperor Frederick III. the right to bear arms +for himself and his brothers, and about the same time he began to act +as the banker of the Habsburgs, a connexion destined to bring fame and +fortune to his house. Under the lead of Jakob, who had been trained for +business in Venice, the Fuggers were interested in silver mines in Tirol +and copper mines in Hungary, while their trade in spices, wool and silk +extended to almost all parts of Europe. Their wealth enabled them to +make large loans to the German king, Maximilian I., who pledged to them +the county of Kirchberg, the lordship of Weissenhorn and other lands, +and bestowed various privileges upon them. Jakob built the castle of +Fuggerau in Tirol, and erected the Fuggerei at Augsburg, a collection of +106 dwellings, which were let at low rents to poor people and which +still exist. Jakob Fugger and his two nephews, Ulrich (d. 1525) and +Hieronymus (d. 1536), the sons of Ulrich, died without direct heirs, and +the family was continued by Georg's sons, Raimund (1489-1535) and Anton +(1493-1560), under whom the Fuggers attained the summit of their wealth +and influence. + +Jakob Fugger's florins had contributed largely to the election of +Charles V. to the imperial throne in 1519, and his nephews and heirs +maintained close and friendly relations with the great emperor. In +addition to lending him large sums of money, they farmed his valuable +quicksilver mines at Almaden, his silver mines at Guadalcanal, the great +estates of the military orders which had passed into his hands, and +other parts of his revenue as king of Spain; receiving in return several +tokens of the emperor's favour. In 1530 Raimund and Anton were granted +the imperial dignity of counts of Kirchberg and Weissenhorn, and +obtained full possession of these mortgaged properties; in 1534 they +were given the right of coining money; and in 1541 received rights of +jurisdiction over their lands. During the diet of Augsburg in 1530 +Charles V. was the guest of Anton Fugger at his house in the Weinmarkt, +and the story relates how the merchant astonished the emperor by +lighting a fire of cinnamon with an imperial bond for money due to him. +This incident forms the subject of a picture by Carl Becker which is in +the National Gallery at Berlin. Continuing their mercantile career, the +Fuggers brought the new world within the sphere of their operations, and +also carried on an extensive and lucrative business in farming +indulgences. Moreover, both brothers found time to acquire landed +property, and were munificent patrons of literature and art. When Anton +died he is said to have been worth 6,000,000 florins, besides a vast +amount of property in Europe, Asia and America; and before this time the +total wealth of the family had been estimated at 63,000,000 florins. The +Fuggers were devotedly attached to the Roman Catholic Church, which +benefited from their liberality. Jakob had been made a count palatine +(_Pfalzgraf_) and had received other marks of favour from Pope Leo X., +and several members of the family had entered the church; one, Raimund's +son, Sigmund, becoming bishop of Regensburg. + +In addition to the bishop, three of Raimund Fugger's sons attained some +degree of celebrity. Johann Jakob (1516-1575), was the author of +_Wahrhaftigen Beschreibung des osterreichischen und habsburgischen +Nahmens_, which was largely used by S. von Bircken in his _Spiegel der +Ehren des Erzhauses Osterreich_ (Nuremberg, 1668), and of a _Geheim +Ernbuch des Fuggerischen Geschlechtes_. He was also a patron of art, and +a distinguished counsellor of Duke Albert IV. of Bavaria. After the +death of his son Konstantin, in 1627, this branch of the family was +divided into three lines, which became extinct in 1738, 1795 and 1846 +respectively. Another of Raimund's sons was Ulrich (1526-1584), who, +after serving Pope Paul III. at Rome, became a Protestant. Hated on this +account by the other members of his family, he took refuge in the +Rhenish Palatinate; greatly interested in the Greek classics, he +occupied himself in collecting valuable manuscripts, which he bequeathed +to the university of Heidelberg. Raimund's other son was Georg (d. +1579), who inherited the countships of Kirchberg and Weissenhorn, and +founded a branch of the family which still exists, its present head +being Georg, Count Fugger of Kirchberg and Weissenhorn (b. 1850). + +Anton Fugger left three sons, Marcus (1529-1597), Johann (d. 1598) and +Jakob (d. 1598), all of whom left male issue. Marcus was the author of +a book on horse-breeding, _Wie und wo man ein Gestut von guten edeln +Kriegsrossen aufrichten soll_ (1578), and of a German translation of the +_Historia ecclesiastica_ of Nicephorus Callistus. He founded the +Nordendorf branch of the family, which became extinct on the death of +his grandson, Nicolaus, in 1676. Another grandson of Marcus was Franz +Fugger (1612-1664), who served under Wallenstein during the Thirty +Years' War, and was afterwards governor of Ingolstadt. He was killed at +the battle of St Gotthard on the 1st of August 1664. + +Johann Fugger had three sons, Christoph (d. 1615) and Marcus (d. 1614), +who founded the families of Fugger-Glott and Fugger-Kirchheim +respectively, and Jakob, bishop of Constance from 1604 until his death +in 1626. Christoph's son, Otto Heinrich (1592-1644), was a soldier of +some distinction and a knight of the order of the Golden Fleece. He was +one of the most active of the Bavarian generals during the Thirty Years' +War, and acted as governor of Augsburg, where his rule aroused much +discontent. The family of Kirchheim died out in 1672. That of Glott was +divided into several branches by the sons of Otto Heinrich and of his +brother Johann Ernst (d. 1628). These lines, however, have gradually +become extinct except the eldest line, represented in 1909 by Karl +Ernst, Count Fugger of Glott (b. 1859). Anton Fugger's third son Jakob, +the founder of the family of Wellenburg, had two sons who left issue, +but in 1777 the possessions of this branch of the family were again +united by Anselm Joseph (d. 1793), Count Fugger of Babenhausen. In 1803 +Anselm's son, Anselm Maria (d. 1821), was made a prince of the Holy +Roman Empire, the title of Prince Fugger of Babenhausen being borne by +his direct descendant Karl (b. 1861). On the fall of the empire in 1806 +the lands of the Fuggers, which were held directly of the empire, were +mediatized under Bavaria and Wurttemberg. The heads of the three +existing branches of the Fuggers are all hereditary members of the +Bavarian Upper House. + +Augsburg has many interesting mementoes of the Fuggers, including the +family burial-chapel in the church of St Anna; the Fugger chapel in the +church of St Ulrich and St Afra; the Fuggerhaus, still in the possession +of one branch of the family; and a statue of Johann Jakob Fugger. + + In 1593 a collection of portraits of the Fuggers, engraved by + Dominique Custos of Antwerp, was issued at Augsburg. Editions with 127 + portraits appeared in 1618 and 1620, the former accompanied by a + genealogy in Latin, the latter by one in German. Another edition of + this _Pinacotheca Fuggerorum_, published at Vienna in 1754, includes + 139 portraits. See _Chronik der Familie Fugger vom Jahre 1599_, edited + by C. Meyer (Munich, 1902); A. Geiger, _Jakob Fugger, 1459-1525_ + (Regensburg, 1895); A. Schulte, _Die Fugger in Rom, 1495-1523_ + (Leipzig, 1904); R. Ehrenberg, _Das Zeitalter der Fugger_ (Jena, + 1896); K. Habler, _Die Geschichte der Fuggerschen Handlung in Spanien_ + (Weimar, 1897); A. Stauber, _Das Haus Fugger_ (Augsburg, 1900); and M. + Jansen, _Die Anfange der Fugger_ (Leipzig, 1907). + + + + +FUGITIVE SLAVE LAWS, a term applied in the United States to the Statutes +passed by Congress in 1793 and 1850 to provide for the return of negro +slaves who escaped from one state into another or into a public +territory. A fugitive slave clause was inserted in the Articles of +Confederation of the New England Confederation of 1643, providing for +the return of the fugitive upon the certificate of one magistrate in the +jurisdiction out of which the said servant fled--no trial by jury being +provided for. This seems to have been the only instance of an +inter-colonial provision for the return of fugitive slaves; there were, +indeed, not infrequent escapes by slaves from one colony to another, but +it was not until after the growth of anti-slavery sentiment and the +acquisition of western territory, that it became necessary to adopt a +uniform method for the return of fugitive slaves. Such provision was +made in the Ordinance of 1787 (for the Northwest Territory), which in +Article VI. provided that in the case of "any person escaping into the +same [the Northwest Territory] from whom labor or service is lawfully +claimed in any one of the original states, such fugitive may be lawfully +reclaimed and conveyed to the person claiming his or her labor or +service as aforesaid." An agreement of the sort was necessary to +persuade the slave-holding states to union, and in the Federal +Constitution, Article IV., Section II., it is provided that "no person +held to service or labor in one state, under the laws thereof, escaping +into another, shall, in consequence of any law or regulation therein, be +discharged from such service or labor, but shall be delivered up on +claim of the party to whom such service or labour may be due." + +The first specific legislation on the subject was enacted on the 12th of +February 1793, and like the Ordinance for the Northwest Territory and +the section of the Constitution quoted above, did not contain the word +"slave"; by its provisions any Federal district or circuit judge or any +state magistrate was authorized to decide finally and without a jury +trial the status of an alleged fugitive. The measure soon met with +strong opposition in the northern states, and Personal Liberty Laws were +passed to hamper officials in the execution of the law; Indiana in 1824 +and Connecticut in 1828 providing jury trial for fugitives who appealed +from an original decision against them. In 1840 New York and Vermont +extended the right of trial by jury to fugitives and provided them with +attorneys. As early as the first decade of the 19th century individual +dissatisfaction with the law of 1793 had taken the form of systematic +assistance rendered to negroes escaping from the South to Canada or New +England--the so-called "Underground Railroad."[1] The decision of the +Supreme Court of the United States in the case of _Prigg_ v. +_Pennsylvania_ in 1842 (16 Peters 539), that state authorities could not +be forced to act in fugitive slave cases, but that national authorities +must carry out the national law, was followed by legislation in +Massachusetts (1843), Vermont (1843), Pennsylvania (1847) and Rhode +Island (1848), forbidding state officials to help enforce the law and +refusing the use of state gaols for fugitive slaves. The demand from the +South for more effective Federal legislation was voiced in the second +fugitive slave law, drafted by Senator J.M. Mason of Virginia, and +enacted on the 18th of September 1850 as a part of the Compromise +Measures of that year. Special commissioners were to have concurrent +jurisdiction with the U.S. circuit and district courts and the inferior +courts of Territories in enforcing the law; fugitives could not testify +in their own behalf; no trial by jury was provided; penalties were +imposed upon marshals who refused to enforce the law or from whom a +fugitive should escape, and upon individuals who aided negroes to +escape; the marshal might raise a _posse comitatus_; a fee of $10 was +paid to the commissioner when his decision favoured the claimant and +only $5 when it favoured the fugitive; and both the fact of the escape +and the identity of the fugitive were to be determined on purely _ex +parte_ testimony. The severity of this measure led to gross abuses and +defeated its purpose; the number of abolitionists increased, the +operations of the Underground Railroad became more efficient, and new +Personal Liberty Laws were enacted in Vermont (1850), Connecticut +(1854), Rhode Island (1854), Massachusetts (1855), Michigan (1855), +Maine (1855 and 1857), Kansas (1858) and Wisconsin (1858). These +Personal Liberty Laws forbade justices and judges to take cognizance of +claims, extended the _habeas corpus_ act and the privilege of jury trial +to fugitives, and punished false testimony severely. The supreme court +of Wisconsin went so far (1859) as to declare the Fugitive Slave Law +unconstitutional. These state laws were one of the grievances officially +referred to by South Carolina (in Dec. 1860) as justifying her secession +from the Union. Attempts to carry into effect the law of 1850 aroused +much bitterness. The arrests of Sims and of Shadrach in Boston in 1851; +of "Jerry" M'Henry, in Syracuse, New York, in the same year; of Anthony +Burns in 1854, in Boston; and of the two Garner families in 1856, in +Cincinnati, with other cases arising under the Fugitive Slave Law of +1850, probably had as much to do with bringing on the Civil War as did +the controversy over slavery in the Territories. + +With the beginning of the Civil War the legal status of the slave was +changed by his master's being in arms. General B.F. Butler, in May 1861, +declared negro slaves contraband of war. A confiscation bill was passed +in August 1861 discharging from his service or labour any slave employed +in aiding or promoting any insurrection against the government of the +United States. By an act of the 17th of July 1862 any slave of a +disloyal master who was in territory occupied by northern troops was +declared _ipso facto_ free. But for some time the Fugitive Slave Law was +considered still to hold in the case of fugitives from masters in the +border states who were loyal to the Union government, and it was not +until the 28th of June 1864 that the Act of 1850 was repealed. + + See J.F. Rhodes, _History of the United States from the Compromise of + 1850_, vols. i. and ii. (New York, 1893); and M.G. M'Dougall, + _Fugitive Slaves, 1619-1865_ (Boston, 1891). + + +FOOTNOTE: + + [1] The precise amount of organization in the Underground Railroad + cannot be definitely ascertained because of the exaggerated use of + the figure of railroading in the documents of the "presidents" of the + road, Robert Purvis and Levi Coffin, and of its many "conductors," + and their discussion of the "packages" and "freight" shipped by them. + The system reached from Kentucky and Virginia across Ohio, and from + Maryland across Pennsylvania and New York, to New England and Canada, + and as early as 1817 a group of anti-slavery men in southern Ohio had + helped to Canada as many as 1000 slaves. The Quakers of Pennsylvania + possibly began the work of the mysterious Underground Railroad; the + best known of them was Thomas Garrett (1789-1871), a native of + Pennsylvania, who, in 1822, removed to Wilmington, Delaware, where he + was convicted in 1848 on four counts under the Fugitive Slave Law and + was fined $8000; he is said to have helped 2700 slaves to freedom. + The most picturesque figure of the Underground Railroad was Harriet + Tubman (c. 1820), called by her friend, John Brown, "General" Tubman, + and by her fellow negroes "Moses." She made about a score of trips + into the South, bringing out with her 300 negroes altogether. At one + time a reward of $40,000 was offered for her capture. She was a + mystic, with remarkable clairvoyant powers, and did great service as + a nurse, a spy and a scout in the Civil War. Levi Coffin (1798-1877), + a native of North Carolina (whose cousin, Vestal Coffin, had + established before 1819 a "station" of the Underground near what is + now Guilford College, North Carolina), in 1826 settled in Wayne + County, Ohio; his home at New Garden (now Fountain City) was the + meeting point of three "lines" from Kentucky; and in 1847 he removed + to Cincinnati, where his labours in bringing slaves out of the South + were even more successful. It has been argued that the Underground + Railroad delayed the final decision of the slavery question, inasmuch + as it was a "safety valve"; for, without it, the more intelligent and + capable of the negro slaves would, it is asserted, have become the + leaders of insurrections in the South, and would not have been + removed from the places where they could have done most damage. + Consult William Still, _The Underground Railroad_ (Philadelphia, + 1872), a collection of anecdotes by a negro agent of the Pennsylvania + Anti-Slavery Society, and of the Philadelphia branch of the Railroad; + and the important and scholarly work of Wilbur H. Siebert, _The + Underground Railroad from Slavery to Freedom_ (New York, 1898). + + + + +FUGLEMAN (from the Ger. _Flugelmann_, the man on the _Flugel_ or wing), +properly a military term for a soldier who is selected to act as +"guide," and posted generally on the flanks with the duty of directing +the march in the required line, or of giving the time, &c., to the +remainder of the unit, which conforms to his movements, in any military +exercise. The word is then applied to a ringleader or one who takes the +lead in any movement or concerted movement. + + + + +FUGUE (Lat. _fuga_, flight), in music, the mutual "pursuit" of voices or +parts. It was, up to the end of the 16th century, if not later, the name +applied to two art-forms. (A) _Fuga ligata_ was the exact reproduction +by one or more voices of the statement of a leading part. The +reproducing voice (_comes_) was seldom if ever written out, for all +differences between it and the _dux_ were rigidly systematic; e.g. it +was an exact inversion, or exactly twice as slow, or to be sung +backwards, &c. &c. Hence, a rule or _canon_ was given, often in +enigmatic form, by which the _comes_ was deduced from the _dux_: and so +the term _canon_ became the appropriate name for the form itself, and is +still retained. (B) A composition in which the canonic style was +cultivated without canonic restriction was, in the 16th century, called +_fuga ricercata_ or simply a _ricercare_, a term which is still used by +Bach as a title for the fugues in _Das musikalische Opfer_. + +The whole conception of fugue, rightly understood, is one of the most +important in music, and the reasons why some contrapuntal compositions +are called fugues, while others are not, are so trivial, technically as +well as aesthetically, that we have preferred to treat the subject +separately under the general heading of CONTRAPUNTAL FORMS, reserving +only technical terms for definition here. + +(i.) If in the beginning or "exposition" the material with which the +opening voice accompanies the answer is faithfully reproduced as the +accompaniment to subsequent entries of the subject, it is called a +_countersubject_ (see COUNTERPOINT, under sub-heading _Double +Counterpoint_). Obviously the process may be carried further, the first +countersubject going on to a second when the subject enters in the third +part and so on. The term is also applied to new subjects appearing later +in the fugue in combination (immediate or destined) with the original +subject. Cherubini, holding the doctrine that a fugue cannot have more +than one subject, insists on applying the term to the less prominent of +the subjects of what are commonly called double fugues, i.e. fugues +which begin with two parts and two subjects simultaneously, and so also +with _triple_ and _quadruple fugues_. + +(ii.) _Episodes_ are passages separating the entries of the subject.[1] +Episodes are usually developed from the material of the subject and +countersubjects; they are very rarely independent, but then +conspicuously so. + +(iii.) _Stretto_, the overlapping of subject and answer, is a resource +the possibilities of which may be exemplified by the setting of the +words _omnes generationes_ in Bach's _Magnificat_ (see BACH). + +(iv.) The distinction between _real_ and _tonal_ fugue, which is still +sometimes treated as a thing of great historical and technical +importance, is really a mere detail resulting from the fact that a +violent oscillation between the keys of tonic and dominant is no part of +the function of a fugal exposition, so that the answer is (especially in +its first notes and in points that tend to shift the key) not so much a +transposition of the subject to the key of the dominant as an adaptation +of it from the tonic part to the dominant part of the scale, or vice +versa; in short, the answer is as far as possible _on_ the dominant, not +_in_ the dominant. The modifications this principle produces in the +answer (which have been happily described as resembling +"fore-shortening") are the only distinctive marks of tonal fugue; and +the text-books are half filled with the attempt to reduce them from +matters of ear to rules of thumb, which rules, however, have the merit +(unusual in those of the academic fugue) of being founded on observation +of the practice of great masters. But the same principle as often as not +produces answers that are exact transpositions of the subject; and so +the only kind of real fugue (i.e. fugue with an exact answer) that could +rightly be contrasted with tonal fugue would be that in which the answer +ought to be tonal but is not. It must be admitted that tonal answers are +rare in the modal music of the 16th century, though their melodic +principles are of yet earlier date; still, though tonal fugue does not +become usual until well on in the 17th century, the idea that it is a +separate species is manifestly absurd, unless the term simply means +"fugue in modern tonality or key," whatever the answer may be. + +The term "answer" is usually reserved for those entries of the subject +that are placed in what may be called the "complementary" position of +the scale, whether they are "tonally" modified or not. Thus the order of +entries in the exposition of the first fugue of the _Wohltemp_. _Klav_. +is subject, answer, answer, subject; a departure from the usual rule +according to which subject and answer are strictly alternate in the +exposition. + +In conclusion we may remind the reader of the most accurate as well as +the most vivid description ever given of the essentials of a fugue, in +the famous lines in _Paradise Lost_, book xi. + + "His volant touch, + Instinct through all proportions, low and high, + Fled and pursued transverse the resonant fugue." + +It is hard to realize that this description of organ-music was written +in no classical period of instrumental polyphony, but just half-way +between the death of Frescobaldi and the birth of Bach. Every word is a +definition, both retrospective and prophetic; and in "transverse" we see +all that Sir Frederick Gore Ouseley expresses in his popular distinction +between the "perpendicular" or homophonic style in which harmony is +built up in chords, and the "horizontal" or polyphonic style in which it +is woven in threads of independent melody. (D. F. T.) + + +FOOTNOTE: + + [1] An episode occurring during the exposition is sometimes called + _codetta_, a distinction the uselessness of which at once appears on + an analysis of Bach's 2nd fugue in the _Wohltemp_. _Klav_. (the term + codetta is more correctly applied to notes filling in a gap between + subject and its first answer, but such a gap is rare in good + examples). + + + + +FUHRICH, JOSEPH VON (1800-1876), Austrian painter, was born at Kratzau +in Bohemia on the 9th of February 1800. Deeply impressed as a boy by +rude pictures adorning the wayside chapels of his native country, his +first attempt at composition was a sketch of the Nativity for the +festival of Christmas in his father's house. He lived to see the day +when, becoming celebrated as a composer of scriptural episodes, his +sacred subjects were transferred in numberless repetitions to the +roadside churches of the Austrian state, where humble peasants thus +learnt to admire modern art reviving the models of earlier ages. Fuhrich +has been fairly described as a "Nazarene," a romantic religious artist +whose pencil did more than any other to restore the old spirit of Durer +and give new shape to countless incidents of the gospel and scriptural +legends. Without the power of Cornelius or the grace of Overbeck, he +composed with great skill, especially in outline. His mastery of +distribution, form, movement and expression was considerable. In its +peculiar way his drapery was perfectly cast. Essentially creative as a +landscape draughtsman, he had still no feeling for colour; and when he +produced monumental pictures he was not nearly so successful as when +designing subjects for woodcuts. Fuhrich's fame extended far beyond the +walls of the Austrian capital, and his illustrations to Tieck's +_Genofeva_, the Lord's Prayer, the Triumph of Christ, the Road to +Bethlehem, the Succession of Christ according to Thomas a Kempis, the +Prodigal Son, and the verses of the Psalter, became well known. His +Prodigal Son, especially, is remarkable for the fancy with which the +spirit of evil is embodied in a figure constantly recurring, and like +that of Mephistopheles exhibiting temptation in a human yet demoniacal +shape. Fuhrich became a pupil at the Academy of Prague in 1816. His +first inspiration was derived from the prints of Durer and the Faust of +Cornelius, and the first fruit of this turn of study was the Genofeva +series. In 1826 he went to Rome, where he added three frescoes to those +executed by Cornelius and Overbeck in the Palazzo Massimi. His subjects +were taken from the life of Tasso, and are almost solitary examples of +his talent in this class of composition. In 1831 he finished the Triumph +of Christ now in the Raczynski palace at Berlin. In 1834 he was made +custos and in 1841 professor of composition in the Academy of Vienna. +After this he completed the monumental pictures of the church of St +Nepomuk, and in 1854-1861 the vast series of wall paintings which cover +the inside of the Lerchenfeld church at Vienna. In 1872 he was pensioned +and made a knight of the order of Franz Joseph; 1875 is the date of his +illustrations to the Psalms. He died on the 13th of March 1876. + + His autobiography was published in 1875, and a memoir by his son Lucas + in 1886. + + + + +FUJI (Fuji-san, Fujiyama, Fusiyama), a celebrated mountain of Japan, +standing W.S.W. of Tokyo, its base being about 70 m. by rail from that +city. It rises to a height of 12,395 ft. and its southern slopes reach +the shore of Suruga Bay. It is a cone of beautifully simple form, the +more striking to view because it stands isolated; but its summit is not +conical, being broken by a crater some 2000 ft. in diameter, for Fuji is +a quiescent volcano. Small outbursts of steam are still to be observed +at some points. An eruption is recorded so lately as the first decade of +the 18th century. The mountain is the resort of great numbers of +pilgrims (see also JAPAN). + + + + +FU-KIEN (formerly MIN), a south-eastern province of China, bounded N. by +the province of Cheh-kiang, S. by that of Kwang-tung, W. by that of +Kiang-si and E. by the sea. It occupies an area of 53,480 sq. m. and its +population is estimated at 20,000,000. The provincial capital is Fuchow +Fu, and it is divided into eleven prefectures, besides that ruled over +by the prefect of the capital city. Fu-kien is generally mountainous, +being overspread by the Nan-shan ranges, which run a general course of +N.E. and S.W. The principal river is the Min, which is formed by the +junction, in the neighbourhood of the city of Yen-p'ing Fu, of three +rivers, namely, the Nui-si, which takes its rise in the mountains on the +western frontier in the prefecture of Kien-ning Fu, the Fuh-tun Ki, the +source of which is found in the district of Kwang-tsih in the north-west +of the province, and the Ta-shi-ki (Shao Ki), which rises in the +mountains in the western district of Ning-hwa. From Yen-p'ing Fu the +river takes a south-easterly course, and after passing along the south +face of the city of Fuchow Fu, empties itself into the sea about 30 m. +below that town. Its upper course is narrow and rocky and abounds in +rapids, but as it approaches Fuchow Fu the channel widens and the +current becomes slow and even. Its depth is very irregular, and it is +navigable only by native boats of a small class. Two other rivers flow +into the sea near Amoy, neither of which, however, is navigable for any +distance from its mouth owing to the shallows and rapids with which they +abound. Thirty-five miles inland from Amoy stands the city of Chang +Chow, famous for the bridge which there spans the Kin-lung river. This +bridge is 800 ft. long, and consists of granite monoliths stretching +from one abutment to another. The soil of the province is, as its name, +"Happy Establishment," indicates, very productive, and the scenery is of +a rich and varied character. Most of the hills are covered with verdure, +and the less rugged are laid out in terraces. The principal products of +the province are tea, of which the best kind is that known as Bohea, +which takes its name, by a mispronunciation, from the Wu-e Mountains, in +the prefecture of Kien-ning Fu, where it is grown; grains of various +kinds, oranges, plantins, lichis, bamboo, ginger, gold, silver, lead, +tin, iron, salt (both marine and rock), deers' horns, beeswax, sugar, +fish, birds' nests, medicine, paper, cloth, timber, &c. Fu-kien has +three open ports, Fuchow Fu opened in 1842, Amoy opened to trade in the +same year and Funing. The latter port was only opened to foreign trade +in 1898, but in 1904 it imported and exported goods to the value of +L7668 and L278,160 respectively. + + + + +FUKUI, a town of Japan in the province of Echizen, Nippon, near the west +coast, 20 m. N. by E. of Wakasa Bay. It lies in a volcanic district much +exposed to earthquakes, and suffered severely during the disturbances of +1891-1892, when a chasm over 40 m. long was opened across the Neo valley +from Fukui to Katabira. But Fukui subsequently revived, and is now in a +flourishing condition, with several local industries, especially the +manufacture of paper, and an increasing population exceeding 50,000. +Fukui has railway communication. There are ruins of a castle of the +Daimios of Echizen. + + + + +FUKUOKA, a town on the north-west coast of the island of Kiushiu, Japan, +in the province of Chikuzen, 90 m. N.N.E. of Nagasaki by rail. Pop. +about 72,000. With Hakata, on the opposite side of a small coast stream, +it forms a large centre of population, with an increasing export trade +and several local industries. Of these the most important is +silk-weaving, and Hakata especially is noted for its durable silk +fabrics. Fukuoka was formerly the residence of the powerful daimio of +Chikuzen, and played a conspicuous part in the medieval history of +Japan; the renowned temple of Yeiyas in the district was destroyed by +fire during the revolution of 1868. There are several other places of +this name in Japan, the most important being Fukuoka in the province of +Mutsu, North Nippon, a railway station on the main line from Tokyo to +Aimori Ura Bay. Pop. about 5000. + + + + +FULA (FULBE, FELLATAH or PEULS), a numerous and powerful African people, +spread over an immense region from Senegal nearly to Darfur. Strictly +they have no country of their own, and nowhere form the whole of the +population, though nearly always the dominant native race. They are most +numerous in Upper Senegal and in the countries under French sway +immediately south of Senegambia, notably Futa Jallon. Farther east they +rule, subject to the control of the French, Segu and Massena, countries +on both banks of the upper Niger, to the south-west of Timbuktu. The +districts within the great bend of the Niger have a large Fula +population. East of that river Sokoto and its tributary emirates are +ruled by Fula princes, subject to the control of the British Nigerian +administration. Fula are settled in Bornu, Bagirmi, Wadai and the upper +Nile Valley,[1] but have no political power in those countries. Their +most southerly emirate is Adamawa, the country on both sides of the +upper Benue. In this vast region of distribution the Fula populations +are most dense towards the west and north, most scattered towards the +east and south. Originally herdsmen in the western and central Sudan, +they extended their sway east of the Niger, under the leadership of +Othman Dan Fodio, during the early years of the 19th century, and having +subdued the Hausa states, founded the empire of Sokoto with the vassal +emirates of Kano, Gando, Nupe, Adamawa, &c. + +The question of the ethnic affinities of the Fula has given rise to an +enormous amount of speculation, but the most reasonable theory is that +they are a mixture of Berber and Negro. This is now the most generally +accepted theory. Certainly there is no reason to connect them with the +ancient Egyptians. In the district of Senegal known as Fuladugu or "Fula +Land," where the purest types of the race are found, the people are of a +reddish brown or light chestnut colour, with oval faces, ringlety or +even smooth hair, never woolly, straight and even aquiline noses, +delicately shaped lips and regular features quite differentiating them +from the Negro type. Like most conquering races the Fula are, however, +not of uniform physique, in many districts approximating to the local +type. They nevertheless maintain throughout their widespread territory a +certain national solidarity, thanks to common speech, traditions and +usages. The ruling caste of the Fula differs widely in character from +the herdsmen of the western Sudan. The latter are peaceable, inoffensive +and abstemious. They are mainly monogamous, and by rigidly abstaining +from foreign marriages have preserved racial purity. The ruling caste in +Nigeria, on the other hand, despise their pastoral brethren, and through +generations of polygamy with the conquered tribes have become more +Negroid in type, black, burly and coarse featured. Love of luxury, pomp +and finery is their chief characteristic. Taken as a whole, the Fula +race is distinguished by great intelligence, frankness of disposition +and strength of character. As soldiers they are renowned almost +exclusively as cavalry; and the race has produced several leaders +possessed of much strategical skill. Besides the ordinary Negro weapons, +they use iron spears with leatherbound handles and swords. They are +generally excellent rulers, stern but patient and just. The Nigerian +emirs acquired, however, an evil reputation during the 19th century as +slave raiders. They have long been devout Mahommedans, and mosques and +schools exist in almost all their towns. Tradition says that of old +every Fula boy and girl was a scholar; but during the decadence of their +power towards the close of the 19th century education was not highly +valued. Power seems to have somewhat spoilt this virile race, but such +authorities as Sir Frederick Lugard believe them still capable of a +great future. + +The Fula language has as yet found no place in any African linguistic +family. In its rudiments it is akin to the Hamito-Semitic group. It +possesses two grammatical genders, not masculine and feminine, but the +human and the non-human; the adjective agrees in assonance with its +noun, and euphony plays a great part in verbal and nominal inflections. +In some ways resembling the Negro dialects, it betrays non-Negroid +influences in the use of suffixes. The name of the people has many +variations. Fulbe or Fula (sing. Pullo, Peul) is the Mandingan name, +Follani the Hausa, Fellatah the Kanuri, Fullan the Arab, and Fulde on +the Benue. Like the name Abate, "white," given them in Kororofa, all +these seem to refer to their light reddish hue. + + See F. Ratzel, _History of Mankind_ (English ed., London, 1896-1898); + Sir F. Lugard, "Northern Nigeria," in _Geographical Journal_ (July + 1904); Grimai de Guirodon, _Les Puls_ (1887); E.A. Brackenbury, _A + Short Vocabulary of the Fulani Language_ (Zungeru, 1907); the articles + NIGERIA and SOKOTO and authorities there cited. + + +FOOTNOTE: + + [1] Sir Wm. Wallace in a report on Northern Nigeria ("Colonial + Office" series, No. 551, 1907) calls attention to the exodus "of + thousands of Fulani of all sorts, but mostly Mellawa, from the French + Middle Niger," and states that the majority of the emigrants are + settling in the Nile valley. + + + + +FULCHER (or FOUCHER) OF CHARTRES (1058-c. 1130), French chronicler, was +a priest who was present at the council of Clermont in 1095, and +accompanied Robert II., duke of Normandy, on the first crusade in 1096. +Having spent some time in Italy and taken part in the fighting on the +way to the Holy Land, he became chaplain to Baldwin, who was chosen king +of Jerusalem in 1100, and lived with Baldwin at Edessa and then at +Jerusalem. He accompanied this king on several warlike expeditions, but +won more lasting fame by writing his _Historia Hierosolymitana_ or +_Gesta Francorum Jerusalem expugnantium_, one of the most trustworthy +sources for the history of the first crusade. In its final form it is +divided into three books, and covers the period between the council of +Clermont and 1127, and the author only gives details of events which he +himself had witnessed. It was used by William of Tyre. Fulcher died +after 1127, probably at Jerusalem. He has been confused with Foucher of +Mongervillier (d. 1171), abbot of St-Pere-en-Vallee at Chartres, and +also with another person of the same name who distinguished himself at +the siege of Antioch in 1098. + + The _Historia_, but in an incomplete form, was first published by J. + Bongars in the _Gesta Dei per Francos_ (Hanover, 1611). The best + edition is in tome iii. of the _Recueil des historiens des croisades, + Historiens occidentaux_ (Paris, 1866); and there is a French + translation in tome xxiv. of Guizot's _Collection des memoires + relatifs a l'histoire de France_ (Paris, 1823-1835). + + See H. von Sybel, _Geschichte des ersten Kreuzzuges_ (Leipzig, 1881); + and A. Molinier, _Les Sources de l'histoire de France_, tome ii. + (Paris, 1902). + + + + +FULDA, a town and episcopal see of Germany, in the Prussian province of +Hesse-Nassau, between the Rhon and the Vogel-Gebirge, 69 m. N.E. from +Frankfort-on-Main on the railway to Bebra. Although irregularly built +the town is pleasantly situated, and contains two fine squares, on one +of which stands a fine statue of St Boniface. The present cathedral was +built at the beginning of the 18th century on the model of St Peter's at +Rome, but it has an ancient crypt, which contains the bones of St +Boniface and was restored in 1892. Opposite the cathedral is the former +monastery of St Michael, now the episcopal palace. The Michaelskirche, +attached to it, is a small round church built, in imitation of the Holy +Sepulchre, in 822 and restored in 1853. Of other buildings may be +mentioned the Library, with upwards of 80,000 printed books and many +valuable MSS., the stately palace with its gardens and orangery, the +former Benedictine nunnery (founded 1625, and now used as a seminary), +and the Minorite friary (1238) now used as a furniture warehouse. Among +the secular buildings are the fine _Schloss_, the _Bibliothek_, the town +hall and the post office. There are several schools, a hospital founded +in the 13th century, and some new artillery barracks. Many industries +are carried on in Fulda. These include weaving and dyeing, the +manufacture of linen, plush and other textiles and brewing. There are +also railway works in the town. A large trade is done in cattle and +grain, many markets being held here. Fine views are obtained from +several hills in the neighbourhood, among these being the Frauenberg, +the Petersberg and the Kalvarienberg. + +Fulda owes its existence to its famous abbey. It became a town in 1208, +and during the middle ages there were many struggles between the abbots +and the townsfolk. During the Peasants' War it was captured by the +rebels and during the Seven Years' War by the Hanoverians. It came +finally into the possession of Prussia in 1866. From 1734 to 1804 Fulda +was the seat of a university, and latterly many assemblies of German +bishops have been held in the town. + +The great Benedictine abbey of Fulda occupies the place in the +ecclesiastical history of Germany which Monte Cassino holds in Italy, St +Gall in South Germany, Corvey in Saxony, Tours in France and Iona in +Scotland. Founded in 744 at the instigation of St Boniface by his pupil +Sturm, who was the first abbot, it became the centre of a great +missionary work. It was liberally endowed with land by the princes of +the Carolingian house and others, and soon became one of the most famous +and wealthy establishments of its kind. About 968 the pope declared that +its abbot was primate of all the abbots in Germany and Gaul, and later +he became a prince of the Empire. Fulda was specially famous for its +school, which was the centre of the theological learning of the early +middle ages. Among the teachers here were Alcuin, Hrabanus Maurus, who +was abbot from 822 to 842, and Walafrid Strabo. Early in the 10th +century the monastery was reformed by introducing monks from Scotland, +who were responsible for restoring in its old strictness the Benedictine +rule. Later the abbey lost some of its lands and also its high position, +and some time before the Reformation the days of its glory were over. +Johann von Henneberg, who was abbot from 1529 to 1541, showed some +sympathy with the teaching of the reformers, but the Counter-Reformation +made great progress here under Abbot Balthasar von Dernbach. Gustavus +Adolphus gave the abbey as a principality to William, landgrave of +Hesse, but William's rule only lasted for ten years. In 1752 the abbot +was raised to the rank of a bishop, and Fulda ranked as a +prince-bishopric. This was secularized in 1802, and in quick succession +it belonged to the prince of Orange, the king of France and the +grand-duchy of Frankfort. In 1816 the greater part of the principality +was ceded by Prussia to Hesse-Cassel, a smaller portion being united +with Bavaria. Sharing the fate of Hesse-Cassel, this larger portion was +annexed by Prussia in 1866. In 1829 a new bishopric was founded at +Fulda. + + For the town see A. Hartmann, _Zeitgeschichte von Fulda_ (Fulda, + 1895); J. Schneider, _Fuhrer durch die Stadt Fulda_ (Fulda, 1899); and + _Chronik von Fulda und dessen Umgebungen_ (1839). For the history of + the abbey see Gegenbaur, _Das Kloster Fulda im Karolinger Zeitalter_ + (Fulda, 1871-1874); Arndt, _Geschichte des Hochstifts Fulda_ (Fulda, + 1860); and the _Fuldaer Geschichtsblatter_ (1902 fol.). + + + + +FULGENTIUS, FABIUS PLANCIADES, Latin grammarian, a native of Africa, +flourished in the first half of the 6th (or the last part of the 5th) +century A.D. He is to be distinguished from Fulgentius, bishop of Ruspe +(468-533), to whom he was probably related, and also from the bishop's +pupil and biographer, Fulgentius Ferrandus. Four extant works are +attributed to him. (1) _Mythologiarum libri iii._, dedicated to a +certain Catus, a presbyter of Carthage, containing 75 myths briefly +told, and then explained in the mystical and allegorical manner of the +Stoics and Neoplatonists. For this purpose the author generally invokes +the aid of etymologies which, borrowed from the philosophers, are highly +absurd. As a Christian, Fulgentius sometimes (but less frequently than +might have been expected) quotes the Bible by the side of the +philosophers, to give a Christian colouring to the moral lesson. (2) +_Expositio Vergilianae continentiae (continentia_ = contents), a sort of +appendix to (1), dedicated to Catus. The poet himself appears to the +author and explains the twelve books of the _Aeneid_ as a picture of +human life. The three words _arma_ (= virtus), _vir_ (= sapientia), +_primus_ (= princeps) in the first line represent respectively +_substantia corporalis, sensualis, ornans_. Book i. symbolizes the birth +and early childhood of man (the shipwreck of Aeneas denotes the peril of +birth), book vi. the plunge into the depths of wisdom. (3) _Expositio +sermonum antiquorum_, explanations of 63 rare and obsolete words, +supported by quotations (sometimes from authors and works that never +existed). It is much inferior to the similar work of Nonius, with which +it is often edited. (4) _Liber absque litteris de aetatibus mundi et +hominis_. In the MS. heading of this work, the name of the author is +given as Fabius Claudius Gordianus Fulgentius (Claudius is the name of +the father, and Gordianus that of the grandfather of the bishop, to whom +some attribute the work). The title _Absque litteris_ indicates that one +letter of the alphabet is wholly omitted in each successive book (A in +bk. i., B in bk. ii.). Only 14 books are preserved. The matter is +chiefly taken from sacred history. In addition to these, Fulgentius +speaks of early poetical attempts after the manner of Anacreon, and of a +work called _Physiologus_, dealing with medical questions, and including +a discussion of the mystical signification of the numbers 7 and 9. +Fulgentius is a representative of the so-called late African style, +taking for his models Apuleius, Tertullian and Martianus Capella. His +language is bombastic, affected and incorrect, while the lengthy and +elaborate periods make it difficult to understand his meaning. + + See the edition of the four works by R. Helm (1898, Teubner series); + also M. Zink, _Der Mytholog Fulgentius_ (1867); E. Jungmann, "De + Fulgentii aetate et scriptis," in _Acta Societatis Philologae + Lipsiensis_, i. (1871); A. Ebert, _Allgemeine Geschichte der Litt. des + Mittelalters_, i.; article "Fulgentius" by C.F. Bohr in Ersch and + Gruber's _Allgemeine Encyklopadie_; Teuffel-Schwabe, _History of Roman + Literature_ (Eng. trans.). + + + + +FULGINIAE (mod. _Foligno_), an ancient town of Umbria, Italy, on the +later line of the Via Flaminia, 15 m. S. of Nuceria. It appears to have +been of comparatively late origin, inasmuch as it had no city walls, +but, in imperial times especially, owing to its position on the new line +of the Via Flaminia, it must have increased in importance as being the +point of departure of roads to Perusia and to Picenum over the pass of +Plestia. It appears to have had an amphitheatre, and three bridges over +the Topino are attributed to the Roman period. Three miles to the N. +lies the independent community of Forum Flaminii, the site of which is +marked by the church of S. Giovanni Profiamma, at or near which the +newer line of the Via Flaminia rejoined the older. It was no doubt +founded by the builder of the road, C. Flaminius, consul in 220 B.C. +(See FOLIGNO and FLAMINIA, VIA.) (T. As.) + + + + +FULGURITE (from Lat. _fulgur_, lightning), in petrology, the name given +to rocks which have been fused on the surface by lightning, and to the +characteristic holes in rocks formed by the same agency. When lightning +strikes the naked surfaces of rocks, the sudden rise of temperature may +produce a certain amount of fusion, especially when the rocks are dry +and the electricity is not readily conducted away. Instances of this +have been observed on Ararat and on several mountains in the Alps, +Pyrenees, &c. A thin glassy crust, resembling a coat of varnish, is +formed; its thickness is usually not more than one-eighth of an inch, +and it may be colourless, white or yellow. When examined under the +microscope, it usually shows no crystallization, and contains minute +bubbles due to the expansion of air or other gases in the fused +pellicle. Occasionally small microliths may appear, but this is uncommon +because so thin a film would cool with extreme rapidity. The minerals of +the rock beneath are in some cases partly fused, but the more refractory +often appear quite unaffected. The glass has arisen from the melting of +the most fusible ingredients alone. + +Another type of fulgurite is commonest in dry sands and takes the shape +of vertical tubes which may be nearly half an inch in diameter. +Generally they are elliptical in cross section, or flattened by the +pressure exerted by the surrounding sand on the fulgurite at a time when +it was still very hot and plastic. These tubes are often vertical and +may run downwards for several feet through the sand, branching and +lessening as they descend. Tubular perforations in hard rocks have been +noted also, but these are short and probably follow original cracks. The +glassy material contains grains of sand and many small round or +elliptical cavities, the long axes of which are radial. Minerals like +felspar and mica are fused more readily than quartz, but analysis shows +that some fulgurite glasses are very rich in silica, which perhaps was +dissolved in the glass rather than simply fused. The central cavity of +the tube and the bubbles in its walls point to the expansion of the +gases (air, water, &c.) in the sand by sudden and extreme heating. Very +fine threads of glass project from the surface of the tube as if fused +droplets had been projected outwards with considerable force. Where the +quartz grains have been greatly heated but not melted they become white +and semi-opaque, but where they are in contact with the glass they +usually show partial solution. Occasionally crystallization has begun +before the glass solidified, and small microliths, the nature of which +is undeterminable, occur in streams and wisps in the clear hyaline +matrix. (J. S. F.) + + + + +FULHAM, a western metropolitan borough of London, England, bounded N.W. +by Hammersmith, N.E. by Kensington, E. by Chelsea, and S.E., S. and S.W. +by the river Thames. Pop. (1901) 137,289. The principal thoroughfares +are Fulham Palace Road running S. from Hammersmith, Fulham Road and +King's Road, W. from Chelsea, converging and leading to Putney Bridge +over the Thames; North End Road between Hammersmith and Fulham Roads; +Lillie Road between South Kensington and Fulham Palace Road; and +Wandsworth Bridge Road leading S. from New King's Road to Wandsworth +Bridge. In the north Fulham includes the residential district known as +West Kensington, and farther south that of Walham Green. The manor house +or palace of the bishops of London stands in grounds, beautifully +planted and surrounded by a moat, believed to be a Danish work, near the +river west of Putney Bridge. Its oldest portion is the picturesque +western quadrangle, built by Bishop Fitzjames (1506-1522). The parish +church of All Saints, between the bridge and the grounds, was erected in +1881 from designs by Sir Arthur Blomfield. The fine old monuments from +the former building, dating from the 16th to the 18th centuries, are +mostly preserved, and in the churchyard are the memorials of several +bishops of London and of Theodore Hook (1841). The public recreation +grounds include the embankment and gardens between the river and the +palace grounds, and there are also two well-known enclosures used for +sports within the borough. Of these Hurlingham Park is the headquarters +of the Hurlingham Polo Club and a fashionable resort; and Queen's Club, +West Kensington, has tennis and other courts for the use of members, and +is also the scene of important football matches, and of the athletic +meetings between Oxford and Cambridge Universities, and those between +the English and American Universities held in England. In Seagrave Road +is the Western fever hospital. The parliamentary borough of Fulham +returns one member. The borough council consists of a mayor, 6 aldermen +and 36 councillors. Area, 1703.5 acres. + +Fulham, or in its earliest form _Fullanham_, is uncertainly stated to +signify "the place" either "of fowls" or "of dirt." The manor is said to +have been given to Bishop Erkenwald about the year 691 for himself and +his successors in the see of London, and Holinshed relates that the +Bishop of London was lodging in his manor place in 1141 when Geoffrey de +Mandeville, riding out from the Tower of London, took him prisoner. At +the Commonwealth the manor was temporarily out of the bishops' hands, +being sold to Colonel Edmund Harvey. There is no record of the first +erection of a parish church, but the first known rector was appointed in +1242, and a church probably existed a century before this. The earliest +part of the church demolished in 1881, however, did not date farther +back than the 15th century. In 879 Danish invaders, sailing up the +Thames, wintered at Fulham and Hammersmith. Near the former wooden +Putney Bridge, built in 1729 and replaced in 1886, the earl of Essex +threw a bridge of boats across the river in 1642 in order to march his +army in pursuit of Charles I., who thereupon fell back on Oxford. +Margravine Road recalls the existence of Bradenburg House, a riverside +mansion built by Sir Nicholas Crispe in the time of Charles I., used as +the headquarters of General Fairfax in 1647 during the civil wars, and +occupied in 1792 by the margrave of Bradenburg-Anspach and Bayreuth and +his wife, and in 1820 by Caroline, consort of George IV. + + + + +FULK, king of Jerusalem (b. 1092), was the son of Fulk IV., count of +Anjou, and his wife Bertrada (who ultimately deserted her husband and +became the mistress of Philip I. of France). He became count of Anjou in +1109, and considerably added to the prestige of his house. In particular +he showed himself a doughty opponent to Henry I. of England, against +whom he continually supported Louis VI. of France, until in 1127 Henry +won him over by betrothing his daughter Matilda to Fulk's son Geoffrey +Plantagenet. Already in 1120 Fulk had visited the Holy Land, and become +a close friend of the Templars. On his return he assigned to the order +of the Templars an annual subsidy, while he also maintained two knights +in the Holy Land for a year. In 1128 he was preparing to return to the +East, when he received an embassy from Baldwin II., king of Jerusalem, +who had no male heir to succeed him, offering his daughter Melisinda in +marriage, with the right of eventual succession to the kingdom. Fulk +readily accepted the offer; and in 1129 he came and was married to +Melisinda, receiving the towns of Acre and Tyre as her dower. In 1131, +at the age of thirty-nine, he became king of Jerusalem. His reign is not +marked by any considerable events: the kingdom which had reached its +zenith under Baldwin II., and did not begin to decline till the capture +of Edessa in the reign of Baldwin III., was quietly prosperous under his +rule. In the beginning of his reign he had to act as regent of Antioch, +and to provide a husband, Raymund of Poitou, for the infant heiress +Constance. But the great problem with which he had to deal was the +progress of the atabeg Zengi of Mosul. In 1137 he was beaten near Barin, +and escaping into the fort was surrounded and forced to capitulate. A +little later, however, he greatly improved his position by strengthening +his alliance with the vizier of Damascus, who also had to fear the +progress of Zengi (1140); and in this way he was able to capture the +fort of Banias, to the N. of Lake Tiberias. Fulk also strengthened the +kingdom on the south; while his butler, Paganus, planted the fortress of +Krak to the south of the Dead Sea, and helped to give the kingdom an +access towards the Red Sea, he himself constructed Blanche Garde and +other forts on the S.W. to overawe the garrison of Ascalon, which was +still held by the Mahommedans, and to clear the road towards Egypt. +Twice in Fulk's reign the eastern emperor, John Comnenus, appeared in +northern Syria (1137 and 1142); but his coming did not affect the king, +who was able to decline politely a visit which the emperor proposed to +make to Jerusalem. In 1143 he died, leaving two sons, who both became +kings, as Baldwin III. and Amalric I. + +Fulk continued the tradition of good statesmanship and sound +churchmanship which Baldwin I. and Baldwin II. had begun. William of +Tyre speaks of him as a fine soldier, an able politician, and a good son +of the church, and only blames him for partiality to his friends, and a +forgetfulness of names and faces, which placed him at a disadvantage and +made him too dependent on his immediate intimates. Little, perhaps, need +be made of these censures: the real fault of Fulk was his neglect to +envisage the needs of the northern principalities, and to head a +combined resistance to the rising power of Zengi of Mosul. + + His reign in Jerusalem is narrated by R. Rohricht (_Geschichte des + Konigreichs Jerusalem_, Innsbruck, 1898), and has been made the + subject of a monograph by G. Dodu (_De Fulconis Hierosolymitani + regno_, Paris, 1894). (E. Br.) + + + + +FULK (d. 900), archbishop of Reims, and partisan of Charles the Simple +in his struggle with Odo, count of Paris, was elected to the see as +archbishop in 883 upon the death of Hincmar. In 887 he was engaged in a +struggle with the Normans who invaded his territories. Upon the +deposition of Charles the Fat he sided with Charles the Simple in his +contest for the West Frankish dominions against Count Odo of Paris, and +crowned him king in his own metropolitan church at Reims after most of +the nobles had gone over to Odo (893). Upon the death of Odo he +succeeded in having Charles recognized as king by a majority of the West +Frankish nobility. In 892 he obtained special privileges for his +province from Pope Formosus, who promised that thereafter, when the +archbishopric became vacant, the revenues should not be enjoyed by +anyone while the vacancy existed, but should be reserved for the new +incumbent, provided the election took place within the canonical limit +of three months. From 898 until his death he held the office of +chancellor, which for some time afterwards was regularly filled by the +archbishop of Reims. In his efforts to keep the wealthy abbeys and +benefices of the church out of the hands of the nobles, he incurred the +hatred of Baldwin, count of Flanders, who secured his assassination on +the 17th of June 900, a crime which the weak Carolingian monarch left +unpunished. + + Fulk left some letters, which are collected in Migne, _Patrologia + Latina_, vol. cxxxi. 11-14. + + + + +FULKE, WILLIAM (1538-1589), Puritan divine, was born in London and +educated at Cambridge. After studying law for six years, he became a +fellow at St John's College, Cambridge, in 1564. He took a leading part +in the "vestiarian" controversy, and persuaded the college to discard +the surplice. In consequence he was expelled from St. John's for a +time, but in 1567 he became Hebrew lecturer and preacher there. After +standing unsuccessfully for the headship of the college in 1569, he +became chaplain to the earl of Leicester, and received from him the +livings of Warley, in Essex, and Dennington in Suffolk. In 1578 he was +elected master of Pembroke Hall, Cambridge. As a Puritan +controversialist he was remarkably active; in 1580 the bishop of Ely +appointed him to defend puritanism against the Roman Catholics, Thomas +Watson, ex-bishop of Lincoln (1513-1584), and John Feckenham, formerly +abbot of Westminster, and in 1581 he was one of the disputants with the +Jesuit, Edmund Campion, while in 1582 he was among the clergy selected +by the privy council to argue against any papist. His numerous polemical +writings include _A Defense of the sincere true Translations of the +holie Scriptures into the English tong_ (London, 1583), and confutations +of Thomas Stapleton (1535-1598), Cardinal Allen and other Roman Catholic +controversialists. + + + + +FULK NERRA (c. 970-1040), count of Anjou, eldest son of Count Geoffrey +I., "Grisegonelle" (Grey Tunic) and Adela of Vermandois, was born about +970 and succeeded his father in the countship of Anjou on the 21st of +July 987. He was successful in repelling the attacks of the count of +Rennes and laying the foundations of the conquest of Touraine (see +ANJOU). In this connexion he built a great number of strong castles, +which has led in modern times to his being called "the great builder." +He also founded several religious houses, among them the abbeys of +Beaulieu, near Loches (c. 1007), of Saint-Nicholas at Angers (1020) and +of Ronceray at Angers (1028), and, in order to expiate his crimes of +violence, made three pilgrimages to the Holy Land (in 1002-1003, c. 1008 +and in 1039). On his return from the third of these journeys he died at +Metz in Lorraine on the 21st of June 1040. By his first marriage, with +Elizabeth, daughter of Bouchard le Venerable, count of Vendome, he had a +daughter, Adela, who married Boon of Nevers and transmitted to her +children the countship of Vendome. Elizabeth having died in 1000, Fulk +married Hildegarde of Lorraine, by whom he had a son, Geoffrey Martel +(q.v.), and a daughter Ermengarde, who married Geoffrey, count of +Gatinais, and was the mother of Geoffrey "le Barbu" (the Bearded) and of +Fulk "le Rechin" (see ANJOU). + + See Louis Halphen, _Le Comte d'Anjou au XI^e siecle_ (Paris, 1906). + The biography of Fulk Nerra by Alexandre de Salies, _Histoire de + Foulques Nerra_ (Angers, 1874) is confused and uncritical. A very + summary biography is given by Celestin Port, _Dictionnaire historique, + geographique et biographique de Maine-et-Loire_ (3 vols., + Paris-Angers, 1874-1878), vol. ii. pp. 189-192, and there is also a + sketch in Kate Norgate, _England under the Angevin Kings_ (2 vols., + London, 1887), vol. i. ch. iii. (L. H.*) + + + + +FULLEBORN, GEORG GUSTAV (1769-1803), German philosopher, philologist and +miscellaneous writer, was born at Glogau, Silesia, on the 2nd of March +1769, and died at Breslau on the 6th of February 1803. He was educated +at the University of Halle, and was made doctor of philosophy in +recognition of his thesis _De Xenophane, Zenone et Gorgia_. He took +diaconal orders in 1791, but almost immediately became professor of +classics at Breslau. His philosophical works include annotations to +Garve's translation of the _Politics_ of Aristotle (1799-1800), and a +large share in the _Beitrage zur Geschichte der Philosophie_ (published +in twelve parts between 1791 and 1799), in which he collaborated with +Forberg, Reinhold and Niethammer. In philology he wrote _Encyclopaedia +philologica sive primae lineae Isagoges in antiquorum studia_ (1798; 2nd +ed., 1805); _Kurze Theorie des lateinischen Stils_ (1793); _Leitfaden +der Rhetorik_ (1802); and an annotated edition of the _Satires_ of +Persius. Under the pseudonym "Edelwald Justus" he published several +collections of popular tales--_Bunte Blatter_ (1795); _Kleine Schriften +zur Unterhaltung_ (1798); _Nebenstunden_ (1799). After his death were +published _Taschenbuch fur Brunnengaste_ (1806) and _Kanzelreden_ +(1807). He was a frequent contributor to the press, where his writings +were very popular. + + See Schummel, _Gedachtnisrede_ (1803) and _Garve und Fulleborn_; + Meusel, _Gelehrtes Teutschland_, vol. ii. + + + + +FULLER, ANDREW (1754-1815), English Baptist divine, was born on the 6th +of February 1754, at Wicken in Cambridgeshire. In his boyhood and youth +he worked on his father's farm. In his seventeenth year he became a +member of the Baptist church at Soham, and his gifts as an exhorter met +with so much approval that, in the spring of 1775, he was called and +ordained as pastor of that congregation. In 1782 he removed to Kettering +in Northamptonshire, where he became friendly with some of the most +eminent ministers of the denomination. Before leaving Soham he had +written the substance of a treatise in which he had sought to counteract +the prevailing Baptist hyper-Calvinism which, "admitting nothing +spiritually good to be the duty of the unregenerate, and nothing to be +addressed to them in a way of exhortation excepting what related to +external obedience," had long perplexed his own mind. This work he +published, under the title _The Gospel worthy of all Acceptation_, soon +after his settlement in Kettering; and although it immediately involved +him in a somewhat bitter controversy which lasted for nearly twenty +years, it was ultimately successful in considerably modifying the views +prevalent among English dissenters. In 1793 he published a treatise, +_The Calvinistic and Socinian systems examined and compared as to their +moral tendency_, in which he rebutted the accusation of antinomianism +levelled by the Socinians against those who over-emphasized the +doctrines of free grace. This work, along with another against Deism, +entitled _The Gospel its own Witness_, is regarded as the production on +which his reputation as a theologian mainly rests. Fuller also published +an admirable _Memoir of the Rev. Samuel Pearce_, of Birmingham, and a +volume of _Expository Lectures in Genesis_, besides a considerable +number of smaller pieces, chiefly sermons and pamphlets, which were +issued in a collected form after his death. He was a man of forceful +character, more prominent on the practical side of religion than on the +devotional, and accordingly not pre-eminently successful in his local +ministry. His great work was done in connexion with the Baptist +Missionary Society, formed at Kettering in 1792, of which he was +secretary until his death on the 7th of May 1815. Both Princeton and +Yale, U.S.A., conferred on him the degree of D. D., but he never used +it. + + Several editions of his collected works have appeared, and a _Memoir_, + principally compiled from his own papers, was published about a year + after his decease by Dr Ryland, his most intimate friend and coadjutor + in the affairs of the Baptist mission. There is also a biography by + the Rev. J.W. Morris (1816); and his son prefixed a memoir to an + edition of his chief works in Bohn's Standard Library (1852). + + + + +FULLER, GEORGE (1822-1884), American figure and portrait painter, was +born at Deerfield, Massachusetts, in 1822. At the age of twenty he +entered the studio of the sculptor H.K. Brown, at Albany, New York, +where he drew from the cast and modelled heads. Having attained some +proficiency he went about the country painting portraits, settling at +length in Boston, where he studied the works of the earlier Americans, +Stuart, Copley and Allston. After three years in that city, and twelve +in New York, where in 1857 he was elected a member of the National +Academy of Design, he went to Europe for a brief visit and for study. +During all this time his work had received little recognition and +practically no financial encouragement, and on his return he settled on +the family farm at Deerfield, where he continued to work in his own way +with no thought of the outside world. In 1876, however, he was forced by +pressing needs to dispose of his work, and he sent some pictures to a +dealer in Boston, where he met with immediate success, financial and +artistic, and for the remaining eight years of his life he never lacked +patrons. He died in Boston on the 21st of March 1884. He was a poetic +painter, and a dreamer of delicate fancies and quaint, intangible phases +of nature, his canvases being usually enveloped in a brown mist that +renders the outlines vague. Among his noteworthy canvases are: "The +Turkey Pasture," "Romany Girl," "And she was a Witch," "Nydia," +"Winifred Dysart" and "The Quadroon." + + + + +FULLER, MARGARET, Marchioness Ossoli (1810-1850), American authoress, +eldest child of Timothy Fuller (1778-1835), a lawyer and politician of +some eminence, was born at Cambridgeport, Massachusetts, on the 23rd of +May 1810. Her education was conducted by her father, who, she states, +made the mistake of thinking to "gain time by bringing forward the +intellect as early as possible," the consequence being "a premature +development of brain that made her a youthful prodigy by day, and by +night a victim of spectral illusions, nightmare and somnambulism." At +six years she began to read Latin, and at a very early age she had +selected as her favourite authors Shakespeare, Cervantes and Moliere. +Soon the great amount of study exacted of her ceased to be a burden, and +reading became a habit and a passion. Having made herself familiar with +the masterpieces of French, Italian and Spanish literature, she in 1833 +began the study of German, and within the year had read some of the +masterpieces of Goethe, Korner, Novalis and Schiller. + +After her father's death in 1835 she went to Boston to teach languages, +and in 1837 she was chosen principal teacher in the Green Street school, +Providence, Rhode Island, where she remained till 1839. From this year +until 1844 she stayed at different places in the immediate neighbourhood +of Boston, forming an intimate acquaintance with the colonists of Brook +Farm, and numbering among her closest friends R.W. Emerson, Nathaniel +Hawthorne and W.H. Channing. In 1839 she published a translation of +Eckermann's _Conversations with Goethe_, which was followed in 1842 by a +translation of the correspondence between Karoline von Gunderode and +Bettina von Arnim, entitled _Gunderode_. Aided by R.W. Emerson and +George Ripley, she in 1840 started _The Dial_, a poetical and +philosophical magazine representing the opinions and aims of the New +England Transcendentalists. This journal she continued to edit for two +years, and while in Boston she also conducted conversation classes for +ladies in which philosophical and social subjects were discussed with a +somewhat over-accentuated earnestness. These meetings may be regarded as +perhaps the beginning of the modern movement in behalf of women's +rights. R.W. Emerson, who had met her as early as 1836, thus describes +her appearance: "She was then twenty-six years old. She had a face and +frame that would indicate fulness and tenacity of life. She was rather +under the middle height; her complexion was fair, with strong fair hair. +She was then, as always, carefully and becomingly dressed, and of +ladylike self-possession. For the rest her appearance had nothing +prepossessing. Her extreme plainness, a trick of incessantly opening and +shutting her eyelids, the nasal tone of her voice, all repelled; and I +said to myself we shall never get far." On better acquaintance this +unprepossessing exterior seemed, however, to melt away, and her +inordinate self-esteem to be lost in the depth and universality of her +sympathy. She possessed an almost irresistible power of winning the +intellectual and moral confidence of those with whom she came in +contact, and "applied herself to her companion as the sponge applies +itself to water." She obtained from each the best they had to give. It +was indeed more as a conversationalist than as a writer that she earned +the title of the Priestess of Transcendentalism. It was her intimate +friends who admired her most. Smart and pungent though she is as a +writer, the apparent originality of her views depends more on +eccentricity than either intellectual depth or imaginative vigour. In +1844 she removed to New York at the desire of Horace Greeley to write +literary criticism for _The Tribune_, and in 1846 she published a +selection from her articles on contemporary authors in Europe and +America, under the title _Papers on Literature and Art_. The same year +she paid a visit to Europe, passing some time in England and France, and +finally taking up her residence in Italy. There she was married in +December 1847 to the marquis Giovanni Angelo Ossoli, a friend of +Mazzini. During 1848-1849 she was present with her husband in Rome, and +when the city was besieged she, at the request of Mazzini, took charge +of one of the two hospitals while her husband fought on the walls. In +May 1850, along with her husband and infant son, she embarked at Leghorn +for America, but when they had all but reached their destination the +vessel was wrecked on Fire Island beach on the 16th of June, and the +Ossolis were among the passengers who perished. + + _Life Without and Life Within_ (Boston, 1860) is a collection of + essays, poems, &c., supplementary to her _Collected Works_, printed in + 1855. See the _Autobiography of Margaret Fuller Ossoli_, with + additional memoirs by J.F. Clarke, R.W. Emerson and W.H. Channing (2 + vols., Boston, 1852); also _Margaret Fuller (Marchesa Ossoli)_, by + Julia Ward Howe (1883), in the "Eminent Women" series; _Margaret + Fuller Ossoli_ (Boston, 1884), by Thomas Wentworth Higginson in the + "American Men of Letters" series, which is based largely on unedited + material; and _The Love Letters of Margaret Fuller, 1845-1846_ (London + and New York, 1903), with an introduction by Julia Ward Howe. + + + + +FULLER, MELVILLE WESTON (1833-1910), American jurist, chief justice of +the Supreme Court of the United States, was born at Augusta, Maine, on +the 11th of February 1833. After graduating at Bowdoin College in 1853 +he spent a year at the Harvard Law School, and in 1855 began the +practice of law at Augusta, where he was an associate-editor of a +Democratic paper, _The Age_, and served in the city council and as city +attorney. In 1856 he removed to Chicago, Illinois, where he continued to +practise until 1888, rising to a high position at the bar of the +Northwest. For some years he was active in Democratic politics, being a +member of the Illinois Constitutional Convention in 1862 and of the +State House of Representatives from 1863 to 1865. He was a delegate to +various National conventions of his party, and in that of 1876 placed +Thomas A. Hendricks in nomination for the presidency. In 1888, by +President Cleveland's appointment, he succeeded Morrison R. Waite as +chief-justice of the Supreme Court of the United States. In 1899 he was +appointed by President McKinley a member of the arbitration commission +at Paris to settle the Venezuela-British Guiana boundary dispute. + + + + +FULLER, THOMAS (1608-1661), English divine and historian, eldest son of +Thomas Fuller, rector of Aldwincle St Peter's, Northamptonshire, was +born at his father's rectory and was baptized on the 19th of June 1608. +Dr John Davenant, bishop of Salisbury, was his uncle and godfather. +According to Aubrey, Fuller was "a boy of pregnant wit." At thirteen he +was admitted to Queens' College, Cambridge, then presided over by Dr +John Davenant. His cousin, Edward Davenant, was a tutor in the same +college. He was apt and quick in study; and in Lent 1624-1625 he became +B.A. and in July 1628 M.A. Being overlooked in an election of fellows of +his college, he was removed by Bishop Davenant to Sidney Sussex College, +November 1628. In 1630 he received from Corpus Christi College the +curacy of St Benet's, Cambridge. + +Fuller's quaint and humorous oratory soon attracted attention. He +published in 1631 a poem on the subject of David and Bathsheba, entitled +_David's Hainous Sinne, Heartie Repentance, Heavie Punishment_. In June +of the same year his uncle gave him a prebend in Salisbury, where his +father, who died in the following year, held a canonry. The rectory of +Broadwindsor, Dorsetshire, then in the diocese of Bristol, was his next +preferment (1634); and on the 11th of June 1635 he proceeded B.D. At +Broadwindsor he compiled _The Historie of the Holy Warre_ (1639), a +history of the crusades, and _The Holy State and the Prophane State_ +(1642). This work describes the holy state as existing in the family and +in public life, gives rules of conduct, model "characters" for the +various professions and profane biographies. It was perhaps the most +popular of all his writings. He was in 1640 elected proctor for Bristol +in the memorable convocation of Canterbury, which assembled with the +Short Parliament. On the sudden dissolution of the latter he joined +those who urged that convocation should likewise dissolve as usual. That +opinion was overruled; and the assembly continued to sit by virtue of a +royal writ. Fuller has left in his _Church History_ a valuable account +of the proceedings of this synod, for sitting in which he was fined +L200, which, however, was never exacted. His first published volume of +sermons appeared in 1640 under the title of _Joseph's party-coloured +Coat_, which contains many of his quaint utterances and odd conceits. +His grosser mannerisms of style, derived from the divines of the former +generation, disappeared for the most part in his subsequent discourses. + +About 1640 he had married Eleanor, daughter of Hugh Grove of Chisenbury, +Wiltshire. She died in 1641. Their eldest child, John, baptized at +Broadwindsor by his father, 6th June 1641, was afterwards rector of +Sidney Sussex College, edited the _Worthies of England_, 1662, and +became rector of Great Wakering, Essex, where he died in 1687. + +At Broadwindsor, early in the year 1641, Thomas Fuller, his curate Henry +Sanders, the church wardens, and others, nine persons altogether, +certified that their parish, represented by 242 grown-up male persons, +had taken the Protestation ordered by the speaker of the Long +Parliament. Fuller was not formally dispossessed of his living and +prebend on the triumph of the Presbyterian party, but he relinquished +both preferments about this time. For a short time he preached with +success at the Inns of Court, and thence removed, at the invitation of +the master of the Savoy (Dr Balcanqual) and the brotherhood of that +foundation, to be lecturer at their chapel of St Mary Savoy. Some of the +best discourses of the witty preacher were delivered at the Savoy to +audiences which extended into the chapel-yard. In one he set forth with +searching and truthful minuteness the hindrances to peace, and urged the +signing of petitions to the king at Oxford, and to the parliament, to +continue their care in advancing an accommodation. In his _Appeal of +Injured Innocence_ Fuller says that he was once deputed to carry a +petition to the king at Oxford. This has been identified with a petition +entrusted to Sir Edward Wardour, clerk of the pells, Dr Dukeson, "Dr +Fuller," and four or five others from the city of Westminster and the +parishes contiguous to the Savoy. A pass was granted by the House of +Lords, on the 2nd of January 1643, for an equipage of two coaches, four +or six horses and eight or ten attendants. On the arrival of the +deputation at Uxbridge, on the 4th of January, officers of the +Parliamentary army stopped the coaches and searched the gentlemen; and +they found upon the latter "two scandalous books arraigning the +proceedings of the House," and letters with ciphers to Lord Viscount +Falkland and the Lord Spencer. Ultimately a joint order of both Houses +remanded the party; and Fuller and his friends suffered a brief +imprisonment. The Westminster Petition, notwithstanding, reached the +king's hands; and it was published with the royal reply (see J.E. +Bailey, _Life of Thomas Fuller_, pp. 245 _et seq._). When it was +expected, three months later, that a favourable result would attend the +negotiations at Oxford, Fuller preached a sermon at Westminster Abbey, +on the 27th of March 1643, on the anniversary of Charles I.'s accession, +on the text, "Yea, let him take all, so my Lord the King return in +peace." On Wednesday, the 26th of July, he preached on church +reformation, satirizing the religious reformers, and maintaining that +only the Supreme Power could initiate reforms. + +He was now obliged to leave London, and in August 1643 he joined the +king at Oxford. He lived in a hired chamber at Lincoln College for 17 +weeks. Thence he put forth a witty and effective reply to John +Saltmarsh, who had attacked his views on ecclesiastical reform. Fuller +subsequently published by royal request a sermon preached on the 10th of +May 1644, at St Mary's, Oxford, before the king and Prince Charles, +called _Jacob's Vow_. + +The spirit of Fuller's preaching, always characterized by calmness and +moderation, gave offence to the high royalists, who charged him with +lukewarmness in their cause. To silence unjust censures he became +chaplain to the regiment of Sir Ralph Hopton. For the first five years +of the war, as he said, when excusing the non-appearance of his _Church +History_, "I had little list or leisure to write, fearing to be made a +history, and shifting daily for my safety. All that time I could not +live to study, who did only study to live." After the defeat of Hopton +at Cheriton Down, Fuller retreated to Basing House. He took an active +part in its defence, and his life with the troops caused him to be +afterwards regarded as one of "the great cavalier parsons." In his +marches with his regiment round about Oxford and in the west, he devoted +much time to the collection of details, from churches, old buildings, +and the conversation of ancient gossips, for his _Church-History_ and +_Worthies of England_. He compiled in 1645 a small volume of prayers and +meditations,--the _Good Thoughts in Bad Times_,--which, set up and +printed in the besieged city of Exeter, whither he had retired, was +called by himself "the first fruits of Exeter press." It was inscribed +to Lady Dalkeith, governess to the infant princess, Henrietta Anne (b. +1644), to whose household he was attached as chaplain. The corporation +gave him the Bodleian lectureship on the 21st of March 1645/6, and he +held it until the 17th of June following, soon after the surrender of +the city to the parliament. _The Fear of losing the Old Light_ (1646) +was his farewell discourse to his Exeter friends. Under the Articles of +Surrender Fuller made his composition with the government at London, his +"delinquency" being that he had been present in the king's garrisons. In +_Andronicus, or the Unfortunate Politician_ (1646), partly authentic and +partly fictitious, he satirized the leaders of the Revolution; and for +the comfort of sufferers by the war he issued (1647) a second devotional +manual, entitled _Good Thoughts in Worse Times_, abounding in fervent +aspirations, and drawing moral lessons in beautiful language out of the +events of his life or the circumstances of the time. In grief over his +losses, which included his library and manuscripts (his "upper and +nether millstone"), and over the calamities of the country, he wrote his +work on the _Cause and Cure of a Wounded Conscience_ (1647). It was +prepared at Boughton House in his native county, where he and his son +were entertained by Edward Lord Montagu, who had been one of his +contemporaries at the university and had taken the side of the +parliament. + +For the next few years of his life Fuller was mainly dependent upon his +dealings with booksellers, of whom he asserted that none had ever lost +by him. He made considerable progress in an English translation from the +MS. of the _Annales_ of his friend Archbishop Ussher. Amongst his +benefactors it is curious to find Sir John Danvers of Chelsea, the +regicide. Fuller in 1647 began to preach at St Clement's, Eastcheap, and +elsewhere in the capacity of lecturer. While at St Clement's he was +suspended; but speedily recovering his freedom, he preached wherever he +was invited. At Chelsea, where also he occasionally officiated, he +covertly preached a sermon on the death of Charles I., but he did not +break with his Roundhead patrons. James Hay, 2nd earl of Carlisle, made +him his chaplain, and presented him in 1648 or 1649 to the curacy of +Waltham Abbey. His possession of the living was in jeopardy on the +appointment of Cromwell's "Tryers"; but he evaded their inquisitorial +questions by his ready wit. He was not disturbed at Waltham in 1655, +when the Protector's edict prohibited the adherents of the late king +from preaching. Lionel, 3rd earl of Middlesex, who lived at Copt Hall, +near Waltham, gave him what remained of the books of the lord treasurer +his father; and through the good offices of the marchioness of Hertford, +part of his own pillaged library was restored to him. Fuller was thus +able to prosecute his literary labours, producing successively his +descriptive geography of the Holy Land, called _A Pisgah-Sight of +Palestine_ (1650), and his _Church-History of Britain_ (1655), from the +birth of Jesus Christ until the year 1648. With the _Church-History_ was +printed _The History of the University of Cambridge since the Conquest_ +and _The History of Waltham Abbey_. These works were furthered in no +slight degree by his connexion with Sion College, London, where he had a +chamber, as well for the convenience of the press as of his city +lectureships. The _Church-History_ was angrily attacked by Dr P. Heylyn, +who, in the spirit of High-Churchmanship, wished, as he said, to +vindicate the truth, the church and the injured clergy. About 1652 +Fuller married his second wife, Mary Roper, youngest sister of Thomas, +Viscount Baltinglass, by whom he had several children. At the Oxford Act +of 1657, Robert South, who was _Terrae filius_, lampooned Fuller, whom +he described in this _Oratio_ as living in London, ever scribbling and +each year bringing forth new _folia_ like a tree. At length, continues +South, the _Church-History_ came forth with its 166 dedications to +wealthy and noble friends; and with this huge volume under one arm, and +his wife (said to be little of stature) on the other, he ran up and +down the streets of London, seeking at the houses of his patrons +invitations to dinner, to be repaid by his dull jests at table. + +His last and best patron was George Berkeley, 1st Earl Berkeley +(1628-1698), of Cranford House, Middlesex, whose chaplain he was, and +who gave him Cranford rectory (1658). To this nobleman Fuller's reply to +Heylyn's _Examen Historicum_, called _The Appeal of Injured Innocence_ +(1659), was inscribed. At the end of the _Appeal_ is an epistle "to my +loving friend Dr Peter Heylyn," conceived in the admirable Christian +spirit which characterized all Fuller's dealings with controversialists. +"Why should _Peter_," he asked, "fall out with _Thomas_, both being +disciples to the same Lord and Master? I assure you, sir, whatever you +conceive to the contrary, I am cordial to the cause of the English +Church, and my hoary hairs will go down to the grave in sorrow for her +sufferings." + +In _An Alarum to the Counties of England and Wales_ (1660) Fuller argued +for a free and full parliament--free from force, as he expressed it, as +well as from abjurations or previous engagements. _Mixt Contemplations +in Better Times_ (1660), dedicated to Lady Monk, tendered advice in the +spirit of its motto, "Let your moderation be known to all men: the Lord +is at hand." There is good reason to suppose that Fuller was at the +Hague immediately before the Restoration, in the retinue of Lord +Berkeley, one of the commissioners of the House of Lords, whose last +service to his friend was to interest himself in obtaining him a +bishopric. _A Panegyrick to His Majesty on his Happy Return_ was the +last of Fuller's verse-efforts. On the 2nd of August, by royal letters, +he was admitted D.D. at Cambridge. He resumed his lectures at the Savoy, +where Samuel Pepys heard him preach; but he preferred his conversation +or his books to his sermons. Fuller's last promotion was that of +chaplain in extraordinary to Charles II. In the summer of 1661 he +visited the west in connexion with the business of his prebend, which +had been restored to him. On Sunday, the 12th of August, while preaching +at the Savoy, he was seized with typhus fever, and died at his new +lodgings in Covent Garden on the 16th of August. He was buried in +Cranford church, where a mural tablet was afterwards set up on the north +side of the chancel, with an epitaph which contains a conceit worthy of +his own pen, to the effect that while he was endeavouring (viz. in _The +Worthies_) to give immortality to others, he himself attained it. + +Fuller's wit and vivacious good-humour made him a favourite with men of +both sides, and his sense of humour kept him from extremes. Probably +Heylyn and South had some excuse for their attitude towards his very +moderate politics. "By his particular temper and management," said +Echard (_Hist. of England_, iii. 71), "he weathered the late great storm +with more success than many other great men." He was known as "a perfect +walking library." The strength of his memory was proverbial, and some +amusing anecdotes are connected with it. + +His writings were the product of a highly original mind. He had a +fertile imagination and a happy faculty of illustration. Antithetic and +axiomatic sentences abound in his pages, embodying literally the wisdom +of the many in the wit of one. He was "quaint," and something more. +"Wit," said Coleridge, in a well-known eulogy, "was the stuff and +substance of Fuller's intellect. It was the element, the earthen base, +the material which he worked in; and this very circumstance has +defrauded him of his due praise for the practical wisdom of the +thoughts, for the beauty and variety of the truths, into which he shaped +the stuff. Fuller was incomparably the most sensible, the least +prejudiced, great man of an age that boasted a galaxy of great men" +(_Literary Remains_, vol. ii. (1836), pp. 389-390). This opinion was +formed after the perusal of the _Church-History_. That work and _The +History of the Worthies of England_ are unquestionably Fuller's greatest +efforts. They embody the collections of an entire life; and since his +day they have been the delight of many readers. The _Holy State_ has +taken rank amongst the best books of "characters." Charles Lamb made +some selections from Fuller, and had a profound admiration for the +"golden works" of the "dear, fine, silly old angel." Since Lamb's time, +mainly through the appreciative criticisms of S.T. Coleridge, Robert +Southey and others, Fuller's works have received much attention. + + There is an elaborate account of the life and writings of Fuller by + William Oldys in the _Biographia Britannica_, vol. iii. (1750), based + on Fuller's own works and the anonymous _Life of ... Dr Thomas Fuller_ + (1661; reprinted in a volume of selections by A.L.J. Gosset, 1893). + The completest account of him is _The Life of Thomas Fuller, with + Notices of his Books, his Kinsmen and his Friends_ (1874), by J.E. + Bailey, who gives a detailed bibliography (pp. 713-762) of his works. + _The Worthies of England_ was reprinted by John Nichols (1811) and by + P.A. Nuttall (1840). His _Collected Sermons_ were edited by J.E. + Bailey and W.E.A. Axon in 1891. Fuller's quaint wit lends itself to + selection, and there are several modern volumes of extracts from his + works. + + + + +FULLER, WILLIAM (1670-c. 1717), English impostor, was born at Milton in +Kent on the 20th of September 1670. His paternity is doubtful, but he +was related to the family of Herbert. After 1688 he served James II.'s +queen, Mary of Modena, and the Jacobites, seeking at the same time to +gain favour with William III.; and after associating with Titus Oates, +being imprisoned for debt and pretending to reveal Jacobite plots, the +House of Commons in 1692 declared he was an "imposter, cheat and false +accuser." Having stood in the pillory he was again imprisoned until +1695, when he was released; and at this time he took the opportunity to +revive the old and familiar story that Mary of Modena was not the mother +of the prince of Wales. In 1701 he published his autobiographical _Life +of William Fuller_ and some _Original Letters of the late King James_. +Unable to prove the assertions made in his writings he was put in the +pillory, whipped and fined. He died, probably in prison, about 1717. +Fuller's other writings are _Mr William Fuller's trip to Bridewell, with +a full account of his barbarous usage in the pillory; The sincere and +hearty confession of Mr William Fuller_ (1704); and _An humble appeal to +the impartial judgment of all parties in Great Britain_ (1716). + + He must be distinguished from WILLIAM FULLER (1608-1675), dean of St + Patrick's (1660), bishop of Limerick (1663), and bishop of Lincoln + (1667), the friend of Samuel Pepys; and also from William Fuller (c. + 1580-1659), dean of Ely and later dean of Durham. + + + + +FULLER'S EARTH (Ger. _Walkererde_, Fr. _terre a foulon_, _argile +smectique_)--so named from its use by fullers as an absorbent of the +grease and oil of cloth,--a clay-like substance, which from its +variability is somewhat difficult to define. In colour it is most often +greenish, olive-green or greenish-grey; on weathering it changes to a +brown tint or it may bleach. As a rule it falls to pieces when placed in +water and is not markedly plastic; when dry it adheres strongly to the +tongue; since, however, these properties are possessed by many clays +that do not exhibit detergent qualities, the only test of value lies in +the capacity to absorb grease or clarify oil. Fuller's earth has a +specific gravity of 1.7-2.4, and a shining streak; it is usually +unctuous to the touch. Microscopically, it consists of minute +irregular-shaped particles of a mineral that appears to be the result of +a chloritic or talcose alteration of a felspar. The small size of most +of the grains, less than .07 mm., makes their determination almost +impossible. Chemical analysis shows that the peculiar properties of this +earth are due to its physical rather than its chemical nature. + + The following analyses of the weathered and unweathered condition of + the earth from Nutfield, Surrey, represent the composition of one of + the best known varieties:-- + + Blue Earth (dried at 100 deg. C.). + + Insoluble residue 69.96 | Insoluble residue-- + Fe2O3 2.48 | SiO2 62.81 + Al2O3 3.46 | Al2O3 3.46 + CaO 5.87 | Fe2O3 1.30 + MgO 1.41 | CaO 1.53 + P2O5 0.27 | MgO 0.86 + SO3 0.05 | ----- + NaCl 0.05 | 69.96 + K2O 0.74 | ===== + H2O (combined) 15.57 | + ----- | + 99.86 | + ===== | + + + Yellow Earth (dried at 100 deg. C.). + + Insoluble residue 76.13 | Insoluble residue-- + Fe2O3 2.41 | SiO2 59.37 + Al2O3 1.77 | Al2O3 10.05 + CaO 4.31 | Fe2O3 3.86 + MgO 1.05 | CaO 1.86 + P2O5 0.14 | MgO 1.04 + SO3 0.07 | ----- + NaCl 0.14 | 76.18 + K2O 0.84 | ===== + H2O (combined) 13.19 | + ------ | + 100.05 | + ====== | + + (Analysis by P.G. Sanford, _Geol. Mag._, 1889, 6, pp. 456, 526.) + + Of other published analyses, not a few show a lower silica content + (44%, 50%), along with a higher proportion of alumina (11%, 23%). + +Fuller's earth may occur on any geological horizon; at Nutfield in +Surrey, England, it is in the Cretaceous formations; at Midford near +Bath it is of Jurassic age; at Bala, North Wales, it occurs in +Ordovician strata; in Saxony it appears to be the decomposition product +of a diabasic rock. In America it is found in California in rocks +ranging from Cretaceous to Pleistocene age; in S. Dakota, Custer county +and elsewhere a yellow, gritty earth of Jurassic age is worked; in +Florida and Georgia occurs a brittle, whitish earth of Oligocene age. +Other deposits are worked in Arkansas, Texas, Colorado, Massachusetts +and South Carolina. + +Fuller's earth is either mined or dug in the open according to local +circumstances. It is then dried in the sun or by artificial heat and +transported in small lumps in sacks. In other cases it is ground to a +fine powder after being dried; or it is first roughly ground and made +into a slurry with water, which is allowed to carry off the finer from +the coarser particles and deposit them in a creamy state in suitable +tanks. After consolidation this fine material is dried artificially on +drying floors, broken into lumps, and packed for transport. The use of +fuller's earth for cleansing wool and cloth has greatly decreased, but +the demand for the material is as great or greater than it ever was. It +is now used very largely in the filtration of mineral oils, and also for +decolourizing certain vegetable oils. It is employed in the formation of +certain soaps and cleansing preparations. + +The term "Fuller's Earth" has a special significance in geology, for it +was applied by W. Smith in 1799 to certain clays in the neighbourhood of +Bath, and the use of the expression is still retained by English +geologists, either in this form or in the generalized "Fullonian." The +Fullonian lies at the base of the Great Oolite or Bathonian series, but +its palaeontological characters place it between that series and the +underlying Inferior Oolite. The zonal fossils are _Perisphinctes +arbustigerus_ and _Macrocephalus subcontractus_ with _Ostrea acuminata_, +_Rhynchonella concinna_ and _Goniomya angulifera_. The formation is in +part the equivalent of the "Vesulien" of J. Marcou (Vesoul in +Haute-Saone). In Dorsetshire and Somersetshire, where it is best +developed, it is represented by an Upper Fuller's Earth Clay, the +Fuller's Earth Rock (an impersistent earthy limestone, usually +fossiliferous), and the Lower Fuller's Earth Clay. Commercial fuller's +earth has been obtained only from the Upper Clay. In eastern +Gloucestershire and northern Oxfordshire the Fuller's Earth passes +downwards without break into the Inferior Oolite; northward it dies out +about Chipping Norton in Oxfordshire and passes laterally into the +Stonesfield Slates series; in the midland counties it may perhaps be +represented by the "Upper Estuarine Series." In parts of Dorsetshire the +clays have been used for brickmaking and the limestone (rock) for local +buildings. + + See H.B. Woodward, "Jurassic Rocks of Great Britain," vol. iv. (1894), + _Mem. Geol. Survey_ (London). [J. A. H.] + + + + +FULLERTON, LADY GEORGIANA CHARLOTTE (1812-1885), English novelist and +philanthropist, youngest daughter of the 1st Earl Granville, was born at +Tixall Hall in Staffordshire on the 23rd of September 1812. In 1833 she +married Alexander George Fullerton, then an Irish officer in the guards. +After living in Paris for some eight years she and her husband +accompanied Lord Granville to Cannes and thence to Rome. In 1843 her +husband entered the Roman Catholic church, and in the following year +Lady Georgiana Fullerton published her first novel, _Ellen Middleton_, +which attracted W.E. Gladstone's attention in the _English Review_. In +1846 she entered the Roman Catholic church. The death of her only son in +1854 plunged her in grief, and she continued to wear mourning until the +end of her life. In 1856 she became one of the third order of St +Francis, and thenceforward devoted herself to charitable work. In +conjunction with Miss Taylor she founded the religious community known +as "The Poor Servants of the Mother of God Incarnate," and she also took +an active part in bringing to England the sisters of St Vincent of Paul. +Her philanthropic work is described in Mrs Augustus Craven's work _Lady +Georgiana Fullerton, sa vie et ses [oe]uvres_ (Paris, 1888), which was +translated into English by Henry James Coleridge. She died at +Bournemouth on the 19th of January 1885. Among her other novels were +_Grantley Manor_ (1847), _Lady Bird_ (1852), and _Too Strange not to be +True_ (1864). + + + + +FULMAR, from the Gaelic _Fulmaire_, the _Fulmarus glacialis_ of modern +ornithologists, one of the largest of the petrels (_Procellariidae_) of +the northern hemisphere, being about the size of the common gull (_Larus +canus_) and not unlike it in general coloration, except that its +primaries are grey instead of black. This bird, which ranges over the +North Atlantic, is seldom seen on the European side below lat. 53 deg. +N., but on the American side comes habitually to lat. 45 deg. or even +lower. In the Pacific it is represented by a scarcely separable form, +_F. glupischa_. It has been commonly believed to have two +breeding-places in the British Islands, namely, St Kilda and South +Barra; but, according to Robert Gray (_Birds of the West of Scotland_, +p. 499), it has abandoned the latter since 1844, though still breeding +in Skye. Northward it established itself about 1838 on Myggenaes Holm, +one of the Faeroes, while it has several stations off the coast of +Iceland and Spitsbergen, as well as at Bear Island. Its range towards +the pole seems to be only bounded by open water, and it is the constant +attendant upon all who are employed in the whale and seal fisheries, +showing the greatest boldness in approaching boats and ships, and +feeding on the offal obtained from them. By British seamen it is +commonly called the "molly mawk"[1] (corrupted from _Mallemuck_), and is +extremely well known to them, its flight, as it skims over the waves, +first with a few beats of the wings and then gliding for a long way, +being very peculiar. It only visits the land to deposit its single white +egg, which is laid on a rocky ledge, where a shallow nest is made in the +turf and lined with a little dried grass. Many of its breeding-places +are a most valuable property to those who live near them and take the +eggs and young, which, from the nature of the locality, are only to be +had at a hazardous risk of life. In St Kilda a large number of the young +are killed in one week of August, the only time when, by the custom of +the community, they are allowed to be taken. These, after the oil is +extracted from them, serve the islanders with food for the winter. The +oil has been chemically analysed and found to be a fish-oil, and to +possess nearly all the qualities of that obtained from the liver of the +cod, with a lighter specific gravity. It, however, has an extremely +strong scent, which is said by those who have visited St Kilda to +pervade every thing and person on the island, and is certainly retained +by an egg or skin of the bird for many years. Whenever a live example is +seized in the hand it ejects a considerable quantity of this oil from +its mouth. + + +FOOTNOTE: + + [1] A name misapplied in the southern hemisphere to _Diomedea + melanophrys_, one of the albatrosses. + + + + +FULMINIC ACID, HCNO or H2C2N2O2, an organic acid isomeric with cyanic +and cyanuric acids; its salts, termed fulminates, are very explosive and +are much employed as detonators. The free acid, which is obtained by +treating the salts with acids, is an oily liquid smelling like prussic +acid; it is very explosive, and the vapour is poisonous to about the +same degree as that of prussic acid. The first fulminate prepared was +the "fulminating silver" of L.G. Brugnatelli, who found in 1798 that if +silver be dissolved in nitric acid and the solution added to spirits of +wine, a white, highly explosive powder was obtained. This substance is +to be distinguished from the black "fulminating silver" obtained by +C.L. Berthollet in 1788 by acting with ammonia on precipitated silver +oxide. The next salt to be obtained was the mercuric salt, which was +prepared in 1799 by Edward Charles Howard, who substituted mercury for +silver in Brugnatelli's process. A similar method is that of J. von +Liebig (1823), who heated a mixture of alcohol, nitric acid and mercuric +nitrate; the salt is largely manufactured by processes closely +resembling the last. A laboratory method is to mix solutions of sodium +nitromethane, CH2:NO(ONa), and mercuric chloride, a yellow basic salt +being formed at the same time. Mercuric fulminate is less explosive than +the silver salt, and forms white needles (with 1/2H2O) which are +tolerably soluble in water. The use of mercuric fulminate as a detonator +dates from about 1814, when the explosive cap was invented. It is still +the commonest detonator, but it is now usually mixed with other +substances; the British service uses for percussion caps 6 parts of +fulminate, 6 of potassium chlorate and 4 of antimony sulphide, and for +time fuses 4 parts of fulminate, 6 of potassium chlorate and 4 of +antimony sulphide, the mixture being damped with a shellac varnish; for +use in blasting, a home office order of 1897 prescribes a mixture of 4 +parts of fulminate and 1 of potassium chlorate. In 1900 Bielefeldt found +that a fulminate placed on top of an aromatic nitro compound, such as +trinitrotoluene, formed a useful detonator; this discovery has been +especially taken advantage of in Germany, in which country detonators of +this nature are being largely employed. Tetranitromethylaniline (tetryl) +has also been employed (Brit. Pat. 13340 of 1905). It has been proposed +to replace fulminate by silver azoimide (Wohler & Matter, Brit. Pat. +4468 of 1908), and by lead azoimide (Hyronimus, Brit. Pat. 1819 of +1908). + + The constitution of fulminic acid has been investigated by many + experimenters, but apparently without definitive results. The + researches of Liebig (1823), Liebig and Gay-Lussac (1824), and of + Liebig again in 1838 showed the acid to be isomeric with cyanic acid, + and probably (HCNO)2, since it gave mixed and acid salts. Kekule, in + 1858, concluded that it was nitroacetonitrile, NO2.CH2.CN, a view + opposed by Steiner (1883), E. Divers and M. Kawakita (1884), R. Scholl + (1890), and by J.U. Nef (1894), who proposed the formulae: + + C : N.OH / N : CH CH : N.O + .. O / . . C : N.OH. + C : N.OH, \ N : C.OH, CH : N.0, + + Steiner, Divers, Scholl, Nef. + + The formulae of Kekule, Divers and Armstrong have been discarded, and + it remains to be shown whether Nef's carbonyloxime formula (or the + bimolecular formula of Steiner) or Scholl's glyoxime peroxide formula + is correct. There is some doubt as to the molecular formula of + fulminic acid. The existence of double salts, and the observations of + L. Wohler and K. Theodorovits (_Ber._, 1905, 38, p. 345), that only + compounds containing two carbon atoms yielded fulminates, points to + (HCNO)2; on the other hand, Wohler (_loc. cit._ p. 1351) found that + cryoscopic and electric conductivity measurements showed sodium + fulminate to be NaCNO. Nef based his formula, which involves bivalent + carbon, on many reactions; in particular, that silver fulminate with + hydrochloric acid gave salts of formylchloridoxime, which with water + gave hydroxylamine and formic acid, thus + + // NOAg // N.OH + C : NO OAg -> HC // --> HC // --> H.CO2H + H2N.OH, + \ Cl \ OH + + and also on the production from sodium nitromethane and mercuric + chloride, thus CH2 : NO.Ohg --> H2O + C : NOhg(hg = 1/2Hg). H. Wieland + and F.C. Palazzo (1907) support this formula, finding that methyl + nitrolic acid, NO2.CH : N.OH, yielded under certain conditions + fulminic acid, and vice versa (Palazzo, 1907). M.Z. Jowitschitsch + (_Ann._, 1906, 347, p. 233) inclines to Scholl's formula; he found + that the synthetic silver salt of glyoxime peroxide resembled silver + fulminate in yielding hydroxylamine with hydrochloric acid, but + differed in being less explosive, and in being soluble in nitric acid. + H. Wieland and his collaborators regard "glyoxime peroxide" as an + oxide of furazane (q.v.), and have shown that a close relationship + exists between the nitrile oxides, furoxane, and fulminic acid (see + Ann. Rep., London Chem. Soc., 1909, p. 84). _Fulminuric acid_, + (HCNO)3, obtained by Liebig by boiling mercuric fulminate with water, + was synthesized in 1905 by C. Ulpiani and L. Bernardini (_Gazetta_, + iii. 35, p. 7), who regard it as NO2.CH(CN).CO.NH2. It deflagrates at + 145 deg., and forms a characteristic cuprammonium salt. + + The early history of mercuric fulminate and a critical account of its + application as a detonator is given in _The Rise and Progress of the + British Explosives Industry_ (International Congress of Applied + Chemistry, 1909). The manufacture and modern aspects are treated in + Oscar Guttmann, _The Manufacture of Explosives_, and _Manufacture of + Explosives, Twenty Years' Progress_ (1909). + + + + +FULTON, ROBERT (1765-1815), American engineer, was born in 1765 in +Little Britain (now Fulton, Lancaster county), Pa. His parents were +Irish, and so poor that they could afford him only a very scanty +education. At an early age he was bound apprentice to a jeweller in +Philadelphia, but subsequently adopted portrait and landscape painting +as his profession. In his twenty-second year, with the object of +studying with his countryman, Benjamin West, he went to England, and +there became acquainted with the duke of Bridgewater, Earl Stanhope and +James Watt. Partly by their influence he was led to devote his