Chapter 9

See theMemoirby W. W. Currie, his son (1831).

CURRY.(1) (Through the O. Fr.correier, from Late Lat.conredare, to make ready, prepare; a later form of the French iscourroyer, and modern French iscorroyer), to dress a horse by rubbing down and grooming with a comb; to dress and prepare leather already tanned. The currier pares off roughnesses and inequalities, makes the leather soft and pliable, and gives it the necessary surface and colour (seeLeather). The word “currier,” though early confused in origin with “to curry,” is derived from the Late Lat.coriarius, a leather dresser, fromcorium, hide. The phrase “to curry favour,” to flatter or cajole, is a 16th century corruption of “to curry favel,”i.e.a chestnut horse. This older phrase is an adaptation of an Old French proverbial expressionestriller fauvel, and is paralleled in German by the similarden fahlen Hengst streichen. A chestnut or fallow horse seems to have been taken as typical of deceit and trickery, at least since the appearance of a French satirical beast romance theRoman de fauvel(1310), the hero of which is a counterpart of Reynard the Fox (q.v.).

(2) A name applied to a great variety of seasoned dishes, especially those of Indian origin. The word is derived from the Tamilkari, a sauce or relish for rice. In the East, where the staple food of the people consists of a dish of rice, wheaten cakes, or some other cereal, some kind of relish is required to lend attraction to this insipid food; and that is the special office of curry. In India the following are employed as ingredients in curries: anise, coriander, cumin, mustard and poppy seeds; allspice, almonds, assafoetida, butter or ghee, cardamoms, chillies, cinnamon, cloves, cocoa-nut and cocoanut milk and oil, cream and curds, fenugreek, the tender unripe fruit ofBuchanania lancifolia, cheroonjie nuts (the produce of another species,B. latifolia), garlic and onions, ginger, lime-juice, vinegar, the leaves ofBergera Koenigii(the curry-leaf tree), mace, mangoes, nutmeg, pepper, saffron, salt, tamarinds and turmeric.

The cumin and coriander seeds are generally used roasted. The various materials are cleaned, dried, ground, sifted, thoroughly mixed and bottled. In the East the spices are ground freshly every day, which gives the Indian curry its superiority in flavour over dishes prepared with the curry-powders of the European market.

CURSOR, LUCIUS PAPIRIUS,Roman general, five times consul and twice dictator. In 325 he was appointed dictator to carry on the second Samnite War. His quarrel with Q. Fabius Maximus Rullianus, hismagister equitum, is well known. The latter had engaged the enemy against the orders of Cursor, by whom he was condemned to death, and only the intercession of his father, the senate and the people, saved his life. Cursor treated his soldiers with such harshness that they allowed themselves to be defeated; but after he had regained their good-will by more lenient treatment and lavish promises of booty, they fought with enthusiasm and gained a complete victory. After the disaster of the Caudine Forks, Cursor to some extent wiped out the disgrace by compelling Luceria (which had revolted) to surrender. He delivered the Roman hostages who were held in captivity in the town, recovered the standards lost at Caudium, and made 7000 of the enemy pass under the yoke. In 309, when the Samnites again rose, Cursor was appointed dictator for the second time, and gained a decisive victory at Longula, in honour of which he celebrated a magnificent triumph. Cursor’s strictness was proverbial; he was a man of immense bodily strength, while his bravery wasbeyond dispute. He was surnamed Cursor from his swiftness of foot.

Livy viii., ix.; Aurelius Victor,De viris illustribus, 31; Eutropius ii. 8. 9.

His son of the same name, also a distinguished general, completed the subjection of Samnium (272). He set up a sun-dial, the first of its kind in Rome, in the temple of Quirinus.

Livy x. 39-47; Pliny,Nat. Hist., vii. 60.

CURSOR MUNDI,an English poem in the Northern dialect dating from the 13th century. It is a religious epic of 24,000 lines “over-running” the history of the world as related in the Old and New Testaments. “Cursur o werld man aght it call, For almast it over-rennes all.” The author explains in his prologue his reasons for undertaking the work. Men desire to read old romances of Alexander, Julius Caesar, Greece, Troy, Brut, Arthur, of Tristram, Sweet Ysoude and others. But better than tales of love is the story of the Virgin who is man’s best lover, therefore in her honour he will write this book, founded on the steadfast ground of the Holy Trinity. He writes in English for the love of English people of merry England, so that those who know no French may understand. The history is treated under seven ages. The first four include the period from the creation of the world to the successors of Solomon, the fifth deals with Mary and the birth and childhood of Jesus, the sixth with the lives of Christ and the chief apostles, and with the finding of the holy cross, and the seventh with Doomsday. Four short poems follow, more in some MSS. The bulk of the poem is written in rhyming couplets of short lines of four accents, and maintains a fair level throughout. The narrative is enlivened by many legends and much entertaining matter drawn from various sources; and the numerous transcripts of it prove that it was able to hold its own against profane romance.

The chief sources of the compilation have been identified by Dr Haenisch. For the Old Testament history the author draws largely from theHistoria scholasticaof Peter Comestor; for the history of the Virgin he often translates literally from Wace’sÉtablissement de la fête de la conception Notre Dame; the parables of the king and four daughters, and of the castle of Love and Grace, are taken from “Sent Robert bok” (1.9516), that is, from theChasteau d’Amourof Robert Grosseteste, bishop of Lincoln; other sources are the apocryphal gospels of Matthew and Nicodemus, a southern English poem on the Assumption of Our Lady, attributed by the writer ofCursor mundito Edmund Rich of Pontigny, the Vulgate, theLegenda aureaof Jacobus de Voragine, and theDe vita et morte sanctorumof Isidore of Seville. The original of the section on the invention of the holy cross is still to seek. In its general plan the work is similar to theLivre de sapienceof Herman de Valenciennes.

