the remaining terms (o^-aj) mf + (a 2 2 - a s ) m a a + . . indicate tangents which are in the limit the lines drawn to the several summits, that is, we have (a^ - a t ) m^ summits on the curve l ]=0, &c.,
There is of course a precisely similar theory as regards line- coordinates ; taking IIj, n a &c., to be rational and integral func tions of the co-ordinates (f,7j,0 we connect with the ultimate curve n^HIjj^. . =0, and consider as belonging to it certain lines, which for the moment may be called "axes" tangents to the component curves n^Oj n 2 = respectively. Considering an equation in point-coordinates, we may hare among the component curves right lines, and if in order to put these in evidence we take the equation to be LjT]. . Pj a i. . . =0, where L x = is a right line, P 1 = 0acurve of the second or any higher order, then the curve will contain as part of itself summits not exhibited in this equation, but the corresponding line-equation will be A^i. . H^i . = 0, where A t = 0, . . are the equations of the summits in question, n x = 0, &c., are the line-equations corresponding to the several point-equations P! = O, &c. ; and this curve will contain as part of itself axes not exhibited by this equation, but which are the lines L x = 0, . . of the equation iu point-coordinates.
In conclusion a little may be said as to curves of double curvature, otherwise twisted curves, or curves in space. The analytical theory by Cartesian coordinates was first con sidered by Clairaut, Recherches sur les courbes ct double courbure (Paris, 1731). Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the aame time rotates about the line ; the point is a point, the line a tangent, and the plane an osculating plane, of the curve ; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression. Analogous to the order and class of a plane curve we have the order, rank, and class, of the system (assumed to be a geometrical one), viz., if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of the system, then m, r, n are the order, rank, and class respec tively. The system has singularities, and there exist be tween m, r, n and the numbers of the several singularities equations analogous to Pliicker s equations for a plane curve. It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U = 0, V = 0; the two equations represent surfaces, inter secting in a curve ; but there are curves which are not the complete intersection of any two surfaces ; thus we have the cubic in space, or skew cubic, which is the residual in tersection of two quadric surfaces which have a line in common ; the equations U = 0, V = Oof the two quadric surfaces represent the cubic curve, not by itself, but together with the line.
CURZOLA (Slavonic, Karkar), a city of Dalmatia, Austria, the capital of an island of the same name in the Adriatic, which is situated between 42 50 and 43 1 N. lat. and 16 40 and 17 20 E. long, and has a length of about 25 miles, with an average breadth of 4 miles. The city is about 55 miles north of Ragusa. It is regularly built, and, besides the old cathedral, the loggia or council chambers, and the palace of its former Venetian governors, it possesses the noble mansion of the Arnieri, and other specimens of the domestic architecture of the 15th and 16th centuries, and still retains the massive walls and towers that were erected in 1420. Its principal industry is the building of the boats for which it is famed throughout the Adriatic. Originally, as it would seem, a Phoenician settlement, Curzola was afterwards colo nized by Greeks from Cnidus ; but nothing is known of its earlier history. The present name is a corrup tion of Corcyra Nigra, or KcpKvpa MeAatra, the designa tion by which it was known to the Greeks and Romans. In 997 it came under the suzerainty of Venice, and it was one of the earliest cities in Dalmatia to receive municipal rights. In 1571 it defended itself so gallantly against the Turks under Uluch Ali of Algiers that it obtained the de signation fidelissima. Population, 2200.
CUSA, Nicolas de [Nicolaus Cusanus] cardinal (1401-1464), was the son of a poor fisherman named Krypffs or Krebs, and derived the name by which he is known from the place of his birth, Cues or Cusa, on the Moselle, in the archbishopric of Treves. In his youth he was employed in the service of Count Ulrich of Manderscheid, who, seeing in him evidence of exceptional ability, sent him to study at the school of the Brothers of the Common Life at Deventer, and afterwards at the uni versity of Padua, where he took his doctor s degree in law in his twenty-third year. Failing in his first cause he abandoned the legal profession, and resolved to enter the church. After filling several subordinate offices he became archdeacon of Lie"ge. He was a member of the Council of Basel, and dedicated to the assembled fathers a work entitled De Concordantia Catholica, in which he main tained the superiority of councils over popes, and assailed the false decretals and the story of the donation of Con- stantine. A few years later, however, he had reversed his position, and zealously defended the supremacy of the Pope. He was intrusted with various missions in the in terests of Catholic unity, the most important being to Con stantinople, to endeavour to bring about a union of the Eastern and Western churches. In 1448 he was raised by Pope Nicolas V. to the dignity of cardinal; and in 1450 he was appointed bishop of Brixen against the wish of the Archduke Sigismund, who opposed the reforms the new bishop sought to introduce into the diocese. In 1451 he was sent to Germany and the Netherlands to check ecclesi astical abuses and bring back the monastic life to the original rule of poverty, chastity, and obedience, a mission which be discharged with well-tempered firmness. Soon afterwards his dispute with the Archduke Sigismund in his own diocese was brought to a point by his claiming certain dues of the bishopric, which the temporal prince had appropriated. Upon this the bishop was imprisoned by the archduke, who, in his turn, was excommunicated by the Pope. These extreme measures were not persisted in ; but the dispute remained unsettled at the time of the bishop s death, which occurred at Lodi in Umbria on the llth August 1464. In 1459 he had acted as governor of Rome during the absence of his friend Pope Pius II. at the assembly of princes at Milan ; and he wrote his Crebratio Alcorani, a treatise against Mahometanism, iu support of the expedition against the Turks proposed at that assembly. Some time before bis death he had founded a hospital in his native place for thirty-three poor persons, the number being that of the years of the earthly life of Christ. To this institution he left his valuable library.
cal much more than in his political or ecclesiastical activity. As in religion he is entitled to be called one of the " Refor mers before the Reformation," so in philosophy he was one of those who broke with scholasticism while it was still the orthodox system. In his principal work, De docta ignorantia (1440), supplemented by De Conjccturis libri duo published in the same year, he maintains that all human knowledge is mere conjecture, and that man s wisdom is to recognize his ignorance. From scepticism he escapes by accepting the doctrine of the mystics that God can be apprehended by intuition (intuitio, speculatio), an exalted
state of the intellect in which all limitations disappear.