attention +to engineering, especially in connexion with canal construction; he +obtained an English patent in 1794 for superseding canal locks by +inclined planes, and in 1796 he published a _Treatise on the Improvement +of Canal Navigation_. He then took up his residence in Paris, where he +projected the first panorama ever exhibited in that city, and +constructed a submarine boat, the "Nautilus," which was tried in Brest +harbour in 1801 before a commission appointed by Napoleon I., and by the +aid of which he was enabled to blow up a small vessel with a torpedo. It +was at Paris also in 1803 that he first succeeded in propelling a boat +by steam-power, thus realizing a design which he had conceived ten years +previously. Returning to America he continued his experiments with +submarine explosives, but failed to convince either the English, French +or United States governments of the adequacy of his methods. With steam +navigation he had more success. In association with Robert R. Livingston +(q.v.), who in 1798 had been granted the exclusive right to navigate the +waters of New York state with steam-vessels, he constructed the +"Clermont," which, engined by Boulton & Watt of Birmingham, began to ply +on the Hudson between New York and Albany in 1807. The privilege +obtained by Livingston in 1798 was granted jointly to Fulton and +Livingston in 1803, and by an act passed in 1808 the monopoly was +secured to them and their associates for a period depending on the +number of steamers constructed, but limited to a maximum of thirty +years. In 1814-1815, on behalf of the United States government, he +constructed the "Fulton," a vessel of 38 tons with central +paddle-wheels, which was the first steam warship. He died at New York on +the 24th of February 1815. Among Fulton's inventions were machines for +spinning flax, for making ropes, and for sawing and polishing marble. + + See C.D. Colden, _Life of Robert Fulton_ (New York, 1817); Robert H. + Thurston, _History of the Growth of the Steam-Engine_ (New York, + 1878); George H. Preble, _Chronological History of Steam Navigation_ + (Philadelphia, 1883); and Mrs A.C. Sutcliffe, _Robert Fulton and the + Clermont_ (New York, 1909). + + + + +FULTON, a city and the county-seat of Callaway county, Missouri, U.S.A., +25 m. N.E. of Jefferson City. Pop. (1890) 4314; (1900) 4883 (1167 +negroes); (1910) 5228. It is served by the Chicago & Alton railway. The +city has an important stock market and manufactures fire-brick and +pottery. At Fulton are the Westminster College (Presbyterian, founded in +1853), the Synodical College for Young Women (Pres., founded in 1871), +the William Woods College for Girls (Christian Church, 1890), and the +Missouri school for the deaf (1851). Here, too, is a state hospital for +the insane (1847), the first institution of the kind in Missouri. The +place was laid out as a town in 1825 and named Volney, but in honour of +Robert Fulton the present name was adopted a little later. Fulton was +incorporated in 1859. + + + + +FULTON, a city of Oswego county, New York, U.S.A., on the right bank of +the Oswego river, about 10 m. S. by E. of Oswego. Pop. (1900) 5281; +(1905, state census) 8847; (1910) 10,480. Fulton is served by the +Delaware, Lackawanna & Western, the New York Central & Hudson River, and +the New York, Ontario & Western railways, by electric railway to Oswego +and Syracuse and by the Oswego Canal. The city has a Carnegie library. +Ample water-power is furnished by the Oswego river, which here flows in +a series of rapids, and the manufactures are many in kind. On the 3rd of +July 1756, on an island (afterward called Battle Island) 4 m. N. of the +present city of Fulton, a British force of about 300 under Captain John +Bradstreet (1711-1774) defeated an attacking force of French and Indians +(numbering about 700) under De Villiers. Soon after this, Bradstreet +built a fort within the present limits of Fulton. The first civilian +settler came in 1793, and the first survey (which included only a part +of the subsequent village) was made in 1815. Fulton was incorporated as +a village in 1835, and in April 1902 was combined with the village of +Oswego Falls (pop. in 1900, 2925) and was chartered as a city. + + + + +FUM, or FUNJ HWANG, one of the four symbolical creatures which in +Chinese mythology are believed to keep watch and ward over the Celestial +Empire. It was begotten by fire, was born in the Hill of the Sun's Halo, +and its body bears inscribed on it the five cardinal virtues. It has the +breast of a goose, the hindquarters of a stag, a snake's neck, a fish's +tail, a fowl's forehead, a duck's down, the marks of a dragon, the back +of a tortoise, the face of a swallow, the beak of a cock, is about six +cubits high, and perches only on the woo-tung tree. The appearance of +Fum heralds an age of universal virtue. Its figure is that which is +embroidered on the dresses of some mandarins. + + + + +FUMARIC AND MALEIC ACIDS, two isomeric unsaturated acids of composition +C4H4O4. _Fumaric acid_ is found in fumitory (_Fumaria officinalis_), in +various fungi (_Agaricus piperatus_, &c.), and in Iceland moss. It is +obtained by heating malic acid alone to 150 deg. C., or by heating it +with hydrochloric acid (V. Dessaignes, _Jahresb_., 1856, p. 463) or with +a large quantity of hydrobromic acids (A. Kekule, _Ann._, 1864, 130, p. +21). It may also be obtained by boiling monobromsuccinic acid with +water; by the action of dichloracetic acid and water on silver malonate +(T. Komnenos, _Ann._, 1883, 218, p. 169); by the cyanide synthesis from +acetylene di-iodide; and by heating maleic acid to 210 deg. C. (Z. +Skraup, _Monats. f. Chemie_, 1891, 12, p. 112). It crystallizes in small +prisms or needles, and is practically insoluble in cold water. It +sublimes to some extent at about 200 deg. C., being partially converted +into maleic anhydride and water, the reaction becoming practically +quantitative if dehydrating agents be used. Reducing agents (zinc and +caustic alkali, hydriodic acid, sodium amalgam, &c.) convert it into +succinic acid. Bromine converts it into dibromsuccinic acid. Potassium +permanganate oxidizes it to racemic acid (A. Kekule and R. Anschutz, +_Ber._, 1881, 14, p. 713). By long-continued heating with caustic soda +at 100 deg. C. it is converted into inactive malic acid. + +_Maleic acid_ is obtained by distilling malic or fumaric acids; by +heating fumaric acid with acetyl chloride to 100 deg. C; or by the +hydrolysis of trichlorphenomalic acid ([beta]-trichloraceto-acrylic +acid) [A. Kekule, _Ann._, 1884, 223, p. 185]. It crystallizes in +monoclinic prisms, which are easily soluble in water, melt at 130 deg. +C., and boil at 160 deg. C., decomposing into water and maleic +anhydride. When heated with concentrated hydrobromic or hydriodic acids, +it is converted into fumaric acid. It yields an anilide; oxidation +converts it into mesotartaric acid. Maleic anhydride is obtained by +distilling fumaric acid with phosphorus pentoxide. It forms triclinic +crystals which melt at 60 deg. C. and boil at 196 deg. C. + + Both acids are readily esterified by the action of alkyl halides on + their silver salts, and the maleic ester is readily transformed into + the fumaric ester by warming with iodine, the same result being + obtained by esterification of maleic acid in alcoholic solution by + means of hydrochloric acid. Both acids yield acetylene by the + electrolysis of aqueous solutions of their alkali salts, and on + reduction both yield succinic acid, whilst by the addition of + hydrobromic acid they both yield monobromsuccinic acid (R. Fittig, + _Ann._, 1877, 188, p. 98). From these results it follows that the two + acids are structurally identical, and the isomerism has consequently + to be explained on other grounds. This was accomplished by W. + Wislicenus ["Uber die raumliche Anordnung der Atome," &c., _Trans, of + the Saxon Acad. of Sciences_ (Math. Phys. Section), 1887, p. 14] by an + extension of the van't Hoff hypothesis (see STEREO-ISOMERISM). The + formulae of the acids are written thus: + + HC.CO2H HC.CO2H + .. Maleic acid. .. Fumaric acid. + HC.CO2H HO2C.C.H + + These account for maleic acid readily yielding an anhydride, whereas + fumaric acid does not, and for the behaviour of the acids towards + bromine, fumaric acid yielding ordinary dibromsuccinic acid, and + maleic acid the isomeric isodibromsuccinic acid. + + + + +FUMAROLE, a vent from which volcanic vapours issue, named indirectly +from the Lat. _fumariolum_, a smoke-hole. The vapours from fumaroles +were studied first by R.W. Bunsen, on his visit to Iceland, and +afterwards by H. Sainte-Claire Deville and other chemists and geologists +in France, who examined the vapours from Santorin, Etna, &c. The hottest +vapours issue from dry fumaroles, at temperatures of at least 500 deg. +C., and consist chiefly of anhydrous chlorides, notably sodium chloride. +The acid fumaroles yield vapours of lower temperature (300 deg. to 400 +deg.) containing much water vapour, with hydrogen chloride and sulphur +dioxide. The alkaline fumaroles are still cooler, though above 100 deg., +and evolve ammonium chloride with other vapours. Cold fumaroles, below +100 deg., discharge principally aqueous vapour, with carbon dioxide, and +perhaps hydrogen sulphide. The fumaroles of Mont Pele in Martinique +during the eruption of 1902 were examined by A. Lacroix, and the vapours +analysed by H. Moissan, who found that they consisted chiefly of water +vapour, with hydrogen chloride, sulphur, carbon dioxide, carbon +monoxide, methane, hydrogen, nitrogen, oxygen and argon. These vapours +issued at a temperature of about 400 deg. Armand Gautier has pointed out +that these gases are practically of the same composition as those which +he obtained on heating granite and certain other rocks. (See VOLCANO). + + + + +FUMIGATION (from Lat. _fumigare_, to smoke), the process of producing +smoke or fumes, as by burning sulphur, frankincense, tobacco, &c., +whether as a ceremony of incantation, or for perfuming a room, or for +purposes of disinfection or destruction of vermin. In medicine the term +has been used of the exposure of the body, or a portion of it, to fumes +such as those of nitre, sal-ammoniac, mercury, &c.; fumigation, by the +injection of tobacco smoke into the great bowel, was a recognized +procedure in the 18th century for the resuscitation of the apparently +drowned. "Fumigated" or "fumed" oak is oak which has been darkened by +exposure to ammonia vapour. + + + + +FUMITORY, in botany, the popular name for the British species of +_Fumaria_, a genus of small, branched, often climbing annual herbs with +much-divided leaves and racemes of small flowers. The flowers are +tubular with a spurred base, and in the British species are pink to +purplish in colour. They are weeds of cultivation growing in fields and +waste places. _F. capreolata_ climbs by means of twisting petioles. In +past times fumitory was in esteem for its reputed cholagogue and other +medicinal properties; and in England, boiled in water, milk or whey, it +was used as a cosmetic. The root of the allied species (_Corydalis cava_ +or _tuberosa_) is known as _radix aristolochia_, and has been used +medicinally for various cutaneous and other disorders, in doses of 10 to +30 grains. Some eleven alkaloids have been isolated from it. The herbage +of _Fumaria officinalis_ and _F. racemosa_ is used in China under the +name of _Tsze-hwa-ti-ting_ as an application for glandular swellings, +carbuncles and abscesses, and was formerly valued in jaundice, and in +cases of accidental swallowing of the beard of grain (see F. Porter +Smith, _Contrib. towards the Mat. Medica ... of China_, p. 99, 1871). +The name fumitory, Latin _fumus terrae_, has been supposed to be derived +from the fact that its juice irritates the eyes like smoke (see Fuchs, +_De historia stirpium_, p. 338, 1542); but _The Grete Herball_, cap. +clxix., 1529, fol., following the _De simplici medicina_ of Platearius, +fo. xciii. (see in _Nicolai Praepositi dispensatorium ad aromatarios_, +1536), says: "It is called Fumus terre fume or smoke of the erthe +bycause it is engendred of a cours fumosyte rysynge frome the erthe in +grete quantyte lyke smoke: this grosse or cours fumosyte of the erthe +wyndeth and wryeth out: and by workynge of the ayre and sonne it turneth +into this herbe." + + + + +FUNCHAL, the capital of the Portuguese archipelago of the Madeiras; on +the south coast of Madeira, in 32 deg. 37' N. and 16 deg. 54' W. Pop. +(1900) 20,850. Funchal is the see of a bishop, in the archiepiscopal +province of Lisbon; it is also the administrative centre of the +archipelago, and the residence of the governor and foreign consuls. The +city has an attractive appearance from the sea. Its whitewashed houses, +in their gardens full of tropical plants, are built along the curving +shore of Funchal Bay, and on the lower slopes of an amphitheatre of +mountains, which form a background 4000 ft. high. Numerous country +houses (_quintas_), with terraced gardens, vineyards and sugar-cane +plantations occupy the surrounding heights. Three mountain streams +traverse the city through deep channels, which in summer are dry, owing +to the diversion of the water for irrigation. A small fort, on an +isolated rock off shore, guards the entrance to the bay, and a larger +and more powerfully armed fort crowns an eminence inland. The chief +buildings include the cathedral, Anglican and Presbyterian churches, +hospitals, opera-house, museum and casino. There are small public +gardens and a meteorological observatory. In the steep and narrow +streets, which are lighted by electricity, wheeled traffic is +impossible; sledges drawn by oxen, and other primitive conveyances are +used instead (see MADEIRA). In winter the fine climate and scenery +attract numerous invalids and other visitors, for whose accommodation +there are good hotels; many foreigners engaged in the coal and wine +trades also reside here permanently. The majority of these belong to the +British community, which was first established here in the 18th century. +Funchal is the headquarters of Madeiran industry and commerce (see +MADEIRA). It has no docks and no facilities for landing passengers or +goods; vessels are obliged to anchor in the roadstead, which, however, +is sheltered from every wind except the south. Funchal is connected by +cable with Carcavellos (for Lisbon), Porthcurnow (for Falmouth, England) +and St Vincent in the Cape Verde Islands (for Pernambuco, Brazil). + + + + +FUNCTION,[1] in mathematics, a variable number the value of which +depends upon the values of one or more other variable numbers. The +theory of functions is conveniently divided into (I.) Functions of Real +Variables, wherein real, and only real, numbers are involved, and (II.) +Functions of Complex Variables, wherein complex or imaginary numbers are +involved. + + +I. FUNCTIONS OF REAL VARIABLES + +1. _Historical._--The word function, defined in the above sense, was +introduced by Leibnitz in a short note of date 1694 concerning the +construction of what we now call an "envelope" (_Leibnizens +mathematische Schriften_, edited by C.I. Gerhardt, Bd. v. p. 306), and +was there used to denote a variable length related in a defined way to a +variable point of a curve. In 1698 James Bernoulli used the word in a +special sense in connexion with some isoperimetric problems (Joh. +Bernoulli, _Opera_, t. i. p. 255). He said that when it is a question of +selecting from an infinite set of like curves that one which best +fulfils some function, then of two curves whose intersection determines +the thing sought one is always the "line of the function" (_Linea +functionis_). In 1718 John Bernoulli (_Opera_, t. ii. p. 241) defined a +"function of a variable magnitude" as a quantity made up in any way of +this variable magnitude and constants; and in 1730 (Opera, t. iii. p. +174) he noted a distinction between "algebraic" and "transcendental" +functions. By the latter he meant integrals of algebraic functions. The +notation [f](x) for a function of a variable x was introduced by +Leonhard Euler in 1734 (_Comm. Acad. Petropol._ t. vii. p. 186), in +connexion with the theorem of the interchange of the order of +differentiations. The notion of functionality or functional relation of +two magnitudes was thus of geometrical origin; but a function soon came +to be regarded as an analytical expression, not necessarily an algebraic +expression, containing the variable or variables. Thus we may have +rational integral algebraic functions such as _ax_^2 + _bx_ + c, or +rational algebraic functions which are not integral, such as + + a1x^n + a2x^(n - 1) + ... + a_n + -------------------------------, + b1x^m + b2x^(m - 1) + ... + b_m + +or irrational algebraic functions, such as [root]x, or, more generally +the algebraic functions that are determined implicitly by an algebraic +equation, as, for instance, + + [f]_n(x, y) + [f]_(n - 1) (x, y) + ... + [f]0 = 0 + +where [f]_n(x, y), ... mean homogeneous expressions in x and y having +constant coefficients, and having the degrees indicated by the suffixes, +and [f]0 is a constant. Or again we may have trigonometrical functions, +such as sin x and tan x, or inverse trigonometrical functions, such as +sin^(-1)x, or exponential functions, such as e^x and a^x, or logarithmic +functions, such as log x and log (1 + x). We may have these functional +symbols combined in various ways, and thus there arises a great number +of functions. Further we may have functions of more than one variable, +as, for instance, the expression xy/(x^2 + y^2), in which both x and y +are regarded as variable. Such functions were introduced into analysis +somewhat unsystematically as the need for them arose, and the later +developments of analysis led to the introduction of other classes of +functions. + +2. _Graphic Representation._--In the case of a function of one variable +x, any value of x and the corresponding value y of the function can be +the co-ordinates of a point in a plane. To any value of x there +corresponds a point N on the axis of x, in accordance with the rule that +x is the abscissa of N. The corresponding value of y determines a point +P in accordance with the rule that x is the abscissa and y the ordinate +of P. The ordinate y gives the value of the function which corresponds +to that value of the variable x which is specified by N; and it may be +described as "the value of the function at N." Since there is a +one-to-one correspondence of the points N and the numbers x, we may also +describe the ordinate as "the value of the function at x." In simple +cases the aggregate of the points P which are determined by any +particular function (of one variable) is a curve, called the "graph of +the function" (see S 14). In like manner a function of two variables +defines a surface. + +3. _The Variable._--Graphic methods of representation, such as those +just described, enabled mathematicians to deal with irrational values of +functions and variables at the time when there was no theory of +irrational numbers other than Euclid's theory of incommensurables. In +that theory an irrational number was the ratio of two incommensurable +geometric magnitudes. In the modern theory of number irrational numbers +are defined in a purely arithmetical manner, independent of the +measurement of any quantities or magnitudes, whether geometric or of any +other kind. The definition is effected by means of the system of +_ordinal_ numbers (see NUMBER). When this formal system is established, +the theory of measurement may be founded upon it; and, in particular, +the co-ordinates of a point are defined as numbers (not lengths), which +are assigned in accordance with a rule. This rule involves the +measurement of lengths. The theory of functions can be developed without +any reference to graphs, or co-ordinates or lengths. The process by +which analysis has been freed from any consideration of measurable +quantities has been called the "arithmetization of analysis." In the +theory so developed, the variable upon which a function depends is +always to be regarded as a number, and the corresponding value of the +function is also a number. Any reference to points or co-ordinates is to +be regarded as a picturesque mode of expression, pointing to a possible +application of the theory to geometry. The development of "arithmetized +analysis" in the 19th century is associated with the name of Karl +Weierstrass. + +All possible values of a variable are numbers. In what follows we shall +confine our attention to the case where the numbers are real. When +complex numbers are introduced, instead of real ones, the theory of +functions receives a wide extension, which is accompanied by appropriate +limitations (see below, II. Functions of Complex Variables). The set of +all real numbers forms a _continuum_. In fact the notion of a +one-dimensional continuum first becomes precise in virtue of the +establishment of the system of real numbers. + +4. _Domain of a Variable.--Theory of Aggregates._--The notion of a +"variable" is that of a number to which we may assign at pleasure any +one of the values that belong to some chosen set, or _aggregate_, of +numbers; and this set, or aggregate, is called the "domain of the +variable." This domain may be an "interval," that is to say it may +consist of two terminal numbers, all the numbers between them and no +others. When this is the case the number is said to be "continuously +variable." When the domain consists of all real numbers, the variable is +said to be "unrestricted." A domain which consists of all the real +numbers which exceed some fixed number may be described as an "interval +unlimited towards the right"; similarly we may have an interval +"unlimited towards the left." + + In more complicated cases we must have some rule or process for + assigning the aggregate of numbers which constitute the domain of a + variable. The methods of definition of particular types of aggregates, + and the theorems relating to them, form a branch of analysis called + the "theory of aggregates" (_Mengenlehre, Theorie des ensembles, + Theory of sets of points_). The notion of an "aggregate" in general + underlies the system of ordinal numbers. An aggregate is said to be + "infinite" when it is possible to effect a one-to-one correspondence + of all its elements to some of its elements. For example, we may make + all the integers correspond to the even integers, by making 1 + correspond to 2, 2 to 4, and generally n to 2n. The aggregate of + positive integers is an infinite aggregate. The aggregates of all + rational numbers and of all real numbers and of points on a line are + other examples of infinite aggregates. An aggregate whose elements are + real numbers is said to "extend to infinite values" if, after any + number N, however great, is specified, it is possible to find in the + aggregate numbers which exceed N in absolute value. Such an aggregate + is always infinite. The "neighbourhood of a number (or point) a for a + positive number h" is the aggregate of all numbers (or points) x for + which the absolute value of x - a denoted by |x - a|, does not exceed + h. + +5. _General Notion of Functionality._--A function of one variable was +for a long time commonly regarded as the ordinate of a curve; and the +two notions (1) that which is determined by a curve supposed drawn, and +(2) that which is determined by an analytical expression supposed +written down, were not for a long time clearly distinguished. It was for +this reason that Fourier's discovery that a single analytical expression +is capable of representing (in different parts of an interval) what +would in his time have been called different functions so profoundly +struck mathematicians (S 23). The analysts who, in the middle of the +19th century, occupied themselves with the theory of the convergence of +Fourier's series were led to impose a restriction on the character of a +function in order that it should admit of such representation, and thus +the door was opened for the introduction of the general notion of +functional dependence. This notion may be expressed as follows: We have +a variable number, y, and another variable number, x, a domain of the +variable x, and a rule for assigning one or more definite values to y +when x is any point in the domain; then y is said to be a "function" of +the variable x, and x is called the "argument" of the function. +According to this notion a function is, as it were, an indefinitely +extended table, like a table of logarithms; to each point in the domain +of the argument there correspond values for the function, but it remains +arbitrary what values the function is to have at any such point. + + For the specification of any particular function two things are + requisite: (1) a statement of the values of the variable, or of the + aggregate of points, to which values of the function are to be made to + correspond, i.e. of the "domain of the argument"; (2) a rule for + assigning the value or values of the function that correspond to any + point in this domain. We may refer to the second of these two + essentials as "the rule of calculation." The relation of functions to + analytical expressions may then be stated in the form that the rule of + calculation is: "Give the function the value of the expression at any + point at which the expression has a determinate value," or again more + generally, "Give the function the value of the expression at all + points of a definite aggregate included in the domain of the + argument." The former of these is the rule of those among the earlier + analysts who regarded an analytical expression and a function as the + same thing, and their usage may be retained without causing confusion + and with the advantage of brevity, the analytical expression serving + to specify the domain of the argument as well as the rule of + calculation, e.g. we may speak of "the function 1/x." This function is + defined by the analytical expression 1/x at all points except the + point x = 0. But in complicated cases separate statements of the + domain of the argument and the rule of calculation cannot be dispensed + with. In general, when the rule of calculation is determined as above + by an analytical expression at any aggregate of points, the function + is said to be "represented" by the expression at those points. + + When the rule of calculation assigns a single definite value for a + function at each point in the domain of the argument the function is + "uniform" or "one-valued." In what follows it is to be understood that + all the functions considered are one-valued, and the values assigned + by the rule of calculation real. In the most important cases the + domain of the argument of a function of one variable is an interval, + with the possible exception of isolated points. + +6. _Limits._--Let [f](x) be a function of a variable number x; and let a +be a point such that there are points of the domain of the argument x in +the neighbourhood of a for any number h, however small. If there is a +number L which has the property that, after any positive number +[epsilon], however small, has been specified, it is possible to find a +positive number h, so that |L - [f](x)| < [epsilon] for all points x of +the domain (other than a) for which |x-a| < h, then L is the "limit of +[f](x) at the point a." The condition for the existence of L is that, +after the positive number [epsilon] has been specified, it must be +possible to find a positive number h, so that |[f](x') - [f](x)| < +[epsilon] for all points x and x' of the domain (other than a) for which +|x - a| < h and |x' - a| < h. + +It is a fundamental theorem that, when this condition is satisfied, +there exists a perfectly definite number L which is the limit of [f](x) +at the point a as defined above. The limit of [f](x) at the point a is +denoted by Lt_(x = a)[f](x), or by lim_(x = a)[f](x). + + If [f](x) is a function of one variable x in a domain which extends to + infinite values, and if, after [epsilon] has been specified, it is + possible to find a number N, so that |[f](x') - [f](x)| <[epsilon] for + all values of x and x' which are in the domain and exceed N, then + there is a number L which has the property that |[f](x) - L| < + [epsilon] for all such values of x. In this case [f](x) has a limit L + at x = [oo]. In like manner [f](x) may have a limit at x = -[oo]. This + statement includes the case where the domain of the argument consists + exclusively of positive integers. The values of the function then form + a "sequence," u1, u2, ... u_n, ..., and this sequence can have a limit + at n = [oo]. + + The principle common to the above definitions and theorems is called, + after P. du Bois Reymond, "the general principle of convergence to a + limit." + + It must be understood that the phrase "x = [oo]" does not mean that x + takes some particular value which is infinite. There is no such value. + The phrase always refers to a limiting process in which, as the + process is carried out, the variable number x increases without limit: + it may, as in the above example of a sequence, increase by taking + successively the values of all the integral numbers; in other cases it + may increase by taking the values that belong to any domain which + "extends to infinite values." + + A very important type of limits is furnished by _infinite series_. + When a sequence of numbers u1, u2, ... u_n, ... is given, we may form + a new sequence s1, s2, ... s_n, ... from it by the rules s1 = u1, s2 = + u1 + u2, ... s_n = u1 + u2 + ... + u_n or by the equivalent rules s1 = + u, s_n - s_(n - 1) = u_n(n = 2, 3, ...). If the new sequence has a + limit at n = [oo], this limit is called the "sum of the infinite + series" u1 + u2 + ..., and the series is said to be "convergent" (see + SERIES). + + A function which has not a limit at a point a may be such that, if a + certain aggregate of points is chosen out of the domain of the + argument, and the points x in the neighbourhood of a are restricted to + belong to this aggregate, then the function has a limit at a. For + example, sin(1/x) has limit zero at 0 if x is restricted to the + aggregate 1/[pi], 1/2[pi], ... 1/n[pi], ... or to the aggregate + 1/2[pi], 2/5[pi], ... n/(n^2 + 1)[pi], ..., but if x takes all values + in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again, + there may be a limit at a if the points x in the neighbourhood of a + are restricted by the condition that x - a is positive; then we have a + "limit on the right" at a; similarly we may have a "limit on the left" + at a point. Any such limit is described as a "limit for a restricted + domain." The limits on the left and on the right are denoted by [f](a + - 0) and [f](a + 0). + + The limit L of [f](x) at a stands in no necessary relation to the + value of [f](x) at a. If the point a is in the domain of the argument, + the value of [f](x) at a is assigned by the rule of calculation, and + may be different from L. In case [f](a) = L the limit is said to be + "attained." If the point a is not in the domain of the argument, there + is no value for [f](x) at a. In the case where [f](x) is defined for + all points in an interval containing a, except the point a, and has a + limit L at a, we may arbitrarily annex the point a to the domain of + the argument and assign to [f](a) the value L; the function may then + be said to be "extrinsically defined." The so-called "indeterminate + forms" (see INFINITESIMAL CALCULUS) are examples. + +7. _Superior and Inferior Limits; Infinities._--The value of a function +at every point in the domain of its argument is finite, since, by +definition, the value can be assigned, but this does not necessarily +imply that there is a number N which exceeds all the values (or is less +than all the values). It may happen that, however great a number N we +take, there are among the values of the function numbers which exceed N +(or are less than -N). + +If a number can be found which is greater than every value of the +function, then either ([alpha]) there is one value of the function +which exceeds all the others, or ([beta]) there is a number S which +exceeds every value of the function but is such that, however small a +positive number [epsilon] we take, there are values of the function +which exceed S -[epsilon]. In the case ([alpha]) the function has a +greatest value; in case ([beta]) the function has a "superior limit" S, +and then there must be a point a which has the property that there are +points of the domain of the argument, in the neighbourhood of a for any +h, at which the values of the function differ from S by less than +[epsilon]. Thus S is the limit of the function at a, either for the +domain of the argument or for some more restricted domain. If a is in +the domain of the argument, and if, after omission of a, there is a +superior limit S which is in this way the limit of the function at a, if +further [f](a) = S, then S is the greatest value of the function: in +this case the greatest value is a limit (at any rate for a restricted +domain) which is attained; it may be called a "superior limit which is +attained." In like manner we may have a "smallest value" or an "inferior +limit," and a smallest value may be an "inferior limit which is +attained." + + All that has been said here may be adapted to the description of + greatest values, superior limits, &c., of a function in a restricted + domain contained in the domain of the argument. In particular, the + domain of the argument may contain an interval; and therein the + function may have a superior limit, or an inferior limit, which is + attained. Such a limit is a _maximum_ value or a _minimum_ value of + the function. + + Again, if, after any number N, however great, has been specified, it + is possible to find points of the domain of the argument at which the + value of the function exceeds N, the values of the function are said + to have an "infinite superior limit," and then there must be a point a + which has the property that there are points of the domain, in the + neighbourhood of a for any h, at which the value of the function + exceeds N. If the point a is in the domain of the argument the + function is said to "tend to become infinite" at a; it has of course a + finite value at a. If the point a is not in the domain of the argument + the function is said to "become infinite" at a; it has of course no + value at a. In like manner we may have a (negatively) infinite + inferior limit. Again, after any number N, however great, has been + specified and a number h found, so that all the values of the + function, at points in the neighbourhood of a for h, exceed N in + absolute value, all these values may have the same sign; the function + is then said to become, or to tend to become, "determinately + (positively or negatively) infinite"; otherwise it is said to become + or to tend to become, "indeterminately infinite." + + All the infinities that occur in the theory of functions are of the + nature of variable finite numbers, with the single exception of the + infinity of an infinite aggregate. The latter is described as an + "actual infinity," the former as "improper infinities." There is no + "actual infinitely small" corresponding to the actual infinity. The + only "infinitely small" is zero. All "infinite values" are of the + nature of superior and inferior limits which are not attained. + +8. _Increasing and Decreasing Functions._--A function [f](x) of one +variable x, defined in the interval between a and b, is "increasing +throughout the interval" if, whenever x and x' are two numbers in the +interval and x' > x, then [f](x') > [f](x); the function "never +decreases throughout the interval" if, x' and x being as before, [f](x') +> [f](x). Similarly for decreasing functions, and for functions which +never increase throughout an interval. A function which either never +increases or never diminishes throughout an interval is said to be +"monotonous throughout" the interval. If we take in the above definition +b > a, the definition may apply to a function under the restriction that +x' is not b and x is not a; such a function is "monotonous within" the +interval. In this case we have the theorem that the function (if it +never decreases) has a limit on the left at b and a limit on the right +at a, and these are the superior and inferior limits of its values at +all points within the interval (the ends excluded); the like holds +_mutatis mutandis_ if the function never increases. If the function is +monotonous throughout the interval, [f](b) is the greatest (or least) +value of [f](x) in the interval; and if [f](b) is the limit of [f](x) on +the left at b, such a greatest (or least) value is an example of a +superior (or inferior) limit which is attained. In these cases the +function tends continually to its limit. + + These theorems and definitions can be extended, with obvious + modifications, to the cases of a domain which is not an interval, or + extends to infinite values. By means of them we arrive at sufficient, + but not necessary, criteria for the existence of a limit; and these + are frequently easier to apply than the general principle of + convergence to a limit (S 6), of which principle they are particular + cases. For example, the function represented by x log (1/x) + continually diminishes when 1/e > x > 0 and x diminishes towards + zero, and it never becomes negative. It therefore has a limit on the + right at x = 0. This limit is zero. The function represented by x sin + (1/x) does not continually diminish towards zero as x diminishes + towards zero, but is sometimes greater than zero and sometimes less + than zero in any neighbourhood of x = 0, however small. Nevertheless, + the function has the limit zero at x = 0. + +9. _Continuity of Functions._--A function [f](x) of one variable x is +said to be continuous at a point a if (1) [f](x) is defined in an +interval containing a; (2) [f](x) has a limit at a; (3) [f](a) is equal +to this limit. The limit in question must be a limit for continuous +variation, not for a restricted domain. If [f](x) has a limit on the +left at a and [f](a) is equal to this limit, the function may be said to +be "continuous to the left" at a; similarly the function may be +"continuous to the right" at a. + +A function is said to be "continuous throughout an interval" when it is +continuous at every point of the interval. This implies continuity to +the right at the smaller end-value and continuity to the left at the +greater end-value. When these conditions at the ends are not satisfied +the function is said to be continuous "within" the interval. By a +"continuous function" of one variable we always mean a function which is +continuous throughout an interval. + + The principal properties of a continuous function are: + + 1. The function is practically constant throughout sufficiently small + intervals. This means that, after any point a of the interval has been + chosen, and any positive number [epsilon], however small, has been + specified, it is possible to find a number h, so that the difference + between any two values of the function in the interval between a-h and + a + h is less than [epsilon]. There is an obvious modification if a is + an end-point of the interval. + + 2. The continuity of the function is "uniform." This means that the + number h which corresponds to any [epsilon] as in (1) may be the same + at all points of the interval, or, in other words, that the numbers h + which correspond to [epsilon] for different values of a have a + positive inferior limit. + + 3. The function has a greatest value and a least value in the + interval, and these are superior and inferior limits which are + attained. + + 4. There is at least one point of the interval at which the function + takes any value between its greatest and least values in the interval. + + 5. If the interval is unlimited towards the right (or towards the + left), the function has a limit at [oo] (or at -[oo]). + +10. _Discontinuity of Functions._--The discontinuities of a function of +one variable, defined in an interval with the possible exception of +isolated points, may be classified as follows: + +(1) The function may become infinite, or tend to become infinite, at a +point. + +(2) The function may be undefined at a point. + +(3) The function may have a limit on the left and a limit on the right +at the same point; these may be different from each other, and at least +one of them must be different from the value of the function at the +point. + +(4) The function may have no limit at a point, or no limit on the left, +or no limit on the right, at a point. + + In case a function [f](x), defined as above, has no limit at a point + a, there are four limiting values which come into consideration. + Whatever positive number h we take, the values of the function at + points between a and a + h (a excluded) have a superior limit (or a + greatest value), and an inferior limit (or a least value); further, as + h decreases, the former never increases and the latter never + decreases; accordingly each of them tends to a limit. We have in this + way two limits on the right--the inferior limit of the superior limits + in diminishing neighbourhoods, and the superior limit of the inferior + limits in diminishing neighbourhoods. These are denoted by /{[f](a + + o)} and {[f](a + 0)}/, and they are called the "limits of + indefiniteness" on the right. Similar limits on the left are denoted + by /{[f](a - 0)} and {[f](a - 0)}/. Unless [f](x) becomes, or tends to + become, infinite at a, all these must exist, any two of them may be + equal, and at least one of them must be different from [f](a), if + [f](a) exists. If the first two are equal there is a limit on the + right denoted by [f](a + 0); if the second two are equal, there is a + limit on the left denoted by [f](a - 0). In case the function becomes, + or tends to become, infinite at a, one or more of these limits is + infinite in the sense explained in S 7; and now it is to be noted + that, e.g. the superior limit of the inferior limits in diminishing + neighbourhoods on the right of a may be negatively infinite; this + happens if, after any number N, however great, has been specified, it + is possible to find a positive number h, so that all the values of the + function in the interval between a and a + h (a excluded) are less + than -N; in such a case [f](x) tends to become negatively infinite + when x decreases towards a; other modes of tending to infinite limits + may be described in similar terms. + +11. _Oscillation of Functions._--The difference between the greatest and +least of the numbers [f](a), /{[f](a + 0)}, {[f](a + 0)}/, /{[f](a - +0)}, {[f](a - 0)}/, when they are all finite, is called the +"oscillation" or "fluctuation" of the function [f](x) at the point a. +This difference is the limit for h = 0 of the difference between the +superior and inferior limits of the values of the function at points in +the interval between a - h and a + h. The corresponding difference for +points in a finite interval is called the "oscillation of the function +in the interval." When any of the four limits of indefiniteness is +infinite the oscillation is infinite in the sense explained in S 7. + + For the further classification of functions we divide the domain of + the argument into partial intervals by means of points between the + end-points. Suppose that the domain is the interval between a and b. + Let intermediate points x1, x2 ... x{n - 1}_, be taken so that b > + X_(n - 1) > x_(n - 2) ... > X1 > a_. We may devise a rule by which, as + n increases indefinitely, all the differences b - x_(n - 1), x_(n - 1) + - x_(n - 2), ... x1 - a tend to zero as a limit. The interval is then + said to be divided into "indefinitely small partial intervals." + + A function defined in an interval with the possible exception of + isolated points may be such that the interval can be divided into a + set of finite partial intervals within each of which the function is + monotonous (S 8). When this is the case the sum of the oscillations of + the function in those partial intervals is finite, provided the + function does not tend to become infinite. Further, in such a case the + sum of the oscillations will remain below a fixed number for any mode + of dividing the interval into indefinitely small partial intervals. A + class of functions may be defined by the condition that the sum of the + oscillations has this property, and such functions are said to have + "restricted oscillation." Sometimes the phrase "limited fluctuation" + is used. It can be proved that any function with restricted + oscillation is capable of being expressed as the sum of two monotonous + functions, of which one never increases and the other never diminishes + throughout the interval. Such a function has a limit on the right and + a limit on the left at every point of the interval. This class of + functions includes all those which have a finite number of maxima and + minima in a finite-interval, and some which have an infinite number. + It is to be noted that the class does not include all continuous + functions. + +12. _Differentiable Function._--The idea of the differentiation of a +continuous function is that of a process for measuring the rate of +growth; the increment of the function is compared with the increment of +the variable. If _[f](x)_ is defined in an interval containing the point +a, and _a - k_ and _a + k_ are points of the interval, the expression + + [f](a + h) - [f](a) + ------------------- (1) + h + +represents a function of h, which we may call [phi](h), defined at all +points of an interval for h between -k and k except the point 0. Thus +the four limits /[phi](+0), [phi](+0)/, /[phi](-0), [phi](-0)/ exist, +and two or more of them may be equal. When the first two are equal +either of them is the "progressive differential coefficient" of [f](x) +at the point a; when the last two are equal either of them is the +"regressive differential coefficient" of [f](x) at a; when all four are +equal the function is said to be "differentiable" at a, and either of +them is the "differential coefficient" of [f](x) at a, or the "first +derived function" of [f](x) at a. It is denoted by d[f](x) / dx or by +[f]'(x). In this case [phi](h) has a definite limit at h = 0, or is +determinately infinite at h = 0 (S 7). The four limits here in question +are called, after Dini, the "four derivates" of [f](x) at a. In +accordance with the notation for derived functions they may be denoted +by + + ---------- ---------- + [f]' + (a), [f]' + (a), [f]' - (a), f' - (a). + --------- -------- + + A function which has a finite differential coefficient at all points + of an interval is continuous throughout the interval, but if the + differential coefficient becomes infinite at a point of the interval + the function may or may not be continuous throughout the interval; on + the other hand a function may be continuous without being + differentiable. This result, comparable in importance, from the point + of view of the general theory of functions, with the discovery of + Fourier's theorem, is due to G.F.B. Riemann; but the failure of an + attempt made by Ampere to prove that every continuous function must be + differentiable may be regarded as the first step in the theory. + Examples of analytical expressions which represent continuous + functions that are not differentiable have been given by Riemann, + Weierstrass, Darboux and Dini (see S 24). The most important theorem + in regard to differentiable functions is the "theorem of intermediate + value." (See INFINITESIMAL CALCULUS.) + +13. _Analytic Function._--If [f](x) and its first n differential +coefficients, denoted by[f]'(x), [f]''(x), ... [f](^n)(x), are +continuous in the interval between a and a + h, then + + h^2 + [f](a + h) = [f](a) + h[f]'(a) + --- f''(a) + ... + 2! + + h^(n - 1) + + --------- [f]^(n - 1)(a) + R_n, + (n - 1)! + +where R_n may have various forms, some of which are given in the article +INFINITESIMAL CALCULUS. This result is known as "Taylor's theorem." + +When Taylor's theorem leads to a representation of the function by means +of an infinite series, the function is said to be "analytic" (cf. S 21). + +14. _Ordinary Function._--The idea of a curve representing a continuous +function in an interval is that of a line which has the following +properties: (1) the co-ordinates of a point of the curve are a value x +of the argument and the corresponding value y of the function; (2) at +every point the curve has a definite tangent; (3) the interval can be +divided into a finite number of partial intervals within each of which +the function is monotonous; (4) the property of monotony within partial +intervals is retained after interchange of the axes of co-ordinates x +and y. According to condition (2) y is a continuous and differentiable +function of x, but this condition does not include conditions (3) and +(4): there are continuous partially monotonous functions which are not +differentiable, there are continuous differentiable functions which are +not monotonous in any interval however small; and there are continuous, +differentiable and monotonous functions which do not satisfy condition +(4) (cf. S 24). A function which can be represented by a curve, in the +sense explained above, is said to be "ordinary," and the curve is the +graph of the function (S2). All analytic functions are ordinary, but not +all ordinary functions are analytic. + +15. _Integrable Function._--The idea of integration is twofold. We may +seek the function which has a given function as its differential +coefficient, or we may generalize the question of finding the area of a +curve. The first inquiry leads directly to the indefinite integral, the +second directly to the definite integral. Following the second method we +define "the definite integral of the function [f](x) through the +interval between a and b" to be the limit of the sum + + _n + \ [f](x'_r)(x_r - x_(r - 1)) + /_ + 1 + +when the interval is divided into ultimately indefinitely small partial +intervals by points x1, x2, ... x_(n - 1). Here x'_r denotes any point +in the rth partial interval, x0 is put for a, and x_n for b. It can be +shown that the limit in question is finite and independent of the mode +of division into partial intervals, and of the choice of the points such +as x'_r, provided (1) the function is defined for all points of the +interval, and does not tend to become infinite at any of them; (2) for +any one mode of division of the interval into ultimately indefinitely +small partial intervals, the sum of the products of the oscillation of +the function in each partial interval and the difference of the +end-values of that partial interval has limit zero when n is increased +indefinitely. When these conditions are satisfied the function is said +to be "integrable" in the interval. The numbers a and b which limit the +interval are usually called the "lower and upper limits." We shall call +them the "nearer and further end-values." The above definition of +integration was introduced by Riemann in his memoir on trigonometric +series (1854). A still more general definition has been given by +Lebesgue. As the more general definition cannot be made intelligible +without the introduction of some rather recondite notions belonging to +the theory of aggregates, we shall, in what follows, adhere to Riemann's +definition. + + We have the following theorems:-- + + 1. Any continuous function is integrable. + + 2. Any function with restricted oscillation is integrable. + + 3. A discontinuous function is integrable if it does not tend to + become infinite, and if the points at which the oscillation of the + function exceeds a given number [sigma], however small, can be + enclosed in partial intervals the sum of whose breadths can be + diminished indefinitely. + + These partial intervals must be a set chosen out of some complete set + obtained by the process used in the definition of integration. + + 4. The sum or product of two integrable functions is integrable. + + As regards integrable functions we have the following theorems: + + 1. If S and I are the superior and inferior limits (or greatest and + least values) of [f](x) in the interval between a and b, [int] [a to + b] [f](x)dx is intermediate between S(b - a) and I(b - a). + + 2. The integral is a continuous function of each of the end-values. + + 3. If the further end-value b is variable, and if [int] [a to x] + [f](x)_dx_ = F(x), then if [f](x) is continuous at b, F(x) is + differentiable at b, and F'(b) = [f](b). + + 4. In case [f](x) is continuous throughout the interval F(x) is + continuous and differentiable throughout the interval, and F'(x) = + [f](x) throughout the interval. + + 5. In case [f]'(x) is continuous throughout the interval between a and + b, + _ + / b + | [f]'(x)dx = [f](b) - [f](a). + _/a + + 6. In case [f](x) is discontinuous at one or more points of the + interval between a and b, in which it is integrable, + _ + / x + | [f](x)dx + _/a + + is a function of x, of which the four derivates at any point of the + interval are equal to the limits of indefiniteness of [f](x) at the + point. + + 7. It may be that there exist functions which are differentiable + throughout an interval in which their differential coefficients are + not integrable; if, however, F(x) is a function whose differential + coefficient, F'(x), is integrable in an interval, then + _ + / x + F(x) = | F'(x)dx + const., + _/a + + where a is a fixed point, and x a variable point, of the interval. + Similarly, if any one of the four derivates of a function is + integrable in an interval, all are integrable, and the integral of + either differs from the original function by a constant only. + + The theorems (4), (6), (7) show that there is some discrepancy between + the indefinite integral considered as the function which has a given + function as its differential coefficient, and as a definite integral + with a variable end-value. + + We have also two theorems concerning the integral of the product of + two integrable functions [f](x) and [phi](x); these are known as "the + first and second theorems of the mean." The first theorem of the mean + is that, if [phi](x) is one-signed throughout the interval between a + and b, there is a number M intermediate between the superior and + inferior limits, or greatest and least values, of [f](x) in the + interval, which has the property expressed by the equation + _ _ + / b / b + M | [phi](x)dx = | [f](x)[phi](x)dx. + _/a _/a + + The second theorem of the mean is that, if [f](x) is monotonous + throughout the interval, there is a number [xi] between a and b which + has the property expressed by the equation + _ _ _ + / b /[xi] / b + | [f](x)[phi](x)dx = [f](a) | [phi](x)dx + [f](b) | [phi](x)dx. + _/a _/a _/[xi] + + (_See_ FOURIER'S SERIES.) + +16. _Improper Definite Integrals._--We may extend the idea of +integration to cases of functions which are not defined at some point, +or which tend to become infinite in the neighbourhood of some point, and +to cases where the domain of the argument extends to infinite values. If +c is a point in the interval between a and b at which [f](x) is not +defined, we impose a restriction on the points x'_r of the definition: +none of them is to be the point c. This comes to the same thing as +defining [int] [a to b] [f](x)dx to be + _ _ + / c-[epsilon] / b + Lt | [f](x)dx + Lt | [f](x)dx, (1) + _/a _/c+[epsilon]' + [epsilon]=0 [epsilon]'=0 + +where, to fix ideas, b is taken > a, and [epsilon] and [epsilon]' are +positive. The same definition applies to the case where [f](x) becomes +infinite, or tends to become infinite, at c, provided both the limits +exist. This definition may be otherwise expressed by saying that a +partial interval containing the point c is omitted from the interval of +integration, and a limit taken by diminishing the breadth of this +partial interval indefinitely; in this form it applies to the cases +where c is a or b. + +Again, when the interval of integration is unlimited to the right, or +extends to positively infinite values, we have as a definition + _ _ + / [oo] / h + | [f](x)dx = Lt | [f](x)dx, + _/a _/a + h=[oo] + +provided this limit exists. Similar definitions apply to + _ _ + /-[oo] / [oo] + | [f](x)dx and to | [f](x)dx. + _/a _/-[oo] + +All such definite integrals as the above are said to be "improper." For +example, [int] {0 to [oo]} (sin x / x)dx is improper in two ways. It +means + _ + / h sin x + Lt Lt | ----- dx, + h=[oo] [epsilon]=0 _/[epsilon] x + +in which the positive number [epsilon] is first diminished indefinitely, +and the positive number h is afterwards increased indefinitely. + +The "theorems of the mean" (S 15) require modification when the +integrals are improper (see FOURIER'S SERIES). + +When the improper definite integral of a function which becomes, or +tends to become, infinite, exists, the integral is said to be +"convergent." If [f](x) tends to become infinite at a point c in the +interval between a and b, and the expression (1) does not exist, then +the expression [int] [a to b][f](x)_dx_, which has no value, is called a +"divergent integral, "and it may happen that there is a definite value +for + _ _ _ _ + | / c-[epsilon] / b | + Lt | | [f](x) dx + | [f](x) dx | + |_ _/a _/c+[epsilon]' _| + +provided that [epsilon] and [epsilon]' are connected by some definite +relation, and both, remaining positive, tend to limit zero. The value of +the above limit is then called a "principal value" of the divergent +integral. Cauchy's principal value is obtained by making [epsilon]' = +[epsilon], i.e. by taking the omitted interval so that the infinity is +at its middle point. A divergent integral which has one or more +principal values is sometimes described as "semi-convergent." + +17. _Domain of a Set of Variables._--The numerical continuum of n +dimensions (C_n) is the aggregate that is arrived at by attributing +simultaneous values to each of n variables x1, x2, ... x_n, these values +being any real numbers. The elements of such an aggregate are called +"points," and the numbers x1, x2 ... x_n the "co-ordinates" of a point. +Denoting in general the points (x1, x2, ... x_n) and (x'1, x'2 ... x'_n) +by x and x', the sum of the differences |x1 - x'1| + |x2 - x'2| + ... + +|x_n - x'_n| may be denoted by |x - x'| and called the "difference of +the two points." We can in various ways choose out of the continuum an +aggregate of points, which may be an infinite aggregate, and any such +aggregate can be the "domain" of a "variable point." The domain is said +to "extend to an infinite distance" if, after any number N, however +great, has been specified, it is possible to find in the domain points +of which one or more co-ordinates exceed N in absolute value. The +"neighbourhood" of a point a for a (positive) number h is the aggregate +constituted of all the points x, which are such that the "difference" +denoted by |x - a| < h. If an infinite aggregate of points does not +extend to an infinite distance, there must be at least one point a, +which has the property that the points of the aggregate which are in the +neighbourhood of a for any number h, however small, themselves +constitute an infinite aggregate, and then the point a is called a +"limiting point" of the aggregate; it may or may not be a point of the +aggregate. An aggregate of points is "perfect" when all its points are +limiting points of it, and all its limiting points are points of it; it +is "connected" when, after taking any two points a, b of it, and +choosing any positive number [epsilon], however small, a number m and +points x', x", ... x^(m) of the aggregate can be found so that all the +differences denoted by |x' - a|, |x" - x'|, ... |b - x^(m)| are less +than [epsilon]. A perfect connected aggregate is a _continuum_. This is +G. Cantor's definition. + + The definition of a continuum in C_n leaves open the question of the + number of dimensions of the continuum, and a further explanation is + necessary in order to define arithmetically what is meant by a + "homogeneous part" H_n of C_n. Such a part would correspond to an + interval in C1, or to an area bounded by a simple closed contour in + C2; and, besides being perfect and connected, it would have the + following properties: (1) There are points of C_n, which are not + points of H_n; these form a complementary aggregate H'_n. (2) There + are points "within" H_n; this means that for any such point there is a + neighbourhood consisting exclusively of points of H_n. (3) The points + of H_n which do not lie "within" H_n are limiting points of H'_n; they + are not points of H'_n, but the neighbourhood of any such point for + any number h, however small, contains points within H_n and points of + H'_n: the aggregate of these points is called the "boundary" of H_n. + (4) When any two points a, b within H_n are taken, it is possible to + find a number [epsilon] and a corresponding number m, and to choose + points x', x", ... x^m, so that the neighbourhood of a for [epsilon] + contains x', and consists exclusively of points within H_n, and + similarly for x' and x", x" and x"', ... x^m and b. Condition (3) + would exclude such an aggregate as that of the points within and upon + two circles external to each other and a line joining a point on one + to a point on the other, and condition (4) would exclude such an + aggregate as that of the points within and upon two circles which + touch externally. + +18. Functions of Several Variables.--A function of several variables +differs from a function of one variable in that the argument of the +function consists of a set of variables, or is a variable point in a C_n +when there are n variables. The function is definable by means of the +domain of the argument and the rule of calculation. In the most +important cases the domain of the argument is a homogeneous part H_n of +C_n with the possible exception of isolated points, and the rule of +calculation is that the value of the function in any assigned part of +the domain of the argument is that value which is assumed at the point +by an assigned analytical expression. The limit of a function at a point +a is defined in the same way as in the case of a function of one +variable. + + We take a positive fraction [epsilon] and consider the neighbourhood + of a for h, and from this neighbourhood we exclude the point a, and we + also exclude any point which is not in the domain of the argument. + Then we take x and x' to be any two of the retained points in the + neighbourhood. The function [f] has a limit at a if for any positive + [epsilon], however small, there is a corresponding h which has the + property that |[f](x') - [f](x)| < [epsilon], whatever points x, x' in + the neighbourhood of a for h we take (a excluded). For example, when + there are two variables x1, x2, and both are unrestricted, the domain + of the argument is represented by a plane, and the values of the + function are correlated with the points of the plane. The function has + a limit at a point a, if we can mark out on the plane a region + containing the point a within it, and such that the difference of the + values of the function which correspond to any two points of the + region (neither of the points being a) can be made as small as we + please in absolute value by contracting all the linear dimensions of + the region sufficiently. When the domain of the argument of a function + of n variables extends to an infinite distance, there is a "limit at + an infinite distance" if, after any number [epsilon], however small, + has been specified, a number N can be found which is such that + |[f](x') - [f](x)| < [epsilon], for all points x and x' (of the + domain) of which one or more co-ordinates exceed N in absolute value. + In the case of functions of several variables great importance + attaches to limits for a restricted domain. The definition of such a + limit is verbally the same as the corresponding definition in the case + of functions of one variable (S 6). For example, a function of x1 and + x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1 + without limit, keeping x2 constant, and afterwards diminish x2 without + limit. Expressed in geometrical language, this process amounts to + approaching the origin along the axis of x2. The definitions of + superior and inferior limits, and of maxima and minima, and the + explanations of what is meant by saying that a function of several + variables becomes infinite, or tends to become infinite, at a point, + are almost identical verbally with the corresponding definitions and + explanations in the case of a function of one variable (S 7). The + definition of a continuous function (S 9) admits of immediate + extension; but it is very important to observe that a function of two + or more variables may be a continuous function of each of the + variables, when the rest are kept constant, without being a continuous + function of its argument. For example, a function of x and y may be + defined by the conditions that when x = 0 it is zero whatever value y + may have, and when x [/=] 0 it has the value of sin {4tan^(-1)(y/x)}. + When y has any particular value this function is a continuous function + of x, and, when x has any particular value this function is a + continuous function of y; but the function of x and y is discontinuous + at (x = 0, y = 0). + +19. _Differentiation and Integration._--The definition of partial +differentiation of a function of several variables presents no +difficulty. The most important theorems concerning differentiable +functions are the "theorem of the total differential," the theorem of +the interchangeability of the order of partial differentiations, and the +extension of Taylor's theorem (see INFINITESIMAL CALCULUS). + +With a view to the establishment of the notion of integration through a +domain, we must define the "extent" of the domain. Take first a domain +consisting of the point a and all the points x for which |x - a| < 1/2h, +where h is a chosen positive number; the extent of this domain is h^n, n +being the number of variables; such a domain may be described as +"square," and the number h may be called its "breadth"; it is a +homogeneous part of the numerical continuum of n dimensions, and its +boundary consists of all the points for which |x - a| = 1/2h. Now the +points of any domain, which does not extend to an infinite distance, may +be assigned to a finite number m of square domains of finite breadths, +so that every point of the domain is either within one of these square +domains or on its boundary, and so that no point is within two of the +square domains; also we may devise a rule by which, as the number m +increases indefinitely, the breadths of all the square domains are +diminished indefinitely. When this process is applied to a homogeneous +part, H, of the numerical continuum C_n, then, at any stage of the +process, there will be some square domains of which all the points +belong to H, and there will generally be others of which some, but not +all, of the points belong to H. As the number m is increased +indefinitely the sums of the extents of both these categories of square +domains will tend to definite limits, which cannot be negative; when the +second of these limits is zero the domain H is said to be "measurable," +and the first of these limits is its "extent"; it is independent of the +rule adopted for constructing the square domains and contracting their +breadths. The notion thus introduced may be adapted by suitable +modifications to continua of lower dimensions in C_n. + + The integral of a function f(x) through a measurable domain H, which + is a homogeneous part of the numerical continuum of n dimensions, is + defined in just the same way as the integral through an interval, the + extent of a square domain taking the place of the difference of the + end-values of a partial interval; and the condition of integrability + takes the same form as in the simple case. In particular, the + condition is satisfied when the function is continuous throughout the + domain. The definition of an integral through a domain may be adapted + to any domain of measurable extent. The extensions to "improper" + definite integrals may be made in the same way as for a function of + one variable; in the particular case of a function which tends to + become infinite at a point in the domain of integration, the point is + enclosed in a partial domain which is omitted from the integration, + and a limit is taken when the extent of the omitted partial domain is + diminished indefinitely; a divergent integral may have different + (principal) values for different modes of contracting the extent of + the omitted partial domain. In applications to mathematical physics + great importance attaches to convergent integrals and to principal + values of divergent integrals. For example, any component of magnetic + force at a point within a magnet, and the corresponding component of + magnetic induction at the same point are expressed by different + principal values of the same divergent integral. Delicate questions + arise as to the possibility of representing the integral of a function + of n variables through a domain H_n, as a repeated integral, of + evaluating it by successive integrations with respect to the variables + one at a time and of interchanging the order of such integrations. + These questions have been discussed very completely by C. Jordan, and + we may quote the result that all the transformations in question are + valid when the function is continuous throughout the domain. + +20. _Representation of Functions in General._--We have seen that the +notion of a function is wider than the notion of an analytical +expression, and that the same function may be "represented" by one +expression in one part of the domain of the argument and by some other +expression in another part of the domain (S 5). Thus there arises the +general problem of the representation of functions. The function may be +given by specifying the domain of the argument and the rule of +calculation, or else the function may have to be determined in +accordance with certain conditions; for example, it may have to satisfy +in a prescribed domain an assigned differential equation. In either case +the problem is to determine, when possible, a single analytical +expression which shall have the same value as the function at all points +in the domain of the argument. For the representation of most functions +for which the problem can be solved recourse must be had to limiting +processes. Thus we may utilize infinite series, or infinite products, or +definite integrals; or again we may represent a function of one variable +as the limit of an expression containing two variables in a domain in +which one variable remains constant and another varies. An example of +this process is afforded by the expression Lt_y = [oo]xy/(x^2y + 1), +which represents a function of x vanishing at x = 0 and at all other +values of x having the value of 1/x. The method of series falls under +this more general process (cf. S 6). When the terms u1, u2, ... of a +series are functions of a variable x, the sum s_n of the first n terms +of the series is a function of x and n; and, when the series is +convergent, its sum, which is Lt_n = [oo]s_n, can represent a function +of x. In most cases the series converges for some values of x and not +for others, and the values for which it converges form the "domain of +convergence." The sum of the series represents a function in this +domain. + + The apparently more general method of representation of a function of + one variable as the limit of a function of two variables has been + shown by R. Baire to be identical in scope with the method of series, + and it has been developed by him so as to give a very complete account + of the possibility of representing functions by analytical + expressions. For example, he has shown that Riemann's totally + discontinuous function, which is equal to 1 when x is rational and to + 0 when x is irrational, can be represented by an analytical + expression. An infinite process of a different kind has been adapted + to the problem of the representation of a continuous function by T. + Broden. He begins with a function having a graph in the form of a + regular polygon, and interpolates additional angular points in an + ordered sequence without limit. The representation of a function by + means of an infinite product falls clearly under Baire's method, while + the representation by means of a definite integral is analogous to + Broden's method. As an example of these two latter processes we may + cite the Gamma function [[Gamma](x)] defined for positive values of x + by the definite integral + _ + / [oo] + | e^(-t)t^(x - 1)dt, + _/0 + + or by the infinite product + + / x \ + L t_(n = [oo]) n^x/x (1 + x)(1 + 1/2x) ... ( 1 + ----- ). + \ n - 1 / + + The second of these expressions avails for the representation of the + function at all points at which x is not a negative integer. + +21. _Power Series._--Taylor's theorem leads in certain cases to a +representation of a function by an infinite series. We have under +certain conditions (S 13) + + _n-1 + \ (x - a)^r + [f](x) = [f](a) + /_ --------- [f]^(r) (a) + R_n; + r=1 r! + +and this becomes + + _[oo] + \ (x - a)^r + [f](x) = [f](a) + /_ --------- [f]^(r) (a), + r=1 r! + +provided that ([alpha]) a positive number k can be found so that at all +points in the interval between a and a + k (except these points) [f](x) +has continuous differential coefficients of all finite orders, and at a +has progressive differential coefficients of all finite orders; ([beta]) +Cauchy's form of the remainder R_n, viz. [(x - a)^n / (n - 1)!] (1 - +[theta])^(n - 1)[f]^n {a + [theta](x - a)}, has the limit zero when n +increases indefinitely, for all values of [theta] between 0 and 1, and +for all values of x in the interval between a and a + k, except possibly +a + k. When these conditions are satisfied, the series (1) represents +the function at all points of the interval between a and a + k, except +possibly a + k, and the function is "analytic" (S 13) in this domain. +Obvious modifications admit of extension to an interval between a and a +- k, or between a - k and a + k. When a series of the form (1) +represents a function it is called "the Taylor's series for the +function." + +Taylor's series is a power series, i.e. a series of the form + + _[oo] + \ a_n (x - a)^n. + /_ + n=0 + + As regards power series we have the following theorems: + + 1. If the power series converges at any point except a there is a + number k which has the property that the series converges absolutely + in the interval between a - k and a + k, with the possible exception + of one or both end-points. + + 2. The power series represents a continuous function in its domain of + convergence (the end-points may have to be excluded). + + 3. This function is analytic in the domain, and the power series + representing it is the Taylor's series for the function. + + The theory of power series has been developed chiefly from the point + of view of the theory of functions of complex variables. + +22. _Uniform Convergence._--We shall suppose that the domain of +convergence of an infinite series of functions is an interval with the +possible exception of isolated points. Let [f](x) be the sum of the +series at any point x of the domain, and [f]_n(x) the sum of the first n ++ 1 terms. The condition of convergence at a point a is that, after any +positive number [epsilon], however small, has been specified, it must be +possible to find a number n so that |[f]_m(a) - [f]_p(a)| < [epsilon] +for all values of m and p which exceed n. The sum, [f](a), is the limit +of the sequence of numbers [f]_n(a) at n = [oo]. The convergence is +said to be "uniform" in an interval if, after specification of +[epsilon], the same number n suffices at all points of the interval to +make |[f](x) - [f]_m(x)| < [epsilon] for all values of m which exceed n. +The numbers n corresponding to any [epsilon], however small, are all +finite, but, when [epsilon] is less than some fixed finite number, they +may have an infinite superior limit (S 7); when this is the case there +must be at least one point, a, of the interval which has the property +that, whatever number N we take, [epsilon] can be taken so small that, +at some point in the neighbourhood of a, n must be taken > N to make +|[f](x) - f_m(x)| < [epsilon] when m > n; then the series does not +converge uniformly in the neighbourhood of a. The distinction may be +otherwise expressed thus: Choose a first and [epsilon] afterwards, then +the number n is finite; choose [epsilon] first and allow a to vary, then +the number n becomes a function of a, which may tend to become infinite, +or may remain below a fixed number; if such a fixed number exists, ho +wever small [epsilon] may be, the convergence is uniform. + + For example, the series sin x - 1/2 sin 2x + {1/3} sin 3x - ... is + convergent for all real values of x, and, when [pi] > x > -[pi] its + sum is 1/2x; but, when x is but a little less than [pi], the number of + terms which must be taken in order to bring the sum at all near to the + value of 1/2x is very large, and this number tends to increase + indefinitely as x approaches [pi]. This series does not converge + uniformly in the neighbourhood of x = [pi]. Another example is + afforded by the series + + _[oo] nx (n + 1)x + \ ---------- - ---------------- , + /_ n^2x^2 + 1 (n + 1)^2x^2 + 1 + n=0 + + of which the remainder after n terms is nx/(n^2x^2 + 1). If we put x = + 1/n, for any value of n, however great, the remainder is 1/2; and the + number of terms required to be taken to make the remainder tend to + zero depends upon the value of x when x is near to zero--it must, in + fact, be large compared with 1/x. The series does not converge + uniformly in the neighbourhood of x = 0. + +As regards series whose terms represent continuous functions we have the +following theorems: + +(1) If the series converges uniformly in an interval it represents a +function which is continuous throughout the interval. + +(2) If the series represents a function which is discontinuous in an +interval it cannot converge uniformly in the interval. + +(3) A series which does not converge uniformly in an interval may +nevertheless represent a function which is continuous throughout the +interval. + +(4) A power series converges uniformly in any interval contained within +its domain of convergence, the end-points being excluded. + +(5) If [Sigma] (r=0 to [oo]) [f]_r(x) = [f](x) converges uniformly in +the interval between a and b + + _ _[oo] _ + / b \ / b + | [f](x)dx = /_ | [f]_r(x)dx, + _/ a r=0 _/a + +or a series which converges uniformly may be integrated term by term. + +(6) If [Signa] (r=0 to [oo]) [f]'_r(x) converges uniformly in an +interval, then [Signa] (r=o to [oo]) [f]_r(x) converges in the interval, +and represents a continuous differentiable function, [phi](x); in fact +we have + + _[oo] + [phi]'(x) = \ [f]'_r(x), + /_ + r=0 + +or a series can be differentiated term by term if the series of derived +functions converges uniformly. + +A series whose terms represent functions which are not continuous +throughout an interval may converge uniformly in the interval. If +[Signa] (r=0 to [oo]) [f]_r(x) = [f](x), is such a series, and if all +the functions [f]_r(x) have limits at a, then [f](x) has a limit at a, +which is [Signa] (r=0 a=0 to [oo]) Lt [f]_r(x). A similar theorem holds +for limits on the left or on the right. + +23. _Fourier's Series._--An extensive class of functions admit of being +represented by series of the form + + _[oo] / n[pi]x n[pi]x \ + a0 + \ ( a_n cos ------ + b_n sin ------ ), (i.) + /_ \ c c / + n=1 + +and the rule for determining the coefficients a_n, b_n of such a series, +in order that it may represent a given function [f](x) in the interval +between -c and c, was given by Fourier, viz. we have + _ _ + 1 / c 1 / c n[pi]x + a0 = --- | [f](x)dx, a_n = -- | [f](x)cos ------ dx, + 2c _/-c c _/-c c + _ + / c 1 n[pi]x + b_n = | -- [f](x)sin ------ dx. + _/-c c c + +The interval between -c and c may be called the "periodic interval," and +we may replace it by any other interval, e.g. that between 0 and 1, +without any restriction of generality. When this is done the sum of the +series takes the form + + _ _r=n + / 1 \ [f](z)cos {2r[pi](z - x)} dz, + Lt | /_ + n=[oo] _/0 r=-n + + +and this is + _ + / 1 sin {(2n + 1)(z - x)[pi]} + Lt | [f](z) ------------------------ dz. (ii.) + n=[oo] _/0 sin {(z - x)[pi]} + +Fourier's theorem is that, if the periodic interval can be divided into +a finite number of partial intervals within each of which the function +is ordinary (S 14), the series represents the function within each of +those partial intervals. In Fourier's time a function of this character +was regarded as completely arbitrary. + + By a discussion of the integral (ii.) based on the Second Theorem of + the Mean (S 15) it can be shown that, if [f](x) has restricted + oscillation in the interval (S 11), the sum of the series is equal to + 1/2{[f](x + 0) + [f](x - 0)} at any point x within the interval, and + that it is equal to 1/2{[f]( + 0) + [f](1 - 0} at each end of the + interval. (See the article FOURIER'S SERIES.) It therefore represents + the function at any point of the periodic interval at which the + function is continuous (except possibly the end-points), and has a + definite value at each point of discontinuity. The condition of + restricted oscillation includes all the functions contemplated in the + statement of the theorem and some others. Further, it can be shown + that, in any partial interval throughout which [f](x) is continuous, + the series converges uniformly, and that no series of the form (i), + with coefficients other than those determined by Fourier's rule, can + represent the function at all points, except points of discontinuity, + in the same periodic interval. The result can be extended to a + function [f](x) which tends to become infinite at a finite number of + points a of the interval, provided (1) [f](x) tends to become + determinately infinite at each of the points a, (2) the improper + definite integral of [f](x) through the interval is convergent, (3) + [f](x) has not an infinite number of discontinuities or of maxima or + minima in the interval. + +24. _Representation of Continuous Functions by Series._--If the series +for [f](x) formed by Fourier's rule converges at the point a of the +periodic interval, and if [f](x) is continuous at a, the sum of the +series is [f](a); but it has been proved by P. du Bois Reymond that the +function may be continuous at a, and yet the series formed by Fourier's +rule may be divergent at a. Thus some continuous functions do not admit +of representation by Fourier's series. All continuous functions, +however, admit of being represented with arbitrarily close approximation +in either of two forms, which may be described as "terminated Fourier's +series" and "terminated power series," according to the two following +theorems: + +(1) If [f](x) is continuous throughout the interval between 0 and 2[pi], +and if any positive number [epsilon] however small is specified, it is +possible to find an integer n, so that the difference between the value +of [f](x) and the sum of the first n terms of the series for [f](x), +formed by Fourier's rule with periodic interval from 0 to 2[pi], shall +be less than [epsilon] at all points of the interval. This result can be +extended to a function which is continuous in any given interval. + +(2) If [f](x) is continuous throughout an interval, and any positive +number [epsilon] however small is specified, it is possible to find an +integer n and a polynomial in x of the nth degree, so that the +difference between the value of [f](x) and the value of the polynomial +shall be less than [epsilon] at all points of the interval. + +Again it can be proved that, if [f](x) is continuous throughout a given +interval, polynomials in x of finite degrees can be found, so as to form +an infinite series of polynomials whose sum is equal to [f](x) at all +points of the interval. Methods of representation of continuous +functions by infinite series of rational fractional functions have also +been devised. + + Particular interest attaches to continuous functions which are not + differentiable. Weierstrass gave as an example the function + represented by the series [Sigma] (n=0 to [oo]) a^n cos(b^[n] x[pi]), + where a is positive and less than unity, and b is an odd integer + exceeding (1 + (3/2)[pi]) / a. It can be shown that this series is + uniformly convergent in every interval, and that the continuous + function [f](x) represented by it has the property that there is, in + the neighbourhood of any point x0, an infinite aggregate of points x', + having x0 as a limiting point, for which {[f](x') - [f](x0)} / (x' - + x0) tends to become infinite with one sign when x' - x0 approaches + zero through positive values, and infinite with the opposite sign when + x' - x0 approaches zero through negative values. Accordingly the + function is not differentiable at any point. The definite integral of + such a function [f](x) through the interval between a fixed point and + a variable point x, is a continuous differentiable function F(x), for + which F'(x) = [f](x); and, if [f](x) is one-signed throughout any + interval F(x) is monotonous throughout that interval, but yet F(x) + cannot be represented by a curve. In any interval, however small, the + tangent would have to take the same direction for infinitely many + points, and yet there is no interval in which the tangent has + everywhere the same direction. Further, it can be shown that all + functions which are everywhere continuous and nowhere differentiable + are capable of representation by series of the form [Sigma][a]_n + [phi]_n (x), where [Sigma][a]_n is an absolutely convergent series of + numbers, and [phi]_n(x) is an analytic function whose absolute value + never exceeds unity. + +25. _Calculations with Divergent Series._--When the series described in +(1) and (2) of S 24 diverge, they may, nevertheless, be used for the +approximate numerical calculation of the values of the function, +provided the calculation is not carried beyond a certain number of +terms. Expansions in series which have the property of representing a +function approximately when the expansion is not carried too far are +called "asymptotic expansions." Sometimes they are called +"semi-convergent series"; but this term is avoided in the best modern +usage, because it is often used to describe series whose convergence +depends upon the order of the terms, such as the series 1 - 1/2 + 1/3 - +... + + In general, let [f]0(x) + [f]1(x) + ... be a series of functions which + does not converge in a certain domain. It may happen that, if any + number [epsilon], however small, is first specified, a number n can + afterwards be found so that, at a point a of the domain, the value + [f](a) of a certain function [f](x) is connected with the sum of the + first n + 1 terms of the series by the relation |[f](a) - [Sigma] (r=0 + to n) [f]_r(a) | < [epsilon]. It must also happen that, if any number + N, however great, is specified, a number n'(>n) can be found so that, + for all values of m which exceed n', | [Sigma](r=0 to m) [f]_r(a) | > + N. The divergent series [f]0(x) + [f]1(x) + ... is then an asymptotic + expansion for the function f(x) in the domain. + + The best known example of an asymptotic expansion is Stirling's + formula for n! when n is large, viz. + _____ + / + n! = \/(2[pi]) 1/2n^(n + 1/2) e^(-n + [theta] / 12n), + + where [theta] is some number lying between 0 and 1. This formula is + included in the asymptotic expansion for the Gamma function. We have + in fact + + log {[Gamma](x)} = (x - 1/2) log x - x + 1/2 log 2[pi] + [~omega](x), + + where [~omega](x) is the function defined by the definite integral + _ + / [oo] + ~[omega](x) = | {[1 - e^(-t)]^(-1) - t^(-1) - 1/2} t^(-1) e^(-tx)dt. + _/0 + + The multiplier of e^(-tx) under the sign of integration can be + expanded in the power series + + B1 B2 B3 + ---- - ---- t^2 + ---- t^4 - ..., + 2! 4! 6! + + where B1, B2, ... are "Bernoulli's numbers" given by the formula + + _[oo] + \ + B_m = 2.2m! (2[pi])^(-2m) /_ [r^(-2m)]. + r=1 + + When the series is integrated term by term, the right-hand member of + the equation for [~omega](x) takes the form + + B1 1 B2 1 B3 1 + ---- --- - ---- --- + ---- --- - ..., + 1.2 x 3.4 x^3 5.6 x^5 + + This series is divergent; but, if it is stopped at any term, the + difference between the sum of the series so terminated and the value + of [~omega](x) is less than the last of the retained terms. Stirling's + formula is obtained by retaining the first term only. Other well-known + examples of asymptotic expansions are afforded by the descending + series for Bessel's functions. Methods of obtaining such expansions + for the solutions of linear differential equations of the second order + were investigated by G.G. Stokes (_Math. and Phys. Papers_, vol. ii. + p. 329), and a general theory of asymptotic expansions has been + developed by H. Poincare. A still more general theory of divergent + series, and of the conditions in which they can be used, as above, for + the purposes of approximate calculation has been worked out by E. + Borel. The great merit of asymptotic expansions is that they admit of + addition, subtraction, multiplication and division, term by term, in + the same way as absolutely convergent series, and they admit also of + integration term by term; that is to say, the results of such + operations are asymptotic expansions for the sum, difference, + product, quotient, or integral, as the case may be. + +26. _Interchange of the Order of Limiting Operations._--When we require +to perform any limiting operation upon a function which is itself +represented by the result of a limiting process, the question of the +possibility of interchanging the order of the two processes always +arises. In the more elementary problems of analysis it generally happens +that such an interchange is possible; but in general it is not possible. +In other words, the performance of the two processes in different orders +may lead to two different results; or the performance of them in one of +the two orders may lead to no result. The fact that the interchange is +possible under suitable restrictions for a particular class of +operations is a theorem to be proved. + + Among examples of such interchanges we have the differentiation and + integration of an infinite series term by term (S 22), and the + differentiation and integration of a definite integral with respect to + a parameter by performing the like processes upon the subject of + integration (S 19). As a last example we may take the limit of the sum + of an infinite series of functions at a point in the domain of + convergence. Suppose that the series [Sigma] (r=0 to [oo]) [f]_r(x) + represents a function ([f]x) in an interval containing a point a, and + that each of the functions [f]_r(x) has a limit at a. If we first put + x = a, and then sum the series, we have the value [f](a); if we first + sum the series for any x, and afterwards take the limit of the sum at + x = a, we have the limit of [f](x) at a; if we first replace each + function [f]_r(x) by its limit at a, and then sum the series, we may + arrive at a value different from either of the foregoing. If the + function [f](x) is continuous at a, the first and second results are + equal; if the functions [f]_r(x) are all continuous at a, the first + and third results are equal; if the series is uniformly convergent, + the second and third results are equal. This last case is an example + of the interchange of the order of two limiting operations, and a + sufficient, though not always a necessary, condition, for the validity + of such an interchange will usually be found in some suitable + extension of the notion of uniform convergence. + + AUTHORITIES.--Among the more important treatises and memoirs connected + with the subject are: R. Baire, _Fonctions discontinues_ (Paris, + 1905); O. Biermann, _Analytische Functionen_ (Leipzig, 1887); E. + Borel, _Theorie des fonctions_ (Paris, 1898) (containing an + introductory account of the Theory of Aggregates), and _Series + divergentes_ (Paris, 1901), also _Fonctions de variables reelles_ + (Paris, 1905); T.J. I'A. Bromwich, _Introduction to the Theory of + Infinite Series_ (London, 1908); H.S. Carslaw, _Introduction to the + Theory of Fourier's Series and Integrals_ (London, 1906); U. Dini, + _Functionen e. reellen Grosse_ (Leipzig, 1892), and _Serie di Fourier_ + (Pisa, 1880); A. Genocchi u. G. Peano, _Diff.- u. Int.-Rechnung_ + (Leipzig, 1899); J. Harkness and F. Morley, _Introduction to the + Theory of Analytic Functions_ (London, 1898); A. Harnack, _Diff. and + Int. Calculus_ (London, 1891); E.W. Hobson, _The Theory of Functions + of a real Variable and the Theory of Fourier's Series_ (Cambridge, + 1907); C. Jordan, _Cours d'analyse_ (Paris, 1893-1896); L. Kronecker, + _Theorie d. einfachen u. vielfachen Integrale_ (Leipzig, 1894); H. + Lebesgue, _Lecons sur l'integration_ (Paris, 1904); M. Pasch, _Diff.- + u. Int.-Rechnung_ (Leipzig, 1882); E. Picard, _Traite d'analyse_ + (Paris, 1891); O. Stolz, _Allgemeine Arithmetik_ (Leipzig, 1885), and + _Diff.- u. Int.-Rechnung_ (Leipzig, 1893-1899); J. Tannery, _Theorie + des fonctions_ (Paris, 1886); W.H. and G.C. Young, _The Theory of Sets + of Points_ (Cambridge, 1906); Broden, "Stetige Functionen e. reellen + Veranderlichen," _Crelle_, Bd. cxviii.; G. Cantor, A series of memoirs + on the "Theory of Aggregates" and on "Trigonometric series" in _Acta + Math_. tt. ii., vii., and _Math. Ann_. Bde. iv.-xxiii.; Darboux, + "Fonctions discontinues," _Ann. Sci. Ecole normale sup_. (2), t. iv.; + Dedekind, _Was sind u. was sollen d. Zahlen_? (Brunswick, 1887), and + _Stetigkeit u. irrationale Zahlen_ (Brunswick, 1872); Dirichlet, + "Convergence des series trigonometriques," _Crelle_, Bd. iv.; P. Du + Bois Reymond, _Allgemeine Functionentheorie_ (Tubingen, 1882), and + many memoirs in _Crelle_ and in _Math. Ann_.; Heine, + "Functionenlehre," _Crelle_, Bd. lxxiv.; J. Pierpont, _The Theory of + Functions of a real Variable_ (Boston, 1905); F. Klein, "Allgemeine + Functionsbegriff," _Math. Ann_. Bd. xxii.; W.F. Osgood, "On Uniform + Convergence," _Amer. J. of Math_. vol. xix.; Pincherle, "Funzioni + analitiche secondo Weierstrass," _Giorn. di mat_. t. xviii.; + Pringsheim, "Bedingungen d. Taylorschen Lehrsatzes," _Math. Ann_. Bd. + xliv.; Riemann, "Trigonometrische Reihe," _Ges. Werke_ (Leipzig, + 1876); Schoenflies, "Entwickelung d. Lehre v. d. + Punktmannigfaltigkeiten," _Jahresber. d. deutschen Math.-Vereinigung_, + Bd. viii.; Study, Memoir on "Functions with Restricted Oscillation," + _Math. Ann_. Bd. xlvii.; Weierstrass, Memoir on "Continuous Functions + that are not Differentiable," _Ges. math. Werke_, Bd. ii. p. 71 + (Berlin, 1895), and on the "Representation of Arbitrary Functions," + ibid. Bd. iii. p. 1; W.H. Young, "On Uniform and Non-uniform + Convergence," _Proc. London Math. Soc._ (Ser. 2) t. 6. Further + information and very full references will be found in the articles by + Pringsheim, Schoenflies and Voss in the _Encyclopadie der math. + Wissenschaften_, Bde. i., ii. (Leipzig, 1898, 1899). (A. E. H. L.) + + +II.--FUNCTIONS OF COMPLEX VARIABLES + +In the preceding section the doctrine of functionality is discussed with +respect to real quantities; in this section the theory when complex or +imaginary quantities are involved receives treatment. The following +abstract explains the arrangement of the subject matter: (S 1), _Complex +numbers_, states what a complex number is; (S 2), _Plotting of simple +expressions involving complex numbers_, illustrates the meaning in some +simple cases, introducing the notion of conformal representation and +proving that an algebraic equation has complex, if not real, roots; (S +3), _Limiting operations_, defines certain simple functions of a complex +variable which are obtained by passing to a limit, in particular the +exponential function, and the generalized logarithm, here denoted by +[lambda](z); (S 4), _Functions of a complex variable in general_, after +explaining briefly what is to be understood by a region of the complex +plane and by a path, and expounding a logical principle of some +importance, gives the accepted definition of a function of a complex +variable, establishes the existence of a complex integral, and proves +Cauchy's theorem relating thereto; (S 5), _Applications_, considers the +differentiation and integration of series of functions of a complex +variable, proves Laurent's theorem, and establishes the expansion of a +function of a complex variable as a power series, leading, in (S 6), +_Singular points_, to a definition of the region of existence and +singular points of a function of a complex variable, and thence, in (S +7), _Monogenic Functions_, to what the writer believes to be the +simplest definition of a function of a complex variable, that of +Weierstrass; (S 8), _Some elementary properties of single valued +functions_, first discusses the meaning of a pole, proves that a single +valued function with only poles is rational, gives Mittag-Leffler's +theorem, and Weierstrass's theorem for the primary factors of an +integral function, stating generalized forms for these, leading to the +theorem of (S 9), _The construction of a monogenic function with a given +region of existence_, with which is connected (S10), _Expression of a +monogenic function by rational functions in a given region_, of which +the method is applied in (S 11), _Expression of_ (1 - z)^(-1) _by +polynomials_, to a definite example, used here to obtain (S 12), _An +expansion of an arbitrary function by means of a series of polynomials, +over a star region_, also obtained in the original manner of +Mittag-Leffler; (S 13), _Application of Cauchy's theorem to the +determination of definite integrals_, gives two examples of this method; +(S 14), _Doubly Periodic Functions_, is introduced at this stage as +furnishing an excellent example of the preceding principles. The reader +who wishes to approach the matter from the point of view of Integral +Calculus should first consult the section (S 20) below, dealing with +_Elliptic Integrals_; (S 15), _Potential Functions, Conformal +representation in general_, gives a sketch of the connexion of the +theory of potential functions with the theory of conformal +representation, enunciating the Schwarz-Christoffel theorem for the +representation of a polygon, with the application to the case of an +equilateral triangle; (S 16), _Multiple-valued Functions, Algebraic +Functions_, deals for the most part with algebraic functions, proving +the residue theorem, and establishing that an algebraic function has a +definite Order; (S 17), _Integrals of Algebraic Functions_, enunciating +Abel's theorem; (S 18), _Indeterminateness of Algebraic Integrals_, +deals with the periods associated with an algebraic integral, +establishing that for an elliptic integral the number of these is two; +(S 19), _Reversion of an algebraic integral_, mentions a problem +considered below in detail for an elliptic integral; (S 20), _Elliptic +Integrals_, considers the algebraic reduction of any elliptic integral +to one of three standard forms, and proves that the function obtained by +reversion is single-valued; (S 21), _Modular Functions_, gives a +statement of some of the more elementary properties of some functions of +great importance, with a definition of Automorphic Functions, and a hint +of the connexion with the theory of linear differential equations; (S +22), _A property of integral functions, deduced from the theory of +modular functions_, proves that there cannot be more than one value not +assumed by an integral function, and gives the basis of the well-known +expression of the modulus of the elliptic functions in terms of the +ratio of the periods; (S 23), _Geometrical applications of Elliptic +Functions_, shows that any plane curve of deficiency unity can be +expressed by elliptic functions, and gives a geometrical proof of the +addition theorem for the function RN(u); (S 24), _Integrals of Algebraic +Functions in connexion with the theory of plane curves_, discusses the +generalization to curves of any deficiency; (S 25), _Monogenic Functions +of several independent variables_, describes briefly the beginnings of +this theory, with a mention of some fundamental theorems: (S 26), +_Multiply-Periodic Functions and the Theory of Surfaces_, attempts to +show the nature of some problems now being actively pursued. + +Beside the brevity necessarily attaching to the account here given of +advanced parts of the subject, some of the more elementary results are +stated only, without proof, as, for instance: the monogeneity of an +algebraic function, no reference being made, moreover, to the cases of +differential equations whose integrals are monogenic; that a function +possessing an algebraic addition theorem is necessarily an elliptic +function (or a particular case of such); that any area can be +conformally represented on a half plane, a theorem requiring further +much more detailed consideration of the meaning of _area_ than we have +given; while the character and properties, including the connectivity, +of a Riemann surface have not been referred to. The theta functions are +referred to only once, and the principles of the theory of Abelian +Functions have been illustrated only by the developments given for +elliptic functions. + +S 1. _Complex Numbers._--Complex numbers are numbers of the form x + iy, +where x, y are ordinary real numbers, and i is a symbol imagined capable +of combination with itself and the ordinary real numbers, by way of +addition, subtraction, multiplication and division, according to the +ordinary commutative, associative and distributive laws; the symbol i is +further such that i^2 = -1. + + Taking in a plane two rectangular axes Ox, Oy, we assume that every + point of the plane is definitely associated with two real numbers x, y + (its co-ordinates) and conversely; thus any point of the plane is + associated with a single complex number; in particular, for every + point of the axis Ox, for which y = O, the associated number is an + ordinary real number; the complex numbers thus include the real + numbers. The axis Ox is often called the real axis, and the axis Oy + the imaginary axis. If P be the point associated with the complex + variable z = x + iy, the distance OP be called r, and the positive + angle less than 2[pi] between Ox and OP be called [theta], we may + write z = r(cos[theta] + i sin[theta]); then r is called the modulus + or absolute value of z and often denoted by |z| and [theta] is called + the phase or amplitude of z, and often denoted by ph (z); strictly the + phase is ambiguous by additive multiples of 2[pi]. If z' = x' + iy' be + represented by P', the complex argument z' + z is represented by a + point P" obtained by drawing from P' a line equal to and parallel to + OP; the geometrical representation involves for its validity certain + properties of the plane; as, for instance, the equation z' + z = z + + z' involves the possibility of constructing a parallelogram (with OP" + as diagonal). It is important constantly to bear in mind, what is + capable of easy algebraic proof (and geometrically is Euclid's + proposition III. 7), that the modulus of a sum or difference of two + complex numbers is generally less than (and is never greater than) the + sum of their moduli, and is greater than (or equal to) the difference + of their moduli; the former statement thus holds for the sum of any + number of complex numbers. We shall write E(i[theta]) for cos[theta] + + i sin [theta]; it is at once verified that E(i[alpha]). E(i[beta]) = + E[i([alpha] + [beta])], so that the phase of a product of complex + quantities is obtained by addition of their respective phases. + +S 2. _Plotting and Properties of Simple Expressions involving a Complex +Number._--If we put [zeta] = (z-i)/(z + i), and, putting [zeta] = [xi] + +i[eta], take a new plane upon which [xi], [eta] are rectangular +co-ordinates, the equations [xi] = (x^2 + y^2-1)/[x^2 + (y + 1)^2], +[eta] = -2xy/[x^2 + (y + i)^2] will determine, corresponding to any +point of the first plane, a point of the second plane. There is the one +exception of z = -i, that is, x = 0, y = -1, of which the corresponding +point is at infinity. It can now be easily proved that as z describes +the real axis in its plane the point [zeta] describes once a circle of +radius unity, with centre at [zeta] = 0, and that there is a definite +correspondence of point to point between points in the z-plane which are +above the real axis and points of the [zeta]-plane which are interior to +this circle; in particular z = i corresponds to [zeta] = 0. + + Moreover, [zeta] being a rational function of z, both [xi] and [eta] + are continuous differentiable functions of x and y, save when [zeta] + is infinite; writing [zeta] = [f](x, y) = [f](z - iy, y), the fact + that this is really independent of y leads at once to (Pd)f/(Pd)x + + i(Pd)[f]/(Pd)y = 0, and hence to + + (Pd)[xi] (Pd)[eta] (Pd)[xi] (Pd)[eta] (Pd)^2[xi] (Pd)^2[xi] + -------- = ---------, -------- = - ---------, ---------- + ---------- = 0; + (Pd)x (Pd)y' (Pd)y (Pd)x' (Pd)x^2 (Pd)y^2 + + so that [xi] is not any arbitrary function of x, y, and when [xi] is + known [eta] is determinate save for an additive constant. Also, in + virtue of these equations, if [zeta], [zeta]' be the values of [zeta] + corresponding to two near values of z, say z and z', the ratio + ([zeta]'-[zeta])/(z'- z) has a definite limit when z' = z, independent + of the ultimate phase of z'- z, this limit being therefore equal to + (Pd)[zeta]/(Pd)x, that is, (Pd)[xi]/(Pd)x + i(Pd)[eta])/(Pd)x. + Geometrically this fact is interpreted by saying that if two curves in + the z-plane intersect at a point P, at which both the differential + coefficients (Pd)[xi]/(Pd)x, (Pd)[eta]/(Pd)x are not zero, and P', P" + be two points near to P on these curves respectively, and the + corresponding points of the [zeta]-plane be Q, Q', Q", then (1) the + ratios PP"/PP', QQ"/QQ' are ultimately equal, (2) the angle P'PP" is + equal to Q'QQ", (3) the rotation from PP' to PP" is in the same sense + as from QQ' to QQ", it being understood that the axes of [xi], [eta] + in the one plane are related as are the axes of x, y. Thus any diagram + of the z-plane becomes a diagram of the [zeta]-plane with the same + angles; the magnification, however, which is equal to + _ _ + | /(Pd)[xi]\^2 /(Pd)[xi]\^2 | 1/2 + | ( -------- ) + ( -------- ) | + |_ \ (Pd)x / \ (Pd)y / _| + + varies from point to point. Conversely, it appears subsequently that + the expression of any copy of a diagram (say, a map) which preserves + angles requires the intervention of the complex variable. + + As another illustration consider the case when [zeta] is a polynomial + in z, + + [zeta] = p0 z^n + p1 z^(n - 1) + ... + p_n; + + H being an arbitrary real positive number, it can be shown that a + radius R can be found such for every |z| > R we have |[zeta]| > H; + consider the lower limit of |[zeta]| for |z| < R; as [xi]^2 + [eta]^2 + is a real continuous function of x, y for |z| < R, there is a point + (x, y), say (x0, y0), at which |[zeta]| is least, say equal to [rho], + and therefore within a circle in the [zeta]-plane whose centre is the + origin, of radius [rho], there are no points [zeta] representing + values corresponding to |z| < R. But if [zeta]0 be the value of [zeta] + corresponding to (x0, y0), and the expression of [zeta] - [zeta]0 near + z0 = x0 + iy0, in terms of z - z0, be A(z - z0)^m + B(z - z0)^(m + 1) + + ..., where A is not zero, to two points near to (x0, y0), say (x1, + y1) or z1 and z2 = z0 + (z1 - z0)(cos [pi]/m + i sin [pi]/m), will + correspond two points near to [zeta]0, say [zeta]1, and 2[zeta]0 + -[zeta]'1, situated so that [zeta]0 is between them. One of these must + be within the circle ([rho]). We infer then that [rho] = 0, and have + proved that every polynomial in z vanishes for some value of z, and + can therefore be written as a product of factors of the form z - + [alpha], where [alpha] denotes a complex number. This proposition + alone suffices to suggest the importance of complex numbers. + +S 3. _Limiting Operations._--In order that a complex number [zeta] = +[xi] + i[eta] may have a limit it is necessary and sufficient that each +of [xi] and [eta] has a limit. Thus an infinite series w0 + w1 + w2 + +..., whose terms are complex numbers, is convergent if the real series +formed by taking the real parts of its terms and that formed by the +imaginary terms are both convergent. The series is also convergent if +the real series formed by the moduli of its terms is convergent; in that +case the series is said to be absolutely convergent, and it can be shown +that its sum is unaltered by taking the terms in any other order. +Generally the necessary and sufficient condition of convergence is that, +for a given real positive [epsilon], a number m exists such that for +every n > m, and every positive p, the batch of terms w_n + w_(n + 1) + +... + w_(n + p) is less than [epsilon] in absolute value. If the terms +depend upon a complex variable z, the convergence is called _uniform_ +for a range of values of z, when the inequality holds, for the same +[epsilon] and m, for all the points z of this range. + + The infinite series of most importance are those of which the general + term is a_nz^n, wherein a_n is a constant, and z is regarded as + variable, n = 0, 1, 2, 3, ... Such a series is called a power series, + if a real and positive number M exists such that for z = z0 and every + n, |a_n z0^n| < M, a condition which is satisfied, for instance, if + the series converges for z = z0, then it is at once proved that the + series converges absolutely for every z for which |z| < |z0|, and + converges uniformly over every range |z| < r' for which r' < |z0|. To + every power series there belongs then a circle of convergence within + which it converges absolutely and uniformly; the function of z + represented by it is thus continuous within the circle (this being the + result of a general property of uniformly convergent series of + continuous functions); the sum for an interior point z is, however, + continuous with the sum for a point z0 on the circumference, as z + approaches to z0 provided the series converges for z = z0, as can be + shown without much difficulty. Within a common circle of convergence + two power series [Sigma] a_n z^n, [Sigma] b_n z^n can be multiplied + together according to the ordinary rule, this being a consequence of a + theorem for absolutely convergent series. If r1 be less than the + radius of convergence of a series [Sigma] a_nz^n and for |z| = r1, the + sum of the series be in absolute value less than a real positive + quantity M, it can be shown that for |z| = r1 every term is also less + than M in absolute value, namely, |a_n| < Mr1^(-n). If in every + arbitrarily small neighbourhood of z = 0 there be a point for which + two converging power series [Sigma]a_nz^n, [Sigma]b_n z^n agree in + value, then the series are identical, or a_n = b_n; thus also if + [Sigma]a_nz^n vanish at z = 0 there is a circle of finite radius about + z = 0 as centre within which no other points are found for which the + sum of the series is zero. Considering a power series [f](z) = + [Sigma]a_nz^n of radius of convergence R, if |z0| < R and we put z = + z0 + t with |t| < R-|z0|, the resulting series [Sigma]a_n (z0 + t)^n + may be regarded as a double series in z0 and t, which, since |z0| + t + < R, is absolutely convergent; it may then be arranged according to + powers of t. Thus we may write [f](z) = [Sigma]A_n t^n; hence A0 = + [f](z0), and we have [[f](z0 + t) - [f](z0)]/t = [Sigma](n=1) A_n + t^(n-l), wherein the continuous series on the right reduces to A1 for + t = 0; thus the ratio on the left has a definite limit when t = 0, + equal namely to A1 or [Sigma]na_nz0^(n - 1). In other words, the + original series may legitimately be differentiated at any interior + point z0 of its circle of convergence. Repeating this process we find + [f](z0 + t) = [Sigma]t^n [f]^(n) (z0)/n!, where [f]^(n) (z0) is the + nth differential coefficient. Repeating for this power series, in t, + the argument applied about z = 0 for [Sigma]a_n z^n, we infer that for + the series [f](z) every point which reduces it to zero is an isolated + point, and of such points only a finite number lie within a circle + which is within the circle of convergence of [f](z). + + Perhaps the simplest possible power series is e^z = exp(z) = 1 + + z^2/2! + z^3/3! + ... of which the radius of convergence is infinite. + By multiplication we have exp(z).exp(z^1) = exp(z + z^1). In + particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy). + Now the functions + + U0 = sin y, V0 = 1 - cos y, U1 = y - sin y, + V1 = 1/2y^2 - 1 + cos y, U2 = (1/6)y^3 - y + sin y, + V2 = (1/24)y^4 - 1/2y^2 + 1 - cos y, ... + + all vanish for y = 0, and the differential coefficient of any one + after the first is the preceding one; as a function (of a real + variable) is increasing when its differential coefficient is positive, + we infer, for y positive, that each of these functions is positive; + proceeding to a limit we hence infer that + + cos y = 1 - 1/2y^2 + (1/24)y^4 - ..., sin y = y - (1/6)y^3 + + (1/120)y^5 - ..., + + for positive, and hence, for all values of y. We thus have exp(iy) = + cos y + i sin y, and exp (z) = exp (x).