Of the author nothing is known. In the Cotton MS. Vespasian (A III.) the name of the owner William Cosyn is given (for particulars of this family, which is mentioned in Lincolnshire records as early as 1276, see Dr H. Hupe in the E.E.T.S. ed. ofCursor mundi, vol. i. p. 124 *). The date of the book was placed by Dr J. A. H. Murray (The Dialect of the Southern Counties of Scotland, 1873, p. 30) in the last quarter of the 13th century, and the place of writing near Durham. Dr Hupe (loc. cit.p. 186 *) gives good reasons for believing that the author was a Lincolnshire man, who wrote between 1260 and 1290, although the Cotton MS. probably belongs to the late 14th century. In the Göttingen MS. there are lines (17099-17110) desiring the reader to pray for John of Lindbergh, “that this bock gart dight,” and cursing anybody who shall steal it. Lindberg is probably Limber Magna, near Ulceby, in north Lincolnshire. Dr Hupe hazards an identification of the author with this John of Lindberg, who may have been a member of the Cistercian Abbey of Lindberg; but this is improbable.

Cursor mundiwas edited for the Early English Text Society in 1874-1893 by Dr Richard Morris in parallel columns from four MSS.:—Cotton Vespasian A III., British Museum; Fairfax MS. 14, in the Bodleian library, Oxford; MS. theol. 107 at Göttingen; and MS. R. 3.8 in Trinity College, Cambridge. The edition includes a “Preface” by the editor, “An Inquiry into the Sources of theCursor mundi” (1885), by Dr Haenisch, an essay “On the Filiation and the Text of the MSS. ofCursor mundi” (1885), by Dr H. Hupe, “Cursor Studies and Criticisms on the Dialects of its MSS.” (1888), by Dr Hupe and a glossary by Dr Max Kaluza.

CURTAIN,a screen of any textile material, running by means of rings fixed to a rod or pole. Curtains are now used chiefly to cover windows and doors, but for many centuries every bed of importance was surrounded by them, and sometimes, as in France, the space thus screened off was much larger than the actual bed and was called theruelle. The curtain is very ancient—indeed the absence of glass and ill-fitting windows long made it a necessity. Originally single curtains were used; it would appear that it was not until the 17th century that they were employed in pairs. Curtains are made in an infinite variety of materials and styles; when placed over a door they are usually calledportières. In fortification the “curtain” is that part of the enceinte which lies between two bastions, towers, gates, &c.

The word comes into English through the O. Fr.cortineorcourtinefrom the Late Lat.cortina. According to Du Cange (Glossarium, s.v. “Cortis”) this is a diminutive ofcortis, an enclosed space, a court. It is used in the various senses of the English “curtain.” Classical Latin had also a wordcortina, meaning a caldron or round kettle. It was very rarely applied to round objects generally. In the Vulgatecortinais used of the curtains of the tabernacle (Exodus xxvi). There is some difficulty in connecting the classical and the Late Latin words. The earliest use in English is, according to theNew English Dictionary, for the hangings of a bed.

CURTANA(a latinized form of the A.-Fr.curtein, from Lat.curtus, shortened), the pointless sword of mercy, known also as Edward the Confessor’s sword, borne at the coronation of the kings of England between the two pointed swords of temporal and spiritual justice (seeRegalia).

CURTEA DE ARGESH(Rumanian,Curtea de Arges; also writtenCurtea d’Argesh,Curtea d’Ardges,ArgishandArdjish), the capital of the department of Argesh, Rumania; situated on the right bank of the river Argesh, where it flows through a valley of the lower Carpathians; and on the railway from Pitesci to the Rothenthurm Pass. Pop. (1900) 4210. The city is one of the oldest in Rumania. According to tradition it was founded early in the 14th century by Prince Radu Negru, succeeding Câmpulung as capital of Walachia. Hence its nameCurtea, “the court.” It contains a few antique churches, and was created a bishopric at the close of the 18th century.

The cathedral of Curtea de Argesh, by far the most famous building in Rumania, stands in the grounds of a monastery, 1½ m. N. of the city. It resembles a very large and elaborate mausoleum, built in Byzantine style, with Moorish arabesques. In shape it is oblong, with a many-sided annexe at the back. In the centre rises a dome, fronted by two smaller cupolas; while a secondary dome, broader and loftier than the central one, springs from the annexe. Each summit is crowned by an inverted pear-shaped stone, bearing a triple cross, emblematic of the Trinity. The windows are mere slits; those of the tambours, or cylinders, on which the cupolas rest, are curved, and slant at an angle of 70°, as though the tambours were leaning to one side. Between the pediment and the cornice a thick corded moulding is carried round the main building. Above this comes a row of circular shields, adorned with intricate arabesques, while bands and wreaths of lilies are everywheresculpturedon the windows, balconies, tambours and cornices, adding lightness to the fabric. The whole is raised on a platform 7 ft. high, and encircled by a stone balustrade. Facing the main entrance is a small open shrine, consisting of a cornice and dome upheld by four pillars. The cathedral is faced with pale grey limestone, easily chiselled, but hardening on exposure. The interior is of brick, plastered and decorated with frescoes. Close by stands a large royal palace, Moorish in style. The archives of the cathedral were plundered by Magyars and Moslems, but several inscriptions, Greek, Slav and Ruman, are left. One tablet records that the founder was Prince Neagoe Bassarab (1512-1521); another that Prince John Raducompleted the work in 1526. A third describes the repairs executed in 1681 by Prince Sherban Cantacuzino; a fourth, the restoration, in 1804, by Joseph, the first bishop. Between 1875 and 1885 the cathedral was reconstructed; and in 1886 it was reconsecrated. Its legends have inspired many Rumanian poets, among them the celebrated V. Alexandri (1821-1890). One tradition describes how Neagoe Bassarab, while a hostage in Constantinople, designed a splendid mosque for the sultan, returning to build the cathedral out of the surplus materials. Another version makes him employ one Manole or Manoli as architect. Manole, being unable to finish the walls, the prince threatened him and his assistant with death. At last Manole suggested that they should follow the ancient custom of building a living woman into the foundations; and that she who first appeared on the following morning should be the victim. The other masons warned their families, and Manole was forced to sacrifice his own wife. Thus the cathedral was built except the roof. So arrogant, however, did the masons become, that the prince bade remove the scaffolding, and all, save Manole, perished of hunger. He fell to the ground, and a spring of clear water, which issued from the spot, is still called after him.

CURTESY(a variant of “courtesy,”q.v.), in law, the life interest which a husband has in certain events in the lands of which his wife was in her lifetime actually seised for an estate of inheritance. As to the historical origin of the custom and the meaning of the word there is considerable doubt. It has been said to be an interest peculiar to England and to Scotland, hence called the “curtesy of England” and the “curtesy of Scotland”; but this is erroneous, for it is found also in Germany and France. TheMirroir des Justicesascribes it to Henry I. K. E. Digby (Hist. Real Prop.chap. iii.) says that it is connected with curia, and has reference either to the attendance of the husband as tenant of the lands at the lord’s court, or to mean simply that the husband is acknowledged tenant by the courts of England (tenens per legem Angliae). The requisites necessary to make tenancy by the curtesy are: (1) a legal marriage; (2) an estate in possession of which the wife must have been actually seised; (3) issue born alive and during the mother’s existence, though it is immaterial whether the issue live or die, or whether it is born before or after the wife’s seisin; in the case of gavelkind lands the husband has a right to curtesy, whether there is issue born or not; but the curtesy extends only to a moiety of the wife’s lands and ceases if the husband marries again. The issue must have been capable of inheriting as heir to the wife,e.g.if a wife were seised of lands in tail male the birth of a daughter would not entitle the husband to a tenancy by curtesy; (4) the title to the tenancy vests only on the death of the wife. The Married Women’s Property Act 1882 has not affected the right of curtesy so far as relates to the wife’s undisposed-of realty (Hopev.Hope, 1892, 2 Ch. 336), and the Settled Land Act 1884, s. 8, provides that for the purposes of the Settled Land Act 1882 the estate of a tenant by curtesy is to be deemed an estate arising under a settlement made by the wife.

See Pollock and Maitland,Hist. Eng. Law; K. E. Digby,Hist. Real Prop.; Goodeve,Real Property.

CURTILAGE(Med. Lat.curtilagium, fromcurtileorcortile, a court or yard, cf. “court”), the area of land which immediately surrounds a dwelling-house and its yard and outbuildings. In feudal times every castle with its dependent buildings was protected by a surrounding wall, and all the land within the wall was termed the curtilage; but the modern legal interpretation of the word,i.e.what area is enclosed by the curtilage, depends upon the circumstances of each individual case, such as the terms of the grant or deed which passes the property, or upon what is held to be a convenient amount of land for the occupation of the house, &c. The importance of the word in modern law depends on the fact that the curtilage marks the limit of the premises in which housebreaking can be committed.

CURTIN ANDREW GREGG(1817-1894), American political leader, was born at Bellefonte, Centre county, Pennsylvania, on the 22nd of April 1817, the son of a native of Ireland who was a pioneer iron manufacturer in Pennsylvania. He graduated from the law department of Dickinson College in 1837, was admitted to the bar in 1839, and successfully practised his profession. Entering politics as a Whig, he was chairman of the Whig state central committee in 1854, and from 1855 to 1858 was secretary of the commonwealth. In this capacity he was alsoex officiothe superintendent of common schools, and rendered valuable services to his state in perfecting and expanding the free public school system, and in establishing state normal schools. Upon the organization of the Republican party he became one of its leaders in Pennsylvania, and in October 1860 was chosen governor of the state on its ticket, defeating Henry D. Foster, the candidate upon whom the Douglas and Breckinridge Democrats and the Constitutional Unionists had united, by 32,000 votes, after a spirited campaign which was watched with intense interest by the entire country as an index of the result of the ensuing presidential election. During the Civil War he was one of the closest and most constant advisers of President Lincoln, and one of the most efficient, most energetic and most patriotic of the “war governors” of the North. Pennsylvania troops were the first to reach Washington after the president’s call, and from first to last the state, under Governor Curtin’s guidance, furnished 387,284 officers and men to the Northern armies. One of his wisest and most praiseworthy acts Was the organization of the famous “Pennsylvania Reserves,” by means of which the state was always able to fill at once its required quota after each successive call. In raising funds and equipping and supplying troops the governor showed great energy and resourcefulness, and his plans and organizations for caring for the needy widows and children of Pennsylvania soldiers killed in battle, and for aiding and removing to their homes the sick and wounded were widely copied throughout the North. He was re-elected governor in 1863 and served until January 1867. He was United States minister to Russia from 1869 until 1872, when he returned to America and took part in the Liberal Republican revolt against President U. S. Grant. In 1872-1873 he was a member of the state constitutional convention. Subsequently he joined the Democratic party and was a representative in Congress from 1881 to 1887. He died at his birthplace, Bellefonte, Pennsylvania, on the 7th of October 1894.