(cos y + i sin y). In other + words, the modulus of exp (z) is exp (x) and the phase is y. Hence + also + + exp(z + 2[pi]i) = exp(x) [cos (y + 2[pi]) + i sin(y + 2[pi])], + + which we express by saying that exp (z) has the period 2[pi]i, and + hence also the period 2k[pi]i, where k is an arbitrary integer. From + the fact that the constantly increasing function exp (x) can vanish + only for x = 0, we at once prove that exp (z) has no other periods. + + Taking in the plane of z an infinite strip lying between the lines y = + 0, y = 2[pi] and plotting the function [zeta] = exp (z) upon a new + plane, it follows at once from what has been said that every complex + value of [zeta] arises when z takes in turn all positions in this + strip, and that no value arises twice over. The equation [zeta] = + exp(z) thus defines z, regarded as depending upon [zeta], with only an + additive ambiguity 2k[pi]i, where k is an integer. We write z = + [lambda]([zeta]); when [zeta] is real this becomes the logarithm of + [zeta]; in general [lambda]([zeta]) = log |[zeta]| + i ph ([zeta]) + + 2k[pi]i, where k is an integer; and when [zeta] describes a closed + circuit surrounding the origin the phase of [zeta] increases by 2[pi], + or k increases by unity. Differentiating the series for [zeta] we have + d[zeta]/dz = [zeta], so that z, regarded as depending upon [zeta], is + also differentiable, with dz/d[zeta] = [zeta]^(-1). On the other hand, + consider the series [zeta] - 1 - 1/2([zeta] - 1)^2 + 1/3([zeta] - 1)^3 + - ...; it converges when [zeta] = 2 and hence converges for |[zeta] - + 1| < 1; its differential coefficient is, however, 1 - ([zeta] - 1) + + ([zeta] - 1)^2 - ..., that is, (1 + [zeta] - 1)^(-1). Wherefore if + [phi]([zeta]) denote this series, for |[zeta] - 1| < 1, the difference + [lambda]([zeta]) - [phi]([zeta]), regarded as a function of [xi] and + [eta], has vanishing differential coefficients; if we take the value + of [lambda]([zeta]) which vanishes when [zeta] = 1 we infer thence + that for |[zeta] - 1| < 1, [lambda]([zeta]) = [Sigma][n = 1] [(-1)^(n + - 1)]/n ([zeta] - 1)^n. It is to be remarked that it is impossible for + [zeta] while subject to |[zeta] - 1| < 1 to make a circuit about the + origin. For values of [zeta] for which |[zeta] - 1| [not less than] 1, + we can also calculate [lambda]([zeta]) with the help of infinite + series, utilizing the fact that [lambda]([zeta][zeta]') = + [lambda]([zeta]) + [lambda]([zeta]'). + + The function [lambda]([zeta]) is required to define [zeta]^a when + [zeta] and a are complex numbers; this is defined as exp + [a[lambda]([zeta])], that is as [Sigma] (n=0) a^n[[lambda] + ([zeta])]^n/n!. When a is a real integer the ambiguity of + [lambda]([zeta]) is immaterial here, since exp [a[lambda]([zeta]) + + 2ka[pi]i] = exp[a[lambda]([zeta])]; when a is of the form 1/q, where q + is a positive integer, there are q values possible for [zeta]^(1/q), + of the form exp [1/q [lambda]([zeta])] exp(2k[pi]i/q), with k = 0, 1, + ... q - 1, all other values of k leading to one of these; the qth + power of any one of these values is [zeta]; when a = p/q, where p, q + are integers without common factor, q being positive, we have + [zeta]^(p/q) = ([zeta]^(1/q))^p. The definition of the symbol [zeta]^a + is thus a generalization of the ordinary definition of a power, when + the numbers are real. As an example, let it be required to find the + meaning of i^i; the number i is of modulus unity and phase 1/2[pi]; + thus [lambda](i) = i(1/2[pi] + 2k[pi]); thus + + i^i = exp(-1/2[pi] - 2k[pi]) = exp(-1/2[pi]) exp(-2k[pi]), + + is always real, but has an infinite number of values. + + The function exp (z) is used also to define a generalized form of the + cosine and sine functions when z is complex; we write, namely, cos z = + 1/2[exp(iz) + exp(-iz)] and sin z = -1/2i[exp(iz) - exp(-iz)]. It will + be found that these obey the ordinary relations holding when z is + real, except that their moduli are not inferior to unity. For example, + cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity. + +S4. _Of Functions of a Complex Variable in General._--We have in what +precedes shown how to generalize the ordinary rational, algebraic and +logarithmic functions, and considered more general cases, of functions +expressible by power series in z. With the suggestions furnished by +these cases we can frame a general definition. So far our use of the +plane upon which z is represented has been only illustrative, the +results being capable of analytical statement. In what follows this +representation is vital to the mode of expression we adopt; as then the +properties of numbers cannot be ultimately based upon spatial +intuitions, it is necessary to indicate what are the geometrical ideas +requiring elucidation. + + Consider a square of side a, to whose perimeter is attached a definite + direction of description, which we take to be counter-clockwise; + another square, also of side a, may be added to this, so that there is + a side common; this common side being erased we have a composite + region with a definite direction of perimeter; to this a third square + of the same size may be attached, so that there is a side common to it + and one of the former squares, and this common side may be erased. If + this process be continued any number of times we obtain a region of + the plane bounded by one or more polygonal closed lines, no two of + which intersect; and at each portion of the perimeter there is a + definite direction of description, which is such that the region is on + the left of the describing point. Similarly we may construct a region + by piecing together triangles, so that every consecutive two have a + side in common, it being understood that there is assigned an upper + limit for the greatest side of a triangle, and a lower limit for the + smallest angle. In the former method, each square may be divided into + four others by lines through its centre parallel to its sides; in the + latter method each triangle may be divided into four others by lines + joining the middle points of its sides; this halves the sides and + preserves the angles. When we speak of a _region_ of the plane in + general, unless the contrary is stated, we shall suppose it capable of + being generated in this latter way by means of a finite number of + triangles, there being an upper limit to the length of a side of the + triangle and a lower limit to the size of an angle of the triangle. We + shall also require to speak of a _path_ in the plane; this is to be + understood as capable of arising as a limit of a polygonal path of + finite length, there being a definite direction or sense of + description at every point of the path, which therefore never meets + itself. From this the meaning of a closed path is clear. The boundary + points of a region form one or more closed paths, but, in general, it + is only in a limiting sense that the interior points of a closed path + are a region. + + There is a logical principle also which must be referred to. We + frequently have cases where, about every interior or boundary, point + z0 of a certain region a circle can be put, say of radius r0, such + that for all points z of the region which are interior to this circle, + for which, that is, |z - z0| < r0, a certain property holds. Assuming + that to r0 is given the value which is the upper limit for z0, of the + possible values, we may call the points |z - z0| < r0, the + neighbourhood belonging to or _proper_ to z0, and may speak of the + property as the property (z, z0). The value of r0 will in general vary + with z0; what is in most cases of importance is the question whether + the lower limit of r0 for all positions is zero or greater than zero. + (A) This lower limit is certainly greater than zero provided the + property (z, z0) is of a kind which we may call extensive; such, + namely, that if it holds, for some position of z0 and all positions of + z, within a certain region, then the property (z, z1) holds within a + circle of radius R about any interior point z1 of this region for all + points z for which the circle |z - z1| = R is within the region. Also + in this case r0 varies continuously with z0. (B) Whether the property + is of this extensive character or not we can prove that the region can + be divided into a finite number of sub-regions such that, for every + one of these, the property holds, (1) for _some_ point z0 within or + upon the boundary of the sub-region, (2) for _every_ point z within or + upon the boundary of the sub-region. + + We prove these statements (A), (B) in reverse order. To prove (B) let + a region for which the property (z, z0) holds for all points z and + some point z0 of the region, be called _suitable_: if each of the + triangles of which the region is built up be suitable, what is desired + is proved; if not let an unsuitable triangle be subdivided into four, + as before explained; if one of these subdivisions is unsuitable let it + be again subdivided; and so on. Either the process terminates and then + what is required is proved; or else we obtain an indefinitely + continued sequence of unsuitable triangles, each contained in the + preceding, which converge to a point, say [zeta]; after a certain + stage all these will be interior to the proper region of [zeta]; this, + however, is contrary to the supposition that they are all unsuitable. + + We now make some applications of this result (B). Suppose a definite + finite real value attached to every interior or boundary point of the + region, say [f](x, y). It may have a finite upper limit H for the + region, so that no point (x, y) exists for which [f](x, y) > H, but + points (x, y) exist for which [f](x, y) > H - [epsilon], however small + [epsilon] may be; if not we say that its upper limit is infinite. + There is then at least one point of the region such that, for points + of the region within a circle about this point, the upper limit of + [f](x, y) is H, however small the radius of the circle be taken; for + if not we can put about every point of the region a circle within + which the upper limit of [f](x, y) is less than H; then by the result + (B) above the region consists of a finite number of sub-regions within + each of which the upper limit is less than H; this is inconsistent + with the hypothesis that the upper limit for the whole region is H. A + similar statement holds for the lower limit. A case of such a function + [f](x, y) is the radius r0 of the neighbourhood proper to any point + z0, spoken of above. We can hence prove the statement (A) above. + + Suppose the property (z, z0) extensive, and, if possible, that the + lower limit of r0 is zero. Let then [zeta] be a point such that the + lower limit of r0 is zero for points z0 within a circle about [zeta] + however small; let r be the radius of the neighbourhood proper to + [zeta]; take z0 so that |z0 - [zeta]| < 1/2r; the property (z, z0), + being extensive, holds within a circle, centre z0, of radius r - |z0 - + [zeta]|, which is greater than |z0 - [zeta]|, and increases to r as + |z0 - [zeta]| diminishes; this being true for all points z0 near + [zeta], the lower limit of r0 is not zero for the neighbourhood of + [zeta], contrary to what was supposed. This proves (A). Also, as is + here shown that r0 [ = >] r - |z0-[zeta]|, may similarly be shown that + r [=>] r0 - |z0 - [zeta]|. Thus r0 differs arbitrarily little from r + when |z0-[zeta]| is sufficiently small; that is, r0 varies + continuously with z0. Next suppose the function [f](x, y), which has a + definite finite value at every point of the region considered, to be + continuous but not necessarily real, so that about every point z0, + within or upon the boundary of the region, [eta] being an arbitrary + real positive quantity assigned beforehand, a circle is possible, so + that for all points z of the region interior to this circle, we have + |[f](x, y) - [f](x0, y0)| < 1/2[eta], and therefore (x', y') being any + other point interior to this circle, |[f](x', y') - [f](x, y)| < + [eta]. We can then apply the result (A) obtained above, taking for the + neighbourhood proper to any point z0 the circular area within which, + for any two points (x, y), (x', y'), we have |[f](x', y') - [f](x, y)| + < [eta]. This is clearly an extensive property. Thus, a number r is + assignable, greater than zero, such that, for any two points (x, y), + (x', y') within a circle |z - z0| = r about any point z0, we have + |[f](x', y') - [f](x, y)| < [eta], and, in particular, |[f](x, y) - + [f](x0, y0)| < [eta], where [eta] is an arbitrary real positive + quantity agreed upon beforehand. + + Take now any path in the region, whose extreme points are z0, z, and + let z1, ... z_(n - 1) be intermediate points of the path, in order; + denote the continuous function [f](x, y) by [f](z), and let [f]_r + denote any quantity such that |[f]_r - [f](z_r)| [=<] |[f](z_(r + 1)) + - [f](z_r)|; consider the sum + + (z1 - z0)[f]0 + (z2 - z1)[f]1 + ... + (z - z_(n - 1))[f](n - 1). + + By the definition of a path we can suppose, n being large enough, that + the intermediate points z1, ... z_(n - 1) are so taken that if z_i, + z_(i + 1) be any two points intermediate, in order, to z_r and z_(r + + 1), we have |z_(i + i) - z_i| < |z_(r + 1) - z_r|; we can thus suppose + |z1 - z0|, |z2 - z1|, ... |z - z_(n - 1)|all to converge constantly + to zero. This being so, we can show that the sum above has a definite + limit. For this it is sufficient, as in the case of an integral of a + function of one real variable, to prove this to be so when the + convergence is obtained by taking new points of division intermediate + to the former ones. If, however, z_(r, 1), z_(r, 2), ... z_(r, m - 1) + be intermediate in order to z_r and z_(r + 1), and |[f]_(r, i) - + [f](z_(r, i))| < |[f](z_(r, i + 1)) - [f](z_(r, i))|, the difference + between [Sigma](z_(r + 1) - z_r)[f]_r and + + [Sigma]{(z_(r, 1) - z_r)[f]_(r, 0) + (z_(r, 2) - z{r, 1})[f]_(r, 1) + + ... + (z_(r + 1) - z_(r, m - 1))[f]_(r, m - 1)}, + + which is equal to + + [Sigma]_r [Sigma]_i (z_(r, i + 1) - z_(r, i))([f]_(r, i) - [f]_r), + + is, when |z_(r + 1) - z_r| is small enough, to ensure |[f](z_(r + + 1)) - [f](z_r)| < [eta], less in absolute value than + + [Sigma]2[eta] [Sigma] |z_(r, i + 1) - z{r, i}|, + + which, if S be the upper limit of the perimeter of the polygon from + which the path is generated, is < 2[eta]S, and is therefore + arbitrarily small. + + The limit in question is called [int](z_0 to z) [f](z)dz. In + particular when [f](z) = 1, it is obvious from the definition that its + value is z - z0; when [f](z) = z, by taking [f]_r = 1/2(z_(r + 1) - + z_r), it is equally clear that its value is 1/2(z^2 - z0^2); these + results will be applied immediately. + + Suppose now that to every interior and boundary point z0 of a certain + region there belong two definite finite numbers [f](z0), F(z0), such + that, whatever real positive quantity [eta] may be, a real positive + number [epsilon] exists for which the condition + + | [f](z) - [f](z0) | + | ---------------- - F(z0) | < [eta], + | z-z0 | + + which we describe as the condition (z, z0), is satisfied for every + point z, within or upon the boundary of the region, satisfying the + limitation |z - z0| < [epsilon]. Then [f](z0) is called a + differentiable function of the complex variable z0 over this region, + its differential coefficient being F(z0). The function [f](z0) is thus + a continuous function of the real variables x0, y0, where z0 = x0 + + iy0, over the region; it will appear that F(z0) is also continuous and + in fact also a differentiable function of z0. + + Supposing [eta] to be retained the same for all points z0 of the + region, and [sigma]0 to be the upper limit of the possible values of + [epsilon] for the point z0, it is to be presumed that [sigma]0 will + vary with z0, and it is not obvious as yet that the lower limit of the + values of [sigma]0 as z0 varies over the region may not be zero. We + can, however, show that the region can be divided into a finite number + of sub-regions for each of which the condition (z, z0), above, is + satisfied for all points z, within or upon the boundary of this + sub-region, for an appropriate position of z0, within or upon the + boundary of this sub-region. This is proved above as result (B). + + Hence it can be proved that, for a differentiable function [f](z), the + integral [int](z_1 to z) [f](z)dz has the same value by whatever path + within the region we pass from z1 to z. This we prove by showing that + when taken round a closed path in the region the integral + [int][f](z)dz vanishes. Consider first a triangle over which the + condition (z, z0) holds, for some position of z0 and every position of + z, within or upon the boundary of the triangle. Then as + + [f](z) = [f](z0) + (z - z0)F(z0) + [eta][theta](z - z0), where |[theta]| < 1, + + we have + _ _ _ _ + / / / / + |[f](z)dz = [[f](z0) - z0F(z0)] |dz + F(z0) |zdz + [eta] |[theta](z - z0)dz, + _/ _/ _/ _/ + + which, as the path is closed, is [eta] [int][theta](z-z0)dz. Now, from + the theorem that the absolute value of a sum is less than the sum of + the absolute values of the terms, this last is less, in absolute + value, than [eta]ap, where a is the greatest side of the triangle and + p is its perimeter; if [Delta] be the area of the triangle, we have + [Delta] = 1/2ab sin C > ([alpha]/[pi])ba, where [alpha] is the least + angle of the triangle, and hence a(a + b + c) < 2a(b + c) < + 4[pi][Delta]/[alpha]; the integral [int][f](z)dz round the perimeter + of the triangle is thus < 4[pi][eta][Delta]/[alpha]. Now consider any + region made up of triangles, as before explained, in each of which the + condition (z, z0) holds, as in the triangle just taken. The integral + [int][f](z)dz round the boundary of the region is equal to the sum of + the values of the integral round the component triangles, and thus + less in absolute value than 4[pi][eta]K/[alpha], where K is the whole + area of the region, and [alpha] is the smallest angle of the component + triangles. However small [eta] be taken, such a division of the region + into a finite number of component triangles has been shown possible; + the integral round the perimeter of the region is thus arbitrarily + small. Thus it is actually zero, which it was desired to prove. Two + remarks should be added: (1) The theorem is proved only on condition + that the closed path of integration belongs to the region at every + point of which the conditions are satisfied. (2) The theorem, though + proved only when the region consists of triangles, holds also when the + boundary points of the region consist of one or more closed paths, no + two of which meet. + + Hence we can deduce the remarkable result that the value of [f](z) at + any interior point of a region is expressible in terms of the value of + [f](z) at the boundary points. For consider in the original region the + function [f](z)/(z - z0), where z0 is an interior point: this + satisfies the same conditions as [f](z) except in the immediate + neighbourhood of z0. Taking out then from the original region a small + regular polygonal region with z0 as centre, the theorem holds for the + remaining portion. Proceeding to the limit when the polygon becomes a + circle, it appears that the integral [int] dz[f](z)/(z - z0) round the + boundary of the original region is equal to the same integral taken + counter-clockwise round a small circle having z0 as centre; on this + circle, however, if z - z0 = rE(i[theta]), dz/(z - z0) = id[theta], + and [f](z) differs arbitrarily little from f(z0) if r is sufficiently + small; the value of the integral round this circle is therefore, + ultimately, when r vanishes, equal to 2[pi]i[f](z0). Hence [f](z0) = 1 + / 2[pi]i [int] [dt[f](t)/(t - z0)], where this integral is round the + boundary of the original region. From this it appears that + _ + [f](z) - [f](z0) 1 / dt[f](t) + F(z0) = lim. ---------------- = ------ | --------- + z - z0 2[pi]i _/ (t-z0)^2 + + also round the boundary of the original region. This form shows, + however, that F(z0) is a continuous, finite, differentiable function + of z0 over the whole interior of the original region. + +S 5. _Applications._--The previous results have manifold applications. + + (1) If an infinite series of differentiable functions of z be + uniformly convergent along a certain path lying with the region of + definition of the functions, so that S(2) = u0(z) + u1(z) + ... + u_(n + - 1)(z) + R_n(z), where |R_n(z)| < [epsilon] for all points of the + path, we have + _ _ _ _ _ + /z /z /z /z /z + | S(z)dz = | u0(z)dz + | u1(z)dz + ... + | u_(n - 1)(z)dz + | R_n(z)dz, + _/z0 _/z0 _/z0 _/z0 _/z0 + + wherein, in absolute value, [int](z_0 to z) R_n(z)dz < [epsilon]L, if + L be the length of the path. Thus the series may be integrated, and + the resulting series is also uniformly convergent. + + (2) If [f](x, y) be definite, finite and continuous at every point of + a region, and over any closed path in the region [int][f](x, y)dz = 0, + then [psi](z) = [int](z_0 to z) [f](x, y)dz, for interior points z0, + z, is a differentiable function of z, having for its differential + coefficient the function [f](x, y), which is therefore also a + differentiable function of z at interior points. + + (3) Hence if the series u0(z) + u1(z) + ... to [oo] be uniformly + convergent over a region, its terms being differentiable functions of + z, then its sum S(z) is a differentiable function of z, whose + differential coefficient, given by (1 / 2[pi]i) [int] (2[pi]i / + (t - z)^2), is obtainable by differentiating the series. This theorem, + unlike (1), does not hold for functions of a real variable. + + (4) If the region of definition of a differentiable function [f](z) + include the region bounded by two concentric circles of radii r, R, + with centre at the origin, and z0 be an interior point of this region, + _ _ + 1 / [f](t)dt 1 / [f](t)dt + [f](z0) = ------ | -------- - ------ | --------, + 2[pi]i _/R t - z0 2[pi]i _/r t - z0 + + where the integrals are both counter-clockwise round the two + circumferences respectively; putting in the first (t - z0)^(-1) = + [Sigma]_(n=0) z0^n/t^(n + 1), and in the second (t - z0)^(-1) = + [Sigma]_(n=0) t^n/z0^(n + 1), we find [f](z0) = [Sigma] (-[oo] to + [oo]) A_nz0^n, wherein A_n = (1 / 2[pi]i) [int] [f(t) / t^(n + 1)] dt, + taken round any circle, centre the origin, of radius intermediate + between r and R. Particular cases are: ([alpha]) when the region of + definition of the function includes the whole interior of the outer + circle; then we may take r = 0, the coefficients A_n for which n < 0 + all vanish, and the function [f](z0) is expressed for the whole + interior |z0| < R by a power series [Sigma] (0 to [oo]) A_n z0^n. In + other words, _about every interior point c of the region of definition + a differentiable function of z is expressible by a power series in z - + c; a very important result. + + ([beta]) If the region of definition, though not including the origin, + extends to within arbitrary nearness of this on all sides, and at the + same time the product z^m [f](z) has a finite limit when |z| + diminishes to zero, all the coefficients A_n for which n < -m vanish, + and we have + + f(z0) = A_(-m) z0^(-m) + A_(-m + 1) z0^(-m + 1) + + ... + A_(-1) z0^(-1) + A0 + A1z0 ... to [oo]. + + Such a case occurs, for instance, when [f](z) = cosec z, the number m + being unity. + +S 6. _Singular Points._--The _region of existence_ of a differentiable +function of z is an unclosed aggregate of points, each of which is an +interior point of a neighbourhood consisting wholly of points of the +aggregate, at every point of which the function is definite and finite +and possesses a unique finite differential coefficient. Every point of +the plane, not belonging to the aggregate, which is a limiting point of +points of the aggregate, such, that is, that points of the aggregate lie +in every neighbourhood of this, is called a _singular point_ of the +function. + + About every interior point z0 of the region of existence the function + may be represented by a power series in z-z0, and the series converges + and represents the function over any circle centre at z0 which + contains no singular point in its interior. This has been proved + above. And it can be similarly proved, putting z = 1/[zeta], that if + the region of existence of the function contains all points of the + plane for which |z| > R, then the function is representable for all + such points by a power series in z^(-1) or [zeta]; in such case we say + that the region of existence of the function contains the point z = + [oo]. A series in z^(-1) has a finite limit when |z| = [oo]; a series + in z cannot remain finite for all points z for which |z| > R; for if, + for |z| = R, the sum of a power series [Sigma]a_n z^n in z is in + absolute value less than M, we have |a_n| < Mr(-n), and therefore, if + M remains finite for all values of r however great, a_n = 0. Thus the + region of existence of a function if it contains all finite points of + the plane cannot contain the point z = [oo]; such is, for instance, + the case of the function exp (z) = [Sigma]z^n/n!. This may be regarded + as a particular case of a well-known result (S 7), that the + circumference of convergence of any power series representing the + function contains at least one singular point. As an extreme case + functions exist whose region of existence is circular, there being a + singular point in every arc of the circumference, however small; for + instance, this is the case for the functions represented for |z| < 1 + by the series [Sigma]_(n=0) z^m, where m = n^2, the series + [Sigma]_(n=0) z^m where m = n!, and the series [Sigma](n=1 to 0) + z^m/(m + 1)(m + 2) where m = a^n, a being a positive integer, although + in the last case the series actually converges for every point of the + circle of convergence |z| = 1. If z be a point interior to the circle + of convergence of a series representing the function, the series may + be rearranged in powers of z - z0; as z0 approaches to a singular + point of the function, lying on the circle of convergence, the radii + of convergence of these derived series in z - z0 diminish to zero; + when, however, a circle can be put about z0, not containing any + singular point of the function, but containing points outside the + circle of convergence of the original series, then the series in z - + z0 gives the value of the function for these external points. If the + function be supposed to be given only for the interior of the original + circle, by the original power series, the series in z - z0 converging + beyond the original circle gives what is known as an _analytical + continuation_ of the function. It appears from what has been proved + that the value of the function at all points of its region of + existence can be obtained from its value, supposed given by a series + in one original circle, by a succession of such processes of + analytical continuation. + +S 7. _Monogenic Functions._--This suggests an entirely different way of +formulating the fundamental parts of the theory of functions of a +complex variable, which appears to be preferable to that so far followed +here. + + Starting with a convergent power series, say in powers of z, this + series can be arranged in powers of z - z0, about any point z0 + interior to its circle of convergence, and the new series converges + certainly for |z - z0| < r - |z0|, if r be the original radius of + convergence. If for every position of z0 this is the greatest radius + of convergence of the derived series, then the original series + represents a function existing only within its circle of convergence. + If for some position of z0 the derived series converges for |z - z0| < + r - |z0| + D, then it can be shown that for points z, interior to the + original circle, lying in the annulus r - |z0| < |z - z0| < r - |z0| + + D, the value represented by the derived series agrees with that + represented by the original series. If for another point z1 interior + to the original circle the derived series converges for |z - z1| < r - + |z1| + E, and the two circles |z - z0| = r - |z0| + D, |z - z1| = r - + |z1| + E have interior points common, lying beyond |z| = r, then it + can be shown that the values represented by these series at these + common points agree. Either series then can be used to furnish an + analytical continuation of the function as originally defined. + Continuing this process of continuation as far as possible, we arrive + at the conception of the function as defined by an aggregate of power + series of which every one has points of convergence common with some + one or more others; the whole aggregate of points of the plane which + can be so reached constitutes the region of existence of the function; + the limiting points of this region are the points in whose + neighbourhood the derived series have radii of convergence diminishing + indefinitely to zero; these are the singular points. The circle of + convergence of any of the series has at least one such singular point + upon its circumference. So regarded the function is called a + _monogenic_ function, the epithet having reference to the single + origin, by one power series, of the expressions representing the + function; it is also sometimes called a _monogenic analytical_ + function, or simply an _analytical_ function; all that is necessary to + define it is the value of the function and of all its differential + coefficients, at some one point of the plane; in the method previously + followed here it was necessary to suppose the function differentiable + at every point of its region of existence. The theory of the + integration of a monogenic function, and Cauchy's theorem, that + [int][f](z)dz = 0 over a closed path, are at once deducible from the + corresponding results applied to a single power series for the + interior of its circle of convergence. There is another advantage + belonging to the theory of monogenic functions: the theory as + originally given here applies in the first instance only to single + valued functions; a monogenic function is by no means necessarily + single valued--it may quite well happen that starting from a + particular power series, converging over a certain circle, and + applying the process of analytical continuation over a closed path + back to an interior point of this circle, the value obtained does not + agree with the initial value. The notion of basing the theory of + functions on the theory of power series is, after Newton, largely due + to Lagrange, who has some interesting remarks in this regard at the + beginning of his _Theorie des fonctions analytiques_. He applies the + idea, however, primarily to functions of a real variable for which the + expression by power series is only of very limited validity; for + functions of a complex variable probably the systematization of the + theory owes most to Weierstrass, whose use of the word monogenic is + that adopted above. In what follows we generally suppose this point of + view to be regarded as fundamental. + +S 8. _Some Elementary Properties of Single Valued Functions._--A _pole_ +is a singular point of the function [f](z) which is not a singularity of +the function 1/[f](z); this latter function is therefore, by the +definition, capable of representation about this point, z0, by a series +[[f](z)]^(-1 ) = [Sigma]a_n (z - z0)^n. If herein a0 is not zero we can +hence derive a representation for [f](z) as a power series about z0, +contrary to the hypothesis that z0 is a singular point for this +function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... a_(m - 1) = 0, +but a_m [+-] 0. Then [[f](z)]^(-1) = (z - z0)^m [a_m + a_(m + 1)(z - z0) ++ ...], and hence (z - z0)^m [f](z) = a_m^(-1) + [Sigma]b_n (z - z0)^n, +namely, the expression of [f](z) about z = z0 contains a finite number +of negative powers of z - z0 and a (finite or) infinite number of +positive powers. Thus a pole is always an isolated singularity. + + The integral [int][f](z)dz taken by a closed circuit about the pole + not containing any other singularity is at once seen to be 2[pi]iA1, + where A1 is the coefficient of (z - z0)^(-1) in the expansion of + [f](z) at the pole; this coefficient has therefore a certain + uniqueness, and it is called the _residue of [f](z) at the pole_. + Considering a region in which there are no other singularities than + poles, all these being interior points, _the integral (1 / 2[pi]i) + [int][f](z)dz round the boundary of this region is equal to the sum + of the residues at the included poles_, a very important result. Any + singular point of a function which is not a pole is called an + _essential singularity_; if it be isolated the function is capable, in + the neighbourhood of this point, of approaching arbitrarily near to + any assigned value. For, the point being isolated, the function can be + represented, in its neighbourhood, as we have proved, by a series + [Sigma] (-[oo] to [oo]) a_n(z - z0)^n; it thus cannot remain finite in + the immediate neighbourhood of the point. The point is necessarily an + isolated essential singularity also of the function {[f](z) - A}^(-1) + for if this were expressible by a power series about the point, so + would also the function [f](z) be; as {[f](z) - A}^(-1) approaches + infinity, so does [f](z) approach the arbitrary value A. Similar + remarks apply to the point z = [oo], the function being regarded as a + function of [zeta] = z^(-1). In the neighbourhood of an essential + singularity, which is a limiting point also of poles, the function + clearly becomes infinite. For an essential singularity which is not + isolated the same result does not necessarily hold. + +A single valued function is said to be an _integral_ function when it +has no singular points except z = [oo]. Such is, for instance, an +integral polynomial, which has z = [oo] for a pole, and the functions +exp (z) which has z = [oo] as an essential singularity. A function which +has no singular points for finite values of z other than poles is called +a _meromorphic_ function. If it also have a pole at z = [oo] it is a +_rational_ function; for then, if a1, ... a_s be its finite poles, of +orders m1; m2, ... m_s, the product (z - a1)^m1 ... (z - a_s)^m_s[f](z) +is an integral function with a pole at infinity, capable therefore, for +large values of z, of an expression (z^ - 1)^(-m) [Sigma]_(r=0) a_r(z^ - +1)^r; thus (z - a1)^m1 ... (z - a_s)^m_s[f](z) is capable of a form +[Sigma]_(r=0) b_r z^r, but z^(-m) [Sigma]_(r=0) b_r z^r remains finite +for z = [oo]. Therefore b_(r + 1) = b_(r + 2) = ... = 0, and[f](z) is a +rational function. + + If for a single valued function F(z) every singular point in the + finite part of the plane is isolated there can only be a finite number + of these in any finite part of the plane, and they can be taken to be + a1, a2, a3, ... with |a1| [=<] |a2| [=<] |a3| ... and limit |a_n| = + [oo]. About a_s the function is expressible as [Sigma] (-[oo] to [oo]) + A_n(z - a_s)^n; let [f]_s(z) = [Sigma] (-[oo] to 1) A^n(z - a_s)^n be + the sum of the negative powers in this expansion. Assuming z = 0 not + to be a singular point, let [f]_s(z) be expanded in powers of z, in + the form [Sigma]_(n=0) C_n z^n, and [mu]_s be chosen so that F_s(z) = + [f]_s(z) - [Sigma] (1 to [mu]_s-1) C_nz^n = [Sigma] ([mu]_s to [oo]) + C_n z^n is, for |z| < r_s < |a_s|, less in absolute value than the + general term [epsilon]_s of a fore-agreed convergent series of real + positive terms. Then the series [phi](z) = [Sigma] (s=1 to [oo]) + F_s(z) converges uniformly in any finite region of the plane, other + than at the points a_s, and is expressible about any point by a power + series, and near a_s, [phi](z) - f_s(z) is expressible by a power + series in z-a_s. Thus F(z) - [phi](z) is an integral function. In + particular when all the finite singularities of F(z) are poles, F(z) + is hereby expressed as the sum of an integral function and a series of + rational functions. The condition |F_s(z)| < [epsilon]_s is imposed + only to render the series [Sigma]F_s(z) uniformly convergent; this + condition may in particular cases be satisfied by a series [Sigma] + G_s(z) where G_s(z) = [f]_s(z) - [Sigma] (1 to [nu]_s-1) C_nz^n and + [nu]_s < [mu]_s. An example of the theorem is the function [pi] cot + [pi]z - z^(-1) for which, taking at first only half the poles, + [f]_s(z) = 1/(z-s); in this case the series [Sigma]F_s(z) where F_s(z) + = (z - s)^-1 + s^-1 is uniformly convergent; thus [pi]cot[pi]z - z^-1 + - [Sigma] (-[oo] to [oo]) [(z - s)^-1 + s^-1], where s = 0 is excluded + from the summation, is an integral function. It can be proved that + this integral function vanishes. + + Considering an integral function [f](z), if there be no finite + positions of z for which this function vanishes, the function + [lambda][[f](z)] is at once seen to be an integral function, [phi](z), + or [f](z) = exp[[phi](z)]; if however great R may be there be only a + finite number of values of z for which [f](z) vanishes, say z = a1, + ... a_m, then it is at once seen that [f](z) = exp [[phi](z)].(z - + a1)^h1...(z - a_m)^h_m, where [phi](z) is an integral function, and + h1, ... h_m are positive integers. If, however, [f](z) vanish for z = + a1, a2 ... where |a1| [=<] |a2| [=<] ... and limit |a_n| = [oo], and + if for simplicity we assume that z - 0 is not a zero and all the zeros + a1, a2, ... are of the first order, we find, by applying the preceding + theorem to the function (1 / [f](z)) (d[f](z) / dz), that [f](z) = + exp[[phi](z)] [Pi] (n=1 to [oo]) {(1 - z/a_n) exp[phi]_n(z)}, where + [phi](z) is an integral function, and [phi]_n(z) is an integral + polynomial of the form + + z z^2 z^s + [phi]_n(z) = --- + ------ + ... + ------. + a_n 2a^2_n sa_n^s + + The number s may be the same for all values of n, or it may increase + indefinitely with n; it is sufficient in any case to take s = n. In + particular for the function sin[pi]x / [pi]x, we have + _ _ + sin[pi]x [oo] | / x \ / x \ | + -------- = [Pi] | ( 1 - -- ) exp ( -- ) |, + [pi]x -[oo] |_ \ n / \ n / _| + + where n = 0 is excluded from the product. Or again we have + _ _ + 1 [oo] | / x \ / x \ | + ---------- = xe^C_x [Pi] | ( 1 - -- ) exp ( - -- ) |, + [Gamma](x) n=1 |_ \ n / \ n / _| + + where C is a constant, and [Gamma](x) is a function expressible when x + is real and positive by the integral [int] (0 to [oo]) + e^(-t) t^(x - 1)dt. + + There exist interesting investigations as to the connexion of the + value of s above, the law of increase of the modulus of the integral + function [f](z), and the law of increase of the coefficients in the + series [f](z) = [Sigma] a_n z^n as n increases (see the bibliography + below under _Integral Functions_). It can be shown, moreover, that an + integral function actually assumes every finite complex value, save, + in exceptional cases, one value at most. For instance, the function + exp (z) assumes every finite value except zero (see below under S 21, + _Modular Functions_). + +The two theorems given above, the one, known as Mittag-Leffler's +theorem, relating to the expression as a sum of simpler functions of a +function whose singular points have the point z = [oo] as their only +limiting point, the other, Weierstrass's factor theorem, giving the +expression of an integral function as a product of factors each with +only one zero in the finite part of the plane, may be respectively +generalized as follows:-- + + I. If a1, a2, a3, ... be an infinite series of isolated points having + the points of the aggregate (c) as their limiting points, so that in + any neighbourhood of a point of (c) there exists an infinite number of + the points a1, a2, ..., and with every point a_i there be associated a + polynomial in (z - a_i)^-1, say g_i; then there exists a single valued + function whose region of existence excludes only the points (a) and + the points (c), having in a point a_i a pole whereat the expansion + consists of the terms g_i, together with a power series in z - a_i; + the function is expressible as an infinite series of terms g_i - + [gamma]_i, where [gamma]_i is also a rational function. + + II. With a similar aggregate (a), with limiting points (c), suppose + with every point a_i there is associated a positive integer r_i. Then + there exists a single valued function whose region of existence + excludes only the points (c), vanishing to order r_i at the point a_i, + but not elsewhere, expressible in the form + + [oo] / a_n - c_n \^r_n + [Pi] ( 1 - --------- ) exp(g_n), + n=1 \ z - c_n / + + where with every point a_n is associated a proper point c_n of (c), + and + + _[mu]_n + \ 1 / a_n - c_n \^s + g_n = r_n /_ -- ( --------- ), + s=1 s \ z - c_n / + + [mu]_n being a properly chosen positive integer. + + If it should happen that the points (c) determine a path dividing the + plane into separated regions, as, for instance, if a_n = R(1 - n^-1) + exp(i[pi] [root]2.n), when (c) consists of the points of the circle + |z| = R, the product expression above denotes different monogenic + functions in the different regions, not continuable into one another. + +S 9. _Construction of a Monogenic Function with a given Region of +Existence._--A series of isolated points interior to a given region can +be constructed in infinitely many ways whose limiting points are the +boundary points of the region, or are boundary points of the region of +such denseness that one of them is found in the neighbourhood of every +point of the boundary, however small. Then the application of the last +enunciated theorem gives rise to a function having no singularities in +the interior of the region, but having a singularity in a boundary point +in every small neighbourhood of every boundary point; this function has +the given region as region of existence. + +S 10. _Expression of a Monogenic Function by means of Rational Functions +in a given Region._--Suppose that we have a region R0 of the plane, as +previously explained, for all the interior or boundary points of which z +is finite, and let its boundary points, consisting of one or more closed +polygonal paths, no two of which have a point in common, be called C0. +Further suppose that all the points of this region, including the +boundary points, are interior points of another region R, whose boundary +is denoted by C. Let z be restricted to be within or upon the boundary +of C0; let a, b, ... be finite points upon C or outside R. Then when b +is near enough to a, the fraction (a - b)/(z - b) is arbitrarily small +for all positions of z; say + + | a - b | + | ----- | < [epsilon], for |a - b| < [eta]; + | z - b | + +the rational function of the complex variable t, + _ _ + 1 | / a - b \^n | + ----- |1 - ( ----- ) |, + t - a |_ \ t - b / _| + +in which n is a positive integer, is not infinite at t = a, but has a +pole at t = b. By taking n large enough, the value of this function, for +all positions z of t belonging to R0, differs as little as may be +desired from (t - a)^-1. By taking a sum of terms such as + + _ _ _ + \ { 1 | / a - b \^n | }^p + F = /_ A_p { ----- |1 - ( ----- ) | }, + { t - a |_ \ t - b / _| } + +we can thus build a rational function differing, in value, in R0, as +little as may be desired from a given rational function + _ + \ + [f] = /_ A_p (t - a)^(-p), + +and differing, outside R or upon the boundary of R, from [f], in the +fact that while [f] is infinite at t = a, F is infinite only at t = b. +By a succession of steps of this kind we thus have the theorem that, +given a rational function of t whose poles are outside R or upon the +boundary of R, and an arbitrary point c outside R or upon the boundary +of R, which can be reached by a finite continuous path outside R from +all the poles of the rational function, we can build another rational +function differing in R0 arbitrarily little from the former, whose poles +are all at the point c. + + Now any monogenic function [f](t) whose region of definition includes + C and the interior of R can be represented at all points z in R[0] by + _ + 1 / [f](t)dt + [f](z) = ------ | --------, + 2[pi]i _/ t - z + + where the path of integration is C. This integral is the limit of a + sum + _ + 1 \ [f](t_i) (t_(i + 1) - t_i) + S = ------ /_ --------------------------, + 2[pi]i t_i - z + + where the points t_i are upon C; and the proof we have given of the + existence of the limit shows that the sum S converges to [f](z) + uniformly in regard to z, when z is in R0, so that we can suppose, + when the subdivision of C into intervals t_(i + 1) - t_i, has been + carried sufficiently far, that + + | S - [f](z) | < [epsilon], + + for all points z of R0, where [epsilon] is arbitrary and agreed upon + beforehand. The function S is, however, a rational function of z with + poles upon C, that is external to R0. We can thus find a rational + function differing arbitrarily little from S, and therefore + arbitrarily little from [f](z), for all points z of R0, with poles at + arbitrary positions outside R0 which can be reached by finite + continuous curves lying outside R from the points of C. + + In particular, to take the simplest case, if C0, C be simple closed + polygons, and [GAMMA] be a path to which C approximates by taking the + number of sides of C continually greater, we can find a rational + function differing arbitrarily little from [f](z) for all points of R0 + whose poles are at one finite point c external to [GAMMA]. By a + transformation of the form t - c = r^-1, with the appropriate change + in the rational function, we can suppose this point c to be at + infinity, in which case the rational function becomes a polynomial. + Suppose [epsilon]1, [epsilon]2, ... to be an indefinitely continued + sequence of real positive numbers, converging to zero, and P_r to be + the polynomial such that, within C0, |P_r - [f](z)| < [epsilon]_r; + then the infinite series of polynomials + + P1(z) + {P2(z) - P1(z)} + {P3(z) - P2(z)} + ..., + + whose sum to n terms is P_n(z), converges for all finite values of z + and represents [f](z) within C0. + + When C consists of a series of disconnected polygons, some of which + may include others, and, by increasing indefinitely the number of + sides of the polygons C, the points C become the boundary points + [Gamma] of a region, we can suppose the poles of the rational + function, constructed to approximate to [f](z) within R0, to be at + points of [Gamma]. A series of rational functions of the form + + H1(z) + {H2(z) - H1(z)} + {H3(z) - H2(z)} + ... + + then, as before, represents [f](z) within R0. And R0 may be taken to + coincide as nearly as desired with the interior of the region bounded + by [Gamma]. + +S 11. _Expression of (1 - z)^(-1) by means of Polynomials. +Applications._--We pursue the ideas just cursorily explained in some +further detail. + + Let c be an arbitrary real positive quantity; putting the complex + variable [zeta] = [xi] + i[eta], enclose the points [zeta] = l, [zeta] + = 1 + c by means of (i.) the straight lines [eta] = [+-]a, from [xi] = + l to [xi] = 1 + c, (ii.) a semicircle convex to [zeta] = 0 of equation + ([xi] - 1)^2 + [eta]^2 = a^2, (iii.) a semicircle concave to [zeta] = + 0 of equation ([xi] - 1 - c)^2 + [eta]^2 = a^2. The quantities c and a + are to remain fixed. Take a positive integer r so that 1/r (c/a) is + less than unity, and put [sigma] = 1/r (c/a). Now take + + c1 = 1 + c/r, c2 = 1 + 2c/r, ... c_r = 1 + c; + + if n1, n2, ... n_r, be positive integers, the rational function + _ _ + 1 | / c1 - 1 \^n1 | + ---------- |1 - ( ----------- ) | + 1 - [zeta] |_ \ c1 - [zeta] / _| + + is finite at [zeta] = 1, and has a pole of order n1 at [zeta] = c1; + the rational function + _ _ _ _ + 1 | / c1 - 1 \^n1 | | / c2 - c1 \^n2 |^n1 + ---------- |1 - ( ------------ ) | |1 - ( ----------- ) | + 1 - [zeta] |_ \ c1 - [zeta] / _| |_ \ c2 - [zeta]/ _| + + is thus finite except for [zeta] = c2, where it has a pole of order + n1n2; finally, writing + + / c_s - c_(s-1) \^n_s + x_s = ( ------------- ), + \ c_s - [zeta] / + + the rational function + + U = (1 - [zeta])^(-1) (1 - x1)(1 - x2)^n1 (1 - x3)^n1n2 ... + (1 - x_r)^(n1n2 ... n_(r - 1)) + + has a pole only at [zeta] = 1 + c, of order n1n2 ... n_r. + + The difference (1 - [zeta])^(-1) - U is of the form (1 - + [zeta])^(-1)P, where P, of the form + + 1 - (1 - [rho]1)(1 - [rho]2)...(1 - [rho]_k), + + in which there are equalities among [rho]1, [rho]2, ... [rho]_k, is of + the form + + [Sigma][rho]1 - [Sigma][rho]1[rho]2 + [Sigma][rho]1[rho]2[rho]3 - ...; + + therefore, if |r_i| = |[rho]_i|, we have + + |P| < [Sigma]r1 + [Sigma]r1r2 + [Sigma]r1r2r3 + ... < + (1 + r1)(1 + r2)...(1 + r_k) - 1; + + now, so long as [zeta] is without the closed curve above described + round [zeta] = 1, [zeta] = 1 + c, we have + + | 1 | 1 |c_m - c_(m-1)| c/r + |----------| < ---, |-------------| < --- < [sigma], + |1 - [zeta]| a |c_m - [zeta] | a + + and hence + + |(1 - [zeta])^(-1) - U| < a^(-1) {(1 + [sigma]^n1) (1 + [sigma]^n2)^n1 + (1 + [sigma]^n3)^n1n2 ... (1 + [sigma]^n_r)^(n1n2 ... n_(r-1)) - 1}. + + Take an arbitrary real positive [epsilon], and [mu], a positive + number, so that [epsilon]^[mu] - 1 < [epsilon]a, then a value of n1 + such that [sigma]^n1 < [mu]/(1 + [mu]) and therefore [sigma]^n1 / (1 - + [sigma]^n1 < [mu], and values for n2, n3 ... such that[sigma]^n2 < + 1/n1 [sigma]^2n1, [sigma]^n3 < 1/n1n2 [sigma]^{3n1, ... [sigma]^n_r} < + 1/(n1...n_(r - 1)} [sigma]^n_rn1; then, as 1 + x < e^x, we have + |(-[zeta])^(-1) - U| < a^-1 {exp([sigma]^n1 + n1[sigma]^n2 + + n1n2[sigma]^n3 + ... + n1n2...n_(r - 1)[sigma]^n_r) - 1}, and + therefore less than + + a^(-1) {exp([sigma]^n1 + [sigma]^2n1 + ... + [sigma]^n_rn1) - 1}, + + which is less than + _ _ + 1 | / [sigma]^n1 \ | + -- |exp ( -------------- ) - 1 | + a |_ \1 - [sigma]^n1/ _| + + and therefore less than [epsilon]. + + The rational function U, with a pole at [zeta] = 1 + c, differs + therefore from (1 - [zeta])^(-1), for all points outside the closed + region put about [zeta] = 1, [zeta] = l + c, by a quantity numerically + less than [epsilon]. So long as a remains the same, r and [sigma] will + remain the same, and a less value of [epsilon] will require at most an + increase of the numbers n1, n2, ... n_r; but if a be taken smaller it + may be necessary to increase r, and with this the complexity of the + function U. + + Now put + + c[zeta] (c + 1)z + z = --------------, [zeta] = --------; + c + 1 - [zeta] c + z + + thereby the points [zeta] = 0, 1, 1 + c become the points z = 0, 1, + [oo], the function (1 - z)^(-1) being given by (1-z)^(-1) = c(c + + 1)^(-1)(1 - [zeta])^(-1) + (c + 1)^(-1); the function U becomes a + rational function of z with a pole only at z = [oo], that is, it + becomes a polynomial in z, say [(c + 1)/c]H - 1/c, where H is also a + polynomial in z, and + _ _ + 1 c | 1 | + ----- - H = ----- | ---------- - U |; + 1 - z c + 1 |_1 + [zeta] _| + + the lines [eta] = [+-]a become the two circles expressed, if z = x + + iy, by + + c(c + 1) + (x + c)^2 + y^2 = [+-] -------- y, + a + + the points ([eta] = 0, [xi] = 1 - a), ([eta] = 0, [xi] = 1 + c + a) + become respectively the points (y = 0, x = c(1 - a)/(c + a), (y = 0, x + = -c(l + c + a)/a), whose limiting positions for a = 0 are + respectively (y = 0, x = 1), (y = 0, x = -[oo]). The circle (x + c)^2 + + y^2 = c(c + 1)y/a can be written + + (x + c)^2 (x + c)^4 + y = --------- + --------- {[mu] + [root][[mu]^2 - (x + c)^2]}^(-2), + 2[mu] 2[mu] + + where [mu] = 1/2c(c + 1)/a; its ordinate y, for a given value of x, + can therefore be supposed arbitrarily small by taking a sufficiently + small. + + We have thus proved the following result; taking in the plane of z any + finite region of which every interior and boundary point is at a + finite distance, however short, from the points of the real axis for + which 1 =< x =< [oo], we can take a quantity a, and hence, with an + arbitrary c, determine a number r; then corresponding to an arbitrary + [epsilon]_s, we can determine a polynomial P_s, such that, for all + points interior to the region, we have + + |(1 - z^(-1)) - P_s| < [epsilon]_s; + + thus the series of polynomials + + P1 + (P2 - P1) + (P3 - P2) + ..., + + constructed with an arbitrary aggregate of real positive numbers + [epsilon]1, [epsilon]2, [epsilon]3, ... with zero as their limit, + converges uniformly and represents (1-z)^(-1) for the whole region + considered. + + S 12. _Expansion of a Monogenic Function in Polynomials, over a Star + Region._--Now consider any monogenic function [f](z) of which the + origin is not a singular point; joining the origin to any singular + point by a straight line, let the part of this straight line, produced + beyond the singular point, lying between the singular point and z = + [oo], be regarded as a barrier in the plane, the portion of this + straight line from the origin to the singular point being erased. + Consider next any finite region of the plane, whose boundary points + constitute a path of integration, in a sense previously explained, of + which every point is at a finite distance greater than zero from each + of the barriers before explained; we suppose this region to be such + that any line joining the origin to a boundary point, when produced, + does not meet the boundary again. For every point x in this region R + we can then write + _ + / dt [f](t) + 2[pi]i[f](x) = | -- -----------, + _/ t 1 - xt^(-1) + + where [f](x) represents a monogenic branch of the function, in case it + be not everywhere single valued, and t is on the boundary of the + region. Describe now another region R0 lying entirely within R, and + let x be restricted to be within R0 or upon its boundary; then for any + point t on the boundary of R, the points z of the plane for which + zt^(-1) is real and positive and equal to or greater than 1, being + points for which |z| = |t| or |z| > |t|, are without the region R0, + and not infinitely near to its boundary points. Taking then an + arbitrary real positive [epsilon] we can determine a polynomial in + xt^(-1), say P(xt^(-1)), such that for all points x in R0 we have + + |[1 - xt^(-1)]^(-1) - P[xt^(-1)]| < [epsilon]; + + the form of this polynomial may be taken the same for all points t on + the boundary of R, and hence, if E be a proper variable quantity of + modulus not greater than [epsilon], + _ _ + | / dt | | / dt | + |2[pi]i[f](x) - | --[f](t)P(xt^(-1))| = | | --[f](t)E| <= [epsilon]LM, + | _/ t | | _/ t | + + where L is the length of the path of integration, the boundary of R, + and M is a real positive quantity such that upon this boundary + |t^(-1)[f](t)| < M. If now + + P(xt^(-1)) = c0 + c1xt^(-1) + ... + c_mx^mt^(-m), + + and + _ + 1 / + ------ | t^(-r-1)[f](t)dt = [mu]_r, + 2[pi]i _/ + + this gives + + |[f](x) - {c0[mu]0 + c1[mu]1x + ... + c_m[mu]_mx^m}| =< [epsilon]LM/2[pi], + + where the quantities [mu]0, [mu]1, [mu]2, ... are the coefficients in + the expansion of [f](x) about the origin. + + If then an arbitrary finite region be constructed of the kind + explained, excluding the barriers joining the singular points of + [f](x) to x = [oo], it is possible, corresponding to an arbitrary real + positive number [sigma], to determine a number m, and a polynomial + Q(x), of order m, such that for all interior points of this region + + |[f](x) - Q(x)| < [sigma]. + + Hence as before, within this region [f](x) can be represented by a + series of polynomials, converging uniformly; when [f](x) is not a + single valued function the series represents one branch of the + function. + + The same result can be obtained without the use of Cauchy's integral. + We explain briefly the character of the proof. If a monogenic function + of t, [phi](t) be capable of expression as a power series in t-x about + a point x, for |t - x| =< [rho], and for all points of this circle + |[phi](t)| < g, we know that |[phi]^(n)(x)| < g[rho]^(-n)(n!). Hence, + taking |z| < 1/3[rho], and, for any assigned positive integer [mu], + taking m so that for n > m we have ([mu] + n)^[mu] < (3/2)^n, we have + + |[phi]^(([mu] + n))(x)z^n| [phi]^([mu] + n)(x) + |------------------------| < -------------------([mu] + n)^([mu])|z|^n + | n! | ([mu] + n)! + + g /3 \ /[rho]\ g + < ---------------- ( -- )^n ( ----- )^n < --------------, + [rho]^([mu] + n) \2 / \ 3 / [rho]^[mu] 2^n + + and therefore + + _m + \ [phi]^([mu] + n)(x) + [phi]^([mu])(x + z) = /_ ------------------- z^n + [epsilon]_[mu], + n=0 n! + + where + + g _[oo] 1 g + |[epsilon]_([mu])| < ---------- \ --- < -------------- + [rho]^[mu] /_ 2^n [rho]^[mu] 2^m + n=m+1 + + Now draw barriers as before, directed from the origin, joining the + singular point of [phi](z) to z = [oo], take a finite region excluding + all these barriers, let [rho] be a quantity less than the radii of + convergence of all the power series developments of [phi](z) about + interior points of this region, so chosen moreover that no circle of + radius [rho] with centre at an interior point of the region includes + any singular point of [phi](z), let g be such that |[phi](z)| < g for + all circles of radius [rho] whose centres are interior points of the + region, and, x being any interior point of the region, choose the + positive integer n so that 1/n |x| 1/3 - [rho]; then take the points + a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... a_n = x; it is supposed that the + region is so taken that, whatever x may be, all these are interior + points of the region. Then by what has been said, replacing x, z + respectively by 0 and x/n, we have + + _m1 [phi]^([mu] + [lambda]1)(0) /x \^[lambda]1 + [phi]^([mu]) (a{1}) = \ --------------------------- ( -- ) + [alpha]_[mu] + /_ [lambda]{1}! \n / + [lambda]1=0 + + with + + |[alpha]{[mu]}| < g/[rho]^[mu] 2^m1, + + provided ([mu] + m1 + 1)^[mu] < (3/2)^(m1+1); in fact for [mu] =< + 2n^(2n-2) it is sufficient to take m1 = n^2n; by another application + of the same inequality, replacing x, z respectively by a1 and x/n, we + have + + _ m2 [phi]^([mu]+[lambda]2)(a1) /x \^[lambda]2 + [phi]^([mu])(a2) = \ -------------------------- ( -- ) + [beta]'_[mu], + /_ [lambda]{2}! \n / + [lambda]2=0 + + where + + |[beta]'[mu]| < g/[rho]^[mu] 2^m2 + + provided ([mu] + m2 + 1)^[mu] < {3/2}^(m2 + 1); we take m2 = n^(2n - + 2), supposing [mu] < 2^(2n - 4). So long as [lambda]2 =< = m} =< n^(2n + - 2) and [mu] < 2n^(2n - 4) we have [mu] + [lambda]{2} < 2n^(2n - 2), + and we can use the previous inequality to substitute here for + [phi]^([mu] + [lambda]2) (a1). When this is done we find + + _ m2 _ m1 [phi]^([mu] + [lambda]1 + [lambda]2)(0) + [phi]^([mu])(a2) = \ \ --------------------------------------- + /_ /_ [lambda]1! [lambda]2! + [lambda]2=0 [lambda]1=0 + + /x \ ^[lambda]1 + [lambda]2 + ( -- ) + [beta]_[mu], + \n / + + where |[beta]_[mu]| < 2g/[rho]^[mu] 2^(m2), the numbers m1, m2 being + respectively n^2n and n^(2n - 2). + + Applying then the original inequality to [phi]^([mu]) (a3) = + [phi]^([mu]) (a2 + x/n), and then using the series just obtained, we + find a series for [phi]^([mu]) (a3). This process being continued, we + finally obtain + + _ m1 _ m2 _ m_n + [phi](x) = \ \ ... \ [phi]^(h)(0) /x \^h + /_ /_ /_ ----------- ( -- ) + [epsilon], + [lambda]1=0 [lambda]2=0 [lambda]_n=0 K \n / + + where h = [lambda]1 + [lambda]2 + ... + [lambda]_n , K = [lambda]1! + [lambda]2! ... [lambda]_n!, m1 = n^(2n), m1 = n^(2n - 2), ... , m1 = + n^2, |[epsilon]| < 2g/2^(m_n). + + By this formula [phi](x) is represented, with any required degree of + accuracy, by a polynomial, within the region in question; and thence + can be expressed as before by a series of polynomials converging + uniformly (and absolutely) within this region. + +S 13. _Application of Cauchy's Theorem to the Determination of Definite +Integrals._--Some reference must be made to a method whereby real +definite integrals may frequently be evaluated by use of the theorem of +the vanishing of the integral of a function of a complex variable round +a contour within which the function is single valued and non singular. + + We are to evaluate an integral [int][a to b] [f](x)dx; we form a + closed contour of which the portion of the real axis from x = a to x = + b forms a part, and consider the integral [int][f](z)dz round this + contour, supposing that the value of this integral can be determined + along the curve forming the completion of the contour. The contour + being supposed such that, within it, [f](z) is a single valued and + finite function of the complex variable z save at a finite number of + isolated interior points, the contour integral is equal to the sum of + the values of [int][f](z)dz taken round these points. Two instances + will suffice to explain the method. (1) The integral [int][0 to [oo]] + (tan x)/x dx is convergent if it be understood to mean the limit when + [epsilon], [zeta], [sigma], ... all vanish of the sum of the integrals + + _1/2[pi]-[epsilon] _(3/2)[pi]-[zeta] _(5/2)[pi]-[sigma] + / tan x / tan x / tan x + | ----- dx, | ----- dx, | ----- dx, ... + _/ 0 x _/1/2[pi]+[epsilon] x _/(3/2)[pi]+[zeta] x + + Now draw a contour consisting in part of the whole of the positive and + negative real axis from x = -n[pi] to x = + n[pi], where n is a + positive integer, broken by semicircles of small radius whose centres + are the points x = [+-]1/2[pi], x = [+-]3/4[pi], ... , the contour + containing also the lines x = n[pi] and x = -n[pi] for values of y + between 0 and n[pi] tan [alpha], where [alpha] is a small fixed angle, + the contour being completed by the portion of a semicircle of radius + n[pi] sec [alpha] which lies in the upper half of the plane and is + terminated at the points x = [+-]n[pi], y = n[pi] tan [alpha]. Round + this contour the integral [int](tan z /z) dz has the value zero. The + contributions to this contour integral arising from the semicircles of + centres -1/2(2s - 1)[pi], + 1/2(2s - 1)[pi], supposed of the same + radius, are at once seen to have a sum which ultimately vanishes when + the radius of the semicircles diminishes to zero. The part of the + contour lying on the real axis gives what is meant by 2 [int][0 to + n[pi]](tan x / x) dx. The contribution to the contour integral from + the two straight portions at x = [+-]n[pi] is + + _n[pi] tan [alpha] + / / tan iy tan iy \ + | idy ( ---------- - ----------- ) + _/ 0 \n[pi] + iy -n[pi] + iy / + + where i tan iy, = -[exp(y) - exp(-y)]/[exp(y) + exp(-y)], is a real + quantity which is numerically less than unity, so that the + contribution in question is numerically less than + + _n[pi] tan [alpha] + / 2n[pi] + | dy ---------------, that is than 2[alpha]. + _/ 0 n^2[pi]^2 + y^2 + + Finally, for the remaining part of the contour, for which, with R = + n[pi] sec [alpha], we have z = R(cos [theta] + i sin [theta]) = + RE(i[theta]), we have + + dz + -- = id[theta], i tan z = + z + + exp(-R sin [theta]) E(iR cos [theta]) - exp(R sin [theta]) E(-iR cos [theta]) + -----------------------------------------------------------------------------; + exp(-R sin [theta]) E(iR cos [theta]) + exp(R sin [theta]) E(-iR cos [theta]) + + when n and therefore R is very large, the limit of this contribution + to the contour integral is thus + _ + / [pi]-[alpha] + - | d[theta] = -([pi] - 2[alpha]). + _/ [alpha] + + Making n very large the result obtained for the whole contour is + _ + / [oo] tan x + 2 | ----- dx - ([pi] - 2[alpha]) - 2[alpha][epsilon] = 0; + _/ 0 x + + where [epsilon] is numerically less than unity. Now supposing [alpha] + to diminish to zero we finally obtain + _ + / [oo] tan x [pi] + | ----- dx = ---- + _/ 0 x 2 + + (2) For another case, to illustrate a different point, we may take the + integral + _ + / z^(a-1) + | ------- dz, + _/ 1 + z + + wherein a is real quantity such that 0 < a < 1, and the contour + consists of a small circle, z = rE(i[theta]), terminated at the points + x = r cos [alpha], y = [+-] r sin [alpha], where [alpha] is small, of + the two lines y = [+-] r sin [alpha] for r cos [alpha] =< x =< R cos + [beta], where R sin [beta] = r sin [alpha], and finally of a large + circle z = RE(i[phi]), terminated at the points x = R cos [beta], y = + [+-] R sin [beta]. We suppose [alpha] and [beta] both zero, and that + the phase of z is zero for r cos a =< x =< R cos [beta], y = r sin + [alpha] = R sin [beta]. Then on r cos [alpha] =< x =< R cos [beta], y + = -r sin [alpha], the phase of z will be 2[pi], and z^([alpha] - 1) + will be equal to x^([alpha] - 1) exp (2[pi]i(a - 1)), where x is real + and positive. The two straight portions of the contour will thus + together give a contribution + + _ + / R cos [beta] x^(a - 1) + [1 - exp (2[pi]i[alpha])] | --------- dx. + _/ r cos [alpha] 1 + x + + It can easily be shown that if the limit of z[f](z) for z = 0 is zero, + the integral [int] [f](z)dz taken round an arc, of given angle, of a + small circle enclosing the origin is ultimately zero when the radius + of the circle diminishes to zero, and if the limit of z[f](z) for z = + [oo] is zero, the same integral taken round an arc, of given angle, of + a large circle whose centre is the origin is ultimately zero when the + radius of the circle increases indefinitely; in our case with [f](z) = + z^([alpha] - 1)/(1 + z), we have z[f](z) = z^a/(1 + z), which, for 0 < + a < 1, diminishes to zero both for z = 0 and for z = [oo]. Thus, + finally the limit of the contour integral when r = 0, R = [oo] is + _ + / [oo] x^([alpha] - 1) + [1 - exp (2[pi]i[alpha])] | --------------- dx. + _/ 0 1 + x + + Within the contour [f](z) is single valued, and has a pole at z = 1; + at this point the phase of z is [pi] and z^(a - 1) is exp [i[pi](a - + 1)] or - exp (i[pi]a); this is then the residue of [f](z) at z = -1; + we thus have + _ + / [oo] x^(a - 1) + [1 - exp (2[pi]ia)] | --------- dx = -2[pi]i exp (i[pi]a), + _/ 0 1 + x + + that is + _ + / [oo] x^(a - 1) + | --------- dx = [pi] cosec (a[pi]). + _/ 0 1 + x + +S 14. _Doubly Periodic Functions._--An excellent illustration of the +preceding principles is furnished by the theory of single valued +functions having in the finite part of the plane no singularities but +poles, which have two periods. + + Before passing to this it may be convenient to make here a few remarks + as to the periodicity of (single valued) monogenic functions. To say + that [f](z) is periodic is to say that there exists a constant [omega] + such that for every point z of the interior of the region of existence + of [f](z) we have [f](z + [omega]) = [f](z). This involves, + considering all existing periods [omega] = [rho] + i[sigma], that + there exists a lower limit of [rho]^2 + [sigma]^2 other than zero; for + otherwise all the differential coefficients of [f](z) would be zero, + and [f](z) a constant; we can then suppose that not both [rho] and + [sigma] are numerically less than [epsilon], where [epsilon] > + [sigma]. Hence, if g be any real quantity, since the range (-g, ... g) + contains only a finite number of intervals of length [epsilon], and + there cannot be two periods [omega] = [rho] + i[sigma] such that + [mu][epsilon] =< [rho] < ([mu] + 1)[epsilon], [nu][epsilon] =< [sigma] + < ([nu] + 1)[epsilon], where [mu], [nu] are integers, it follows that + there is only a finite number of periods for which both [rho] and + [sigma] are in the interval (-g ... g). Considering then all the + periods of the function which are real multiples of one period + [omega], and in particular those periods [lambda][omega] wherein 0 < + [lambda] =< 1, there is a lower limit for [lambda], greater than zero, + and therefore, since there is only a finite number of such periods for + which the real and imaginary parts both lie between -g and g, a least + value of [lambda], say [lambda]0. If [Omega] = [lambda]0[omega] and + [lambda] = M[lambda]0 + [lambda]', where M is an integer and 0 [< = ] + [lambda]' < [lambda]0, any period [lambda][omega] is of the form + M[Omega] + [lambda]'[omega]; since, however, [Omega], M[Omega] and + [lambda][omega] are periods, so also is [lambda]'[omega], and hence, + by the construction of [lambda]0, we have [lambda]' = 0; thus all + periods which are real multiples of [omega] are expressible in the + form M[Omega] where M is an integer, and [Omega] a period. + + If beside [omega] the functions have a period [omega]' which is not a + real multiple of [omega], consider all existing periods of the form + [mu][omega] + [nu][omega]' wherein [mu], [nu] are real, and of these + those for which 0 [< = ] [mu] =< 1, 0 < [nu] =< 1; as before there is + a least value for [nu], actually occurring in one or more periods, say + in the period [Omega]' = [mu]0[omega] + [nu]0[omega]'; now take, if + [mu][omega] + [nu][omega]' be a period, [nu] = N'[nu]0 + [nu]', where + N' is an integer, and 0 =< [nu]' < [nu]0; thence [mu][omega] + + [nu][omega]' = [mu][omega] + N'([Omega]' - [mu]0[omega]) + + [nu]'[omega]'; take then [mu] - N[mu]0 = N[lambda]0 + [lambda]', where + N is an integer and [lambda]0 is as above, and 0 =< [lambda]' < + [lambda]0; we thus have a period N[Omega] + N'[Omega]' + + [lambda]'[omega] + [nu]'[omega]', and hence a period [lambda]'[omega] + + [nu]'[omega]', wherein [lambda]' < [lambda]0, [nu]' < [nu]0; hence + [nu]' = 0 and [lambda]' = 0. All periods of the form [mu][omega] + + [nu][omega]' are thus expressible in the form N[Omega] + N'[Omega]', + where [Omega], [Omega]' are periods and N, N' are integers. But in + fact any complex quantity, P + iQ, and in particular any other + possible period of the function, is expressible, with [mu], [nu] real, + in the form [mu][omega] + [nu][omega]'; for if [omega] = [rho] + + i[sigma], [omega]' = [rho]' + i[sigma]', this requires only P = + [mu][rho] + [nu][rho]', Q = [mu][sigma] + [nu][sigma]', equations + which, since [omega]'/[omega] is not real, always give finite values + for [mu] and [nu]. + + It thus appears that if a single valued monogenic function of z be + periodic, either all its periods are real multiples of one of them, + and then all are of the form M[Omega], where [Omega] is a period and M + is an integer, or else, if the function have two periods whose ratio + is not real, then all its periods are expressible in the form N[Omega] + + N'[Omega]', where [Omega], [Omega]' are periods, and N, N' are + integers. In the former case, putting [zeta] = 2[pi]iz/[Omega], and + the function [f](z) = [phi]([zeta]), the function [phi]([zeta]) has, + like exp ([zeta]), the period 2[pi]i, and if we take t = exp([zeta]) + or [zeta] = [lambda](t) the function is a single valued function of t. + If then in particular [f](z) is an integral function, regarded as a + function of t, it has singularities only for t = 0 and t = [oo], and + may be expanded in the form [Sigma](-[oo] to [oo]) a_nt^n. + + Taking the case when the single valued monogenic function has two + periods [omega], [omega]' whose ratio is not real, we can form a + network of parallelograms covering the plane of z whose angular points + are the points c + m[omega] + m'[omega]', wherein c is some constant + and m, m' are all possible positive and negative integers; choosing + arbitrarily one of these parallelograms, and calling it the primary + parallelogram, all the values of which the function is at all capable + occur for points of this primary parallelogram, any point, z', of the + plane being, as it is called, _congruent_ to a definite point, z, of + the primary parallelogram, z' - z being of the form m[omega] + + m'[omega]', where m, m' are integers. Such a function cannot be an + integral function, since then, if, in the primary parallelogram + |[f](z)| < M, it would also be the case, on a circle of centre the + origin and radius R, that |[f](z)| < M, and therefore, if + [Sigma]a_nz^n be the expansion of the function, which is valid for an + integral function for all finite values of z, we should have |a_n| < + MR^(-n), which can be made arbitrarily small by taking R large enough. + The function must then have singularities for finite values of z. + + We consider only functions for which these are poles. Of these there + cannot be an infinite number in the primary parallelogram, since then + those of these poles which are sufficiently near to one of the + necessarily existing limiting points of the poles would be arbitrarily + near to one another, contrary to the character of a pole. Supposing + the constant c used in naming the corners of the parallelograms so + chosen that no pole falls on the perimeter of a parallelogram, it is + clear that the integral 1/(2[pi]i) [int][f](z)dz round the perimeter + of the primary parallelogram vanishes; for the elements of the + integral corresponding to two such opposite perimeter points as z, z + + [omega] (or as z, z + [omega]') are mutually destructive. This + integral is, however, equal to the sum of the residues of [f](z) at + the poles interior to the parallelogram. Which sum is therefore zero. + There cannot therefore be such a function having only one pole of the + first order in any parallelogram; we shall see that there can be such + a function with two poles only in any parallelogram, each of the first + order, with residues whose sum is zero, and that there can be such a + function with one pole of the second order, having an expansion near + this pole of the form (z - a)^(-2) + (power series in z - a). + + Considering next the function [phi](z) = [[f](z)]^(-1) d[f](z)/dz, it + is easily seen that an ordinary point of [f](z) is an ordinary point + of [phi](z), that a zero of order m for [f](z) in the neighbourhood of + which [f](z) has a form, (z - a)^m multiplied by a power series, is a + pole of [phi](z) of residue m, and that a pole of [f](z) of order n is + a pole of [phi](z) of residue -n; manifestly [phi](z) has the two + periods of [f](z). We thus infer, since the sum of the residues of + [phi](z) is zero, that for the function [f](z), the sum of the orders + of its vanishing at points belonging to one parallelogram, [Sigma]m, + is equal to the sum of the orders of its poles, [Sigma]n; which is + briefly expressed by saying that the number of its zeros is equal to + the number of its poles. Applying this theorem to the function + [f](z) - A, where A is an arbitrary constant, we have the result, that + the function [f](z) assumes the value A in one of the parallelograms + as many times as it becomes infinite. Thus, by what is proved above, + every conceivable complex value does arise as a value for the doubly + periodic function [f](z) in any one of its parallelograms, and in fact + at least twice. The number of times it arises is called the _order_ of + the function; the result suggests a property of rational functions. + + Consider further the integral [int] z [f]'(z)/[f](z) dz, where [f]'(z) + = d[f](z)/dz taken round the perimeter of the primary parallelogram; + the contribution to this arising from two opposite perimeter points + such as z and z + [omega] is of the form -[omega] [int] z + [f]'(z)/[f](z) dz, which, as z increases from z0 to z0 + [omega]', + gives, if [lambda] denote the generalized logarithm, - + [omega]{[lambda][[f](z0 + [omega]')] - [lambda][[f](z0)]}, that is, + since [f](z0 + [omega]') = [f](z0), gives 2[pi]iN[omega], where N is + an integer; similarly the result of the integration along the other + two opposite sides is of the form 2[pi]iN'[omega]', where N' is an + integer. The integral, however, is equal to 2[pi]i times the sum of + the residues of z[f]'(z)/[f](z) at the poles interior to the + parallelogram. For a zero, of order m, of [f](z) at z = a, the + contribution to this sum is 2[pi]ima, for a pole of order n at z = b + the contribution is -2[pi]inb; we thus infer that [Sigma]ma - + [Sigma]nb = N[omega] + N'[omega]'; this we express in words by saying + that the sum of the values of z where [f](z) = 0 within any + parallelogram is equal to the sum of the values of z where [f](z) = + [oo] save for integral multiples of the periods. By considering + similarly the function [f](z) - A where A is an arbitrary constant, we + prove that each of these sums is equal to the sum of the values of z + where the function takes the value A in the parallelogram. + +We pass now to the construction of a function having two arbitrary +periods [omega], [omega]' of unreal ratio, which has a single pole of +the second order in any one of its parallelograms. + + For this consider first the network of parallelograms whose corners + are the points [Omega] = m[omega] + m'[omega]', where m, m' take all + positive and negative integer values; putting a small circle about + each corner of this network, let P be a point outside all these + circles; this will be interior to a parallelogram whose corners in + order may be denoted by z0, z0 + [omega], z0 + [omega] + [omega]', z0 + + [omega]'; we shall denote z0, z0 + [omega] by A0, B0; this + parallelogram [Pi]0 is surrounded by eight other parallelograms, + forming with [Pi]0 a larger parallelogram [Pi]1, of which one side, + for instance, contains the points z0 - [omega] - [omega]', z0 - + [omega]', z0 - [omega]' + [omega], z0 - [omega]' + 2[omega], which we + shall denote by A1, B1, C1, D1. This parallelogram [Pi]1 is surrounded + by sixteen of the original parallelograms, forming with [Pi]1 a still + larger parallelogram [Pi]2 of which one side, for instance, contains + the points z0 - 2[omega] - 2[omega]', z0 - [omega] - 2[omega]', z0 - + 2[omega]', z0 + [omega] - 2[omega]', z0 + 2[omega] - 2[omega]', z0 + + 3[omega] - 2[omega]', which we shall denote by A2, B2, C2, D2, E2, F2. + And so on. Now consider the sum of the inverse cubes of the distances + of the point P from the corners of all the original parallelograms. + The sum will contain the terms + + 1 / 1 1 1 \ / 1 1 1 \ + S0 = ----- + ( ----- + ----- + ----- ) + ( ----- + ----- + ... + ----- ) + ... + PA0^3 \PA1^3 PB1^3 PC1^3/ \PA2^3 PB2^3 PE2^3/ + + and three other sets of terms, each infinite in number, formed in a + similar way. If the perpendiculars from P to the sides A0B0, A1B1C1, + A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0 is at + most equal to + + 1 3 5 2n + 1 + --- + --------- + ---------- + ... + ---------- + ... + p^3 (p + q)^3 (p + 2q)^3 (p + nq)^3 + + of which the general term is ultimately, when n is large, in a ratio + of equality with 2q^(-3)n^(-2), so that the series S0 is convergent, + as we know the sum [Sigma]n^(-2) to be; this assumes that p[/ = ]0; if + P be on A0B0 the proof for the convergence of S0 - 1/PA0^3, is the + same. Taking the three other sums analogous to S0 we thus reach the + result that the series + + [phi](z) = -2[Sigma](z - [Omega])^(-3), + + where [Omega] is m[omega] + m'[omega]', and m, m' are to take all + positive and negative integer values, and z is any point outside small + circles described with the points [Omega] as centres, is _absolutely + convergent_. Its sum is therefore independent of the order of its + terms. By the nature of the proof, which holds for all positions of z + outside the small circles spoken of, the series is also clearly + _uniformly convergent_ outside these circles. Each term of the series + being a monogenic function of z, the series may therefore be + differentiated and integrated outside these circles, and represents a + monogenic function. It is clearly periodic with the periods [omega], + [omega]'; for [phi](z + [omega]) is the same sum as [phi](z) with the + terms in a slightly different order. Thus [phi](z + [omega]) = + [phi](z) and [phi](z + [omega]') = [phi](z). + + Consider now the function + _ _ _ + 1 / z | 2 | + [f](z) = --- + | | [phi](z) + -- | dz, + z^2 _/ 0 |_ z^3 _| + + where, for the subject of integration, the area of uniform convergence + clearly includes the point z = 0; this gives + + d[f](z) + ------- = [phi](z) + dz + + and + _ _ + 1 | 1 1 | + [f](z) = --- + [Sigma]' | -------------- - -------- |, + z^2 |_ (z - [Omega])^2 [Omega]^2 _| + + wherein [Sigma]' is a sum excluding the term for which m = 0 and m' = + 0. Hence [f](z + [omega]) - [f](z) and [f](z + [omega]') - [f](z) are + both independent of z. Noticing, however, that, by its form, [f](z) is + an even function of z, and putting z = -1/2[omega], z = -1/2[omega]' + respectively, we infer that also [f](z) has the two periods [omega] + and [omega]'. In the primary parallelogram [Pi]0, however, [f](z) is + only infinite at z = 0 in the neighbourhood of which its expansion is + of the form z^(-2) + (power series in z). Thus [f](z) is such a doubly + periodic function as was to be constructed, having in any + parallelogram of periods only one pole, of the second order. + +It can be shown that any single valued meromorphic function of z with +[omega] and [omega]' as periods can be expressed rationally in terms of +[f](z) and [phi](z), and that [[phi](z)]^2 is of the form 4[[f](z)]^3 + +A[f](z) + B, where A, B are constants. + + To prove the last of these results, we write, for |z| < |[Omega]|, + + 1 1 2z 3z^2 + --------------- - --------- = --------- + --------- + ..., + (z - [Omega])^2 [Omega]^2 [Omega]^3 [Omega]^4 + + and hence, if [Sigma]'[Omega]^(-2n) = [sigma]_n, since + [Sigma]'[Omega]^(-(2n - 1)) = 0, we have, for sufficiently small z + greater than zero, + + [f](z) = z^(-2) + 3[sigma]2.z^2 + 5[sigma]3.z^4 + ... + + and + + [phi](z) = -2z^(-3) + 6[sigma]2.z + 20[sigma]3.z^3 + ...; + + using these series we find that the function + + F(z) = [[phi](z)]^2 - 4[[f](z)]^3 + 60[sigma]2[f](z) + 140[sigma]3 + + contains no negative powers of z, being equal to a power series in z^2 + beginning with a term in z^2. The function F(z) is, however, doubly + periodic, with periods [omega], [omega]', and can only be infinite + when either [f](z) or [phi](z) is infinite; this follows from its form + in [f](z) and [phi](z); thus in one parallelogram of periods it can be + infinite only when z = 0; we have proved, however, that it is not + infinite, but, on the contrary, vanishes, when z = 0. Being, + therefore, never infinite for finite values of z it is a constant, and + therefore necessarily always zero. Putting therefore [f](z) = [zeta] + and [phi](z) = d[zeta]/dz we see that + + dz + ------- = (4[zeta]^3 - 60[sigma]2[zeta] - 140[sigma]3)^(-1/2) + d[zeta] + + Historically it was in the discussion of integrals such as + _ + / + | d[zeta](4[zeta]^3 - 60[sigma]2.[zeta] - 140[sigma]3)^(-1/2), + _/ + + regarded as a branch of Integral Calculus, that the doubly periodic + functions arose. As in the familiar case + _ + / [zeta] + z = | (1 - [zeta]^2)^(-1/2) d[zeta], + _/ 0 + + where [zeta] = sin z, it has proved finally to be simpler to regard + [zeta] as a function of z. We shall come to the other point of view + below, under S 20, _Elliptic Integrals_. + +To prove that any doubly periodic function F(z) with periods [omega], +[omega]', having poles at the points z = a1, ... z = a_m of a +parallelogram, these being, for simplicity of explanation, supposed to +be all of the first order, is rationally expressible in terms of +[phi](z) and [f](z), and we proceed as follows:-- + + Consider the expression + + ([zeta], 1)_m + [eta]([zeta], 1)_(m - 2) + [Phi](z) = ------------------------------------------ + ([zeta]- A1)([zeta] - A2)...([zeta] - A_m) + + where A_s = [f](a_s), [zeta] is an abbreviation for [f](z) and [eta] + for [phi](z), and ([zeta], 1)_m, ([zeta], 1)_(m - 2), denote integral + polynomials in [zeta], of respective orders m and m - 2, so that there + are 2m unspecified, homogeneously entering, constants in the + numerator. It is supposed that no one of the points a1, ... a_m is one + of the points m[omega] + m'[omega]' where f(z) = [oo]. The function + [Phi](z) is a monogenic function of z with the periods [omega], + [omega]', becoming infinite (and having singularities) only when (1) + [zeta] = [oo] or (2) one of the factors [zeta] - A_s is zero. In a + period parallelogram including z = 0 the first arises only for z = 0; + since for [zeta] = [oo], [eta] is in a finite ratio to [zeta]^(3/2); + the function [Phi](z) for [zeta] = [oo] is not infinite provided the + coefficient of [zeta]^m in ([zeta], 1)_m is not zero; thus [Phi](z) is + regular about z = 0. When [zeta] - A_s = 0, that is [f](z) = f(a_s), + we have z = [+-]a_s + m[omega] + m'[omega]', and no other values of z, m + and m' being integers; suppose the unspecified coefficients in the + numerator so taken that the numerator vanished to the first order in + each of the m points -a1, -a2, ... -a_m; that is, if [phi](a_s) = B_s, + and therefore [phi](-a_s) = -B_s, so that we have the m relations + + (A_s, 1)_m - B_s(A_s, 1)_(m - 2) = 0; + + then the function [Phi](z) will only have the m poles a1, ... a_m. + Denoting further the m zeros of F(z) by a1', ... a_m', putting + [f](a_s') = A_s', [phi](a_s') = B_s', suppose the coefficients of the + numerator of [Phi](z) to satisfy the further m-1 conditions + + (A_s', 1)_m + B_s'(A_s',1)_(m - 2) = 0 + + for s = 1, 2, ... (m - 1). The ratios of the 2m coefficients in the + numerator of [Phi](z) can always be chosen so that the m + (m - 1) + linear conditions are all satisfied. Consider then the ratio + + F(z)/[Phi](z); + + it is a doubly periodic function with no singularity other than the + one pole a_m'. It is therefore a constant, the numerator of [Phi](z) + vanishing spontaneously in a_m'. We have + + F(z) = A[Phi](z), + + where A is a constant; by which F(z) is expressed rationally in terms + of [f](z) and [phi](z), as was desired. + + When z = 0 is a pole of F(z), say of order r, the other poles, each of + the first order, being a1, ... a_m, similar reasoning can be applied + to a function + + ([zeta], 1)_h + [eta]([zeta], 1)_k + ----------------------------------, + ([zeta] - A1)...([zeta] - A_m) + + where h, k are such that the greater of 2h - 2m, 2k + 3 - 2m is equal + to r; the case where some of the poles a1, ... a_m are multiple is to + be met by introducing corresponding multiple factors in the + denominator and taking a corresponding numerator. We give a solution + of the general problem below, of a different form. + + One important application of the result is the theorem that the + functions [f](z + t), [phi](z + t), which are such doubly periodic + function of z as have been discussed, can each be expressed, so far as + they depend on z, rationally in terms of [f](z) and [phi](z), and + therefore, so far as they depend on z and t, rationally in terms of + [f](z), [f](t), [phi](z) and [phi](t). It can in fact be shown, by + reasoning analogous to that given above, that + _ _ + | [phi](z) - [phi](t) |^2 + [f](z + t) + [f](z) + [f](t) = 1/4 | ------------------- |. + |_ [f](z) - [f](t) _| + + This shows that if F(z) be any single valued monogenic function which + is doubly periodic and of meromorphic character, then F(z + t) is an + algebraic function of F(z) and F(t). Conversely any single valued + monogenic function of meromorphic character, F(z), which is such that + F(z + t) is an algebraic function of F(z) and F(t), can be shown to be + a doubly periodic function, or a function obtained from such by + degeneration (in virtue of special relations connecting the + fundamental constants). + + The functions [f](z), [phi](z) above are usually denoted by RN(z), + RN'(z); further the fundamental differential equation is usually + written + + (RN'z)^2 = 4(RNz)^3 - g2RNz - g3, + + and the roots of the cubic on the right are denoted by e1, e2, e3; for + the odd function, RN'z, we have, for the congruent arguments + -1/2[omega]and 1/2[omega], RN'(1/2[omega]) = -RN'(-1/2[omega]) = + -RN'(1/2[omega]), and hence RN'(1/2[omega]) = 0; hence we can take e1 + = RN(1/2[omega]), e2 = RN(1/2[omega] + 1/2[omega]'), e3 = + RN(1/2[omega]). It can then be proved that [RN(z) - e1][RN(z + + 1/2[omega]) - e1] = (e1 - e2)(e1 - e3), with similar equations for the + other half periods. Consider more particularly the function RN(z) - + e1; like RN(z) it has a pole of the second order at z = 0, its + expansion in its neighbourhood being of the form z^(-2)(1 - e1z^2 + + Az^4 + ...); having no other pole, it has therefore either two zeros, + or a double zero in a period parallelogram ([omega], [omega]'). In + fact near its zero 1/2[omega] its expansion is (x - 1/2[omega]) + RN'(1/2[omega]) + 1/2(z - 1/2[omega])^2 RN"(1/2[omega]) + ...; we have + seen that RN'(1/2[omega]) = 0; thus it has a zero of the second order + wherever it vanishes. Thus it appears that the square root [RN(z) - + e1]^1/2, if we attach a definite sign to it for some particular value + of z, is a single valued function of z; for it can at most have two + values, and the only small circuits in the plane which could lead to + an interchange of these values are those about either a pole or a + zero, neither of which, as we have seen, has this effect; the function + is therefore single valued for any circuit. Denoting the function, for + a moment, by [f]1(z), we have [f]1(z + [omega]) = [+-][f]1(z), [f]1(z + + [omega]') = [+-][f]1(z); it can be seen by considerations of + continuity that the right sign in either of these equations does not + vary with z; not both these signs can be positive, since the function + has only one pole, of the first order, in a parallelogram ([omega], + [omega]'); from the expansion of [f]1(z) about z = 0, namely z^(-1) (1 + - 1/2e1z^2 + ...), it follows that [f]1(z) is an odd function, and + hence [f]1(-1/2[omega]') = -[f]1(1/2[omega]'), which is not zero since + [[f]1(1/2[omega]')]^2 = e3 - e1, so that we have [f]1(z + [omega]') = + -[f]1(z); an equation f1(z + [omega]) = -[f]1(z) would then give + [f]1(z + [omega] + [omega]') = [f]1(z), and hence [f]1(1/2[omega] + + 1/2[omega]') = [f]1(-1/2[omega] - 1/2[omega]'), of which the latter is + -[f]1(1/2[omega] + 1/2[omega]'); this would give [f]1(1/2[omega] + + 1/2[omega]') = 0, while [[f]1 (1/2[omega] + 1/2[omega]')]^2 = e2 - e1. + We thus infer that [f]1(z + [omega]) = [f]1(z), [f]1(z + [omega]') = + -[f]1(z), [f]1(z + [omega] + [omega]') = -[f]1(z). The function + [f]1(z) is thus doubly periodic with the periods [omega] and + 2[omega]'; in a parallelogram of which two sides are [omega] and + 2[omega]' it has poles at z = 0, z = [omega]' each of the first order, + and zeros of the first order at z = 1/2[omega], z = 1/2[omega] + + [omega]'; it is thus a doubly periodic function of the second order + with two different poles of the first order in its parallelogram + ([omega], 2[omega]'). We may similarly consider the functions [f]2(z) + = [RN(z) - e2]^1/2, [f]3(z) = [RN(z) - e3]^1/2; they give + + [f]2(z + [omega] + [omega]') = [f]2(z), [f]2(z + [omega]) = -[f]2(z), [f]2(z + [omega]') = -[f]2(z), + + [f]3(z + [omega]') = [f]3z, [f]3(z + [omega]) = -[f]3(z), [f]3(z + [omega] + [omega]') = -[f]3(z). + + Taking u = z(e1 - e3)^1/2, with a definite determination of the + constant (e1 - e3)^1/2, it is usual, taking the preliminary signs so + that for z = 0 each of z[f]1(z), z[f]2(z), z[f]3(z) is equal to + 1, + to put + + (e1 - e3)^1/2 [f]1(z) f2(z) + sn(u) = -------------, cn(u) = -------, dn(u) = -----, + [f]3(z) [f]3(z) f3(z) + + k^2 = (e2 - e3)/(e1 - e3), K = 1/2[omega](e1 - e3)^1/2, iK' = 1/2[omega]'(e1 - e3)^1/2; + + thus sn(u) is an odd doubly periodic function of the second order with + the periods 4K, 2iK, having poles of the first order at u = iK', u = + 2K + iK', and zeros of the first order at u = 0, u = 2K; similarly + cn(u), dn(u) are even doubly periodic functions whose periods can be + written down, and sn^2(u) + cn^2(u) = 1, k^2sn^2(u) + dn^2(u) = 1; if + x = sn(u) we at once find, from the relations given here, that + + du + -- = [(1 - x^2) (1 - k^2x^2)]^(-1/2); + dx + + if we put x = sin[phi] we have + + du + ------ = [1 - k^2sin^2 [phi]]^(-1/2), + d[phi] + + and if we call [phi] the amplitude of u, we may write [phi] = am(u), x + = sin.am(u), which explains the origin of the notation sn(u). + Similarly cn(u) is an abbreviation of cos.am(u), and dn(u) of + [Delta]am(u), where [Delta]([phi]) meant (1 - k^2sin^2 [phi])^1/2. The + addition equation for each of the functions [f]1(z), [f]2(z), [f]3(z) + is very simple, being + + /(Pd) (Pd) \ [f](z) + [f](t) [f](z)[f]'(t) - [f](t)[f]'(z) + [f](z + t) = 1/2( ----- + ----- ) log --------------- = -----------------------------, + \(Pd)z (Pd)i/ [f](z) - [f](t) [f]^2(z) - [f]^2(t) + + where f1'(z) means d[f]1(z)/dz, which is equal to -[f]2(z).[f]3(z), + and [f]^2(z) means [[f](z)]^2. This may be verified directly by + showing, if R denote the right side of the equation, that (Pd)R/(Pd)z + = (Pd)R/(Pd)t; this will require the use of the differential equation + + [[f]1'^(z)]^2 = [[f]1^2(z) + e1 - e2] [[f]1^2(z) + e1 - e3], + + and in fact we find + + / (Pd)^2 (Pd)^2\ + ( ------- - ------ ) log [[f](z) + [f](t)] = [f]^2(z) - [f]^2(t) = + \(Pd)z^2 dt^2 / + + / (Pd)^2 (Pd)^2\ + ( ------- - ------ ) log [[f](z) - [f](t)]; + \(Pd)z^2 dt^2 / + + hence it will follow that R is a function of z + t, and R is at once + seen to reduce to [f](z) when t = 0. From this the addition equation + for each of the functions sn(u), cn(u), dn(u) can be deduced at once; + if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1), + sn(u2), cn(u2), dn(u2), they can be put into the forms + + sn(u1 + u2) = (s1c2d2 + s2c1d1)/D, + + cn(u1 + u2) = (c1c2 - s1s2d1d2)/D, + + dn(u1 + u2) = (d1d2 - k^2s1s2c1c2)/D, + + where + + D = 1 - k^2s1^2s2^2. + + The introduction of the function [f]1(z) is equivalent to the + introduction of the function RN(z; [omega], 2[omega]') constructed + from the periods [omega], 2[omega]' as was RN(z) from [omega] and + [omega]'; denoting this function by RN1(z) and its differential + coefficient by RN'1(z), we have in fact + + RN'1(z) + [f]1(z) = 1/2 ---------------------- + RN1([omega]') - RN1(z) + + as we see at once by considering the zeros and poles and the limit of + z[f]1(z) when z = 0. In terms of the function RN1(z) the original + function RN(z) is expressed by + + RN(z) = RN1(z) + RN1(z + [omega]') - RN1([omega]'), + + as a consideration of the poles and expansion near z = 0 will show. + + A function having [omega], [omega]' for periods, with poles at two + arbitrary points a, b and zeros at a', b', where a' + b' = a + b save + for an expression m[omega] + m'[omega]', in which m, m' are integers, + is a constant multiple of + + {RN[z - 1/2(a' + b')] - RN[a' - 1/2(a' + b')]} / {RN[z - 1/2(a + b)] - RN[a - 1/2(a + b)]}; + + if the expansion of this function near z = a be + _ + [lambda](z - a)^(-1) + [mu] + \ [mu]_n(z - a)^n, + /_ + n = 1 + + the expansion near z = b is + _ + -[lambda](z - b)^(-1) + [mu] + \ (-1)^n [mu]_n (z - b)^n, + /_ + n = 1 + + as we see by remarking that if z'- b = -(z - a) the function has the + same value at z and z'; hence the differential equation satisfied by + the function is easily calculated in terms of the coefficients in the + expansions. + + From the function RN(z) we can obtain another function, termed the + Zeta-function; it is usually denoted by [zeta](z), and defined by + _ _ _ + 1 / [pi] | 1 | _ / 1 1 z \ + [zeta](z) -- = | | --- - RN(z) |dz = \ ' ( ----------- + ------- + --------- ), + z _/ 0 |_ z^2 _| /_ \z - [Omega] [Omega] [Omega]^2/ + + for which as before we have equations + + [zeta](z + [omega]) = [zeta](z) + 2[pi]i[eta], + [zeta](z + [omega]') = [zeta](z) + 2[pi]i[eta]', + + where 2[eta], 2[eta]' are certain constants, which in this case do not + both vanish, since else [zeta](z) would be a doubly periodic function + with only one pole of the first order. By considering the integral + _ + / + | [zeta](z)dz + _/ + + round the perimeter of a parallelogram of sides [omega], [omega]' + containing z = 0 in its interior, we find [eta][omega]' - + [eta]'[omega] = 1, so that neither of [eta], [eta]' is zero. We have + [zeta]'(z) = -RN(z). From [zeta](z) by means of the equation + _ _ _ + [sigma](z) { / z | 1 | } + ---------- = exp { | | [zeta](x) - -- |dz } = + z { _/ 0 |_ z _| } + _ _ + | / 2 \ / z z^2 \ | + [Pi]' | ( 1 - ------- ) exp ( ------- + ---------- ) |, + |_ \ [Omega]/ \[Omega] 2[Omega]^2/ _| + + we determine an integral function [sigma](z), termed the + Sigma-function, having a zero of the first order at each of the points + z = [Omega]; it can be seen to satisfy the equations + + [sigma](z + [omega]) + -------------------- = -exp [2[pi] i[eta](z + 1/2[omega])], + [sigma](z) + + [sigma](z + [omega]') + --------------------- = -exp [2[pi] i[eta]'(z + 1/2[omega]')]. + [sigma](z) + + By means of these equations, if a1 + a2 + ... + a_m = a'1 + a'2 + ... + + a'_m, it is readily shown that + + [sigma](z - a'1)[sigma](z - a'2) ... [sigma](z - a'_m) + ------------------------------------------------------ + [sigma](z - a1[sigma](z - a2) ... [sigma](z - a_m) + + is a doubly periodic function having a1, ... a_m as its simple poles, + and a'1, ... a'_m as its simple zeros. Thus the function [sigma](z) + has the important property of enabling us to write any meromorphic + doubly periodic function as a product of factors each having one zero + in the parallelogram of periods; these form a generalization of the + simple factors, z - a, which have the same utility for rational + functions of z. We have [zeta](z) = [sigma]'(z)/[sigma](z). + + The functions [zeta](z), RN(z) may be used to write any meromorphic + doubly periodic function F(z) as a sum of terms having each only one + pole; for if in the expansion of F(z) near a pole z = a the terms with + negative powers of z-a be + + A1(z - a)^(-1) + A2(z - a){-2} + ... + A_(m + 1)(z - a)^(-(m + 1)), + + then the difference + + A_(m + 1) + F(z) - A1[zeta](z - a) - A2[Fraktur](z - a)- ... + ---------(-1)^m RN^(m - 1)(z - a) + m! + + will not be infinite at z = a. Adding to this a sum of further terms + of the same form, one for each of the poles in a parallelogram of + periods, we obtain, since the sum of the residues A is zero, a doubly + periodic function without poles, that is, a constant; this gives the + expression of F(z) referred to. The indefinite integral [int]F(z)dz + can then be expressed in terms of z, functions RN(z - a) and their + differential coefficients, functions [zeta](z - a) and functions + log[sigma](z - a). + +S 15. _Potential Functions. Conformal Representation in +General._--Consider a circle of radius a lying within the region of +existence of a single valued monogenic function, u + iv, of the complex +variable z, = x + iy, the origin z = 0 being the centre of this circle. +If z = rE(i[phi]) = r(cos [phi] + i sin [phi]) be an internal point of +this circle we have + _ + 1 / (U + iV) + u + iv = ------ | -------- dt, + 2[pi]i _/ t - z + +where U + iV is the value of the function at a point of the +circumference and t = aE(i[theta]); this is the same as + _ + 1 / (U + iV) [1 - (r/a)E(i[theta] - i[phi])] + u + iv = ----- | ------------------------------------------ d[theta]. + 2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi]) + +If in the above formula we replace z by the external point +(a^2/r)E(i[phi]) the corresponding contour integral will vanish, so that +also + + _ + 1 / (U + iV) [(r/a)^2 - (r/a)E(i[theta] - i[phi])] + 0 = ----- | ---------------------------------------------- d[theta]; + 2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi]) + +hence by subtraction we have + _ + 1 / U(a^2 - r^2) + u = ----- | ------------------------------------- d[theta], + 2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]) + +and a corresponding formula for v in terms of V. If O be the centre of +the circle, Q be the interior point z, P the point aE(i[theta]) of the +circumference, and [omega] the angle which QP makes with OQ produced, +this integral is at once found to be the same as + _ _ + 1 / 1 / + u = ---- | Ud[omega] - ----- | Ud[theta] + [pi] _/ 2[pi] _/ + +of which the second part does not depend upon the position of z, and the +equivalence of the integrals holds for every arc of integration. + + Conversely, let U be any continuous real function on the + circumference, U0 being the value of it at a point P0 of the + circumference, and describe a small circle with centre at P0 cutting + the given circle in A and B, so that for all points P of the arc AP0B + we have |U - U0| < [epsilon], where [epsilon] is a given small real + quantity. Describe a further circle, centre P0 within the former, + cutting the given circle in A' and B', and let Q be restricted to lie + in the small space bounded by the arc A'P0B' and this second circle; + then for all positions of P upon the greater arc AB of the original + circle QP^2 is greater than a definite finite quantity which is not + zero, say QP^2 > D^2. Consider now the integral + _ + 1 / (a^2 - r^2) + u' = ----- | U ------------------------------------ d[theta], = + 2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi] + + _ _ + 1 / 1 / + ---- | Ud[omega] - ----- | Ud[theta], + [pi] _/ 2[pi] _/ + + which we evaluate as the sum of two, respectively along the small arc + AP0B and the greater arc AB. It is easy to verify that, for the whole + circumference, + _ + 1 / (a^2 - r^2) + U0 = ----- | U0 ------------------------------------ d[theta] = + 2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi] + + _ _ + 1 / 1 / + ---- | U0d[omega] - ----- | U0d[theta]. + [pi] _/ 2[pi] _/ + + Hence we can write + _ _ + 1 / 1 / + u' - U0 = ----- | (U - U0) d[omega] - ----- | (U - U0) d[theta] + + 2[pi] _/AP0B 2[pi] _/AP0B + + _ + 1 / (a^2 - r^2) + ----- | (U - U0) ----------- d[theta]. + 2[pi] _/AB QP^2 + + If the finite angle between QA and QB be called [Phi] and the finite + angle AOB be called [Theta], the sum of the first two components is + numerically less than + + [epsilon] + --------- ([Phi] + [Theta]). + 2[pi] + + If the greatest value of |(U - U0)| on the greater arc AB be called H, + the last component is numerically less than + + H + --- (a^2 - r^2), + D^2 + + of which, when the circle, of centre P0, passing through A'B' is + sufficiently small, the factor a^2 - r^2 is arbitrarily small. Thus it + appears that u' is a function of the position of Q whose limit, when + Q, interior to the original circle, approaches indefinitely near to + P0, is U0. From the form + _ _ + 1 / 1 / + u' = ---- | Ud[omega] - ----- | Ud[theta], + [pi] _/ 2[pi] _/ + + since the inclination of QP to a fixed direction is, when Q varies, P + remaining fixed, a solution of the differential equation + + (Pd)^2[psi] (Pd)^2 + ----------- + ------- = 0, + (Pd)x^2 (Pd)y^2 + + where z, = x + iy, is the point Q, we infer that u' is a + differentiable function satisfying this equation; indeed, when r < a, + we can write + _ + 1 / (a^2 - r^2) + ----- | U ------------------------------------- d[theta] + 2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]) + _ _ _ + 1 / | r r^2 | + = ----- | U | 1 + 2 -- cos ([theta] - [phi]) + 2 --- cos 2([theta] - [phi]) + ...