See William H. Egle’sLife and Times of Andrew Gregg Curtin(Philadelphia, 1896), which contains chapters written by A. K. McClure, Jno. Russell Young, Wayne McVeagh, Fitz John Porter and others.

CURTIS, GEORGE TICKNOR(1812-1894), American lawyer, legal writer and constitutional historian, was born in Watertown, Massachusetts, on the 28th of November 1812. He graduated at Harvard in 1832, was admitted to the bar in 1836, and practised in Worcester, Boston, New York and Washington, appearing before the United States Supreme Court in many important cases, including the Dred Scott case, in which he argued the constitutional question for Scott, and the “legal tender” cases. In Boston he was for many years the United States commissioner, and in this capacity, despite the vigorous protests of the abolitionists and his own opposition to slavery, ordered the return to his owner of the famous fugitive slave, Themas Sims, in 1852. He was the nephew and close friend of George Ticknor, the historian of Spanish literature, and his association with his uncle was influential in developing his scholarly tastes; while his other personal friendships with eminent Bostonians during the period of conservative Whig ascendancy in Massachusetts politics were of direct influence upon his political opinions and published estimates. He is best known as the author ofA History of the Origin, Formation and Adoption of the Constitution of the United States, with Notices of its principal Framers(1854), republished, with many additions, asThe Constitutional History of the United States from their Declaration of Independence to the Close of their Civil War(2 vols., 1889-1896). This history, which had been watched in its earlier progress by Daniel Webster, may be said to present the old Federalist or “Webster-Whig” view of the formation and powers of the Constitution; and it was natural that Curtis should follow it witha voluminousLife of Daniel Webster(2 vols., 1870), the most valuable biography of that statesman. Both these works are characterized by solidity and comprehensiveness rather than by rhetorical attractiveness or literary perspective. In his later years Mr Curtis, like so many of the followers of Webster, turned towards the Democratic party; and he wrote, among other works of minor importance, an exculpatory life of President James Buchanan (2 vols., 1883) and two vindications of General George B. McClellan’s career (1886 and 1887). He died in New York on the 28th of March 1894.

In addition to the works above mentioned he published:Digest of the English and American Admiralty Decisions(1839);Rights and Duties of Merchant Seamen(1841), which elicited the hearty praise of Justice Joseph Story;Law of Patents(1849);Equity Precedents(1850);Commentaries on the Jurisprudence, Practice and Peculiar Jurisdiction of the Courts of the United States(1854-1858);Creation or Evolution: A Philosophical Inquiry(1887); and a novel,John Chambers: A Tale of the Civil War in America(1889).

His brother,Benjamin Robbins Curtis(1809-1874), also an eminent jurist, was born on the 4th of November 1809, in Watertown, Massachusetts, graduated at Harvard in 1829, studied law at Cambridge and at Northfield, Mass., where, after his admission to the bar in 1832, he practised law for two years, and then in Boston in 1834-1851. In 1851, being then a member of the lower house of the Massachusetts legislature, he was on the 22nd of September appointed to the Supreme Court of the United States, where he gained his greatest fame in 1857 by his dissenting opinion in the Dred Scott case, in which he argued that the Missouri Compromise was constitutional, and that negroes could become citizens. His argument was immediately published as an anti-slavery document. On the 1st of September 1857 he resigned from the Supreme Court and resumed his private practice. In 1868 he was one of the counsel for President Andrew Johnson in his impeachment trial, and opened for the defence in a remarkable two-days’ speech. He died at Newport, Rhode Island, on the 15th of September 1874. He preparedDecisions of the Supreme Court(22 vols.) and aDigestof its decisions down to 1854.

A Memoir of Benjamin Robbins Curtis, with Some of his Professional and Miscellaneous Papers, edited by his son Benjamin R. Curtis, was published at Boston in 1879, theMemoirbeing by George Ticknor Curtis.

CURTIS, GEORGE WILLIAM(1824-1892), American man of letters, was born in Providence, Rhode Island, on the 24th of February 1824, of old New England stock. His mother died when he was two years old. At six he was sent with his elder brother to school in Jamaica Plain, Massachusetts, where he remained for five years. Then, his father having again married happily, the boys were brought home to Providence, where they stayed till, in 1839, their father removed to New York. Three years later, Curtis, being allowed to determine for himself his course of life, and being in sympathy with the spirit of the so-called Transcendental movement, became a boarder at the community of Brook Farm. He was accompanied by his brother, James Burrill Curtis, whose influence upon him was strong and helpful. He remained there for two years, brought into stimulating and serviceable relations with many interesting men and women. Then came two years, passed partly in New York, partly in Concord in order mainly to be in the friendly neighbourhood of Emerson, and then followed four years spent in Europe, Egypt and Syria.