| d[theta] + 2[pi] _/ |_ a a^2 _| + + = a0 + a1x + b1y + a2(x^2 - y^2) + 2b2xy + ..., + + where + + _ _ _ + 1 / 1 / U cos[theta] 1 / U sin[theta] + a0 = ----- | Ud[theta], a1 = ---- | ------------ d[theta], b1 = ---- | ------------ d[theta], + 2[pi] _/ [pi] _/ a [pi] _/ a + + _ _ + 1 / U cos 2[theta] 1 / U sin 2[theta] + a2 = ---- | -------------- d[theta], b2 = ---- | -------------- d[theta]. + [pi] _/ a^2 [pi] _/ a^2 + + In this series the terms of order n are sums, with real coefficients, + of the various integral polynomials of dimension n which satisfy the + equation (Pd)^2[psi]/(Pd)x^2 + (Pd)^2[psi]/(Pd)y^2; the series is thus + the real part of a power series in z, and is capable of + differentiation and integration within its region of convergence. + + Conversely we may suppose a function, P, defined for the interior of a + finite region R of the plane of the real variables x, y, capable of + expression about any interior point x0, y0 of this region by a power + series in x - x0, y - y0, with real coefficients, these various series + being obtainable from one of them by continuation. For any region R0 + interior to the region specified, the radii of convergence of these + power series will then have a lower limit greater than zero, and hence + a finite number of these power series suffice to specify the function + for all points interior to R0. Each of these series, and therefore the + function, will be differentiable; suppose that at all points of R0 the + function satisfies the equation + + (Pd)^2P (Pd)P^2 + ------- + ------- = 0, + (Pd)x^2 (Pd)y^2 + + we then call it a monogenic potential function. From this, save for an + additive constant, there is defined another potential function by + means of the equation + _ + /(x, y) /(Pd)P (Pd)P \ + Q = | ( ----- dy - ----- dx ). + _/ \(Pd)x (Pd)y / + + The functions P, Q, being given by a finite number of power series, + will be single valued in R0, and P + iQ will be a monogenic function + of z within R0. In drawing this inference it is supposed that the + region R0 is such that every closed path drawn in it is capable of + being deformed continuously to a point lying within R0, that is, is + _simply connected_. + + Suppose in particular, c being any point interior to R0, that P + approaches continuously, as z approaches to the boundary of R, to the + value log r, where r is the distance of c to the points of the + perimeter of R. Then the function of z expressed by + + [zeta] = (z - c) exp (-P - iQ) + + will be developable by a power series in (z - z0) about every point z0 + interior to R0, and will vanish at z = c; while on the boundary of R + it will be of constant modulus unity. Thus if it be plotted upon a + plane of [zeta] the boundary of R will become a circle of radius unity + with centre at [zeta] = 0, this latter point corresponding to z = c. A + closed path within R0, passing once round z = c, will lead to a closed + path passing once about [zeta] = 0. Thus every point of the interior + of R will give rise to one point of the interior of the circle. The + converse is also true, but is more difficult to prove; in fact, the + differential coefficient d[zeta]/dz does not vanish for any point + interior to R. This being assumed, we obtain a conformal + representation of the interior of the region R upon the interior of a + circle, in which the arbitrary interior point c of R corresponds to + the centre of the circle, and, by utilizing the arbitrary constant + arising in determining the function Q, an arbitrary point of the + boundary of R corresponds to an arbitrary point of the circumference + of the circle. + + There thus arises the problem of the determination of a real monogenic + potential function, single valued and finite within a given arbitrary + region, with an assigned continuous value at all points of the + boundary of the region. When the region is circular this problem is + solved by the integral 1/[pi] [int] Ud[omega] - 1/[pi] [int] Ud[theta] + previously given. When the region is bounded by the outermost portions + of the circumferences of two overlapping circles, it can hence be + proved that the problem also has a solution; more generally, consider + a finite simply connected region, whose boundary we suppose to consist + of a single closed path in the sense previously explained, ABCD; + joining A to C by two non-intersecting paths AEC, AFC lying within the + region, so that the original region may be supposed to be generated by + the overlapping regions AECD, CFAB, of which the common part is AECF; + suppose now the problem of determining a single valued finite + monogenic potential function for the region AECD with a given + continuous boundary value can be solved, and also the same problem for + the region CFAB; then it can be shown that the same problem can be + solved for the original area. Taking indeed the values assigned for + the original perimeter ABCD, assume arbitrarily values for the path + AEC, continuous with one another and with the values at A and C; then + determine the potential function for the interior of AECD; this will + prescribe values for the path CFA which will be continuous at A and C + with the values originally proposed for ABC; we can then determine a + function for the interior of CFAB with the boundary values so + prescribed. This in its turn will give values for the path AEC, so + that we can determine a new function for the interior of AECD. With + the values which this assumes along CFA we can then again determine a + new function for the interior of CFAB. And so on. It can be shown that + these functions, so alternately determined, have a limit representing + such a potential function as is desired for the interior of the + original region ABCD. There cannot be two functions with the given + perimeter values, since their difference would be a monogenic + potential function with boundary value zero, which can easily be shown + to be everywhere zero. At least two other methods have been proposed + for the solution of the same problem. + + A particular case of the problem is that of the conformal + representation of the interior of a closed polygon upon the upper half + of the plane of a complex variable t. It can be shown without much + difficulty that if a, b, c, ... be real values of t, and [alpha], + [beta], [gamma], ... be n real numbers, whose sum is n - 2, the + integral + + _ + / + z = | (t - a)^([alpha] - 1) (t - b)^([beta] - 1) ... dt, + _/ + + as t describes the real axis, describes in the plane of z a polygon of + n sides with internal angles equal to [alpha][pi], [beta][pi], ..., + and, a proper sign being given to the integral, points of the upper + half of the plane of t give rise to interior points of the polygon. + Herein the points a, b, ... of the real axis give rise to the corners + of the polygon; the condition [Sigma][alpha] = n - 2 ensures merely + that the point t = [oo] does not correspond to a corner; if this + condition be not regarded, an additional corner and side is introduced + in the polygon. Conversely it can be shown that the conformal + representation of a polygon upon the half plane can be effected in + this way; for a polygon of given position of more than three sides it + is necessary for this to determine the positions of all but three of + a, b, c, ...; three of them may always be supposed to be at arbitrary + positions, such as t = 0, t = 1, t = [oo]. + + As an illustration consider in the plane of z = x + iy, the portion of + the imaginary axis from the origin to z = ih, where h is positive and + less than unity; let C be this point z = ih; let BA be of length unity + along the positive real axis, B being the origin and A the point z = + 1; let DE be of length unity along the negative real axis, D being + also the origin and E the point z = -1; let EFA be a semicircle of + radius unity, F being the point z = i. If we put [zeta] = [(z^2 + + h^2)/(1 + h^2z^2)]^1/2, with [zeta] = 1 when z = 1, the function is + single valued within the semicircle, in the plane of z, which is slit + along the imaginary axis from the origin to z = ih; if we plot the + value of [zeta] upon another plane, as z describes the continuous + curve ABCDE, [zeta] will describe the real axis from [zeta] = 1 to + [zeta] = -1, the point C giving [zeta] = 0, and the points B, D giving + the points [zeta] = [+-]h. Near z = 0 the expansion of [zeta] is + [zeta] - h = z^2 1 - h^4 / 2h + ..., or [zeta] + h = -z^2 (1 - h^4)/2h + + ...; in either case an increase of 1/2[pi] in the phase of z gives + an increase of [pi] in the phase of [zeta] - h or [zeta] + h. Near z = + ih the expansion of [zeta] is [zeta] = (z - ih)^1/2 [2ih/(1 - + h^4)]^1/2 + ..., and an increase of 2[pi] in the phase of z - ih also + leads to an increase of [pi] in the phase of [zeta]. Then as z + describes the semicircle EFA, [zeta] also describes a semicircle of + radius unity, the point z = i becoming [zeta] = i. There is thus a + conformal representation of the interior of the slit semicircle in the + z-plane, upon the interior of the whole semicircle in the + [zeta]-plane, the function + + z = [([zeta]^2 - h^2) / (1 - h^2[zeta]^2)]^1/2 + + being single valued in the latter semicircle. By means of a + transformation t = ([zeta] + 1)^2/([zeta] - 1)^2, the semicircle in + the plane of [zeta] can further be conformably represented upon the + upper half of the whole plane of t. + + As another illustration we may take the conformal representation of an + equilateral triangle upon a half plane. Taking the elliptic function + RN(u) for which RN'^2(u) = 4RN^3(u) - 4, so that, with [epsilon] = exp + (2/3[pi]i), we have e1 = 1, e2 = [epsilon]^2, e3 = [epsilon], the half + periods may be taken to be + _ _ + / [oo] dt / [oo] dt + 1/2[omega] = | --------------, 1/2[omega]' = | -------------- = 1/2[epsilon][omega]; + _/ 1 2(t^3 - 1)^1/2 _/ e3 2(t^3 - 1)^1/2 + + drawing the equilateral triangle whose vertices are O, of argument O, + A of argument [omega], and B of argument [omega] + [omega]' = + -[epsilon]^2[omega], and the equilateral triangle whose angular points + are O, B and C, of argument [omega]', let E, of argument + {1/3}(2[omega] + [omega]'), and D, of argument 1/3([omega] + + 2[omega]'), be the centroids of these triangles respectively, and let + BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut + OC, BC, OB in F, G, H respectively; then if u = [xi] + i[eta] be any + point of the interior of the triangle OEH and v = [epsilon]u0 = + [epsilon]([xi] - i[eta]) be any point of the interior of the triangle + OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB, + EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by + -[epsilon]v, [omega] + [epsilon]u, [omega] - [eta]^2v, [omega] + + [omega]' + [epsilon]^2u, [omega] + [omega]' - v, [omega] + [omega]' - + u, [omega] + [omega]' + [epsilon]v, [omega]' - [epsilon]u, [omega]' + + [epsilon]^2v, -[epsilon]^2u. Further, when u is real, since the term + -2(u + m[omega] + m'[epsilon]^2[omega])^(-3), which is the conjugate + complex of -2(u + m[omega] + m'[epsilon]^2[omega])^3, arises in the + infinite sum which expresses RN'(u), namely as -2(u + [mu][omega] + + [mu]'[epsilon][omega])^(-3), where [mu] = m - m', [mu]' = -m', it + follows that RN'(u) is real; in a similar way we prove that RN'(u) is + pure imaginary when u is pure imaginary, and that RN'(u) = + RN'([epsilon]u) = RN'([epsilon]^2u), as also that for v = [epsilon]u0, + RN'(v) is the conjugate complex of RN'(u). Hence it follows that the + variable + + t = 1/2iRN'(u) + + takes each real value once as u passes along the perimeter of the + triangle ODE, being as can be shown respectively [oo], 1, 0, -1 at O, + D, H, E, and takes every complex value of imaginary part positive once + in the interior of this triangle. This leads to + _ + / [oo] + u = 1/3i | (t^2 - 1)^(-2/3) dt + _/ t + + in accordance with the general theory. + + It can be deduced that [tau] = t^2 represents the triangle ODH on the + upper half plane of [tau], and [zeta] = {i-[tau]^(-1)}^(1/2) + represents similarly the triangle OBD. + +S 16. _Multiple valued Functions. Algebraic Functions._--The +explanations and definitions of a monogenic function hitherto given have +been framed for the most part with a view to single valued functions. +But starting from a power series, say in z - c, which represents a +single value at all points of its circle of convergence, suppose that, +by means of a derived series in z - c', where c' is interior to the +circle of convergence, we can continue the function beyond this, and +then by means of a series derived from the first derived series we can +make a further continuation, and so on; it may well be that when, after +a closed circuit, we again consider points in the first circle of +convergence, the value represented may not agree with the original +value. One example is the case z^(1/2), for which two values exist for +any value of z; another is the generalized logarithm [Lambda] (z), for +which there is an infinite number of values. In such cases, as before, +the region of existence of the function consists of all points which can +be reached by such continuations with power series, and the singular +points, which are the limiting points of the point-aggregate +constituting the region of existence, are those points in whose +neighbourhood the radii of convergence of derived series have zero for +limit. In this description the point z = [oo] does not occupy an +exceptional position, a power series in z - c being transformed to a +series in 1/z when z is near enough to c by means of z - c = c(1 - +cz^(-1)) [1 - (1 - cz^(-1))]^(-1), and a series in 1/z to a series in z +- c, when z is near enough to c, by means of + + 1 1 / z - c \^(-1) + -- = -- ( 1 + ----- ). + z c \ c / + + The commonest case of the occurrence of multiple valued functions is + that in which the function s satisfies an algebraic equation [f](s, z) + = p_0s^n + p1s^(n - 1) + ... + p_n = 0, wherein p0, p1, ... p_n are + integral polynomials in z. Assuming [f](s, z) incapable of being + written as a product of polynomials rational in s and z, and excepting + values of z for which the polynomial coefficient of s^n vanishes, as + also the values of z for which beside [f](s, z) = 0 we have also + (Pd)f(s, z)/(Pd)s = 0, and also in general the point z = [oo], the + roots of this equation about any point z = c are given by n power + series in z-c. About a finite point z = c for which the equation + (Pd)f(s, z)/(Pd)s = 0 is satisfied by one or more of the roots s of + [f](s, z) = 0, the n roots break up into a certain number of cycles, + the r roots of a cycle being given by a set of power series in a + radical (z - c)^(1/r), these series of the cycle being obtainable from + one another by replacing (z - c)^(1/r) by [omega](z - r)^(1/r), where + [omega], equal to exp (2[pi]ih/r), is one of the rth roots of unity. + Putting then z - c = t^r we may say that the r roots of a cycle are + given by a single power series in t, an increase of 2[pi] in the phase + of t giving an increase of 2[pi]r in the phase of z - c. This single + series in t, giving the values of s belonging to one cycle in the + neighbourhood of z = c when the phase of z-c varies through 2[pi]r, is + to be looked upon as defining a single _place_ among the aggregate of + values of z and s which satisfy [f](s, z) = 0; two such places may be + at the same _point_ (z = c, s = d) without coinciding, the + corresponding power series for the neighbouring points being + different. Thus for an ordinary value of z, z = c, there are n places + for which the neighbouring values of s are given by n power series in + z-c; for a value of z for which (Pd)f(s, z)/(Pd)s = 0 there are less + than n places. Similar remarks hold for the neighbourhood of z = [oo]; + there may be n places whose neighbourhood is given by n power series + in z^(-1) or fewer, one of these being associated with a series in t, + where t = (z^(-1))^(1/r); the sum of the values of r which thus arise + is always n. In general, then, we may say, with t of one of the forms + (z-c), (z-c)^(1/r), z^(-1), (z^(-1))^(1/r). that the neighbourhood of + any place (c, d) for which [f](c, d) = 0 is given by a pair of + expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular + case of a) power series vanishing for t = 0, and Q(t) is a power + series vanishing for t = 0, and t vanishes at (c, d), the expression + z-c being replaced by z^(-1) when c is infinite, and similarly the + expression s-d by s^(-1) when d is infinite. The last case arises when + we consider the finite values of z for which the polynomial + coefficient of s^n vanishes. Of such a pair of expressions we may + obtain a continuation by writing t = t0 + [lambda]1[tau] + + [lambda]2[tau]^2 + .. , where [tau] is a new variable and [lambda]1 is + not zero; in particular for an ordinary finite place this equation + simply becomes t = t0 + [tau]. It can be shown that all the pairs of + power series z = c + P(t), s = d + Q(t) which are necessary to + represent all pairs of values of z, s satisfying the equation [f](s, + z) = 0 can be obtained from one of them by this process of + continuation, a fact which we express by saying that the equation + [f](s, z) = 0 defines a _monogenic algebraic construct_. With less + accuracy we may say that an irreducible algebraic equation [f](s, z) = + 0 determines a single monogenic function s of z. + + Any rational function of z and s, where [f](s, z) = 0, may be + considered in the neighbourhood of any place (c, d) by substituting + therein z = c + P(t), s = d + Q(t); the result is necessarily of the + form t^m H(t), where H(t) is a power series in t not vanishing for t = + 0 and m is an integer. If this integer is positive, the function is + said to vanish to order m at the place; if this integer is negative, = + -[mu], the function is infinite to order [mu] at the place. More + generally, if A be an arbitrary constant, and, near (c, d), R(s, z) - + A is of the form t^mH(t), where m is positive, we say that R(s, z) + becomes m times equal to A at the place; if R(s, z) is infinite of + order [mu] at the place, so also is R(s, z) - A. It can be shown that + the sum of the values of m at all the places, including the places z = + [oo], where R(s, z) vanishes, which we call the number of zeros of + R(s, z) on the algebraic construct, is finite, and equal to the sum of + the values of [mu] where R (s, z) is infinite, and more generally + equal to the sum of the values of m where R(s, z) = A; this we express + by saying that a rational function R(s, z) takes any value (including + [oo]) the same number of times on the algebraic construct; this number + is called the _order_ of the rational function. + + That the total number of zeros of R (s, z) is finite is at once + obvious, these values being obtainable by rational elimination of s + between [f](s, z) = 0, R(s, z) = 0. That the number is equal to the + total number of infinities is best deduced by means of a theorem which + is also of more general utility. Let R(s, z) be any rational function + of s, z, which are connected by [f](s, z) = 0; about any place (c, d) + for which z = c + P(t), s = d + Q(t), expand the product + + dz + R(s, z) -- + dt + + in powers of t and pick out the coefficient of t^(-1). There is only a + finite number of places of this kind. The theorem is that the sum of + these coefficients of t^(-1) is zero. This we express by + _ _ + | dz | + |R(s, z) -- | = 0. + |_ dt _|t^(-1) + + The theorem holds for the case n = 1, that is, for rational functions + of one variable z; in that case, about any finite point we have z - c + = t, and about z = [oo] we have z^(-1) = t, and therefore dz/dt = + -t^(-2); in that case, then, the theorem is that in any rational + function of z, + + _ / A1 A2 A_m \ + \ ( ----- + --------- + ... + --------- ) + Pz^h + Qz^(h - 1) + ... + R, + /_ \z - a (z - a)^2 (z - a)^m / + + the sum [Sigma]A1 of the sum of the residues at the finite poles is + equal to the coefficient of 1/z in the expansion, in ascending powers + of 1/z, about z = [oo]; an obvious result. In general, if for a finite + place of the algebraic construct associated with [f](s, z) = 0, whose + neighbourhood is given by z = c + t^r, s = d + Q(t), there be a + coefficient of t^(-1) in R(s, z)dz/dt, this will be r times the + coefficient of t^(-r) in R(s, z) or R[d + Q(t), c + t^r], namely will + be the coefficient of t^(-r) in the sum of the r series obtainable + from R[d + Q(t), c + t^r] by replacing t by [omega]t, where [omega] is + an rth root of unity; thus the sum of the coefficients of t^(-1) in + R(s, z)dz/dt for all the places which arise for z = c, and the + corresponding values of s, is equal to the coefficient of (z - c)^(-1) + in R(s1, z) + R(s2z) + ... + R(s_n, z), where s1, ... s_n are the n + values of s for a value of z near to z = c; this latter sum [Sigma] + R(s_i, z) is, however, a rational function of z only. Similarly, near + z = [oo], for a place given by z^(-1) = t^r, s = d + Q(t), or s^(-1) = + Q(t), the coefficient of t^(-1) in R(s, z)dz/dt is equal to -r times + the coefficient of t^r in R[d + Q(t), t^(-r)], that is equal to the + negative coefficient of z^(-l) in the sum of the r series R[d + + Q([omega]t), t^(-r)], so that, as before, the sum of the coefficients + of t^(-1) in R(s, z)dz/dt at the various places which arise for z = + [oo] is equal to the negative coefficient of z^(-1) in the same + rational function of z, [Sigma] R(s_i, z). Thus, from the + corresponding theorem for rational functions of one variable, the + general theorem now being proved is seen to follow. + + Apply this theorem now to the rational function of s and z, + + 1 dR(s, z) + ------- -------; + R(s, z) dz + + at a zero of R(s, z) near which R(s, z) = t^mH(t), we have + + 1 dR(s, z) dz d + ------- ------- -- = -- {[lambda] [R(s, z)]} + R(s, z) dz dt dt + + where [lambda] denotes the generalized logarithmic function, that is + equal to + + mt^(-1) + power series in t; + + similarly at a place for which R(s, z) = t^(-[mu]) K(t); the theorem + _ _ + | 1 dR(s, z) dz | + | ------- -------- -- | t^(-1) = 0 + |_ R(s, z) dz dt _| + + thus gives [Sigma]m = [Sigma][mu], or, in words, the total number of + zeros of R(s, z) on the algebraic construct is equal to the total + number of its poles. The same is therefore true of the function R(s, + z) - A, where A is an arbitrary constant; thus the number in question, + being equal to the number of poles of R(s, z) - A, is equal also to + the number of times that R(s, z) = A on the algebraic construct. + + We have seen above that all single valued doubly periodic meromorphic + functions, with the same periods, are rational functions of two + variables s, z connected by an equation of the form s^2 = 4z^3 + Az + + B. Taking account of the relation connecting these variables s, z with + the argument of the doubly periodic functions (which was above denoted + by z), it can then easily be seen that the theorem now proved is a + generalization of the theorem proved previously establishing for a + doubly periodic function a definite _order_. There exists a + generalization of another theorem also proved above for doubly + periodic functions, namely, that the sum of the values of the argument + in one parallelogram of periods for which a doubly periodic function + takes a given value is independent of that value; this generalization, + known as Abel's Theorem, is given S 17 below. + +S 17. _Integrals of Algebraic Functions._--In treatises on Integral +Calculus it is proved that if R(z) denote any rational function, an +indefinite integral [int]R(z)dz can be evaluated in terms of rational +and logarithmic functions, including the inverse trigonometrical +functions. In generalization of this it was long ago discovered that if +s^2 = az^2 + bz + c and R(s, z) be any rational function of s, z any +integral [int]R(s, z)dz can be evaluated in terms of rational functions +of s, z and logarithms of such functions; the simplest case is +[int]s^(-1)dz or [int](az^2 + bz + c)^(-1/2)dz. More generally if f(s, +z) = 0 be such a relation connecting s, z that when [theta] is an +appropriate rational function of s and z both s and z are rationally +expressible, in virtue of [f](s, z) = 0 in terms of [theta], the +integral [int]R(s, z)dz is reducible to a form [int]H ([theta])d[theta], +where H([theta]) is rational in [theta], and can therefore also be +evaluated by rational functions and logarithms of rational functions of +s and z. It was natural to inquire whether a similar theorem holds for +integrals [int]R(s, z)dz wherein s^2 is a cubic polynomial in z. The +answer is in the negative. For instance, no one of the three integrals + + _ _ _ + / dz / zdz / dz + | --, | ---, | -------- + _/ s _/ s _/ (z - c)s + +can be expressed by rational and logarithms of rational functions of s +and z; but it can be shown that every integral [int]R(s, z)dz can be +expressed by means of integrals of these three types together with +rational and logarithms of rational functions of s and z (see below +under S 20, Elliptic Integrals). A similar theorem is true when s^2 = +quartic polynomial in z; in fact when s^2 = A(z - a)(z - b)(z - c)(z - +d), putting y = s(z - a)^(-2), x = (z - a)^(-1), we obtain y2 = cubic +polynomial in x. Much less is the theorem true when the fundamental +relation [f](s, z) = 0 is of more general type. There exists then, +however, a very general theorem, known as _Abel's Theorem_, which may be +enunciated as follows: Beside the rational function R(s, z) occurring in +the integral [int]R(s, z)dz, consider another rational function H(s, z); +let (a1), ... (a_m) denote the places of the construct associated with +the fundamental equation [f](s, z) = 0, for which H(s, z) is equal to +one value A, each taken with its proper multiplicity, and let (b1), ... +(b_m) denote the places for which H(s, z) = B, where B is another value; +then the sum of the m integrals [int] [(b_i) to (a_i)] R(s, z)dz is +equal to the sum of the coefficients of t^(-1) in the expansions of the +function + + dz / H(s, z) - B \ + R(s, z) -- [lambda] ( ----------- ), + dt \ H(s, z) - A / + +where [lambda] denotes the generalized logarithmic function, at the +various places where the expansion of R(s, z)dz/dt contains negative +powers of t. This fact may be obtained at once from the equation + _ _ + | 1 dz | + | -------------- R(s, z) -- | = 0, + |_ H(s, z) - [mu] dt _|t^(-1) + +wherein [mu] is a constant. (For illustrations see below, under S 20, +Elliptic Integrals.) + +S 18. _Indeterminateness of Algebraic Integrals._--The theorem that the +integral [int][a to x] [f](z)dz is independent of the path from a to z, +holds only on the hypothesis that any two such paths are equivalent, +that is, taken together from the complete boundary of a region of the +plane within which [f](z) is finite and single valued, besides being +differentiable. Suppose that these conditions fail only at a finite +number of isolated points in the finite part of the plane. Then any path +from a to z is equivalent, in the sense explained, to any other path +together with closed ~~ paths beginning and ending at the arbitrary +point a each enclosing one or more of the exceptional points, these +closed paths being chosen, when [f](z) is not a single valued function, +so that the final value of [f](z) at a is equal to its initial value. It +is necessary for the statement that this condition may be capable of +being satisfied. + + For instance, the integral [int][1 to z] z^(-1)dz is liable to an + additive indeterminateness equal to the value obtained by a closed + path about z = 0, which is equal to 2[pi]i; if we put u = [int][1 to + z] z^(-1)dz and consider z as a function of u, then we must regard + this function as unaffected by the addition of 2[pi]i to its argument + u; we know in fact that z = exp (u) and is a single valued function of + u, with the period 2[pi]i. Or again the integral [int][0 to z] (1 + + z^2)^(-1)dz is liable to an additive indeterminateness equal to the + value obtained by a closed path about either of the points z = [+-]i; + thus if we put u = [int][0 to z] (1 + z^2)^(-1)dz, the function z of u + is periodic with period [pi], this being the function tan (u). Next we + take the integral u = [int][(0) to (z)] (1 - z^2)^(-1/2)dz, agreeing + that the upper and lower limits refer not only to definite values of + z, but to definite values of z each associated with a definite + determination of the sign of the associated radical (1 - z^2)^(-1/2). + We suppose 1 + z, 1 - z each to have phase zero for z = 0; then a + single closed circuit of z = -1 will lead back to z = 0 with (l - + z^2)^1/2 = -1; the additive indeterminateness of the integral, + obtained by a closed path which restores the initial value of the + subject of integration, may be obtained by a closed circuit containing + both the points [+-]1 in its interior; this gives, since the integral + taken about a vanishing circle whose centre is either of the points z + = [+-] 1 has ultimately the value zero, the sum + + _ _ _ _ + / -1 dz / 0 dz / 1 dz / 0 dz + | ----------- + | -------------- + | -------------- + | -------------, + _/ 0 (1-z^2)^1/2 _/-1 -(1 - z^2)^1/2 _/ 0 -(1 - z^2)^1/2 _/ 1 (1 - z^2)^1/2 + + where, in each case, (1 - z^2)^1/2 is real and positive; that is, it + gives + _ + / 1 dz + -4 | ------------- + _/ 0 (1 - z^2)^1/2 + + or 2[pi]. Thus the additive indeterminateness of the integral is of + the form 2k[pi], where k is an integer, and the function z of u, which + is sin (u), has 2[pi] for period. Take now the case + _ + / (z) dz + u = | ------------------------------------, + _/ (z0) [root]{(z - a)(z - b)(z - c)(z - d)} + + adopting a definite determination for the phase of each of the factors + z - a, z - b, z - c, z - d at the arbitrary point z0, and supposing + the upper limit to refer, not only to a definite value of z, but also + to a definite determination of the radical under the sign of + integration. From z0 describe a closed loop about the point z = a, + consisting, suppose, of a straight path from z0 to a, followed by a + vanishing circle whose centre is at a, completed by the straight path + from a to z0. Let similar loops be imagined for each of the points b, + c, d, no two of these having a point in common. Let A denote the value + obtained by the positive circuit of the first loop; this will be in + fact equal to twice the integral taken from z0 along the straight path + to a; for the contribution due to the vanishing circle is ultimately + zero, and the effect of the circuit of this circle is to change the + sign of the subject of integration. After the circuit about a, we + arrive back at z0 with the subject of integration changed in sign; let + B, C, D denote the values of the integral taken by the loops enclosing + respectively b, c and d when in each case the initial determination of + the subject of integration is that adopted in calculating A. If then + we take a circuit from z0 enclosing both a and b but not either c or + d, the value obtained will be A - B, and on returning to z0 the + subject of integration will have its initial value. It appears thus + that the integral is subject to an additive indeterminateness equal to + any one of the six differences such as A - B. Of these there are only + two linearly independent; for clearly only A - B, A - C, A - D are + linearly independent, and in fact, as we see by taking a closed + circuit enclosing all of a, b, c, d, we have A - B + C - D = 0; for + there is no other point in the plane beside a, b, c, d about which the + subject of integration suffers a change of sign, and a circuit + enclosing all of a, b, c, d may by putting z = 1/[zeta] be reduced to + a circuit about [zeta] = 0 about which the value of the integral is + zero. The general value of the integral for any position of z and the + associated sign of the radical, when we start with a definite + determination of the subject of integration, is thus seen to be of the + form u0 + m(A - B) + n(A - C), where m and n are integers. The value + of A - B is independent of the position of z0, being obtainable by a + single closed positive circuit about a and b only; it is thus equal to + twice the integral taken once from a to b, with a proper initial + determination of the radical under the sign of integration. Similar + remarks to the above apply to any integral [int]H(z)dz, in which H(z) + is an algebraic function of z; in any such case H(z) is a rational + function of z and a quantity s connected therewith by an irreducible + rational algebraic equation [f](s, z) = 0. Such an integral [f]K(z, + s)dz is called an Abelian Integral. + +S 19. _Reversion of an Algebraic Integral._--In a limited number of +cases the equation u = [int] [z0 to z] H(z)dz, in which H(z) is an +algebraic function of z, defines z as a single valued function of u. +Several cases of this have been mentioned in the previous section; from +what was previously proved under S 14, _Doubly Periodic Functions_, it +appears that it is necessary for this that the integral should have at +most two linearly independent additive constants of indeterminateness; +for instance, for an integral + _ + / z + u = | [(z - a)(z - b)(z - c)(z - d)(z - e)(z - f)]^(-1/2) dz, + _/ z0 + +there are three such constants, of the form A - B, A - C, A - D, which +are not connected by any linear equation with integral coefficients, and +z is not a single valued function of u. + +S 20. _Elliptic Integrals._--An integral of the form [int] R(z, s)dz, +where s denotes the square root of a quartic polynomial in z, which may +reduce to a cubic polynomial, and R denotes a rational function of z and +s, is called an _elliptic integral_. + + To each value of z belong two values of s, of opposite sign; starting, + for some particular value of z, with a definite one of these two + values, the sign to be attached to s for any other value of z will be + determined by the path of integration for z. When z is in the + neighbourhood of any finite value z0 for which the radical s is not + zero, if we put z - z0 = t, we can find s - s0 = a power series in t, + say s = s0 + Q(t); when z is in the neighbourhood of a value, a, for + which s vanishes, if we put z = a + t^2, we shall obtain s = tQ(t), + where Q(t) is a power series in t; when z is very large and s^2 is a + quartic polynomial in z, if we put z^(-1) = t, we shall find s^(-1) = + t^2Q(t); when z is very large and s^2 is a cubic polynomial in z, if we + put z^(-1) = t^2, we shall find s^(-l) = t^3Q(t). By means of + substitutions of these forms the character of the integral [int] R(z, + s)dz may be investigated for any position of z; in any case it takes a + form [int] [Ht^(-m) + Kt^(-m + 1) + ... + Pt^(-1) + R + St + ...]dt + involving only a finite number of negative powers of t in the subject + of integration. Consider first the particular case [int] s^(-1)dz; it + is easily seen that neither for any finite nor for infinite values of + z can negative powers of t enter; the integral is _everywhere finite_, + and is said to be of _the first kind_; it can, moreover, be shown + without difficulty that no integral [int] R(z, s)dz, save a constant + multiple of [int] s^(-1)dz, has this property. Consider next, s^2 being + of the form a0z^4 + 4a1z^3 + ..., wherein a0 may be zero, the integral + [int] {a0z^2 + 2a1z) s^(-1)dz; for any finite value of z this integral + is easily proved to be everywhere finite; but for infinite values of z + its value is of the form At^(-1) + Q(t), where Q(t) is a power series; + denoting by [root]a0 a particular square root of a0 when a0 is not + zero, the integral becomes infinite for z = [oo] for both signs of s, + the value of A being + [root]a0 or - [root]a0 according as s is + [root]a0.z^2 (1 + [2a1/a0] z^(-1) + ...) or is the negative of this; + hence the integral J1 = [int] ([a0z^2 + 2a1z / s] + [root]a0)dz becomes + infinite when z is infinite, for the former sign of s, its infinite + term being 2[root]a0 t^(-1) or 2a0.z, but does not become infinite for + z infinite for the other sign of s. When a0 = 0 the signs of s for z = + [oo] are not separated, being obtained one from the other by a circuit + of z about an infinitely large circle, and the form obtained + represents an integral becoming infinite as before for z = [oo], its + infinite part being 2[root]a1.t^(-1) or 2[root]a1.[root]z. Similarly + if z0 be any finite value of z which is not a root of the polynomial + [f](z) to which s^2 is equal, and s0 denotes a particular one of the + determinations of s for z = z0, the integral + _ + / / s0^2 + 1/2(z - z0) [f]'(z0) s0 \ + J2 = | ( --------------------------- + --------- ) dz, + _/ \ (z - z0)^2s (z - z0)^2 / + + wherein [f]'(z) = d[f](z)/dz, becomes infinite for z = z0, s = s0, but + not for z = z0, s = -s0. its infinite term in the former case being + the negative of 2s0(z - z0). For no other finite or infinite value of + z is the integral infinite. If z = [theta] be a root of [f](z), in + which case the corresponding value of s is zero, the integral + _ + / dz + J3 = 1/2[f]'([theta]) | -------------- + _/ (z - [theta])s + + becomes infinite for z = 0, its infinite part being, if z - [theta] = + t^2, equal to -[[f]'([theta])] 1/2t^(-1): and this integral is not + elsewhere infinite. In each of these cases, of the integrals J1, J2, + J3, the subject of integration has been chosen so that when the + integral is written near its point of infinity in the form + [int][At^(-2) + Bt^(-1) + Q(t)]dt, the coefficient B is zero, so that + the infinity is of algebraic kind, and so that, when there are two + signs distinguishable for the critical value of z, the integral + becomes infinite for only one of these. An integral having only + algebraic infinities, for finite or infinite values of z, is called an + integral of the _second kind_, and it appears that such an integral + can be formed with only one such infinity, that is, for an infinity + arising only for one particular, and arbitrary, pair of values (s, z) + satisfying the equation s^2 = [f](z), this infinity being of the first + order. A function having an algebraic infinity of the mth order (m > + 1), only for one sign of s when these signs are separable, at (1) z = + [oo], (2) z = z0, (3) z = a, is given respectively by (s d/dz)^(m - + 1)J1, (s d/dz)^(m - 1) J2, (s d/dz )^(m - 1) J3, as we easily see. If + then we have any elliptic integral having algebraic infinities we can, + by subtraction from it of an appropriate sum of constant multiples of + J1, J2, J3 and their differential coefficients just written down, + obtain, as the result, an integral without algebraic infinities. But, + in fact, if J, J^1 denote any two of the three integrals J1, J2, J3, + there exists an equation AJ + BJ' + C[f]s^(-1)dz = rational function + of s, z, where A, B, C are properly chosen constants. For the rational + function + + s + s0 + ------ + z [root]a0 + z - z0 + + is at once found to become infinite for (z0, s0), not for (z0, -s0), + its infinite part for the first point being 2s/(z - z0), and to become + infinite for z infinitely large, and one sign of s only when these are + separable, its infinite part there being 2z [root] a0 or 2 [root] a1 + [root] z when a0 = 0. It does not become infinite for any other pair + (z, s) satisfying the relation s^2 = [f](z); this is in accordance + with the easily verified equation + _ + s + s0 / dz + ------- + z [root]a0 - J1 + J2 + (a0z0^2 + 2a1z0) | -- = 0; + z - z^0 _/ s + + and there exists the analogous equation + _ + s / dz + ----------- + z [root]a0 - J1 + J3 + (a0[theta]^2 + 2a1[theta]) | -- = 0. + z - [theta] _/ s + + Consider now the integral + _ + / /s + s0 \ dz + P = | ( --------- + z [root]a0 ) --; + _/ \z - z0 / 2s + + this is at once found to be infinite, for finite values of z, only for + (z0, s0), its infinite part being log (z - z0), and for z = [oo], for + one sign of s only when these are separable, its infinite part being + -log t, that is -log z when a0 /= 0, and -log (z^1/2) when a0 = 0. + And, if [f]([theta]) = 0, the integral + + _ + / / s \ dz + P1 = | ( ----------- + z [root]a0 ) -- + _/ \z - [theta] / 2s + + is infinite at z = [theta], s = 0 with an infinite part log t, that is + log (z - [theta])^1/2, is not infinite for any other finite value of z, + and is infinite like P for z = [oo]. An integral possessing such + logarithmic infinities is said to be of the third kind. + + Hence it appears that any elliptic integral, by subtraction from it of + an appropriate sum formed with constant multiples of the integral J3 + and the rational functions of the form (s d/dz)^(m - 1) J1 with + constant multiples of integrals such as P or P1, with constant + multiples of the integral u = [int]s^(-1)dz, and with rational + functions, can be reduced to an integral H becoming infinite only for + z = [oo], for one sign of s only when these are separable, its + infinite part being of the form A log t, that is, A log z or A log + (z^1/2). Such an integral H = [int]R(z, s)dz does not exist, however, + as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z) + are rational functions of z, and examining the forms possible for + these in order that the integral may have only the specified infinity. + An analogous theorem holds for rational functions of z and s; there + exists no rational function which is finite for finite values of z and + is infinite only for z = [oo] for one sign of s and to the first order + only; but there exists a rational function infinite in all to the + first order for each of two or more pairs (z, s), however they may be + situated, or infinite to the second order for an arbitrary pair (z, + s); and any rational function may be formed by a sum of constant + multiples of functions such as + + s + s0 s + ------ + z [root]a0 or ----------- + z [root]a0 + z - z0 z - [theta] + + and their differential coefficients. + + The consideration of elliptic integrals is therefore reducible to that + of the three + _ _ _ + / dz / /a0z^2 + 2a1z \ / /s + s0 \ dz + u = | --, J = | ( ------------ + z [root]a0 ) dz, P = | ( ------ + z [root]a0 )-- + _/ s _/ \ s / _/ \z - z0 / 2s + + respectively of the first, second and third kind. Now the equation s^2 + = a0z^4 + ... = a0(z - [theta]) (z - [phi]) (z - [psi]) (z - [chi]), + by putting + + y = 2s(z - [theta])^(-2) [a0([theta] - [phi]) ([theta] - [psi]) ([theta] - [chi])]^(-1/2) + + 1 1 / 1 1 1 \ + x = ----------- + -- ( --------------- + -------------- + --------------- ) + z - [theta] 3 \[theta] - [phi] [theta] - [psi] [theta] - [chi]/ + + is at once reduced to the form y^2 = 4x^3- g2x - g3 = 4(x - e1)(x - + e2(x - e3), say; and these equations enable us to express s and z + rationally in terms of x and y. It is therefore sufficient to consider + three elliptic integrals + _ _ _ + / dx / xdx / y + y0 dx + u = | --, J = | ---, P = | ------ --. + _/ y _/ y _/ x - x0 2y + + Of these consider the first, putting + _ + / ([oo]) dx + u = | --, + _/ (x) y + + where the limits involve not only a value for x, but a definite sign + for the radical y. When x is very large, if we put x^(-1) = t^2, y^(-1) + = 2t^3(1 - 1/4 g2t^4 - 1/4 g3t^6)^(-1/2), we have + _ + / t / 1 \ 1 + u = | ( 1 + -- g2t^4 + ... )dt = t + -- g2t^5 + ..., + _/ 0 \ 8 / 40 + + whereby a definite power series in u, valid for sufficiently small + value of u, is found for t, and hence a definite power series for x, + of the form + + x = u^(-2) + (1/20)g2u^2 + ... + + Let this expression be valid for 0 < |u| < R, and the function defined + thereby, which has a pole of the second order for u = 0, be denoted by + [phi](u). In the range in question it is single valued and satisfies + the differential equation + + [[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3; + + in terms of it we can write x = [phi](u), y = -[phi]'(u), and, + [phi]'(u) being an odd function, the sign attached to y in the + original integral for x = [oo] is immaterial. Now for any two values + u, v in the range in question consider the function + _ _ + | [phi]'(u) - [phi]'(v) |^2 + F(u, v) = 1/4 | --------------------- | - [phi](u) - [phi](v); + |_ [phi](u) - [phi](v) _| + + it is at once seen, from the differential equation, to be such that + (Pd)F/(Pd)u = (Pd)F/(Pd)v; it is therefore a function of u + v; + supposing |u + v| < R we infer therefore, by putting v = 0, that + _ _ + | [phi]'(u) - [phi]'(v)] |^2 + [phi](u + v) = 1/4 | --------------------- | - [phi](u) - [phi](v). + |_ [phi](u) - [phi](v) _| + + By repetition of this equation we infer that if u1, ... u_n be any + arguments each of which is in absolute value less than R, whose sum is + also in absolute value less than R, then [phi](u1 + ... + u_n) is a + rational function of the 2n functions [phi](u_s), [phi]'(u_s); and + hence, if |u| < R, that + + [phi](u) = H [[phi](u/n), [phi]'(u/n)], + + where H is some rational function of the arguments [phi](u/n), + [phi]'(u/n). In fact, however, so long as |u/n| < R, each of the + functions [phi](u/n), [phi]'(u/n) is single valued and without + singularity save for the pole at u = 0; and a rational function of + single valued functions, each of which has no singularities other than + poles in a certain region, is also a single valued function without + singularities other than poles in this region. We infer, therefore, + that the function of u expressed by H[[phi](u/n), [phi]'(u/n)] is + single valued and without singularities other than poles so long as + |u| < nR; it agrees with [phi](u) when |u| < R, and hence furnishes a + continuation of this function over the extended range |u| < nR. + Moreover, from the method of its derivation, it satisfies the + differential equation [[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3. + This equation has therefore one solution which is a single valued + monogenic function with no singularities other than poles for any + finite part of the plane, having in particular for u = 0, a pole of + the second order; and the method adopted for obtaining this near u = 0 + shows that the differential equation has no other such solution. This, + however, is not the only solution which is a single valued meromorphic + function, a the functions [phi](u + [alpha]), wherein [alpha] is + arbitrary, being such. Taking now any range of values of u, from u = + 0, and putting for any value of u, x = [phi](u), y = -[phi]'(u), so + that y^2 = 4x^3 - g2x - g3, we clearly have + _ + / ([oo]) dx + u = | --; + _/ (x, y) y + + conversely if x0 = [phi](u0), y0 = -[phi]'(u0) and [xi], [eta] be any + values satisfying [eta]2 = 4[xi]^2 - g2[xi] - g3, which are + sufficiently near respectively to x0, y0, while v is defined by + _ + / ([xi], [eta]) d[xi] + v - u0 = -| -----, + _/ (x0, y0) [eta] + + then [xi], [eta] are respectively [phi](v) and -[phi]'(v); for this + equation leads to an expansion for [xi]-x0 in terms of v = u0 and only + one such expansion, and this is obtained by the same work as would be + necessary to expand [phi](v) when v is near to u0; the function + [phi](u) can therefore be continued by the help of this equation, from + v = u0, provided the lower limit of |[xi] - x0| necessary for the + expansions is not zero in the neighbourhood of any value (x0, y0). In + fact the function [phi](u) can have only a finite number of poles in + any finite part of the plane of u; each of these can be surrounded by + a small circle, and in the portion of the finite part of the plane of + u which is outside these circles, the lower limit of the radii of + convergence of the expansions of [phi](u) is greater than zero; the + same will therefore be the case for the lower limit of the radii |[xi] + - x0| necessary for the continuations spoken of above provided that + the values of ([xi], [eta]) considered do not lead to infinitely + increasing values of v; there does not exist, however, any definite + point ([xi]0, [eta]0) in the neighbourhood of which the integral [int] + [([xi], [eta]) to (x0, y0)] d[xi]/[eta] increases indefinitely, it is + only by a path of infinite length that the integral can so increase. + We infer therefore that if ([xi],[eta]) be any point, where [eta]2 = + 4[xi]^3 - g2[xi] - g3, and v be defined by + _ + / ([oo]) dx + v = | -- , + _/ ([xi], [eta]) y + + then [xi] = [phi](v) and [eta] = -[phi]'(v). Thus this equation + determines ([xi], [eta]) without ambiguity. In particular the additive + indeterminatenesses of the integral obtained by closed circuits of the + point of integration are periods of the function [phi](u); by + considerations advanced above it appears that these periods are sums + of integral multiples of two which may be taken to be + _ _ + / [oo] dx / [oo] dx + [omega] = 2| --, [omega]' = 2 | --; + _/ e1 y _/ e3 y + + these quantities cannot therefore have a real ratio, for else, being + periods of a monogenic function, they would, as we have previously + seen, be each integral multiples of another period; there would then + be a closed path for (x, y), starting from an arbitrary point (x0, + y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3, + 0), ([oo], [oo]), which leads back to the initial point (x0, y0), + which is impossible. On the whole, therefore, it appears that the + function [phi](u) agrees with the function RN(u) previously discussed, + and the discussion of the elliptic integrals can be continued in the + manner given under S 14, _Doubly Periodic Functions_. + +S 21. _Modular Functions._--One result of the previous theory is the +remarkable fact that if + _ _ + / [oo] dx / [oo] dx + [omega] = 2| --, [omega]' = 2 | --, + _/ e1 y _/ e3 y + +where y^2 = 4(x - e1) (x - e2) (x - e3), then we have + + e1 = (1/2[omega])^(-2) + [Sigma]' {[(m + 1/2)[omega] + m'[omega]']^(-2) - + [m[omega] + m'[omega]']^(-2)}, + +and a similar equation for e3, where the summation refers to all integer +values of m and m' other than the one pair m = 0, m' = 0. This, with +similar results, has led to the consideration of functions of the +complex ratio [omega]'/[omega]. + + It is easy to see that the series for RN(u), u^(-2) + [Sigma] [(u + + m[omega] + m'[omega]')^2-(m[omega] + m'[omega]')^2], is unaffected by + replacing [omega], [omega]' by two quantities [Omega], [Omega]' equal + respectively to p[omega] + q[omega]', p'[omega]' + q'[omega]', where + p, q, p', q' are any integers for which pq' - p'q = [+-]1; further it + can be proved that all substitutions with integer coefficients [Omega] + = p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', wherein pq' + - p'q = 1, can be built up by repetitions of the two particular + substitutions ([Omega] = -[omega]', [Omega]' = [omega]), ([Omega] = + [omega], [Omega]' = [omega] + [omega]'). Consider the function of the + ratio [omega]'/[omega] expressed by + + h = -RN (1/2[omega]') / RN(1/2[omega]); + + it is at once seen from the properties of the function RN(u) that by + the two particular substitutions referred to we obtain the + corresponding substitutions for h expressed by + + h' = 1/h, h' = 1 - h; + + thus, by all the integer substitutions [Omega] = p[omega] + q[omega]', + [Omega]' = p'[omega] + q'[omega]', in which pq' - p'q = 1, the + function h can only take one of the six values h, 1/h, 1 - h, 1/(1 - + h), h/(h - 1), (h - 1)/h, which are the roots of an equation in + [theta], + + (1 - [theta] + [theta]^2)^3 (1 - h + h^2)^3 + --------------------------- = ---------------; + [theta]^2(1 - [theta])^2 h^2(1 - h)^2 + + the function of [tau], = [omega]'/[omega], expressed by the right + side, is thus unaltered by every one of the substitutions [tau]' = (p' + + q'[tau] / p + q[tau]), wherein p, q, p', q' are integers having pq' + - p'q = 1. If the imaginary part [sigma], of [tau], which we may write + [tau] = [rho] + i[sigma], is positive, the imaginary part of [tau]', + which is equal to [sigma](pq' - p'q)/[(p + q[rho])^2 + q^2[sigma]^2], + is also positive; suppose [sigma] to be positive; it can be shown that + the upper half of the infinite plane of the complex variable [tau] can + be divided into regions, all bounded by arcs of circles (or straight + lines), no two of these regions overlapping, such that any + substitution of the kind under consideration, [tau]' = (p' + + q'[tau])/(p + q[tau]) leads from an arbitrary point [tau], of one of + these regions, to a point [tau]' of another; taking [tau] = [rho] + + i[sigma], one of these regions may be taken to be that for which -1/2 + < [rho] < 1/2, [rho]^2 + [sigma]^2 > 1, together with the points for + which [rho] is negative on the curves limiting this region; then every + other region is obtained from this so-called fundamental region by one + and only one of the substitutions [tau] = (p' + q'[tau])/(p + q[tau]), + and hence by a definite combination of the substitutions [tau]' = + -1/[tau], [tau]' = 1 + [tau]. Upon the infinite half plane of [tau], + the function considered above, + + 4 [RN^2(1/2[omega]) + RN(z(1/2[omega]) RN(1/2[omega]') + RN^2 (1/2[omega]')]^3 + z([tau]) = -- ---------------------------------------------------------------------------- + 27 RN^2(1/2[omega]) RN^2(1/2[omega]') [RN(1/2[omega]) + RN(1/2[omega]')]^2 + + is a single valued monogenic function, whose only essential + singularities are the points [tau]' = (p' + q'[tau])/(p + q[tau]) for + which [tau] = [oo], namely those for which [tau]' is any real rational + value; the real axis is thus a line over which the function z([tau]) + cannot be continued, having an essential singularity in every arc of + it, however short; in the fundamental region, z([tau]) has thus only + the single essential singularity, r = [rho] + i[sigma], where [sigma] + = [oo]; in this fundamental region z([tau]) takes any assigned complex + value just once, the relation z([tau]') = z([tau]) requiring, as can + be shown, that [tau]' is of the form (p' + q'[tau])/(p + q[tau]), in + which p, q, p', q' are integers with pq' - p'q = 1; the function + z([tau]) has thus a similar behaviour in every other of the regions. + The division of the plane into regions is analogous to the division of + the plane, in the case of doubly periodic functions, into + parallelograms; in that case we considered only functions without + essential singularities, and in each of the regions the function + assumed every complex value twice, at least. Putting, as another + function of [tau], J([tau]) = z([tau])[z([tau]) - 1], it can be shown + that J([tau]) = 0 for [tau] = exp (2/3[pi]i), that J([tau]) = 1 for + [tau] = i, these being values of [tau] on the boundary of the + fundamental region; like z([tau]) it has an essential singularity for + [tau] = [rho] + i[sigma], [sigma] = + [oo]. In the theory of linear + differential equations it is important to consider the inverse + function [tau](J); this is infinitely many valued, having a cycle of + three values for circulation of J about J = 0 (the circuit of this + point leading to a linear substitution for [tau] of period 3, such as + [tau]' = -(1 + [tau])^(-1)), having a cycle of two values about J = 1 + (the circuit leading to a linear substitution for [tau] of period 2, + such as [tau]' = -[tau]^(-1)), and having a cycle of infinitely many + values about J = [oo] (the circuit leading to a linear substitution + for [tau] which is not periodic, such as [tau]' = 1 + [tau]). These + are the only singularities for the function [tau](J). Each of the + functions + _ _ + | RN(1/2[omega]) + 2RN(1/2[omega]') |^(1/8) + [J([tau])]^(1/3), [J([tau])-1]^1/2, | - --------------------------------- | , + |_ RN(1/2[omega]) - RN(1/2)[omega]') _| + + beside many others (see below), is a single valued function of [tau], + and is expressible without ambiguity in terms of the single valued + function of [tau], + + /i[pi][tau]\ [oo] + [eta]([tau]) = exp( ---------- ) [Pi] [1-exp (2i[pi]n[tau])], + \ 12 / n=1 + + /i[pi][tau]\ _[oo] + = exp(----------- ) \ (-1)^m exp [(3m^2 + m) i[pi][tau]]. + \ 12 / /_ + m = -[oo] + + It should be remarked, however, that [eta]([tau]) is not unaltered by + all the substitutions we have considered; in fact + + [eta](-[tau]^(-1)) = (-i[tau])1/2[eta]([tau]), [eta](1 + [tau]) = + exp (1/12 i[pi]) [eta]([tau]). + + The aggregate of the substitutions [tau]' = (p' + q'[tau])/(p + + q[tau]), wherein p, q, p', q' are integers with pq' - p'q = 1, + represents a _Group_; the function J([tau]), unaltered by all these + substitutions, is called a _Modular Function_. More generally any + function unaltered by all the substitutions of a group of linear + substitutions of its variable is called an _Automorphic Function_. A + rational function, of its variable h, of this character, is the + function (1 - h + h^2)^3 h^(-2)(1 - h)^(-2) presenting itself + incidentally above; and there are other rational functions with a + similar property, the group of substitutions belonging to any one of + these being, what is a very curious fact, associable with that of the + rotations of one of the regular solids, about an axis through its + centre, which bring the solid into coincidence with itself. Other + automorphic functions are the double periodic functions already + discussed; these, as we have seen, enable us to solve the algebraic + equation y^2 = 4x^3 - g2x - g3 (and in fact many other algebraic + equations, see below, under S 23, _Geometrical Applications of + Elliptic Functions_) in terms of single valued functions x = RN(u), y + = -RN'(u). A similar utility, of a more extended kind, belongs to + automorphic functions in general; but it can be shown that such + functions necessarily have an infinite number of essential + singularities except for the simplest cases. + + The modular function J([tau]) considered above, unaltered by the group + of linear substitutions [tau]' = (p' + q'[tau]) / (p + q[tau]), where + p, q, p', q' are integers with pq' - p'q = 1, may be taken as the + independent variable x of a differential equation of the third order, + of the form + + s''' 3 /s''\^2 1 - a^2 1 - [beta]^2 [alpha]^2 + [beta]^2 - [gamma]^2 - 1 + ---- - -- ( --- ) = ---------- + ------------ + ------------------------------------, + s' 2 \ s'/ 2(x - 1)^2 2x^2 2x(x - 1) + + where s' = ds/dx, &c., of which the dependent variable s is equal to + [tau]. A differential equation of this form is satisfied by the + quotient of two independent integrals of the linear differential + equation of the second order satisfied by the hypergeometric + functions. If the solution of the differential equation for s be + written s([alpha], [beta], [gamma], x), we have in fact [tau] = s(1/2, + 1/3, 0, J). If we introduce also the function of [tau] given by + + 2RN (1/2[omega]') + f V(1/2[omega]) + [lambda] = -----------------------------------, + RN(1/2[omega]') - RN(1/2[omega]) + + we similarly have [tau] = s(0, 0, 0, [lambda]); this function [lambda] + is a single valued function of [tau], which is also a modular + function, being unaltered by a group of integral substitutions also of + the form [tau]' = (p' + q'[tau])/(p + q[tau]), with pq' - p'q = 1, but + with the restriction that p' and q are even integers, and therefore p + and q' are odd integers. This group is thus a subgroup of the general + modular group, and is in fact of the kind called a self-conjugate + subgroup. As in the general case this subgroup is associated with a + subdivision of the plane into regions of which any one is obtained + from a particular region, called the fundamental region, by a + particular one of the substitutions of the subgroup. This fundamental + region, putting [tau] = [rho] + i[sigma], may be taken to be that + given by -1 < [rho] < 1, ([rho] + 1/2)^2 + [sigma]^2 > 1/4, ([rho] - + 1/2)^2 + [sigma]^2 > 1/4, and is built up of six of the regions which + arose for the general modular group associated with J([tau]). Within + this fundamental region, [lambda] takes every complex value just once, + except the values [lambda] = 0, 1, [oo], which arise only at the + angular points [tau] = 0, [tau] = [oo], [tau] = -1 and the equivalent + point [tau] = 1; these angular points are essential singularities for + the function [lambda]([tau]). For [lambda]([tau]) as for J([tau]), the + region of existence is the upper half plane of [tau], there being an + essential singularity in every length of the real axis, however short. + + + If, beside the plane of [tau], we take a plane to represent the values + of [lambda], the function [tau] = s(0, 0, 0, [lambda]) being + considered thereon, the values of [tau] belonging to the interior of + the fundamental region of the [tau]-plane considered above, will + require the consideration of the whole of the [lambda]-plane taken + once with the exception of the portions of the real axis lying between + -[oo] and 0 and between 1 and + [oo], the two sides of the first + portion corresponding to the circumferences of the [tau]-plane + expressed by ([rho] + 1/2)^2 + [sigma]^2 = 1/4, ([rho] - 1/2)^2 + + [sigma]^2 = 1/4, while the two sides of the latter portion, for which + [lambda] is real and > 1, correspond to the lines of the [tau]-plane + expressed by [rho] = [+-]1. The line for which [lambda] is real, + positive and less than unity corresponds to the imaginary axis of the + [tau]-plane, lying in the interior of the fundamental region. All the + values of [tau] = s(0, 0, 0, [lambda]) may then be derived from those + belonging to the fundamental region of the [tau]-plane by making + [lambda] describe a proper succession of circuits about the points + [lambda] = 0, [lambda] = 1; any such circuit subjects [tau] to a + linear substitution of the subgroup of [tau] considered, and + corresponds to a change of [tau] from a point of the fundamental + region to a corresponding point of one of the other regions. + +S 22. _A Property of Integral Functions deduced from the Theory of +Modular Functions._--Consider now the function exp(z), for finite values +of z; for such values of z, exp(z) never vanishes, and it is impossible +to assign a closed circuit for z in the finite part of the plane of z +which will make the function [lambda] = exp(z) pass through a closed +succession of values in the plane of [lambda] having [lambda] = 0 in its +interior; the function s[0, 0, 0, exp(z)], however z vary in the finite +part of the plane, will therefore never be subjected to those linear +substitutions imposed upon s(0, 0, 0, [lambda]) by a circuit of [lambda] +about [lambda] = 0; more generally, if [phi](z) be an integral function +of z, never becoming either zero or unity for finite values of z, the +function [lambda] = [phi](z), however z vary in the finite part of the +plane, will never make, in the plane of [lambda], a circuit about either +[lambda] = 0 or [lambda] = 1, and s(0, 0, 0, [lambda]), that is s[0, 0, +0, [phi](z)], will be single valued for all finite values of z; it will +moreover remain finite, and be monogenic. In other words, s[0, 0, 0, +[phi](z)] is also an integral function--whose imaginary part, moreover, +by the property of s(0, 0, 0, [lambda]), remains positive for all finite +values of z. In that case, however, exp{is[0, 0, 0, [phi](z)]} would +also be an integral function of z with modulus less than unity for all +finite values of z. If, however, we describe a circle of radius R in the +z plane, and consider the greatest value of the modulus of an integral +function upon this circle, this certainly increases indefinitely as R +increases. We can infer therefore that _an integral function [phi](z) +which does not vanish for any finite value of z, takes the value unity +and hence_ (by considering the function A^(-1)[phi](z)) _takes every +other value for some definite value of z_; or, an integral function for +which both the equations [phi](z) = A, [phi](z) = B are unsatisfied by +definite values of z, does not exist, A and B being arbitrary constants. + + A similar theorem can be proved in regard to the values assumed by the + function [phi](z) for points z of modulus greater than R, however + great R may be, also with the help of modular functions. In general + terms it may be stated that it is a very exceptional thing for an + integral function not to assume every complex value an infinite number + of times. + + Another application of modular functions is to prove that the function + s([alpha], [beta], [gamma], [lambda]) is a single valued function of + [tau] = s(0, 0, 0, [lambda]); for, putting [tau]' = ([tau] - i)/([tau] + + i), the values of [tau]' which correspond to the singular points + [lambda] = 0, 1, [oo] of s([alpha], [beta], [gamma], [lambda]), though + infinite in number, all lie on the circumference of the circle + |[tau]'| = 1, within which therefore s([alpha], [beta], [gamma], x) is + expressible in a form [Sigma] [n = 0 to [oo]] a_n[tau]'^n. More + generally any monogenic function of [lambda] which is single valued + save for circuits of the points [lambda] = 0, 1, [oo], is a single + valued function of [tau] = s(0, 0, 0, [lambda]). Identifying [lambda] + with the square of the modulus in Legendre's form of the elliptical + integral, we have [tau] = iK'/K, where + + _ _ + /1 dt /1 dt + K = | ----------------------------------, K' = | ---------------------------------------; + _/0 [root][1 - t^2] [1 - [lambda]t^2] _/0 [root][1 - t^2] [1 - (1 - [lambda])t^2] + + functions such as [lambda]^1/4, (1 - [lambda])^1/4, [[lambda](1 - + [lambda])]^1/4, which have only [lambda] = 0, 1, [oo] as singular + points, were expressed by Jacobi as power series in q = + e^(i[pi][tau]), and therefore, at least for a limited range of values + of [tau], as single valued functions of [tau]; it follows by the + theorem given that any product of a root of [lambda] and a root of 1 - + [lambda] is a single valued function of [tau]. More generally the + differential equation + + d^2y dy + x(1 - x) ---- + [[gamma] - ([alpha] + [beta] + 1)x] -- -[alpha][beta][gamma] = 0 + dx^2 dx + + may be solved by expressing both the independent and dependent + variables as single valued functions of a single variable [tau], the + expression for the independent variable being x = [lambda]([tau]). + +S 23. _Geometrical Applications of Elliptic Functions._--Consider any +irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a +form that the equation represents a plane curve of order n with 1/2n(n +-3) double points; taking upon this curve n-3 arbitrary fixed points, +draw through these and the double points the most general curve of order +n -2; this will intersect [f] in n(n - 2) - n(n - 3) - (n - 3) = 3 +other points, and will contain homogeneously at least 1/2(n - 1)n - +1/2n(n -3) - (n - 3) = 3 arbitrary constants, and so will be of the form +[lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 + ... = 0, wherein +[lambda]3, [lambda]4, ... are in general zero. Put now [xi] = +[phi]1/[phi], [eta] = [phi]2/[phi] and eliminate x, y between these +equations and [f](x, y) = 0, so obtaining a rational irreducible +equation F([xi], [eta]) = 0, representing a further plane curve. To any +point (x, y) of [f] will then correspond a definite point ([xi], [eta]) +of F. + + For a general position of (x, y) upon [f] the equations [phi]1(x', + y')/[phi](x', y') = [phi]1(x, y)/[phi](x, y), [phi]2(x', y')/[phi](x', + y') = [phi]2(x, y)/[phi](x, y), subject to [f](x', y') = 0, will have + the same number of solutions (x', y'); if their only solution is x' = + x, y' = y, then to any position ([xi],[eta]) of F will conversely + correspond only one position (x, y) of [f]. If these equations have + another solution beside (x, y), then any curve [lambda][phi] + + [lambda]1[phi]1 + [lambda]2[phi]2 = 0 which passes (through the double + points of [f] and) through the n - 2 points of [f] constituted by the + fixed n-3 points and a point (x0, y0), will necessarily pass through a + further point, say (x0', y0'), and will have only one further + intersection with [f]; such a curve, with the n - 2 assigned points, + beside the double points, of [f], will be of the form [mu][psi] + + [mu]1[psi]1 + ... = 0, where [mu]2, [mu]3, ... are generally zero; + considering the curves [psi] + t[psi]1 = 0, for variable t, one of + these passes through a further arbitrary point of [f], by choosing t + properly, and conversely an arbitrary value of t determines a single + further point of [f]; the co-ordinates of the points of [f] are thus + rational functions of a parameter t, which is itself expressible + rationally by the co-ordinates of the point; it can be shown + algebraically that such a curve has not 1/2(n - 3)n but 1/2(n - 3)n + + 1 double points. We may therefore assume that to every point of F + corresponds only one point of [f], and there is a birational + transformation between these curves; the coefficients in this + transformation will involve rationally the co-ordinates of the n-3 + fixed points taken upon [f], that is, at the least, by taking these to + be consecutive points, will involve the co-ordinates of one point of + [f], and will not be rational in the coefficients of [f] unless we can + specify a point of [f] whose co-ordinates are rational in these. The + curve F is intersected by a straight line a[xi] + b[eta] + c = 0 in as + many points as the number of unspecified intersections of [f] with + a[phi] + b[phi]1 + c[phi]2 = 0, that is, 3; or F will be a cubic + curve, without double points. + + Such a cubic curve has at least one point of inflection Y, and if a + variable line YPQ be drawn through Y to cut the curve again in P and + Q, the locus of a point R such that YR is the harmonic mean of YP and + YQ, is easily proved to be a straight line. Take now a triangle of + reference for homogeneous co-ordinates XYZ, of which this straight + line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation + of the cubic curve will then be of the form + + ZY^2 = aX^3 + bX^2Z + cXZ^2 + dZ^3; + + by putting X equal to [lambda]X + [mu]Z, that is, choosing a suitable + line through Y to be X = 0, and choosing [lambda] properly, this is + reduced to the form + + ZY^2 = 4X^3 - g2XZ^2 -g3Z^3, + + of which a representation is given, valid for every point, in terms of + the elliptic functions RN(u), RN'(u), by taking X = ZRN(u), Y = + ZRN'(u). The value of u belonging to any point is definite save for + sums of integral multiples of the periods of the elliptic functions, + being given by + _ + / (x) ZdX - XdZ + u = | ---------, + _/ ([oo]) ZY + + where ([oo]) denotes the point of inflection. + + It thus appears that the co-ordinates of any point of a plane curve, + [f], of order n with 1/2(n - 3)n double points are expressible as + elliptic functions, there being, save for periods, a definite value of + the argument u belonging to every point of the curve. It can then be + shown that if a variable curve, [phi], of order m be drawn, passing + through the double points of the curve, the values of the argument u + at the remaining intersections of [phi] with [f], have a sum which is + unaffected by variation of the coefficients of [phi], save for + additive aggregates of the periods. In virtue of the birational + transformation this theorem can be deduced from the theorem that if + any straight line cut the cubic y^2 = 4x^3 - g2x - g3, in points (u1), + (u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general + theorem is a corollary from Abel's theorem proved under S 17, + _Integrals of Algebraic Functions_. To prove the result directly for + the cubic we remark that the variation of one of the intersections (x, + y) of the cubic with the straight line y = mx + n, due to a variation + [delta]m, [delta]n in m and n, is obtained by differentiation of the + equation for the three abscissae, namely the equation + + F(x) = 4x^3 - g2x - g3 - (mx + n)^2 = 0, + + and is thus given by + + dx x[delta]m + [delta]n + -- = 2 --------------------, + y F'(x) + + and the sum of three such fractions as that on the right for the three + roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the + straight line considered; if in particular this become the inflexional + tangent each of u1, u2, u3 vanishes. It may be remarked in passing + that x1 + x2 + x3 = 1/4m^2, and hence is 1/4{(y1 - y2)/(x1 - x2)}^2; so + that we have another proof of the addition equation for the function + RN(u). From this theorem for the cubic curve many of its geometrical + properties, as for example those of its inflections, the properties of + inscribed polygons, of the three kinds of corresponding points, and + the theory of residuation, are at once obvious. And similar results + hold for the curve of order n with 1/2(n - 3)n double points. + +S 24. _Integrals of Algebraic Functions in Connexion with the Theory of +Plane Curves._--The developments which have been explained in connexion +with elliptic functions may enable the reader to appreciate the vastly +more extensive theory similarly arising for any algebraical +irrationality, [f](x, y) = o. + + The algebraical integrals [int] R(x, y)dx associated with this may as + before be divided into those of the _first kind_, which have no + infinities, those of the _second kind_, possessing only algebraical + infinities, and those of the _third kind_, for which logarithmic + infinities enter. Here there is a certain number, p, greater than + unity, of linearly independent integrals of the first kind; and this + number p is unaltered by any birational transformation of the + fundamental equation [f](x, y) = 0; a rational function can be + constructed with poles of the first order at p + 1 arbitrary positions + (x, y), satisfying [f](x, y) = 0, but not with a fewer number unless + their positions are chosen properly, a property we found for the case + p = 1; and p is the number of linearly independent curves of order n-3 + passing through the double points of the curve of order n expressed by + [f](x, y) = 0. Again any integral of the second kind can be expressed + as a sum of p integrals of this kind, with poles of the first order at + arbitrary positions, together with rational functions and integrals of + the first kind; and an integral of the second kind can be found with + one pole of the first order of arbitrary position, and an integral of + the third kind with two logarithmic infinities, also of arbitrary + position; the corresponding properties for p = 1 are proved above. + + There is, however, a difference of essential kind in regard to the + inversion of integrals of the first kind; if u = [int] R(x, y)dx be + such an integral, it can be shown, in common with all algebraic + integrals associated with [f](x, y) = 0, to have 2p linearly + independent additive constants of indeterminateness; the upper limit + of the integral cannot therefore, as we have shown, be a single valued + function of the value of the integral. The corresponding theorem, if + [int] R_i(x, y)dx denote one of the integrals of the first kind, is + that the p equations + _ _ + / / + | R_i (x1, y1)dx1 + ... + | Ri (x_p, y_p) dx_p = u_i, + _/ _/ + + determine the rational symmetric functions of the p positions (x1, + y1), ... (x_p, y_p) as single valued functions of the p variables, u1, + ... u_p. It is thus necessary to enter into the theory of functions of + several independent variables; and the equation [f](x, y) = 0 is thus + not, in this way, capable of solution by single valued functions of + one variable. That solution in fact is to be sought with the help of + automorphic functions, which, however, as has been remarked, have, for + p > 1, an infinite number of essential singularities. + +S 25. _Monogenic Functions of Several Independent Variables._--A +monogenic function of several independent complex variables u_i, ... u_p +is to be regarded as given by an aggregate of power series all +obtainable by continuation from any one of them in a manner analogous to +that before explained in the case of one independent variable. The +singular points, defined as the limiting points of the range over which +such continuation is possible, may either be _poles_, or _polar points +of indetermination_, or _essential singularities_. + + A pole is a point (u1^(0), ... u_p^(0)) in the neighbourhood of which + the function is expressible as a quotient of converging power series + in u1 - u1^(0) ... u_p - u_p^(0); of these the denominator series D + must vanish at (u1^(0), ... u_p^(0)), since else the fraction is + expressible as a power series and the point is not a singular point, + but the numerator series N must not also vanish at (u1^(0), ... + u_p^(0)), or if it does, it must be possible to write D = M0, N = MN0, + where M is a converging power series vanishing at (u1^(0), ... + u_p^(0)), and N0 is a converging power series, in (u1 - u1^(0) ... u_p + - u_p^(0)), not so vanishing. A polar point of indetermination is a + point about which the function can be expressed as a quotient of two + converging power series, both of which vanish at the point. As in such + a simple case as (Ax + By) / (ax + by), about x = 0, y = 0, it can be + proved that then the function can be made to approach to any + arbitrarily assigned value by making the variables u1, ... u_p + approach to u1^(0), ... u_p^(0) by a proper path. It is the necessary + existence of such polar points of indetermination, which in case p > 2 + are not merely isolated points, which renders the theory essentially + more difficult than that of functions of one variable. An essential + singularity is any which does not come under one of the two former + descriptions and includes very various possibilities. A point at + infinity in this theory is one for which any one of the variables u1, + ... u_p is indefinitely great; such points are brought under the + preceding definitions by means of the convention that for u_i^(0) = + [oo], the difference u_i - u_i^(0) is to be understood to stand for + u_i^(-1) . This being so, a single valued function of u1, ... u_p + without essential singularities for infinite or finite values of the + variables can be shown, by induction, to be, as in the case of p = 1, + necessarily a rational function of the variables. A function having no + singularities for finite values of all the variables is as before + called an integral function; it is expressible by a power series + converging for all finite values of the variables; a single valued + function having for finite values of the variables no singularities + other than poles or polar points of indetermination is called a + meromorphic function; as for p = 1 such a function can be expressed as + a quotient of two integral functions having no common zero point other + than the points of indetermination of the function; but the proof of + this theorem is difficult. + + The single valued functions which occur, as explained above, in the + inversion of algebraic integrals of the first kind, for p > 1, are + meromorphic. They must also be periodic, unaffected that is when the + variables u1, ... u_p are _simultaneously_ increased each by a proper + constant, these being the additive constants of indeterminateness for + the p integrals [int] R_i(x, y)dx arising when (x, y) makes a closed + circuit, the same for each integral. The theory of such single valued + meromorphic periodic functions is simpler than that of meromorphic + functions of several variables in general, as it is sufficient to + consider only finite values of the variables; it is the natural + extension of the theory of doubly periodic functions previously + discussed. It can be shown to reduce, though the proof of this + requires considerable developments of which we cannot speak, to the + theory of a single integral function of u1, ... u_p, called the _Theta + Function_. This is expressible as a series of positive and negative + integral powers of quantities exp (c1u1), exp (c2u2), ... exp (c_p + u_p), wherein c1, ... c_p are proper constants; for p = 1 this theta + function is essentially the same as that above given under a different + form (see S 14, _Doubly Periodic Functions_), the function [sigma](u). + In the case of p = 1, all meromorphic functions periodic with the same + two periods have been shown to be rational functions of two of them + connected by a single algebraic equation; in the same way all + meromorphic functions of p variables, periodic with the same sets of + simultaneous periods, 2p sets in all, can be shown to be expressible + rationally in terms of p + 1 such periodic functions connected by a + single algebraic equation. Let x1, ... x_p, y denote p + 1 such + functions; then each of the partial derivatives dx_i/(Pd)u_i will + equally be a meromorphic function of the same periods, and so + expressible rationally in terms of x1, ... x_p, y; thus there will + exist p equations of the form + + dx_i = R1 du1 + ... + R_p du_p, + + and hence p equations of the form + + du_i = H_(i, 1)dx1 + ... + H_(i, p)dx_p, + + wherein H_(i, j) are rational functions of x1, ... x_p, y, these being + connected by a fundamental algebraic (rational) equation, say [f](x1, + ... x_p, y) = 0. This then is the generalized form of the + corresponding equation for p = 1. + +S 26. _Multiply-Periodic Functions and the Theory of Surfaces._--The +theory of algebraic integrals [int] R(x, y)dx, wherein x, y are +connected by a rational equation [f](x, y) = 0, has developed +concurrently with the theory of algebraic curves; in particular the +existence of the number p invariant by all birational transformations is +one result of an extensive theory in which curves capable of birational +correspondence are regarded as equivalent; this point of view has made +possible a general theory of what might otherwise have remained a +collection of isolated theorems. + + In recent years developments have been made which point to a similar + unity of conception as possible for surfaces, or indeed for algebraic + constructs of any number of dimensions. These developments have been + in two directions, at first followed independently, but now happily + brought into the most intimate connexion. On the analytical side, E. + Picard has considered the possibility of classifying integrals of the + form [int](Rds + Sdy), belonging to a surface [f](x, y, z) = 0, + wherein R and S are rational functions of x, y, z, according as they + are (1) everywhere finite, (2) have poles, which then lie along curves + upon the surface, or (3) have logarithmic infinities, also then lying + along curves, and has brought the theory to a high degree of + perfection. On the geometrical side A. Clebsch and M. Noether, and + more recently the Italian school, have considered the geometrical + characteristics of a surface which are unaltered by birational + transformation. It was first remarked that for surfaces of order n + there are associated surfaces of order n-4, having properties in + relation thereto analogous to those of curves of order n-3 for a plane + curve of order n; if such a surface [f](x, y, z) = 0 have a double + curve with triple points triple also for the surface, and [phi](x, y, + z) = 0 be a surface of order n - 4 passing through the double curve, + the double integral + _ _ + / / [phi] dx dy + | | ----------- + _/ _/ (Pd)f/(Pd)z + + is everywhere finite; and, the most general everywhere finite integral + of this form remains invariant in a birational transformation of the + surface [f], the theorem being capable of generalization to algebraic + constructs of any number of dimensions. The number of linearly + independent surfaces of order n - 4, possessing the requisite + particularity in regard to the singular lines and points of the + surface, is thus a number invariant by birational transformation, and + the equality of these numbers for two surfaces is a necessary + condition of their being capable of such transformation. The number of + surfaces of order m having the assigned particularity in regard to the + singular points and lines of the fundamental surface can be given by a + formula for a surface of given singularity; but the value of this + formula for m = n - 4 is not in all cases equal to the actual number + of surfaces of order n - 4 with the assigned particularity, and for a + cone (or ruled surface) is in fact negative, being the negative of the + deficiency of the plane section of the cone. Nevertheless this number + for m = n - 4 is also found to be invariant for birational + transformation. This number, now denoted by p_a, is then a second + invariant of birational transformation. The former number, of actual + surfaces of order n - 4 with the assigned particularity in regard to + the singularities of the surface, is now denoted by p_g. The + difference p_g - p_a, which is never negative, is a most important + characteristic of a surface. When it is zero, as in the case of the + general surface of order n, and in a vast number of other ordinary + cases, the surface is called regular. + + On a plane algebraical curve we may consider linear series of sets of + points, obtained by the intersection with it of curves [lambda][phi] + + [lambda]1[phi]1 + ... = 0, wherein [lambda], [lambda]1, ... are + variable coefficients; such a series consists of the sets of points + where a rational function of given poles, belonging to the construct + [f](x, y) = 0, has constant values. And we may consider series of sets + of points determined by variable curves whose coefficients are + algebraical functions, not necessarily rational functions, of + parameters. Similarly on a surface we may consider linear systems of + curves, obtained by the intersection with the given surface of + variable surfaces [lambda][phi] + [lambda]1[phi]1 + ... = 0, and may + consider algebraic systems, of which the individual curve is given by + variable surfaces whose coefficients are algebraical, not necessarily + rational, functions of parameters. Of a linear series upon a plane + curve there are two numbers manifestly invariant in birational + transformation, the _order_, which is the number of points forming a + set of the series, and the _dimension_, which is the number of + parameters [lambda]1/[lambda], [lambda]2/[lambda], ... entering + linearly in the equation of the series. The series is _complete_ when + it is not contained in a series of the same order but of higher + dimension. So for a linear system of curves upon a surface, we have + three invariants for birational transformation; the _order_, being in + the number of variable intersections of two curves of the system, the + _dimension_, being the number of linear parameters [lambda]1/[lambda], + [lambda]2/[lambda], ... in the equation for the system, and the + _deficiency_ of the individual curves of the system. Upon any curve of + the linear system the other curves of the system define a linear + series, called the _characteristic_ series; but even when the linear + system is complete, that is, not contained in another linear system of + the same order and higher dimension, it does not follow that the + characteristic series is complete; it may be contained in a series + whose dimension is greater by p_g - p_a than its own dimension. When + this is so it can be shown that the linear system of curves is + contained in an algebraic system whose dimension is greater by p_g - + p_a than the dimension of the linear system. The extra p = p_g - p_a + variable parameters so entering may be regarded as the independent + co-ordinates of an algebraic construct [f](y, x1, ... x_p) = 0; this + construct has the property that its co-ordinates are single valued + meromorphic functions of p variables, which are periodic, possessing + 2p systems of periods; the p variables are expressible in the forms + _ + / + u_i = | R1(x, y) dx1 + ... + R_p(x, y) dx_p, + _/ + + wherein R_i(x, y) denotes a rational function of x1, ... x_p and y. + The original surface has correspondingly p integrals of the form + [int](R dx + S dy), wherein R, S are rational in x, y, z, which are + everywhere finite; and it can be shown that it has no other such + integrals. From this point of view, then, the number p, = p_g - p_a + is, for a surface, analogous to the deficiency of a plane curve; + another analogy arises in the comparison of the theorems: for a plane + curve of zero deficiency there exists no algebraic series of sets of + points which does not consist of sets belonging to a linear series; + for a surface for which p_g - p_a = 0 there exists no algebraic system + of curves not contained in a linear system. + + But whereas for a plane curve of deficiency zero, the co-ordinates of + the points of the curve are rational functions of a single parameter, + it is not necessarily the case that for a surface having p_g - p_a = 0 + the co-ordinates of the points are rational functions of two + parameters; it is necessary that p_g - p_a = 0, but this is not + sufficient. For surfaces, beside the p_g linearly independent surfaces + of order n - 4 having a definite particularity at the singularities of + the surface, it is useful to consider surfaces of order k(n - 4), also + having each a definite particularity at the singularities, the number + of these, not containing the original surface as component, which are + linearly independent, is denoted by P_k. It can then be stated that a + sufficient condition for a surface to be rational consists of the two + conditions p_a = 0, P2 = 0. More generally it becomes a problem to + classify surfaces according to the values of the various numbers which + are invariant under birational transformation, and to determine for + each the simplest form of surface to which it is birationally + equivalent. Thus, for example, the hyperelliptic surface discussed by + Humbert, of which the co-ordinates are meromorphic functions of two + variables of the simplest kind, with four sets of periods, is + characterized by p_g = 1, p_a = -1; or again, any surface possessing a + linear system of curves of which the order exceeds twice the + deficiency of the individual curves diminished by two, is reducible by + birational transformation to a ruled surface or is a rational surface. + But beyond the general statement that much progress has already been + made in this direction, of great interest to the student of the theory + of functions, nothing further can be added here. + + BIBLIOGRAPHY.--The learner will find a lucid introduction to the + theory in E. Goursat, _Cours d'analyse mathematique_, t. ii. (Paris, + 1905), or, with much greater detail, in A.R. Forsyth, _Theory of + Functions of a Complex Variable_ (2nd ed., Cambridge, 1900); for + logical rigour in the more difficult theorems, he should consult W.F. + Osgood, _Lehrbuch der Functionentheorie_, Bd. i. (Leipzig, 1906-1907); + for greater precision in regard to the necessary quasi-geometrical + axioms, beside the indications attempted here, he should consult W.H. + Young, _The Theory of Sets of Points_ (Cambridge, 1906), chs. + viii.-xiii., and C. Jordan, _Cours d'analyse_, t. i. (Paris, 1893), + chs. i., ii.; a comprehensive account of the _Theory of Functions of + Real Variables_ is by E.W. Hobson (Cambridge, 1907). Of the theory + regarded as based after Weierstrass upon the theory of power series, + there is J. Harkness and F. Morley, _Introduction to the Theory of + Analytic Functions_ (London, 1898), an elementary treatise; for the + theory of the convergence of series there is also T.J. I'A. Bromwich, + _An Introduction to the Theory of Infinite Series_ (London, 1908); but + the student should consult the collected works of Weierstrass (Berlin, + 1894 ff.), and the writings of Mittag-Leffler in the early volumes of + the _Acta mathematica_; earlier expositions of the theory of functions + on the basis of power series are in C. Meray, _Lecons nouvelles sur + l'analyse infinitesimale_ (Paris, 1894), and in Lagrange's books on + the Theory of Functions. An account of the theory of potential in its + applications to the present theory is found in most treatises; in + particular consult E. Picard, _Traite d'analyse_, t. ii. (Paris, + 1893). For elliptic functions there is an introductory book, P. Appell + and E. Lacour, _Principes de la theorie des fonctions elliptiques et + applications_ (Paris, 1897), beside the treatises of G.H. Halphen, + _Traite des fonctions elliptiques et de leurs applications_ (three + parts, Paris, 1886 ff.), and J. Tannery et J. Molk, _Elements de la + theorie des fonctions elliptiques_ (Paris, 1893 ff.); a book, A.G. + Greenhill, _The Applications of Elliptic Functions_ (London, 1892), + shows how the functions enter in problems of many kinds. For modular + functions there is an extensive treatise, F. Klein and R. Fricke, + _Theorie der elliptischen Modulfunctionen_ (Leipzig, 1890); see also + the most interesting smaller volume, F. Klein, _Uber das Ikosaeder_ + (Leipzig, 1884) (also obtainable in English). For the theory of + Riemann's surface, and algebraic integrals, an interesting + introduction is P. Appeil and E. Goursat, _Theorie des fonctions + algebriques et de leurs integrales_; for Abelian functions see also H. + Stahl, _Theorie der Abel'schen Functionen_ (Leipzig, 1896), and H.F. + Baker, _An Introduction to the Theory of Multiply Periodic Functions_ + (Cambridge, 1907), and H.F. Baker, _Abel's Theorem and the Allied + Theory, including the Theory of the Theta Functions_ (Cambridge, + 1897); for theta functions of one variable a standard work is C.G. + Jacobi, _Fundamenta nova, &c._ (Konigsberg, 1828); for the general + theory of theta functions, consult W. Wirtinger, _Untersuchungen uber + Theta-Functionen_ (Leipzig, 1895). For a history of the theory of + algebraic functions consult A. Brill and M. Noether, _Die Entwicklung + der Theorie der algebraischen Functionen in alterer und neuerer Zeit, + Bericht der deutschen Mathematiker-Vereinigung_ (1894); and for a + special theory of algebraic functions, K. Hensel and G. Landsberg, + _Theorie der algebraischen Function u.s.w._ (Leipzig, 1902). The + student will, of course, consult also Riemann's and Weierstrass's + _Ges. Werke_. For the applications to geometry in general an important + contribution, of permanent value, is E. Picard and G. Simart, _Theorie + des fonctions algebriques de deux variables independantes_ (Paris, + 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable + summary by MM. Castelnuovo and Enriques, _Sur quelques resultats + nouveaux dans la theorie des surfaces algebriques_, containing many + references to the numerous memoirs to be found, for the most part, in + the transactions of scientific societies and the mathematical journals + of Italy. + + Beside the books above enumerated there exists an unlimited number of + individual memoirs, often of permanent importance and only + imperfectly, or too elaborately, reproduced in the pages of the + volumes in which the student will find references to them. The German + _Encyclopaedia of Mathematics_, and the Royal Society's _Reference + Catalogue of Current Scientific Literature, Pure Mathematics_, + published yearly, should also be consulted. (H. F. Ba.) + + +FOOTNOTE: + + [1] The word "function" (from Lat. _fungi_, to perform) has many + uses, with the fundamental sense of an activity special or proper to + an office, business or profession, or to an organ of an animal or + plant, the definite work for which the organ is an apparatus. From + the use of the word, as in the Italian _funzione_, for a ceremony of + the Roman Church, "function" is often employed for a public ceremony + of any kind, and loosely of a social entertainment or gathering. + + + + +FUNDY, BAY OF, an inlet of the North Atlantic, separating New Brunswick +from Nova Scotia. It is 145 m. long and 48 m. wide at the mouth, but +gradually narrows towards the head, where it divides into Chignecto Bay +to the north, which subdivides into Shepody Bay and Cumberland Basin +(the French Beaubassin), and Minas Channel, leading into Minas Basin, to +the east and south. Off its western shore opens Passamaquoddy Bay, a +magnificent sheet of deep water with good anchorage, receiving the +waters of the St Croix river and forming part of the boundary between +New Brunswick and the state of Maine, The Bay of Fundy is remarkable for +the great rise and fall of the tide, which at the head of the bay has +been known to reach 62 ft. In Passamaquoddy Bay the rise and fall is +about 25 ft., which gradually increases toward the narrow upper reaches. +At spring tides the water in the Bay of Fundy is 19 ft. higher than it +is in Bay Verte, in Northumberland Strait, only 15 m. distant. Though +the bay is deep, navigation is rendered dangerous by the violence and +rapidity of the tide, and in summer by frequent fogs. At low tide, at +such points as Moncton or Amherst, only an expanse of red mud can be +seen, and the tide rushes in a bore or crest from 3 to 6 ft. in height. +Large areas of fertile marshes are situated at the head of the bay, and +the remains of a submerged forest show that the land has subsided in the +latest geological period at least 40 ft. The bay receives the waters of +the St Croix and St John rivers, and has numerous harbours, of which the +chief are St Andrews (on Passamaquoddy Bay) and St John in New +Brunswick, and Digby and Annapolis (on an inlet known as Annapolis +Basin) in Nova Scotia. It was first explored by the Sieur de Monts (d. +c. 1628) in 1604 and named by him La Baye Francaise. + + + + +FUNERAL RITES, the ceremonies associated with different methods of +disposing of the dead. (See also BURIAL AND BURIAL ACTS; CEMETERY; and +CREMATION.) In general we have little record, except in their tombs, of +races which, in a past measured not merely by hundreds but by thousands +of years, occupied the earth; and exploration of these often furnishes +our only clue to the religions, opinions, customs, institutions and arts +of long vanished societies. In the case of the great culture folks of +antiquity, the Babylonians, Egyptians, Hindus, Persians, Greeks and +Romans, we have, besides their monuments, the evidence of their +literatures, and so can know nearly as much of their rites as we do of +our own. The rites of modern savages not only help us to interpret +prehistoric monuments, but explain peculiarities in our own rituals and +in those of the culture folks of the past of which the significance was +lost or buried under etiological myths. We must not then confine +ourselves to the rites of a few leading races, neglecting their less +fortunate brethren who have never achieved civilization. It is better to +try to classify the rites of all races alike according as they embody +certain leading conceptions of death, certain fears, hopes, beliefs +entertained about the dead, about their future, and their relations with +the living. + + The main ideas, then, underlying funeral rites may roughly be + enumerated as follows: + + 1. The pollution or taboo attaching to a corpse. + + 2. Mourning. + + 3. The continued life of the dead as evinced in the housing and + equipment of the dead, in the furnishing of food for them, and in the + orientation and posture assigned to the body. + + 4. Communion with the dead in a funeral feast and otherwise. + + 5. Sacrifice for the dead and expiation of their sins. + + 6. Death witchery. + + 7. Protection of the dead from ghouls. + + 8. Fear of ghosts. + +1. A dead body is unclean, and the uncleanness extends to things and +persons which touch it. Hence the Jewish law (Num. v. 2) enacted that +"whoever is unclean by the dead shall be put outside the camp, that they +defile not the camp in the midst whereof the Lord dwells." Such persons +were unclean until the even, and might not eat of the holy things unless +they bathed their flesh in water. A high priest might on no account "go +in to any dead body" (Lev. xxi. 11). Why a corpse is so widely tabooed +is not certain; but it is natural to see one reason in the corruption +which in warm climates soon sets in. The common experience that where +one has died another is likely to do so may also have contributed, +though, of course, there was no scientific idea of infection. The old +Persian scriptures are full of this taboo. He who has touched a corpse +is "powerless in mind, tongue and hand" (_Zend Avesta_ in _Sacred Books +of the East_, pt. i. p. 120), and the paralysis is inflicted by the +innumerable _drugs_ or evil spirits which invest a corpse. Fire and +earth, being alike creations of the good and pure god Ahuramazda, a +body must not be burned or buried; and so the ancient Persians and their +descendants the Parsees build Dakmas or "towers of silence" on hill-tops +far from human habitations. Inside these the corpses are laid on a +flagged terrace which drains into a central pit. Twice a year the bones, +picked clean by dogs and birds of prey, are collected in the pit, and +when it is full another tower is built. In ancient times perhaps the +bodies of the magi or priests alone were exposed at such expense; the +common folk were covered with wax and laid in the earth, the wax saving +the earth from pollution. In Rome and Greece the corpse was buried by +night, lest it should pollute the sunlight; and a trough of water was +set at the door of the house of death that men might purify themselves +when they came out, before mixing in general society. Priests and +magistrates in Rome might not meet or look on a corpse, for they were +thereby rendered unclean and incapable of fulfilling their official +duties without undergoing troublesome rites of purification. At a Roman +funeral, when the remains had been laid in the tomb, all present were +sprinkled with lustral water from a branch of olive or laurel called +_aspergillum_; and when they had gone home they were asperged afresh and +stepped over a fire. The house was also swept out with a broom, probably +lest the ghost of the dead should be lying about the floor. Many races, +to avoid pollution, destroy the house and property of the deceased. Thus +the Navahos pull down the hut in which he died, leaving its ruins on the +ground; but if it be an expensive hut, a shanty is extemporized +alongside, into which the dying man is transferred before death. No one +will use the timbers of a hut so ruined. A burial custom of the Solomon +Islands, noted by R.H. Codrington (_The Melanesians_, p. 255), may be +dictated by the same scruple. There "the mourners having hung up a dead +man's arms on his house make great lamentations; all remains afterwards +untouched, the house goes to ruin, mantled, as time goes on, with the +vines of the growing yams, a picturesque and indeed, perhaps, a touching +sight; for these things are not set up that they may in a ghostly manner +accompany their former owner." H. Oldenberg (_Religion des Veda_, p. +426) describes how Hindus shave themselves and cut off their nails after +a death, at the same time that they wash, renew the hearth fire, and +furnish themselves with new vessels. For the hair and nails may harbour +pollution, just as the medieval Greeks believed that evil spirits could +lurk in a man's beard (Leo Allatius, _De opinionibus quorundam +Graecorum_). The dead man's body is shorn and the nails cut for a +kindred reason; for it must be purified as much as can be before it is +burned as an offering on the pyre and before he enters on a new sphere +of existence. + +2. We are accustomed to regard mourning costume as primarily an outward +sign of our grief. Originally, however, the special garb seems to have +been intended to warn the general public that persons so attired were +unclean. In ancient Rome mourners stayed at home and avoided all feasts +and amusements; laying aside gold, purple and jewels, they wore black +dresses called _lugubria_ or even skins. They cut neither hair nor +beard, nor lighted fire. Under the emperors women began to wear white. +On the west coast of Africa negroes wear white, on the Gold Coast red. +The Chinese wear hemp, which is cheap, for mourning dress must as a rule +be destroyed when the season of grief is past to get rid of the taboo. +Among the Aruntas of Australia the wives of a dead man smear themselves +with white pipe-clay until the last ceremonies are finished, sometimes +adding ashes--this not to conceal themselves from the ghost (which may +partly be the aim of some mourning costumes), but to show the ghost that +they are duly sorrowing for their loss. These widows must not talk +except on their hands for a whole year. "Among the Maoris," says Frazer +(_Golden Bough_, i. 323), "anyone who had handled a corpse, helped to +convey it to the grave, or touched a dead man's bones; was cut off from +all intercourse and almost all communication with mankind. He could not +enter any house, or come into contact with any person or thing, without +utterly bedevilling them. He might not even touch food with his hands, +which had become so frightfully tabooed or unclean as to be quite +useless. Food would be set for him on the ground, and he would then sit +or kneel down, and, with his hands carefully held behind his back, would +gnaw at it as best he could." Often a degraded outcast was kept in a +village to feed mourners. Such a taboo is strictly similar to those +which surround a sacred chief or his property, a menstruous woman or a +homicide, rendering them dangerous to themselves and to all who approach +them. + +3. Primitive folk cannot conceive of a man's soul surviving apart from +his body, nor of another life as differing from this, and the dead must +continue to enjoy what they had here. Accordingly the Patagonians kill +horses at the grave that the dead may ride to _Alhuemapu_, or country of +the dead. After a year they collect a chief's bones, arrange them, tie +them together and dress them in his best garments with beads and +feathers. Then they lay him with his weapons in a square pit, round +which dead horses are placed set upright on their feet by stakes. As +late as 1781 in Poland F. Casimir's horse was slain and buried with him. +In the Caucasus a Christian lady's jewels are buried with her. The +Hindus used to burn a man's widow on his pyre, because he could not do +without her; and St Boniface commends the self-sacrifice of the Wend +widows who in his day burned themselves alive on their husbands' pyres. + +The tumuli met with all over the north of Europe (in the Orkneys alone +2000 remain) are regular houses of the dead, models of those they +occupied in life. The greater the dignity of the deceased, the loftier +was his barrow. Silbury hill is 170 ft. high; the tomb of Alyattes, +father of Croesus, was a fourth of a league round; the Pyramids are +still the largest buildings in existence; at Oberea in Tahiti is a +barrow 267 ft. long, 87 wide and 44 high. Some Eskimo just leave a dead +man's body in his house, and shut it up, often leaving by his side a +dog's head to guide him on his last journey, along with his tools and +kayak. The Sea Dyaks set a chief adrift in his war canoe with his +weapons. So in Norse story Hake "was laid wounded on a ship with the +dead men and arms; the ship was taken out to sea and set on fire." The +Viking was regularly buried in his ship or boat under a great mound. He +sailed after death to Valhalla. In the ship was laid a stone as anchor +and the tools, clothes, weapons and treasures of the dead. The +Egyptians, whose land was the gift of the river Nile, equally believed +that the dead crossed over water, and fashioned the hearse in the form +of a boat. Hence perhaps was derived the Greek myth of Charon and the +Styx, and the custom, which still survives in parts of Europe, of +placing a coin in the mouth of the dead with which to pay the ferryman. +The Egyptians placed in the tomb books of a kind to guide the dead to +the next world. The Copts in a later age did the same, and to this +custom we owe the recovery in Egypt of much ancient literature. The +Armenians till lately buried with a priest his missal or gospel. + +In Egyptian entombments of the XIIth to the XIVth dynasties were added +above the sepulchres what Professor Petrie terms soul-houses, viz. small +models of houses furnished with couch and table, &c., for the use of the +_ka_ or double whenever it might wish to come above ground and partake +of meats and drinks. They recall, in point of size, the hut-urns of the +Etruscans, but the latter had another use, for they contain incinerated +remains. Etruscan tombs, like those of Egypt and Asia Minor, were made +to resemble the dwelling-houses of the living, and furnished with +coffered ceilings, panelled walls, couches, stools, easy chairs with +footstools attached, all hewn out of the living rock (Dennis, _Cities +and Cemeteries of Etruria_, vol i. p. lxx.). + +Of the old Peruvian mummies in the Kircherian Museum at Rome, several +are of women with babies in their arms, whence it is evident that a +mother had her suckling buried with her; it would console her in the +next world and could hardly survive her in this. The practice of burying +ornaments, tools and weapons with the dead characterizes the inhumations +of the Quaternary epoch, as if in that dim and remote age death was +already regarded as the portal of another life closely resembling this. +The cups, tools, weapons, ornaments and other articles deposited with +the dead are often carefully broken or turned upside down and inside +out; for the soul or _manes_ of objects is liberated by such fracture or +inversion and so passes into the dead man's use and possession. For the +same reason where the dead are burned, their properties are committed to +the flames. The ghost of the warrior has a ghostly sword and buckler to +fight with and a ghostly cup to drink from, and he is also nourished by +the impalpable odour and reek of the animal victims sacrificed over his +grave. Instead of valuable objects cheap images and models are often +substituted; and why not, if the mere ghosts of the things are all that +the wraith can enjoy? Thus Marco Polo (ii. 76) describes how in the land +of Kinsay (Hang-chau) "the friends and relations make a great mourning +for the deceased, and clothe themselves in hempen garments, and follow +the corpse, playing on a variety of instruments and singing hymns to +their idols. And when they come to the burning place they take +representations of things cut out of parchment, such as caparisoned +horses, male and female slaves, camels, armour, suits of cloth of gold +(and money), in great quantities, and these things they put on the fire +along with the corpse so that they are all burned with it. And they tell +you that the dead man shall have all these slaves and animals of which +the effigies are burned, alive in flesh and blood, and the money in +gold, at his disposal in the next world; and that the instruments which +they have caused to be played at his funeral, and the idol hymns that +have been chaunted shall also be produced again to welcome him in the +next world." The manufacture of such paper _simulacra_ for consumption +at funerals is still an important industry in Chinese cities. The +ancient Egyptians, assured that a man's _ka_ or double shall revivify +his body, took pains to guard the flesh from corruption, steeping the +corpse in natron and stuffing it with spices. A body so prepared is +called a mummy (q.v.), and the custom was already of a hoary antiquity +in 3200 B.C., when the oldest dated mummy we have was made. The bowels, +removed in the process, were placed in jars over the corpse in the tomb, +together with writing tablets, books, musical instruments, &c., of the +dead. Cemeteries also remain full of mummies of crocodiles, cats, fish, +cows and other sacred animals. The Greeks settled in Egypt learned to +mummify their dead, but the custom was abhorrent to the Jews, although +the Christian belief in the resurrection of the flesh must have been +formed to a large extent under Egyptian influence. Half the superiority +of the Jewish to other ancient religions lay in this, that it prescribed +no funeral rites other than the simplest inhumation. + +The dead all over the world and from remote antiquity have been laid not +anyhow in the earth, but with the feet and face towards the region in +which their future will be spent; the Samoans and Fijians towards the +far west whither their souls have preceded them; the Guarayos with head +turned eastwards because their god Tamoi has in that quarter "his happy +hunting grounds where the dead will meet again" (Tylor, _Prim. Cult._ +ii. 422). The legend is that Christ was buried with His head to the +west, and the church follows the custom, more ancient than itself, of +laying the dead looking to the East, because that is the attitude of +prayer, and because at the last trump they will hurry eastwards. So in +Eusebius (_Hist. Eccl._ 430.19) a martyr explains to his pagan judge +that the heavenly Jerusalem, the fatherland of the pious, lay exactly in +the east at the rising place of the sun. Where the body is laid out +straight it is difficult to discern the presence of any other idea than +that it is at rest. In Scandinavian barrows, e.g. in the one opened at +Goldhavn in 1830, the skeletons have been found seated on a low stone +bench round the wall of the grave chamber facing its opening, which +always looks south or east, never north. Here the dead were continuing +the drinking bouts they enjoyed on earth. + +The Peruvians mummified their dead and placed them jointed and huddled +up with knees to chin, looking toward the sunset, with the hands held +before the face. In the oldest prehistoric tombs along the Nile the +bodies are doubled up in the same position. It would seem as if in these +and numerous other similar cases the dead were deliberately given in +their graves the attitude of a foetus in the womb, and, as Dr Budge +remarks (_Egyptian Ideas of the Future Life_, London, 1899, p. 162), "we +may perhaps be justified in seeing in this custom the symbol of a hope +that, as the child is born from this position into the world, so might +the deceased be born into the life beyond the grave." The late +Quaternary skeletons of the Mentone cave were laid in a layer of +ferrugineous earth specially laid down for them, and have contracted a +red colour therefrom. Many other prehistoric skeletons found in Italy +have a reddish colour, perhaps for the same reason, or because, as often +to-day, the bones were stripped of flesh and painted. Ambrose relates +that the skeletons of the martyrs Gervasius and Protasius, which he +found and deposited A.D. 386 under the altar of his new basilica in +Milan, were _mirae magnitudinis ut prisca aetas ferebat_, and were also +coloured red. He imagined the red to be the remains of the martyrs' +blood! _Hic sanguis clamat coloris indicio._ Salomon Reinach has rightly +divined that what Ambrose really hit upon was a prehistoric tomb. Red +earth was probably chosen as a medium in which to lay a corpse because +demons flee from red. Sacred trees and stones are painted red, and for +the most solemn of their rites savages bedaub themselves with red clay. +It is a favourite taboo colour. + +4. A feast is an essential feature of every primitive funeral, and in +the Irish "wake" it still survives. A dead man's soul or double has to +be fed at the tomb itself, perhaps to keep it from prowling about the +homes of the survivors in search of victuals; and such food must also be +supplied to the dead at stated intervals for months or years. Many races +leave a narrow passage or tube open down to the cavity in which the +corpse lies, and through it pour down drinks for the dead. Traces of +such tubes are visible in the prehistoric tombs of the British Isles. +However, such provision of food is not properly a funeral feast unless +the survivors participate. In the Eastern churches and in Russia the +departed are thus fed on the ninth, twelfth and fortieth days from +death. "Ye appease the shades of the dead with wine and meals," was the +charge levelled at the Catholics by the 4th-century Manichaeans, and it +has hardly ceased to be true even now after the lapse of sixteen +centuries. The funeral feast proper, however, is either a meal of +communion with or in the dead, which accompanies interment, or a banquet +off the flesh of victims slain in atonement of the dead man's sins. Some +anthropologists see in the common meal held at the grave "the pledge and +witness of the unity of the kin, the chief means, if not of making, at +least of repairing and renewing it."[1] The flesh provided at these +banquets is occasionally that of the dead man himself; Herodotus and +Strabo in antiquity relate this of several half-civilized races in the +East and West, and a similar story is told by Marco Polo of certain +Tatars. Nor among modern savages are funeral feasts off the flesh of the +dead unknown, and they seem to be intended to effect and renew a +sacramental union or kinship of the living with the dead. The Uaupes in +the Amazons incinerate a corpse a month after death, pound up the ashes, +and mix them with their fermented drink. They believe that the virtues +of the dead will thus be passed on to his survivors. The life of the +tribe is kept inside the tribe and not lost. Such cannibal sacraments, +however, are rare, and, except in a very few cases, the evidence for +them weak. The slaying and eating of animal victims, however, at the +tomb is universal and bears several meanings, separately or all at once. +The animals may be slain in order that their ghosts may accompany the +deceased in his new life. This significance we have already dwelt upon. +Or it is believed that the shade feeds upon them, as the shades came up +from Hades and lapped up out of a trench the blood of the animals slain +by Ulysses. The survivors by eating the flesh of a victim, whose blood +and soul the dead thus consume, sacramentally confirm the mystic tie of +blood kinship with the dead. Or lastly, the victim may be offered for +the sins of the dead. His sins are even supposed to be transferred into +it and eaten by the priest. Such expiatory sacrifices of animals for the +dead survive in the Christian churches of Armenia, Syria and of the East +generally. Their vicarious character is emphasized in the prayers which +accompany them, but the popular understanding of them probably combines +all the meanings above enumerated. It has been suggested by Robertson +Smith (_Religion of the Semites_, 336) that the world-wide customs of +tearing the hair, rending the garments, and cutting and wounding the +body were originally intended to establish a life-bond between the dead +and the living. The survivors, he argues, in leaving portions of their +hair and garments, and yet more by causing their own blood to stream +over the corpse from self-inflicted wounds, by cutting off a finger and +throwing it into the grave, leave what is eminently their own with the +dead, so drawing closer their tie with him. Conversely, many savages +daub themselves with the blood and other effluences of their dead +kinsmen, and explain their custom by saying that in this way a portion +of the dead is incorporated in themselves. Often the survivors, +especially the widows, attach the bones or part of them to their persons +and wear them, or at least keep them in their houses. The retention of +the locks of the deceased and of parts of his dress is equally common. +There is also another side to such customs. Having in their possession +bits of the dead, and being so far in communion with him, the survivors +are surer of his friendship. They have ensured themselves against ghosts +who are apt to be by nature envious and mischievous. But whatever their +original significance, the tearing of cheeks and hair and garments and +cutting with knives are mostly expressions of real sorrow, and, as +Robertson Smith remarks, of deprecation and supplication to an angry god +or spirit. It must not be supposed that the savage or ancient man feels +less than ourselves the poignancy of loss. + +6. Death-witchery has close parallels in the witch and heretic hunts of +the Christians, but, happily for us, only flourishes to-day among +savages. Sixty % of the deaths which occur in West Africa are, according +to Miss Mary Kingsley--a credible witness--believed to be due to +witchcraft and sorcery. The blacks regard old age or effusion of blood +as the sole legitimate causes of death. All ordinary diseases are in +their opinion due to private magic on the part of neighbours, just as a +widespread epidemic marks the active hatred "of some great outraged +nature spirit, not of a mere human dabbler in devils."[2] Similarly in +Christian countries an epidemic is set down to the wrath of a God +offended by the presence of Jews, Arians and other heretics. The duty of +an African witch-doctor is to find out who bewitched the deceased, just +as it was of an inquisitor to discover the heretic. Every African +post-mortem accordingly involves the murder of the person or persons who +bewitched the dead man and caused him to die. The death-rate by these +means is nearly doubled; but, since the use of poison against an +obnoxious neighbour is common, the right person is occasionally +executed. It is also well for neighbours not to quarrel, for, if they do +and one of them dies of smallpox, the other is likely to be slain as a +witch, and his lungs, liver and spleen impaled on a pole at the entrance +of the village. It is the same case with the Australian blacks: "no such +thing as natural death is realized by the native; a man who dies has of +necessity been killed by some other man, or perhaps even by a woman, and +sooner or later that man or woman will be attacked. In the normal +condition of the tribe every death meant the killing of another +individual."[3] + +7. Lastly, a primitive interment guards against the double risk of the +ghost haunting the living and of ghouls or vampires taking possession of +the corpse. The latter end is likely to be achieved if the body is +cremated, for then there is no nidus to harbour the demon; but whether, +in the remote antiquity to which belong many barrows containing +incinerated remains, this motive worked, cannot be ascertained. The +Indo-European race seems to have cremated at an early epoch, perhaps +before the several races of East and West separated. In Christian +funeral rites many prayers are for the protection of the body from +violation by vampires, and it would seem as if such a motive dictated +the architectural solidity of some ancient tombs. Christian graves were +for protection regularly sealed with the cross; and the following is a +characteristic prayer from the old Armenian rite for the burial of a +layman: + + "Preserve, Almighty Lord, this man's spirit with all saints and with + all lovers of Thy holy name. And do Thou seal and guard the sepulchre + of Thy servant, Thou who shuttest up the depths and sealest them with + Thy almighty right hand ... so let the seal of Thy Lordship abide + unmoved upon this man's dwelling-place and upon the shrine which + guards Thy servant. And _let not any filthy and unclean devil dare to + approach him, such as assail the body and souls of the heathen_, who + possess not the birth of the holy font, and have not the dread seal + laid upon their graves." + +A terrible and revolting picture of the superstitious belief in ghouls +which violate Christian tombs is given by Leo Allatius (who held it) in +his tract _De opinionibus quorundam Graecorum_ (Paris, 1646). It was +probably the fear of such demonic assaults on the dead that inspired the +insanitary custom of burying the dead under the floors of churches, and +as near as possible to the altar. In the Greek Church this practice was +happily forbidden by the code of Justinian as well as by the older law +in the case of churches consecrated with _Encaenia_ and deposition of +relics. In the Armenian Church the same rule holds, and Ephrem Syrus in +his testament particularly forbade his body to be laid within a church. +Such prohibitions, however, are a witness to the tendency in question. + +The custom of lighting candles round a dead body and watching at its +side all night was originally due to the belief that a corpse, like a +person asleep, is specially liable to the assaults of demons. The +practice of tolling a bell at death must have had a similar origin, for +it was a common medieval belief that the sound of a consecrated bell +drives off the demons which when a man dies gather near in the air to +waylay his fleeting soul. For a like reason the consecrated bread of the +Eucharist was often buried with believers, and St Basil is said to have +specially consecrated a Host to be placed in his coffin. + +8. Some of the rites described under the previous heads may be really +inspired by the fear of the dead haunting the living, but it must be +kept in mind that the taboo attaching to a dead body is one thing and +fear of a ghost another. A corpse is buried or burned, or scaffolded on +a tree, a tower or a house-top, in order to get it out of the way and +shield society from the dangerous infection of its taboo; but ghosts +_qua_ ghosts need not be feared and a kinsman's ghost usually is not. On +the contrary, it is fed and consoled with everything it needs, is asked +not to go away but to stay, is in a thousand ways assured of the sorrow +and sympathy of the survivors. Even if the body be eaten, it is merely +to keep the soul of the deceased inside the circle of kinsmen, and +Strabo asserts that the ancient Irish and Massagetae regarded it as a +high honour to be so consumed by relatives. In Santa Cruz in Melanesia +they keep the bones for arrow heads and store a skull in a box and set +food before it "saying that this is the man himself" (R.H. Codrington, +_The Melanesians_, p. 264), or the skull and jaw bone are kept and "are +called _mangite_, which are _saka_, hot with spiritual power, and by +means of which the help of the _lio'a_, the powerful ghost of the man +whose relics these are, can be obtained" (ibid. p. 262). Here we have +the savage analogue to Christian relics. So the Australian natives make +pointing sticks out of the small bones of the arm, with which to bewitch +enemies. + +We may conclude then that in the most primitive societies, where +blood-kinship is the only social tie and root of social custom it is the +shades, not of kinsmen, but of strangers, who as such are enemies, that +are dangerous and uncanny. In more developed societies, however, all +ghosts alike are held to be so; and if a ghost walks it is because its +body has not been properly interred or because its owner was a +malefactor. Still, even allowing for this, it remains true that for a +friendly ghost the proper place is the grave and not the homes of the +living, and accordingly the Aruntas with cries of _Wah! Wah!_ with +wearing of fantastic head-dresses, wild dancing and beating of the air +with hands and weapons "drive the spirit away from the old camp which it +is supposed to haunt," and which has been set fire to, and hunt it at a +run into the grave prepared, and there stamp it down into the earth. +"The loud shouting of the men and women shows him that they do not wish +to be frightened by him in his present state, and that they will be +angry with him if he does not rest." (Spencer and Gillen, _Native +Tribes of Central Australia_, p. 508). In Mesopotamia cemeteries have +been discovered where the sepulchral jars were set upside down, clearly +by way of hindering the ghosts from escaping into the upper world. In +the Dublin museum we see specimens of ancient Celtic tombs showing the +same peculiarity. For a like reason perhaps the name of the dead must +among the Aruntas not be uttered, nor the grave approached, by certain +classes of kinsmen. The same repugnance to naming the dead exists all +over the world, and leads survivors who share the dead man's name to +adopt another, at least for a time. If the dead man's name was that of a +plant, tree, animal or stream, that too is changed. Here is a potent +cause of linguistic change, that also renders any historical tradition +impossible. The survivors seem to fear that the ghost will come when he +hears his name called; but it also hangs together with the taboo which +hedges round the dead as it does kings, chieftains and priests. + + AUTHORITIES.--B. Spencer and F.J. Gillen, _The Native Tribes of + Central Australia_ (London, 1899); F.B. Jevons, _Introduction to + History of Religion_ (London, 1896); E.S. Hartland, _The Legend of + Perseus_, vol. ii.; J.G. Frazer, _The Golden Bough_ (London, 1900); + L.W. Faraday, "Custom and Belief in the Icelandic Sagas," in + _Folk-lore_, vol. xvii. No. 4; E.B. Tylor, _Primitive Culture_ + (London, 1903); E.A. W. Budge, _The Mummy_ (Cambridge, 1893); C. + Royer, "Les Rites funeraires aux epoques prehistoriques," _Revue + d'anthropologie_ (1876); Forrer, _Uber die Totenbestattung bei den + Pfahlbauern_ (Ausland, 1885); J. Lubbock, _Origin of Civilization_ + (London, 1875) and _Prehistoric Times_ (London, 1865); L.A. Muratori, + "De antiquis Christianorum sepulchris," _Anecd. Graeca_ (Padua, 1709); + Onaphr. Panvinius, _De ritu sepeliendi mortuos apua veteres + Christianos_, reprinted in Volbeding's _Thesaurus_ (Leipzig, 1841). + (F. C. C.) + + +FOOTNOTES: + + [1] E.S. Hartland, _Legend of Perseus_ (1895), ii. 278. + + [2] Mary Kingsley, _West African Studies_ (1901), p. 178. + + [3] B. Spencer and F.J. Gillen, _The Native Tribes of Central + Australia_ (1899), p. 48. + + + + +FUNGI (pl. of Lat. _fungus_, a mushroom), the botanical name covering in +the broad sense all the lower cellular Cryptogams devoid of chlorophyll, +which arise from spores, and the thallus of which is either unicellular +or composed of branched or unbranched tubes or cell-filaments (hyphae) +with apical growth, or of more or less complex wefted sheets or +tissue-like masses of such (mycelium). The latter may in certain cases +attain large dimensions, and even undergo cell-divisions in their +interior, resulting in the development of true tissues. The spores, +which may be uni- or multicellular, are either abstricted free from the +ends of hyphae (acrogenous), or formed from segments in their course +(_chlamydospores_) or from protoplasm in their interior (endogenous). +The want of chlorophyll restricts their mode of life--which is rarely +aquatic--since they are therefore unable to decompose the carbon dioxide +of the atmosphere, and renders them dependent on other plants or +(rarely) animals for their carbonaceous food-materials. These they +obtain usually in the form of carbohydrates from the dead remains of +other organisms, or in this or other forms from the living cells of +their hosts; in the former case they are termed saprophytes, in the +latter parasites. While some moulds (_Penicillium_, _Aspergillus_) can +utilize almost any organic food-materials, other fungi are more +restricted in their choice--e.g. insect-parasites, horn- and +feather-destroying fungi and parasites generally. It was formerly the +custom to include with the Fungi the Schizomycetes or Bacteria, and the +Myxomycetes or Mycetozoa; but the peculiar mode of growth and division, +the cilia, spores and other peculiarities of the former, and the +emission of naked amoeboid masses of protoplasm, which creep and fuse to +streaming plasmodia, with special modes of nutrition and spore-formation +of the latter, have led to their separation as groups of organisms +independent of the true Fungi. On the other hand, lichens, previously +regarded as autonomous plants, are now known to be dual organisms--fungi +symbiotic with algae. + +The number of species in 1889 was estimated by Saccardo at about 32,000, +but of these 8500 were so-called _Fungi imperfecti_--i.e. forms of which +we only know certain stages, such as conidia, pycnidia, &c., and which +there are reasons for regarding as merely the corresponding stages of +higher forms. Saccardo also included about 400 species of Myxomycetes +and 650 of Schizomycetes. Allowing for these and for the cases, +undoubtedly not few, where one and the same fungus has been described +under different names, we obtain Schroeter's estimate (in 1892) of +20,000 species. In illustration of the very different estimates that +have been made, however, may be mentioned that of De Bary in 1872 of +150,000 species, and that of Cooke in 1895 of 40,000, and Massee in 1899 +of over 50,000 species, the fact being that no sufficient data are as +yet to hand for any accurate census. As regards their geographical +distribution, fungi, like flowering plants, have no doubt their centres +of origin and of dispersal; but we must not forget that every exchange +of wood, wheat, fruits, plants, animals, or other commodities involves +transmission of fungi from one country to another; while the migrations +of birds and other animals, currents of air and water, and so forth, are +particularly efficacious in transmitting these minute organisms. Against +this, of course, it may be argued that parasitic forms can only go where +their hosts grow, as is proved to be the case by records concerning the +introduction of _Puccinia malvacearum_, _Peronospora viticola_, +_Hemileia vastatrix_, &c. Some fungi--e.g. moulds and yeasts--appear to +be distributed all over the earth. That the north temperate regions +appear richest in fungi may be due only to the fact that North America +and Europe have been much more thoroughly investigated than other +countries; it is certain that the tropics are the home of very numerous +species. Again, the accuracy of the statement that the fleshy Agaricini, +Polyporei, _Pezizae_, &c., are relatively rarer in the tropics may +depend on the fact that they are more difficult to collect and remit for +identification than the abundantly recorded woody and coriaceous forms +of these regions. When we remember that many parts of the world are +practically unexplored as regards fungi, and that new species are +constantly being discovered in the United States, Australia and northern +Europe--the best explored of all--it is clear that no very accurate +census of fungi can as yet be made, and no generalizations of value as +to their geographical distribution are possible. + +The existence of fossil fungi is undoubted, though very few of the +identifications can be relied on as regards species or genera. They +extend back beyond the Carboniferous, where they occur as hyphae, &c., +preserved in the fossil woods, but the best specimens are probably those +in amber and in siliceous petrifactions of more recent origin. + +[Illustration: FIG. 1.--1, _Peronospora parasitica_ (De Bary). Mycelium +with haustoria (h); 2, _Erysiphe_; A and B, mycelium (m), with haustoria +(h). (After De Bary.)] + + _Organs._--Individual hyphae or their branches often exhibit + specializations of form. In many Basidiomycetes minute branches arise + below the septa; their tips curve over the outside of the latter, and + fuse with the cell above just beyond it, forming a _clamp-connexion_. + Many parasitic hyphae put out minute lateral branches, which pierce + the cell-wall of the host and form a peg-like (_Trichosphaeria_), + sessile (_Cystopus_), or stalked (_Hemileia_), knot-like, or a more or + less branched (_Peronospora_) or coiled (_Protomyces_) haustorium. In + _Rhizopus_ certain hyphae creep horizontally on the surface of the + substratum, and then anchor their tips to it by means of a tuft of + short branches (_appressorium_), the walls of which soften and gum + themselves to it, then another branch shoots out from the tuft and + repeats the process, like a strawberry-runner. Appressoria are also + formed by some parasitic fungi, as a minute flattening of the tip of a + very short branch (_Erysiphe_), or the swollen end of any hypha which + comes in contact with the surface of the host (_Piptocephalis_, + _Syncephalis_), haustoria piercing in each case the cell-wall below. + In _Botrytis_ the appressoria assume the form of dense tassels of + short branches. In _Arthrobotrys_ side-branches of the mycelium sling + themselves around the host (_Tylenchus_) much as tendrils round a + support. + + Many fungi (_Phallus_, _Agaricus_, _Fumago_, &c.) when strongly + growing put out ribbon-like or cylindrical cords, or sheet-like + mycelial plates of numerous parallel hyphae, all growing together + equally, and fusing by anastomoses, and in this way extend long + distances in the soil, or over the surfaces of leaves, branches, &c. + These mycelial strands may be white and tender, or the outer hyphae + may be hard and black, and very often the resemblance of the + subterranean forms to a root is so marked that they are termed + rhizomorphs. The outermost hyphae may even put forth thinner hyphae, + radiating into the soil like root-hairs, and the convergent tips may + be closely appressed and so divided by septa as to resemble the + root-apex of a higher plant (_Armillaria mellea_). + + _Sclerotia._--Fungi, like other plants, are often found to store up + large quantities of reserve materials (oil, glycogen, carbohydrates, + &c.) in special parts of their vegetative tissues, where they lie + accumulated between a period of active assimilation and one of renewed + activity, forming reserves to be consumed particularly during the + formation of large fructifications. These reserve stores may be packed + away in single hyphae or in swollen cells, but the hyphae containing + them are often gathered into thick cords or mycelial strands + (_Phallus_, mushroom, &c.), or flattened and anastomosing ribbons and + plates, often containing several kinds of hyphae (_Merulius + lacrymans_). In other cases the strands undergo differentiation into + an outer layer with blackened, hardened cell-walls and a core of + ordinary hyphae, and are then termed rhizomorphs (_Armillaria + mellea_), capable not only of extending the fungus in the soil, like + roots, but also of lying dormant, protected by the outer casing. Such + aggregations of hyphae frequently become knotted up into dense masses + of interwoven and closely packed hyphae, varying in size from that of + a pin's head or a pea (_Peziza_, _Coprinus_) to that of a man's fist + or head, and weighing 10 to 25 lb. or more (_Polyporus Mylittae_, _P. + tumulosus_, _Lentinus Woermanni_, _P. Sapurema_, &c.). The interwoven + hyphae fuse and branch copiously, filling up all interstices. They + also undergo cutting up by numerous septa into short cells, and these + often divide again in all planes, so that a pseudoparenchyma results, + the walls of which may be thickened and swollen internally, or + hardened and black on the exterior. In many cases the swollen + cell-walls serve as reserves, and sometimes the substance is so + thickly deposited in strata as to obliterate the lumen, and the hyphae + become nodular (_Polyporus sacer_, _P. rhinoceros_, _Lentinus + Woermanni_). The various sclerotia, if kept moist, give rise to the + fructifications of the fungi concerned, much as a potato tuber does to + a potato plant, and in the same way the reserve materials are + consumed. They are principally Polyporei, Agaricini, Pezizae; none are + known among the Phycomycetes, Uredineae or Ustilagineae. The functions + of mycelial strands, rhizomorphs and sclerotia are not only to collect + and store materials, but also to extend the fungus, and in many cases + similar strands act as organs of attack. The same functions of storage + in advance of fructification are also exercised by the stromata so + common in Ascomycetes. + + _Tissue Differentiations._--The simpler mycelia consist of hyphae all + alike and thin-walled, or merely differing in the diameter of the + branches of various orders, or in their relations to the environment, + some plunging into the substratum like roots, others remaining on its + surface, and others (aerial hyphae) rising into the air. Such hyphae + may be multicellular, or they may consist of simple tubes with + numerous nuclei and no septa (_Phycomycetes_), and are then + non-cellular. In the more complex tissue-bodies of higher fungi, + however, we find considerable differences in the various layers or + strands of hyphae. + + An epidermis-like or cortical protective outer layer is very common, + and is usually characterized by the close septation of the densely + interwoven hyphae and the thickening and dark colour of their outer + walls (sclerotia, _Xylaria_, &c.). Fibre-like hyphae with the lumen + almost obliterated by the thick walls occur in mycelial cords + (_Merulius_). Latex-tubes abound in the tissues of _Lactarius_, + _Stereum_, _Mycena_, _Fistulina_, filled with white or coloured milky + fluids, and Istvanffvi has shown that similar tubes with fluid or oily + contents are widely spread in other Hymenomycetes. Sometimes fatty oil + or watery sap is found in swollen hyphal ends, or such tubes contain + coloured sap. Cystidia and paraphyses may be also classed here. In + _Merulius lacrymans_ Hartig has observed thin-walled hyphae with large + lumina, the septa of which are perforated like those of sieve-tubes. + + As regards its composition, the cell-wall of fungi exhibits variations + of the same kind as those met with in higher plants. While the + fundamental constituent is a cellulose in many Mucorini and other + Phycomycetes, in others bodies like pectose, callose, &c., commonly + occur, and Wisselingh's researches show that chitin, a gluco-proteid + common in animals, forms the main constituent in many cases, and is + probably deposited directly as such, though, like the other + substances, it may be mixed with cellulose. As in other cell-walls, so + here the older membranes may be altered by deposits of various + substances, such as resin, calcium oxalate, colouring matters; or more + profoundly altered throughout, or in definite layers, by + lignification, suberization (_Trametes_, _Daedalea_), or swelling to a + gelatinous mucilage (_Tremella_, _Gymnosporangium_), while + cutinization of the outer layers is common. One of the most striking + alterations of cell-walls is that termed _carbonization_, in which the + substance gradually turns black, hard and brittle, as if charred--e.g. + _Xylaria_, _Ustulina_, some sclerotia. At the other extreme the + cell-walls of many lichen-fungi are soft and colourless, but turn blue + in iodine, as does starch. The young cell-wall is always tenuous and + flexible, and may remain so throughout, but in many cases thickenings + and structural differentiations, as well as the changes referred to + above, alter the primary wall considerably. Such thickening may be + localized, and _pits_ (e.g. _Uredospores_, septa of Basidiomycetes), + _spirals_, _reticulations_, _rings_, &c. (capillitium fibres of + _Podaxon_, _Calostoma_, _Battarrea_), occur as in the vessels of + higher plants, while sculptured networks, pittings and so forth are as + common on fungus-spores as they are on pollen grains. + + _Cell-Contents._--The cells of fungi, in addition to protoplasm, + nuclei and sap-vacuoles, like other vegetable cells, contain formed + and amorphous bodies of various kinds. Among those directly visible to + the microscope are oil drops, often coloured (_Uredineae_) crystals of + calcium oxalate (_Phallus_, _Russula_), proteid crystals (_Mucor_, + _Pilobolus_, &c.) and resin (Polyporei). The oidia of Erysipheae + contain fibrosin bodies and the hyphae of Saprolegnieae cellulin + bodies, but starch apparently never occurs. Invisible to the + microscope, but rendered visible by reagents, are glycogen, _Mucor_, + Ascomycetes, yeast, &c. In addition to these cell-contents we have + good indirect evidence of the existence of large series of other + bodies, such as proteids, carbohydrates, organic acids, alkaloids, + enzymes, &c. These must not be confounded with the numerous substances + obtained by chemical analysis of masses of the fungus, as there is + often no proof of the manner of occurrence of such bodies, though we + may conclude with a good show of probability that some of them also + exist preformed in the living cell. Such are sugars (glucose, mannite, + &c.), acids (acetic, citric and a whole series of lichen-acids), + ethereal oils and resinous bodies, often combined with the intense + colours of fungi and lichens, and a number of powerful alkaloid + poisons, such as muscarin (_Amanita_), ergotin (_Claviceps_), &c. + + Among the enzymes already extracted from fungi are _invertases_ + (yeasts, moulds, &c.), which split cane-sugar and other complex sugars + with hydrolysis into simpler sugars such as dextrose and levulose; + _diastases_, which convert starches into sugars (_Aspergillus_, &c.); + _cytases_, which dissolve cellulose similarly (_Botrytis_, &c.); + _peptases_, using the term as a general one for all enzymes which + convert proteids into peptones and other bodies (_Penicillium_, &c.); + lipases, which break up fatty oils (_Empusa_, _Phycomyces_, &c.); + oxydases, which bring about the oxidations and changes of colour + observed in _Boletus_, and _zymase_, extracted by Buchner from yeast, + which brings about the conversion of sugar into alcohol and + carbon-dioxide. That such enzymes are formed in the protoplasm is + evident from the behaviour of hyphae, which have been observed to + pierce cell-membranes, the chitinous coats of insects, artificial + collodion films and layers of wax, &c. That a fungus can secrete more + than one enzyme, according to the materials its hyphae have to attack, + has been shown by the extraction of diastase, inulase, trehalase, + invertase, maltase, raffinase, malizitase, emulsin, trypsin and lipase + from _Aspergillus_ by Bourquelot, and similar events occur in other + fungi. The same fact is indicated by the wide range of organic + substances which can be utilized by _Penicillium_ and other moulds, + and by the behaviour of parasitic fungi which destroy various + cell-contents and tissues. Many of the coloured pigments of fungi are + fixed in the cell-walls or excreted to the outside (_Peziza + aeruginosa_). Matruchot has used them for staining the living + protoplasm of other fungi by growing the two together. Striking + instances of coloured mycella are afforded by _Corticium sanguineum_, + blood-red; _Elaphomyces Leveillei_, yellow-green; _Chlorosplenium + aeruginosum_, verdigris green; and the _Dematei_, brown or black. + + _Nuclei._--Although many fungi have been regarded as devoid of nuclei, + and all have not as yet been proved to contain them, the numerous + investigations of recent years have revealed them in the cells of all + forms thoroughly examined, and we are justified in concluding that the + nucleus is as essential to the cell of a fungus as to that of other + organisms. The hyphae of many contain numerous, even hundreds of + nuclei (Phycomycetes); those of others have several (_Aspergillus_) in + each segment, or only two (_Exoascus_) or one (_Erysiphe_) in each + cell. Even the isolated cells of the yeast plant have each one + nucleus. As a rule the nuclei of the mycelium are very minute (1.5-2 + [mu] in _Phycomyces_), but those of many asci and spores are large and + easily rendered visible. As with other plants, so in fungi the + essential process of fertilization consists in the fusion of two + nuclei, but owing to the absence of well-marked sexual organs from + many fungi, a peculiar interest attaches to certain nuclear fusions in + the vegetative cells or in young spores of many forms. Thus in + Ustilagineae the chlamydospores, and in Uredineae the teleutospores, + each contain two nuclei when young, which fuse as the spores mature. + In young asci a similar fusion of two nuclei occurs, and also in + basidia, in each case the nucleus of the ascus or of the basidium + resulting from the fusion subsequently giving rise by division to the + nuclei of the ascospores and basidiospores respectively. The + significance of these fusions will be discussed under the various + groups. Nuclear division is usually accompanied by all the essential + features of karyokinesis. + + _Spores._--No agreement has ever been arrived at regarding the + consistent use of the term spore. This is apparently owing to the + facts that too much has been attempted in the definition, and that + differences arise according as we aim at a morphological or a + physiological definition. Physiologically, any cell or group of cells + separated off from a hypha or unicellular fungus, and capable of + itself growing out--germinating--to reproduce the fungus, is a spore; + but it is evident that so wide a definition does not exclude the + ordinary vegetative cells of sprouting fungi, such as yeasts, or small + sclerotium like cell-aggregates of forms like _Coniothecium_. + Morphologically considered, spores are marked by peculiarities of + form, size, colour, place of origin, definiteness in number, mode of + preparation, and so forth, such that they can be distinguished more or + less sharply from the hyphae which produce them. The only + physiological peculiarity exhibited in common by all spores is that + they germinate and initiate the production of a new fungus-plant. + Whether a spore results from the sexual union of two similar gametes + (zygospore) or from the fertilization of an egg-cell by the protoplasm + of a male organ (oospore); or is developed asexually as a motile + (zoospore) or a quiescent body cut off from a hypha (conidium) or + developed along its course (oidium or chlamydospore), or in its + protoplasm (endospore), are matters of importance which have their + uses in the classification and terminology of spores, though in many + respects they are largely of academic interest. + + [Illustration: FIG. 2.--_Peronospora parasitica_ (De Bary). + Conidiophore with conidia.] + + Klebs has attempted to divide spores into three categories as follows: + (1) kinospores, arising by relatively simple cell-divisions and + subserving rapid dissemination and propagation, e.g. zoospores, + conidia, endogonidia, stylospores, &c.; (2) paulospores, due to simple + rearrangement of cell-contents, and subserving the persistence of the + fungus through periods of exigency, e.g. gemmae, chlamydospores, + resting-cells, cysts, &c.; (3) carpospores, produced by a more or less + complex formative process, often in special fructifications, and + subserving either or both multiplication and persistence, e.g. + zygospores, oospores, brand-spores, aecidiospores, ascospores, + basidiospores, &c. Little or nothing is gained by these definitions, + however, which are especially physiological. In practice these various + kinds of spores of fungi receive further special names in the separate + groups, and names, moreover, which will appear, to those unacquainted + with the history, to have been given without any consistency or regard + to general principles; nevertheless, for ordinary purposes these names + are far more useful in most cases, owing to their descriptive + character, than the proposed new names, which have been only partially + accepted. + + _Sporophores._--In some of the simpler fungi the spores are not borne + on or in hyphae which can be distinguished from the vegetative parts + or mycelium, but in the vast majority of cases the sporogenous hyphae + either ascend free into the air or radiate into the surrounding water + as distinct branches, or are grouped into special columns, cushions, + layers or complex masses obviously different in colour, consistency, + shape and other characters from the parts which gather up and + assimilate the food-materials. The term "receptacle" sometimes applied + to these spore-bearing hyphae is better replaced by sporophore. The + sporophore is obsolete when the spore-bearing hyphae are not sharply + distinct from the mycelium, simple when the constituent hyphae are + isolated, and compound when the latter are conjoined. The chief + distinctive characters of the sporogenous hyphae are their + orientation, usually vertical; their limited apical growth; their + peculiar branching, form, colour, contents, consistency; and their + spore-production. According to the characters of the last, we might + theoretically divide them into conidiophores, sporangiophores, + gametophores, oidiophores, &c.; but since the two latter rarely occur, + and more than one kind of spore or spore-case may occur on a + sporophore, it is impossible to carry such a scheme fully into + practice. + + A simple sporophore may be merely a single short hypha, the end of + which stops growing and becomes cut off as a conidium by the formation + of a septum, which then splits and allows the conidium to fall. More + generally the hypha below the septum grows forwards again, and repeats + this process several times before the terminal conidium falls, and so + a chain of conidia results, the oldest of which terminates the series + (_Erysiphe_); when the primary branch has thus formed a basipetal + series, branches may arise from below and again repeat this process, + thus forming a tuft (_Penicillium_). Or the primary hypha may first + swell at its apex, and put forth a series of short peg-like branches + (_sterigmata_) from the increased surface thus provided, each of which + develops a similar basipetal chain of conidia (_Aspergillus_), and + various combinations of these processes result in the development of + numerous varieties of exquisitely branched sporophores of this type + (_Botrytis_, _Botryosporium_, _Verticillium_, &c.). + + [Illustration: FIG. 3.--_Cystopus candidus_. + + A. a, Conidia. + b, Conidiophores. + c, Conidium emitting zoospores. + d, Free zoospore. + B.og, Oogonium. + os, Oosphere. + an, Antheridium. + C. Formation of zoospores by oospores. + z, Free zoospores. (After De Bary.)] + + A second type is developed as follows: the primary hypha forms a + septum below its apex as before, and the terminal conidium, thus + abstricted, puts out a branch at its apex, which starts as a mere + point and rapidly swells to a second conidium; this repeats the + process, and so on, so that we now have a chain of conidia developed + in acropetal succession, the oldest being below, and, as in + _Penicillium_, &c., branches put forth lower down may repeat the + process (_Hormodendron_). In all these cases we may speak of simple + conidiophores. The simple sporophore does not necessarily terminate in + conidia, however. In _Mucor_, for example, the end of the primary + hypha swells into a spheroidal head (sporangium), the protoplasm of + which undergoes segmentation into more or less numerous globular + masses, each of which secretes an enveloping cell-wall and becomes a + spore (endospore), and branched systems of sporangia may arise as + before (_Thamnidium_). Such may be termed sporangiophores. In + _Sporodinia_ the branches give rise also to short branches, which meet + and fuse their contents to form zygospores. In Peronospora, + Saprolegnia, &c., the ends of the branches swell up into sporangia, + which develop zoospores in their interior (zoosporangia), or their + contents become oospheres, which may be fertilized by the contents of + other branches (antheridia) and so form egg-cases (oogonia). Since in + such cases the sporophore bears sexual cells, they may be conveniently + termed gametophores. + + Compound sporophores arise when any of the branched or unbranched + types of spore-bearing hyphae described above ascend into the air in + consort, and are more or less crowded into definite layers, cushions, + columns or other complex masses. The same laws apply to the individual + hyphae and their branches as to simple sporophores, and as long as the + conidia, sporangia, gametes, &c., are borne on their external + surfaces, it is quite consistent to speak of these as compound + sporophores, &c., in the sense described, however complex they may + become. Among the simplest cases are the sheet-like aggregates of + sporogenous hyphae in _Puccinia_, _Uromyces_, &c., or of basidia in + _Exobasidium_, _Corticium_, &c., or of asci in _Exoascus_, + _Ascocorticium_, &c. In the former, where the layer is small, it is + often termed a sorus, but where, as in the latter, the sporogenous + layer is extensive, and spread out more or less sheet-like on the + supporting tissues, it is more frequently termed a hymenium. Another + simple case is that of the columnar aggregates of sporogenous hyphae + in forms like _Stilbum_, _Coremium_, &c. These lead us to cases where + the main mass of the sporophore forms a supporting tissue of closely + crowded or interwoven hyphae, the sporogenous terminal parts of the + hyphae being found at the periphery or apical regions only. Here we + have the cushion-like type (stroma) of _Nectria_ and many + Pyrenomycetes, the clavate "receptacle" of _Clavaria_, &c., passing + into the complex forms met with in _Sparassis_, _Xylaria_, + _Polyporei_, and _Agaricini_, &c. In these cases the compound + sporophore is often termed the hymenophore, and its various parts + demand special names (pileus, stipes, gills, pores, &c.) to denote + peculiarities of distribution of the hymenium over the surface. + + Other series of modifications arise in which the tissues corresponding + to the stroma invest the sporogenous hyphal ends, and thus enclose the + spores, asci, basidia, &c., in a cavity. In the simplest case the + stroma, after bearing its crop of conidia or oidia, develops + ascogenous branches in the loosened meshes of its interior (e.g. + _Onygena_). Another simple case is where the plane or slightly convex + surface of the stroma rises at its margins and overgrows the + sporogenous hyphal ends, so that the spores, asci, &c., come to lie in + the depression of a cavity--e.g. _Solenia_, _Cyphella_--and even + simpler cases are met with in _Mortierella_, where the zygospore is + invested by the overgrowth of a dense mat of closely branching hyphae, + and in _Gymnoascus_, where a loose mat of similarly barren hyphae + covers in the tufts of asci as they develop. + + In such examples as the above we may regard the hymenium (_Solenia_, + _Cyphella_), zygospores, or asci as truly invested by later growth, + but in the vast majority of cases the processes which result in the + enclosure of the spores, asci, &c., in a "fructification" are much + more involved, inasmuch as the latter is developed in the interior of + hyphal tissues, which are by no means obviously homologous with a + stroma. Thus in _Penicillium_, _Eurotium_, _Erysiphe_, &c., hyphal + ends which are the initials of ascogenous branches, are invested by + closely packed branches at an early stage of development, and the asci + develop inside what has by that time become a complete investment. + Whether a true sexual process precedes these processes or not does not + affect the present question, the point being that the resulting + spheroidal "fructification" (cleistocarp, perithecium) has a definite + wall of its own not directly comparable with a stroma. In other cases + (_Hypomyces_, _Nectria_) the perithecia arise on an already mature + stroma, while yet more numerous examples can be given (_Poronia_, + _Hypoxylon_, _Claviceps_, &c.) where the perithecia originate below + the surface of a stroma formed long before. Similarly with the various + types of conidial or oidial "fructifications," termed pycnidia, + spermogonia, aecidia, &c. In the simplest of these cases--e.g. + _Fumago_--a single mycelial cell divides by septa in all three planes + until a more or less solid clump results. Then a hollow appears in the + centre owing to the more rapid extension of the outer parts, and into + this hollow the cells lining it put forth short sporogenous branches, + from the tips of which the spores (stylospores, conidia, spermatia) + are abstricted. In a similar way are developed the pycnidia of + _Cicinnobolus_, _Pleospora_, _Cucurbitaria_, _Leptosphaeria_ and + others. In other cases (_Diplodia_, _Aecidium_, &c.) conidial or + oidial "fructifications" arise by a number of hyphae interweaving + themselves into a knot, as if they were forming a Sclerotium. The + outer parts of the mass then differentiate as a wall or investment, + and the interior becomes a hollow, into which hyphal ends grow and + abstrict the spores. Much more complicated are the processes in a + large series of "fructifications," where the mycelium first develops a + densely packed mass of hyphae, all alike, in which labyrinths of + cavities subsequently form by separation of hyphae in the previously + homogeneous mass, and the hymenium covers the walls of these cavities + and passages as with a lining layer. Meanwhile differences in + consistency appear in various strata, and a dense outer protective + layer (peridium), soft gelatinous layers, and so on are formed, the + whole eventually attaining great complexity--e.g. puff-balls, + earth-stars and various _Phalloideae_. + + _Spore-Distribution._--Ordinary conidia and similarly abstricted dry + spores are so minute, light and numerous that their dispersal is + ensured by any current of air or water, and we also know that rats and + other burrowing animals often carry them on their fur; similarly with + birds, insects, slugs, worms, &c., on claws, feathers, proboscides, + &c., or merely adherent to the slimy body. In addition to these + accidental modes of dispersal, however, there is a series of + interesting adaptations on the part of the fungus itself. Passing over + the locomotor activity of zoospores (_Pythium_, _Peronospora_, + _Saprolegnia_) we often find spores held under tension in sporangia + (_Pilobolus_) or in asci (_Peziza_) until ripe, and then forcibly shot + out by the sudden rupture of the sporangial wall under the pressure of + liquid behind--mechanism comparable to that of a pop-gun, if we + suppose air replaced by watery sap. Even a single conidium, held tense + to the last moment by the elastic cell-wall, may be thus shot forward + by a spurt of liquid under pressure in the hypha abstricting it (e.g. + _Empusa_), and similarly with _basidiospores_ (_Coprinus_, _Agaricus_, + &c.). A more complicated case is illustrated by _Sphaerobolus_, where + the entire mass of spores, enclosed in its own peridium, is suddenly + shot up into the air like a bomb from a mortar by the elastic + retroversion of a peculiar layer which, up to the last moment, + surrounded the bomb, and then suddenly splits above, turns inside out, + and drives the former as a projectile from a gun. Gelatinous or + mucilaginous degenerations of cell-walls are frequently employed in + the interests of spore dispersal. The mucilage surrounding endospores + of _Mucor_, conidia of _Empusa_, &c., serves to gum the spore to + animals. Such gums are formed abundantly in pycnidia, and, absorbing + water, swell and carry out the spores in long tendrils, which emerge + for days and dry as they reach the air, the glued spores gradually + being set free by rain, wind, &c. In oidial chains (_Sclerotinia_) a + minute double wedge of wall-substance arises in the middle lamella + between each pair of contiguous oidia, and by its enlargement splits + the separating lamella. These disjunctors serve as points of + application for the elastic push of the swelling spore-ends, and as + the connecting outer lamella of cell-wall suddenly gives way, the + spores are jerked asunder. In many cases the slimy masses of spermatia + (_Uredineae_), conidia (_Claviceps_), basidiospores (_Phallus_, + _Coprinus_), &c., emit more or less powerful odours, which attract + flies or other insects, and it has been shown that bees carry the + fragrant oidia of _Sclerotinia_ to the stigma of _Vaccinium_ and + infect it, and that flies carry away the foetid spores of _Phallus_, + just as pollen is dispersed by such insects. Whether the strong odour + of trimethylamine evolved by the spores of _Tilletia_ attracts insects + is not known. + + The recent observations and exceedingly ingenious experiments of Falck + have shown that the sporophores of the Basidiomycetes--especially the + large sporophores of such forms as _Boletus_, _Polyporus_--contain + quantities of reserve combustible material which are burnt up by the + active metabolism occurring when the fruit-body is ripe. By this means + the temperature of the sporophore is raised and the difference between + it and the surrounding air may be one of several degrees. As a result + convection currents are produced in the air which are sufficient to + catch the basidiospores in their fall and carry them, away from the + regions of comparative atmospheric stillness near the ground, to the + upper air where more powerful air-currents can bring about their wide + distribution. + +_Classification._--It has been accepted for some time now that the +majority of the fungi proper fall into three main groups, the +Phycomycetes, Ascomycetes and Basidiomycetes, the Schizomycetes and +Myxomycetes (Mycetozoa) being considered as independent groups not +coming under the true fungi. + +The chief schemes of classification put forward in detail have been +those of P.A. Saccardo (1882-1892), of Oskar Brefeld and Von Tavel +(1892), of P.E.L. Van Tieghem (1893) and of J. Schroeter (1892). The +scheme of Brefeld, which was based on the view that the Ascomycetes and +Basidiomycetes were completely asexual and that these two groups had +been derived from one division (Zygomycetes) of the Phycomycetes, has +been very widely accepted. The recent work of the last twelve years has +shown, however, that the two higher groups of fungi exhibit distinct +sexuality, of either a normal or reduced type, and has also rendered +very doubtful the view of the origin of these two groups from the +Phycomycetes. The real difficulty of classification of the fungi lies in +the polyphyletic nature of the group. There is very little doubt that +the primitive fungi have been derived by degradation from the lower +algae. It appears, however, that such a degradation has occurred not +only once in evolution but on several occasions, so that we have in the +Phycomycetes not a series of naturally related forms, but groups which +have arisen perfectly independently of one another from various groups +of the algae. It is also possible in the absence of satisfactory +intermediate forms that the Ascomycetes and Basidiomycetes have also +been derived from the algae independently of the Phycomycetes, and +perhaps of one another. + +A natural classification on these lines would obviously be very +complicated, so that in the present state of our knowledge it will be +best to retain the three main groups mentioned above, bearing in mind +that the Phycomycetes especially are far from being a natural group. The +following gives a tabular survey of the scheme adopted in the present +article: + + A. PHYCOMYCETES. Alga-like fungi with unicellular thallus and + well-marked sexual organs. + + CLASS I.--Oomycetes. Mycelium usually well developed, but sometimes + poor or absent. Sexual reproduction by oogonia and antheridia; asexual + reproduction by zoospores or conidia. + + 1. Monoblepharidineae. Mycelium present, antheridia with + antherozoids, oogonium with single oosphere: Monoblepharidaceae. + + 2. Peronosporineae. Mycelium present; antheridia but no + antherozoids; oogonia with one or more oospheres: Peronosporaceae, + Saprolegniaceae. + + 3. Chytridineae. Mycelium poorly developed or absent; oogonia and + antheridia (without antherozoids) known in some cases; zoospores + common: Chytridiaceae. Ancylistaceae. + + CLASS II.--Zygomycetes. Mycelium well developed; sexual reproduction + by zygospores; asexual reproduction by sporangia and conidia. + + 1. Mucorineae. Sexual reproduction as above, asexual by sporangia or + conidia or both: Mucoraceae. Mortierellaceae, Chaetocladiaceae, + Piptocephalidaceae. + + 2. Entomophthorineae. Sexual reproduction typical but with sometimes + inequality of the fusing gametes (gametangia ?): Entomophthoraceae. + + B. HIGHER FUNGI. Fungi with segmental thallus; sexual reproduction + sometimes with typical antheridia and oogonia (ascogonia) but usually + much reduced. + + CLASS I.--Ustilaginales. Forms with septate thallus, and reproduction + by chlamydospores which on germination produce sporidia; sexuality + doubtful. + + CLASS II.--Ascomycetes. Thallus septate; spores developed in special + type of sporangium, the ascus, the number of spores being usually + eight. Sexual reproduction sometimes typical, usually reduced. + + Exoascineae, Saccharomycetineae, Perisporinea, Discomycetes, + Pyrenomycetes, Tuberineae, Laboulbeniineae. + + CLASS III.--Basidiales. Thallus septate. Conidia (basidiospores) borne + in fours on a special conidiophore, the basidium. Sexual reproduction + always much reduced. + + 1. Uredineae. Life-history in some cases very complex and with + well-marked sexual process and alternation of generations, in others + much reduced; basidium (promycelium) derived usually from a + thick-walled spore (teleutospore). + + 2. Basidiomycetes. Life-history always very simple, no well-marked + alternation of generations; basidium borne directly on the mycelium. + + (A) Protobasidiomycetes. Basidia septate. Auriculariaceae, + Pilacreaceae, Tremellinaceae. + + (B) Autobasidiomycetes. Basidia non-septate. Hymenomycetes, + Gasteromycetes. + +A. PHYCOMYCETES.--Most of the recent work of importance in this group +deals with the cytology of sexual reproduction and of spore-formation, +and the effect of external conditions on the production of reproductive +organs. + + _Monoblepharidaceae_ consists of a very small group of aquatic forms + living on fallen twigs in ponds and ditches. Only one genus, + _Monoblepharis_, can certainly be placed here, though a somewhat + similar genus, _Myrioblepharis_, with a peculiar multiciliate zoospore + like that of Vaucheria, is provisionally placed in the same group. + _Monoblepharis_ was first described by Cornu in 1871, but from that + time until 1895 when Roland Thaxter described several species from + America the genus was completely lost sight of. _Monoblepharis_ has + oogonia with single oospheres and antheridia developing a few amoeboid + uniciliate antherozoids; these creep to the opening of the oogonium + and then swim in. The resemblance between this genus and _Oedogonium_ + among the algae is very striking, as is also that of _Myrioblepharis_ + and _Vaucheria_. + + _Peronosporaceae_ are a group of endophytic parasites--about 100 + species--of great importance as comprising the agents of "damping off" + disease (_Pythium_), vine-mildew (_Plasmopara_), potato disease + (Phytophthora), onion-mildew (_Peronospora_). _Pythium_ is a + semi-aquatic form attacking seedlings which are too plentifully + supplied with water; its hyphae penetrate the cell-walls and rapidly + destroy the watery tissues of the living plant; then the fungus lives + in the dead remains. When the free ends of the hyphae emerge again + into the air they swell up into spherical bodies which may either fall + off and behave as conidia, each putting out a germ-tube and infecting + the host; or the germ-tube itself swells up into a zoosporangium which + develops a number of zoospores. In the rotting tissues branches of the + older mycelium similarly swell up and form antheridia and oogonia + (fig. 4). The contents of the antheridium are not set free, but that + organ penetrates the oogonium by means of a narrow outgrowth, the + fertilizing tube, and a male nucleus then passes over into the single + oosphere, which at first multinucleate becomes uninucleate before + fertilization. _Pythium_ is of interest as illustrating the dependence + of zoospore-formation on conditions and the indeterminate nature of + conidia. The other genera are more purely parasitic; the mycelium + usually sends haustoria into the cells of the host and puts out + branched, aerial conidiophores through the stomata, the branches of + which abstrict numerous "conidia"; these either germinate directly or + their contents break up into zoospores (fig. 5). The development of + the "conidia" as true conidial spores or as zoosporangia may occur in + one and the same species (_Cystopus candidus_, _Phytophthora + infestans_) as in _Pythium_ described above; in other cases the direct + conidial germination is characteristic of genera--e.g. _Peronospora_; + while others emit zoospores--e.g. _Plasmopara_, &c. In _Cystopus_ + (_Albugo_) the "conidia" are abstricted in basipetal chain-like series + from the ends of hyphae which come to the surface in tufts and break + through the epidermis as white pustules. Each "conidium" contains + numerous nuclei and is really a zoosporangium, as after dispersal it + breaks up into a number of zoospores. The Peronosporaceae reproduce + themselves sexually by means of antheridia and oogonia as described in + _Pythium_. In _Cystopus Bliti_ the oosphere contains numerous nuclei, + and all the male nuclei from the antheridium pass into it, the male + and female nuclei then fusing in pairs. We thus have a process of + "multiple fertilization"; the oosphere really represents a large + number of undifferentiated gametes and has been termed a coenogamete. + Between _Cystopus Bliti_ on the one hand and _Pythium de Baryanum_ on + the other a number of cytologically intermediate forms are known. The + oospore on germination usually gives origin to a zoosporangium, but + may form directly a germ tube which infects the host. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 4.--Fertilization of the Peronosporeae. After Wager. + + 1, _Peronospora parasitica_. Young multinucleate oogonium (og) and + antheridium (an). + + 2, _Albugo candida_. Oogonium with the central uninucleate oosphere + and the fertilizing tube (a) of the antheridium which introduces the + male nucleus. + + 3, The same. Fertilized egg-cell (o) surrounded by the periplasm + (p).] + + [Illustration: FIG. 5.--_Phytophthora infestans_. Fungus of Potato + Disease. + + A, B, Section of Leaf of Potato with sporangiophores of + _Phytophthora infestans_ passing through the stomata D, on the under + surface of the leaf. + + E, Sporangia. + + F, G, H, J, Further development of the sporangia. + + K, Germination of the zoospores formed in the sporangia. + + L, M, N, Fertilization of the oogonium and development of the + oospore in _Peronospora_.] + + _Saprolegniaceae_ are aquatic forms found growing usually on dead + insects lying in water but occasionally on living fish (e.g. the + salmon disease associated with _Saprolegnia ferax_). The chief genera + are _Saprolegnia, Achlya, Pythiopsis, Dictyuchus, Aplanes._ Motile + zoospores which escape from the zoosporangium are present except in + Aplanes. The sexual reproduction shows all transitions between forms + which are normally sexual, like the Peronosporaceae, to forms in which + no antheridium is developed and the oospheres develop + parthenogenetically. The oogonia, unlike the Peronosporaceae, contain + more than one oosphere. Klebs has shown that the development of + zoosporangia or of oogonia and pollinodia respectively in + _Saprolegnia_ is dependent on the external conditions; so long as a + continued stream of suitable food-material is ensured the mycelium + grows on without forming reproductive organs, but directly the + supplies of nitrogenous and carbonaceous food fall below a certain + degree of concentration sporangia are developed. Further reduction of + the supplies of food effects the formation of oogonia. This explains + the sequence of events in the case of a _Saprolegnia_-mycelium + radiating from a dead fly in water. Those parts nearest the fly and + best supplied develop barren hyphae only; in a zone at the periphery, + where the products of putrefaction dissolved in the water form a + dilute but easily accessible supply, the zoosporangia are developed in + abundance; oogonia, however, are only formed in the depths of this + radiating mycelium, where the supplies of available food materials are + least abundant. + + _Chytridineae._--These parasitic and minute, chiefly aquatic, forms + may be looked upon as degenerate Oomycetes, since a sexual process and + feeble unicellular mycelium occur in some; or they may be regarded as + series of primitive forms leading up to higher members. There is no + means of deciding the question. They are usually included in + Oomycetes, but their simple structure, minute size, usually uniciliate + zoospores, and their negative characters would justify their retention + as a separate group. It contains less than 200 species, chiefly + parasitic on or in algae and other water-plants or animals, of various + kinds, or in other fungi, seedlings, pollen and higher plants. They + are often devoid of hyphae, or put forth fine protoplasmic filaments + into the cells of their hosts. After absorbing the cell-contents of + the latter, which it does in a few hours or days, the fungus puts out + a sporangium, the contents of which break up into numerous minute + swarm-spores, usually one-ciliate, rarely two-ciliate. Any one of + these soon comes to rest on a host-cell, and either pierces it and + empties its contents into its cavity, where the further development + occurs (_Olpidium_), or merely sends in delicate protoplasmic + filaments (_Rhizophydium_) or a short hyphal tube of, at most, two or + three cells, which acts as a haustorium, the further development + taking place outside the cell-wall of the host (_Chytridium_). In some + cases resting spores are formed inside the host (_Chytridium_), and + give rise to zoosporangia on germination. In a few species a sexual + process is described, consisting in the conjugation of similar cells + (_Zygochytrium_) or the union of two dissimilar ones (_Polyphagus_). + In the development of distinct antheridial and oogonial cells the + allied Ancylistineae show close alliances to _Pythium_ and the + Oomycetes. On the other hand, the uniciliate zoospores of _Polyphagus_ + have slightly amoeboid movements, and in this and the + pseudopodium-like nature of the protoplasmic processes, such forms + suggest resemblances to the Myxomycetes. Opinions differ as to whether + the Chytridineae are degraded or primitive forms, and the group still + needs critical revision. Many new forms will doubtless be discovered, + as they are rarely collected on account of their minuteness. Some + forms cause damping off of seedlings--e.g. _Olpidium Brassicae_; + others discoloured spots and even tumour-like swellings--e.g. + _Synchytium Scabiosae_, _S. Succisae_, _Urophlyctis_, &c., on higher + plants. Analogies have been pointed out between Chytridiaceae and + unicellular algae, such as Chlorosphaeraceae, Protococcaceae, + "Palmellaceae," &c., some of which are parasitic, and suggestions may + be entertained as to possible origin from such algae. + + The _Zygomycetes_, of which about 200 species are described, are + especially important from a theoretical standpoint, since they + furnished the series whence Brefeld derived the vast majority of the + fungi. They are characterized especially by the zygospores, but the + asexual organs (sporangia) exhibit interesting series of changes, + beginning with the typical sporangium of _Mucor_ containing numerous + endospores, passing to cases where, as in _Thamnidium_, these are + accompanied with more numerous small sporangia (sporangioles) + containing few spores, and thence to _Chaetocladium_ and + _Piptocephalis_, where the sporangioles form but one spore and fall + and germinate as a whole; that is to say, the monosporous sporangium + has become a conidium, and Brefeld regarded these and similar series + of changes as explaining the relation of ascus to conidium in higher + fungi. According to his view, the ascus is in effect the sporangium + with several spores, the conidium the sporangiole with but one spore, + and that not loose but fused with the sporangiole wall. On this basis, + with other interesting morphological comparisons, Brefeld erected his + hypothesis, now untenable, that the Ascomycetes and Basidiomycetes + diverge from the Zygomycetes, the former having particularly + specialized the ascus (sporangial) mode of reproduction, the latter + having specialized the conidial (indehiscent one-spored sporangiole) + mode. In addition to sporangia and the conidial spores referred to, + some Mucorini show a peculiar mode of vegetative reproduction by means + of gemmae or chlamydospores--i.e. short segments of the hyphae become + stored with fatty reserves and act as spores. The gemmae formed on + submerged Mucors may bud like a yeast, and even bring about alcoholic + fermentation in a saccharine solution. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 6.--_Mucor Mucedo._ Different stages in the formation and + germination of the zygospore. (After Brefeld, 1-4. 5 from v. Tavel, + _Pilze_.) + + 1, Two conjugating branches in contact. + + 2, Septation of the conjugating cells (a) from the suspensors (b). + + 3, More advanced stage, the conjugating cells (a) are still distinct + from one another; the warty thickenings of their walls have + commenced to form. + + 4, Ripe zygospore (b) between the suspensors (a). + + 5, Germinating zygospore with a germ-tube bearing a sporangium.] + + The segments of the hyphae in this group usually contain several + nuclei. At the time of sporangial formation the protoplasm with + numerous nuclei streams into the swollen end of the sporangiophore and + there becomes cut off by a cell-wall to form the sporangium. The + protoplasm then becomes cut up by a series of clefts into a number of + smaller and smaller pieces which are unicellular in _Pilobolus_, + multicellular in _Sporodinia_. These then become surrounded by a + cell-wall and form the spores. This mode of spore-formation is totally + different from that in the ascus; hence one of the difficulties of the + acceptance of Brefeld's view of the homology of ascus and sporangium. + The cytology of zygospore-formation is not known in detail; the + so-called gametes which fuse are multinucleate and are no doubt of the + nature of gametangia. The fate of these nuclei is doubtful, probably + they fuse in pairs (fig. 6). + + Blakeslee has lately made some very important observations of the + Zygomycetes. It is well known that while in some forms, e.g. + _Spordinia_, zygospores are easily obtained, in others, e.g. most + species of _Mucor_, they are very erratic in their appearance. This + has now been explained by Blakeslee, who finds that the Mucorinae can + be divided into two groups, termed homothallic and heterothallic + respectively. In the first group zygospores can arise by the union of + branches from the _same_ mycelium and so can be produced by the growth + from a single spore; this group includes _Spordinia grandis_, + _Spinellus fusiger_, some species of _Mucor_, &c. The majority of + forms, however, fall into the heterothallic group, in which the + association of branches from two mycelia _different in nature_ is + necessary for the formation of zygospores. These structures cannot + then be produced from the product of a single spore nor even from the + thalli derived from _any_ two spores. The two kinds of thalli + Blakeslee considers to have a differentiation of the nature of sex and + he distinguishes them as ( + ) and (-) forms; the former being usually + distinguished by a somewhat greater luxuriance of growth. + + The classification of the Mucorini depends on the prevalence and + characters of the conidia, and of the sporangia and zygospores--e.g. + the presence or absence of a columella in the former, the formation of + an investment round the latter. Most genera are saprophytes, but + some--_Chaetocladium_, _Piptocephalis_--are parasites on other + Mucorini, and one or two are associated casually with the rotting of + tomatoes and other fruits, bulbs, &c., the fleshy parts of which are + rapidly destroyed if once the hyphae gain entrance. Even more + important is the question of mycosis in man and other animals, + referred to species of _Mucor_, and investigated by Lucet and + Costantin. Klebs has concluded that transpiration is the important + factor in determining the formation of sporangia, while + zygote-development depends on totally different conditions; these + results have been called in question by Falck. + + The _Entomophthoraceae_ contain three genera, _Empusa_, + _Entomophthora_ and _Basidiobolus_. The two first genera consist of + forms which are parasitic on insects. _Empusa Muscae_ causes the + well-known epidemic in house-flies during the autumn; the dead, + affected flies are often found attached to the window surrounded by a + white halo of conidia. _B. ranarum_ is found in the alimentary canal + of the frog and growing on its excrement. In these three genera the + conidia are cast off with a jerk somewhat in the same way as the + sporangium of _Pilobolus_. + +B. HIGHER FUNGI.--Now that Brefeld's view of the origin of these forms +from the Zygomycetes has been overthrown, the relationship of the higher +and lower forms of fungi is left in obscurity. The term _Eumycetes_ is +sometimes applied to this group to distinguish them from the +Phycomycetes, but as the same name is also applied to the fungi as a +whole to differentiate them from the Mycetozoa and Bacteria, the term +had best be dropped. The Higher Fungi fall into three groups: the +_Ustilaginales_, of doubtful position, and the two very sharply marked +groups _Basidiales_ and _Ascomycetes_. + +[Illustration: From Vine's _Students' Text Book of Botany_, by +permission of Swan Sonnenschein & Co. + +FIG. 7.--Germinating resting-gonidia. A, of _Ustilago receptaculorum_; +B, of _Tilletia Caries_. + + sp, The gonidium. + + pm, The promycelium. + + d, The sporidia: in B the sporidia have coalesced in pairs at v.] + + I. _Ustilaginales._--This includes two families Ustilaginaceae (smuts) + and Tilletiaceae (bunts). The bunts and smuts which damage our grain + and fodder plants comprise about 400 species of internal parasites, + found in all countries on herbaceous plants, and especially on + Monocotyledons. They are remarkable for their dark spores developed in + gall-like excrescences on the leaves, stems, &c., or in the fruits of + the host. The discovery of the yeast-conidia of these fungi, and their + thorough investigation by Brefeld, have thrown new lights on the + group, as also have the results elucidating the nature of the ordinary + dark spores--smuts, bunt, &c.--which by their mode of origin and + development are chlamydospores. When the latter germinate a slender + "promycelium" is put out; in _Ustilago_ and its allies this is + transversely septate, and bears lateral conidia (sporidia); in + _Tilletia_ and its allies non-septate, and bears a terminal tuft of + conidia (sporidia) (fig. 7). Brefeld regarded the promycelium as a + kind of _basidium_, bearing lateral or terminal conidia (comparable to + _basidiospores_), but since the number of basidiospores is not fixed, + and the basidium has not yet assumed very definite morphological + characters, Brefeld termed the group _Hemibasidii_, and regarded them + as a half-way stage in the evolution of the true Basidiomycetes from + Phycomycetes, the _Tilletia_ type leading to the true basidium + (Autobasidium), the _Ustilago_ type to the protobasidium, with lateral + spores; but this view is based on very poor evidence, so that it is + best to place these forms as a separate group, the _Ustilaginales_. + The yeast-conidia, which bud off from the conidia or their resulting + mycelium when sown in nutrient solutions, are developed in successive + crops by budding exactly as in the yeast plant, but they cannot + ferment sugar solutions. It is the rapid spread of these yeast-conidia + in manure and soil waters which makes it so difficult to get rid of + smuts, &c., in the fields, and they, like the ordinary conidia, + readily infect the seedling wheat, oats, barley or other cereals. + Infection in these cases occurs in the seedling at the place where + root and shoot meet, and the infecting hypha having entered the plant + goes on living in it and growing up with it as if it had no parasitic + action at all. When the flowers form, however, the mycelium sends + hyphae into the young ovaries and rapidly replaces the stores of sugar + and starch, &c., which would have gone to make the grain, by the + soot-like mass of spores so well known as smut, &c. These spores + adhere to the grain, and unless destroyed, by "steeping" or other + treatment, are sown with it, and again produce sporidia and + yeast-conidia which infect the seedlings. In other species the + infection occurs through the style of the flower, but the fungus after + reaching the ovule develops no further during that year but remains + dormant in the embryo of the seed. On germination, however, the fungus + behaves in the same way as one which has entered in the seedling + stage. The cytology of these forms is very little known; Dangeard + states that there is a fusion of two nuclei in the chlamydospore, but + this requires confirmation. Apart from this observation there is no + other trace of sexuality in the group. + + II. _Ascomycetes._--This, except in the case of a few of the simpler + forms, is a very sharply marked group characterized by a special type + of sporangium, the ascus. In the development of the ascus we find two + nuclei at the base which fuse together to form the single nucleus of + the young _ascus_. The single nucleus divides by three successive + divisions to form eight nuclei lying free in the protoplasm of the + ascus. Then by a special method, described first by Harper, a mass of + protoplasm is cut out round each nucleus; thus eight uninucleate + ascospores are formed by free-cell formation. The protoplasm remaining + over is termed _epiplasm_ and often contains glycogen (fig. 8). In + some cases nuclear division is carried further before spore-formation + occurs, and the number of spores is then 16, 32 and 64, &c.; in a few + cases the number of spores is less than eight by abortion of some of + the eight nuclei. The ascus is thus one of the most sharply + characterized structures among the fungi. + + In some forms we find definite male and female sexual organs + (_Sphaerotheca_, _Pyronema_, &c.), in others the antheridium is + abortive or absent, but the ascogonium (oogonium) is still present and + the female nuclei fuse in pairs (_Lachnea stercorea_, _Humaria + granulata_, _Ascobolus furfuraceus_); while in other forms ascogonium + and antheridium are both absent and fusion occurs between vegetative + nuclei (_Humaria rutilans_, and probably the majority of other forms). + In other cases the sexual fusion is apparently absent altogether, as + in _Exoascus_. In the first case (fig. 9) we have a true sexual + process, while in the second and third cases we have a _reduced_ + sexual process in which the fusion of other nuclei has replaced the + fusion of the normal male and female nuclei. It is to be noted that + all the forms exhibit the fusion of nuclei in the ascus, so that those + with the normal or reduced sexual process described above have two + nuclear fusions in their life-history. The advantage or significance + of the second (ascus) fusion is not clearly understood. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 8.--Development of the Ascus. + + A-C, _Pyronema confluens_. (After Harper.) + + D, Young ascus of _Boudiera_ with eight spores. (After Claussen.)] + + The group of the Hemiasci was founded by Brefeld to include forms + which were supposed to be a connecting link between Phycomycetes and + Ascomycetes. As mentioned before, the connexion between these two + groups is very doubtful, and the derivation of the ascus from an + ordinary sporangium of the Zygomycetes cannot be accepted. The + majority of the forms which were formerly included in this group have + been shown to be either true Phycomycetes (like _Ascoidea_) or true + Ascomycetes (like _Thelebolus_). _Eremascus_ and _Dipodascus_, which + are often placed among the Hemiasci, possibly do not belong to the + Ascomycetes series at all. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + Fig. 9.--_Sphaerotheca Castagnei_. Fertilization and Development of + the Perithecium. (After Harper.) + + 1, Oogonium (og) with the antheridial branch (az) applied to its + surface + + 2, Separation of antheridium (an). + + 3, Passage of the antheridial nucleus towards that of the oogonium. + + 4, Union of the nuclei. + + 5, Fertilized oogonium surrounded by two layers of hyphae derived + from the stalk-cell (st). + + 6, The multicellular ascogonium derived by division from the + oogonium; the terminal cell with the two nuclei (as) gives rise to + the ascus.] + + _Exoascaceae_ are a small group of doubtful extent here used to + include _Exoascus_, _Taphrina_, _Ascorticium_ and _Endomyces_. The + mycelium is very much reduced in extent. The asci are borne directly + on the mycelium and are therefore fully exposed, being devoid from the + beginning of any investment. The _Taphrineae_, which include + _Exoascus_ and _Taphrina_, are important parasites--e.g. pocket-plums + and witches' brooms on birches, &c., are due to their action (fig. + 10). _Exoascus_ and _Ascorticium_ present interesting parallels to + _Exobasidium_ and _Corticium_ among the Basidiomycetes. + + _Saccharomycetaceae_ include the well-known yeasts which belong mainly + to the genus _Saccharomyces_. They are characterized by their + unicellular nature, their power of rapid budding, their capacity for + fermenting various sugars, and their power of forming endogenous + spores. The sporangium with its endogenous spores has been compared + with an ascus, and on these grounds the group is placed among the + Ascomycetes--a very doubtful association. The group has attained an + importance of late even beyond that to which it was brought by + Pasteur's researches on alcoholic fermentation, chiefly owing to the + exact results of the investigations of Hansen, who first applied the + methods of pure cultures to the study of these organisms, and showed + that many of the inconsistencies hitherto existing in the literature + were due to the coexistence in the cultures of several species or + races of yeasts morphologically almost indistinguishable, but + physiologically very different. About fifty species of _Saccharomyces_ + are described more or less completely, but since many of these cannot + be distinguished by the microscope, and some have been found to + develop physiological races or varieties under special conditions of + growth, the limits are still far too ill-defined for complete + botanical treatment of the genus. A typical yeast is able to develop + new cells by budding when submerged in a saccharine solution, and to + ferment the sugar--i.e. so to break up its molecules that, apart from + small quantities used for its own substance, masses of it out of all + proportion to the mass of yeast used become resolved into other + bodies, such as carbon dioxide and alcohol, the process requiring + little or no oxygen. Brefeld regards the budding process as the + formation of conidia. Under other conditions, of which the temperature + is an important one, the nucleus in the yeast-cell divides, and each + daughter-nucleus again, and four spores are formed in the mother cell, + a process obviously comparable to the typical development of + ascospores in an ascus. Under yet other conditions the quiescent + yeast-cells floating on the surface of the fermented liquor grow out + into elongated sausage-shaped or cylindrical cells and branching + cell-series, which mat together into mycelium-like veils. At the + bottom of the fermented liquor the cells often obtain fatty contents + and thick walls, and behave as resting cells (chlamydospores). The + characters employed by experts for determining a species of yeast are + the sum of its peculiarities as regards form and size: the shapes, + colours, consistency, &c., of the colonies grown on certain definite + media; the optimum temperature for spore-formation, and for the + development of the "veils"; and the behaviour as regards the various + sugars. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 10.--_Taphrina Pruni._ Transverse section through the epidermis + of an infected plum. Four ripe asci, a1, a2, with eight spores, a3, + a4, with yeast-like conidia abstricted from the spores. After + Sadebeck. + + st, Stalk-cells of the asci. + + m, Filaments of the mycelium cut transversely. + + cut, Cuticle. + + sp, Epidermis.] + + The following summary of some of the principal characteristics of + half-a-dozen species will serve to show how such peculiarities can be + utilized for systematic purposes: + + +---------------------+---------------------------------+----------------------------------+----------------------------------------+ + | | Optimum | Characters of | | + | Species. | Temperature for +----------+-----------+-----------+ Sugars Fermented and | + | +----------------+----------------+ Fermenta-| Cells. | Spores. | Products, &c. | + | | Spores. | Veils. | tion | | | | + +---------------------+----------------+----------------+----------+-----------+-----------+----------------------------------------+ + |_S. cereviseae I_. | 30 deg. | 20 deg.-28 deg.| High | Rounded | Globoid | / Inverts maltose and saccharose and | + |_S. Pastorianus I_ | 27 deg.-5 deg. | 26 deg.-28 deg.| Low | Rounded | Globoid |< form alcohol 4-6 vol. %. | + |_S. ellipsoideus_ | 25 deg. | 33 deg.-34 deg.| Low | Rounded | Globoid | \ | + | | | | | | | | + |_S. anomalus_ | 28 deg.-31 deg.| ? | High | Elliptical| Hat-shaped| Ditto, and evolves a fragrant ether. | + | | | | | | | + |_S. Ludwigii_ | 30 deg.-31 deg.| ? | ? | Elongated | Globoid | Will not invert maltose. | + | | | | | | | + |_S. membranaefaciens_| 30 deg. | ? | High | Elongated | Globoid | Inverts neither maltose nor saccharose.| + +---------------------+----------------+----------------+----------+-----------+-----------+----------------------------------------+ + + Two questions of great theoretical importance have been raised over + and over again in connexion with yeasts, namely, (1) the morphological + one as to whether yeasts are merely degraded forms of higher fungi, as + would seem implied by their tendency to form elongated, hypha-like + cells in the veils, and their development of "ascospores" as well as + by the wide occurrence of yeast-like "sprouting forms" in other fungi + (e.g. _Mucor_, Exoasci, Ustilagineae, higher Ascomycetes and + Basidiomycetes); and (2) the question as to the physiological nature + and meaning of fermentation. With regard to the first question no + satisfactory proof has as yet been given that Saccharomycetes are + derivable by culture from any higher form, the recent statements to + that effect not having been confirmed. At the same time there are + strong grounds for insisting on the resemblances between _Endomyces_, + a hyphal fungus bearing yeast-like asci, and such a form as + _Saccharomyces anomalus_. Concerning the second question, the recent + investigations of Buchner and others have shown that a ferment + (zymase) can be extracted from yeast-cells which causes sugar to break + up into carbon dioxide and alcohol. It has since been shown by Buchner + and Albert that yeast-cells which have been killed by alcohol and + ether, or with acetone, still retain the enzyme. Such material is far + more active than the zymase obtained originally by Buchner from the + expressed juice of yeast-cells. Thus alcoholic fermentation is brought + into line with the other fermentations. + + _Schizosaccharomyces_ includes a few species in which the cells do not + "bud" but become elongated and then divide transversely. In the + formation of sporangia two cells fuse together by means of outgrowths, + in a manner very similar to that of _Spirogyra_; sometimes, however, + the wall between two cells merely breaks down. The fused cell becomes + a sporangium, and in it eight spores are developed. In certain cases + single cells develop parthenogenetically, without fusion, each cell + producing, however, only four spores. In _Zygosaccharomyces_ described + by Barker (1901) we have a form of the usual sprouting type, but here + again there is a fusion of two cells to form a sporangium. + + _Cytology._--The study of the nucleus of yeast-cells is rendered + difficult by the presence of other deeply staining granules termed by + Guillermond _metachromatic granules_. These have often been mistaken + for nuclei and have to be carefully distinguished by differential + stains. In the process of budding the nucleus divides apparently by a + process of direct division. In the formation of spores the nucleus of + the cell divides, the protoplasm collects round the nuclei to form the + spores by free-cell formation; the protoplasm (epiplasm) not used in + this process becomes disorganized. A fusion of nuclei was originally + described by Jansens and Leblanc, but it was observed neither by Wager + nor Guillermond and is probably absent. In _Schizosaccharomyces_ and + _Zygosaccharomyces_, however, we have a fusion of nuclei in connexion + with the conjugation of cells which precedes sporangium-formation. The + theory may be put forward that the ordinary forms have been derived + from sexual forms like _Schizosaccharomyces_ and _Zygosaccharomyces_ + by a loss of sexuality, the sporangium being formed + parthenogenetically without any nuclear fusion. This suggests a + possible relationship to _Eremascus_, which can only doubtfully be + placed in the Ascomycetes (_vide supra_). + + _Carpoascomycetes._--The other divisions of the Ascomycetes may be + distinguished as Carpoascomycetes because they do not bear the asci + free on the mycelium but enclosed in definite fruit bodies or + ascocarps. The ascocarps can be distinguished into two portions, a + mass of sterile or vegetative hyphae forming the main mass of the + fruit body, and surrounding the fertile ascogenous hyphae which bear + at their ends the asci. When the ascogonium (female organ) is present + the ascogenous hyphae arise from it, with or without its previous + fusion with an antheridium. In other cases the ascogenous hyphae arise + directly from the vegetative hyphae. In connexion with this condition + of reduction a fusion of nuclei has been observed in _Humaria + rutilans_ and is probably of frequent occurrence. The asci may be + derived from the terminal cell of the branches of the ascogenous + hyphae, but usually they are derived from the penultimate cell, the + tip curving over to form the so-called crozier. By this means the + ascus cell is brought uppermost, and after the fusion of the two + nuclei it develops enormously and produces the ascospores. The + ascospores escape from the asci in various ways, sometimes by a + special ejaculation-mechanism. The Ascomycetes, at least the + Carpoascomycetes, exhibit a well-marked alternation of sexual and + asexual generations. The ordinary mycelium is the gametophyte since it + bears the ascogonia and antheridia when present; the ascogenous hyphae + with their asci represent the sporophyte since they are derived from + the fertilized ascogonium. The matter is complicated by the apogamous + transition from gametophyte to sporophyte in the absence of the + ascogonium; also by the fact that there are normally two fusions in + the life-history as mentioned earlier. If there are two fusions one + would expect two reductions, and Harper has suggested that the + division of the nuclei into eight in the ascus, instead of into four + spores as in most reduction processes, is associated with a _double_ + reduction process in the ascus. Miss Fraser in _Humaria rutilans_ + finds two reductions: a normal synaptic reduction in the first nuclear + division of the ascus, and a peculiar reduction division termed + _brachymeiosis_ in the third ascus division. + + Various types of ascocarp are characteristic of the different + divisions of the Carpoascomycetes: the cleistothecium, apothecium and + perithecium. + + _Perisporineae._--This includes two chief families, Erysiphaceae and + Perisporiaceae. They are characterized by an ascocarp without any + opening to the exterior, the ascospores being set free by the decay or + rupture of the ascocarp wall; such a fruit-body is termed a + _cleistothecium_ (cleistocarp). The Erysiphaceae are a sharply marked + group of forms which live as parasites. They form a superficial + mycelium on the surface of the plant, the hyphae not usually + penetrating the tissues but merely sending haustoria into the + epidermal cells. Only in rare cases is the mycelium intercellular. + Owing to their appearance they go by the popular name of mildews. + _Sphaerotheca Humuli_ is the well known hop-mildew, _Sphaerotheca + Mors-Uvae_ is the gooseberry mildew, the recent advent of which has + led to special legislation in Great Britain to prevent its spreading, + as when rampant it makes the culture of gooseberries impossible. + _Erysiphe_, _Uncinula_ and _Phyllactinia_ are other well-known genera. + The form of the fruit body, the difference and the nature of special + outgrowths upon it--the appendages--are characteristic of the various + genera. Besides peritheca the members of the Erysiphaceae possess + conidia borne in simple chains. De Bary brought forward very strong + evidence for the origin of the ascocarp in _Sphaerotheca_ and + _Erysiphe_ by a sexual process, but Harper in 1895 was the first to + prove conclusively, by the observation of the nuclear fusion, that + there was a definite fertilization in _Sphaerotheca Humuli_ by the + fusion of a male (antheridial) nucleus with a female, ascogonial + (oogonial) nucleus. Since then Harper has shown that the same process + occurs in _Erysiphe_ and _Phyllactinia_. + + [Illustration: FIG. 11.--Development of _Eurotium repens_. (After De + Bary.) + + A, Small portion of mycelium with conidiophore (c), and archicarp + (as). + + B, The spiral archicarp (as), with the antheridium (p). + + D, The same, beginning to be surrounded by the hyphae forming the + perithecium wall. + + D, The perithecium. + + E, F, Sections of young perithecia. + + w, Parietal cells. + + f, Pseudo-parenchyma. + + as, Ascogonium. + + G, An ascus. + + H, An ascospore.] + + The Perisporiaceae are saprophytic forms, the two chief genera being + _Aspergillus_ and _Penicillium_. The blue-green mould _P. crustaceum_ + and the green mould _A. herbariorium_ ( = _Eurotium herbariorum_) are + extraordinarily widely distributed, moulds being found on almost any + food-material which is exposed to the air. They have characteristic + conidiophores bearing numerous conidia, and also cleistothecia which + are spherical in form and yellowish in colour. The latter arise from + the crown of a spirally coiled archicarp (bearing an ascogonium at its + end) and a straight antheridium. Vegetative hyphae then grow up and + surround these and enclose them in a continuous sheath of plectenchyma + (fig. 11). It has lately been shown by Fraser and Chambers that in + _Eurotium_ both ascogonium and antheridium contain a number of nuclei + (i.e. are coenogametes), but that the antheridium disorganizes without + passing its contents into the ascogonium. There is apparently a + reduced sexual process by the fusion of the ascogonial (female) nuclei + in pairs. _Aspergillus Oryzae_ plays an important part in + saccharifying the starch of rice, maize, &c., by means of the abundant + diastase it secretes, and, in symbiosis with a yeast which ferments + the sugar formed, has long been used by the Japanese for the + preparation of the alcoholic liquor sake. The process has now been + successfully introduced into European commerce. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 12.--_Peziza aurantiaca._ (After Krombholz, nat. size.)] + + [Illustration: FIG. 13.--_Ascobolus furfuraceus._ Diagrammatic section + of the fructification. (After Janczewski.) + + m, Mycelium. + + c, Archicarp. + + l, Pollinodium. + + s, Ascogenous filaments. + + a, Asri. + + r, p, The sterile tissue from which the paraphyses h spring.] + + _Discomycetes._--Used in its widest sense this includes the + Hysteriaceae, Phacidiaceae, Helvellaceae, &c. The group is + characterized in general by the possession of an ascocarp which, + though usually a completely closed structure during the earlier stages + of development, at maturity opens out to form a bowl or saucer-shaped + organ, thus completely exposing the layer of asci which forms the + hymenium. Such an ascocarp goes by the name of _apothecium_. Owing to + the shape of the fruit-body many of these forms are known as + cup-fungi, the cup or apothecium often attaining a large size, + sometimes several inches across (fig. 12). Functional male and female + organs have been shown to exist in _Pyronema_ and _Boudiera_; in + _Lachnea stercorea_ both ascogonia and antheridia are present, but the + antheridium is non-functional, the ascogonial (female) nuclei fusing + in pairs; this is also the case in _Humaria granulata_ and _Ascobolus + furfuraceus_, where the antheridium is entirely absent. In _H. + rutilans_, however, both sexual organs are absent and the ascogenous + hyphae arise apogamously from the ordinary hyphae of the mycelim. In + all these cases the ascogonium and antheridium contain numerous + nuclei; they are to be looked upon as gametangia in which there is no + differentiation of gametes, and since they act as single gametes they + are termed coenogametes. In some forms as in _Ascobolus_ the + ascogonium is multicellular, the various cells communicating by pores + in the transverse walls (fig. 13). + + In the Helvellaceae there is no apothecium but a large irregular fruit + body which at maturity bears the asci on its surface. The development + is only slightly known, but there is some evidence for believing that + the fruit-body is closed in its very early stages. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 14.--Perithecium of Podospora fimiseda in longitudinal section + After v. Tavel. + + s, Asci. + + a, Paraphyses. + + e, Periphyses. + + m, Mycelial hyphae.] + + The genus _Peziza_ (in its widest sense) may be taken as the type of + the group. Most of them grow on living plants or on dead vegetable + remains, very often on fallen wood; a number, however, are found + growing on earth which is rich in humus. The genus _Sclerotinia_ may + be mentioned here; a number of forms have been investigated by + Woronin. The conidia are fragrant and are carried by bees to the + stigma of the bilberry; here they germinate with the pollen and the + hyphae pass with the pollen tubes down the style; the former infect + the ovules and produce sclerotia, therein reducing the fruits to a + mummified condition. From the sclerotia later the apothecium develops. + One species, _S. heteroica_, is _heteroecious_; the ascospores + infecting the leaves of _Vaccinium uliginosum_, while the conidia + which then arise infect only _Ledum palustre_. This is the only case + of heteroecism known in the vegetable kingdom outside the Uredineae. + + _Pyrenomycetes._--This is an extraordinarily large and varied group of + forms which mostly live parasitically or saprophytically on vegetable + tissue, but a few are parasitic on insect-larvae. The group is + characterized by a special type of ascocarp, the _perithecium_. This + is typically of a flask-shaped form opening with a small pore at the + top. The asci live at the bottom often mixed with paraphyses, while + the upper "neck" of the flask is lined with special hyphae, the + periphyses, which aid in the ejection of the spores (fig. 14). The + simpler forms bear the perithecia directly on the mycelium, but the + more highly developed forms often bear them on a special mycelial + development--the stroma, which is often of large size and special + shape and colour, and of dense consistence. The cytological details of + development of the perithecia are not well known; most of them appear + to develop their ascogenous hyphae in an apogamous way without any + connexion with an ascogonium. Besides the special ascocarps, accessory + reproductive organs are known in the majority of cases in the form of + conidia. + + _Tuberineae._--These are a small group of fungi including the + well-known truffles. They are found living saprophytically (in part + parasitically) underground in forests. The asci are developed in the + large dense fruit bodies (cleistothecia) and the spores escape by the + decay of the wall. The fruit-body is of complicated structure, but its + early stages of development are not known. Many of the fruit-bodies + have a pleasant flavour and are eaten under the name of truffles + (_Tuber brumale_ and other species). The exact life-history of the + truffle is not known. + + _Laboulbeniineae_ are a group of about 150 species of fungi found on + insects, especially beetles, and principally known from the researches + of Thaxter in America. The plant is a small, dark brown, erect + structure (receptacle) of a few cells, and 1-10 mm. high, attached to + the insect by the lowermost end (foot), and easily mistaken for a hair + or similar appendage of the insect. The receptacle ends above in + appendages, each consisting of one or a few cells, some of which are + the male organs, others the female organs, and others again may be + barren hairs. The male organ (antheridium) consists of a few cells, + the terminal one of which either abstricts from its end, or emits from + its interior the non-motile spermatia, reminding us of those of the + Florideae. The female organ is essentially a flask-shaped structure; + the neck of the flask growing out as the trichogyne, and the belly + composed of an axial carpogenic cell surrounded by investing cells, + and with one cell (trichophoric) between it and the trichogyne. These + three elements--trichogyne, trichophoric cell, and carpogenic + cell--are regarded as the procarp. The spermatia have been shown by + Thaxter to fuse with the trichogyne, after which the axial cell below + (carpogenic cell) undergoes divisions, and ultimately forms asci + containing ascospores, while cells investing this form a perithecium, + the whole structure reminding us essentially of the fructification of + a Pyrenomycete. Many modifications in details occur, and the plants + may be dioecious. No injury is done to the infested insects. It has + lately been shown that there is a fusion of nuclei in connexion with + ascus formation, so that there can be no doubt of the position of this + extraordinary group of plants among the Ascomycetes. The various cells + of these organisms are connected by large pits which are traversed by + thick protoplasmic threads connecting one cell with the next. In this + point and in their method of fertilization the Laboulbeniineae suggest + a possible relationship of Ascomycetes and the Red Algae. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 15.--_Armillaria mellea._ (After Ruhland.) + + A, Young basidium with the two primary nuclei. + + B, After fusion of the two nuclei. _Hypholoma appendiculatum_. + + C, A basidium before the four nuclei derived from the secondary + nucleus of the basidium have passed into the four basidiospores. + + D, Passage of a nucleus through the sterigma into the basidiospore.] + + _Basidiales._--This very large group of plants is characterized by the + possession of a special type of conidiophore--the basidium, which + gives its name to the group. The basidium is a unicellular or + multicellular structure from which four basidiospores arise as + outgrowths; it starts as a binucleate structure, but soon, like the + ascus, becomes uninucleate by the fusion of the two nuclei. Then two + successive nuclear divisions occur resulting in the formation of four + nuclei which later migrate respectively into the four basidiospores + (fig. 15). The Basidiales are further characterized by the complete + loss of normal sexuality, but at some time or other in the + life-history there takes place an association of two nuclei in a cell; + the two nuclei are derived from separate cells or possibly in some + cases are sister nuclei of the same cell. The two nuclei when once + associated are termed "conjugate" nuclei, and they always divide at + the same time, a half of each passing into each cell. This conjugate + condition is finally brought to a close by the nuclear fusion in the + basidium. Between the nuclear association and the nuclear fusion in + the basidium many thousands of cell generations may be intercalated. + This nuclear association of equivalent nuclei apparently represents a + reduced sexual process (like the fusion of female nuclei in _Humaria + granulata_ and of vegetative nuclei in _H. rutilans_, among the + Ascomycetes) in which, however, the actual fusion (normally, in a + sexual process, occurring immediately after association) is delayed + until the formation of the basidium. During the tetrad division in the + basidium nuclear reduction occurs. There is thus in all the Basidiales + an alternation of generations, obscured, however, by the apogamous + transition from the gametophyte to sporophyte. The sporophyte may be + considered to begin at the stage of nuclear association and end with + the nuclear reduction in the basidium. + + [Illustration: FIG. 16.