Curtis returned from Europe in 1850, handsome, attractive, accomplished, ambitious of literary distinction. He instantly plunged into the whirl of life in New York, obtained a place on the staff of theTribune, entered the field as a popular lecturer, set himself to work on a volume published in the spring of 1851, under the title ofNile Notes of a Howadji, and became a favourite in society. He wrote much forPutnam’s Magazine, of which he was associate editor; and a number of volumes, composed of essays written for that publication and forHarper’s Monthly, came in rapid succession from his pen. The chief of these were thePotiphar Papers(1853), a satire on the fashionable society of the day; andPrue and I(1856), a pleasantly sentimental, fancifully tender and humorous study of life. In 1855 he married Miss Anna Shaw. Not long after his marriage he became, through no fault of his own, deeply involved in debt owing to the failure ofPutnam’s Magazine; and his high sense of honour compelled him to devote the greater part of his earnings for many years to the discharge of obligations for which he had become only by accident responsible, and from which he might have freed himself by legal process. In the period just preceding the Civil War other interests became subordinate to those of national concern. Curtis made his first important speech on the questions of the day at Wesleyan University in 1856; he engaged actively in the presidential campaign of that year, and was soon recognized not only as an effective public speaker, but also as one of the ablest, most high-minded, and most trustworthy leaders of public opinion. In 1863 he became the political editor ofHarper’s Weekly, and no other journal exercised during the war and after it a more important part in shaping public opinion. His writing was always clear, direct, forcible; his fairness of mind and sweetness of temper were invincible. He never became a mere partisan, and never failed to apply the test of moral principle to political measures. From month to month he contributed toHarper’s Monthly, under the title of “The Easy Chair,” brief essays on topics of social and literary interest, charming in style, touched with delicate humour and instinct with generous spirit. His service to the Republican party was such, that more than once he was offered nominations to office of high distinction, and might have been sent as minister to England; but he refused all offers of the kind, feeling that he could render more essential service to the country as editor and public speaker.

In 1871 he was appointed by President Grant chairman of the commission to report on the reform of the civil service. The report which he wrote was the foundation of every effort since made for the purification and regulation of the service and for the destruction of political patronage. From that time till his death Curtis was the leader in this reform, and to his sound judgment, his vigorous presentation of the evils of the corrupt prevailing system, and his untiring efforts, the progress of the reform is mainly due. He was president of the National Civil Service Reform League and of the New York Civil Service Reform Association. In 1884 he refused to support the nomination of James G. Blaine as candidate for the presidency, and thus broke with the Republican party, of which he had been one of the founders and leaders. From that time he stood as the typical independent in politics. In April 1892 he delivered at Baltimore his eleventh annual address as president of the National Civil Service Reform League, and in May he appeared for the last time in public, to repeat in New York an admirable address on James Russell Lowell, which he had first delivered in Brooklyn on the 22nd of the preceding February, the anniversary of Lowell’s birth. On the 31st of the following August he died. He was a man of consistent virtue, whose face and figure corresponded with the traits and stature of his soul. The grace and charm of his manner were the expression of his nature. Of the Americans of his time few were more widely beloved, and the respect in which he was held was universal.

SeeGeorge William Curtis, by Edward Cary, in the “American Men of Letters” series (Boston, 1894), an excellent biography; “An Epistle to George William Curtis,” by James Russell Lowell (1874-1887), in Lowell’sPoems;George William Curtis, a Commemorative Address delivered before The Century Association, 17th December 1892, by Parke Godwin (New York, 1893);Orations and Addresses by George William Curtis, edited by Charles Eliot Norton (3 vols. New York, 1894).

(C. E. N.)

CURTIUS, ERNST(1814-1896), German archaeologist and historian, was born at Lübeck on the 2nd of September 1814. On completing his university studies he was chosen by C. A. Brandis to accompany him on a journey to Greece for the prosecution of archaeological researches. Curtius then became Otfried Müller’s companion in his exploration of the Peloponnese, and on Müller’s death in 1840 returned to Germany. In 1844 he became an extraordinary professor at the university of Berlin, and in the same year was appointed tutor to Prince Frederick William (afterwards the Emperor Frederick III.)—a post which he held till 1850. After holding a professorship at Göttingen andundertaking a further journey to Greece in 1862, Curtius was appointed (in 1863) ordinary professor at Berlin. In 1874 he was sent to Athens by the German government, and concluded an agreement by which the excavations at Olympia (q.v.) were entrusted exclusively to Germany. Curtius died at Berlin on the 11th of July 1896. His best-known work is hisHistory of Greece(1857-1867, 6th ed. 1887-1888; Eng. trans. by A. W. Ward, 1868-1873). It presented in an attractive style what were then the latest results of scholarly research, but was criticized as wanting in erudition. It is now superseded (seeGreece:History, Ancient, § Bibliography). His other writings are chiefly archaeological. The most important are:Die Akropolis von Athen(1844);Naxos(1846);Peloponnesos, eine historisch-geographische Beschreibung der Halbinsel(1851);Olympia(1852);Die Ionier vor der ionischen Wanderung(1855);Attische Studien(1862-1865);Ephesos(1874);Die Ausgrabungen zu Olympia(1877, &c.);Olympia und Umgegend(edited by Curtius and F. Adler, 1882);Olympia; Die Ergebnisse der von dem deutschen Reich veranstalteten Ausgrabung(with F. Adler, 1890-1898);Die Stadtgeschichte von Athen(1891);Gesammelte Abhandlungen(1894). His collected speeches and lectures were published under the title ofAltertum und Gegenwart(5th ed., 1903 foll.), to which a third volume was added under the title ofUnter drei Kaisern(2nd ed., 1895).

A full list of his writings will be found in L. Gurlitt,Erinnerungen an Ernst Curtius(Berlin, 1902); see also article by O. Kern inAllgemeine deutsche Biographie, xlvii. (1903), to which may be addedErnst Curtius. Ein Lebensbild in Briefen, by F. Curtius (1903); T. Hodgkin,Ernest Curtius(1905).

His brother,Georg Curtius(1820-1885), philologist, was born at Lübeck on the 16th of April 1820. After an education at Bonn and Berlin he was for three years a schoolmaster in Dresden, until (in 1845) he returned to Berlin University asprivat-docent. In 1849 he was placed in charge of the Philological Seminary at Prague, and two years later was appointed professor of classical philology in Prague University. In 1854 he removed from Prague to a similar appointment at Kiel, and again in 1862 from Kiel to Leipzig. He died at Hermsdorf on the 12th of August 1885. His philological theories exercised a widespread influence. The more important of his publications are:Die Sprachvergleichung in ihrem Verhältniss zur classischen Philologie(1845; Eng. trans. by F. H. Trithen, 1851);Sprachvergleichende Beiträge zur griechischen und lateinischen Grammatik(1846);Grundzüge der griechischen Etymologie(1858-1862, 5th ed. 1879);Das Verbum der griechischen Sprache(1873). The last two works have been translated into English by A. S. Wilkins and E. B. England. From 1878 till his death Curtius was general editor of theLeipziger Studien zur classischen Philologie. HisGriechische Schulgrammatik, first published in 1852, has passed through more than twenty editions, and has been edited in English. In his last work,Zur Kritik der neuesten Sprachforschung(1885), he attacks the views of the “new” school of philology.

Opusculaof Georg Curtius were edited after his death by E. Windisch (Kleine Schriften von E. C., 1886-1887). For further information consult articles by R. Meister inAllgemeine deutsche Biographie, xlvii. (1903), and by E. Windisch in C. Bursian’sBiographisches Jahrbuch für Alterthumskunde(1886).

CURTIUS, MARCUS,a legendary hero of ancient Rome. It is said that in 362B.C.a deep gulf opened in the forum, which the seers declared would never close until Rome’s most valuable possession was thrown into it. Then Curtius, a youth of noble family, recognizing that nothing was more precious than a brave citizen, leaped, fully armed and on horseback, into the chasm, which immediately closed again. The spot was afterwards covered by a marsh called the Lacus Curtius. Two other explanations of the name Lacus Curtius are given: (1) a Sabine general, Mettius (or Mettus) Curtius, hard pressed by the Romans under Romulus, leaped into a swamp which covered the valley afterwards occupied by the forum, and barely escaped with his life; (2) in 445B.C.the spot was struck by lightning, and enclosed as sacred by the consul, Gaius Curtius. It was marked by an altar which was removed to make room for the games in celebration of Caesar’s funeral (Pliny,Nat. Hist.xv. 77), but restored by Augustus (cf. Ovid,Fasti, vi. 403), in whose time there was apparently nothing but a dry well. The altar seems to have been restored early in the 4th centuryA.D.In April 1904, on the N. side of the Via Sacra and 20 ft. N.W. of the Equus Domitiani, remains of the buildings were discovered.

See Livy i. 12, vii. 6; Dion Halic. ii. 42; Varro,De lingua Latina, v. 148; Ch. Hülsen,The Roman Forum(Eng. trans. of 2nd ed., J. B. Carter, 1906); O. Gilbert,Geschichte und Topographie der Stadt Rom im Altertum, i. (1883), 334-338.

CURTIUS RUFUS, QUINTUS,biographer of Alexander the Great. Of his personal history nothing is known, nor can his date be fixed with certainty. Modern authorities regard him as a rhetorician who flourished during the reign of Claudius (A.D.41-54). His work (De Rebus gestis Alexandri Magni) originally consisted of ten books, of which the first two are entirely lost, and the remaining eight are incomplete. Although the work is uncritical, and shows the author’s ignorance of geography, chronology and military matters, it is written in a picturesque style.

There are numerous editions: (text) T. Vogel (1889), P. H. Damste (1897), E. Hedicke (1908); (with notes), T. Vogel (1885 and later), M. Croiset (1885), H. W. Reich (1895), C. Lebaigue (1900), T. Stangl (1902). There is an English translation by P. Pratt (1821). See S. Dosson,Étude sur Quinte-Curce, sa vie, et ses œuvres(1887) a valuable work; F. von Schwarz,Alexander des Grossen Feldzüge in Turkestan(1893), a commentary on Arrian and Curtius based upon the author’s personal knowledge of the topography; C. Wachsmuth,Einleitung in das Studium der alten Geschichte(1895), p. 574, cf. p. 567, note 2; Schwarz, “Curtius Rufus” No. 31 in Pauly-Wissowa (1901).

CURULE(Lat.currus, “chariot”), in Roman antiquities, the epithet applied to the chair of office,sella curulis, used by the “curule” or highest magistrates and also by the emperors. This chair seems to have been originally placed in the magistrate’s chariot (hence the name). It was inlaid with ivory or in some cases made of it, had curved legs but no back, and could be folded up like a camp-stool. In English the word is used in the general sense of “official.” (SeeConsul,PraetorandAedile.)

CURVE(Lat.curvus, bent), a word commonly meaning a shape represented by a line bending continuously out of the straight without making an angle, but only properly to be defined in its geometrical sense in the terms set out below. This subject is treated here from an historical point of view, for the purpose of showing how the different leading ideas were successively arrived at and developed.

1. A curve is a line, or continuous singly infinite system of points. We consider in the first instance, and chiefly, a plane curve described according to a law. Such a curve may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former point of view, it is the locus of all the points which satisfy a given condition; in the latter, it is the locus of a point moving subject to a given condition. Thus the most simple and earliest known curve, the circle, is the locus of all the points at a given distance from a fixed centre, or else the locus of a point moving so as to be always at a given distance from a fixed centre. (The straight line and the point are not for the moment regarded as curves.)

Next to the circle we have the conic sections, the invention of them attributed to Plato (who lived 430-347B.C.); the original definition of them as the sections of a cone was by the Greek geometers who studied them soon replaced by a proper definitionin planolike that for the circle, viz. a conic section (or as we now say a “conic”) is the locus of a point such that its distance from a given point, the focus, is in a given ratio to its (perpendicular) distance from a given line, the directrix; or it is the locus of a point which moves so as always to satisfy the foregoing condition. Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.