--_Puccinia graminis._ + + A, Mass of teleutospores (t) on a leaf of couch-grass. + + e, Epidermis ruptured. + + b, Sub-epidermal fibres. (After De Bary.) + + B, Part of vertical section through leaf of Berberis vulgaris, with + a, aecidium fruits, p, peridium, and sp, spermogonia. (After Sachs.) + + C, Mass of uredospores (ur), with one teleutospore (t). + + sh, Sub-hymenial hyphae. (After De Bary.)] + + _Uredineae._--This is a large group of about 2000 forms. They are all + intercellular parasites living mostly on the leaves of higher plants. + Owing to the presence of oily globules of an orange-yellow or + rusty-red colour in their hyphae and spores they are termed + Rust-Fungi. They are distinguished from the other fungi and the rest + of the Basidiales by the great variety of the spores and the great + elaboration of the life-history to be found in many cases. Five + different kinds of spores may be present--teleutospores, sporidia ( = + basidiospores), aecidiospores, spermatia and uredospores (fig. 16). + The teleutospore, with the sporidia which arise from it, is always + present, and the division into genera is based chiefly on its + characters. The teleutospore puts forth on germination a four-celled + structure, the promycelium or basidium, and this bears later four + sporidia or basidiospores, one on each cell. When the sporidia infect + a plant the mycelium so produced gives origin to aecidiospores and + spermatia; the aecidiospores on infection produce a mycelium which + bears uredospores and later teleutospores. This is the life-history of + the most complicated forms, of the so-called _eu_ forms. In the + _opsis_ forms the uredospores are absent, the mycelium from the + aecidiospores producing directly the teleutospores. In _brachy_ and + _hemi_ the aecidiospores are absent, the mycelium from the sporidia + giving origin directly to the uredospores; the former possess + spermatia, in the latter they are absent. In _lepto_ and _micro_ forms + both aecidiospores and uredospores are absent, the sporidia producing + a mycelium which gives rise directly to teleutospores; in the _lepto_ + forms the teleutospores can germinate directly, in the _micro_ forms + only after a period of rest. We have thus a series showing a + progressive reduction in the complexity of the life-history, the + _lepto_ and _micro_ forms having a life-history like that of the + Basidiomycetes. The _eu_ and _opsis_ forms may exhibit the remarkable + phenomenon of heteroecism, i.e. the dependence of the fungus on two + distinct host-plants for the completion of the life-history. + Heteroecism is very common in this group and is now known in over one + hundred and fifty species. In all cases of heteroecism the sporidia + infect one host leading to the production of aecidiospores and + spermatia (if present), while the aecidiospores are only able to + infect another host on which the uredospores (if present) and the + teleutospores are developed. A few examples are appended: + + +-----------------------------+------------------+------------------+ + | Species. | Teleutospores on | Aecidiospores on | + +-----------------------------+------------------+------------------+ + | _Coleosporium Senecionis_ | _Pinus_ | _Senecio_ | + | _Melampsora Rostrupi_ | _Populus_ | _Mecurialis_ | + | _Pucciniastrum Goeppertiana_| _Vaccinium_ | _Abies_ | + | _Gymnosporangium Sabinae_ | _Juniperus_ | _Pyrus_ | + | _Uromyces Pisi_ | _Pisum, &c._ | _Euphorbia_ | + | _Puccinia graminis_ | _Triticum, &c._ | _Berberis_ | + | _P. dispersa_ | _Secale, &c._ | _Anchusa_ | + | _P. coronata_ | _Agrostis_ | _Rhamnus_ | + | _P. Ari-Phalaridis_ | _Phalaris_ | _Arum_ | + | _P. Caricis_ | _Carex_ | _Urtica_ | + | _Cronartium Ribicola_ | _Ribes_ | _Pinus_ | + | _Chrysomyxa Rhododendri_ | _Rhododendron_ | _Picea_ | + +-----------------------------+------------------+------------------+ + + Some of the Uredineae also exhibit the peculiarity of the development + of biologic forms within a single morphological species, sometimes + termed specialization of parasitism; this will be dealt with later + under the section Physiology. + + [Illustration: From Strasburger's _Lehrbuch der Botanik_, by + permission of Gustav Fischer. + + FIG. 17.--_Phragmidium Violaceum._ (After Blackman.) + + A, Portion of a young aecidium. + + st, Sterile cell. + + a, Fertile cells; at a2 the passage of a nucleus from the adjoining + cell is seen. + + B, Formation of the first spore-mother-cell (sm), from the basal + cell (a) of one of the rows of spores. + + C, A further stage in which from sm1 the first aecidiospore (a) and + the intercalary cell (z) have arisen. + + sm2, The second spore-mother-cell. + + D, Ripe aecidiospore.] + + _Cytology of Uredineae._--The study of the nuclear behaviour of the + cells of the Uredineae has thrown great light on the question of + sexuality. This group like the rest of the Basidiales exhibits an + association of nuclei at some point in its life-history, but unlike + the case of the Basidiomycetes the point of association in the + Uredineae is very well defined in all those forms which possess + aecidiospores. We find thus that in the _eu_ and _opsis_ forms the + association of nuclei takes place at the base of the aecidium which + produces the aecidiospores. There we find an association of nuclei + either by the fusion of two similar cells as described by Christmann + or by the migration of the nucleus of a vegetative cell into a special + cell of the aecidium. After this association the nuclei continue in + the conjugate condition so that the aecidiospores, the + uredospore-bearing mycelium, the uredospores and the young + teleutospores all contain two paired nuclei in their cells (fig. 17). + Before the teleutospore reaches maturity the nuclei fuse, and the + uninucleate condition then continues again until aecidium formation. + In the _hemi_, _brachy_, _micro_ and _lepto_ forms, which possess no + aecidium, we find that the association takes place at various points + in the ordinary mycelium but always before the formation of the + uredospores in the _hemi_ and _brachy_ forms, and before the formation + of teleutospores in _micro_ and _lepto_ form. Whether the association + of nuclei in the ordinary mycelium takes place by the migration of a + nucleus from one cell to another or whether two daughter nuclei become + conjugate in one cell, is not yet clear. The most reasonable + interpretation of the spermatia is that they are abortive male cells. + They have never been found to cause infection, and they have not the + characters of conidia; the large size of their nuclei, the reduction + of their cytoplasm and the absence of reserve material and their thin + cell wall all point to their being male gametes. Although in the forms + without aecidia the two generations are not sharply marked off from + one another, we may look up the generation with single nuclei in the + cells as the gametophyte and that with conjugate nuclei as the + sporophyte. The subjoined diagram will indicate the relationship of + the forms. + + _Basidiomycetes._--This group is characterized by its greatly reduced + life-history as compared with that of the _eu_ forms among the + Uredineae. All the forms have the same life-history as the _lepto_ + forms of that group, so that there is no longer any trace of sexual + organs. There is also a further reduction in that the basidium is not + derived from a teleutospore but is borne directly on the mycelium. + Formerly, before the relationship of promycelium and basidium were + understood, the Uredineae were considered as quite independent of the + Basidiomycetes. Later, however, these Uredineae were placed as a mere + subdivision of the Basidiomycetes. Although the Uredineae clearly lead + on to the Basidiomycetes, yet owing to their retaining in many cases + definite traces of sexual organs they are clearly a more primitive + group. Their marked parasitic habit also separates them off, so that + they are best included with the Basidiomycetes in a larger cohort + which may be called Basidiales. Most of Basidiomycetes are + characterized by the large sporophore on which the basidia with its + basidiospores are borne. + + [Illustration: From _Annals of Botany_, by permission of the Clarendon + Press. + + FIG. 18.] + + It must be clearly borne in mind that though the Basidiomycetes show + no traces of differentiated sexual organs yet, like the _micro_ and + _lepto_ forms of the Uredineae, they still show (in the association of + nuclei and later fusion of nuclei in the basidium), a reduced + fertilization which denotes their derivation, through the Uredineae, + from more typically sexual forms. No one has yet made out in any form + the exact way in which the association of nuclei takes place in the + group. The mycelium is always found to contain conjugate nuclei before + the formation of basidia, but the point at which the conjugate + condition arises seems very variable. Miss Nichols finds that it + occurs very soon after the germination of the spore in _Coprinus_, but + no fusion of cells or migration of nuclei was to be observed. + + _Protobasidiomycetes._--This, by far the smaller division of + Basidiomycetes, includes those forms which have a septate basidium. + There are three families--Auriculariaceae, Pilacreaceae and + Tremellinaceae. The first named contains a small number of forms with + the basidium divided like the promycelium of the Uredineae. They are + characterized by their gelatinous consistence and large size of their + sporophore. _Hirneola_ (_Auricularia_) _Auricula-Judae_ is the + well-known Jew's Ear, so named from the resemblance of the sporophore + to a human ear. + + [Illustration: FIG. 19.--_Amanita muscaria_. + + A, The young plant. + + B, The mature plant. + + C, Longitudinal section of mature plant. + + p, The _pileus_. + + g, The gills. + + a, The _annulus_, or remnant of _velum partiale_, + + v, Remains of _volva_ or _velum universale_. + + s, The stalk.] + + The Pilacreaceae are a family found by Brefeld to contain the genus + _Pilacre_. _P. Petersii_ has a transversely divided basidium as in + _Auriculariaceae_, but the basidia are surrounded with a peridium-like + sheath. The _Tremellinaceae_ are characterized by the possession of + basidia which are divided by two _vertical_ walls at right angles to + one another. From each of the four segments in the case of _Tremella_ + a long outgrowth arises which reaches to the surface of the hymenium + and bears the basidiospores. In _Dacryomyces_ only two outgrowths and + two spores are produced. + + _Autobasidiomycetes._--In this by far the larger division of the + Basidiomycetes the basidia are undivided and the four basidiospores + are borne on short sterigmata nearly always at the apex of the + basidium. The group may be divided into two main divisions, + _Hymenomycetes_ and _Gasteromycetes_. + + _Hymenomycetes_ are a very large group containing over 11,000 species, + most of which live in soil rich in humus or on fallen wood or stems, a + few only being parasites. In the simplest forms (e.g. _Exobasidium_) + the basidia are borne directly on the ordinary mycelium, but in the + majority of cases the basidia are found developed in layers (hymenium) + on special sporophores of characteristic form in the various groups. + In these sporophores (such as the well-known toadstools and mushrooms + where the ordinary vegetative mycelium is underground) we have + structures specially developed for bearing the basidiospores and + protecting them from rain, &c., and for the distribution of the + spores--see earlier part of article on distribution of spores (figs. + 19 and 20). The underground mycelium in many cases spreads wider and + wider each year, often in a circular manner, and the sporophores + springing from it appear in the form of a ring--the so-called fairy + rings. _Armillaria melleus_ and _Polyporus annosus_ are examples of + parasitic forms which attack and destroy living trees, while _Merulius + lacrymans_ is the well-known "dry rot" fungus. + + [Illustration: FIG. 20.--_Agaricus mucidus_. Portion of hymenium. s, + Sporidia; st, sterigmata; g, sterile cells; c, cystidium, with + operculum o.] + + _Gasteromycetes_ are characterized by having closed sporophores or + fruit-bodies which only open after the spores are ripe and then often + merely by a small pore. The fruit-bodies are of very various shapes, + showing a differentiation into an outer _peridium_ and an inner + spore-bearing mass, the _gleba_. The gleba is usually differentiated + into a number of chambers which are lined directly by the hymenium + (basidial layer), or else the chambers contain an interwoven mass of + hyphae, the branches of which bear the basidia. By the breaking down + of the inner tissues the spores often come to lie as a loose powdery + mass in the interior of the hollow fruit-body, mixed sometimes with a + capillitium. The best-known genera are _Bovista, Lycoperdon_ + (puff-ball) _Scleroderma, Geaster_ (earth-star, q.v.). In the + last-named genus the peridium is double and the outer layer becomes + ruptured and spreads out in the form of star-shaped pieces; the inner + layer, however, merely opens at the apex by a small pore. + + The most complex members of the Gasteromycetes belong to the + _Phalloideae_, which is sometimes placed as a distinct division of the + Autobasidiomycetes. _Phallus impudicus_, the stink-horn, is + occasionally found growing in woods in Britain. The fruit-body before + it ruptures may reach the size of a hen's egg and is white in colour; + from this there grows out a hollow cylindrical structure which can be + distinguished at the distance of several yards by its disgusting + odour. It is highly poisonous. + +_Physiology._--The physiology of the fungi comes under the head of that +of plants generally, and the works of Pfeffer, Sachs, Vines, Darwin and +Klebs may be consulted for details. But we may refer generally here to +certain phenomena peculiar to these plants, the life-actions of which +are restricted and specialized by their peculiar dependence on organic +supplies of carbon and nitrogen, so that most fungi resemble the +colourless cells of higher plants in their nutrition. Like these they +require water, small but indispensable quantities of salts of potassium, +magnesium, sulphur and phosphorus, and supplies of carbonaceous and +nitrogenous materials in different stages of complexity in the different +cases. Like these, also, they respire oxygen, and are independent of +light; and their various powers of growth, secretion, and general +metabolism, irritability, and response to external factors show similar +specific variations in both cases. It is quite a mistake to suppose +that, apart from the chlorophyll function, the physiology of the +fungus-cell is fundamentally different from that of ordinary +plant-cells. Nevertheless, certain biological phenomena in fungi are +especially pronounced, and of these the following require particular +notice. + + _Parasitism._--Some fungi, though able to live as saprophytes, + occasionally enter the body of living plants, and are thus termed + facultative parasites. The occasion may be a wound (e.g. _Nectria_, + _Dasyscypha_, &c.), or the enfeeblement of the tissues of the host, or + invigoration of the fungus, the mycelium of which then becomes strong + enough to overcome the host's resistance (_Botrytis_). Many fungi, + however, cannot complete their life-history apart from the host-plant. + Such _obligate_ parasites may be epiphytic (_Erysipheae_), the + mycelium remaining on the outside and at most merely sending haustoria + into the epidermal cells, or endophytic (_Uredineae_, _Ustilagineae_, + &c.), when the mycelium is entirely inside the organs of the host. An + epiphytic fungus is not necessarily a parasite, however, as many + saprophytes (moulds, &c.) germinate and develop a loose mycelium on + living leaves, but only enter and destroy the tissues after the leaf + has fallen; in some cases, however, these saprophytic epiphytes can do + harm by intercepting light and air from the leaf (_Fumago_, &c.), and + such cases make it difficult to draw the line between saprophytism and + parasitism. Endophytic parasites may be intracellular, when the fungus + or its mycelium plunges into the cells and destroys their contents + directly (_Olpidium_, _Lagenidium_, _Sclerotinia_, &c.), but they are + far more frequently intercellular, at any rate while young, the + mycelium growing in the lacunae between the cells (_Peronospora_, + _Uredineae_) into which it may send short (_Cystopus_), or long and + branched (_Peronospora Calotheca_) haustoria, or it extends in the + middle lamella (_Ustilago_), or even in the solid substance of the + cell-wall (_Botrytis_). No sharp lines can be drawn, however, since + many mycelia are intercellular at first and subsequently become + intracellular (_Ustilagineae_), and the various stages doubtless + depend on the degrees of resistance which the host tissues are able to + offer. Similar gradations are observed in the direct effect of the + parasite on the host, which may be local (_Hemileia_) when the + mycelium never extends far from the point of infection, or general + (_Phytophthora_) when it runs throughout the plant. Destructive + parasites rapidly ruin the whole plant-body (_Pythium_), whereas + restrained parasites only tax the host slightly, and ill effects may + not be visible for a long time, or only when the fungus is epidemic + (_Rhytisma_). A parasite may be restricted during a long + incubation-period, however, and rampant and destructive later + (_Ustilago_). The latter fact, as well as the extraordinary + fastidiousness, so to speak, of parasites in their choice of hosts or + of organs for attack, point to reactions on the part of the + host-plant, as well as capacities on that of the parasite, which may + be partly explained in the light of what we now know regarding enzymes + and chemotropism. Some parasites attack many hosts and almost any + tissue or organ (_Botrytis cinerea_), others are restricted to one + family (_Cystopus Candidus_) or genus (_Phytophthora infestans_) or + even species (_Pucciniastrum Padi_), and it is customary to speak of + root-parasites, leaf-parasites, &c., in expression of the fact that a + given parasite occurs only on such organs--e.g. _Dematophora necatrix_ + on roots, _Calyptospora Goeppertiana_ on stems, _Ustilago Scabiosae_ + in anthers, _Claviceps purpurea_ in ovaries, &c. Associated with these + relations are the specializations which parasites show in regard to + the age of the host. Many parasites can enter a seedling, but are + unable to attack the same host when older--e.g. _Pythium_, + _Phytophthora omnivora_. + + _Chemotropism._--Taken in conjunction with Pfeffer's beautiful + discovery that certain chemicals exert a distinct attractive influence + on fungus hyphae (_chemotropism_), and the results of Miyoshi's + experimental application of it, the phenomena of enzyme-secretion + throw considerable light on the processes of infection and parasitism + of fungi. Pfeffer showed that certain substances in definite + concentrations cause the tips of hyphae to turn towards them; other + substances, though not innutritious, repel them, as also do nutritious + bodies if too highly concentrated. Marshall Ward showed that the + hyphae of _Botrytis_ pierce the cell-walls of a lily by secreting a + cytase and dissolving a hole through the membrane. Miyoshi then + demonstrated that if _Botrytis_ is sown in a lamella of gelatine, and + this lamella is superposed on another similar one to which a + chemotropic substance is added, the tips of the hyphae at once turn + from the former and enter the latter. If a thin cellulose membrane is + interposed between the lamellae, the hyphae nevertheless turn + chemotropically from the one lamella to the other and pierce the + cellulose membrane in the process. The hyphae will also dissolve their + way through a lamella of collodion, paraffin, parchment paper, + elder-pith, or even cork or the wing of a fly, to do which it must + excrete very different enzymes. If the membrane is of some impermeable + substance, like gold leaf, the hyphae cannot dissolve its way through, + but the tip finds the most minute pore and traverses the barrier by + means of it, as it does a stoma on a leaf We may hence conclude that a + parasitic hyphae pierces some plants or their stomata and refuses to + enter others, because in the former case there are chemotropically + attractive substances present which are absent from the latter, or are + there replaced by repellent poisonous or protective substances such as + enzymes or antitoxins. + + _Specialization of Parasitism._--The careful investigations of recent + years have shown that in several groups of fungi we cannot be content + to distinguish as units morphologically different species, but we are + compelled to go deeper and analyse further the species. It has been + shown especially in the _Uredineae_ and _Erysiphaceae_ that many forms + which can hardly be distinguished morphologically, or which cannot be + differentiated at all by structural characters, are not really + homogeneous but consist of a number of forms which are sharply + distinguishable by their infecting power. Eriksson found, for example, + that the well-known species _Puccinia graminis_ could be split up into + a number of forms which though morphologically similar were + physiologically distinct. He found that the species really consisted + of six distinct races, each having a more or less narrow range of + grasses on which it can live. The six races he named _P. graminis + Secalis_, _Tritici_, _Avenae_, _Airae_, _Agrostis_, _Poae_. The first + named will grow on rye and barley but not on wheat or oat. The form + _Tritici_ is the least sharply marked and will grow on wheat, barley, + rye and oat but not on the other grasses. The form _Avenae_ will grow + on oat and many grasses but not on the other three cereals mentioned. + The last three forms grow only on the genera _Aira_, _Agrostis_ and + _Poa_ respectively. All these forms have of course their + aecidium-stage on the barberry. The terms biologic forms, biological + species, physiological species, physiological races, specialized forms + have all been applied to these; perhaps the term biologic forms is the + most satisfactory. A similar specialization has been observed by + Marshall Ward in the _Puccinia_ parasitic on species of _Bromus_, and + by Neger, Marchal and especially Salmon in the Erysiphaceae. In the + last-named family the single morphological species _Erysiphe graminis_ + is found growing on the cereals, barley, oat, wheat, rye and a number + of wild grasses (such as _Poa_, _Bromus_, _Dactylis_). On each of + these host-plants the fungus has become specialized so that the form + on barley cannot infect the other three cereals or the wild grasses + and so on. Just as the uredospores and aecidiospores both show these + specialized characters in the case of _Puccinia graminis_ so we find + that both the conidia and ascospores of _E. graminis_ show this + phenomenon. Salmon has further shown in investigating the relation of + _E. graminis_ to various species of the genus, _Bromus_, that certain + species may act as "bridging species," enabling the transfer of a + biologic form to a host-plant which it cannot normally infect. Thus + the biologic form on _B. racemosus_ cannot infect _B. commutatus_. If, + however, conidia from _B. racemosus_ are sown on _B. hordaceus_, the + conidia which develop on that plant are now able to infect _B. + commutatus_; thus _B. hordaceus_ acts as a bridging species. Salmon + also found that injury of a leaf by mechanical means, by heat, by + anaesthetics, &c., would affect the immunity of the plant and allow + infection by conidia which was not able to enter a normal leaf. The + effect of the abnormal conditions is probably to stop the production + of, or weaken or destroy the protective enzymes or antitoxins, the + presence of which normally confers immunity on the leaf. + + _Symbiosis._--The remarkable case of life in common first observed in + lichens, where a fungus and an alga unite to form a compound + organism--the lichen--totally different from either, has now been + proved to be universal in these plants, and lichens are in all cases + merely algae enmeshed in the interwoven hyphae of fungi (see LICHENS). + This dualism, where the one constituent (alga) furnishes + carbohydrates, and the other (fungus) ensures a supply of mineral + matters, shade and moisture, has been termed _symbiosis_. Since then + numerous other cases of symbiosis have been demonstrated. Many trees + are found to have their smaller roots invaded by fungi and deformed by + their action, but so far from these being injurious, experiments go to + show that this mycorhiza (fungus-root) is necessary for the well-being + of the tree. This is also the case with numerous other plants of moors + and woodlands--e.g. Ericaceae, Pyrolaceae, Gentianaceae, Orchidaceae, + ferns, &c. Recent experiments have shown that the difficulties of + getting orchid seeds to germinate are due to the absence of the + necessary fungus, which must be in readiness to infect the young + seedling immediately after it emerges from the seed. The well-known + failures with rhododendrons, heaths, &c., in ordinary garden soils are + also explained by the need of the fungus-infected peat for their + roots. The role of the fungus appears to be to supply materials from + the leaf-mould around, in forms which ordinary root-hairs are + incapable of providing for the plant; in return the latter supports + the fungus at slight expense from its abundant stores of reserve + materials. Numerous other cases of symbiosis have been discovered + among the fungi of fermentation, of which those between _Aspergillus_ + and yeast in sake manufacture, and between yeasts and bacteria in + kephir and in the ginger-beer plant are best worked out. For cases of + symbiosis see BACTERIOLOGY. + + AUTHORITIES.--_General_: Engler and Prantl, _Die naturlichen + Pflanzenfamilien_, i. Teil (1892 onwards); Zopf, _Die Pilze_ (Breslau, + 1890); De Bary, _Comparative Morphology of Fungi_, &c. (Oxford, 1887); + von Tafel, _Vergleichende Morphologie der Pilze_ (Jena, 1892); + Brefeld, _Unters. aus dem Gesamtgebiete der Mykologie_, Heft i. 13 + (1872-1905); Lotsy, _Vortrage uber botanische Stammesgeschichte_ + (Jena, 1907). _Distribution_, &c.: Cooke, _Introduction to the Study + of Fungi_ (London, 1895); Felix in _Zeitschr. d. deutsch. geologisch. + Gesellsch._ (1894-1896); Staub, _Sitzungsber. d. bot. Sec. d. Kgl. + ungarischen naturwiss. Gesellsch. zu Budapest_ (1897). _Anatomy_, &c.: + Bommer, "Sclerotes et cordons myceliens," _Mem. de l'Acad. Roy. de + Belg._ (1894); Mangin, "Observ. sur la membrane des mucorinees," + _Journ. de Bot._ (1899); Zimmermann, _Die Morph. und Physiologie des + Pflanzenzellkernes_ (Jena, 1896); Wisselingh, "Microchem. Unters. uber + die Zellwande d. Fungi," _Pringsh. Jahrb._ B. 31, p. 619 (1898); + Istvanffvi, "Unters. uber die phys. Anat. der Pilze," _Prings. Jahrb._ + (1896). _Spore Distribution_: Fulton, "Dispersal of the Spores of + Fungi by Insects," _Ann. Bot._ (1889); Falck, "Die Sporenverbreitung + bei den Basidiomyceten," _Beitr. zur Biol. d. Pflanzen_, ix. (1904). + _Spores and Sporophores_: Zopf, _Die Pilze_; also the works of von + Tafel and Brefeld. _Classification_: van Tieghem, _Journ. de bot._ p. + 77 (1893), and the works of Brefeld, Engler and Prantl, von Tafel, + Saccardo and Lotsy already cited, _Oomycetes_: Wager, "On the + Fertilization of _Peronospora parasitica_," _Ann. Bot._ vol. xiv. + (1900); Stevens, "The Compound Oosphere of _Albugo Bliti_," _Bot. + Gaz._ vol. 28 (1899); "Gametogenesis and Fertilization in _Albugo_," + ibid. vol. 32 (1901); Miyake, "The Fertilization of _Pythium de + Baryanum_," _Ann. of Bot._ vol. xv. (1901); Trow, "On Fertilization in + the Saprolegnieae," _Ann. of Bot._ vol. xviii. (1904); Thaxter, "New + and Peculiar Aquatic Fungi," _Bot. Gaz._ vol. 20 (1895); Lagerheim, + "Unters. uber die Monoblepharideae," _Bih. Svenska Vet. Acad. + Handlingar_, 25. Afd. iii. (1900); Woronin, "Beitrag zur Kenntnis der + Monoblepharideen," _Mem. de l'Acad. Imp. d. Sc. de St-Petersbourg_, 8 + ser. vol. 16 (1902). _Zygomycetes_: Harper, "Cell-division in + Sporangia and Asci," _Ann. Bot._ vol. xiii. (1899); Klebs, _Die + Bedingungen der Fortpflanzung_, &c. (Jena, 1896), and "Zur Physiologie + der Fortpflanzung" _Prings. Jahr._ (1898 and 1899), "Uber _Sporodinia + grandis_," _Bot. Zeit._ (1902); Falck, "Die Bedingungen der + Zygotenbildung bei Sporodinia grandis," Cohn's Beitr. z. Biol. d. + Pflanzen, Bd. 8 (1902); Gruber "Verhalten der Zellkerne in den + Zygosporen von _Sporodinia grandis_," _Ber. d. deutschen bot. Ges._ + Bd. 19 (1901); Blakeslee, "Sexual Reproduction in the Mucorineae," + _Proc. Am. Acad._ (1904); "Zygospore germination in the Mucorineae," + _Annales mycologici_ (1906). _Ustilagineae_: Plowright, _British + Uredineae and Ustilagineae_ (London, 1889); Massee, _British Fungi_ + (Phycomycetes and Ustilagineae) (London, 1891); Brefeld, _Unters. aus + dem Gesamtgeb. der Mykol._ Hefte xi. and xii.; and Falck, "Die + Bluteninfektion bei den Brandpilzen," ibid. Heft xiii. 1905; Dangeard, + "La Reproduction sexuelle des Ustilaginees," C.R., Oct. 9, 1893; + Maire, "Recherches cytologiques et taxonomiques sur les + Basidiomyceten," _Annexe au Bull. de la Soc. Mycol. de France_ (1902). + _Saccharomycetaceae_: Jorgensen, _The Micro-organisms of Fermentation_ + (1899); Barker, _Ann. of Bot._ vol. xiv. (1901); "On Spore-formation + among the Saccharomycetes," _Journ. of the Fed. Institute of Brewing_, + vol. 8 (1902); Guillermond, _Recherches cytologiques sur les levures_ + (Paris, 1902); Hansen, _Centralbl. f. Bakt. u. Parasitenp._ Abt. ii. + Bd. 12 (1904). _Exoascaceae_: Giesenhagen, "_Taphrina, Exoascus, + Magnusiella_" (complete literature given), _Bot. Zeit._ Bd. 7 (1901). + _Erysiphaceae_: Harper, "Die Entwicklung des Perithecium bei + _Sphaerotheca castagnei_," _Ber. d. deut bot Ges._ (1896); "Sexual + Reproduction and the Organization of the Nucleus in certain Mildews," + _Publ. Carnegie Institution_ (Washington, 1906); Blackman & Fraser, + "Fertilization in _Sphaerotheca_," _Ann. of Bot._ (1905). + _Perisporiaceae_: Brefeld, _Untersuchungen aus dem Gesamtgeb. der + Mykol._ Heft 10 (1891); Fraser and Chamber, _Annales mycologici_ + (1907). _Discomycetes_: Harper, "Uber das Verhalten der Kerne bei + Ascomyceten," _Jahr. f. wiss. Bot._ Bd. 29 (1890); "Sexual + Reproduction in _Pyronema confluens_," _Ann. of Bot._ 14 (1900); + Claussen, "Zur Entw. der Ascomyceten," Boudiera, Bot. Zeit. Bd. 63 + (1905); Dangeard, "Sur le _Pyronema confluens_," _Le Botaniste_, 9 + serie (1903) (and numerous papers in same journal earlier and later); + Ramlow, "Zur Entwick. von _Thelebolus stercoren_," _Bot. Zeit._ + (1906); Woronin, "Uber die Sclerotienkrankheit der Vaccineen Beeren," + _Mem. de l'Acad. Imp. des Sciences de St-Petersbourg_, 7 serie, 36 + (1888); Dittrich, "Zur Entwickelungsgeschichte der Helvellineen," + Cohn's _Beitr. z. Biol. d. Pflanzen_ (1892). _Pyrenomycetes_: Fisch, + "Beitr. z. Entwickelungsgeschichte einiger Ascomyceten," _Bot. Zeit._ + (1882); Frank, "Uber einige neue u. weniger bekannte Pflanzkrankh.," + _Landw. Jahrb._ Bd. 12 (1883); Ward, "_Onygena equina_, a + horn-destroying fungus," _Phil. Trans._, vol. 191 (1899); Dawson, "On + the Biology of Poroniapunctata," Ann. of Bot. 14 (1900). _Tuberineae_: + Buchholtz, "Zur Morphologie u. Systematik der Fungi hypogaei," _Ann. + Mycol._ Bd. 1 (1903); Fischer in Engler and Prantl, _Die naturlichen + Pflanzenfamilien_ (1896). _Laboulbeniineae_: Thaxter, "Monograph of + the Laboulbeniaceae," _Mem. Amer. Acad. of Arts and Sciences_, vol. 12 + (1895). _Uredineae_: Eriksson and Henning, _Die Getreideroste_ + (Stockholm, 1896); Eriksson, _Botan. Gaz._ vol. 25 (1896); "On the + Vegetative Life of some Uredineae," Ann. of Bot. (1905); Klebahn, _Die + wirtwechselnden Rostpilze_ (Berlin, 1904); Sapin-Trouffy, "Recherches + histologiques sur la famille des Uredinees," _Le Botaniste_ + (1896-1897); Blackman, "On the Fertilization, Alternation of + Generations and General Cytology of the Uredineae," _Ann. of Bot._ + vol. 18 (1904); Blackman and Fraser, "Further Studies on the Sexuality + of Uredineae," _Ann. of Bot._ vol. 20 (1906); Christman, "Sexual + Reproduction of Rusts," _Ann. of Bot._ vol. 20 (1906); Ward, "The + Brooms and their Rust Fungus," _Ann. of Bot._ vol. 15 (1901). + _Basidiomycetes_: Dangeard, "La Reprod. sexuelle des Basidiomycetes," + _Le Botaniste_ (1894 and 1900); Maire, "Recherches cytologiques et + taxonomiques sur les Basidiomycetes," _Annexe du Bull. de la Soc. + Mycol. de France_ (1902); Moller, "Protobasidiomyceten," _Schimper's + Mitt. aus den Tropen_, Heft 8 (Jena, 1895); Nichols, "The Nature and + Origin of the Binucleated Cells in certain Basidiomycetes," _Trans. + Wisconsin Acad. of Sciences_, vol. 15 (1905); Wager, "The Sexuality of + the Fungi," _Ann. of Bot._ 13 (1899); Woronin, "_Exobasidium + Vaccinii_," _Verh. Naturf. Ges. zu Freiburg_, Bd. 4 (1867). + _Fermentation_: Buchner, "Gahrung ohne Hefezellen," _Bot. Zeit._ Bd. + 18 (1898); Albert, _Cent. f. Bakt._ Bd. 17 (1901); Green, _The + Soluble Ferments and Fermentation_ (Cambridge, 1899). _Parasitism_: + "On some Relations between Host and Parasite," _Proc. Roy. Soc_. vol. + 47 (1890); "A Lily Disease," _Ann. of Botany_, vol. 2 (1888); Eriksson + & Hennings, _Die Getreideroste (vide supra_); Ward, "On the Question + of Predisposition and Immunity in Plants," _Proc. Cambridge Phil. + Soc_. vol. 11 (1902); also _Annals of Bot_. vol. 16 (1902) and vol. 19 + (1905); Neger, "Beitr. z. Biol. d. Erysipheen" _Flora_, Bde. 88 and 90 + (1901-1902); Salmon, "Cultural Experiments with 'Biologic Forms' of + the Erysiphaceae," _Phil. Trans_. (1904); "On Erysiphe graminis and + its adaptative parasitism within the genus, _Bromus_," _Ann. Mycol_. + vol. 11 (1904), also _Ann. of Bot_. vol. 19 (1905). _Symbiosis_: Ward, + "The Ginger-Beer Plant," _Phil. Trans. Roy. Soc_. (1892); "Symbiosis," + _Ann. of Bot_. 13 (1899); Shalk, "Der Sinn der Mykorrhizenbildung," + _Jahrb. f. wiss. Bot_. Bd. 34 (1900); Bernard, "On some Different + Cases of Germination," _Gardener's Chronicle_ (1900); Pierce, _Publ. + Univ. California_ (1900). (H. M. W.; V. H. B.) + + + + +FUNJ (FUNNIYEH, FUNG, FUNGHA), a very mixed negroid race, occupying +parts of Sennar and the hilly country to the south between the White and +Blue Niles. They traditionally come from west of the White Nile and are +affiliated by some to the Kordofan Nubas, by others, more justifiably, +to the negro Shilluks. These Funj, who became the dominant race in +Sennar in the 15th century, almost everywhere assimilated the speech, +religion and habits of the Arabs settled in that region. Until the 19th +century they were one of the most powerful of African peoples in the +eastern Sudan. About the end of the 15th century they overthrew the +kingdom of Aloa, between the two Niles, and conquered the neighbouring +peoples of the Sudan, Nubia and even Kordofan. The Funj had mixed much +with the Arabs before their conquests, and had been converted to Islam. +But they were still in many ways savages, for James Bruce (who traversed +the district in 1772) says that their most famous king, Malek-el-Gahman, +preferred human liver to any other food, and the Belgian traveller E. +Pruyssenaere (1826-1864) found them still performing pagan rites on +their sacred Mount Gula. Ernst Marno declared that as late as 1870 the +most southern branch of the race, the Boruns, a non-Arabic speaking +tribe, were cannibals. The Funj kings were content with levying tribute +on their neighbours, and in this loose way Shendi, Berber and Dongola +were once tributary. The Arab viziers gradually absorbed all power, the +Funj sovereignty becoming nominal; and in 1821 the Egyptians easily +destroyed the Funj domination. To-day the Funj are few, and represent no +real type. They are a bright, hospitable folk. Many of them are skilful +surgeons and go far afield in their work. The fellahin, indeed, call +surgeons "Senaari" (men of Sennar). See further SENNAR AND SUDAN +(Anglo-Egyptian). + + + + +FUNKIA, in botany, a genus of rather handsome, hardy, herbaceous plants +belonging to the natural order Liliaceae, and natives of China and +Japan. They are tuberous, with broadly ovate or heart-shaped leaves and +racemes of white or pale lilac, drooping, funnel-shaped flowers. They +are useful for the borders of a shrubbery, the lawn or rock-work, or may +be grown in pots for the greenhouse. The plants are propagated by +dividing the crowns in autumn or when growth begins in spring. + + + + +FUNNEL (through an O. Fr. _founil_, found in Breton, from Lat. +_infundibulum_, that through which anything is poured, from _fundere_, +to pour), a vessel shaped like a cone having a small tube at the apex +through which powder, liquid, &c., may be easily passed into another +vessel with a small opening. The term is used in metal-casting of the +hole through which the metal is poured into a mould, and in anatomy and +zoology of an _infundibulum_ or funnel-shaped organ. The word is thus +used generally of any shaft or passage to convey light, air or smoke, as +of the chimney of an engine or a steam-boat, or the flue of an ordinary +chimney. It is also used of a shaft or channel in rocks, and in the +decoying of wild-fowl is applied to the cone-shaped passage leading from +a pond and covered with a net, a "funnel-net," into which the birds are +decoyed. + + + + +FUR (connected with O. Fr. _forre_, a sheath or case; so "an outer +covering"), the name specially given to the covering of the skin in +certain animals which are natives of the colder climates, lying +alongside of another and longer covering, called the overhair. The fur +differs from the overhair, in that it is soft, silky, curly, downy and +barbed lengthwise, while the overhair is straight, smooth and +comparatively rigid. These properties of fur constitute its essential +value for felting purposes, and mark its difference from wool and silk; +the first, after some slight preparation by the aid of hot water, +readily unites its fibres into a strong and compact mass; the others can +best be managed by spinning and weaving. + +On the living animal the overhair keeps the fur filaments apart, +prevents their tendency to felt, and protects them from injury--thus +securing to the animal an immunity from cold and storm; while, as a +matter of fact, this very overhair, though of an humbler name, is most +generally the beauty and pride of the pelt, and marks its chief value +with the furrier. We arrive thus at two distinct and opposite uses and +values of fur. Regarded as useful for felt it is denominated staple fur, +while with respect to its use with and on the pelt it is called fancy +fur. + +_History._--The manufacture of fur into a felt is of comparatively +modern origin, while the use of fur pelts as a covering for the body, +for the couch, or for the tent is coeval with the earliest history of +all northern tribes and nations. Their use was not simply a barbarous +expedient to defend man from the rigours of an arctic winter; woven wool +alone cannot, in its most perfect form, accomplish this. The pelt or +skin is requisite to keep out the piercing wind and driving storm, while +the fur and overhair ward off the cold; and "furs" are as much a +necessity to-day among more northern peoples as they ever were in the +days of barbarism. With them the providing of this necessary covering +became the first purpose of their toil; subsequently it grew into an +object of barter and traffic, at first among themselves, and afterwards +with their neighbours of more temperate climes; and with the latter it +naturally became an article of fashion, of ornament and of luxury. This, +in brief, has been the history of its use in China, Tatary, Russia, +Siberia and North America, and at present the employment of fancy furs +among civilized nations has grown to be more extensive than at any +former period. + +The supply of this demand in earlier times led to such severe +competition as to terminate in tribal pillages and even national wars; +and in modern times it has led to commercial ventures on the part of +individuals and companies, the account of which, told in its plainest +form, reads like the pages of romance. Furs have constituted the price +of redemption for royal captives, the gifts of emperors and kings, and +the peculiar badge of state functionaries. At the present day they vie +with precious gems and gold as ornaments and garniture for wealth and +fashion; but by their abundance, and the cheapness of some varieties, +they have recently come within the reach of men of moderate incomes. The +history of furs can be read in Marco Polo, as he grows eloquent with the +description of the rich skins of the khan of Tatary; in the early +fathers of the church, who lament their introduction into Rome and +Byzantium as an evidence of barbaric and debasing luxury; in the +political history of Russia, stretching out a powerful arm over Siberia +to secure her rich treasures; in the story of the French occupation of +Canada, and the ascent of the St Lawrence to Lake Superior, and the +subsequent contest to retain possession against England; in the history +of early settlements of New England, New York and Virginia; in Irving's +_Astoria_; in the records of the Hudson's Bay Company; and in the annals +of the fairs held at Nizhniy Novgorod and Leipzig. Here it may suffice +to give some account of the present condition of the trade in fancy +furs. The collection of skins is now chiefly a matter of private +enterprise. Few, if any, monopolies exist. + +_Natural Supplies._--We are dependent upon the Carnivora, Rodentia, +Ungulata and Marsupialia for our supplies of furs, the first two classes +being by far of the greatest importance. The Carnivora include bears, +wolverines, wolves, raccoons, foxes, sables, martens, skunks, kolinskis, +fitch, fishers, ermines, cats, sea otters, fur seals, hair seals, lions, +tigers, leopards, lynxes, jackals, &c. The Rodentia include beavers, +nutrias, musk-rats or musquash, marmots, hamsters, chinchillas, hares, +rabbits, squirrels, &c. The Ungulata include Persian, Astrachan, +Crimean, Chinese and Tibet lambs, mouflon, guanaco, goats, ponies, &c. +The Marsupialia include opossums, wallabies and kangaroos. These, of +course, could be subdivided, but for general purposes of the fur trade +the above is deemed sufficient. + +The question frequently arises, not only for those interested in the +production of fur apparel, but for those who derive so much comfort and +pleasure from its use, whether the supply of fur-bearing animals is +likely to be exhausted. Although it is a fact that the demand is ever +increasing, and that some of the rarer animals are decreasing in +numbers, yet on the other hand some kinds of furs are occasionally +neglected through vagaries of fashion, which give nature an opportunity +to replenish their source. These respites are, however, becoming fewer +every day, and what were formerly the most neglected kinds of furs are +becoming more and more sought after. The supply of some of the most +valuable, such as sable, silver and natural black fox, sea otter and +ermine, which are all taken from animals of a more or less shy nature, +does very gradually decrease with persistent hunting and the +encroachment of man upon the districts where they live, but the climate +of these vast regions is so cold and inhospitable that the probabilities +of man ever permanently inhabiting them in numbers sufficient to scare +away or exterminate the fur-bearing wild animals is unlikely. Besides +these there are many useful, though commonplace, fur-bearing animals +like mink, musquash, skunk, raccoon, opossum, hamster, rabbit, hares and +moles, that thrive by depredations upon cultivated land. Some of these +are reared upon extensive wild farms. In addition there are domestic +fur-bearing animals, such as Persian, Astrachan and Chinese lambs, and +goats, easily bred and available. + +With regard to the rearing of the Persian lamb, there is a prevalent +idea that the skins of the unborn lamb are frequently used; this, +however, is a mistake. A few such skins have been taken, but they are +too delicate to be of any service. The youngest, known as "broadtails," +are killed when a few days old, but for the well-developed curly fur, +the lambs must be six or seven weeks old. During these weeks their +bodies are covered with leather so that the fur may develop in close, +light and clean curls. The experiment has been tried of rearing rare, +wild, fur-bearing animals in captivity, and although climatic conditions +and food have been precisely as in their natural environment, the fur +has been poor in quality and bad in colour, totally unlike that taken +from animals in the wild state. The sensation of fear or the restriction +of movement and the obtaining of food without exertion evidently prevent +the normal development of the creature. + +In mountainous districts in the more temperate zones some good supplies +are found. Chinchillas and nutrias are obtained from South America, +whence come also civet cats, jaguars, ocelots and pumas. Opossums and +wallabies, good useful furs, come from Australia and New Zealand. The +martens, foxes and otters imported from southern Europe and southern +Asia, are very mixed in quality, and the majority are poor compared with +those of Canada and the north. + +Certain characteristics In the skin reveal to the expert from what +section of territory they come, but in classifying them it is considered +sufficient to mention territories only. + +Some of the poorer sorts of furs, such as hamster, marmot, Chinese goats +and lambs, Tatar ponies, weasels, kaluga, various monkeys, antelopes, +foxes, otters, jackals and others from the warmer zones, which until +recently were neglected on account of their inferior quality of colour, +by the better class of the trade, are now being deftly dressed or dyed +in Europe and America, and good effects are produced, although the lack +of quality when compared with the better furs from colder climates which +possess full top hair, close underwool and supple leathers, is readily +manifest. It is only the pressure of increasing demand that makes +marketable hard pelts with harsh brittle hair of nondescript hue, and +these would, naturally, be the last to attract the notice of dealers. + +As it is impossible that we shall ever discover any new fur-bearing +animals other than those we know, it behoves responsible authorities to +enforce close seasons and restrictions, as to the sex and age, in the +killing for the purpose of equalizing the numbers of the catches. As +evidence of indiscriminate slaughter the case of the American buffaloes +may be cited. At one time thousands of buffalo skins were obtainable and +provided material for most useful coats and rugs for rough wear in cold +regions, but to-day only a herd or so of the animals remain, and in +captivity. + +The majority of animals taken for their fur are trapped or snared, the +gun being avoided as much as possible in order that the coat may be +quite undamaged. Many weary hours are spent in setting baits, traps and +wires, and, frequently, when the hunter retraces his steps to collect +the quarry it is only to find it gone, devoured by some large animal +that has visited his traps before him. After the skins have been +carefully removed--the sooner after death the better for the subsequent +condition of the fur--they are lightly tacked out, pelt outwards, and, +without being exposed to the sun or close contact with a fire, allowed +to dry in a hut or shady place where there is some warmth or movement of +air. With the exception of sealskins, which are pickled in brine, all +raw skins come to the various trade markets simply dried like this. + +_Quality and Colour._--The best fur is obtained by killing animals when +the winter is at its height and the colder the season the better its +quality and colour. Fur skins taken out of season are indifferent, and +the hair is liable to shed itself freely; a good furrier will, however, +reject such faulty specimens in the manufacturing. The finest furs are +obtained from the Arctic and northern regions, and the lower the +latitude the less full and silky the fur, till, at the torrid zone, fur +gives place to harsh hair without any underwool. The finest and closest +wools are possessed by the amphibious Carnivora and Rodentia, viz. +seals, otters, beavers, nutrias and musquash, the beauty of which is not +seen until after the stiff water or top hairs are pulled out or +otherwise removed. In this class of animal the underneath wool of the +belly is thicker than that of the back, while the opposite is true of +those found on the land. The sea otter, one of the richest and rarest of +furs, especially for men's wear, is an exception to this unhairing +process, which it does not require, the hair being of the same length as +the wool, silky and bright, quite the reverse of the case of other +aquatic animals. + +Of sealskins there are two distinct classes, the fur seals and the hair +seals. The latter have no growth of fur under the stiff top hair and are +killed, with few exceptions (generally of the marbled seals), on account +of the oil and leather they yield. The best fur seals are found off the +Alaska coast and down as far south as San Francisco. + +It is found that in densely wooded districts furs are darker in colour +than in exposed regions, and that the quality of wool and hair is softer +and more silky than those from bare tracts of country, where nature +exacts from its creatures greater efforts to secure food, thereby +developing stronger limbs and a consequently coarser body covering. + +As regards density of colour the skunk or black marten has the blackest +fur, and some cats of the domestic kind, specially reared for their fur, +are nearly black. Black bears have occasionally very black coats, but +the majority have a brownish underwool. The natural black fox is a +member of the silver fox family and is very rare, the skins bringing a +high price. Most silver foxes have dark necks and in some the dark shade +runs a quarter, half-way, or three-quarters, or even the whole length of +the skin, but it is rather of a brownish hue. Some Russian sables are of +a very dense bluish brown almost a black, which is the origin +undoubtedly of the term "sables," while some, from one district in +particular, have a quantity of silver hairs, evenly interspersed in the +fur, a peculiarity which has nothing to do with age. The best sea otters +have very dark coats which are highly esteemed, a few with silver hairs +in parts; where these are equally and evenly spread the skins are very +valuable. Otters and beavers that run dark in the hair or wool are more +valuable than the paler ones, the wools of which are frequently touched +with a chemical to produce a golden shade. This is also done with +nutrias after unhairing. The darker sorts of mink, musquash, raccoon +and wolverine are more valuable than the paler skins. + +_Collective Supplies and Sales._--There are ten large American and +Canadian companies with extensive systems for gathering the annual hauls +of skins from the far-scattered trappers. These are the Hudson's Bay +Co., Russian Fur Co., Alaska Commercial Co., North American Commercial +Co., Russian Sealskin Co., Harmony Fur Co., Royal Greenland Fur Co., +American Fur Co., Missouri Co. and Pacific Co. Most of the raw skins are +forwarded to about half-a-dozen brokers in London, who roughly sort them +in convenient lots, issuing catalogues to the traders of the world, and +after due time for examination of the goods by intending purchasers, the +lots are sold by public auction. The principal sales of general furs are +held in London in January and March, smaller offerings being made in +June and October; while the bulk of fur sealskins is sold separately in +December. The Hudson's Bay Co.'s sales take place before the others, +and, as no reserves are placed on any lot, the results are taken as +exactly indicating current values. While many buyers from America and +Russia are personally in attendance at the sales, many more are +represented by London and Leipzig agents who buy for them upon +commission. In addition to the fur skins coming from North America vast +numbers from Russia, Siberia, China, Japan, Australia and South America +are offered during the same periods at public auction. Fairs are also +held in Siberia, Russia and Germany for the distribution of fur skins as +follows:-- + + January: Frankfort-on-the- Small collection of provincial produce, + Oder such as otter, fox, fitch and marten. + + February: Irbit, Siberia General Russian furs. + + Easter: Leipzig, Germany General furs. + + August: Nizhniy Novgorod, Persian lamb and general furs. + Russia + + August: Kiakhta, Siberia Chinese furs and ermine. + + December: Ishim, Siberia Chiefly squirrels. + +Of course there are many transactions, generally in the cheaper and +coarser kinds of furs, used only in central Europe, Russia and Asia +which in no way interest the London market, and there are many direct +consignments of skins from collectors in America and Russia to London, +New York and Leipzig merchants. But the bulk of the fine furs of the +world is sold at the large public trade auction sales in London. The +chief exceptions are the Persian and Astrachan lambs, which are bought +at the Russian fairs, and are dressed and dyed in Leipzig, and the +ermine and Russian squirrels, which are dressed and manufactured into +linings either in Russia or Germany before offered for sale to the +wholesale merchants or manufacturers. + +The annual collection of fur skins varies considerably in quantity +according to the demand and to the good or bad climatic conditions of +the season; and it is impossible to give a complete record, as many +skins are used in the country of their origin or exported direct to +merchants. But a fairly exact statement of the numbers sold in the great +public trade auction sales in London during the year 1905-1906 is +herewith set out. + + _Year ending 31st of March 1906._ Total Number + of Skins. + + Badger 28,634 + Badger, Japanese 6,026 + Bear 18,576 + Beaver 80,514 + Cat, Civet 157,915 + Cat, House 126,703 + " Wild 32,253 + Chinchilla (La Plata), known also as Bastard 43,578 + " Peruvian finest 5,603 + Deer, Chinese 124,355 + Ermine 40,641 + Fisher 5,949 + Fitch 77,578 + Fox, Blue 1,893 + " Cross 10,276 + " Grey 59,561 + " Japanese 81,429 + " Kit 4,023 + " Red 158,961 + " Silver 2,510 + " White 27,463 + Goats, Chinese 261,190 + Hares 41,256 + Kangaroo 7,115 + Kid, Chinese linings and skins equal to 5,080,047 + Kolinsky 114,251 + Lamb, Mongolian linings and skins equal to 214,072 + " Slink " " " 167,372 + " Tibet " " " 794,130 + Leopard 3,574 + Lynx 88,822 + Marmot, linings and skins equal to 1,600,600 + Marten, Baum 4,573 + " Japanese 16,461 + " Stone 12,939 + Mink, Canadian and American 299,254 + " Japanese 360,373 + Mouflon 23,594 + Musk-rat or Musquash, Brown 5,126,339 + " " Black 41,788 + Nutria 82,474 + Opossum, American 902,065 + " Australian 4,161,685 + Otter, River 21,235 + " Sea 522 + Raccoon 310,712 + Sable, Canadian and American 97,282 + " Japanese 556 + " Russian 26,399 + Seals, Fur 77,000 + " Hair 31,943 + Skunk 1,068,408 + Squirrel 194,596 + " Linings each averaging 126 skins 1,982,736 + Tiger 392 + Wallaby 60,956 + Wolf 56,642 + Wolverine 1,726 + Wombat 193,625 + +A brief account of the different qualities of the pelts, with some +general remarks as to their customary uses, follows. The prices quoted +are subject to constant fluctuation and represent purely trade prices +for bulk, and it should be explained that the very great variations are +due to different sizes, qualities and colours, and moreover are only +_first cost_, before skins are dressed and prepared. These preparations +are in some cases expensive, and there is generally a considerable +percentage of waste. The prices cannot be taken as a guide to the +wholesale price of a single and finished skin, but simply as _relative_ +value. + +The fullest and darkest skins of each kind are the most valuable, and, +in cases of bluish grey or white, the fuller, clearer and brighter are +the more expensive. A few albinos are found in every species, but +whatever their value to a museum, they are of little commercial +importance. Some odd lots of skins arrive designated simply as +"sundries," so no classification is possible, and this will account for +the absence of a few names of skins of which the imports are +insignificant in quantity, or are received direct by the wholesale +merchants. + + + _Names, Qualities and Uses of Pelts._[1] + + ASTRACHAN.--See _Lambs_, below. + + BADGER.--Size 2 X 1 ft. American sorts have coarse thick underwool of + a pale fawn or stone colour with a growth of longer black and white + hairs, 3 or 4 in. long; a very durable but clumsy fur. The best skins + are exported to France, Spain and Italy, and used for carriage rugs + and military purposes. Asiatic, including Japanese, skins are more + woolly. Russian and Prussian kinds are coarser and darker, and used + mostly for brush trade. Value 6d. to 19s. + + BEAR, AUSTRALIAN.--See _Wombat_, below. + + BEAR, BLACK.--Size 6 X 3 ft. Fine dark brown underwool with bright + black and flowing top hair 4 in. long. Cubs are nearly as long in the + hair although only about half the size and not only softer and better, + but have the advantage of being very much lighter in pelt. Widely + distributed in North America, the best come from Canada, are costly + and are used for military caps, boas, muffs, trimmings, carriage rugs + and coachmen's capes, and the fur wears exceedingly well. Value 17s. + 6d. to 86s. Those from East India and warm climates are harsh, poor + and only fit for floor rugs. + + BEAR, BROWN.--Size 6 X 3 ft. Similar in quality to the black, but far + more limited in number; the colours range from light yellow to a rich + dark brown. The best come from Hudson Bay territory and are valuable. + Used for muffs, trimmings, boas, and carriage rugs. Inferior sorts, + almost grizzly in effect and some very pale, are found in Europe and + Asia and are mostly used locally. In India there is a species called + Isabelline bear, which was formerly imported to Great Britain, but + does not now arrive in any quantity worth mentioning. Value 10s. 6d. + to 60s., Isabelline sort 10s. 6d. to 78s. + + BEAR, GRIZZLY.--Size 8 X 4 ft. Coarse hair, heavy pelt, mostly dark + yellowish and brown colours, only found in western parts of United + States, Russia and Siberia. Used as carriage rugs and floor rugs, most + durable for latter purpose and of fine effect. They are about half the + value of brown bear. Value 15s. to 54s. + + BEAR, ISABELLINE.--See _Bear_, _Brown_, above. + + BEAR, WHITE.--Size 10 X 5 ft. The largest of all bears. Short close + hair except on flanks, colour white to yellow. An inhabitant of the + Arctic circle, best from Greenland. Used for floor rugs, very durable; + and very white specimens are valuable. Value 20s. to 520s. + + BEAVER. Size 3 X 2 ft. The largest of rodents, it possesses a close + underwool of bluish-brown hue, nearly an inch in depth, with coarse, + bright, black or reddish-brown top hair, 3 in. long. Found widely in + North America. After being unhaired the darkest wools are the most + valuable, although many people prefer the bright, lighter brown tones. + Used for collars, cuffs, boas, muffs, trimmings, coat linings and + carriage aprons, and is of a most durable nature, in addition to + having a rich and good appearance. Value 10s. to 39s. 6d. + + BROADTAIL.--See _Lambs_, below. + + CARACAL.--A small lynx from India, the fur very poor, seldom imported. + + CARACUL.--See _Goats_ and _Lambs_, below. + + CAT, CIVET.--Size 9 X 4-1/2 in., short, thick and dark underwool with + silky black top hair with irregular and unique white markings. It is + similar to skunk, but is much lighter in weight, softer and less full, + without any disagreeable odour. Used for coat linings it is very warm + and durable. A few come from China, but the fur is yellowish-grey, + slightly spotted and worth little. Value 1s. 1d. to 1s. 11d. + + CAT, HOUSE, &C.--18 X 9 in., mostly black and dark brown, imported + from Holland, Bavaria, America and Russia, where they are reared for + their coats. The best, from Holland, are used for coat linings. + Although in colour, weight and warmth they are excellent, the fur is + apt to become loose and to fall off with friction of wear. The black + are known as genet, although the true genet is a spotted wild cat. + Wild sorts of the tabby order are coarser, and not so good and silky + in effect as when domestically reared. Value of the black sorts 2d. to + 3s. Wild 9d. to 14s. Some small wild cats, very poor flat fur of a + pale fawn colour with yellow spots, are imported from Australia and + used for linings. Value 5-1/2d. to 1s. 1d. + + CHEETAH.--Size of a small leopard and similar in colour, but has black + spots in lieu of rings. Only a few are now imported, which are used + for mats. Value 2s. 6d. to 18s. + + CHINCHILLA, PERUVIAN and BOLIVIAN.--Size 12 X 7 in., fur 1 to 1-1/4 + in. deep. Delicate blue-grey with black shadings, one of nature's most + beautiful productions, though not a durable one. Used for ladies' + coats, stoles, muffs, hats and trimmings. Yearly becoming scarcer and + most costly. Value 8s. 6d. to 56s. 8d. + + CHINCHILLA, LA PLATA, incorrectly named and known in the trade as + "bastard chinchilla," size 9 X 4 in., in a similar species, but owing + to lower altitudes and warmer climatic conditions of habitation is + smaller, with shorter and less beautiful fur, the underwool colour + being darker and the top colour less pure. Used exactly as the better + kind, and the picked skins are most effective. As with the best sort + it is not serviceable for constant wear. Value 4s. 2d. to 27s. 6d. + + CHINCHILLONE.--Size 13 X 8 in., obtained also from South America. Fur + is longer and weaker and poorer and yellower than chinchilla. Probably + a crossbred animal, very limited importation. Value 3s. 6d. to 16s. + 8d. + + DEER, CHINESE and EAST INDIAN.--Small, light, pelted skins, the + majority of which are used for mats. Reindeer and other varieties are + of little interest for use other than trophy mats. Thousands are taken + for the leather trade. Value of Chinese 1s. 2d. to 1s. 6d. each. + + DOG.--The only dogs that are used in the fur trade in civilized + countries are those imported from China, which are heavy and coarse, + and only used in the cheaper trade, chiefly for rugs. Value 6d. to 1s. + + DOG WOLF.--See _Wolf_, below. + + ERMINE.--Size 12 X 2-1/2 in. Underwool short and even, with a shade + longer top hair. Pelt light and close in texture, and durable. In the + height of winter the colour is pure white with exception of the tip of + tail, which is quite black. Supplies are obtained from Siberia and + America. Best are from Ishim in Siberia. Used for cloak linings, + stoles, muffs and trimmings, also for embellishment of British state, + parliamentary and legal robes. When this fur is symmetrically spotted + with black lamb pieces it is styled miniver, in which form it is used + at the grand coronation functions of British sovereigns. Value 1s. 3d. + to 8s. 6d. + + FISHER.--Size 30 X 12 in., tail 12 to 18 in. long, the largest of the + martens; has a dark shaded deep underwool with fine, glossy, dark and + strong top hair 2 in. or more long. Best obtained from British + America. The tails are almost black and make up most handsomely into + trimmings, muffs, &c. Tails worked separately in these forms are as + rich and fine and more durable than any other fur suitable for a like + purpose. The fur of the skin itself is something like a dark silky + raccoon, but is not as attractive as the tails. Value 12s. to 46s. + + FITCH.--Size 12 X 3 in., of the marten species, also known as the pole + cat. Yellow underwool 1/3 in. deep, black top hair, 1-1/2 to 1-3/4 in. + long, very fine and open in growth, and not close as in martens. + Largest skins come from Denmark, Holland and Germany. The Russian are + smaller, but more silky and, as now dyed, make a cheap and fair + substitute for sable. They are excellent for linings of ladies' coats, + being of light weight and fairly strong in the pelt. English mayors' + and civic officials' robes are frequently trimmed with this fur in + lieu of sable. Value of the German variety 2s. to 5s. 6d. and of the + Russian 7d. to 1s. 4d. + + FOX, BLUE.--Size 24 X 8 in. Underwool thick and long. Top hair fine + and not so plentiful as in other foxes. Found in Alaska, Hudson Bay + territory, Archangel and Greenland. Although called blue, the colour + is a slaty or drab tone. Those from Archangel are more silky and of a + smoky bluish colour and are the most valuable. These are scarce and + consequently dear. The white foxes that are dyed smoke and celestial + blue are brilliant and totally unlike the browner shades of this fox. + Value 34s. to 195s. + + FOX, COMMON.--The variation of size and quality is considerable, and + the colour is anything from grey to red. In Great Britain the animal + is now only regarded for the sport it provides. On the European + continent, however, some hundreds of thousands of skins, principally + German, Russian and Norwegian, are sold annually, for home use, and + for dyeing and exportation, chiefly to the United States. The + qualities do not compare with those species found in North America and + the Arctic circle. The Asiatic, African and South American varieties + are, with the exception of those taken in the mountains, poorly furred + and usually brittle and therefore of no great service. No commercial + value can be quoted. + + FOX, CROSS.--Size 20 X 7 in., are about as large as the silver and + generally have a pale yellowish or orange tone with some silvery + points and a darkish cross marking on the shoulders. Some are very + similar to the pale red fox from the North-West of America and a few + are exceptionally large. The darkest and best come from Labrador and + Hudson Bay, and the ordinary sorts from the north-west of the United + States and, as with silver and other kinds, the quality is inferior + when taken from warmer latitudes. Value 10s. 6d. to 60s. + + FOX, GREY.--Size 27 X 10 in. Has a close dark drab underwool with + yellowish grizzly, grey, regular and coarse top hair. The majority + used for the trade come from Virginia and the southern and western + parts of the United States. Those from the west are larger than the + average, with more fur of a brighter tone. The fur is fairly + serviceable for carriage rugs, the leather being stout, but its + harshness of quality and nondescript colour does not contribute to + make it a favourite. Value 9d. to 4s. 9d. + + FOX, JAPANESE.--See _Fox, Red_, and _Raccoon_, below. + + FOX, KIT.--Size 20 X 6 in. The underwool is short and soft, as is also + the top hair, which is of very pale grey mixed with some + yellowish-white hair. It is the smallest of foxes, and is found in + Canada and the northern section of the United States. It is similar in + colour and quality to the prairie fox and to many kinds from the + warmer zones, such as from Turkey, eastern Asia and elsewhere. Value + 1s. 3d. to 5s. 6d. + + FOX, RED.--Size 24 X 8 in., though a few kinds are much larger. The + underwool is long and soft and the hair plentiful and strong. It is + found widely in the northern parts of America and in smaller numbers + south of the United States, also in China, Japan and Australia. The + colours vary from pale yellowish to a dark red, some being very + brilliant. Those of Kamschatka are rich and fine in quality. Farther + north, especially near the sea, the fur is coarse. Where the best + coloured skins are not used for carriage rugs they are extensively + dyed, and badger and other white hairs are inserted to resemble silver + fox. They are also dyed a sable colour. The skins, being the strongest + of foxes', both in the fur and pelt, are serviceable. The preparations + in imitation of the natural black and silver sorts are very good and + attractive. Value 1s. to 41s. + + FOX, SILVER. Size 30 X 10 in. Underwool close and fine. Top hair black + to silvery, 3 in. long. The fur upon the necks usually runs dark, + almost black, and in some cases the fur is black half-way down the + length of the skin, in rarer cases three-quarters of the length and, + in the most exceptional instances, the whole length, and when this is + the case they are known as "Natural Black Foxes" and fetch enormous + prices. The even silvery sorts are highly esteemed, and the fur is one + of the most effective and precious. The finest are taken in Labrador. + The farther south they are found, the poorer and coarser the fur. The + brush has invariably a white tip. Value L1 to L320. + + FOX, WHITE.--Size 20 X 7 in. Animals of this species are generally + small in size and inhabit the extreme northern sections of Hudson Bay, + Newfoundland, Greenland, Labrador and Siberia. The Canadian are silky + in nature and inclined to a creamy colour, while the Siberian are more + woolly and rather whiter. Those taken in central Asia near or in + Chinese territory are poorer and yellowish. The underwool in all sorts + is generally of a bluish-grey tone, but the top hair in the depth of + winter is usually full enough in quantity to hide any such variation. + Those skins in which the underwool is quite white are rare and much + more expensive. In summer specimens of this species, as with other + white furred animals, have slightly discoloured coats. The skins that + are not perfectly white are dyed jet black, dark or light smoke, + violet-blue, blue-grey, and also in imitation of the drab shades of + the natural blue. Value 18s. to 66s. + + GENET.--Size 10 X 4 in. The genet proper is a small white spotted cat + found in Europe, but the quantity is too small to be of commercial + interest. The name has been adopted for the black cats used so much in + the trade. (See CATS, above.) Value 1s. to 6s. 6d. + + GOATS.--Size varies greatly. The European, Arabian and East Indian + kinds are seldom used for rugs, the skins are chiefly dressed as + leather for books and furniture, and the kids for boots and gloves, + and the finer wool and hair are woven into various materials. Many + from Russia are dyed black for floor and carriage rugs; the hair is + brittle, with poor underwool and not very durable; the cost, however, + is small. The Chinese export thousands of similar skins in black, grey + and white, usually ready dressed and made into rugs of two skins each. + A great many are dyed black and brown, in imitation of bear, and are + used largely in the western parts of the United States and Canada for + sleigh and carriage rugs. Many are used for their leather. Thousands + of the kids are also dyed black and worked into cross-shaped pieces, + in which shape they are largely exported to Germany, France, Great + Britain and America, and sold by the retail as caracal, kid or + caracul. The grey ones are in good demand for motor coats. The word + caracul has been adopted from the Turkish and signifies black-eared. + See also LAMBS, CARACUL. Value of Chinese white 3s. 6d. to 6s. 6d.: + grey, 4s. to 6s. 9d. + + The Angora from the heights of central Asia Minor has curly, fleecy, + silky, white wool, 4 to 7 in. long. The fur is not used in Great + Britain, as formerly, and the greater quantity, known as mohair, is + now imported for purposes of weaving. This species of goat was some + years since introduced into Cape Colony, but its wool is not so good + as the Asiatic breed. Good business, however, is done with the + product, but chiefly for leather. Value 4s. to 12s. 6d. + + The Mongolian goat has a very soft silk underwool, and after the long + top hair is removed it is dressed and imported and erroneously named + mouflon. The colour is a light fawn, but it is so pale that it lends + itself to be dyed any colour. It was popular some years since in the + cheaper trade, but it is not now much seen in England. Value 2s. to + 6s. + + The Tibet goat is similar to the Angora in the fineness of its wool, + and many are used in the making of cashmere shawls. The Tibet lamb so + largely imported and used for children's wear is often miscalled Tibet + goat. Value 3s. to 7s. 6d. + + GUANACO.--Size 30 X 15 in. Is a species of goat found in Patagonia and + other parts of South America. It has a very long neck and exceedingly + soft woolly fur of a light reddish-fawn colour with very white flanks. + It is usually imported in small quantities, native dressed, and ready + made into rugs. The dressing is hard and brittle. If the skins are + dressed in Europe they afford a very comfortable rug, though a very + marked one in effect. They have a similar wool to the vicuna, but + coarser and redder; both are largely used in South America. Value 1s. + to 4s. 6d. + + HAMSTER.--Size 8 X 3-1/2 in. A destructive rodent, is found in great + numbers in Russia and Germany. The fur is very flat and poor, of a + yellowish pale brown with a little marking of black. Being of a light + weight it is used for linings. Value 3d. to 1s. + + HARE.--Size 24 X 9 in. The common hare of Europe does not much + interest the furrier, the fur being chiefly used by makers of hatters' + felt. The white hares, however, of Russia, Siberia and other regions + in the Arctic circle are very largely used in the cheaper trade of + Europe, America and the British colonies. The fur is of the whitest + when killed in winter, and that upon the flanks of the animal is very + much longer than that upon its back. The flanks are usually cut off + and made into muffs and stoles. The hair is, however, brittle and is + not at all durable. This fur is dyed jet black and various shades of + brown and grey, and manufactured into articles for the small drapers + and for exportation. The North American hares are also dyed black and + brown and used in the same way. Value of white 2d. to 5d. + + JACKAL.--Size 2 to 3 ft. long. Is found in India and north and south + Africa. Indian are light brown and reddish, those from the Cape are + dark grey and rather silvery. Few are imported. Fur generally poor and + harsh, only suitable for carriage rugs. Value 1s. to 3s. 6d. + + JAGUAR.--Size 7 to 10 ft. long. Is found in Mexico and British + Honduras. The markings are an irregular ring formation with a spot in + the centre. Leopards have rings only and cheetahs solid spots. + Suitable only for hearth-rugs. Supply very limited. Value 5s. to 45s. + + KALUGA.--See _Souslik_, below. + + KANGAROO.--The sizes vary considerably, some being huge, others quite + small. The larger varieties, viz. the red and the great, do not + usually interest furriers, the fur being harsh and poor without + underwool. They are tanned for the leather trade. The sorts used for + carriage aprons, coat linings and the outside of motor coats include: + blue kangaroo, bush kangaroo, bridled kangaroo, wallaroo, yellow + kangaroo, rock wallaby, swamp wallaby and short-tailed wallaby. Many + of the swamp sort are dyed to imitate skunk and look well. Generally + the colours are yellowish or brown. Some are dark brown as in the + swamp, which being strong are suitable for motor coats. The rock + wallabies are soft and woolly and often of a pretty bluish tone, and + make moderately useful carriage rugs and perambulator aprons. The + redder and browner sorts are also good for rugs as they are thick in + the pelt. On the European continent many of these are dyed. The best + of the lighter weights are frequently insufficiently strong in the + hair to stand the friction of wear in a coat lining. Value, kangaroo + 9d. to 3s., wallaby 1-1/2 d. to 5s. 3d., wallaroo 1s. to 5s. 6d. + + KIDS.--See _Goats_, above. + + KOLINSKY.--Size 12 X 2-1/2 in. Is one of the marten tribe. The + underwool is short and rather weak, but regular, as is also the top + hair; the colour is usually yellow. They have been successfully dyed + and used as a substitute for sable. They are found in Siberia, Amoor, + China and Japan, but the best are from Siberia. They are light in + weight and therefore suitable for linings of coats. The tails are used + for artists' "sable" brushes. The fur has often been designated as red + or Tatar sable. Value 1s. 6d. to 4s. 6d. + + LAMBS.--The sorts that primarily interest the fur trade in Europe and + America are those from south Russia, Persia and Afghanistan, which are + included under the following wholesale or retail commercial terms: + Persian lamb, broadtail, astrachan, Shiraz, Bokharan and caracul lamb. + With the public the general term astrachan is an old one, embracing + all the above curly sorts; the flatter kinds, as broadtail and caracul + lamb, have always been named separately. The Persian lambs, size 18 X + 9 in., are the finest and the best of them. When dressed and dyed they + should have regular, close and bright curl, varying from a small to a + very large one, and if of equal size, regularity, tightness and + brightness, the value is comparatively a matter of fancy. Those that + are dull and loose, or very coarse and flat in the curl, are of far + less market value. + + All the above enumerated lambs are naturally a rusty black or brown, + and with very few exceptions are dyed a jet black. Lustre, however, + cannot be imparted unless the wool was originally of a silky nature. + Broadtails, size 10 X 5 in., are the very young of the Persian sheep, + and are killed before the wool has time to develop beyond the flat + wavy state which can be best compared to a piece of moire silk. They + are naturally exceedingly light in weight, and those that are of an + even pattern, possessing a lustrous sheen, are costly. There is, + notwithstanding, a great demand for these from the fashionable world, + as not only are they very effective, but being so flat in the wool the + figure of the wearer can be shown as perfectly as in a garment made of + silk. It cannot be regarded as an economical fur, as the pelt is too + delicate to resist hard wear. + + Persian Lamb price 12s. 6d. to 25s. + Broadtail " 10s. " 35s. + + Astrachan, Shiraz and Bokharan lambs, size 22 by 9 in., are of a + coarser, looser curl, and chiefly used for coat linings, while the + Persians are used for outside of garments, collars, cuffs, stoles, + muffs, hats and trimmings and gloves. The so-called caracul lambs, + size 12 X 6 in., are the very young of the astrachan sheep, and the + pick of them are almost as effective as broadtails, although less fine + in the texture. See also remarks as to caracul kid under Goats, above. + + Astrachan price 1s. to 5s. 6d. + Caracul Lamb " 2s. 6d " 10s. 6d. + Shiraz " 4s. 6d " 10s. + Bokharan " 1s. 6d " 3s. 6d. + + Grey lambs, size 24 X 10 in., are obtained from the Crimea and known + in the trade as "crimmers." They are of a similar nature to the + caracul lambs, but looser in curl, ranging from a very light to a dark + grey. The best are the pale bluish greys, and are chiefly used for + ladies' coats, stoles, muffs and hats. Price 2s. to 6s. Mongolian + lambs, size 24 X 15 in., are of a short wavy loose curl, creamy white + colour, and are usually exported from China dressed, the majority + being ready-made into cross-shaped coats or linings. They are used + principally for linings of good evening wraps for ladies. Price 1s. to + 2s. 6d. Slink lambs come from South America and China. The former are + very small and generally those that are stillborn. They have a + particularly thin pelt with very close wool of minute curl. The China + sorts are much larger. The smallest are used for glove linings and the + others for opera cloak linings. Price 1s. to 6s. 6d. + + LEOPARD.--Size 3 to 6 ft. long. There are several kinds, the chief + being the snow or ounce, Chinese, Bengal, Persian, East Indian and + African. The first variety inhabit the Himalayas and are beautifully + covered with a deep soft fur quite long compared to the flat harsh + hair of the Bengal sort. The colours are pale orange and white with + very dark markings, a strong contrast making a fine effect. Most + artists prize these skins above all others. The Chinese are of a + medium orange brown colour, but full in fur. The East Indian are less + full and not so dark. The Bengal are dark and medium in colour, short + and hard hair, but useful for floor rugs, as they do not hold the dust + like the fuller and softer hair of the kinds previously named. They + are also used for drummers' aprons and saddle cloths in the Indian + army. The African are small with pale lemon colour grounds very + closely marked with black spots on the skin, the strong contrast + making a pleasing effect. Occasionally, where something very marked is + wanted, skating jackets and carriage aprons are made from the softest + and flattest of skins, but usually they are made into settee covers, + floor rugs and foot muffs. Value 2s. to 40s. + + LION.--Size 5 to 6 ft. long. These skins are found in Africa, Arabia + and part of India, and are every year becoming scarcer. They are only + used for floor rugs, and the males are more highly esteemed on account + of the set-off of the mane. Value, lions' L10 to L100; lionesses' L5 + to L25. + + LYNX.--Size 45 X 20 in. The underwool is thinner than fox, but the top + hair is fine, silky and flowing, 4 in. long, of a pale grey, slightly + mottled with fine streaks and dark spots. The fur upon the flanks is + longer and white with very pronounced markings of dark spots, and this + part of the skin is generally worked separately from the rest and is + very effective for gown trimmings. Where the colour is of a sandy and + reddish hue the value is far less than where it is of a bluish tone. + They inhabit North America as far south as California, also Norway and + Sweden. Those from the Hudson Bay district and Sweden are the best and + are very similar. Those taken in Central Asia are mostly used locally. + For attire the skins manufactured in Europe are generally dyed black + or brown, in which state it has a similar appearance to dyed fox, but + having less thick underwool and finer hair flows freely. The finest + skins when dyed black are used very largely in America in place of the + dyed black fox so fashionable for mourning wear in Great Britain and + France. The British Hussar busbies are made of the dark brown lynx, + and it is the free silky easy movement of the fur with the least + disturbance in the atmosphere that gives it such a pleasing effect. It + is used for rugs in its natural state and also in Turkey as trimmings + for garments. Value 13s. 6d. to 56s. + + LYNX CAT or BAY LYNX.--Is about half the size and depth of fur of a + lynx proper, and inhabits the central United States. It is a flat and + reddish fur compared to the lynx and is suitable for cheap carriage + aprons. A few come from Canada and are of better quality. Value 5s. to + 15s. + + MARMOT.--Size 18 X 12 in. Is a rodent and is found in considerable + numbers in the south of Prussia. The fur is a yellowish brown and + rather harsh and brittle and has no underwool. Since, however, the + value of all good furs has advanced, dyers and manufacturers have made + very successful efforts with this fur. The Viennese have been + particularly successful, and their method has been to dye the skins a + good brown and then not put in the dark stripes, which exist in sable + and mink, until the garment or article is finished, thus obtaining as + perfectly symmetrical effects as if the articles were made of small + skins instead of large ones. Marmots are also found in North America, + Canada and China; the best, however, come from Russia. It should + always be a cheap fur, having so few good qualities to recommend it. + Value 9d. to 2s. 6d. + + MARTEN, AMERICAN.--See _Sable_, below. + + MARTEN, BAUM.--Size 16 X 5 in. Is sometimes called the pine marten, + and is found in quantity in the wooded and mountainous districts of + Russia, Norway, Germany and Switzerland. It possesses a thick + underwool with strong top hair, and ranges from a pale to a dark + bluish brown. The best, from Norway, are very durable and of good + appearance and an excellent substitute for American sable. The tails + when split into two or three, with small strips of narrow tape so as + to separate the otherwise dense fur, formerly made very handsome sets + of trimmings, ties and muffs, and the probabilities are, as with other + fashions, such use will have its period of revival. Value 6s. to 85s. + + MARTEN, BLACK.--See _Skunk_, below. + + MARTEN, JAPANESE.--Size 16 X 5 in. Is of a woolly nature with rather + coarse top hair and quite yellow in colour. It is dyed for the cheap + trade for boas and muffs, but it is not an attractive fur at the best + of times. It lacks a silky, bright and fresh appearance, and therefore + is unlikely to be in great demand, except where economy is an object. + Value 6s. 6d. to 18s. 6d. + + MARTEN, STONE.--Size and quality similar to the baum; the colour, + however, of the underwool is a stony white and the top hair is very + dark, almost black. They live in rocky and stony districts. Skins of a + pale bluish tone are generally used in their natural state for stoles, + boas and muffs, but the less clear coloured skins are dyed in + beautiful shades similar in density to the dark and valuable sables + from Russia, and are the most effective skins that can be purchased at + a reasonable price. The tails have also been worked, in the manner + explained with regard to the baum marten, as sets of trimmings and in + other forms. Stone martens are found in Russia, Bosnia, Turkey, + Greece, Germany, the Alps and France. The Bosnian and the French are + the best in colour. The Asiatic sorts are less woolly, but being silky + are useful when dyed. There are many from Afghanistan and India which + are too poor to interest the European markets. Value 7s. 6d. to 26s. + + MINK.--Size 16 X 5 in. Is of the amphibious class and is found + throughout North America and in Russia, China and Japan. The underwool + is short, close and even, as is also the top hair, which is very + strong. The best skins are very dark and are obtained from Nova + Scotia. In the central states of America the colour is a good brown, + but in the north-west and south-west the fur is coarse and generally + pale. It is very durable for linings, and is an economical substitute + for sable for coats, capes, boas and trimmings. Values have greatly + increased, and the fur possessing good qualities as to colour and + durability will doubtless always be in good request. The Russian + species is dark but flat and poor in quality, and the Chinese and + Japanese are so pale that they are invariably dyed. These, however, + are of very inferior nature. Value of American 3s. 3d. to 40s., + Japanese 3d. to 2s. 3d. + + MOLE.--Size 3-1/2 X 2-1/2 in. Moles are plentiful in the British Isles + and Europe, and owing to their lovely velvety coats of exquisite blue + shade and to the dearness of other furs are much in demand. Though the + fur is cheap in itself, the expense of dressing and working up these + little skins is considerable, and they possess the unique charm of an + exceptional colour with little weight of pelt; the quality of + resistance to friction is, however, so slight as to make them + expensive in wear. The best are the dark blue from the Fen district of + Cambridgeshire in England. Value 1/2d. to 2d. + + MONGOLIAN LAMBS.--See _Lambs_, above. + + MONKEY, BLACK.--Size 18 X 10 in. Among the species of monkeys only one + interests to any extent the fur trade, and that is the black monkey + taken on the west coast of Africa (_Colobus satanas_). The hair is + very long, very black and bright with no underwool, and the white pelt + of the base of the hair, by reason of the great contrast of colour, is + very noticeable. The skins were in 1850 very fashionable in England + for stoles, muffs and trimmings, and in America also as recently as + 1890. They are now mostly bought for Germany and the continent. Value + 6d. to 1s. 6d. + + MOUFLON.--Size 30 X 15 in. Is a sheep found in Russia and Corsica and + now very little in demand, and but few are imported into Great + Britain. Many Mongolian goats with the long hairs pulled out are sold + as mouflon. Value 4s. to 10s. 6d. + + MUSK-OX.--Size 6 X 3 ft. These animals have a dense coat of fine, long + brown wool, with very long dark brown hair on the head, flanks and + tail, and, in the centre, a peculiar pale oval marking. There is no + other fur that is so thick, and it is eminently suitable for sleighing + rugs, for which purpose it is highly prized in Canada. The musk-ox + inhabits the north part of Greenland and part of Canada, but in very + limited numbers. Value 10s. to 130s. + + MUSQUASH or MUSK-RAT, BROWN and BLACK RUSSIAN.--Size 12 X 8 in. A very + prolific rodent of the amphibious class obtained from Canada and the + United States, similar in habit to the English vole, with a fairly + thick and even brown underwool and rather strong top dark hair of + medium density. It is a very useful fur for men's coat linings and + ladies' driving or motoring coats, being warm, durable and not too + heavy. If the colour were less motley and the joins between the skins + could be made less noticeable, it would be largely in demand for + stoles, ties and muffs. As it is, this fur is only used for these + smaller articles for the cheaper trade. It has, however, of later + years been "unhaired," the underwool clipped very even and then dyed + seal colour, in which way very useful and attractive garments are + supplied at less than half the cost of the cheaper sealskins. They do + not wear as well, however, as the pelt and the wool are not of a + strength comparable to those of sealskin. With care, however, such a + garment lasts sufficiently long to warrant the present outlay. Value + 5-1/2d. to 1s. 9d. + + There is a so-called black variety found in Delaware and New Jersey, + but the number is very small compared to the brown species. They are + excellent for men's coat linings and the outside of ladies' coats, for + stoles, muffs, collars and cuffs. Value 10d. to 3s. 7d. + + The Russian musquash is very small, 7 X 4 in., and is limited in + numbers compared to the brown. Only a few thousands are imported to + London. It is of a very pretty silvery-blue shade of even wool with + very little silky top hair, having silvery-white sides and altogether + a very marked effect. The odour, however, even after dressing is + rather pungent of musk, which is generally an objection. Value 4s. to + 6s. 6d. + + NUTRIA.--Size 20 X 12 in. Is a rodent known in natural history as the + coypu, about half the size of a beaver, and when unhaired has not more + than half, generally less, the depth of fur, which is also not so + close. Formerly the fur was only used for hatters' felt, but with the + rise in prices of furs these skins have been more carefully removed + and--with improved dressing, unhairing and silvering processes--the + best provides a very effective and suitable fur for ladies' coats, + capes, stoles, muffs, hats and gloves, while the lower qualities make + very useful, light-weighted and inexpensive linings for men's or + women's driving coats. It is also dyed sealskin colour, but its woolly + nature renders it less effective than the more silky musquash. They + are obtained from the northern part of South America. Value is. 6d. to + 6s. 6d. + + OCELOT.--Size 36 X 13 in. Is of the nature of a leopard and prettily + marked with stripes and oblong spots. Only a few are now imported from + South America for carriage aprons or mats. The numbers are very + limited. Value 1s. to 2s. 6d. + + OPOSSUM, AMERICAN.--Size 18 X 10 in. Is a marsupial, a class with this + exception not met with out of Australia. The underwool is of a very + close frizzy nature, and nearly white, with long bluish grey mixed + with some black top hair. It is only found in the central sections of + the United States. About 1870 in England it was dyed dark brown or + black and used for boas, muffs and trimmings, but until recently has + been neglected on the continent. With, however, recent experiments in + brown and skunk coloured dyes, it bids fair to become a popular fur. + Value 2-1/2d. to 5s. 6d. + + OPOSSUM, AUSTRALIAN.--Size 16 X 8 in. Is a totally different nature of + fur to the American. Although it has wool and top hair, the latter is + so sparse and fine that the coat may be considered as one of close + even wool. The colour varies according to the district of origin, from + a blue grey to yellow with reddish tones. Those from the neighbourhood + of Sydney are light clear blue, while those from Victoria are dark + iron grey and stronger in the wool. These animals are most prolific + and evidently increasing in numbers. Their fur is pretty, warm and as + yet inexpensive, and is useful for rugs, coat linings, stoles, muffs, + trimmings and perambulator aprons. The worst coloured ones are + frequently dyed black and brown. The most pleasing natural grey come + from Adelaide. The reddest are the cheapest. Value 3-3/4d. to 3s. 6d. + + OPOSSUM, RINGTAILED.--Size 7 X 4 in. Has a very short close and dark + grey wool, some being almost black. There are but a few thousands + imported, and being so flat they are only of use for coat linings, but + they are very warm and light in weight. Value 6d. to 10d. + + OPOSSUM, TASMANIAN (grey and black).--Size 20 X 10 in. Is of a similar + description, but darker and stronger in the wool and larger. Besides + these there are some very rich brown skins which were formerly in such + request in Europe, especially Russia, that undue killing occurred + until 1899, when the government stopped for a time the taking of any + of this class. They are excellent for carriage aprons, being not only + very light in weight and warm, but handsome. Value 2s. 6d. to 8s. 6d. + + OTTER, RIVER.--The size varies considerably, as does the underwool and + the top hair, according to the country of origin. There are few rivers + in the world where they do not live. But it is in the colder northern + regions that they are found in the greatest numbers and with the best + fur or underwool, the top hair, which, with the exception of the + scarce and very rich dark brown specimens they have in common with + most aquatic animals, is pulled out before the skins are manufactured. + Most of the best river otter comes from Canada and the United States + and averages 36 X 18 in. in size. Skins from Germany and China are + smaller, and shorter in the wool. The colours of the under wools of + river otters vary, some being very dark, others almost yellow. Both as + a fur and as a pelt it is extremely strong, but owing to its short and + close wool it is usually made up for the linings, collars and cuffs of + men's coats. A large number of skins, after unhairing, is dyed seal + colour and used in America. Those from hot climates are very poor in + quality. Value 28s. to 118s. + + OTTER, SEA.--Size 50 X 25 in. Possesses one of the most beautiful of + coats. Unlike other aquatic animals the skin undergoes no process of + unhairing, the fur being of a rich dense silky wool with the softest + and shortest of water hairs. The colours vary from pale grey brown to + a rich black, and many have even or uneven sprinkling of white or + silvery-white hairs. The blacker the wool and the more regular the + silver points, the more valuable the skin. Sea otters are, + unfortunately, decreasing in numbers, while the demand is increasing. + The fur is most highly esteemed in Russia and China; in the latter + country it is used to trim mandarins' state robes. In Europe and + America it is much used for collar, long facings and cuffs of a + gentleman's coat; such a set may cost from L200 to L600, and in all + probability will soon cost more. Taking into consideration the size, + it is not so costly as the natural black fox, or the darkest Russian + sable, which is now the most expensive of all. The smaller and young + sea otters of a grey or brown colour are of small value compared to + the large dark and silvery ones. Value L10 to L220. A single skin has + been known to fetch L400. + + OUNCE.--See _Leopard_, above. + + PERSIAN LAMBS.--See _Lambs_, above. + + PLATYPUS.--Size 12 X 8 in. One of the most singular of fur-bearing + animals, being the link between bird and beast. It has fur similar to + otter, is of aquatic habits, being web-footed with spurs of a cock and + the bill of a duck. The skins are not obtained in any numbers, but + being brought over by travellers as curiosities and used for muffs, + collars and cuffs, &c., they are included here for reference. Value + 2s. to 3s. 6d. + + _Pony_ or _Tatar Foal._--Size 36 X 20 in. These skins are of + comparatively recent importation to the civilized world. They are + obtained from the young of the numerous herds of wild horses that roam + over the plains of Turkestan. The coat is usually a shade of brown, + sometimes greyish, fairly bright and with a suggestion of waviness. + Useful for motor coats. Value 3s. to 10s. 6d. + + PUMA.--Size 4-1/2 X 3 ft. Is a native of South America, similar to a + lion in habits and colour of coat. The hair and pelt is, however, of + less strength, and only a few are now used for floor rugs. Value 5s. + to 10s. + + RACCOON.--Size 20 X 12 in. Is an animal varying considerably in size + and in quality and colour of fur, according to the part of North + America in which it is found. In common parlance, it may be described + as a species of wild dog with close affinity to the bear. The + underwool is 1 to 1-1/2 in. deep, pale brown, with long top hairs of a + dark and silvery-grey mixture of a grizzly type, the best having a + bluish tone and the cheapest a yellowish or reddish-brown. A limited + number of very dark and black sorts exist and are highly valued for + trimmings. The very finest skins are chiefly used for stoles and + muffs, and the general run for coachmen's capes and carriage rugs, + which are very handsome when the tails, which are marked with rings of + dark and light fur alternately, are left on. Raccoons are used in + enormous quantities in Canada for men's coats, the fur outside. The + poorer qualities are extensively bought and made up in a similar way + for Austria-Hungary and Germany. These make excellent linings for + coats or footsacks for open driving in very cold climates. The worst + coloured skins are dyed black or brown and are used for British + military busbies, or caps, stoles, boas, muffs and coachmen's capes. + The best skins come from the northern parts of the United States. A + smaller and poorer species inhabits South America, and a very few are + found in the north of India, but these do not interest the European + trade. From Japan a similar animal is obtained in smaller quantities + with very good but longer fur, of yellowish motley light-brown shades. + It is more often imported and sold as Japanese fox, but its + resemblance to the fur of the American raccoon is so marked as to + surely identify it. When dyed dark blue or skunk colour it is + good-looking and is sold widely in Europe. Raccoon skins are also + frequently unhaired, and if the underwool is of good quality the + effect is similar to beaver. It is the most useful fur for use in + America or Russia, having a full quantity of fur which will retain + heat. Value 10d. to 26s. + + SABLE, AMERICAN and CANADIAN.--Size 17 X 5 in. The skins are sold in + the trade sale as martens, but as there are many that are of a very + dark colour and the majority are almost as silky as the Russian sable, + the retail trade has for generations back applied the term of sable to + this fur. The prevailing colour is a medium brown, and many are quite + yellow. The dyeing of these very pale skins has been for so long well + executed that it has been possible to make very good useful and + effective articles of them at a moderate price compared to Russian + sable. The finest skins are found in the East Main and the Esquimaux + Bay, in the Hudson's Bay Company's districts, and the poorest in + Alaska. They are not found very far south of the northern boundary of + the United States. The best skins are excellent in quality, colour and + effect, and wear well. Value 27s. 3d. to 290s. + + SABLE, CHINESE and JAPANESE.--Size 14 X 4-1/2 in. These are similar to + the Amur skins previously referred to, but of much poorer quality and + generally only suitable for linings. The very palest skins are dyed + and made by the Chinese into mandarins' coats, in which form they are + found in the London trade sales, but being overdressed they are + inclined to be loose in the hair and the colour of the dye is not + good. The Japanese kind are imported raw, but are few in numbers, very + pale and require dyeing. Value 15s. to 150s. + + SABLE, RUSSIAN.--Size 15 X 5 in. These skins belong to a species of + marten, very similar to the European and American, but much more silky + in the nature of their fur. They have long been known as "sables," + doubtless owing to the density of colour to which many of them attain, + and they have always been held in the highest esteem by connoisseurs + as possessing a combination of rare qualities. The underwool is close, + fine and very soft, the top hair is regular, fine, silky and flowing, + varying from 1-1/2 to 2-1/2 in. in depth. In colour they range from a + pale stony or yellowish shade to a rich dark brown, almost black with + a bluish tone. The pelts are exceedingly fine and close in texture + and, although of little weight, are very durable, and articles made of + them produce a sensation of warmth immediately they are put upon the + body. + + The Yakutsk, Okhotsk and Kamschatka sorts are good, the last being the + largest and fullest furred, but of less density of colour than the + others. Many from other districts are pale or yellowish brown, and + those from Saghalien are poor in quality. The most valuable are the + darkest from Yakutsk in Siberia, particularly those that have silvery + hairs evenly distributed over the skin. These however are exceedingly + scarce, and when a number are required to match for a large garment, + considerable time may be necessary to collect them. This class of skin + is the most expensive fur in the world, reckoning values by a square + foot unit. + + The Amur skins are paler, but often of a pretty bluish stony tone with + many frequently interspersed silvery hairs. The quality too is lower, + that is, the fur is not so close or deep, but they are very effective, + particularly for close-fitting garments, as they possess the least + appearance of bulk. The paler skins from all districts in Siberia are + now cleverly coloured or "topped," that is, just the tips of the hair + are stained dark, and it is only an expert who can detect them from + perfectly natural shades. If this colouring process is properly + executed it remains fairly fast. Notwithstanding the reported rights + of the Russian imperial authorities over some regions with respect to + these and other valuable fur-bearing animals, there are in addition to + the numbers regularly sent to the trade auction sales in London many + good parcels of raw skins to be easily bought direct, provided price + is not the first consideration. Value 25s. to 980s. + + SEAL, FUR.--Sizes range from 24 X 15 in. to 55 X 25 in., the width + being taken at the widest part of the skin after preparation. The + centre of the skin between the fins is very narrow and the skins taper + at each end, particularly at the tail. The very small pups are of a + beautiful quality, but too tiny to make into garments, and, as the aim + of a good furrier is to avoid all lateral or cross seams, skins are + selected that are the length of the garment that is to be made. The + most useful skins for coats are the large pups 42 in. long, and the + quality is very good and uniform. The largest skins, known in the + trade as "wigs," which range up to 8 ft. in length, are uneven and + weak in the fur, and hunters do not seek to obtain them. The supply of + the best sort is chiefly from the North Pacific, viz. Pribilof + Islands, Alaska, north-west coast of America, Copper Island of the + Aleutian group near to Kamschatka, Robben Island and Japan. Other + kinds are taken from the South Pacific and South Atlantic Oceans, + around Cape Horn, the Falkland Islands up to Lobos Islands at the + entrance of the La Plata river, off the Cape of Good Hope and Crozet + Isles. With, however, the exception of the pick of the Lobos Island + seals the fur of the southern sea seals is very poor and only suitable + for the cheapest market. Formerly many skins were obtained from New + Zealand and Australia, but the importation is now small and the + quality not good. The preparation of seal skin occupies a longer time + than any other fur skin, but its fine rich effect when finished and + its many properties of warmth and durability well repay it. Value 10s. + to 232s. + + SEAL, HAIR.--There are several varieties of these seals in the seas + stretching north from Scotland, around Newfoundland, Greenland and the + north-west coast of America, and they are far more numerous than fur + seals. Generally they have coarse rigid hair and none possess any + underwool. They are taken principally for the oil and leather they + yield. Some of the better haired sorts are dyed black and brown and + used for men's motor coats when quite a waterproof garment is wanted, + and they are used also for this quality in China. The young of the + Greenland seals are called whitecoats on account of the early growth + being of a yellowish white colour; the hair is 3/4 to 1 in. long, and + at this early stage of their life is soft compared to that of the + older seals. These fur skins are dyed black or dark brown and are used + for military caps and hearth-rugs. Value 2s. to 15s. There are fewer + hair seals in the southern than in the northern seas. + + SHEEP.--Vary much in size and in quality of wool. Many of the domestic + kind in central and northern Europe and Canada are used for drivers' + and peasants' coat linings, &c. In Great Britain many coats of the + home-reared sheep, having wools two and a half to five inches long, + are dyed various colours and used as floor rugs. Skins with very short + wool are dyed black and used for military saddle-cloths. The bulk, + however, is used in the wool trade. The Hungarian peasants are very + fond of their natural brown sheep coats, the leather side of which is + not lined, but embellished by a very close fancy embroidery, worked + upon the leather itself; these garments are reversible, the fur being + worn inside when the weather is cold. Chinese sheep are largely used + for cheap rugs. Value of English sheep from 3s. to 10s. + + SKUNK or BLACK MARTEN.