The Greek geometers invented other curves; in particular, the conchoid (q.v.), which is the locus of a point such that its distance from a given line, measured along the line drawn throughit to a fixed point, is constant; and the cissoid (q.v.), which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point. Obviously the number of such geometrical or kinematical definitions is infinite. In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively. Every curve thus arbitrarily defined has its own properties; and there was not any principle of classification.

2.Cartesian Co-ordinates.—The principle of classification first presented itself in theGéometrieof Descartes (1637). The idea was to represent any curve whatever by means of a relation between the co-ordinates (x, y) of a point of the curve, or say to represent the curve by means of its equation. (SeeGeometry:Analytical.)

Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (x, y).

Observe that the distinctive feature is in theexclusiveuse of such determination of a curve by means of its equation. The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x²/a² + y²/b² = 1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

3.Order of a Curve.—We obtain from the equation the notion of an algebraical as opposed to a transcendental curve, viz. an algebraical curve is a curve having an equation F(x, y) = 0 where F(x, y) is a rational and integral function of the co-ordinates (x, y); and in what follows we attend throughout (unless the contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational and integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of agiven order, viz. the order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy − 1 = 0 is a curve of the second order.

It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

We hence divide curves according to their order, viz. a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax + by + c = 0, or say (*x, y, 1) = 0; or by an equation of the second order, ax² + 2hxy + by² + 2fy + 2gx + c = 0, say (*x, y, 1)² = 0; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.

A curve of the order m has for its equation (*x, y, 1)m= 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is ½(m + 1)(m + 2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as ½(m + 1)(m + 2) − 1, that is, ½m(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through ½m(m + 3) points.

It is to be remarked that an equation maybreak up; thus a quadric equation may be (ax + by + c)(a′x + b′y + c′) = 0, breaking up into the two equations ax + by + c = 0, a′x + b′y + c′ = 0, viz. the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. But it is animproperquadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.

4.Intersections of Curves.—The intersections of two curves are obtained by combining their equations; viz. the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).

Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n = 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.

The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry—and it in fact plays an all-important part in modern geometry. As in algebra we say that an equation of themth order hasmroots, viz. we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of themth order is met by an arbitrary line inmpoints, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of themth order, as a curve which is met by an arbitrary line in m points (that is, of course, inm, and not more thanm, points).

The theorem of themintersections has been stated in regard to anarbitraryline; in fact, for particular lines the resultant equation may be or appear to be of an order less thanm; for instance, takingm= 2, if the hyperbola xy − 1 = 0 be cut by the line y = β, the resultant equation in x is βx − 1 = 0, and there is apparently only the intersection (x = 1/β, y = β); but the theorem is, in fact, true for every line whatever: a curve of the order m meets every line whatever in precisely m points. We have, in the case just referred to, to take account of a point at infinity on the line y = β; the two intersections are the point (x = 1/β, y = β), and the point at infinity on the line y = β.

It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is necessary to take account of these various circumstances.

5.Line at Infinity.—The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line,i.e.this projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection isalwaysa line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane.1

Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it inmpoints, real or imaginary, distinct or coincident.

Descartes in theGéométriedefined and considered the remarkable curves called after him the ovals of Descartes, or simply Cartesians, which will be again referred to. The next important work, founded on theGéométrie, was Sir Isaac Newton’sEnumeratio linearum tertii ordinis(1706), establishing a classification of cubic curves founded chiefly on the nature of their infinite branches, which was in some details completed by James Stirling (1692-1770), Patrick Murdoch (d. 1774) and Gabriel Cramer; the work also contains the remarkable theorem (to be again referred to), that there are five kinds of cubic curves giving by their projections every cubic curve whatever. Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin’s memoir, “De linearum geometricarum proprietatibus generalibus Tractatus” (posthumous, say 1746, published in the 6th edition of hisAlgebra). We have in it a particular kind ofcorrespondenceof two points on a cubic curve, viz. two points correspond to each other when the tangents at the two points again meet the cubic in the same point.

6.Reciprocal Polars. Intersections of Circles. Duality. Trilinear and Tangential Co-ordinates.—The Géométrie descriptive, by Gaspard Monge, was written in the year 1794 or 1795 (7th edition, Paris, 1847), and in it we have stated,in planowith regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz. “Given a surface of the second order and a circumscribed conic surface which touches it ... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point.” The theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry. It is in Charles Julian Brianchon’s memoir “Sur les surfaces du second degré” (Jour. Polyt.t. vi. 1806) shown how for any given position of the summit the plane of contact is determined, or reciprocally; say the plane XY is determined when the point P is given, or reciprocally; and it is noticed that when P is situate in the interior of the surface the plane XY does not cut the surface; that is, we have a real plane XY intersecting the surface in the imaginary curve of contact of the imaginary circumscribed cone having for its summit a given real point P inside the surface.

Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other. The term “pole” was first used by François Joseph Servois, and “polar” by Joseph Diez Gergonne (Gerg.t. i. and iii., 1810-1813); and from the theorem we have the method of reciprocal polars for the transformation of geometrical theorems, used already by Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by various writers in the earlier volumes of Gergonne. We are here concerned with the method less in itself than as leading to the general notion of duality.

Bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by L. Gaultier, on the graphical construction of circles and spheres (Jour. Polyt.t. ix., 1813). The well-known theorem as to radical axes may be stated as follows. Consider two circles partially drawn so that it does not appear whether the circles, if completed, would or would not intersect in real points, say two arcs of circles; then we can, by means of a third circle drawn so as to intersect in two real points each of the two arcs, determine a right line, which, if the complete circles intersect in two real points, passes through the points, and which is on this account regarded as a line passing through two (real or imaginary) points of intersection of the two circles. The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the “radical axis of the two circles”; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles.

The intersections which lie on the radical axis are two out of the four intersections of the two circles. The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Jean Victor Poncelet’sTraité des propriétés projectives(1822), where we find (p. 49) the statement, “deux circles placés arbitrairement sur un plan ... ont idéalement deux points imaginaires communs à l’infini”; that is, a circlequacurve of the second order is met by the line infinity in two points; but, more than this, they are the same two points for any circle whatever. The points in question have since been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J; and we have thus the circle characterized as a conic which passes through the two circular points at infinity; the number of conditions thus imposed upon the conic is = 2, and there remain three arbitrary constants, which is the right number for the circle. Poncelet throughout his work makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars.

Poncelet’s two memoirsSur les centres des moyennes harmoniquesandSur la théorie générale des polaires réciproques, although presented to the Paris Academy in 1824, were only published (Crelle, t. iii. and iv., 1828, 1829) subsequent to the memoir by Gergonne,Considérations philosophiques sur les élémens de la science de l’étendue(Gerg.t. xvi., 1825-1826). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find “dans la géométrie plane, à chaque théorème il en répond nécessairement un autre qui s’en déduit en échangeant simplement entre eux les deux motspointsetdroites; tandis que dans la géométrie de l’espace ce sont les motspointsetplansqu’il faut échanger entre eux pour passer d’un théorème à son corrélatif”; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns. There was a reclamation as to priority by Poncelet in theBulletin universelreprinted with remarks by Gergonne (Gerg.t. xix., 1827), and followed by a short paper by Gergonne,Rectifications de quelques théorèmes, &c., which is important as first introducing the wordclass. We find in it explicitly the two correlative definitions: “a plane curve is said to be of themth degree (order) when it has with a linemreal or ideal intersections,” and “a plane curve is said to be of themth class when from any point of its plane there can be drawn to itmreal or ideal tangents.”

It may be remarked that in Poncelet’s memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the orderm, or say the class of the curve, is in general and at most =m(m− 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively. It is to be understood that the theorem extends to all points or lines, drawn or not drawn; thus if in the first figure there are any number of points on a line drawn or not drawn, the corresponding lines in the second figure, produced if necessary, must meet in a point. And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the orderm, any line meets it inmpoints; and hence from the corresponding point in the second figure there must be to the corresponding curvemtangents; that is, the corresponding curve must be of the classm.

Trilinear co-ordinates (seeGeometry:Analytical) were first used by E. E. Bobillier in the memoirEssai sur un nouveau mode de recherche des propriétés de l’étendue(Gerg.t. xviii., 1827-1828). It is convenient to use these rather than Cartesian co-ordinates. We represent a curve of the order m by an equation (*x, y, z)m= 0, the function on the left hand being a homogeneous rational and integral function of the ordermof the three co-ordinates (x, y, z); clearly the number of constants is the same as for the equation (*x, y, 1)m= 0 in Cartesian co-ordinates.

The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in theMémoire de géométrie sur deux principes généraux de la science, la dualité et l’homographie, which forms a sequel to theAperçu historique sur l’origine et le développement des méthodes en géométrie(Mém. de Brux.t. xi., 1837).

We now come to Julius Plücker; his “six equations” were given in a short memoir inCrelle(1842) preceding his great work, theTheorie der algebraischen Curven(1844). Plücker first gave a scientific dual definition of a curve, viz.; “A curve is a locus generated by a point, and enveloped by a line—the point moving continuously along the line, while the line rotates continuously about the point”; the point is a point (ineunt.) of the curve, the line is a tangent of the curve. And, assuming the above theory of geometrical imaginaries, a curve such thatmof its points are situate in an arbitrary line is said to be of the orderm; a curve such thatnof its tangents pass through an arbitrary point is said to be of the classn; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne. Thus the line is a curve of the order 1 and class 0; and corresponding dually thereto, we have the point as a curve of the order 0 and class 1.

Plücker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates). (SeeGeometry:Analytical.) The Cartesian co-ordinates (x, y) and trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (ξ, η, ζ) are line-co-ordinates for determining the position of a line. It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-co-ordinates; viz. taking ξx + ηy + ζz = 0 to be the equation of a line, we say that (ξ, η, ζ) are the line-co-ordinates of this line. A linear relation aξ + bη + cζ = 0 between these co-ordinate determines a point, viz. the point whose point-co-ordinates are (a, b, c); in fact, the equation in question aξ + bη + cζ = 0 expresses that the equation ξx + ηy + ζz = 0, where (x, y, z) are current point-co-ordinates, is satisfied on writing therein x, y, z = a, b, c; or that the line in question passes through the point (a, b, c). Thus (ξ, η, ζ) are the line-co-ordinates of any line whatever; but when these, instead of being absolutely arbitrary, are subject to the restriction aξ + bη + cζ = 0, this obliges the line to pass through a point (a, b, c); and the last-mentioned equation aξ + bη + cζ = 0 is considered as the line-equation of this point.

A line has only a point-equation, and a point has only a line-equation; but any other curve has a point-equation and also a line-equation; the point-equation (*x, y, z)m= 0 is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (*ξ, η, ζ)n= 0 is the relation which is satisfied by the line-co-ordinates (ξ, η, ζ) of each line (tangent) of the curve.

There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in demonstrating by point-co-ordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem; that is, we do not demonstrate the one theorem, and then (as by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (x, y, z.) instead of meaning point-co-ordinates may mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem.

7.Singularities of a Curve. Plücker’s Equations.—The above dual generation explains the nature of the singularities of a plane curve. The ordinary singularities, arranged according to a cross division, are

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