--Size 15 X 8 in. The underwool is full and + fairly close with glossy, flowing top hair about 2-1/2 in. long. The + majority have two stripes of white hair, extending the whole length of + the skin, but these are cut out by the manufacturing furrier and sold + to the dealers in pieces for exportation. The animals are found widely + spread throughout North and South America. The skins which are of the + greatest interest to the European trade are those from North America, + the South American species being small, coarse and generally brown. + The best skins come from Ohio and New York. If it were not for its + disagreeable odour, skunk would be worth much more than the usual + market value, as it is naturally the blackest fur, silky in appearance + and most durable. The improved dressing processes have to a large + extent removed the naturally pungent scent. The fur is excellent for + stoles, boas, collars, cuffs, muffs and trimmings. Value 1s. 6d. to + 11s. + + SOUSLIK.--Size 7 in. X 2-1/4. Is a small rodent found in the south of + Russia and also in parts of America. It has very short hair and is a + poor fur even for the cheapest linings, which is the only use to which + the skin could be put. It is known as kaluga when imported in + ready-made linings from Russia where the skins are dressed and worked + in an inferior way. Value 1d. to 3d. + + SQUIRREL.--Size 10 X 5 in. This measurement refers to the Russian and + Siberian sorts, which are the only kind imported for the fur. The + numerous other species are too poor in their coats to attract notice + from fur dealers. The back of the Russian squirrel has an even close + fur varying from a clear bluish-grey to a reddish-brown, the bellies + in the former being of a flat quality and white, in the latter + yellowish. The backs are worked into linings separately, as are the + bellies or "locks." The pelts, although very light, are tough and + durable, hence their good reputation for linings for ladies' walking + or driving coats. The best skins also provide excellent material for + coats, capes, stoles, ties, collars, cuffs, gloves, muffs, hoods and + light-weight carriage aprons. The tails are dark and very small, and + when required for ends of boas three or four are made as one. Value + per skin from 2-1/2d. to 1s. 1d. + + TIBET LAMB.--Size 27 X 13 in. These pretty animals have a long, very + fine, silky and curly fleece of a creamy white. The majority are + consigned to the trade auction sales in London ready dressed and + worked into cross-shaped coats, and the remainder, a fourth of the + total, come as dressed skins. They are excellent for trimmings of + evening mantles and for children's ties, muffs and perambulator + aprons. The fur is too long and bulky for linings. Value per skin from + 4s. 6d. to 8s. 6d. + + TIGER.--Size varies considerably, largest about 10 ft. from nose to + root of tail. Tigers are found throughout India, Turkestan, China, + Mongolia and the East Indies. The coats of the Bengal kind are short + and of a dark orange brown with black stripes, those from east or + further India are similar in colour, but longer in the hair, while + those from north of the Himalayas and the mountains of China are not + only huge in size, but have a very long soft hair of delicate orange + brown with very white flanks, and marked generally with the blackest + of stripes. The last are of a noble appearance and exceedingly scarce. + They all make handsome floor rugs. + + Value of the Indian from L3 to L15. + " " Chinese " L10 to L65. + + VICUNA is a species of long-necked sheep native to South America, + bearing some resemblance to the guanaco, but the fur is shorter, + closer and much finer. The colour is a pale golden-brown and the fur + is held in great repute in South America for carriage rugs. The supply + is evidently small as the prices are high. There is scarcely a + commercial quotation in London, few coming in except from private + sources. 2s. 6d. to 5s. 6d. may be considered as the average value. + + WALLABY.--See _Kangaroo_, above. + + WALLAROO.--See _Kangaroo_, above. + + WOLF.--Size 50 X 25 in. Is closely allied to the dog tribe and, like + the jackals, is found through a wide range of the world,--North and + South America, Europe and Asia. Good supplies are available from North + America and Siberia and a very few from China. The best are the full + furred ones of a very pale bluish-grey with fine flowing black top + hair, which are obtained from the Hudson Bay district. Those from the + United States and Asia are harsher in quality and browner. A few black + American specimens come into the market, but usually the quality is + poor compared to the lighter furred animal. The Siberian is smaller + than the North American and the Russian still smaller. Besides the + wolf proper a large number of prairie or dog wolves from America and + Asia are used for cheaper rugs. In size they are less than half that + of a large wolf and are of a motley sandy colour. Numbers of the + Russian are retained for home use. The finest wolves are very light + weighted and most suitable for carriage aprons, in fact, ideal for the + purpose, though lacking the strength of some other furs. + + Wolves value 2s. 6d. to 64s. + Dog wolves " 1s. to 2s. 6d. + + WOLVERINE.--Size 16 X 18 in. Is native to America, Siberia, Russia and + Scandinavia and generally partakes of the nature of a bear. The + underwool is full and thick with strong and bright top hair about + 2-1/2 in. long. The colour is of two or three shades of brown in one + skin, the centre being an oval dark saddle, edged as it were with + quite a pale tone and merging to a darker one towards the flanks. This + peculiar character alone stamps it as a distinguished fur, in addition + to which it has the excellent advantage of being the most durable fur + for carriage aprons, as well as the richest in colour. It is not + prolific, added to which it is very difficult to match a number of + skins in quality as well as colour. Hence it is an expensive fur, but + its excellent qualities make it valuable. The darkest of the least + coarse skins are worth the most. Prices from 6s. to 37s. + + WOMBAT, KOALA or Australian Bear.--Size 20 X 12 in. Has light grey or + brown close thick wool half an inch deep without any top hair, with a + rather thick spongy pelt. It is quite inexpensive and only suitable + for cheap rough coats, carriage rugs, perambulator aprons and linings + for footbags. The coats are largely used in western America and + Canada. Value 3d. to 1s. 8-1/2d. + +_Preparing and Dressing._--A furrier or skin merchant must possess a +good eye for colour to be successful, the difference in value on this +subtle matter solely (in the rarer precious sorts, especially sables, +natural black, silver and blue fox, sea otters, chinchillas, fine mink, +&c.) being so considerable that not only a practised but an intuitive +sense of colour is necessary to accurately determine the exact merits of +every skin. In addition to this a knowledge is required of what the +condition of a pelt should be; a good judge knows by experience whether +a skin will turn out soft and strong, after dressing, and whether the +hair is in the best condition of strength and beauty. The dressing of +the pelt or skin that is to be preserved for fur is totally different to +the making of leather; in the latter tannic acid is used, but never +should be with a fur skin, as is so often done by natives of districts +where a regular fur trade is not carried on. The results of applying +tannic acid are to harden the pelt and discolour and weaken the fur. The +best methods for dressing fur skins are those of a tawer or currier, the +aim being to retain all the natural oil in the pelt, in order to +preserve the natural colour of the fur, and to render the pelt as supple +as possible. Generally the skins are placed in an alkali bath, then by +hand with a blunt wooden instrument the moisture of the pelt is worked +out and it is drawn carefully to and fro over a straight, dull-edged +knife to remove any superfluous flesh and unevenness. Special grease is +then rubbed in and the skin placed in a machine which softly and +continuously beats in the softening mixture, after which it is put into +a slowly revolving drum, fitted with wooden paddles, partly filled with +various kinds of fine hard sawdust according to the nature of the furs +dealt with. This process with a moderate degree of heat thoroughly +cleans it of external greasy matter, and all that is necessary before +manufacturing is to gently tap the fur upon a leather cushion stuffed +with horsehair with smooth canes of a flexibility suited to the strength +of the fur. After dressing most skins alter in shape and decrease in +size. + +With regard to the merits of European dressing, it may be fairly taken +that English, German and French dressers have specialities of +excellence. In England, for instance, the dressing of sables, martens, +foxes, otters, seals, bears, lions, tigers and leopards is first rate; +while with skunk, mink, musquash, chinchillas, beavers, lambs and +squirrels, the Germans show better results, particularly in the last. +The pelt after the German dressing is dry, soft and white, which is due +to a finishing process where meal is used, thus they compare favourably +with the moister and consequently heavier English finish. In France they +do well with cheaper skins, such as musquash, rabbit and hare, which +they dye in addition to dressing. Russian dressing is seldom reliable; +not only is there an unpleasant odour, but in damp weather the pelts +often become clammy, which is due to the saline matter in the dressing +mixture. Chinese dressing is white and supple, but contains much powder, +which is disagreeable and difficult to get rid of, and in many instances +the skin is rendered so thin that the roots of the fur are weakened, +which means that it is liable to shed itself freely, when subject to +ordinary friction in handling or wearing. American and Canadian dressing +is gradually improving, but hitherto their results have been inferior to +the older European methods. + +In the case of seal and beaver skins the process is a much more +difficult one, as the water or hard top hairs have to be removed by hand +after the pelt has been carefully rendered moist and warm. With seal +skins the process is longer than with any other fur preparation and the +series of processes engage many specialists, each man being constantly +kept upon one section of the work. The skins arrive simply salted. After +being purchased at the auction sales they are washed, then stretched +upon a hoop, when all blubber and unnecessary flesh is removed, and the +pelt is reduced to an equal thickness, but not so thin as it is finally +rendered. Subsequently the hard top hairs are taken out as in the case +of otters and beavers and the whole thoroughly cleaned in the revolving +drums. The close underwool, which is of a slightly wavy nature and +mostly of a pale drab colour, is then dyed by repeated applications of a +rich dark brown colour, one coat after another, each being allowed to +thoroughly dry before the next is put on, till the effect is almost a +lustrous black on the top. The whole is again put through the cleaning +process and evenly reduced in thickness by revolving emery wheels, and +eventually finished off in the palest buff colour. + +The English dye for seals is to-day undoubtedly the best; its +constituents are more or less of a trade secret, but the principal +ingredients comprise gall nuts, copper dust, camphor and antimony, and +it would appear after years of careful watching that the atmosphere and +particularly the water of London are partly responsible for good and +lasting results. The Paris dyers do excellent work in this direction, +but the colour is not so durable, probably owing to a less pure water. +In America of late, strides have been made in seal dyeing, but +preference is still given to London work. In Paris, too, they obtain +beautiful results in the "topping" or colouring Russian sables and the +Germans are particularly successful in dyeing Persian lambs black and +foxes in all blue, grey, black and smoke colours and in the insertion of +white hairs in imitation of the real silver fox. Small quantities of +good beaver are dyed in Russia occasionally, and white hairs put in so +well that an effect similar to sea otter is obtained. + +The process of inserting white hairs is called in the trade "pointing, +"and is either done by stitching them in with a needle or by adhesive +caoutchouc. + +The Viennese are successful in dyeing marmot well, and their cleverness +in colouring it with a series of stripes to represent the natural +markings of sable which has been done after the garments have been made, +so as to obtain symmetry of lines, has secured for them a large trade +among the dealers of cheap furs in England and the continent. + +_Manufacturing Methods and Specialities._--In the olden times the +Skinners' Company of the city of London was an association of furriers +and skin dressers established under royal charter granted by Edward III. +At that period the chief concern of the body was to prevent buyers from +being imposed upon by sellers who were much given to offering old furs +as new; a century later the Skinners' Company received other charters +empowering them to inspect not only warehouses and open markets, but +workrooms. In 1667 they were given power to scrutinize the preparing of +rabbit or cony wool for the wool trade and the registration of the then +customary seven years' apprenticeship. To-day all these privileges and +powers are in abeyance, and the interest that they took in the fur trade +has been gradually transferred to the leather-dressing craft. + +The work done by English furriers was generally good, but since about +1865 has considerably improved on account of the influx of German +workmen, who have long been celebrated for excellent fur work, being In +their own country obliged to satisfy officially appointed experts and to +obtain a certificate of capacity before they can be there employed. The +French influence upon the trade has been, and still is, primarily one of +style and combination of colour, bad judgment in which will mar the +beauty of the most valuable furs. It is a recognized law among +high-class furriers that furs should be simply arranged, that is, that +an article should consist of one fur or of two furs of a suitable +contrast, to which lace may be in some cases added with advantage. As +illustrative of this, it may be explained that any brown tone of fur +such as sable, marten, mink, black marten, beaver, nutria, &c., will go +well upon black or very dark-brown furs, while those of a white or grey +nature, such as ermine, white lamb, chinchilla, blue fox, silver fox, +opossum, grey squirrel, grey lamb, will set well upon seal or black +furs, as Persian lamb, broadtail, astrachan, caracul lamb, &c. White is +also permissible upon some light browns and greys, but brown motley +colours and greys should never be in contrast. One neutralizes the other +and the effect is bad. The qualities, too have to be considered--the +fulness of one, the flatness of the other, or the coarseness or fineness +of the furs. The introduction of a third fur in the same garment or +indiscriminate selection of colours of silk linings, braids, buttons, +&c., often spoils an otherwise good article. + +With regard to the natural colours of furs, the browns that command the +highest prices are those that are of a bluish rather than a reddish +tendency. With greys it is those that are bluish, not yellow, and with +white those that are purest, and with black the most dense, that are +most esteemed and that are the rarest. + +Perhaps for ingenuity and the latest methods of manipulating skins in +the manufacturing of furs the Americans lead the way, but as fur cutters +are more or less of a roving and cosmopolitan character the larger fur +businesses in London, Berlin, Vienna, St Petersburg, Paris and New York +are guided by the same thorough and comparatively advanced principles. + +During the period just mentioned the tailors' methods of scientific +pattern cutting have been adopted by the leading furriers in place of +the old chance methods of fur cutters, so that to-day a fur garment may +be as accurately and gracefully fitted as plush or velvet, and with all +good houses a material pattern is fitted and approved before the skins +are cut. + +Through the advent of German and American fur sewing-machines since +about 1890 fur work has been done better and cheaper. There are, +however, certain parts of a garment, such as the putting in of sleeves +and placing on of collars, &c., that can only be sewn by hand. For +straight seams the machines are excellent, making as neat a seam as is +found in glove work, unless, of course, the pelts are especially heavy, +such as bears and sheep rugs. + +A very great feature of German and Russian work is the fur linings +called rotondes, sacques or plates, which are made for their home use +and exportation chiefly to Great Britain, America and France. + +In Weissenfels, near Leipzig, the dressing of Russian grey squirrel and +the making it into linings is a gigantic industry, and is the principal +support of the place. After the dressing process the backs of the +squirrels are made up separately from the under and thinner white and +grey parts, the first being known as squirrel-back and the other as +squirrel-lock linings. A few linings are made from entire skins and +others are made from the quite white pieces, which in some instances are +spotted with the black ear tips of the animals to resemble ermine. The +smaller and uneven pieces of heads and legs are made up into linings, so +there is absolutely no waste. Similar work is done in Russia on almost +as extensive a scale, but neither the dressing nor the work is so good +as the German. + +The majority of heads, gills or throats, sides or flanks, paws and +pieces of skins cut up in the fur workshops of Great Britain, America +and France, weighing many tons, are chiefly exported to Leipzig, and +made up in neighbouring countries and Greece, where labour can be +obtained at an alarmingly low rate. Although the sewing, which is +necessarily done by hand, the sections being of so unequal and tortuous +a character, is rather roughly executed, the matching of colours and +qualities is excellent. The enormous quantities of pieces admit of good +selection and where odd colours prevail in a lining it is dyed. Many +squirrel-lock linings are dyed blue and brown and used for the outside +of cheap garments. They are of little weight, warm and effective, but +not of great durability. + +The principal linings are as follows: Sable sides, sable heads and paws, +sable gills, mink sides, heads and gills, marten sides, heads and gills, +Persian lamb pieces and paws, caracul lamb pieces or paws, musquash +sides and heads, nutria sides, genet pieces, raccoon sides or flanks, +fox sides, kolinski whole skins, and small rodents as kaluga and +hamster. The white stripes cut out of skunks are made into rugs. + +Another great source of inexpensive furs is China, and for many years +past enormous quantities of dressed furs, many of which are made up in +the form of linings and Chinese loose-shaped garments, have been +imported by England, Germany and France for the lower class of business; +the garments are only regarded as so much fur and are reworked. With, +however, the exception of the best white Tibet lambs, the majority of +Chinese furs can only be regarded as inferior material. While the work +is often cleverly done as to matching and manipulation of the pelt which +is very soft, there are great objections in the odour and the +brittleness or weakness of the fur. One of the most remarkable results +of the European intervention in the Boxer rising in China (1900) was the +absurd price paid for so-called "loot" of furs, particularly in +mandarins' coats of dyed and natural fox skins and pieces, and natural +ermine, poor in quality and yellowish in colour; from three to ten times +their value was paid for them when at the same time huge parcels of +similar quality were warehoused in the London docks, because purchasers +could not be found for them. + +With regard to Japanese furs, there is little to commend them. The best +are a species of raccoon usually sold as fox, and, being of close long +quality of fur, they are serviceable for boas, collars, muffs and +carriage aprons. The sables, martens, minks and otters are poor in +quality, and all of a very yellow colour and they are generally dyed for +the cheap trade. A small number of very pretty guanaco and vicuna +carriage rugs are imported into Europe, and many come through travellers +and private sources, but generally they are so badly dressed that they +are quite brittle upon the leather side. Similar remarks are applicable +to opossum rugs made in Australia. From South Africa a quantity of +jackal, hyena, fox, leopard and sheep karosses, i.e. a peculiarly shaped +rug or covering used by native chiefs, is privately brought over. The +skins are invariably tanned and beautifully sewn, the furs are generally +flat in quality and not very strong in the hair, and are retained' more +as curiosities than for use as a warm covering. + +_Hatters' Furs and Cloths and Shawls._--The hat trade is largely +interested in the fur piece trade, the best felt hats being made from +beaver and musquash wool and the cheaper sorts from nutria, hare and +rabbit wools. For weaving, the most valuable pieces are mohair taken +from the angora and vicuna. They are limited in quantity and costly, and +the trade depends upon various sorts of other sheep and goat wools for +the bulk of its productions. + +_Frauds and Imitations._--The opportunities for cheating in the fur +trade are very considerable, and most serious frauds have been +perpetrated in the selling of sables that have been coloured or +"topped"; that is, just the tips of the hairs stained dark to represent +more expensive skins. It is only by years of experience that some of +these colourings can be detected. Where the skins are heavily dyed it is +comparatively easy to see the difference between a natural and a dyed +colour, as the underwool and top hair become almost alike and the +leather is also dark, whereas in natural skins the base of the underwool +is much paler than the top, or of a different colour, and the leather Is +white unless finished in a pale reddish tone as is sometimes the case +when mahogany sawdust is used in the final cleaning. As has been +explained, sable is a term applied for centuries past to the darker +sorts of the Russian Siberian martens, and for years past the same term +has been bestowed by the retail trade upon the American and Canadian +martens. The baum and stone martens caught in France, the north of +Turkey and Norway are of the same family, but coarser in underwool and +the top hair is less in quantity and not so silky. The kolinski, or as +it is sometimes styled Tatar sable, is the animal, the tail of which +supplies hair for artists' brushes. This is also of the marten species +and has been frequently offered, when dyed dark, as have baum and stone +martens, as Russian sables. Hares, too, are dyed a sable colour and +advertised as sable. The fur, apart from a clumsy appearance, is so +brittle, however, as to be of scarcely any service whatever. + +Among the principal imitations of other furs is musquash, out of which +the top hair has been pulled and the undergrowth of wool clipped and +dyed exactly the same colour as is used for seal, which is then offered +as seal or red river seal. Its durability, however, is far less than +that of seal. Rabbit is prepared and dyed and frequently offered as +"electric sealskin." Nutria also is prepared to represent sealskin, and +in its natural colour, after the long hairs are plucked out, it is sold +as otter or beaver. The wool is, however, poor compared to the otter and +beaver, and the pelt thin and in no way comparable to them in strength. +White hares are frequently sold as white fox, but the fur is weak, +brittle and exceedingly poor compared to fox and possesses no thick +underwool. Foxes, too, and badger are dyed a brownish black, and white +hairs inserted to imitate silver fox, but the white hairs are too coarse +and the colour too dense to mislead any one who knows the real article. +But if sold upon its own merits, pointed fox is a durable fur. + +Garments made of sealskin pieces and Persian lamb pieces are frequently +sold as if they were made of solid skins, the term "pieces" being simply +suppressed. The London Chamber of Commerce have issued to the British +trade a notice that any misleading term in advertising and all attempts +at deception are illegal, and offenders are liable under the Merchandise +Marks Act 1887. + + The most usual misnaming of manufactured furs is as follow:-- + + Musquash, pulled and dyed Sold as seal. + Nutria, pulled and dyed Sold as seal. + Nutria, pulled and natural Sold as beaver. + Rabbit, sheared and dyed Sold as seal or electric seal. + Otter, pulled and dyed Sold as seal. + Marmot, dyed Sold as mink or sable. + Fitch, dyed Sold as sable. + Rabbit, dyed Sold as sable or French sable. + Hare, dyed Sold as sable, or fox, or lynx. + Musquash, dyed Sold as mink or sable. + Wallaby, dyed Sold as skunk. + White Rabbit Sold as ermine. + White Rabbit, dyed Sold as chinchilla. + White Hare, dyed or natural Sold as fox, foxaline, and + other similar names. + Goat, dyed Sold as bear, leopard, &c. + Dyed manufactured articles of + all kinds Sold as "natural." + White hairs inserted in foxes + and sables Sold as real or natural furs. + Kids Sold as lamb or broadtails. + American sable Sold as real Russian sable. + Mink Sold as sable. + +_The Preservation of Furs._--For many years raw sealskins have been +preserved in cold storage, but it is only within a recent period, owing +to the difficulty there was in obtaining the necessary perfectly dry +atmosphere, that dressed and made-up furs have been preserved by +freezing. Furs kept in such a condition are not only immune from the +ravages of the larvae of moth, but all the natural oils in the pelt and +fur are conserved, so that its colour and life are prolonged, and the +natural deterioration is arrested. Sunlight has a tendency to bleach +furs and to encourage the development of moth eggs, therefore continued +exposure is to be avoided. When furs are wetted by rain they should be +well shaken and allowed to dry in a current of air without exposure to +sun or open fire. + +Where a freezing store for furs is not accessible, furs should be well +shaken and afterwards packed in linen and kept in a perfectly cool dry +place, and examined in the summer at periods of not less than five +weeks. Naphthalene and the usual malodorous powders are not only very +disagreeable, but quite useless. Any chemical that is strong enough to +destroy the life in a moth egg would also be sufficiently potent to +injure the fur itself. In England moth life is practically continuous +all the year round, that is, as regards those moths that attack furs, +though the destructive element exists to a far greater extent during +spring and summer. + + _Comparative Durability of Various Furs and Weight of Unlined Skins + per Square Foot._ + + The following estimates of durability refer to the use of fur when + made up "hair outside" in garments or stoles, not as a lining. The + durability of fur used as linings, which is affected by other + conditions, is set forth separately. Otter, with its water hairs + removed, the strongest of furs for external use, is, in this table, + taken as the standard at 100 and other furs marked accordingly:-- + + + _The Precious Furs._ + + +---------------------------------------+-------------+------------+ + | | Points of | [2]Weight | + | | Durability. | in oz. per | + | | | sq. ft. | + +---------------------------------------+-------------+------------+ + | Sable | 60 | 2-1/2 | + | Sea | 75 | 3 | + | Fox, Silver or Black | 40 | 3 | + | " White | 20 | 3 | + | Ermine | 25 | 1-1/4 | + | Chinchilla | 15 | 1-1/2 | + | Sea-otter (for stoles or collars) | 100 | 4-1/4 | + +---------------------------------------+-------------+------------+ + + + _The Less Valuable Furs._ + + +----------------------------------------+-------------+------------+ + | | Points of | Weight | + | | Durability. | in oz. per | + | | | sq. ft. | + +----------------------------------------+-------------+------------+ + | Sable "topped," i.e. top hairs coloured| 55 | 2-1/2 | + | " tinted, i.e. fur all coloured. | 50 | 2-1/2 | + | Baum Marten, natural | 65 | 2-3/4 | + | " " tinted | 45 | 2-3/4 | + | Stone Marten | 40 | 2-3/4 | + | Nutria | 27 | 3-1/4 | + | Musquash, natural | 37 | 3-1/4 | + | " water hairs removed, sheared | | | + | and seal finished. | 33 | 3-1/4 | + | Skunk | 70 | 2-3/4 | + | Mink | 70 | 3-1/4 | + | Lynx, natural | 25 | 2-3/4 | + | " tinted black | 20 | 2-3/4 | + | Marmot, tinted | 10 | 3 | + | Fox, tinted black | 25 | 3 | + | " " blue | 20 | 3 | + | Opossum | 37 | 3 | + | Otter (with water hairs) | 100 | 4 | + | " (water hairs removed) | 95 | 3-15/16 | + | Beaver (water hairs cut level with fur)| 90 | 4 | + | " (water hairs removed). | 85 | 3-15/16 | + | Moleskin | 7 | 1-3/4 | + | Persian Lamb | 65 | 3-1/4 | + | Grey " | 30 | 3-1/4 | + | Broadtail | 15 | 2-1/4 | + | Caracul Kid | 10 | 3-1/4 | + | " Lamb | 15 | 3-1/4 | + | Squirrel | 25 | 1-3/4 | + | Hare | 5 | 1-3/4 | + | Rabbit | 5 | 2-1/4 | + +----------------------------------------+-------------+------------+ + + + _Quantities of Fur needed, in Square Feet._ + + The "Paris Model" figure is the basis of these estimates for ladies' + garments, the standard measurements being height 5 ft. 6 in., waist 23 + in., bust 38 in. + + Sq. Ft. + (approximate). + + Straight stole 1/2 length (just below the waist line) 2-3/4 + Straight stole 3/4 length (just below the knee) 3-3/4 + Stole, broad enough at the neck to cover the top of + arm 3/4 length 5 + The same, full length (to hem of skirt) 6 + Eton jacket, without collar 13 + Plain cape, 15 in. long 6-1/2 + Deep cape, 30 in. long 15 + Full cape with broad stole front, 3/4 length 15 + Inverness cape (to knee) 25 + Double-breasted, straight, semi-fitting coat, covering + hips 16 + Double-breasted sacque jacket, 36 in. long, full sleeves 20 + Same, 30 in. long 18 + Same, 22 in. long 15 + Long, full, shawl cape with points at back and front, + well below knee 15 + Shorter shawl cape 16 + Motoring or driving coat, 3/4 length 22 + Motoring or driving coat, full length 27 + + + _Weight and Durability of Furs for Men's Coat Linings._ + + Otter with the water hairs removed, the strongest fur suited for + linings, is here taken as the standard. + + +---------------------------------+-----------+-------------+ + | | Points of | Weight | + | |Durability.| in oz. per | + | | | sq. ft. | + +---------------------------------+-----------+-------------+ + | Otter (the water hairs removed) | 100 | 3-15/16 | + | Beaver " " | 90 | 3-15/16 | + | Mink | 90 | 3-1/4 | + | Sealskin | 75 | 3 | + | Raccoon | 75 | 4-1/2 | + | Persian lamb or astrachan | 70 | 3-1/4 | + | Sable | 65 | 2-1/2 | + | Musquash | 55 | 3-1/2 | + | Nutria | 40 | 3-1/4 | + | Grey Opossum | 40 | 3 | + | Wallaby | 30 | 3-3/4 | + | Squirrel | 30 | 1-3/4 | + | Hamster | 15 | 1-1/4 | + | Rabbit | 10 | 2-1/4 | + +---------------------------------+-----------+-------------+ + + + _Durability and Weight of Linings for Ladies' Coats or Wraps._ + + Sable gills, the strongest fur suited for ladies' linings, is taken as + the standard. + + +---------------------------------+-----------+-------------+ + | | Points of | Weight | + | |Durability.| in oz. per | + | | | sq. ft. | + +---------------------------------+-----------+-------------+ + | Sable gills | 100 | 2-7/8 | + | Sable | 85 | 2-1/2 | + | Sable paws | 64 | 1-5/8 | + | Ermine | 57 | 1-1/4 | + | Squirrel back | 50 | 1-3/4 | + | Squirrel heads | 36 | 2-1/2 | + | Squirrel lock | 21 | 1-3/16 | + | Hamster | 10 | 1-1/4 | + | Rabbit | 7 | 2-1/4 | + +---------------------------------+-----------+-------------+ + + + _Durability and Weight of Motoring Furs made up with Fur outside._ + + Otter with the water hairs, the strongest fur suited for motoring + garments, is taken as the standard. + + +-------------------------------------+-----------+-------------+ + | | Points of | Weight | + | |Durability.| in oz. per | + | | | sq. ft. | + +-------------------------------------+-----------+-------------+ + | Otter (with water hairs) | 100 | 4 | + | Sealskin, marble | 80 | 3 | + | "Hair Sealskin" (tinted) with water | | | + | hairs (a special variety of seal) | 75 | 3-1/4 | + | Raccoon | 65 | 4-1/2 | + | Russian Pony | 35 | 2-5/8 | + +-------------------------------------+-----------+-------------+ + + + _Durability and Weight of Furs for Rugs and Foot-sacks._ + + +-------------------------------------+-----------+-------------+ + | | Points of | Weight | + | |Durability.| in oz. per | + | | | sq. ft. | + +-------------------------------------+-----------+-------------+ + | Wolverine | 100 | 6 | + | Bear (black or brown natural) | 94 | 7 | + | Bear (tinted black) | 88 | 7-1/2 | + | Beaver | 88 | 4 | + | Raccoon | 77 | 4-1/2 | + | Opossum | 61 | 3 | + | Wolf | 50 | 6-1/2 | + | Jackal | 27 | 4-1/2 | + | Australian Bear | 16 | 6 | + | Goat | 11 | 4-1/6 | + +-------------------------------------+-----------+-------------+ + + Wolverine, the strongest fur suited for rugs and foot-sacks, is taken + as the standard. + + For a rug about 20 to 25 sq. ft. of fur are needed, for a foot-sack + 14-1/2. (W. S. P.) + + +FOOTNOTES: + + [1] The measurements given are from nose to root of tail of average + large sizes after the dressing process, which has a shrinking + tendency. The depths of fur quoted are the greatest, but there are + plenty of good useful skins possessing a lesser depth. + + [2] Stout, old-fashioned boxcloth is almost the only cloth that + (after a soft, heavy lining has been added to it) affords even + two-thirds as much protection against cold as does fur. It weighs + 4.273 oz. per sq. ft. more than the heaviest of coat-furs, and is so + rigid as to be uncomfortable, while the subtileness of fur makes it + "kind" to the body. + + + + +FURAZANES (_furo_--a.a'--_diazoles_), organic compounds obtained by +heating the glyoximes (dioximes of ortho-diketones) with alkalis or +ammonia. Dimethylfurazane is prepared by heating dimethylglyoxime with +excess of ammonia for six hours at 165 deg. C. (L. Wolff, _Ber._, 1895, +28, p. 70). It is a liquid (at ordinary temperature) which boils at 156 +deg. C. (744 mm.). Potassium permanganate oxidizes it first to +methylfurazane-carboxylic acid and then to furazanedicarboxylic acid. +Methyl-ethylfurazane and diphenylfurazane are also known. By warming +oxyfurazane acetic acid with excess of potassium permanganate to 100 +deg. C. oxyfurazanecarboxylic acid is obtained (A. Hantzsch and J. +Urbahn, _Ber._, 1895, 28, p. 764). It crystallizes in prisms, which melt +at 175 deg. C. Furazanecarboxylic acid is prepared by the action of a +large excess of potassium permanganate on a hot solution of +furazanepropionic acid. It melts at 107 deg. C, and dissolves in caustic +soda, with a deep yellow colour and formation of nitrosocyanacetic acid +(L. Wolff and P.F. Ganz, _Ber._, 1891, 24, p. 1167). Furoxane is an +oxide of furazane, considered by H. Wieland to be identical with +glyoxime peroxide; Kekules dibromnitroacetonitrile is dibromfuroxane. + +The formulae of the compounds above mentioned are: + + HC:N\ CH3.C:N\ HC:N\ HC--CH \ + . O . O . O .. . O + HC:N/ CH3.C:N/ HO2C.C:N/ N.O.N / + + Furazane. Dimethyl- Furazanecarboxylic Furoxane. + furazane. acid. + + + + +FURETIERE, ANTOINE (1619-1688), French scholar and miscellaneous writer, +was born in Paris on the 28th of December 1619. He first studied law, +and practised for a time as an advocate, but eventually took orders and +after various preferments became abbe of Chalivoy in the diocese of +Bourges in 1662. In his leisure moments he devoted himself to letters, +and in virtue of his satires--_Nouvelle Allegorique, ou histoire des +derniers troubles arrives au royaume d'eloquence_ (1658); _Voyage de +Mercure_ (1653)--he was admitted a member of the French Academy in 1662. +That learned body had long promised a complete dictionary of the French +tongue; and when they heard that Furetiere was on the point of issuing a +work of a similar nature, they interfered, alleging that he had +purloined from their stores, and that they possessed the exclusive +privilege of publishing such a book. After much bitter recrimination on +both sides the offender was expelled in 1685; but for this act of +injustice he took a severe revenge in his satire, _Couches de +l'academie_ (Amsterdam, 1687). His _Dictionnaire universel_ was +posthumously published in 1690 (Rotterdam, 2 vols.). It was afterwards +revised and improved by the Protestant jurist, Henri Basnage de Beauval +(1656-1710), who published his edition (3 vols.) in 1701; and it was +only superseded by the compilation known as the _Dictionnaire de +Trevoux_ (Paris, 3 vols., 1704; 7th ed., 8 vols., 1771), which was in +fact little more than a reimpression of Basnage's edition. Furetiere is +perhaps even better known as the author of _Le Roman bourgeois_ (1666). +It cast ridicule on the fashionable romances of Mlle de Scudery and of +La Calprenede, and is of interest as descriptive of the ~~ everyday +life of his times. There is no element of burlesque, as in Scarron's +_Roman comique_, but the author contents himself with stringing together +a number of episodes and portraits, obviously drawn from life, without +much attempt at sequence. The book was edited in 1854 by Edward Fournier +and Charles Asselineau and by P. Jannet. + + The _Fureteriana_, which appeared in Paris eight years after + Furetiere's death, which took place on the 14th of May 1688, is a + collection of but little value. + + + + +FURFOOZ, a village some 10 m. from Dinant in the Ardennes, Belgium. +Three caves containing prehistoric remains were here excavated in 1872. +Of these the _Trou de Frontal_ is the most famous. In it were found +human skeletons with brachycephalic skulls, associated with animal +bones, those of the reindeer being particularly plentiful. Among the +skeletons was discovered an oval vase of pottery. The Furfooz type of +mankind is believed to date from the close of the Quaternary age. G. de +Mortillet dates the type in the Robenhausen epoch of the Neolithic +period. His theory is that the bones are those of men of that period +buried in what had been a cave-dwelling of the Madelenian epoch. + + + + +FURFURANE, or FURANE, C4H4O, a colourless liquid boiling at 32 deg. C., +found in the distillation products of pine wood. It was first +synthetically prepared by H. Limpricht (_Ann._, 1873, 165, p. 281) by +distilling barium mucate with soda lime, pyromucic acid C4H3O.CO2H being +formed, which, on further loss of carbon dioxide, yielded furfurane. A. +Henniger (_Ann. chim. phys._, 1886 [2], 7, p. 220), by distilling +erthyrite with formic acid, obtained a dihydrofurfurane + + C4H6(OH)4 + 2H2CO2 = C4H6O + CO+CO2 + 4H2O, + +which, on treatment with phosphorus pentachloride, yielded furfurane. +Furfurane is insoluble in water and possesses a characteristic smell. It +does not react with sodium or with phenylhydrazine, but yields +dye-stuffs with isatin and phenanthrenequinone. It reacts violently with +hydrochloric acid, producing a brown amorphous substance. Methyl and +phenyl derivatives have been prepared by C. Paal (_Ber._, 1884, 17, p. +915). Paal prepared acetonyl acetophenone by condensing sodium +acetoacetate with phenacylbromide, and this substance on dehydration +yields [alpha][alpha]'-phenylmethylfurfurane, the acetonyl acetophenone +probably reacting in the tautomeric "enolic" form, + + CH3.CO.CHNa.COOR + C6H5.CO.CH2Br = CH3.CO.CH(CH2COC6H5).COOR. + +This ester readily hydrolyses, and the acid formed yields acetonyl +acetophenone (by loss of carbon dioxide), which then on dehydration +yields the furfurane derivative, thus + + CH--CH CH--CH + // \\ // \\ + CH3.C C.C6H5 = H2O + CH3.C C.C6H5. + \ / \___ ___/ + OH OH O + +L. Knorr (_Ber._, 1889, 22, p. 158) obtained diacetosuccinic ester by +condensing sodium acetoacetate with iodine, and by dehydrating the ester +he prepared [alpha][alpha]'-dimethylfurfurane [beta][beta]'-dicarboxylic +acid (carbopyrotritaric acid), which on distillation yields +[alpha][alpha]'-dimethylfurfurane as a liquid boiling at 94 deg. C. Paal +also obtained this compound by using monochloracetone in the place of +phenacylbromide. By the distillation of mucic acid or isosaccharic acid, +furfurane-[alpha]-carboxylic acid (pyromucic acid), C4H3O.CO2H, is +obtained; it crystallizes in needles or leaflets, and melts at 134 deg. +C. + +_Furfurol_ (furol), C4H3O.CHO, is the aldehyde of pyromucic acid, and is +formed on distilling bran, sugar, wood and most carbohydrates with +dilute sulphuric acid, or by distilling the pentoses with hydrochloric +acid. It is a colourless liquid which boils at 162 deg. C., and is +moderately soluble in water; it turns brown on exposure to air and has a +characteristic aromatic smell. It shows all the usual properties of an +aldehyde, forming a bisulphite compound, an oxime and a hydrazone; +whilst it can be reduced to the corresponding furfuryl alcohol by means +of sodium amalgam, and oxidized to pyromucic acid by means of silver +oxide. It also shows all the condensation reactions of benzaldehyde +(q.v.); condensing with aldehydes and ketones in the presence of caustic +soda to form more complex aldehydes and ketones with unsaturated side +chains, such as furfuracrolein, C4H3O.CH:CH.CHO, and furfuracetone, +C4H3O.CH:CH.CO.CH3. With alcoholic potassium cyanide It changes to +furoin, C4H3O.CHOH.CO.C4H3O, which can be oxidized to furil, +C4H3O.CO.CO.C4H3O, whilst alcoholic potash converts it into furfuryl +alcohol. With fatty acids and acid anhydrides it gives the "Perkin" +reaction (see CINNAMIC ACID). Furfurol is shown to have its aldehydic +group in the a position, by conversion into furfurpropionic acid, +C4H3O.CH2.CH2.CO2H, which on oxidation by bromine water and subsequent +reduction of the oxidized product is converted into n-pimelic acid, +HO2C(CH2)5CO2H. Furfurol in minute quantities can be detected by the red +colour it forms with a solution of aniline acetate. + + Furfurane--[alpha][alpha]'-dicarboxylic acid or dehydromucic acid, + C4H2O(CO2H)2, is formed when mucic acid is heated with hydrochloric + acid at 100 deg. C. On being heated, it loses carbon dioxide and gives + pyromucic acid. By digesting acetoacetic ester with sodium succinate + and acetic anhydride, methronic acid, C8H8O5, is obtained; for the + constitution of this acid, see L. Knorr, _Ber._, 1889, 22, p. 152, and + R. Fittig, Ann., 1889, 259, p. 166. + + Di- and tetrahydrofurfurane compounds are also known (see A. Lipp, + Ber., 1889, 22, p. 1196; W.H. Perkin, junr. _Journ. Chem. Soc._, 1899, + 57, p. 944; and S. Ruhemann, ibid., 1896, 69, p. 1383). + + + + +FURIES (Lat. _Furiae_, also called DIRAE), in Roman mythology an +adaptation of the Greek Erinyes (q.v.), with whom they are generally +identical. A special aspect of them in Virgil is that of agents employed +by the higher gods to stir up mischief, strife and hatred upon earth. +Mention may here be made of an old Italian deity Furina (or Furrina), +whose worship fell early into disuse, and who was almost forgotten in +the time of Varro. By the mythologists of Cicero's time the name was +connected with the verb _furere_ and the noun _furia_, which in the +plural (not being used in the singular in this sense) was accepted as +the equivalent of the Greek Erinyes. But it is more probably related to +_furvus_, _fuscus_, and signifies one of the spirits of darkness, who +watched over men's lives and haunted their abodes. This goddess had her +own special priest, a grove across the Tiber where Gaius Gracchus was +slain, and a festival on the 25th of July. Authorities differ as to the +existence of more than one goddess called Furina, and their identity +with the Forinae mentioned in two inscriptions found at Rome (_C.I.L._ +vi. 422 and 10,200). + + + + +FURLONG (from the O. Eng. _furlang_, i.e. "furrow-long"), a measure of +length, originally the length of a furrow in the "common field" system. +As the field in this system was generally taken to be a square, 10 acres +in extent, and as the acre varied in different districts and at +different times, the "furlong" also varied. The side of a square +containing 10 statute acres is 220 yds. or 40 poles, which was the +usually accepted length of the furlong. This is also the length of +{1/8}th of the statute mile. "Furlong" was as early as the 9th century +used to translate the Latin _stadium_, 1/8th of the Roman mile. + + + + +FURNACE, a contrivance for the production and utilization of heat by the +combustion of fuel. The word is common to all the Romance tongues, +appearing in more or less modified forms of the Latin _fornax_. But in +all those languages the word has a more extended meaning than in +English, as it covers every variety of heating apparatus; while here, in +addition to furnaces proper, we distinguish other varieties as _ovens_, +_stoves_ and _kilns_. The first of these, in the form _Ofen_, is used in +German as a general term like the French _four_; but in English it has +been restricted to those apparatus in which only a moderate temperature, +usually below a red heat, is produced in a close chamber. Our bakers' +ovens, hot-air ovens or stoves, annealing ovens for glass or metal, &c., +would all be called _fours_ in French and _Ofen_ in German, in common +with furnaces of all kinds. Stove, an equivalent of oven, is from the +German _Stube_, i.e. a heated room, and is commonly so understood; but +is also applied to open fire-places, which appears to be somewhat of a +departure from the original signification. + +Furnaces are constructed according to many different patterns with +varying degrees of complexity in arrangement; but all may be considered +as combining three essential parts, namely, the fire-place in which the +fuel is consumed, the heated chamber, laboratory, hearth or working bed, +as it is variously called, where the heat is applied to the special work +for which the furnace is designed, and the apparatus for producing rapid +combustion by the supply of air under pressure to the fire. In the +simplest cases the functions of two or more of these parts may be +combined into one, as in the smith's forge, where the fire-place and +heating chamber are united, the iron being placed among the coals, only +the air for burning being supplied under pressure from a blowing engine +by a second special contrivance, the tuyere, tuiron, twyer or +blast-pipe; but in the more refined modern furnaces, where great economy +of fuel is an object, the different functions are distributed over +separate and distinct apparatus, the fuel being converted into gas in +one, dried in another, and heated in a third, before arriving at the +point of combustion in the working chamber of the furnace proper. + + Furnaces may be classified according as the products of combustion are + employed (1) only for heating purposes, or (2) both for heating and + bringing about some chemical change. The furnaces employed for + steam-raising or for heating buildings are invariably of the first + type (see BOILER and HEATING), while those employed in metallurgy are + generally of the second. The essential difference in construction is + that in the first class the substances heated do not come into contact + with either the fuel or the furnace gases, whereas in the second they + do. Metallurgical furnaces of the first class are termed crucible, + muffle or retort furnaces, and of the second shaft and reverberatory + furnaces. The following is a detailed subdivision:-- + + (1) Fuel and substance in contact. + (a) Height of furnace greater than diameter = shaft furnaces. + ([alpha]) No blast = kilns. + ([beta]) With blast = blast furnaces. + (b) Height not much greater than diameter = hearth furnaces. + + (2) Substance heated by products of combustion = reverberatory + furnaces. + (a) Charge not melted = roasting or calcining furnaces. + (b) Charge melted = melting furnaces. + + (3) Substance is not directly heated by the fuel or by the products + of combustion. + (a) Heating chamber fixed and forming part of furnace = muffle + furnaces. + (b) Crucible furnaces. + (c) Retort furnaces. + +Another classification may be based upon the nature of the heating +agent, according as it is coal (or some similar combustible) oil, gas or +electricity. In this article the general principles of metallurgical +furnaces will be treated; the subject of gas- and oil-heated furnaces is +treated in the article Fuel, and of the electric furnace in the article +Electrometallurgy. For special furnaces reference should be made to the +articles on the industry concerned, e.g. GLASS, GAS, S MANUFACTURE, &c. + +_Shaft, Blast and Hearth Furnaces._--The blast furnace in its simplest +form is among the oldest, if not the oldest, of metallurgical +contrivances. In the old copper-smelting district of Arabia Petraea, +clay blast-pipes dating back to the earlier dynasties of ancient Egypt +have been found buried in slag heaps; and in India the native smiths and +iron-workers continue to use furnaces of similar types. These, when +reduced to their most simple expression, are mere basin-shaped hollows +in the ground, containing ignited charcoal and the substances to be +heated, the fire being urged by a blast of air blown in through one or +more nozzles from a bellows at or near the top. They are essentially the +same as the smith's forge. This class of furnace is usually known as an +open fire or hearth, and is represented in a more advanced stage of +development by the Catalan, German and Walloon forges formerly used in +the production of malleable iron. + +[Illustration: FIG. 1.--Elevation of Catalan Forge.] + + Fig. 1 represents a Catalan forge. The cavity in the ground is + represented by a pit of square or rectangular section lined with brick + or stone of a kind not readily acted on by heat, about 1-1/2 or 2 ft. + deep, usually somewhat larger above than below, with a tuyere or + blast-pipe of copper penetrating one of the walls near the top, with a + considerable downward inclination, so that the air meets the fuel some + way down. In iron-smelting the ore is laid in a heap upon the fuel + (charcoal) filling up the hearth, and is gradually brought to the + metallic state by the reducing action of the carbon monoxide formed at + the tuyere. The metal sinks through the ignited fuel, forming, in the + hearth, a spongy mass or ball, which is lifted out by the smelters at + the end of each operation, and carried to the forge hammer. The earthy + matters form a fusible glass or slag melt, and collect at the lowest + point of the hearth, whence they are removed by opening a hole pierced + through the front wall at the bottom. The active portion of such a + furnace is essentially that above the blast-pipe, the function of the + lower part being merely the collection of the reduced metal; the fire + may therefore be regarded as burning in an unconfined space, with the + waste of a large amount of its heating power. By continuing the walls + of the hearth above the tuyere, into a shaft or stack either of the + same or some other section, we obtain a furnace of increased capacity, + but with no greater power of consuming fuel, in which the material to + be treated can be heated up gradually by loading it into the stack, + alternately with layers of fuel, the charge descending regularly to + the point of combustion, and absorbing a proportion of the heat of the + flame that went to waste in the open fire. This principle is capable + of very wide extension, the blast furnace being mainly limited in + height by the strength the column of materials or "burden" has to + resist crushing, under the weight due to the head adopted, and the + power of the blowing engine to supply blast of sufficient density to + overcome the resistance of the closely packed materials to the free + passage of the spent gases. The consuming power of the furnace or the + rate at which it can burn the fuel supplied is measured by the number + of tuyeres and their section. + +The development of blast furnaces is practically the development of +iron-smelting. The profile has been very much varied at different times. +The earliest examples were square or rectangular in horizontal section, +but the general tendency of modern practice is to substitute round +sections, their construction being facilitated by the use of specially +moulded bricks which have entirely superseded the sandstone blocks +formerly used. The vertical section, on the other hand, is subject to +considerable variation according to the work to which the furnace is +applied. Where the operation is simply one of fusion, as in the +iron-founder's cupola, in which there is no very great change in volume +in the materials on their descent to the tuyeres, the stack is nearly or +quite straight-sided; but when, as is the case with the smelting of iron +ores with limestone flux, a large proportion of volatile matter has to +be removed in the process, a wall of varying inclination is used, so +that the body of the furnace is formed of two dissimilar truncated +cones, joined by their bases, the lower one passing downwards into a +short, nearly cylindrical, position. For further consideration of this +subject see IRON AND STEEL. + +_Hearth furnaces_ are employed in certain metallurgical operations, e.g. +in the air-reduction process for smelting lead ores. The principle is +essentially that of the Catalan forge. Such furnaces are very wasteful, +and have little to recommend them (see Schnabel, Metallurgy, 1905, vol. +1. p. 409). + +_Reverberatory Furnaces._--Blast furnaces are, from the intimate contact +between the burden to be smelted and the fuel, the least wasteful of +heat; but their use supposes the possibility of obtaining fuel of good +quality and free from sulphur or other substances likely to deteriorate +the metal produced. In all cases, therefore, where it is desired to do +the work out of contact with the solid fuel, the operation of burning or +heat-producing must be performed in a special fire-place or combustion +chamber, the body of flame and heated gas being afterwards made to act +upon the surface of the material exposed in a broad thin layer in the +working bed or laboratory of the furnace by reverberation from the low +vaulted roof covering the bed. Such furnaces are known by the general +name of reverberatory or reverbatory furnaces, also as air or wind +furnaces, to distinguish them from those worked with compressed air or +blast. + +Originally the term cupola was used for the reverberatory furnace, but +in the course of time it has changed its meaning, and is now given to a +small blast furnace such as that used by iron-founders--reverberatory +smelting furnaces in the same trade being called air furnaces. + +[Illustration: FIG. 2.--Longitudinal section of Reverberatory Furnace.] + +[Illustration: FIG. 3.--Reverberatory Furnace (horizontal section).] + +[Illustration: FIG. 4.--Reverberatory Furnace (elevation at flue end).] + + Figs. 2, 3 and 4 represent a reverberatory furnace such as is used for + the fusion of copper ores for regulus, and may be taken as generally + representing its class. The fire-place A is divided from the working + bed B by a low wall C known as the fire bridge, and at the opposite + end there is sometimes, though not invariably, a second bridge of less + height called the flue bridge D. A short diagonal flue or up-take E + conveys the current of spent flame to the chimney F, which is of + square section, diminishing by steps at two or three different + heights, and provided at the top with a covering plate or damper G, + which may be raised or lowered by a chain reaching to the ground, and + serves for regulating the speed of the exhaust gases, and thereby the + draught of air through the fire. Where several furnaces are connected + with the same chimney stack, the damper takes the form of a sliding + plate in the mouth of the connecting flue, so that the draught in one + may be modified without affecting the others. The fire bridge is + partially protected against the intense heat of the body of flame + issuing through the fire arch by a passage to which the air has free + access. The material to be melted is introduced into the furnace from + the hoppers HH through the charging holes in the roof. When melted the + products separate on the bed (which is made of closely packed sand or + other infusible substances), according to their density; the lighter + earthy matters forming an upper layer of slag are drawn out by the + slag hole K at the flue end into an iron wagon or bogie, while the + metal subsides to the bottom of the bed, and at the termination of the + operation is run out by the tap hole L into moulds or granulated into + water. The opposite opening M is the working door, through which the + tool for stirring the charge is introduced. It is covered by a plate + suspended to a lever, similar to that seen in the end elevation (fig. + 4) in front of the slag hole. + +According to the purposes to which they are applied, reverberatory +furnaces may be classed into two groups, namely, fusion or melting +furnaces, and calcining or wasting furnaces, also called calciners. The +former have a very extended application in many branches of industry, +being used by both founders and smelters in the fusion of metals; in the +concentration of poor metallic compounds by fusion into regulus; in the +reduction of lead and tin ores; for refining copper and silver; and for +making malleable iron by the puddling processes and welding. Calcining +furnaces have a less extended application, being chiefly employed in the +conversion of metallic sulphides into oxides by continued exposure to +the action of air at a temperature far below that of fusion, or into +chlorides by roasting with common salt. As some of these substances (for +example, lead sulphide and copper pyrites) are readily fusible when +first heated, but become more refractory as part of the sulphur is +dissipated and oxygen takes its place, it is important that the heat +should be very carefully regulated at first, otherwise the mass may +become clotted or fritted together, and the oxidizing effect of the air +soon ceases unless the fritted masses be broken small again. This is +generally done by making the bed of the furnace very long in proportion +to its breadth and to the fire-grate area, which may be the more easily +done as a not inconsiderable amount of heat is given out during the +oxidation of the ore--such increased length being often obtained by +placing two or even three working beds one above the other, and allowing +the flame to pass over them in order from below upwards. Such calciners +are used especially in roasting zinc blende into zinc oxide, and in the +conversion of copper sulphides into chlorides in the wet extraction +process. In some processes of lead-smelting, where the minerals treated +contain sand, the long calciner is provided with a melting bottom close +to the fire-place, so that the desulphurized ore leaves the furnace as a +glassy slag or silicate, which is subsequently reduced to the metallic +state by fusion with fluxes in blast furnaces. Reverberatory furnaces +play an important part in the manufacture of sodium carbonate; +descriptions and illustrations are given in the article ALKALI +MANUFACTURE. + +_Muffle, Crucible and Retort Furnaces._--A third class of furnaces is so +arranged that the work is done by indirect heating; that is, the +material under treatment, whether subjected to calcination, fusion or +any other process, is not brought in contact either with fuel or flame, +but is raised to the proper temperature by exposure in a chamber heated +externally by the products of combustion. These are known as muffle or +chamber furnaces; and by supposing the crucibles or retorts to represent +similar chambers of only temporary duration, the ordinary pot melting +air furnaces, and those for the reduction of zinc ores or the +manufacture of coal gas, may be included in the same category. These are +almost invariably air furnaces, though sometimes air under pressure is +used, as, for example, in the combustion of small anthracitic coal, +where a current of air from a fan-blower is sometimes blown under the +grate to promote combustion. Types of muffle furnaces are figured in the +article ANNEALING, HARDENING AND TEMPERING. + +_Furnace Materials._--The materials used in the construction of furnaces +are divisible into two classes, namely, ordinary and refractory or +fire-resisting. The former are used principally as casing, walls, +pillars or other supporting parts of the structure, and includes +ordinary red or yellow bricks, clay-slate, granite and most building +stones; the latter are reserved for the parts immediately in contact +with the fuel and flame, such as the lining of the fire-place, the +arches, roof and flues, the lower part if not the whole of the chimney +lining in reverberatory furnaces, and the whole of the internal walls of +blast furnaces. Among such substances are fireclay and firebricks, +certain sandstones, silica in the form of ganister, and Dinas stone and +bricks, ferric oxide and alumina, carbon (as coke and graphite), +magnesia, lime and chromium oxide--their relative importance being +indicated by their order, the last two or three indeed being only of +limited use. + +The most essential point in good fireclays, or in the bricks or other +objects made from them, is the power of resisting fusion at the highest +heat to which they may be exposed. This supposes them to be free from +metallic oxides forming easily fusible compounds with silica, such as +lime or iron, the presence of the former even in comparatively small +proportion being very detrimental. As clays they must be sufficiently +plastic to be readily moulded, but at the same time possess sufficient +stiffness not to contract too strongly in drying, whereby the objects +produced would be liable to be warped or cracked before firing. In most +cases, however, the latter tendency is guarded against, in making up the +paste for moulding, by adding to the fresh clay a certain proportion of +burnt material of the same kind, such as old bricks or potsherds, ground +to a coarse powder. Coke dust or graphite is used for the same purpose +in crucible making (see FIREBRICK). + + The most highly valued fireclays are derived from the Coal Measures. + Among the chief localities are the neighbourhood of Stourbridge in + Worcestershire and Stannington near Sheffield, which supply most of + the materials for crucibles used in steel and brass melting, and the + pots for glass houses; Newcastle-on-Tyne and Glenboig near Glasgow, + where heavy blast furnace and other firebricks, gas retorts, &c., are + made in large quantities. Coarse-grained but very strong firebricks + are also made of the waste of china clay works. + + In Belgium the clay raised at Andenne is very largely used for making + retorts for zinc furnaces. The principal French fireclays are derived + from the Tertiary strata in the south, and more nearly resemble + porcelain clays than those of the Coal Measures. They give wares of + remarkably fine texture and surface, combined with high refractory + character. + + In Germany, Ips and Passau on the Danube, and Gross Almerode in Hesse, + are the best known localities producing fireclay goods, the crucibles + from the last-mentioned place, known as Hessian crucibles, going all + over the world. These, though not showing a great resistance to + extreme heat, are very slightly affected by sudden alternations in + heating, as they may be plunged cold into a strongly heated furnace + without cracking, a treatment to which French and Stourbridge pots + cannot be subjected with safety. + +Plumbago or graphite is largely used in the production of crucibles, not +in the pure state but in admixture with fireclay; the proportion of the +former varies with the quality from 25 to nearly 50%. These are the most +enduring of all crucibles, the best lasting out 70 or 80 meltings in +brass foundries, about 50 with bronze, and 8 to 10 in steel-melting. + +Silica is used in furnace-building in the forms of sand, ganister, a +finely ground sandstone from the Coal Measures of Yorkshire, and the +analogous substance known as Dinas clay, which is really nearly pure +silica, containing at most about 2-1/2% of bases. Dinas clay is found at +various places in the Vale of Neath in South Wales, in the form of a +loose disintegrated sandstone, which is crushed between rollers, mixed +with about 1% of lime, and moulded into bricks that are fired in kilns +at a very high temperature. These bricks are specially used for the +roof, fire arches, and other parts subjected to intense heat in +reverberatory steel-melting furnaces, and, although infusible under +ordinary conditions, are often fairly melted by the heat without fluxing +or corrosion after a certain amount of exposure. Ganister, a slightly +plastic siliceous sand, is similarly used for the lining of Bessemer +steel converters; it is found in the neighbourhood of Sheffield. + +Alumina as a refractory material is chiefly used in the form of bauxite, +but its applications are somewhat special. It has been found to stand +well for the linings of rotatory puddling furnaces, where, under +long-continued heating, it changes into a substance as hard and +infusible as natural emery. In the Paris Exhibition of 1878 bricks very +hard and dense in character, said to be of pure alumina, were exhibited +by Muller & Co. of Paris, as well as bricks of magnesia, the latter +being specially remarkable for their great weight. They are intended for +use at the extreme temperatures obtainable in steel furnaces, or for the +melting of platinum before the oxy-hydrogen blowpipe. For the latter +purpose, however, lime is generally used; but as this substance has only +small stability, it is usually bedded in a casing of firebrick. Oxide of +chromium and chrome iron ore have been proposed as refractory crucible +materials. The former may be used as a bed for melting platinum in the +same way as lime or magnesia, without affecting the quality of the +metal. + +Ferric oxide, though not strictly infusible, is largely used as a +protecting lining for furnaces in which malleable iron is made, a +portion of the ore being reduced and recovered in the process. In an +oxidizing atmosphere it is indifferent to silica, and therefore +siliceous bricks containing a considerable proportion of ferric oxide, +when used in flues of boilers, brewers' coppers, &c. and similar +situations, are perfectly fire-resisting so long as the heated gas +contains a large proportion of unconsumed air. The red firebricks known +as Windsor bricks, which are practically similar in composition to soft +red sandstone, are of this character. + +The electric furnace has led to the discovery of several important +materials, which have been employed as furnace linings. Carborundum +(q.v.) was applied by Engels in 1899, firebricks being washed with +carborundum paste and then baked. Siloxicon, a compound of carbon, +silicon and oxygen, formed from carbon and silica in the electric +furnace, was patented by E.G. Acheson in 1903. It is very refractory, +and is applied by mixing with water and some bond, such as sodium +silicate or gas-tar. An amorphous, soft silicon carbide, also formed in +the electric furnace, was patented by B. Talbot in 1899. For basic +linings, magnesia crystallized in the electric furnace is being +extensively used, replacing dolomite to some extent (see E. Kilburn +Scott, "Refractory Materials for Furnace Linings," _Faraday Soc._, 1906, +p. 289). + + _Furnace Construction._--In the construction of furnaces provision has + to be made for the unequal expansion of the different parts under the + effect of heat. This is especially necessary in the case of + reverberatory furnaces, which are essentially weak structures, and + therefore require to be bound together by complicated systems of tie + rods and uprights or buck staves. The latter are very commonly made of + old flat bottom rails, laid with the flat of the flange against the + wall. Puddling furnaces are usually entirely cased with iron plates, + and blast furnaces with hoops round each course of the stack, or in + those of thinner constructions the firebrick work is entirely enclosed + in a wrought iron casing or jacket. Such parts as may be subjected to + extreme heat and the fretting action of molten material, as the tuyere + and slag breasts of blast furnaces, and the fire bridges and bed + plates of reverberatory furnaces, are often made in cast iron with + double walls, a current of water or air being kept circulating through + the intermediate space. In this way the metal, owing to its high + conductivity and low specific heat as compared to that of water, is + kept at a temperature far below its melting point if the water is + renewed quickly enough. It is of course necessary in such cases that + the circulation shall be perfectly free, in order to prevent the + accumulation of steam under pressure in the interior of the casting. + This method has received considerable extension, notably in + furnace-smelting of iron ores containing manganese, where the entire + hearth is often completely water-cased, and in some lead furnaces + where no firebrick lining is used, the lower part of the furnace stack + being a mere double iron box cooled by water sufficiently to keep a + coating of slag adhering to the inner shell which prevents the metal + from being acted upon. + + _Mechanical Furnaces._--The introduction and withdrawal of the charges + in fusion furnaces is effected by gravitation, the solid masses of raw + ore, fuel and flux being thrown in at the top, and flowing out of the + furnace at the taphole or slag run at the bottom. Vertical kilns, such + as those used for burning limestone, are worked in a similar + manner--the raw stone going in at the top, and the burnt product + falling through holes in the bottom when allowed to do so. With + reverberatory calciners, however, where the work is done upon a + horizontal bed, a considerable amount of hand labour is expended in + raking out the charge when finished, and in drawing slags from fusion + furnaces; and more particularly in the puddling process of refining + iron the amount of manual exertion required is very much greater. To + diminish the item of expenditure on this head, various kinds of + mechanical furnaces have been adopted, all of which can be classified + under three heads of gravitating furnaces, mechanical stirrers and + revolving furnaces. + + 1. In _gravitating furnaces_ the bed is laid at a slope just within + the angle of repose of the charge, which is introduced at the upper + end, and is pushed down the slope by fresh material, when necessary, + in the contrary direction to the flame which enters at the lower end. + Gerstenhofer's pyrites burner is a furnace of this class. It has a + tall vertical chamber heated from below, and traversed by numerous + narrow horizontal cross bars at different heights. The ore in fine + powder is fed in at the top, through a hopper, in a regular thin + stream, by a pair of rollers, and in falling lodges on the flats of + the bars, forming a talus upon each of the height corresponding to the + angle of rest of the material, which is, however, at short intervals + removed to lower levels by the arrival of fresh ore from above. In + this way a very large surface is exposed to the heat, and the ore, if + containing sufficient sulphur to maintain the combustion, is perfectly + burned when it arrives at the bottom; if, however, it is imperfectly + sized or damp, or if it contains much earthy matter, the result is not + very satisfactory. There are many other furnaces in which the same + principle is utilized. + + 2. _Mechanical stirrers_ constitute a second division of mechanical + furnaces, in which the labour of rabbling or stirring the charges is + performed by combinations of levers and wheel-work taking motion from + a rotating shaft, and more or less perfectly imitating the action of + hand labour. They are almost entirely confined to puddling furnaces. + + 3. _Revolving furnaces_, the third and most important division of + mechanical furnaces, are of two kinds. The first of these resemble an + ordinary reverberatory furnace by having a flat bed which, however, + has the form of a circular disk mounted on a central shaft, and + receives a slow movement of rotation from a water-wheel or other + motor, so that every part of the surface is brought successively under + the action of the fire, the charge being stirred and ultimately + removed by passing under a series of fixed scraper arms placed above + the surface at various points. Brunton's calciner, used in the + "burning" of the pyritic minerals associated with tin ore, is a + familiar example of this type. The hearth may either rotate on an + inclined axis, so that the path of its surface is oblique to that of + the flame, or the working part may be a hollow cylinder, between the + fireplace and flue, with its axis horizontal or nearly so, whose inner + surface represents the working bed, mounted upon friction rollers, and + receiving motion from a special steam-engine by means of a central + belt of spur gearing. Furnaces of the second kind were first used in + alkali works for the conversion of sulphate into carbonate of sodium + in the process known as black ash fusion, but have since been applied + to other processes. As calciners they are used in tin mines and for + the chlorination of silver ores. Mechanical furnaces are figured in + the article ALKALI MANUFACTURE. + + _Use of Heated Air._--The calorific intensity of fuel is found to be + very considerably enhanced, if the combustion be effected with air + previously heated to any temperature between that of boiling water and + a dull red heat, the same effect being observed both with solid and + gaseous fuel. The latter, especially when brought to the burning point + at a high temperature, produces a heat that can be resisted by the + most refractory substances only, such as silica, alumina and magnesia. + This is attained in the regenerative furnace of Siemens, detailed + consideration of which belongs more properly to the subject of iron. + + _Economy of Waste Heat._--In every system of artificial heating, the + amount of heat usefully applied is but a small proportion of that + developed by combustion. Even under the most advantageous application, + that of evaporation of water in a steam boiler where the gases of the + fire have to travel through a great length of flues bounded by thin + iron surfaces of great heat-absorbing capacity, the temperature of the + current at the chimney is generally much above that required to + maintain an active draught in the fireplace; and other tubes + containing water, often in considerable numbers, forming the so-called + fuel economizers, may often be interposed between the boiler and the + chimney with marked advantage as regards saving of fuel. In + reverberatory and air furnaces used in the different operations of + iron manufacture, where an extremely high temperature has to be + maintained in spaces of comparatively small extent, such as the beds + of puddling, welding and steel-melting furnaces, the temperature of + the exhaust gases is exceedingly high, and if allowed to pass directly + into the chimney they appear as a great body of flame at the top. It + is now general to save a portion of this heat by passing the flame + through flues of steam boilers, air-heating apparatus, or both--so + that the steam required for the necessary operations of the forge and + heated blast for the furnace itself may be obtained without further + expenditure of fuel. The most perfect method of utilizing the waste + heat hitherto applied is that of the Siemens regenerator, in which the + spent gases are made to travel through chambers, known as regenerators + or recuperators of heat, containing a quantity of thin firebricks + piled into a cellular mass so as to offer a very large heat-absorbing + surface, whereby their temperature is very considerably reduced, and + they arrive at the chimney at a heat not exceeding 300 or 400 degrees. + As soon as the bricks have become red hot, the current is diverted to + an adjacent chamber or pair of chambers, and the acquired heat is + removed by a current of cool gas or air passing towards the furnace, + where it arrives at a temperature sufficiently high to ensure the + greatest possible heating effect in combustion. + + In iron-smelting blast furnaces the waste gases are of considerable + fuel value, and may render important services if properly applied. + Owing to the conditions of the work, which require the maintenance of + a sensibly reducing atmosphere, they contain a very notable proportion + of carbonic oxide, and are drawn off by large wrought iron tubes near + the top of the furnace and conveyed by branch pipes to the different + boilers and air-heating apparatus, which are now entirely heated by + the combustion of such gases, or mixed with air and exploded in gas + engines. Formerly they were allowed to burn to waste at the mouth of a + short chimney place above the furnace top, forming a huge body of + flame, which was one of the most striking features of the Black + Country landscape at night. + + _Laboratory and Portable Furnaces._--Small air-furnaces with hot + plates or sand bath flues were formerly much employed in chemical + laboratories, as well as small blast furnaces for crucibles heated + with charcoal or coke. The use of such furnaces has very considerably + diminished, owing to the general introduction of coal-gas for heating + purposes in laboratories, which has been rendered possible by the + invention of the Bunsen burner, in which the mixture of air and gas + giving the least luminous but most powerfully heating flame is + effected automatically by the effluent gas. These burners, or + modifications of them, have also been applied to muffle furnaces, + which are convenient when only a few assays have to be made--the + furnace being a mere clay shell and soon brought to a working + temperature; but the fuel is too expensive to allow of their being + used habitually or on a large scale. Petroleum, or rather the heavy + oils obtained in tar refineries, having an equal or superior heating + power to coal-gas, may also be used in laboratories for producing high + temperatures. The oil is introduced in a thin stream upon a series of + inclined and channelled bars, where it is almost immediately + volatilized and burnt by air flowing in through parallel orifices. + Furnaces of this kind may be used for melting cast iron or bronze in + small quantities, and were employed by H. Sainte Claire Deville in + experiments in the metallurgy of the platinum group of metals. + + Sefstrom's blast furnace, used in Sweden for the assay of iron ores, + is a convenient form of portable furnace applied to melting in + crucibles. It consists of a sheet-iron cylinder about 8 or 9 in. in + diameter, within which is fixed one of smaller size lined with + fireclay. The space between the two cylinders serves as a heater and + distributor for the blast, which is introduced through the nozzle at + the bottom, and enters the furnace through a series of several small + tuyeres arranged round the inner lining. Charcoal is the fuel used, + and the crucibles stand upon the bottom of the clay lining. When a + large body of fuel is required, the cylinder can be lengthened by an + iron hoop which fits over the top ring. Deville's portable blast + furnace is very similar in principle to the above, but the body of the + furnace is formed of a single cast iron cylinder lined with fireclay, + closed below by a cast iron plate perforated by a ring of small + holes--a hemispherical basin below forming the air-heating chamber. + + + + +FURNEAUX, TOBIAS (1735-1781), English navigator, was born at Swilly near +Plymouth on the 21st of August 1735. He entered the royal navy, and was +employed on the French and African coasts and in the West Indies during +the latter part of the Seven Years' War (1760-1763). He served as second +lieutenant of the "Dolphin" under Captain Samuel Wallis on the latter's +voyage round the globe (August 1766-May 1768); was made a commander in +November 1771; and commanded the "Adventure" which accompanied Captain +Cook (in the "Resolution") in Cook's second voyage. On this expedition +Furneaux was twice separated from his leader (February 8-May 19, 1773; +October 22, 1773-July 14, 1774, the date of his return to England). On +the former occasion he explored a great part of the south and east +coasts of Tasmania, and made the earliest British chart of the same. +Most of his names here survive; Cook, visiting this shore-line on his +third voyage, confirmed Furneaux's account and delineation of it (with +certain minor criticisms and emendations), and named after him the +islands in Banks Straits, opening into Bass's Straits, and the group now +known as the Low Archipelago. After the "Adventure" was finally +separated from the "Resolution" off New Zealand in October 1773, +Furneaux returned home alone, bringing with him Omai of Ulaietea. This +first South Sea Islander seen in the British Isles returned to his home +with Cook in 1776-1777. Furneaux was made a captain in 1775, and +commanded the "Syren" in the British attack of the 28th of June 1776 +upon Charleston, South Carolina. His successful efforts to introduce +domestic animals and potatoes into the South Sea Islands are worthy of +note. He died at Swilly on the 19th of September 1781. + + See Hawkesworth's _Narrative of Wallis' Voyage_; Captain Cook's + _Narrative of his Second Voyage_; also T. Furneaux's life by Rev. + Henry Furneaux in the _Dictionary of National Biography_. + + + + + +FURNES (Flem. _Veurne_), an old-fashioned little town amid the dunes +near the coast in West Flanders, Belgium, about 26 m. S.W. of Bruges. +Pop. (1904) 6099. It is the centre of a considerable area extending to +the French frontier, and its market is an important one for the disposal +of corn, stock, hops and dairy produce. During the Norman raids Furnes +was destroyed, and the present town was built by Baldwin Bras de Fer, +first count of Flanders, about the year 870. At the height of the +prosperity of the Flemish communes in the 14th century there were +dependent on the barony of Furnes not fewer than fifty-two rich +villages, but these have all disappeared, partly no doubt as the +consequence of repeated French invasions down to the end of the 18th +century, but chiefly through the encroachment of the sea followed by the +accumulation of sand along the whole of this portion of the coast. +Furnes contains many curious old houses and the church of St Walburga, +which is a fine survival of the 13th century with some older portions. +The old church and buildings, grouped round the Grand Place, which is +the scene of the weekly market, present a quaint picture which is +perhaps not to be equalled in the country. Near Furnes on the seashore +is the fashionable bathing place called La Panne. + +Furnes one day a year becomes a centre of attraction to all the people +of Flanders. This is the last Sunday in July, when the fete of Calvary +and the Crucifixion is celebrated. Of all popular festivities in Belgium +this is the nearest approach to the old Passion Play. The whole story of +Christ is told with great precision by means of succeeding groups which +typify the different phases of the subject. The people of Furnes pose as +Roman soldiers or Jewish priests, as the apostles or mere spectators, +while the women put on long black veils so that they may figure in the +procession as the just women. + + + + +FURNESS, HORACE HOWARD (1833- ), American Shakespearian scholar, was +born in Philadelphia on the 2nd of November 1833, being the son of +William Henry Furness (1802-1896) minister of the First Unitarian church +in that city, a powerful preacher and writer. He graduated at Harvard in +1854, and was admitted to the bar in 1859, but soon devoted himself to +the study of Shakespeare. He accumulated a collection of illustrative +material of great richness and extent, and brought out in 1871 the first +volume of a new Variorum edition, designed to represent and summarize +the conclusions of the best authorities in all languages--textual, +critical and annotative. The volumes appeared as follows: _Romeo and +Juliet_ (1871); _Macbeth_ (1873) (revised edition, 1903); _Hamlet_ (2 +vols., 1877); _King Lear_ (1880); _Othello_ (1886); _The Merchant of +Venice_ (1888); _As You Like It_ (1890); _The Tempest_ (1892); _A +Midsummer Night's Dream_ (1895); _The Winter's Tale_ (1898); _Much Ado +about Nothing_ (1899); _Twelfth Night_ (1901); _Love's Labour's Lost_ +(1904). The edition has been generally accepted as a thorough and +scholarly piece of work; its chief fault is that, beginning with +_Othello_ (1858), the editor used the First Folio text as his basis, +while in others he makes the text of the Cambridge (Globe) editors his +foundation. His wife, Helen Kate Furness (1837-1883), compiled _A +Concordance to the Poems of Shakespeare_ (1872). + + + + +FURNESS, a district of Lancashire, England, separated from the major +portion of the county by Morecambe Bay. It is bounded S.E. by this inlet +of the Irish Sea, S.W. by the sea, W. by the Duddon estuary and +Cumberland, and N. and E. by Westmorland. Its area is about 250 sq. m. +It forms the greater part of the North Lonsdale parliamentary division +of Lancashire, and contains the parliamentary borough of +Barrow-in-Furness. The surface is almost entirely hilly. The northern +half is included in the celebrated Lake District, and contains such +eminences as the Old Man of Coniston and Wetherlam. Apart from the +Duddon, which forms part of the western boundary, the principal rivers +are the Leven and Crake, flowing southward into a common estuary in +Morecambe Bay. The Leven drains Windermere and the Crake Coniston Lake. +The usage of the term "Lake District," however, tends to limit the name +of Furness in common thought to the district south of the Lakes, where +several of the place-names are suffixed with that of the district, as +Barrow-in-Farness, Dalton-in-Furness, Broughton-in-Furness. Between the +Duddon and Morecambe Bay lies Walney Island, 8 m. in length, and in the +shallow strait between it and the mainland are several smaller islands. +That part of Furness which forms a peninsula between the Leven estuary +and Morecambe Bay, and the Duddon estuary, is rich in hematite iron ore, +which has been worked from very early times. It was known and smelted by +British and Romans, and by the monks of Furness Abbey and Conishead +Priory, both in the district. It was owing to the existence of this ore +that the town of Barrow grew up in the 19th century; at first as a port +from which the ore was exported to South Wales, while later furnaces +were established on the spot, and acquired additional importance on the +introduction of the Bessemer process, which requires a non-phosphoric +ore such as is found here. The hematite is also worked at Ulverston, +Askam, Dalton and elsewhere, but the furnaces now depend in part upon +ore imported from Spain. The supposed extension of the ore under the +sands of the Duddon estuary led to the construction of a sea wall to +facilitate the working. The district is served by the main line of the +Furness railway, from Carnforth (junction with the London & +North-Western railway), passing the pleasant watering-place of Grange, +and approximately following the coast by Ulverston, Dalton and Barrow, +with branches to Lake Side, Windermere, and to Coniston. + + + Furness Abbey. + +Apart from its industrial importance and scenic attractions, Furness has +an especial interest on account of its famous abbey. The ruins of this, +beautifully situated in a wooded valley, are extensive, and mainly of +fine transitional Norman and Early English date, acquiring additional +picturesqueness from the warm colour of the red sandstone of which they +are built. The abbey of Furness, otherwise Furdenesia or the further +_nese_ (promontory), which was dedicated to St Mary, was founded in 1127 +by a small body of monks belonging to the Benedictine order of Savigny. +In 1124 they had settled at Tulketh, near Preston, but migrated in 1127 +to Furness under the auspices of Stephen, count of Boulogne, afterwards +king, at that time lord of the liberty of Furness. In 1148 the +brotherhood joined the Cistercian order. Stephen granted to the monks +the lordship of Furness, and his charter was confirmed by Henry I., +Henry II. and subsequent kings. The abbot's power throughout the +lordship was almost absolute; he had a market and fair at Dalton, was +free from service to the county and wapentake, and held a sheriff's +tourn. By a succession of gifts the abbey became one of the richest in +England and was the largest Cistercian foundation in the kingdom. At the +Dissolution its revenues amounted to between L750 and L800 a year, +exclusive of meadows, pastures, fisheries, mines, mills and salt works, +and the wealth of the monks enabled them to practise a regal +hospitality. The abbot was one of the twenty Cistercian abbots summoned +to the parliament of 1264, but was not cited after 1330, as he did not +hold of the king _in capite per baroniam_. The abbey founded several +offshoot houses, one of the most important being Rushen Abbey in the +Isle of Man. In 1535 the royal commissioners visited the abbey and +reported four of its inmates, including the abbot, for incontinence. In +1536 the abbot was charged with complicity in the Pilgrimage of Grace, +and on the 7th of April 1537, under compulsion, surrendered the abbey to +the king. A few monks were granted pensions, and the abbot was endowed +with the profits of the rectory of Dalton, valued at L33, 6s. 8d. per +annum. In 1540 the estates and revenues were annexed by act of +parliament to the Duchy of Lancaster. About James I.'s reign the site +and territories were alienated to the Prestons of Preston-Patrick, from +whom they descended to the dukes of Devonshire. + +Conishead Priory, near Ulverston, an Augustinian foundation of the reign +of Henry II., has left no remains, but of the priory of Cartmel (1188) +the fine church is still in use. It is a cruciform structure of +transitional Norman and later dates, its central tower having the upper +storey set diagonally upon the lower. The chancel contains some superb +Jacobean carved oak screens, with stalls of earlier date. + + + + +FURNISS, HARRY (1854- ), British caricaturist and illustrator, was +born at Wexford, Ireland, of English and Scottish parents. He was +educated in Dublin, and in his schooldays edited a _Schoolboy's Punch_ +in close imitation of the original. He came to London when he was +nineteen, and began to draw for the illustrated papers, being for some +years a regular contributor to the _Illustrated London News_. His first +drawing in _Punch_ appeared in 1880, and he joined its staff in 1884. He +illustrated Lucy's "Diary of Toby, M.P.," in _Punch_, where his +political caricatures became a popular feature. Among his other +successes were a series of "Puzzle Heads," and his annual "Royal Academy +guy'd." In _Royal Academy Antics_ (1890) he published a volume of +caricatures of the work of leading artists. He resigned from the staff +of _Punch_ in 1894, produced for a short time a weekly comic paper _Lika +Joko_, and in 1898 began a humorous monthly, _Fair Game_; but these were +short-lived. Among the numerous books he illustrated were James Payn's +_Talk of the Town_, Lewis Carroll's _Sylvie and Bruno_, Gilbert a +Beckett's _Comic Blackstone_, G.E. Farrow's _Wallypug Book_, and his own +novel, _Poverty Bay_ (1905). _Our Joe, his great Fight_ (1903), was a +collection of original cartoons. His volume of reminiscences, +_Confessions of a Caricaturist_ (1901), was followed by _Harry Furniss +at Home_ (1904). In 1905 he published _How to draw in Pen and Ink_, and +produced the first number of _Harry Furniss's Christmas Annual_. + + + + +FURNITURE (from "furnish," Fr. _fournir_), a general term of obscure +origin, used to describe the chattels and fittings required to adapt +houses and other buildings for use. Wood, ivory, precious stones, +bronze, silver and gold have been used from the most ancient times in +the construction or for the decoration of furniture. The kinds of +objects required for furniture have varied according to the changes of +manners and customs, as well as with reference to the materials at the +command of the workman, in different climates and countries. Of really +ancient furniture there are very few surviving examples, partly by +reason of the perishable materials of which it was usually constructed; +and partly because, however great may have been the splendour of Egypt, +however consummate the taste of Greece, however luxurious the life of +Rome, the number of household appliances was very limited. The chair, +the couch, the table, the bed, were virtually the entire furniture of +early peoples, whatever the degree of their civilization, and so they +remained until the close of what are known in European history as the +middle ages. During the long empire-strewn centuries which intervened +between the lapse of Egypt and the obliteration of Babylon, the +extinction of Greece and the dismemberment of Rome and the great +awakening of the Renaissance, household comfort developed but little. +The Ptolemies were as well lodged as the Plantagenets, and peoples who +spent their lives in the open air, going to bed in the early hours of +darkness, and rising as soon as it was light, needed but little +household furniture. + +Indoor life and the growth of sedentary habits exercised a powerful +influence upon the development of furniture. From being splendid, or at +least massive, and exceedingly sparse and costly, it gradually became +light, plentiful and cheap. In the ancient civilizations, as in the +periods when our own was slowly growing, household plenishings, save in +the rudest and most elementary forms, were the privilege of the +great--no person of mean degree could have obtained, or would have dared +to use if he could, what is now the commonest object in every house, the +chair (q.v.). Sparse examples of the furniture of Egypt, Nineveh, Greece +and Rome are to be found in museums; but our chief sources of +information are mural and sepulchral paintings and sculptures. The +Egyptians used wooden furniture carved and gilded, covered with splendid +textiles, and supported upon the legs of wild animals; they employed +chests and coffers as receptacles for clothes, valuables and small +objects generally. Wild animals and beasts of the chase were carved upon +the furniture of Nineveh also; the lion, the bull and the ram were +especially characteristic. The Assyrians were magnificent in their +household appointments; their tables and couches were inlaid with ivory +and precious metals. Cedar and ebony were much used by these great +Eastern peoples, and it is probable that they were familiar with +rosewood, walnut and teak. Solomon's bed was of cedar of Lebanon. Greek +furniture was essentially Oriental in form; the more sumptuous varieties +were of bronze, damascened with gold and silver. The Romans employed +Greek artists and workmen and absorbed or adapted many of their +mobiliary fashions, especially in chairs and couches. The Roman tables +were of splendid marbles or rare woods. In the later ages of the empire, +in Rome and afterwards in Constantinople, gold and silver were +plentifully used in furniture; such indeed was the abundance of these +precious metals that even cooking utensils and common domestic vessels +were made of them. + +The architectural features so prominent in much of the medieval +furniture begin in these Byzantine and late Roman thrones and other +seats. These features became paramount as Pointed architecture became +general in Europe, and scarcely less so during the Renaissance. Most of +the medieval furniture, chests, seats, trays, &c., of Italian make were +richly gilt and painted. In northern Europe carved oak was more +generally used. State seats in feudal halls were benches with ends +carved in tracery, backs panelled or hung with cloths (called cloths of +estate), and canopies projecting above. Bedsteads were square frames, +the testers of panelled wood, resting on carved posts. Chests of oak +carved with panels of tracery, or of Italian cypress (when they could be +imported), were used to hold and to carry clothes, tapestries, &c., to +distant castles and manor houses; for house furniture, owing to its +scarcity and cost, had to be moved from place to place. Copes and other +ecclesiastical vestments were kept in chests with ornamental lock plates +and iron hinges. The splendour of most feudal houses depended on +pictorial tapestries which could be packed and carried from place to +place. Wardrobes were rooms fitted for the reception of dresses, as well +as for spices and other valuable stores. Excellent carving in relief was +executed on caskets, which were of wood or of ivory, with painting and +gilding, and decorated with delicate hinge and lock metal-work. The +general subjects of sculpture were taken from legends of the saints or +from metrical romances. Renaissance art made a great change in +architecture, and this change was exemplified in furniture. Cabinets +(q.v.) and panelling took the outlines of palaces and temples. In +Florence, Rome, Venice, Milan and other capitals of Italy, sumptuous +cabinets, tables, chairs, chests, &c., were made to the orders of the +native princes. Vasari (_Lives of Painters_) speaks of scientific +diagrams and mathematical problems illustrated in costly materials, by +the best artists of the day, on furniture made for the Medici family. +The great extent of the rule of Charles V. helped to give a uniform +training to artists from various countries resorting to Italy, so that +cabinets, &c., which were made in vast numbers in Spain, Flanders and +Germany, can hardly be distinguished from those executed in Italy. +Francis I. and Henry VIII. encouraged the revived arts in their +respective dominions. _Pietra dura_, or inlay of hard pebbles, agate, +lapis lazuli, and other stones, ivory carved and inlaid, carved and gilt +wood, marquetry or veneering with thin woods, tortoise-shell, brass, +&c., were used in making sumptuous furniture during the first period of +the Renaissance. Subjects of carving or relief were generally drawn from +the theological and cardinal virtues, from classical mythology, from the +seasons, months, &c. Carved altarpieces and woodwork in churches partook +of the change in style. + +The great period of furniture in almost every country was, however, +unquestionably the 18th century. That century saw many extravagances in +this, as in other forms of art, but on the whole it saw the richest +_floraison_ of taste, and the widest sense of invention. This is the +more remarkable since the furniture of the 17th century has often been +criticized as heavy and coarse. The criticism is only partly justified. +Throughout the first three-quarters of the period between the accession +of James I. and that of Queen Anne, massiveness and solidity were the +distinguishing characteristics of all work. Towards the reign of James +II., however, there came in one of the most pleasing and elegant styles +ever known in England. Nearly a generation before then Boulle was +developing in France the splendid and palatial method of inlay which, +although he did not invent it, is inseparably associated with his name. +We owe it perhaps to the fact that France, as the neighbour of Italy, +was touched more immediately by the Renaissance than England that the +reign of heaviness came earlier to an end in that country than on the +other side of the Channel. But there is a heaviness which is pleasing as +well as one which is forbidding, and much of the furniture made in +England any time after the middle of the 17th century was highly +attractive. If English furniture of the Stuart period be not sought +after to the same extent as that of a hundred years later, it is yet +highly prized and exceedingly decorative. Angularity it often still +possessed, but generally speaking its elegance of form and richness of +upholstering lent it an attraction which not long before had been +entirely lacking. Alike in France and in England, the most attractive +achievements of the cabinetmaker belong to the 18th century--English +Queen Anne and early Georgian work is universally charming; the regency +and the reigns of Louis XV. and XVI. formed a period of the greatest +artistic splendour. The inspiration of much of the work of the great +English school was derived from France, although the gropings after the +Chinese taste and the earlier Gothic manner were mainly indigenous. The +French styles of the century, which began with excessive flamboyance, +closed before the Revolution with a chaste perfection of detail which is +perhaps more delightful than anything that has ever been done in +furniture. In the achievements of Riesener, David Rontgen, Gouthiere, +Oeben and Rousseau de la Rottiere we have the high-water mark of +craftsmanship. The marquetry of the period, although not always +beautiful in itself, was executed with extraordinary smoothness and +finish; the mounts of gilded bronze, which were the leading +characteristic of most of the work of the century, were finished with a +minute delicacy of touch which was until then unknown, and has never +been rivalled since. If the periods of Francis I. and Henry II., of +Louis XIV. and the regency produced much that was sumptuous and even +elegant, that of Louis XVI., while men's minds were as yet undisturbed +by violent political convulsions, stands out as, on the whole, the one +consummate era in the annals of furniture. Times of great achievement +are almost invariably followed directly by those in which no tall +thistles grow and in which every little shrub is magnified to the +dimensions of a forest tree; and the so-called "empire style" which had +begun even while the last monarch of the _ancien regime_ still reigned, +lacked alike the graceful conception and the superb execution of the +preceding style. Heavy and usually uninspired, it was nurtured in +tragedy and perished amid disaster. Yet it is a profoundly interesting +style, both by reason of the classical roots from which it sprang and +the attempt, which it finally reflected, to establish new ideas in every +department of life. Founded upon the wreck of a lingering feudalism it +reached back to Rome and Greece, and even to Egypt. If it is rarely +charming, it is often impressive by its severity. Mahogany, satinwood +and other rich timbers were characteristic of the style of the end of +the 18th century; rosewood was most commonly employed for the choicer +work of the beginning of the 19th. Bronze mounts were in high favour, +although their artistic character varied materially. + +Previously to the middle of the 18th century the only cabinetmaker who +gained sufficient personal distinction to have had his name preserved +was Andre Charles Boulle; beginning with that period France and England +produced many men whose renown is hardly less than that of artists in +other media. With Chippendale there arose a marvellously brilliant +school of English cabinetmakers, in which the most outstanding names are +those of Sheraton, Heppelwhite, Shearer and the Adams. But if the school +was splendid it was lamentably short-lived, and the 19th century +produced no single name in the least worthy to be placed beside these +giants. Whether, in an age of machinery, much room is left for fine +individual execution may be doubted, and the manufacture of furniture +now, to a great extent, takes place in large factories both in England +and on the continent. Owing to the necessary subdivision of labour in +these establishments, each piece of furniture passes through numerous +distinct workshops. The master and a few artificers formerly +superintended each piece of work, which, therefore, was never far +removed from the designer's eye. Though accomplished artists are +retained by the manufacturers of London, Paris and other capitals, there +can no longer be the same relation between the designer and his work. +Many operations in these modern factories are carried on by machinery. +This, though an economy of labour, entails loss of artistic effect. The +chisel and the knife are no longer in such cases guided and controlled +by the sensitive touch of the human hand. + +[Illustration: PLATE I. + + FIG. 1.--Venetian Folding Chair of carved and gilt walnut, leather + back and seat; about 1530. + + FIG. 2.--Oak Arm-chair. English, 17th century. + + FIG. 3.--Arm-chair, solid seat, cane back; about 1660. + + FIG. 4.--Arm-chair, stuffed back and seat; about 1650. + + FIG. 5.--Painted and carved High-Back Chair; about 1660. + + FIG. 6.--Carved Walnut Chairs. English, early 18th century. The + arm-chair is inlaid. + + FIG. 7.--Walnut Chair; about 1710. + + FIG. 8.--Carved Mahogany Chair in the style of Chippendale; 2nd half + of 18th century. + + FIG. 9.--Carved Mahogany Arm-chair, in the style of Chippendale, with + ribbon pattern. + + FIG. 10.--Carved and Inlaid Mahogany Chair, in the style of + Hepplewhite; late 18th century. + + FIG. 11.--Mahogany Chair in the style of Sheraton; about 1780. + + Fig. 12.--Painted and gilt Arm-chair with cane seat, in the style of + Adam; about 1790. + + FIG. 13.--Arm-chair of carved and gilt wood with stuffed back, seat + and arms. French, Louis XV. style. + + FIG. 14.--Mahogany Arm-chair. Empire style, early 19th century, said + to have belonged to the Bonaparte family. + + FIG. 15.--Painted and gilt Beech Chair. English, about 1800.] + +[Illustration: PLATE II. + + FIG. 1.--Front of Oak Coffer with wrought iron bands. French, 2nd half + of 13th century. + + FIG. 2.--English Oak Chest, dated 1637. + + FIG. 3.--Italian (Florentine) Coffer of Wood with gilt arabesque + stucco ornament, about 1480. + + FIG. 4.--Italian "Cassone" or Marriage Coffer, 13th century. Carved + and gilt wood with painted front and ends. + + FIG. 5.--Walnut Table with expanding leaves. Swiss, 17th century. + + FIG. 6.--Oak Gate-Legged Table. English, 17th century. + + FIG. 7.--Writing Table. French, end of Louis XV. period. Riesener + marquetry, ormolu mounts and Sevres plaques. + + FIG. 8.--Painted Satin-Wood Tables, in the style of Sheraton, about + 1790. + + (The above are in the Victoria and Albert Museum, except Fig. 8, which + were in the Bethnal Green Exhibition, 1892.)] + +[Illustration: PLATE III. + + 1. CARVED OAK SIDEBOARD. English, 17th century. Victoria and Albert + Museum. + + 2. CARVED OAK COURT CUPBOARD. English, early 17th century. Victoria + and Albert Museum. + + 3. EBONY CARVED CABINET. The interior decorated with inlaid ivory and + coloured woods; French or Dutch, middle of 17th century. Victoria and + Albert Museum. + + 4. VENEERED CHEST OF DRAWERS. About 1690. Lent to Bethnal Green + Exhibition by Sir Spencer Ponsonby-Fane, G.C.B. + + 5. EBONY ARMOIRE. With tortoise-shell panels inlaid with brass and + other metals, and ormolu mountings. Designed by Berain, and executed + by Andre Boulle. French, Louis XIV. period. Victoria and Albert + Museum. + + 6. GLASS-FRONTED BOOKCASE AND CABINET. Of mahogany. In the style of + Sheraton, about 1790. Lent to the Bethnal Green Exhibition by the late + Vincent J. Robinson, C.I.E.] + +[Illustration: PLATE IV. + + 1. COMMODE OF PINE. With marquetry of brass, ebony, tortoise-shell, + mother-of-pearl, ivory, and green-stained bone. "Boulle" work with + designs in the style of Berain. French, late period of Louis XIV. + + 2. COMMODE. With panels of Japanese lacquer and ormolu mountings, in + the style of Caffieri. French, Louis XV. period. + + 3. TABLE OF KING AND TULIP WOODS. With ormolu mountings. Louis XV. + period. + + 4. ESCRITOIRE A TOILETTE. Formerly belonging to Marie Antoinette. Of + tulip and sycamore woods inlaid with other coloured woods, ormolu + mounts. Louis XV. period. + + 5. FOUR-POST BEDSTEAD. Of oak inlaid with bog-oak and holly, from the + "Inlaid Room" at Sizergh Castle, Westmorland. Latter half of sixteenth + century. + + 6. CARVED AND GILT BEDSTEAD. With blue silk damask coverings and + hangings. French, late 18th century. Louis XVI. period. + + From the Victoria and Albert Museum, S. Kensington.] + +[Illustration: PLATE V. + + _Photo, Mansell & Co._ + + THE "BUREAU DU ROI," MADE FOR LOUIS XV., NOW IN THE LOUVRE. For + description, see DESK.] + +A decided, if not always intelligent, effort to devise a new style in +furniture began during the last few years of the 19th century, which +gained the name of "_l'art nouveau_." Its pioneers professed to be free +from all old traditions and to seek inspiration from nature alone. +Happily nature is less forbidding than many of these interpretations of +it, and much of the "new art" is a remarkable exemplification of the +impossibility of altogether ignoring traditional forms. The style was +not long in degenerating into extreme extravagance. Perhaps the most +striking consequence of this effort has been, especially in England, the +revival of the use of oak. Lightly polished, or waxed, the cheap foreign +oaks often produce very agreeable results, especially when there is +applied to them a simple inlay of boxwood and stained holly, or a modern +form of pewter. The simplicity of these English forms is in remarkable +contrast to the tortured and ungainly outlines of continental seekers +after a conscious and unpleasing "originality." + +Until a very recent period the most famous collections of historic +furniture were to be found in such French museums as the Louvre, Cluny +and the Garde Meuble. Now, however, they are rivalled, if not surpassed, +by the magnificent collections of the Victoria and Albert Museum at +South Kensington, and the Wallace collection at Hertford House, London. +The latter, in conjunction with the Jones bequest at South Kensington, +forms the finest of all gatherings of French furniture of the great +periods, notwithstanding that in the Bureau du Roi the Louvre possesses +the most magnificent individual example in existence. In America there +are a number of admirable collections representative of the graceful and +homely "colonial furniture" made in England and the United States during +the Queen Anne and Georgian periods. + + See also the separate articles in this work on particular forms of + furniture. The literature of the subject has become very extensive, + and it is needless to multiply here the references to books. Perrot + and Chipiez, in their great _Histoire de l'art dans l'antiquite_ (1882 + et seq.) deal with ancient times, and A. de Champeaux, in _Le Meuble_ + (1885), with the middle ages and later period; English furniture is + admirably treated by Percy Macquoid in his _History of English + Furniture_ (1905); and Lady Dilke's _French Furniture in the 18th + Century_ (1901), and Luke Vincent Lockwood's _Colonial Furniture in + America_ (1901), should also be consulted. (J. P. B.) + + + + +FURNIVALL, FREDERICK JAMES (1825-1910), English philologist and editor, +was born at Egham, Surrey, on the 4th of February 1825, the son of a +surgeon. He was called to the bar in 1849, but his attention was soon +diverted to philological studies and social problems. He gave Frederick +Denison Maurice valuable assistance in the Christian Socialist movement, +and was one of the founders of the Working Men's College. For half a +century he indefatigably promoted the study of early English literature, +partly by his own work as editor, and still more efficaciously by the +agency of the numerous learned societies of which he was both founder +and director, especially the Early English Text Society (1864), which +has been of inestimable service in promoting the study of early and +middle English. He also established and conducted the Chaucer, Ballad, +New Shakespeare and Wyclif Societies, and at a later period societies +for the special study of Browning and Shelley. He edited texts for the +Early English Text Society, for the Roxburghe Club and the Rolls Series; +but his most important labours were devoted to Chaucer, whose study he +as an editor greatly assisted by his "Six-Text" edition of the +_Canterbury Tales_, and other publications of the Chaucer Society. He +was the honorary secretary of the Philological Society, and was one of +the original promoters of the Oxford _New English Dictionary_. He +co-operated with its first editor, Herbert Coleridge, and after his +death was for some time principal editor during the preliminary period +of the collection of material. The completion of his half-century of +labour was acknowledged in 1900 by a handsome testimonial, including the +preparation by his friends of a volume of philological essays specially +dedicated to him, _An English Miscellany_ (Oxford, 1901), and a +considerable donation to the Early English Text Society. Dr Furnivall +was always an enthusiastic oarsman, and till the end kept up his +interest in rowing; with John Beesley in 1845 he introduced the new type +of narrow sculling boat, and in 1886 started races on the Thames for +sculling fours and sculling eights. He died on the 2nd of July 1910. + + + + +FURSE, CHARLES WELLINGTON (1868-1904), English painter, born at Staines, +the son of the Rev. C.W. Furse, archdeacon of Westminster, was descended +collaterally from Sir Joshua Reynolds, and in his short span of life +achieved such rare excellence as a portrait and figure painter that he +forms an important link in the chain of British portraiture which +extends from the time when Van Dyck was called to the court of Charles +I. to our own day. His talent was precocious; at the age of seven he +gave indications of it in a number of drawings illustrating Scott's +novels. He entered the Slade school in 1884, winning the Slade +scholarship in the following year, and completed his education at +Julian's _atelier_ in Paris. Hard worker as he was, his activity was +frequently interrupted by spells of illness, for he had developed signs +of consumption when he was still attending the Slade school. An +important canvas called "Cain" was his first contribution (1888) to the +Royal Academy, to the associateship of which he was elected in the year +of his death. For some years before he had been a staunch supporter of +the New English Art Club, to the exhibitions of which he was a regular +contributor. He was married in October 1900 to Katherine, daughter of +John Addington Symonds. His fondness for sport and of an open-air life +found expression in his art and introduced a new, fresh and vigorous +note into portraiture. There is never a suggestion of the studio or of +the fatiguing pose in his portraits. The sitters appear unconscious of +being painted, and are generally seen in the pursuit of their favourite +outdoor sport or pastime, in the full enjoyment of life. Such are the +"Diana of the Uplands," the "Lord Roberts" and "The Return from the +Ride" at the Tate Gallery; the four children in the "Cubbing with the +York and Ainsty," "The Lilac Gown," "Mr and Mrs Oliver Fishing" and the +portrait of Lord Charles Beresford. Most of these pictures, and indeed +nearly all the work completed in the few years of Furse's activity, show +a pronounced decorative tendency. His sense of space, composition and +decorative design can best be judged by his admirable mural decorations +for Liverpool town hall, executed between 1899 and 1902. A memorial +exhibition of Furse's paintings and sketches was held at the Burlington +Fine Arts Club in 1906. + + + + +FURST, JULIUS (1805-1873), German Orientalist, was born of Jewish +parents at Zerkowo in Posen, on the 12th of May 1805. He studied +philosophy and philology at Berlin, and oriental literature at Posen, +Breslau and Halle. In 1857 he was appointed to a lectureship at the +university of Leipzig, and he was promoted to a professorship in 1864, +which he held until his death at Leipzig on the 9th of February 1873. +Among his writings may be mentioned _Lehrgebaude der aramaischen Idiome_ +(Leipzig, 1835); _Librorum sacrorum Veteris Testamenti concordantiae +Hebraicae atque Chaldaicae_ (Leipzig, 1837-1840); _Hebraisches und +chaldaisches Worterbuch_ (1851, English translation by S. Davidson +1867); _Kultur und Literaturgeschichte der Juden in Asien_ (1849). Furst +also edited a valuable _Bibliotheca Judaica_ (Leipzig, 1849-1863), and +was the author of some other works of minor importance. From 1840 to +1851 he was editor of _Der Orient_, a journal devoted to the language, +literature, history and antiquities of the Jews. + + + + +FURSTENBERG, the name of two noble houses of Germany. + +1. The more important is in possession of a mediatized principality in +the district of the Black Forest and the Upper Danube, which comprises +the countship of Heiligenberg, about 7 m. to the N. of the Lake of +Constance, the landgraviates of Stuhlingen and Baar, and the lordships +of Jungnau, Trochtelfingen, Hausen and Moskirch or Messkirch. The +territory is discontinuous; and as it lies partly in Baden, partly in +Wurttemberg, and partly in the Prussian province of Sigmaringen, the +head of the family is an hereditary member of the first chamber of Baden +and of the chamber of peers in Wurttemberg and in Prussia. The relations +of the principality with Baden are defined by the treaty of May 1825, +and its relations with Wurttemberg by the royal declaration of 1839. The +_Stammort_ or ancestral seat of the family is Furstenberg in the Black +Forest, about 13 m. N. of Schaffhausen, but the principal residence of +the present representatives of the main line is at Donaueschingen. + +The family of Furstenberg claims descent from a certain Count Unruoch, a +contemporary of Charlemagne, but their authentic pedigree is only +traceable to Egino II., count of Urach, who died before 1136. In 1218 +his successors inherited the possessions of the house of Zahringen in +the Baar district of the Black Forest, where they built the town and +castle of Furstenberg. Of the two sons of Egino V. of Urach, Conrad, the +elder, inherited the Breisgau and founded the line of the counts of +Freiburg, while the younger, Heinrich (1215-1284), received the +territories lying in the Kinzigthal and Baar, and from 1250 onward +styled himself first lord, then count, of Furstenberg. His territories +were subsequently divided among several branches of his descendants, +though temporarily reunited under Count Friedrich III., whose wife, +Anna, heiress of the last count of Wardenberg, brought him the countship +of Heiligenberg and lordships of Jungnau and Trochtelfingen in 1534. On +Friedrich's death (1559) his territories were divided between his two +sons, Joachim and Christof I. Of these the former founded the line of +Heiligenberg, the latter that of Kinzigthal. The Kinzigthal branch was +again subdivided in the 17th century between the two sons of Christof +II. (d. 1614), the elder, Wratislaw II. (d. 1642), founding the line of +Mosskirch, the younger, Friedrich Rudolf (d. 1655), that of Stuhlingen. +The Heiligenberg branch received an accession of dignity by the +elevation of Count Hermann Egon (d. 1674) to the rank of prince of the +Empire in 1664, but his line became extinct with the death of his son +Prince Anton Egon, favourite of King Augustus the Strong and regent of +Saxony, in 1716. The heads of both the Mosskirch and Stuhlingen lines +were now raised to the dignity of princes of the Empire (1716). The +Mosskirch branch died out with Prince Karl Friedrich (d. 1744); the +territories of the Stuhlingen branch had been divided on the death of +Count Prosper Ferdinand (1662-1704) between his two sons, Joseph Wilhelm +Ernst (1699-1762) and Ludwig August Egon (1705-1759). The first of these +was created prince of the Empire on the 10th of December 1716, and +founded the princely line of the Swabian Furstenbergs; in 1772 he +obtained from the emperor Francis I. for all his legitimate sons and +their descendants the right to bear, instead of the style of landgrave, +that of prince, which had so far been confined to the reigning head of +the family. Ludwig, on the other hand, founded the family of the +landgraves of Furstenberg, who, since their territories lay in Austria +and Moravia, were known as the "cadet line in Austria." The princely +line became extinct with the death of Karl Joachim in 1804, and the +inheritance passed to the Bohemian branch of the Austrian cadet line in +the person of Karl Egon II. (see below). Two years later the +principality was mediatized. + +In 1909 there were two branches of the princely house of Furstenberg: +(1) the main branch, that of Furstenberg-Donaueschingen, the head of +which was Prince Maximilian Egon (b. 1863), who succeeded his cousin +Karl Egon III. in 1896; (2) that of Furstenberg-Konigshof, in Bohemia, +the head of which was Prince Emil Egon (b. 1876), chamberlain and +secretary of legation to the Austro-Hungarian embassy in London (1907). +The cadet line of the landgraves of Furstenberg is now extinct, its last +representative having been the landgrave Joseph Friedrich Ernst of +Furstenberg-Weitra (1860-1896), son of the landgrave Ernst (1816-1889) +by a morganatic marriage. He was not recognized as _ebenburtig_ by the +family. The landgraves of Furstenberg were in 1909 represented only by +the landgravines Theresa (b. 1839) and Gabrielle (b. 1844), daughters +of the landgrave Johann Egon (1802-1879). + +From the days of Heinrich of Urach, a relative and notable supporter of +Rudolph of Habsburg, the Furstenbergs have played a stirring part in +German history as statesmen, ecclesiastics and notably soldiers. There +was a popular saying that "the emperor fights no great battle but a +Furstenberg falls." In the Heiligenberg line the following may be more +particularly noticed. + +FRANZ EGON (1625-1682), bishop of Strassburg, was the elder son of Egon +VII., count of Furstenberg (1588-1635), who served with distinction as a +Bavarian general in the Thirty Years' War. He began life as a soldier in +the imperial service, but on the elevation of his friend Maximilian +Henry of Bavaria to the electorate of Cologne in 1650, he went to his +court and embraced the ecclesiastical career. He soon gained a complete +ascendancy over the weak-minded elector, and, with his brother William +Egon (see below), was mainly instrumental in making him the tool of the +aggressive policy of Louis XIV. of France. Ecclesiastical preferments +were heaped upon him. As a child he had been appointed to a canonry of +Cologne; to these he added others at Strassburg, Liege, Hildesheim and +Spires; he became also suffragan bishop and dean of Cologne and provost +of Hildesheim, and in 1663 bishop of Strassburg. Later he was also +prince-abbot of Luders and Murbach and abbot of Stablo and Malmedy. On +the conclusion of a treaty between the emperor and the elector of +Cologne, on the 11th of May 1674, Franz was deprived of all his +preferments in Germany, and was compelled to take refuge in France. He +was, however, amnestied with his brother William by a special article of +the treaty of Nijmwegen (1679), whereupon he returned to Cologne. After +the French occupation of Strassburg (1681) he took up his residence +there and died on the 1st of April 1682. + +His brother WILLIAM EGON (1629-1704), bishop of Strassburg, began his +career as a soldier in the French service. He went to the court of the +elector of Cologne at the same time as Franz Egon, whose zeal for the +cause of Louis XIV. of France he shared. In 1672 the intrigues of the +two Furstenbergs had resulted in a treaty of offensive alliance between +the French monarchy and the electorate of Cologne, and, the brothers +being regarded by the Imperialists as the main cause of this disaster, +William was seized by imperial soldiers in the monastery of St Pantaleon +at Cologne, hurried off to Vienna and there tried for his life. He was +saved by the intervention of the papal nuncio, but was kept in prison +till the signature of the treaty of Nijmwegen (1679). As a reward for +his services Louis XIV. appointed him bishop of Strassburg in succession +to his brother in 1682, in 1686 obtained for him from Pope Innocent XI. +the cardinal's hat, and in 1688 succeeded in obtaining his election as +coadjutor-archbishop of Cologne and successor to the elector Maximilian +Henry. At the instance of the emperor, however, the pope interposed his +veto; the canons followed the papal lead, and, the progress of the +Allies against Louis XIV. depriving him of all prospect of success, +William Egon retired to France. Here he took up his abode at his abbey +of St Germain des Pres near Paris, where he died on the 10th of April +1704. + +In the Stuhlingen line the most notable was KARL EGON (1796-1854), +prince of Furstenberg, the son of Prince Karl Alois of Furstenberg, a +general in the Austrian service, who was killed at the battle of +Loptingen on the 25th of March 1799. In 1804 he inherited the Swabian +principality of Furstenberg and all the possessions of the family except +the Moravian estates. He studied at Freiburg and Wurzburg, and in 1815 +accompanied Prince Schwarzenberg to Paris as staff-officer. In 1817 he +came of age, and in the following year married the princess Amalie of +Baden. By the mediatization of his principality in 1806 the greater part +of his vast estates had fallen under the sovereignty of the grand-duke +of Baden, and Prince Furstenberg took a conspicuous part in the upper +house of the grand-duchy. In politics he distinguished himself by a +liberalism rare in a great German noble, carrying through by his +personal influence with his peers the abolition of tithes and feudal +dues and stanchly advocating the freedom of the press. He was not less +distinguished by his large charities: among other foundations he +established a hospital at Donaueschingen. For the industrial development +of the country, too, he did much, and proved himself also a notable +patron of the arts. His palace of Donaueschingen, with its collections +of paintings, engravings and coins, was a centre of culture, where +poets, painters and musicians met with princely entertainment. He died +on the 14th of September 1869, and was succeeded by his son Karl Egon +II. (1820-1892), with the death of whose son, Karl Egon III., in 1896, +the title and estates passed to Prince Maximilian Egon, head of the +cadet line of Furstenberg-Purglitz. + + See Munch, _Gesch. des Hauses und des Landes Furstenberg_, 4 vols. + (Aix-la-Chapelle, 1829-1847); S. Riezler, _Gesch. des furstlichen + Hauses Furstenberg bis 1507_ (Tubingen, 1883); _Furstenbergisches + Urkundenbuch_, edited by S. Riezler and F.L. Baumann, vols. i.-vii. + (Tubingen, 1877-1891), continued _s. tit. Mitteilungen aus dem + furstlich. Furstenbergischem Archiv_ by Baumann and G. Tumbult, 2 + vols. (ib. 1899-1902); Stokvis, _Manuel d'histoire_ (Leiden, + 1890-1893); _Almanach de Gotha; Allgemeine deutsche Biographie_. + +2. The second Furstenberg family has its possessions in Westphalia and +the country of the Rhine, and takes its name from the castle of +Furstenberg on the Ruhr. The two most remarkable men whom it has +produced are Franz Friedrich Wilhelm, freiherr von Furstenberg, and +Franz Egon, count von Furstenberg-Stammheim. The former (1728-1810) +became ultimately vicar-general of the prince-bishop of Munster, and +effected a great number of important reforms in the administration of +the country, besides doing much for its educational and industrial +development. The latter (1797-1859) was an enthusiastic patron of art, +who zealously advocated the completion of the Cologne cathedral, and +erected the beautiful church of St Apollinaris near Remagen on the +Rhine. He was a member of the Prussian Upper House in 1849, collaborated +in founding the _Preussisches Wochenblatt_, and was an ardent defender +of Catholic interests. His son, Count Gisbert von Furstenberg-Stammheim +(b. 1836), was in 1909 head of the Rhenish line of the house of +Furstenberg. + + + + +FURSTENWALDE, a town of Germany, in the Prussian province of +Brandenburg, on the right bank of the Spree, and on the railway from +Berlin to Frankfort-on-Oder, 28 m. E. of the former city. Pop. (1905) +20,498. Its beautiful cathedral church contains several old monuments. +The industries are important, including, besides brewing and malting, +manufactures of starch, vinegar, electric lamps and gas-fittings, +stoves, &c., iron-founding and wool-weaving. Furstenwalde is one of the +oldest towns of Brandenburg. From 1385 it was the seat of the bishop of +Lebus, whose bishopric was incorporated with the electorate of Brunswick +in 1595. + + + + +FURTH, a manufacturing town of Germany, in the kingdom of Bavaria, at +the confluence of the Pegnitz with the Regnitz, 5 m. N.W. from Nuremberg +by rail, at the junction of lines to Hof and Wurzburg. Pop. (1885) +35,455; (1905) 60,638. It is a modern town in appearance, with broad +streets and palatial business houses. Of its four Evangelical churches, +the old St Michaeliskirche is a handsome structure; but its chief +edifices are the new town hall, with a tower 175 ft. high and the +magnificent synagogue. The Jews have also a high school, which enjoys a +great reputation. There are besides a classical, a wood-carving and an +agricultural school and a library. Furth is the seat of several +important industries; particularly, the production of chromolithographs +and picture-books, the manufacture of mirrors and mirror-frames, bronze +and gold-leaf wares, pencils, toys, haberdashery, optical instruments, +silver work, turnery, chicory, machinery, fancy boxes and cases, and an +extensive trade is carried on in these goods as also in hops, metals, +wool, groceries and coal. A large annual fair is held at Michaelmas and +lasts for eleven days. The earliest railway in Germany was that between +Nuremberg and Furth (opened on the 7th of December 1835). + +Furth was founded, according to tradition, by Charlemagne, who erected a +chapel there. It was for a time a _Vogtei_ (advocateship) under the +burgraves of Nuremberg, but about 1314 it was bequeathed to the see of +Bamberg, and in 1806 it came into the possession of Bavaria. In 1632 +Gustavus Adolphus besieged it in vain, and in 1634 it was pillaged and +burnt by the Croats. It owes its rise to prosperity to the tolerance it +meted out to the Jews, who found here an asylum from the oppression +under which they suffered in Nuremberg. + + See Fronmuller, _Chronik der Stadt Furth_ (1887). + + + + +FURTWANGLER, ADOLF (1853-1907), German archaeologist, was born at +Freiburg im Breisgau, and was educated there, at Leipzig and at Munich, +where he was a pupil of H. Brunn, whose comparative method in +art-criticism he much developed. He took part in the excavations at +Olympia in 1878, became an assistant in the Berlin Museum in 1880, and +professor at Berlin (1884) and later at Munich. His latest excavation +work was at Aegina. He was a prolific writer, with a prodigious +knowledge and memory, and a most ingenious and confident critic; and his +work not only dominated the field of archaeological criticism but also +raised its standing both at home and abroad. Among his numerous +publications the most important were a volume on the bronzes found at +Olympia, vast works on ancient gems and Greek vases, and the invaluable +_Masterpieces of Greek Sculpture_ (English translation by Eugenie +Strong). He died at Athens on the 10th of October 1907. + + + + +FURZE, GORSE or WHIN; botanical name _Ulex_ (Ger. _Stechginster_, Fr. +_ajonc_), a genus of thorny papilionaceous shrubs, of few species, +confined to west and central Europe and north-west Africa. Common furze, +_U. europaeus_, is found on heaths and commons in western Europe from +Denmark to Italy and Greece, and in the Canaries and Azores, and is +abundant in nearly all parts of the British Isles. It grows to a height +of 2-6 ft.; it has hairy stems, and the smaller branches end each in a +spine; the leaves, sometimes lanceolate on the lowermost branches, are +mostly represented by spines from 2 to 6 lines long, and branching at +their base; and the flowers, about three-quarters of an inch in length, +have a shaggy, yellowish-olive calyx, with two small ovate bracts at its +base, and appear in early spring and late autumn. They are yellow and +sweet-scented and visited by bees. The pods are few-seeded; their +crackling as they burst may often be heard in hot weather. This species +comprises the varieties _vulgaris_, or _U. europaeus_ proper, which has +spreading branches, and strong, many-ridged spines, and _strictus_ +(Irish furze), with erect branches, and slender 4-edged spines. The +other British species of furze is _U. nanus_, dwarf furze, a native of +Belgium, Spain and the west of France; it is a procumbent plant, less +hairy than _U. europaeus_, with smaller and more orange-coloured +flowers, which spring from the primary spines, and have a nearly smooth +calyx, with minute basal bracts. Furze, or gorse, is sometimes employed +for fences. + +Notwithstanding its formidable spines, the young shoots yield a +palatable and nutritious winter forage for horses and cattle. To fit it +for this purpose it must be chopped and bruised to destroy the spines. +This is sometimes done in a primitive and laborious way by laying the +gorse upon a block of wood and beating it with a mallet, flat at one end +and armed with crossed knife-edges at the other, by the alternate use of +which it is bruised and chopped. There are now a variety of machines by +which this is done rapidly and efficiently, and which are in use where +this kind of forage is used to any extent. The agricultural value of +this plant has often been over-rated by theoretical writers. In the case +of very poor, dry soils it does, however, yield much valuable food at a +season when green forage is not otherwise to be had. It is on this +account of importance to dairymen; and to them it has this further +recommendation, that cows fed upon it give much rich milk, which is free +from any unpleasant flavour. To turn it to good account, it must be sown +in drills, kept clean by hoeing, and treated as a regular green crop. If +sown in March, on land fitly prepared and afterwards duly cared for, it +is ready for use in the autumn of the following year. A succession of +cuttings of proper age is obtained for several years from the same +field. It is cut by a short stout scythe, and must be brought from the +field daily; for when put in a heap after being chopped and bruised it +heats rapidly. It is given to horses and cows in combination with +chopped hay or straw. An acre will produce about 2000 faggots of green +two-year-old gorse, weighing 20 lb. each. + +This plant is invaluable in mountain sheep-walks. The rounded form of +the furze bushes that are met with in such situations shows how +diligently the annual growth, as far as it is accessible, is nibbled by +the sheep. The food and shelter afforded to them in snowstorms by +clusters of such bushes is of such importance that the wonder is our +sheep farmers do not bestow more pains to have it in adequate quantity. +Young plants of whin are so kept down by the sheep that they can seldom +attain to a profitable size unless protected by a fence for a few years. +In various parts of England it is cut for fuel. The ashes contain a +large proportion of alkali, and are a good manure, especially for peaty +land. + + + + +FUSARO, LAGO, a lake of Campania, Italy, 1/2 m. W. of Baia, and 1 m. S. +of the acropolis of Cumae. It is the ancient _Acherusia palus_, +separated from the sea on the W. by a line of sandhills. It may have +been the harbour of Cumae in early antiquity. In the 1st century A.D. an +artificial outlet was dug for it at its S. end, with a tunnel, lined +with _opus reticulatum_ and brick, under the hill of Torregaveta. This +hill is covered with the remains of a large villa, which is almost +certainly that of Servilius Vatia, described by Seneca (_Epist._ 55). +There are remains of other villas on the shores of the lake. Oyster +cultivation is carried on there. + + See J. Beloch, _Campanien_ (2nd ed., Breslau, 1890), 188. (T. As.) + + + + +FUSELI, HENRY (1741-1825), English painter and writer on art, of +German-Swiss family, was born at Zurich in Switzerland on the 7th of +February 1741; he himself asserted in 1745, but this appears to have +been a mere whim. He was the second child in a family of eighteen. His +father was John Caspar Fussli, of some note as a painter of portraits +and landscapes, and author of Lives of the _Helvetic Painters_. This +parent destined his son for the church, and with this view sent him to +the Caroline college of his native town, where he received an excellent +classical education. One of his schoolmates there was Lavater, with whom +he formed an intimate friendship. + +After taking orders in 1761 Fuseli was obliged to leave his country for +a while in consequence of having aided Lavater to expose an unjust +magistrate, whose family was still powerful enough to make its vengeance +felt. He first travelled through Germany, and then, in 1765, visited +England, where he supported himself for some time by miscellaneous +writing: there was a sort of project of promoting through his means a +regular literary communication between England and Germany. He became in +course of time acquainted with Sir Joshua Reynolds, to whom he showed +his drawings. By Sir Joshua's advice he then devoted himself wholly to +art. In 1770 he made an art-pilgrimage to Italy, where he remained till +1778, changing his name from Fussli to Fuseli, as more Italian-sounding. +Early in 1779 he returned to England, taking Zurich on his way. He found +a commission awaiting him from Alderman Boydell, who was then organizing +his celebrated Shakespeare gallery. Fuseli painted a number of pieces +for this patron, and about this time published an English edition of +Lavater's work on physiognomy. He likewise gave Cowper some valuable +assistance in preparing the translation of Homer. In 1788 Fuseli married +Miss Sophia Rawlins (who it appears was originally one of his models, +and who proved an affectionate wife), and he soon after became an +associate of the Royal Academy. Two years later he was promoted to the +grade of Academician. In 1799 he exhibited a series of paintings from +subjects furnished by the works of Milton, with a view to forming a +Milton gallery corresponding to Boydell's Shakespeare gallery. The +number of the Milton paintings was forty-seven, many of them very large; +they were executed at intervals within nine years. This exhibition, +which closed in 1800, proved a failure as regards profit. In 1799 also +he was appointed professor of painting to the Academy. Four years +afterwards he was chosen keeper, and resigned his professorship; but he +resumed it in 1810, and continued to hold both offices till his death. +In 1805 he brought out an edition of Pilkington's _Lives of the +Painters_, which, however, did not add much to his reputation. Canova, +when on his visit to England, was much taken with Fuseli's works, and on +returning to Rome in 1817 caused him to be elected a member of the first +class in the Academy of St Luke. Fuseli, after a life of uninterrupted +good health, died at Putney Hill on the 16th of April 1825, at the +advanced age of eighty-four, and was buried in the crypt of St Paul's +cathedral. He was comparatively rich at his death, though his +professional gains had always appeared to be meagre. + +As a painter, Fuseli had a daring invention, was original, fertile in +resource, and ever aspiring after the highest forms of excellence. His +mind was capable of grasping and realizing the loftiest conceptions, +which, however, he often spoiled on the canvas by exaggerating the due +proportions of the parts, and throwing his figures into attitudes of +fantastic and over-strained contortion. He delighted to select from the +region of the supernatural, and pitched everything upon an ideal scale, +believing a certain amount of exaggeration necessary in the higher +branches of historical painting. "Damn Nature! she always puts me out," +was his characteristic exclamation. In this theory he was confirmed by +the study of Michelangelo's works and the marble statues of the Monte +Cavallo, which, when at Rome, he used often to contemplate in the +evening, relieved against a murky sky or illuminated by lightning. But +this idea was by him carried out to an excess, not only in the forms, +but also in the attitudes of his figures; and the violent and +intemperate action which he often displays destroys the grand effect +which many of his pieces would otherwise produce. A striking +illustration of this occurs in his famous picture of "Hamlet breaking +from his Attendants to follow the Ghost": Hamlet, it has been said, +looks as though he would burst his clothes with convulsive cramps in all +his muscles. This intemperance is the grand defect of nearly all +Fuseli's compositions. On the other hand, his paintings are never either +languid or cold. His figures are full of life and earnestness, and seem +to have an object in view which they follow with rigid intensity. Like +Rubens he excelled in the art of setting his figures in motion. Though +the lofty and terrible was his proper sphere, Fuseli had a fine +perception of the ludicrous. The grotesque humour of his fairy scenes, +especially those taken from _A Midsummer-Night's Dream_, is in its way +not less remarkable than the poetic power of his more ambitious works. +As a colourist Fuseli has but small claims to distinction. He scorned to +set a palette as most artists do; he merely dashed his tints recklessly +over it. Not unfrequently he used his paints in the form of a dry +powder, which he rubbed up with his pencil with oil, or turpentine, or +gold size, regardless of the quantity, and depending for accident on the +general effect. This recklessness may perhaps be explained by the fact +that he did not paint in oil till he was twenty-five years of age. +Despite these drawbacks he possessed the elements of a great painter. + +Fuseli painted more than 200 pictures, but he exhibited only a minority +of them. His earliest painting represented "Joseph interpreting the +Dreams of the Baker and Butler"; the first to excite particular +attention was the "Nightmare," exhibited in 1782. He produced only two +portraits. His sketches or designs numbered about 800; they have +admirable qualities of invention and design, and are frequently superior +to his paintings. + +His general powers of mind were large. He was a thorough master of +French, Italian, English and German, and could write in all these +tongues with equal facility and vigour, though he preferred German as +the vehicle of his thoughts. His writings contain passages of the best +art-criticism that English literature can show. The principal work is +his series of _Lectures_ in the Royal Academy, twelve in number, +commenced in 1801. + + Many interesting anecdotes of Fuseli, and his relations to + contemporary artists, are given in his _Life_ by John Knowles, who + also edited his works in 3 vols. 8vo, London, 1831. (W. M. R.) + + + + +FUSEL OIL (from the Ger. _Fusel_, bad spirits), the name applied to the +volatile oily liquids, of a nauseous fiery taste and smell, which are +obtained in the rectification of spirituous liquors made by the +fermentation of grain, potatoes, the marc of grapes, and other +material, and which, as they are of higher boiling point than ethyl +alcohol, occur in largest quantity in the last portions of the +distillate. Besides ethyl or ordinary alcohol, and amyl alcohol, which +are present in them all, there have been found in fusel oil several +other bodies of the C_n H_(2n + 1).OH series, also certain ethers, and +members of the C_n H_(2n + 1).CO2H series of fatty acids. Normal propyl +alcohol is contained in the fusel oil of the marc brandy of the south of +France, and isoprimary butyl alcohol in that of beet-root molasses. The +chief constituent of the fusel oil procured in the manufacture of +alcohol from potatoes and grain, usually known as fusel oil and +potato-spirit, is isoprimary amyl alcohol, or isobutylcarbinol. Ordinary +fusel oil yields also an isomeric amyl alcohol (active amyl alcohol) +boiling at about 128 deg. Variable quantities of fusel oil, less or +greater according to the stage of ripening, exist in commercial spirits +(see SPIRITS). + +Fusel oil and its chief constituent, amyl alcohol, are direct nerve +poisons. In small doses it causes only thirst and headache, with furred +tongue and some excitement. In large doses it is a convulsent poison. +Impure beverages induce all the graver neurotic and visceral disorders +in alcoholism; and, like fusel oil, furfurol and the essence of +absinthe, are convulsent poisons. Pure ethyl alcohol intoxication, +indeed, is rarely seen, being modified in the case of spirits by the +higher alcohols contained in fusel oil. According to Rabuteau the toxic +properties of the higher alcohols increase with their molecular weight +and boiling point. Richet considers that the fusel oil contained in +spirits constitutes the chief danger in the consumption of alcoholic +beverages. The expert can immediately detect the peculiarly virulent +characters of the mixed intoxication due to the consumption of spirits +containing a large percentage of fusel oil. + + + + +FUSIBLE METAL, a term applied to certain alloys, generally composed of +bismuth, lead and tin, which possess the property of melting at +comparatively low temperatures. Newton's fusible metal (named after Sir +Isaac Newton) contains 50 parts of bismuth, 31.25 of lead and 18.75 of +tin; that of Jean Darcet (1725-1801), 50 parts of bismuth with 25 each +of lead and tin; and that of Valentin Rose the elder, 50 of bismuth with +28.1 of lead and 24.1 of tin. These melt between 91 deg. and 95 deg. C. +The addition of cadmium gives still greater fusibility; in Wood's metal, +for instance, which is Darcet's metal with half the tin replaced by +cadmium, the melting point is lowered to 66 deg.-71 deg. C.; while +another described by Lipowitz and containing 15 parts of bismuth, 8 of +lead, 4 of tin and 3 of cadmium, softens at about 55 deg. and is +completely liquid a little above 60 deg. By the addition of mercury to +Darcet's metal the melting point may be reduced so low as 45 deg. These +fusible metals have the peculiarity of expanding as they cool; Rose's +metal, for instance, remains pasty for a considerable range of +temperature below its fusing point, contracts somewhat rapidly from 80 +deg. to 55 deg., expands from 55 deg. to 35 deg., and contracts again +from 35 deg. to 0 deg. For this reason they may be used for taking casts +of anatomical specimens or making _cliches_ from wood-blocks, the +expansion on cooling securing sharp impressions. By suitable +modification in the proportions of the components, a series of alloys +can be made which melt at various temperatures above the boiling point +of water; for example, with 8 parts of bismuth, 8 of lead and 3 of tin +the melting point is 123 deg., and with 8 of bismuth, 30 of lead and 24 +of tin it is 172 deg. With tin and lead only in equal proportions it is +241 deg. Such alloys are used for making the fusible plugs inserted in +the furnace-crowns of steam boilers, as a safeguard in the event of the +water-level being allowed to fall too low. When this happens the plug +being no longer covered with water is heated to such a temperature that +it melts and allows the contents of the boiler to escape into the +furnace. In automatic fire-sprinklers the orifices of the pipes are +closed with fusible metal, which melts and liberates the water when, +owing to an outbreak of fire in the room, the temperature rises above a +predetermined limit. + + + + +FUSILIER, originally (in French about 1670, in English about 1680) the +name of a soldier armed with a light flintlock musket called the fusil; +now a regimental designation. Various forms of flintlock small arms had +been used in warfare since the middle of the 16th century. At the time +of the English civil war (1642-1652) the term "firelock" was usually +employed to distinguish these weapons from the more common matchlock +musket. The special value of the firelock in armies of the 17th century +lay in the fact that the artillery of the time used open powder barrels +for the service of the guns, making it unsafe to allow lighted matches +in the muskets of the escort. Further, a military escort was required, +not only for the protection, but also for the surveillance of the +artillerymen of those days. Companies of "firelocks" were therefore +organized for these duties, and out of these companies grew the +"fusiliers" who were employed in the same way in the wars of Louis XIV. +In the latter part of the Thirty Years' War (1643) fusiliers were simply +mounted troops armed with the fusil, as carabiniers were with the +carbine. But the escort companies of artillery came to be known by the +name shortly afterwards, and the regiment of French Royal Fusiliers, +organized in 1671 by Vauban, was considered the model for Europe. The +general adoption of the flintlock musket and the suppression of the pike +in the armies of Europe put an end to the original special duties of +fusiliers, and they were subsequently employed to a large extent in +light infantry work, perhaps on account of the greater individual +aptitude for detached duties naturally shown by soldiers who had never +been restricted to a fixed and unchangeable place in the line of battle. +The senior fusilier regiment in the British service, the (7th) Royal +Fusiliers (City of London Regiment), was formed on the French model in +1685; the 5th foot (now Northumberland Fusiliers), senior to the 7th in +the army, was not at that time a fusilier regiment. The distinctive +head-dress of fusiliers in the British service is a fur cap, generally +resembling, but smaller than and different in details from, that of the +Foot Guards. + +In Germany the name "fusilier" is borne by certain infantry regiments +and by one battalion in each grenadier regiment. + + + + +FUSION, the term generally applied to the melting of a solid substance, +or the change of state of aggregation from the solid to the liquid. The +term "liquefaction" is frequently employed in the same sense, but is +often restricted to the condensation of a gas or vapour. The converse +process of freezing or solidification, the change from the liquid to the +solid state, is subject to the same laws, and must be considered +together with fusion. The solution of a solid in a foreign liquid, and +the deposition or crystallization of a solid from a solution, are so +closely related to the fusion of a pure substance, that it will also be +necessary to consider some of the analogies which they present. + +1. _General Phenomena._--There are two chief varieties of the process of +fusion, namely, crystalline and amorphous, which are in many ways +distinct, although it is possible to find intermediate cases which +partake of the characteristics of both. The melting of ice may be taken +as a typical case of crystalline fusion. The passage from rigid solid to +mobile liquid occurs at a definite surface without any intermediate +stage or plastic condition. The change takes place at a definite +temperature, the fusing or freezing point (abbreviated F.P.), and +requires the addition of a definite quantity of heat to the solid, which +is called the latent heat of fusion. There is also in general a +considerable change of volume during fusion, which amounts in the case +of ice to a contraction of 9%. Typical cases of amorphous solidification +are those of silica, glass, plastic sulphur, pitch, alcohol and many +organic liquids. In this type the liquid gradually becomes more and more +viscous as the temperature falls, and ultimately attains the rigidity +characteristic of a solid, without any definite freezing point or latent +heat. The condition of the substance remains uniform throughout, if its +temperature is uniform; there is no separation into the two distinct +phases of solid and liquid, and there is no sudden change of volume at +any temperature. + +A change or transition from one crystalline form to another may occur in +the solid state with evolution or absorption of heat at a definite +temperature, and is analogous to the change from solid to liquid, but +usually takes place more slowly owing to the small molecular mobility of +the solid state. Thus rhombic sulphur when heated passes slowly at 95.6 +deg. C. into the monosymmetric form which melts at 120 deg., but if +heated rapidly the rhombic form melts at 114.5. The two forms, rhombic +and monosymmetric, can exist in equilibrium at 95.6 deg., the transition +point at which they have the same vapour pressure. Similarly a solid +solution of carbon in iron, when cooled slowly, passes at about 700 deg. +C., with considerable evolution of heat, into the form of "pearlite," +which is soft when cold, but if rapidly chilled the carbon remains in +solution and the steel is very hard (see also ALLOYS). + +In the case of crystalline fusion it is necessary to distinguish two +cases, the homogeneous and the heterogeneous. In the first case the +composition of the solid and liquid phases are the same, and the +temperature remains constant during the whole process of fusion. In the +second case the solid and liquid phases differ in composition; that of +the liquid phase changes continuously, and the temperature does not +remain constant during the fusion. The first case comprises the fusion +of pure substances, and that of eutectics, or cryohydrates; the second +is the general case of an alloy or a solution. These have been very +fully studied and their phenomena greatly elucidated in recent years. + +There is also a sub-variety of amorphous fusion, which may be styled +colloid or gelatinous, and may be illustrated by the behaviour of +solutions of water in gelatin. Many of these jellies melt at a fairly +definite temperature on heating, and coagulate or set at a definite +temperature on cooling. But in some cases the process is not reversible, +and there is generally marked hysteresis, the temperature of setting and +other phenomena depending on the rate of cooling. This case has not yet +been fully worked out; but it appears probable that in many cases the +jelly possesses a spongy framework of solid, holding liquid in its +meshes or interstices. It might be regarded as a case of "heterogeneous" +amorphous fusion, in which the liquid separates into two phases of +different composition, one of which solidifies before the other. The two +phases cannot, as a rule, be distinguished optically, but it is +generally possible to squeeze out some of the liquid phase when the +jelly has set, which proves that the substance is not really +homogeneous. In very complicated mixtures, such as acid lavas or slags +containing a large proportion of silica, amorphous and crystalline +solidification may occur together. In this case the crystals separate +first during the process of cooling, the mother liquor increases +gradually in viscosity, and finally sets as an amorphous ground-mass or +matrix, in which crystals of different kinds and sizes, formed at +different stages of the cooling, remain embedded. The formation of +crystals in an amorphous solid after it has set is also of frequent +occurrence. It is termed devitrification, but is a very slow process +unless the solid is in a plastic state. + +2. _Homogeneous Crystalline Fusion._--The fusion of a solid of this type +is characterized most clearly by the perfect constancy of temperature +during the process. In fact, the law of constant temperature, which is +generally stated as the first of the so-called "laws of fusion," does +not strictly apply except to this case. The constancy of the F.P. of a +pure substance is so characteristic that change of the F.P. is often one +of the most convenient tests of the presence of foreign material. In the +case of substances like ice, which melt at a low temperature and are +easily obtained in large quantities in a state of purity, the point of +fusion may be very accurately determined by observing the temperature of +an intimate mixture of the solid and liquid while slowly melting as it +absorbs heat from surrounding bodies. But in the majority of cases it is +more convenient to observe the freezing point as the liquid is cooled. +By this method it is possible to ensure perfect uniformity of +temperature throughout the mass by stirring the liquid continuously +during the process of freezing, whereas it is difficult to ensure +uniformity of temperature in melting a solid, however gradually the heat +is supplied, unless the solid can be mixed with the liquid. It is also +possible to observe the F.P. in other ways, as by noting the temperature +at the moment of the breaking of a wire, of the stoppage of a stirrer, +or of the maximum rate of change of volume, but these methods are +generally less certain in their indications than the point of greatest +constancy of temperature in the case of homogeneous crystalline solids. + + _Fusing Points of Common Metals_ + + Mercury -38.8 deg. Antimony 630 deg. + Potassium 62.5 deg. Aluminium 655 deg. + Sodium 95.6 deg. Silver 962 deg. + Tin 231.9 deg. Gold 1064 deg. + Bismuth 269.2 deg. Copper 1082 deg. + Cadmium 320.7 deg. Nickel 1427 deg. + Lead 327.7 deg. Palladium 1535 deg. + Zinc 419.0 deg. Platinum 1710 deg. + +The above table contains some of the most recent values of fusing points +of metals determined (except the first three and the last three) with +platinum thermometers. The last three values are those obtained by +extrapolation with platinum-rhodium and platinum-iridium couples. (See +Harker, _Proc. Roy. Soc._ A 76, p. 235, 1905.) Some doubt has recently +been raised with regard to the value for platinum, which is much lower +than that previously accepted, namely 1775 deg. + +3. _Superfusion, Supersaturation._--It is generally possible to cool a +liquid several degrees below its normal freezing point without a +separation of crystals, especially if it is protected from agitation, +which would assist the molecules to rearrange themselves. A liquid in +this state is said to be "undercooled" or "superfused." The phenomenon +is even more familiar in the case of solutions (e.g. sodium sulphate or +acetate) which may remain in the "metastable" condition for an +indefinite time if protected from dust, &c. The introduction into the +liquid under this condition of the smallest fragment of the crystal, +with respect to which the solution is supersaturated, will produce +immediate crystallization, which will continue until the temperature is +raised to the saturation point by the liberation of the latent heat of +fusion. The constancy of temperature at the normal freezing point is due +to the equilibrium of exchange existing between the liquid and solid. +Unless both solid and liquid are present, there is no condition of +equilibrium, and the temperature is indeterminate. + +It has been shown by H.A. Miers (_Jour. Chem. Soc._, 1906, 89, p. 413) +that for a supersaturated solution in metastable equilibrium there is an +inferior limit of temperature, at which it passes into the "labile" +state, i.e. spontaneous crystallization occurs throughout the mass in a +fine shower. This seems to be analogous to the fine misty condensation +which occurs in a supersaturated vapour in the absence of nuclei (see +VAPORIZATION) when the supersaturation exceeds a certain limit. + + 4. _Effect of Pressure on the F.P._--The effect of pressure on the + fusing-point depends on the change of volume during fusion. Substances + which expand on _freezing_, like ice, have their freezing points + lowered by increase of pressure; substances which expand on _fusing_, + like wax, have their melting points raised by pressure. In each case + the effect of pressure is to retard increase of volume. This effect + was first predicted by James Thomson on the analogy of the effect of + pressure on the boiling point, and was numerically verified by Lord + Kelvin in the case of ice, and later by Bunsen in the case of paraffin + and spermaceti. The equation by which the change of the F.P. is + calculated may be proved by a simple application of the Carnot cycle, + exactly as in the case of vapour and liquid. (See THERMODYNAMICS.) If + L be the latent heat of fusion in mechanical units, v' the volume of + unit mass of the solid, and v" that of the liquid, the work done in + an elementary Carnot cycle of range d[theta] will be dp(v"- v'), if + dp is the increase of pressure required to produce a change d[theta] + in the F.P. Since the ratio of the work-difference or cycle-area to + the heat-transferred L must be equal to d[theta]/[theta], we have the + relation + + d[theta]/dp = [theta](v" - v')/L. (1) + + The sign of d[theta], the change of the F.P., is the same as that of + the change of volume (v" - v'). Since the change of volume seldom + exceeds 0.1 c.c. per gramme, the change of the F.P. per atmosphere is + so small that it is not as a rule necessary to take account of + variations of atmospheric pressure in observing a freezing point. A + variation of 1 cm. in the height of the barometer would correspond to + a change of .0001 deg. C. only in the F.P. of ice. This is far beyond + the limits of accuracy of most observations. Although the effect of + pressure is so small, it produces, as is well known, remarkable + results in the motion of glaciers, the moulding and regelation of ice, + and many other phenomena. It has also been employed to explain the + apparent inversion of the order of crystallization in rocks like + granite, in which the arrangement of the crystals indicates that the + quartz matrix solidified subsequently to the crystals of felspar, + mica or hornblende embedded in it, although the quartz has a higher + melting point. It is contended that under enormous pressure the + freezing points of the more fusible constituents might be raised above + that of the quartz, if the latter is less affected by pressure. Thus + Bunsen found the F.P. of paraffin wax 1.4 deg. C. below that of + spermaceti at atmospheric pressure. At 100 atmospheres the two melted + at the same temperature. At higher pressures the paraffin would + solidify first. The effect of pressure on the silicates, however, is + much smaller, and it is not so easy to explain a change of several + hundred degrees in the F.P. It seems more likely in this particular + case that the order of crystallization depends on the action of + superheated water or steam at high temperatures and pressures, which + is well known to exert a highly solvent and metamorphic action on + silicates. + + 5. _Variation of Latent Heat._--C.C. Person in 1847 endeavoured to + show by the application of the first law of thermodynamics that the + increase of the latent heat per degree should be equal to the + difference (s"- s') between the specific heats of the liquid and + solid. If, for instance, water at 0 deg. C. were first frozen and then + cooled to -t deg. C., the heat abstracted per gramme would be (L'+ + s't) calories. But if the water were first cooled to -t deg. C., and + then frozen at -t deg.C., by abstracting heat L", the heat abstracted + would be L" + s"t. Assuming that the heat abstracted should be the + same in the two cases, we evidently obtain L'- L" = (s"- s')t. This + theory has been approximately verified by Petterson, by observing the + freezing of a liquid cooled below its normal F.P. (_Jour. Chem. Soc._ + 24, p. 151). But his method does not represent the true variation of + the latent heat with temperature, since the freezing, in the case of a + superfused liquid, really takes place at the normal freezing point. A + quantity of heat s"t is abstracted in cooling to -t, (L"- s"t) in + raising to 0 deg. and freezing at 0 deg., and s't in cooling the ice + to -t. The latent heat L" at -t does not really enter into the + experiment. In order to make the liquid freeze at a different + temperature, it is necessary to subject it to pressure, and the effect + of the pressure on the latent heat cannot be neglected. The entropy of + a liquid [phi]" at its F.P. reckoned from any convenient zero [phi]0 + in the solid state may be represented by the expression + + _ + / + [phi]" - [phi]0 = | s'd[theta] / [theta] + L / [theta]. (2) + _/ + + Since [theta]d[phi]"/d[theta] = s", we obtain by differentiation the + relation + + dL / d[theta] = s"- s' + L / [theta], (3) + + which is exactly similar to the equation for the specific heat of a + vapour maintained in the saturated condition. If we suppose that the + specific heats s' and s'' of the solid and liquid at equilibrium + pressure are nearly the same as those ordinarily observed at constant + pressure, the relation (3) differs from that of Person only by the + addition of the term L/[theta]. Since s" is greater than s' in all + cases hitherto investigated, and L/[theta] is necessarily positive, it + is clear that the latent heat of fusion must increase with rise of + temperature, or diminish with fall of temperature. It is possible to + imagine the F.P. so lowered by pressure (positive or negative) that + the latent heat should vanish, in which case we should probably obtain + a continuous passage from the liquid to the solid state similar to + that which occurs in the case of amorphous substances. According to + equation (3), the rate of change of the latent heat of water is + approximately 0.80 calorie per degree at 0 deg. C. (as compared with + 0.50, Person), if we assume s" = 1, and s' = 0.5. Putting (s" - s') = + 0.5 in equation (2), we find L = 0 at -160 deg. C. approximately, but + no stress can be laid on this estimate, as the variation of (s" - s') + is so uncertain. + +6. _Freezing of Solutions and Alloys._--The phenomena of freezing of +heterogeneous crystalline mixtures may be illustrated by the case of +aqueous solutions and of metallic solutions or alloys, which have been +most widely studied. The usual effect of an impurity, such as salt or +sugar in solution in water, is to lower the freezing point, so that no +crystallization occurs until the temperature has fallen below the normal +F.P. of the pure solvent, the depression of F.P. being nearly +proportional to the concentration of the solution. When freezing begins, +the solvent generally separates out from the solution in the pure state. +This separation of the solvent involves an increase in the strength of +the remaining solution, so that the temperature does not remain constant +during the freezing, but continues to fall as more of the solvent is +separated. There is a perfectly definite relation between temperature +and concentration at each stage of the process, which may be represented +in the form of a curve as AC in fig. 1, called the freezing point curve. +The equilibrium temperature, at the surface of contact between the solid +and liquid, depends only on the composition of the liquid phase and not +at all on the quantity of solid present. The abscissa of the F.P. curve +represents the composition of that portion of the original solution +which remains liquid at any temperature. If instead of starting with a +dilute solution we start with a strong solution represented by a point +N, and cool it as shown by the vertical line ND, a point D is generally +reached at which the solution becomes "saturated." The dissolved +substance or "solute" then separates out as the solution is further +cooled, and the concentration diminishes with fall of temperature in a +definite relation, as indicated by the curve CB, which is called the +solubility curve. Though often called by different names, the two curves +AC and CB are essentially of a similar nature. To take the case of an +aqueous solution of salt as an example, along CB the solution is +saturated with respect to salt, along AC the solution is saturated with +respect to ice. When the point C is reached along either curve, the +solution is saturated with respect to both salt and ice. The +concentration cannot vary further, and the temperature remains constant, +while the salt and ice crystallize out together, maintaining the exact +proportions in which they exist in the solution. The resulting solid was +termed a cryohydrate by F. Guthrie, but it is really an intimate mixture +of two kinds of crystals, and not a chemical compound or hydrate +containing the constituents in chemically equivalent proportions. The +lowest temperature attainable by means of a freezing mixture is the +temperature of the F.P. of the corresponding cryohydrate. In a mixture +of salt and ice with the least trace of water a saturated brine is +quickly formed, which dissolves the ice and falls rapidly in +temperature, owing to the absorption of the latent heat of fusion. So +long as both ice and salt are present, if the mixture is well stirred, +the solution must necessarily become saturated with respect to both ice +and salt, and this can only occur at the cryohydric temperature, at +which the two curves of solubility intersect. + +[Illustration: FIG. 1.--F.P. or Solubility Curve: simple case.] + +The curves in fig. 1 also illustrate the simplest type of freezing point +curve in the case of alloys of two metals A and B which do not form +mixed crystals or chemical compounds. The alloy corresponding to the +cryohydrate, possessing the lowest melting point, is called the eutectic +alloy, as it is most easily cast and worked. It generally possesses a +very fine-grained structure, and is not a chemical compound. (See +ALLOYS.) + +[Illustration: FIG. 2.--Cooling Curves of Alloys: typical case.] + +To obtain a complete F.P. curve even for a binary alloy is a laborious +and complicated process, but the information contained in such a curve +is often very valuable. It is necessary to operate with a number of +different alloys of suitably chosen composition, and to observe the +freezing points of each separately. Each alloy should also be analysed +after the process if there is any risk of its composition having been +altered by oxidation or otherwise. The freezing points are generally +best determined by observing the gradual cooling of a considerable mass, +which is well stirred so long as it remains liquid. The curve of cooling +may most conveniently be recorded, either photographically, using a +thermocouple and galvanometer, as in the method of Sir W. +Roberts-Austen, or with pen and ink, if a platinum thermometer is +available, according to the method put in practice by C.T. Heycock and +F.H. Neville. A typical set of curves obtained in this manner is shown +in fig. 2. When the pure metal A in cooling reaches its F.P. the +temperature suddenly becomes stationary, and remains accurately constant +for a considerable period. Often it falls slightly below the F.P. owing +to super-fusion, but rises to the F.P. and remains constant as soon as +freezing begins. The second curve shows the cooling of A with 10% of +another metal B added. The freezing begins at a lower temperature with +the separation of pure A. The temperature no longer remains constant +during freezing, but falls more and more rapidly as the proportion of B +in the liquid increases. When the eutectic temperature is reached there +is a second F.P. or arrest at which the whole of the remaining liquid +solidifies. With 20% of B the first F.P. is further lowered, and the +temperature falls faster. The eutectic F.P. is of longer duration, but +still at the same temperature. For an alloy of the composition of the +eutectic itself there is no arrest until the eutectic temperature is +reached, at which the whole solidifies without change of temperature. +There is a great advantage in recording these curves automatically, as +the primary arrest is often very slight, and difficult to observe in any +other way. + + 7. _Change of Solubility with Temperature._--The lowering of the F.P. + of a solution with increase of concentration, as shown by the F.P. or + solubility curves, may be explained and calculated by equation (1) in + terms of the osmotic pressure of the dissolved substance by analogy + with the effect of mechanical pressure. It is possible in salt + solutions to strain out the salt mechanically by a suitable filter or + "semi-permeable membrane," which permits the water to pass, but + retains the salt. To separate 1 gramme of salt requires the + performance of work PV against the osmotic pressure P, where V is the + corresponding diminution in the volume of the solution. In dilute + solutions, to which alone the following calculation can be applied, + the volume V is the reciprocal of the concentration C of the solution + in grammes per unit volume, and the osmotic pressure P is equal to + that of an equal number of molecules of gas in the same space, and may + be deduced from the usual equation of a gas, + + P = R[theta] / VM = R[theta]C / M, (4) + + where M is the molecular weight of the salt in solution, [theta] the + absolute temperature, and R a constant which has the value 8.32 + joules, or nearly 2 calories, per degree C. It is necessary to + consider two cases, corresponding to the curves CB and AB in fig. 1, + in which the solution is saturated with respect to salt and water + respectively. To facilitate description we take the case of a salt + dissolved in water, but similar results apply to solutions in other + liquids and alloys of metals. + + (a) If unit mass of salt is separated in the solid state from a + saturated solution of salt (curve CB) by forcing out through a + semi-permeable membrane against the osmotic pressure P the + corresponding volume of water V in which it is dissolved, the heat + evolved is the latent heat of saturated solution of the salt Q + together with the work done PV. Writing (Q + PV) for L, and V for (v" + - v') in equation (1), and substituting P for p, we obtain + + Q + PV = V[theta]dP / d[theta], (5) + + which is equivalent to equation (1), and may be established by similar + reasoning. Substituting for P and V in terms of C from equation (4), + if Q is measured in calories, R = 2, and we obtain + + QC = 2[theta]^2dC / d[theta], (6) + + which may be integrated, assuming Q constant, with the result + + 2log_eC" / C' = Q / [theta]' - Q / [theta]", (7) + + where C', C" are the concentrations of the saturated solution + corresponding to the temperatures [theta]' and [theta]". This + equation may be employed to calculate the latent heat of solution Q + from two observations of the solubility. It follows from these + equations that Q is of the same sign as dC/d[theta], that is to say, + the solubility increases with rise of temperature if heat is absorbed + in the formation of the saturated solution, which is the usual case. + If, on the other hand, heat is liberated on solution, as in the case + of caustic potash or sulphate of calcium, the solubility diminishes + with rise of temperature. + + (b) In the case of a solution saturated with respect to ice (curve + AC), if one gramme of water having a volume v is separated by + freezing, we obtain a precisely similar equation to (5), but with L + the latent heat of fusion of water instead of Q, and v instead of V. + If the solution is dilute, we may neglect the external work Pv in + comparison with L, and also the heat of dilution, and may write P/t + for dP/d[theta], where t is the depression of the F.P. below that of + the pure solvent. Substituting for P in terms of V from equation (4), + we obtain + + t = 2[theta]^2v / LVM = 2[theta]^2w / LWM, (8) + + where W is the weight of water and w that of salt in a given volume of + solution. If M grammes of salt are dissolved in 100 of water, w = M + and W = 100. The depression of the F.P. in this case is called by van + 't Hoff the "Molecular Depression of the F.P." and is given by the + simple formula + + t = .02[theta]^2 / L. (9) + + Equation (8) may be used to calculate L or M, if either is known, from + observations of t, [theta] and w/W. The results obtained are + sufficiently approximate to be of use in many cases in spite of the + rather liberal assumptions and approximations effected in the course + of the reasoning. In any case the equations give a simple theoretical + basis with which to compare experimental data in order to estimate the + order of error involved in the assumptions. We may thus estimate the + variation of the osmotic pressure from the value given by the gaseous + equation, as the concentration of the solution or the molecular + dissociation changes. The most uncertain factor in the formula is the + molecular weight M, since the molecule in solution may be quite + different from that denoted by the chemical formula of the solid. In + many cases the molecule of a metal in dilute solution in another metal + is either monatomic, or forms a compound molecule with the solvent + containing one atom of the dissolved metal, in which case the + molecular depression is given by putting the atomic weight for M. In + other cases, as Cu, Hg, Zn, in solution in cadmium, the depression of + the F.P. per atom, according to Heycock and Neville, is only half as + great, which would imply a diatomic molecule. Similarly As and Au in + Cd appear to be triatomic, and Sn in Pb tetratomic. Intermediate cases + may occur in which different molecules exist together in equilibrium + in proportions which vary according to the temperature and + concentration. The most familiar case is that of an electrolyte, in + which the molecule of the dissolved substance is partly dissociated + into ions. In such cases the degree of dissociation may be estimated + by observing the depression of the F.P., but the results obtained + cannot always be reconciled with those deduced by other methods, such + as measurement of electrical conductivity, and there are many + difficulties which await satisfactory interpretation. + + Exactly similar relations to (8) and (9) apply to changes of boiling + point or vapour pressure produced by substances in solution (see + VAPORIZATION), the laws of which are very closely connected with the + corresponding phenomena of fusion; but the consideration of the vapour + phase may generally be omitted in dealing with the fusion of mixtures + where the vapour pressure of either constituent is small. + +[Illustration: FIG. 3.--Solubility Curves of Hydrates.] + +8. _Hydrates._--The simple case of a freezing point curve, illustrated +in fig. 1, is generally modified by the occurrence of compounds of a +character analogous to hydrates of soluble salts, in which the dissolved +substance combines with one or more molecules of the solvent. These +hydrates may exist as compound molecules in the solution, but their +composition cannot be demonstrated unless they can be separated in the +solid state. Corresponding to each crystalline hydrate there is +generally a separate branch of the solubility curve along which the +crystals of the hydrate are in equilibrium with the saturated solution. +At any given temperature the hydrate possessing the least solubility is +the most stable. If two are present in contact with the same solution, +the more soluble will dissolve, and the less soluble will be formed at +its expense until the conversion is complete. The two hydrates cannot be +in equilibrium with the same solution except at the temperature at which +their solubilities are equal, i.e. at the point where the corresponding +curves of solubility intersect. This temperature is called the +"Transition Point." In the case of ZnSO4, as shown in fig. 3, the +heptahydrate, with seven molecules of water, is the least soluble +hydrate at ordinary temperatures, and is generally deposited from +saturated solutions. Above 39 deg. C., however, the hexahydrate, with six +molecules, is less soluble, and a rapid conversion of the hepta- into +the hexahydrate occurs if the former is heated above the transition +point. The solubility of the hexahydrate is greater than that of the +heptahydrate below 39 deg., but increases more slowly with rise of +temperature. At about 80 deg. C. the hexahydrate gives place to the +monohydrate, which dissolves in water with evolution of heat, and +diminishes in solubility with rise of temperature. Intermediate hydrates +exist, but they are more soluble, and cannot be readily isolated. Both +the mono- and hexahydrates are capable of existing in equilibrium with +saturated solutions at temperatures far below their transition points, +provided that the less soluble hydrate is not present in the crystalline +form. The solubility curves can therefore be traced, as in fig. 3, over +an extended range of temperature. The equilibrium of each hydrate with +the solvent, considered separately, would present a diagram of two +branches similar to fig. 1, but as a rule only a small portion of each +curve can be realized, and the complete solubility curve, as +experimentally determined, is composed of a number of separate pieces +corresponding to the ranges of minimum solubility of different hydrates. +Failure to recognize this, coupled with the fact that in strong and +viscous solutions the state of equilibrium is but slowly attained, is +the probable explanation of the remarkable discrepancies existing in +many recorded data of solubility. + + _Transition Points of Hydrates._ + + Na2CrO4.10H2O 19.9 deg. NaBr.2H20 50.7 deg. + Na2SO4.10H2O 32.4 deg. MnCl2.4H2O 57.8 deg. + Na2CO3.10H2O 35.1 deg. Na3PO4.12H2O 73.4 deg. + Na2S2O3.5H2O 48.0 deg. Ba(OH)2.8H2O 77.9 deg. + +The transition points of the hydrates given in the above list (Richards, +_Proc. Amer. Acad._, 1899, 34, p. 277) afford well-marked constant +temperatures which can be utilized as fixed points for experimental +purposes. + +9. _Formation of Mixed Crystals._--An important exception to the general +type already described, in which the addition of a dissolved substance +lowers the F.P. of the solvent, is presented by the formation of mixed +crystals, or "solid solutions," in which the solvent and solute occur +mixed in varying proportions. This isomorphous replacement of one +substance by another, in the same crystal with little or no change of +form, has long been known and studied in the case of minerals and salts, +but the relations between composition and melting-point have seldom been +investigated, and much still remains obscure. In this case the process +of freezing does not necessitate the performance of work of separation +of the constituents of the solution, the F.P. is not necessarily +depressed, and the effect cannot be calculated by the usual formula for +dilute solutions. One of the simplest types of F.P. curve which may +result from the occurrence of mixed crystals is illustrated by the case +of alloys of gold and silver, or gold and platinum, in which the F.P. +curve is nearly a straight line joining the freezing-points of the +constituents. The equilibrium between the solid and liquid, in both of +which the two metals are capable of mixing in all proportions, bears in +this case an obvious and close analogy to the equilibrium between a +mixed liquid (e.g. alcohol and water) and its vapour. In the latter +case, as is well known, the vapour will contain a larger proportion of +the more volatile constituent. Similarly in the case of the formation of +mixed crystals, the liquid should contain a larger proportion of the +more fusible constituent than the solid with which it is in equilibrium. +The composition of the crystals which are being deposited at any moment +will, therefore, necessarily change as solidification proceeds, +following the change in the composition of the liquid, and the +temperature will fall until the last portions of the liquid to solidify +will consist chiefly of the more fusible constituent, at the F.P. of +which the solidification will be complete. If, however, as seems to be +frequently the case, the composition of the solid and liquid phases do +not greatly differ from each other, the greater part of the +solidification will occur within a comparatively small range of +temperature, and the initial F.P. of the alloy will be well marked. It +is possible in this case to draw a second curve representing the +composition of the _solid_ phase which is in equilibrium with the liquid +at any temperature. This curve will not represent the average +composition of the crystals, but that of the outer coating only which is +in equilibrium with the liquid at the moment. H.W.B. Roozeboom (_Zeit. +Phys. Chem._ xxx. p. 385) has attempted to classify some of the possible +cases which may occur in the formation of mixed crystals on the basis of +J.W. Gibbs's thermodynamic potential, the general properties of which +may be qualitatively deduced from a consideration of observed phenomena. +But although this method may enable us to classify different types, and +even to predict results in a qualitative manner, it does not admit of +numerical calculation similar to equation (8), as the Gibbs's function +itself is of a purely abstract nature and its form is unknown. There is +no doubt that the formation of mixed crystals may explain many apparent +anomalies in the study of F.P. curves. The whole subject has been most +fruitful of results in recent years, and appears full of promise for the +future. + + For further details in this particular branch the reader may consult a + report by Neville (_Brit. Assoc. Rep._, 1900), which contains numerous + references to original papers by Roberts-Austen, Le Chatelier, + Roozeboom and others. For the properties of solutions see SOLUTION. + (H. L. C.) + + + + + +FUSSEN, a town of Germany, in the kingdom of Bavaria, at the foot of the +Alps (Tirol), on the Lech, 2500 ft. above the sea, with a branch line to +Oberdorf on the railway to Augsburg. Pop. 4000. It has six Roman +Catholic churches, a Franciscan monastery and a castle. Rope-making is +an important industry. The castle, lying on a rocky eminence, is +remarkable for the peace signed here on the 22nd of April 1745 between +the elector Maximilian III., Joseph of Bavaria and Maria Theresa. Two +miles to the S.E., immediately on the Austrian frontier, romantically +situated on a rock overlooking the Schwanensee, is the magnificent +castle of Hohenschwangau, and a little to the north, on the site of an +old castle, that of Neuschwanstein, built by Louis II. of Bavaria. + + See H. Feistle, _Fussen und Umgebung_ (1898). + + + + +FUST, JOHANN ( ?-1466), early German printer, belonged to a rich and +respectable burgher family of Mainz, which is known to have flourished +from 1423, and to have held many civil and religious offices. The name +was always written Fust, but in 1506 Johann Schoffer, in dedicating the +German translation of Livy to the emperor Maximilian, called his +grandfather Faust, and thenceforward the family assumed this name, and +the Fausts of Aschaffenburg, an old and quite distinct family, placed +Johann Fust in their pedigree. Johann's brother Jacob, a goldsmith, was +one of the burgomasters in 1462, when Mainz was stormed and sacked by +the troops of Count Adolf of Nassau, on which occasion he seems to have +perished (see a document, dated May 8, 1463, published by Wyss in +_Quartalbl. des hist. Vereins fur Hessen_, 1879, p. 24). There is no +evidence that, as is commonly asserted, Johann Fust was a goldsmith, but +he appears to have been a money-lender or banker. On account of his +connexion with Gutenberg (q.v.), he has been represented by some as the +inventor of printing, and the instructor as well as the partner of +Gutenberg, by others as his patron and benefactor, who saw the value of +his discovery and supplied him with means to carry it out, whereas +others paint him as a greedy and crafty speculator, who took advantage +of Gutenberg's necessity and robbed him of the fruits of his invention. +However this may be, the Helmasperger document of November 6, 1455, +shows that Fust advanced money to Gutenberg (apparently 800 guilders in +1450, and another 800 in 1452) for carrying on his work, and that Fust, +in 1455, brought a suit against Gutenberg to recover the money he had +lent, claiming 2020 (more correctly 2026) guilders for principal and +interest. It appears that he had not paid in the 300 guilders a year +which he had undertaken to furnish for expenses, wages, &c., and, +according to Gutenberg, had said that he had no intention of claiming +interest. The suit was apparently decided in Fust's favour, November 6, +1455, in the refectory of the Barefooted Friars of Mainz, when Fust made +oath that he himself had borrowed 1550 guilders and given them to +Gutenberg. There is no evidence that Fust, as is usually supposed, +removed the portion of the printing materials covered by his mortgage to +his own house, and carried on printing there with the aid of Peter +Schoffer, of Gernsheim (who is known to have been a scriptor at Paris in +1449), to whom, probably about 1455,[1] he gave his only daughter Dyna +or Christina in marriage. Their first publication was the Psalter, +August 14, 1457, a folio of 350 pages, the first printed book with a +complete date, and remarkable for the beauty of the large initials +printed each in two colours, red and blue, from types made in two +pieces.[2] The Psalter was reprinted with the same types, 1459 (August +29), 1490, 1502 (Schoffer's last publication) and 1516. Fust and +Schoffer's other works are given below.[3] In 1464 Adolf of Nassau +appointed for the parish of St Quintin three _Baumeisters_ +(master-builders) who were to choose twelve chief parishioners as +assistants for life. One of the first of these "Vervaren," who were +named on May-day 1464, was Johannes Fust, and in 1467 Adam von Hochheim +was chosen instead of "the late" (_selig_) Johannes Fust. Fust is said +to have gone to Paris in 1466 and to have died of the plague, which +raged there in August and September. He certainly was in Paris on the +4th of July, when he gave Louis de Lavernade of the province of Forez, +then chancellor of the duke of Bourbon and first president of the +parliament of Toulouse, a copy of his second edition of Cicero, as +appears from a note in Lavernade's own hand at the end of the book, +which is now in the library of Geneva. But nothing further is known than +that on the 30th of October, probably in 1471, an annual mass was +instituted for him by Peter Schoffer, Conrad Henlif (for Henekes, or +Henckis, Schoffer's partner? who married Fust's widow about 1468[4]) and +Johann Fust (the son), in the abbey-church of St Victor of Paris, where +he was buried; and that Peter Schoffer founded a similar memorial +service for Fust in 1473 in the church of the Dominicans at Mainz +(Bockenheimer, _Gesch. der Stadt Mainz_, iv. 15). + +Fust was formerly often confused with the famous magician Dr Johann +Faust, who, though an historical figure, had nothing to do with him (see +FAUST). + + See further the articles GUTENBERG and TYPOGRAPHY. (J. H. H.) + + +FOOTNOTES: + + [1] This date is uncertain; some place the marriage in 1453 or soon + after, others about 1464. It is probable that Fust alluded to this + relationship when he spoke of Schoffer as _pueri mei_ in the + colophons of Cicero's _De officiis_ of 1465 and 1466. + + [2] This method was patented in England by Solomon Henry in 1780, and + by Sir William Congreve in 1819. + + [3] (3) Durandus, _Rationale divinorum officiorum_ (1459), folio, 160 + leaves; (4) the _Clementine Constitutions_, with the gloss of + Johannes Andreae (1460), 51 leaves; (5) _Biblia Sacra Latina_ (1462), + folio, 2 vols., 242 and 239 leaves, 48 lines to a full page; (6) the + Sixth Book of Decretals, with Andreae's gloss, 17th December 1465, + folio, 141 leaves; (7) Cicero, _De officiis_ (1465). 4to, 88 leaves, + the first edition of a Latin classic and the first book containing + Greek characters, while in the colophon Fust for the first time calls + Schoffer "puerum suum"; (8) the same, 4th February 1466; (9) + _Grammatica rhytmica_ (1466), folio, 11 leaves. They also printed in + 1461-1462 several papal bulls, proclamations of Adolf of Nassau, &c. + Nothing is known to have appeared for three years after the storming + and capture of Mainz in 1462. + + [4] Some confusion in the history of the Fust family has arisen since + the publication of Bernard's _Orig. de l'imprimerie_ (1853). On p. + 262, vol. i. he gave an extract from the correspondence between + Oberlin and Bodmann (now preserved in the Paris Nat. Library), from + which it would appear that Peter Schoffer was the son-in-law, not of + Johann Fust, but of a brother of his, Conrad Fust. Of the latter, + however, no other trace has been found, and he is no doubt a fiction + of F.J. Bodmann, who, partly basing himself on the "Conrad" (Henlif, + or Henckis) mentioned above, added the rest to gratify Oberlin (see + Wyss in _Quartalblatter des hist. Vereins fur Hessen_, 1879, p. 17). + + + + +FUSTEL DE COULANGES, NUMA DENIS (1830-1889), French historian, was born +in Paris on the 18th of March 1830, of Breton descent. After studying at +the Ecole Normale Superieure he was sent to the French school at Athens +in 1853, directed some excavations in Chios, and wrote an historical +account of the island. After his return he filled various educational +offices, and took his doctor's degree with two theses, _Quid Vestae +cultus in institutis veterum privatis publicisque valuerit_ and _Polybe, +ou la Grece conquise par les Romains_ (1858). In these works his +distinctive qualities were already revealed. His minute knowledge of the +language of the Greek and Roman institutions, coupled with his low +estimate of the conclusions of contemporary scholars, led him to go +direct to the original texts, which he read without political or +religious bias. When, however, he had succeeded in extracting from the +sources a general idea that seemed to him clear and simple, he attached +himself to it as if to the truth itself, employing dialectic of the most +penetrating, subtle and even paradoxical character in his deduction of +the logical consequences. From 1860 to 1870 he was professor of history +at the faculty of letters at Strassburg, where he had a brilliant career +as a teacher, but never yielded to the influence exercised by the German +universities in the field of classical and Germanic antiquities. + +It was at Strassburg that he published his remarkable volume _La Cite +antique_ (1864), in which he showed forcibly the part played by religion +in the political and social evolution of Greece and Rome. Although his +making religion the sole factor of this evolution was a perversion of +the historical facts, the book was so consistent throughout, so full of +ingenious ideas, and written in so striking a style, that it ranks as +one of the masterpieces of the French language in the 19th century. By +this literary merit Fustel set little store, but he clung tenaciously to +his theories. When he revised the book in 1875, his modifications were +very slight, and it is conceivable that, had he recast it, as he often +expressed the desire to do in the last years of his life, he would not +have abandoned any part of his fundamental thesis. The work is now +largely superseded. + +Fustel de Coulanges was the most conscientious of men, the most +systematic and uncompromising of historians. Appointed to a lectureship +at the Ecole Normale Superieure in February 1870, to a professorship at +the Paris faculty of letters in 1875, and to the chair of medieval +history created for him at the Sorbonne in 1878, he applied himself to +the study of the political institutions of ancient France. The invasion +of France by the German armies during the war of 1870-71 attracted his +attention to the Germanic invasions under the Roman Empire. Pursuing the +theory of J.B. Dubos, but singularly transforming it, he maintained that +those invasions were not marked by the violent and destructive character +usually attributed to them; that the penetration of the German +barbarians into Gaul was a slow process; that the Germans submitted to +the imperial administration; that the political institutions of the +Merovingians had their origins in the Roman laws at least as much as, if +not more than, in German usages; and, consequently, that there was no +conquest of Gaul by the Germans. This thesis he sustained brilliantly in +his _Histoire des institutions politiques de l'ancienne France_, the +first volume of which appeared in 1874. It was the author's original +intention to complete this work in four volumes, but as the first volume +was keenly attacked in Germany as well as in France, Fustel was forced +in self-defence to recast the book entirely. With admirable +conscientiousness he re-examined all the texts and wrote a number of +dissertations, of which, though several (e.g. those on the Germanic mark +and on the _allodium_ and _beneficium_) were models of learning and +sagacity, all were dominated by his general idea and characterized by a +total disregard for the results of such historical disciplines as +diplomatic. From this crucible issued an entirely new work, less well +arranged than the original, but richer in facts and critical comments. +The first volume was expanded into three volumes, _La Gaule romaine_ +(1891), _L'Invasion germanique et la fin de l'empire_ (1891) and _La +Monarchie franque_ (1888), followed by three other volumes, _L'Alleu et +le domaine rural pendant l'epoque merovingienne_ (1889), _Les Origines +du systeme feodal: le benefice et le patronat ..._ (1890) and _Les +Transformations de la royaute pendant l'epoque carolingienne_ (1892). +Thus, in six volumes, he had carried the work no farther than the +Carolingian period. The result of this enormous labour, albeit worthy of +a great historian, clearly showed that the author lacked all sense of +historical proportion. He was a diligent seeker after the truth, and was +perfectly sincere when he informed a critic of the exact number of +"truths" he had discovered, and when he remarked to one of his pupils a +few days before his death, "Rest assured that what I have written in my +book is the truth." Such superb self-confidence can accomplish much, and +it undoubtedly helped to form Fustel's talent and to give to his style +that admirable concision which subjugates even when it fails to +convince; but a student instinctively distrusts an historian who settles +the most controverted problems with such impassioned assurance. The +dissertations not embodied in his great work were collected by himself +and (after his death) by his pupil, Camille Jullian, and published as +volumes of miscellanies: _Recherches sur quelques problemes d'histoire_ +(1885), dealing with the Roman colonate, the land system in Normandy, +the Germanic mark, and the judiciary organization in the kingdom of the +Franks; _Nouvelles recherches sur quelques problemes d'histoire_ (1891); +and _Questions historiques_ (1893), which contains his paper on Chios +and his thesis on Polybius. + +His life was devoted almost entirely to his teaching and his books. In +1875 he was elected member of the Academie des Sciences Morales, and in +1880 reluctantly accepted the post of director of the Ecole Normale. +Without intervening personally in French politics, he took a keen +interest in the questions of administration and social reorganization +arising from the fall of the imperialist regime and the disasters of the +war. He wished the institutions of the present to approximate more +closely to those of the past, and devised for the new French +constitution a body of reforms which reflected the opinions he had +formed upon the democracy at Rome and in ancient France. But these were +dreams which did not hold him long, and he would have been scandalized +had he known that his name was subsequently used as the emblem of a +political and religious party. He died at Massy (Seine-et-Oise) on the +12th of September 1889. Throughout his historical career--at the Ecole +Normale and the Sorbonne and in his lectures delivered to the empress +Eugenie--his sole aim was to ascertain the truth, and in the defence of +truth his polemics against what he imagined to be the blindness and +insincerity of his critics sometimes assumed a character of harshness +and injustice. But, in France at least, these critics were the first to +render justice to his learning, his talents and his disinterestedness. + + See Paul Guiraud, _Fustel de Coulanges_ (1896); H. d'Arbois de + Jubainville, _Deux Manieres d'ecrire l'histoire: critique de Bossuet, + d'Augustin Thierry et de Fustel de Coulanges_ (1896); and Gabriel + Monod, _Portraits et souvenirs_ (1897). (C. B.*) + + + + +FUSTIAN, a term which includes a variety of heavy woven cotton fabrics, +chiefly prepared for men's wear. It embraces plain twilled cloth called +jean, and cut fabrics similar to velvet, known as velveteen, moleskin, +corduroy, &c. The term was once applied to a coarse cloth made of cotton +and flax; now, fustians are usually of cotton and dyed various colours. +In the reign of Edward III. the name was given to a woollen fabric. The +name is said to be derived from El-Fustat, a suburb of Cairo, where it +was first made; and certainly a kind of cloth has long been known under +that name. In a petition to parliament, _temp._ Philip and Mary, +"fustian of Naples" is mentioned. In the 13th and 14th centuries +priests' robes and women's dresses were made of fustian, but though +dresses are still made from some kinds the chief use is for labourers' +clothes. + + + + +FUSTIC (Fr. _fustoc_, from Arab. _fustuq_, Gr. [Greek: pistake], +pistachio) YELLOW WOOD or OLD FUSTIC, a dye-stuff consisting of the wood +of _Chlorophora tinctoria_, a large tree of the natural order Moraceae, +growing in the West Indies and tropical America. Fustic occurs in +commerce in blocks, which are brown without, and of a brownish-yellow +within. It is sometimes employed for inlaid work. The dye-stuff termed +young fustic or Zante fustic, and also Venetian sumach, is the wood of +_Rhus cotinus_ (fustet, or smoke tree), a southern European and Asiatic +shrub of the natural order Anacardiaceae, called by Gerarde "red +sumach," and apparently the "coccygia" and "cotinus" of Pliny (_Nat. +Hist._ xiii. 41, xvi. 30). Its colouring matter is fisetin, C15H10O6, +which was synthesized by S. von Kostanecki (_Ber._, 1904, 37, p. 384). +(See DYEING.) + + + + +FUTURES, a term used in the produce markets for purchases or sales of +commodities to be completed at a future date, as opposed to cash or +"spot" transactions, which are settled immediately. See MARKET, and (for +a detailed discussion of the question as affecting cotton) COTTON: +_Marketing and Supply_. + + + + +FUX, JOHANN JOSEPH (1660-1741), Austrian musician, was born at +Hirtenfeld (Styria) in 1660. Of his youth and early training nothing is +known. In 1696 he was organist at one of the principal churches of +Vienna, and in 1698 was appointed by the emperor Leopold I. as his +"imperial court-composer," with a salary of about L6 a month. At the +court of Leopold and of his successors Joseph I. and Charles VI., Fux +remained for the rest of his life. To his various court dignities that +of organist at St Stephen's cathedral was added in 1704. He married the +daughter of the government secretary Schnitzbaum. As a proof of the high +favour in which he was held by the art-loving Charles VI., it is told +that at the coronation of that emperor as king of Bohemia in 1723 an +opera, _La Constanza e la Fortezza_, especially composed by Fux for the +occasion, was given at Prague in an open-air theatre. Fux at the time +was suffering from gout, but the emperor had him carried in a litter all +the way from Vienna, and gave him a seat in the imperial box. Fux died +at Vienna on the 13th of February 1741. His life, although passed in the +great world, was eventless, and his only troubles arose from the +intrigues of his Italian rivals at court. Of the numerous operas which +Fux wrote it is unnecessary to speak. They do not essentially differ +from the style of the Italian _opera seria_ of the time. Of greater +importance are his sacred compositions, psalms, motets, oratorios and +masses, the celebrated _Missa Canonica_ amongst the latter. It is an all +but unparalleled _tour de force_ of learned musicianship, being written +entirely in that most difficult of contrapuntal devices--the canon. As a +contrapuntist and musical scholar generally, Fux was unsurpassed by any +of his contemporaries, and his great theoretical work, the _Gradus ad +Parnassum_, long remained by far the most thorough treatment of +counterpoint and its various developments. The title of the original +Latin edition is _Gradus ad Parnassum sive manuductio ad compositionem +musicae regularem, methoda nova ac certa nondum ante tam exacta ordine +in lucem edita, elaborata a Joanne Josepho Fux_ (Vienna, 1715). It was +translated into most European languages during the 18th century, and is +still studied by musicians interested in the history of their art. The +expenses of the publication were defrayed by the emperor Charles VI. + + Fux's biography was published by Ludwig von Kochel (Vienna, 1871). It + is based on minute original research and contains, amongst other + valuable materials, a complete catalogue of the composer's numerous + works. + + + + +FUZE or FUSE, an appliance for firing explosives in blasting operations, +military shells, &c. (see BLASTING and AMMUNITION, S SHELL). The +spelling is not governed by authority, but modern convenience has +dictated the adoption of the "z" by military engineers as a general +rule, in order to distinguish this sense from that of melting by heat +(see below). The word, according to the _New English Dictionary_, is one +of the forms in which the Lat. _fusus_, spindle, has been adapted +through Romanic into English, the ordinary fuze taking the shape of a +spindle-like tube. Similarly the term "fusee" (Fr. _fusee_, spindle full +of tow, Late Lat. _fusata_) is applied to a coned spindle sometimes used +in the wheel train of watches and spring clocks to equalize the action +of the mainspring (see WATCH); and the application of the same term to a +special kind of match may also be due to its resemblance to a spindle. +Again, in heraldry, another form, "fusil," derived through the French +from a Late Lat. diminutive (_fusillus_ or _fusellus_) of this same +_fusus_, is used of a bearing, an elongated lozenge. According to other +etymological authorities, however (see Skeat, _Etym. Dict._, 1898), +"fuze" or "fuse," and "fusee" in the sense of match, are all forms +derived through the Fr. fusil, from Late Lat. _focile_, steel for +striking fire from a flint, from Lat. _focus_, hearth. The Fr. _fusil_ +and English "fusil" were thus transferred to the "firelock," i.e. the +light musket of the 17th century (see FUSILIER). + +In electrical engineering a "fuse" (always so spelled) is a safety +device, commonly consisting of a strip or wire of easily fusible metal, +which melts and thus interrupts the circuit of which it forms part, +whenever that circuit, through some accident or derangement, is caused +to carry a current larger than that for which it is intended. In this +sense the word must be connected with _fusus_, the past participle of +Lat. _fundere_, to pour, whence comes the verb "fuse," to melt by heat, +often used figuratively in the sense of blend, mix. + + + + +FYNE, LOCH, an inlet of the sea, Argyllshire, Scotland. From the head, 6 +m. above Inveraray, to the mouth on the Sound of Bute, it has a +south-westerly and then southerly trend and is 44 m. long, its width +varying from 1/4 m. to 6 m. It receives the Fyne, Shira, Aray and many +other streams, and, on the western side, gives off Lochs Shira, Gair, +Gilp (with Ardrishaig, the Crinan Canal and Lochgilphead) and East +Tarbert (with Tarbert village). The glens debouching on the lake are +Fyne, Shira, Aray, Kinglas and Hell's Glen. The coast generally is +picturesque and in many parts well wooded. All vessels using the Crinan +Canal navigate the loch to and from Ardrishaig, and there are daily +excursions during the season, as far up as Inveraray. There are ferries +at St Catherine's and Otter, and piers at Tarbert, Ardrishaig, Kilmory, +Crarae, Furnace, Inveraray, Strachur and elsewhere. The industries +comprise granite quarrying at Furnace and Crarae, distilling at +Ardrishaig, gunpowder-making at Furnace and Kilfinan, and, above all, +fishing. Haddock, whiting and codling are taken, and the famous "Loch +Fyne herrings" command the highest price in the market. + + + + +FYRD, the name given to the English army, or militia, during the +Anglo-Saxon period (see ARMY, 60). It is first mentioned in the +_Anglo-Saxon Chronicle_ under the date 605. The ealdorman, or sheriff, +of the shire was probably charged with the duty of calling out and +leading the fyrd, which appears always to have retained a local +character, as during the time of the Danish invasions we read of the +fyrd of Kent, of Somerset and of Devon. As attendance at the fyrd was +included in the _trinoda necessitas_ it was compulsory on all holders of +land; but that it was not confined to them is shown by the following +extract from the laws of Ine, king of the West Saxons, dated about 690, +which prescribes the penalty for the serious offence of neglecting the +fyrd: "If a _gesithcund_ man owning land neglect the fyrd, let him pay +120 shillings, and forfeit his land; one not owning land 60 shillings; a +ceorlish man 30 shillings as _fyrdwite_." The fyrd was gradually +superseded by the gathering of the thegns and their retainers, but it +was occasionally called out for defensive purposes even after the Norman +Conquest. + + + + +FYT, JOHANNES (1609-1661), Belgian animal painter, was born at Antwerp +and christened on the 19th of August 1609. He was registered apprentice +to Hans van den Berghe in 1621. Professionally van den Berghe was a +restorer of old pictures rather than a painter of new ones. At twenty +Johannes Fyt entered the gild of St Luke as a master, and from that time +till his death in 1661 he produced a vast number of pictures in which +the bold facility of Snyders is united to the powerful effects of +Rembrandt, and harmonies of gorgeous tone are not less conspicuous than +freedom of touch and a true semblance of nature. There never was such a +master of technical processes as Fyt in the rendering of animal life in +its most varied forms. He may have been less correct in outline, less +bold in action than Snyders, but he was much more skilful and more true +in the reproduction of the coat of deer, dogs, greyhounds, hares and +monkeys, whilst in realizing the plumage of peacocks, woodcocks, ducks, +hawks, and cocks and hens, he had not his equal, nor was any artist even +of the Dutch school more effective in relieving his compositions with +accessories of tinted cloth, porcelain ware, vases and fruit. He was not +clever at figures, and he sometimes trusted for these to the +co-operation of Cornelius Schut or Willeborts, whilst his architectural +backgrounds were sometimes executed by Quellyn. "Silenus amongst Fruit +and Flowers," in the Harrach collection at Vienna, "Diana and her Nymphs +with the Produce of the Chase," in the Belvedere at Vienna, and "Dead +Game and Fruit in front of a Triumphal Arch," belonging to Baron von +Rothschild at Vienna, are specimens of the co-operation respectively of +Schut, Willeborts and Quellyn. They are also Fyt's masterpieces. The +earliest dated work of the master is a cat grabbing at a piece of dead +poultry near a hare and birds, belonging to Baron Cetto at Munich, and +executed in 1644. The latest is a "Dead Snipe with Ducks," of 1660, +sold with the Jager collection at Cologne in 1871. Great power is shown +in the bear and boar hunts at Munich and Ravensworth castle. A "Hunted +Roedeer with Dogs in the Water," in the Berlin Museum, has some of the +life and more of the roughness of Snyders, but lacks variety of tint and +finish. A splendid specimen is the Page and Parrot near a table covered +with game, guarded by a dog staring at a monkey, in the Wallace +collection. With the needle and the brush Fyt was equally clever. He +etched 16 plates, and those representing dogs are of their kind unique. + + + + +FYZABAD, or FAIZABAD, a city, district and division of British India in +the United Provinces. The city stands on the left bank of the river +Gogra, 78 m. by rail E. of Lucknow. Pop. (1901) 75,085. To the E. of +Fyzabad, and now forming a suburb, is the ancient site of Ajodhya +(q.v.). Fyzabad was founded about 1730 by Sa'adat Ali Khan, the first +nawab wazir of Oudh, who built a hunting-lodge here. It received its +present name in the reign of his successor; and Shuja-ud-daula, the +third nawab, laid out a large town and fortified it, and here he was +buried. It was afterwards the residence of the Begums of Oudh, famous in +connexion with the impeachment of Warren Hastings. When the court of +Oudh was removed to Lucknow in 1775 all the leading merchants and +bankers abandoned the place. At the census of 1869 Fyzabad contained +only 37,804 inhabitants; but it is now again advancing in prosperity and +population. On the outbreak of the Mutiny in 1857, the cantonment +contained two regiments of infantry, a squadron of cavalry, and a light +field battery of artillery--all natives. Owing to their threatening +demeanour after the Meerut massacre, many of the European women and +children were sheltered by one of the great landholders of Oudh, and +others were sent to less disturbed parts of the country. The troops +rose, as was anticipated, and although they at first permitted their +officers to take boats and proceed towards Dinapur, a message was +afterwards sent to a rebel force lower down the river to intercept the +fugitives. Of four boats, one, having passed the rebels unnoticed, +succeeded in reaching Dinapur safely. Of those in the other three boats, +one alone escaped. Fyzabad is now a station for European as well as for +native troops. It is the headquarters of a brigade in the 8th division +of the northern army. There is a government college. Sugar-refining and +trade in agricultural produce are important. + +THE DISTRICT OF FYZABAD, lying between the two great rivers Gogra and +Gumti, has an area of 1740 sq. m. It is entirely alluvial and well +wooded, and has a good climate. Pop. (1901) 1,225,374, an increase of +.7% in the decade. The district is traversed throughout its length by +the Oudh and Rohilkhand railway from Lucknow to Benares, with a branch +to Allahabad. Tanda, with a population in 1901 of 19,853, has the +largest production of cotton goods in Oudh. + +The DIVISION OF FYZABAD has an area of 12,113 sq. m., and comprises the +six districts of Fyzabad, Gonda, Bahraich, Sultanpur, Partabgarh and +Bara Banki. Pop. (1901) 6,855,991, an increase of 2% in the decade. + + + + + + +End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th +Edition, Volume 11, Slice 3, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK ENCYC. 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