limit the proper tangents to P1 = 0 each α1 times, the lines to its
nodes each 2α1 times, and the lines to its cusps each 3α1, times;
the remaining terms (α12 − α1)m12 + (α22 − α2)m22 + ... indicate tangents
which are in the limit the lines drawn to the several summits, that is,
we have (α12 − α1)m12 summits on the curve P1 = 0, &c.
There is, of course, a precisely similar theory as regards line-co-ordinates; taking Π1, Π2, &c., to be rational and integral functions of the co-ordinates (ξ, η, ζ) we connect with the ultimate curve Π1α1Π2α2 ... = 0, and consider as belonging to it, certain lines, which for the moment may be called “axes” tangents to the component curves Π1 = 01, Π2 = 0 respectively. Considering an equation in point-co-ordinates, we may have among the component curves right lines, and if in order to put these in evidence we take the equation to be L1γ1 .. P1α1 ... = 0, where L1 = 0 is a right line, P1 = 0 a curve 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 1Λδ1 ... Π1α1 = 0, where Λ1 = 0,... are the equations of the summits in question, Π1 = 0, &c., are the line-equations corresponding to the several point-equations P1 = 0, &c.; and this curve will contain as part of itself axes not exhibited by this equation, but which are the lines L1 = 0,... of the equation in point-co-ordinates.
18. Twisted Curves.—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 co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes à 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 same 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 respectively. The system has singularities, and there exist between m, r, n and the numbers of the several singularities equations analogous to Plücker’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, intersecting 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 intersection of two quadric surfaces which have a line in common; the equations U = 0, V = 0 of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.
Authorities.—In addition to the copious authorities mentioned in the text above, see Gabriel Cramer, Introduction à l’analyse des lignes courbes algébriques (Geneva, 1750). Bibliographical articles are given in the Ency. der math. Wiss. Bd. iii. 2, 3 (Leipzig, 1902–1906); H. C. F. von Mangoldt, “Anwendung der Differential- und Integralrechnung auf Kurven und Flächen,” Bd. iii. 3 (1902); F. R. v. Lilienthal, “Die auf einer Fläche gezogenen Kurven,” Bd. iii. 3 (1902); G. W. Scheffers, “Besondere transcendente Kurven,” Bd. iii. 3 (1903); H. G. Zeuthen, “Abzahlende Methoden,” Bd. iii. 2 (1906); L. Berzolari, “Allgemeine Theorie der höheren ebenen algebraischen Kurven,” Bd. iii. 2 (1906). Also A. Brill and M. Noether, “Die Entwicklung der Theorie der algebraischen Funktionen in älterer und neuerer Zeit” (Jahresb. der deutschen math. ver., 1894); E. Kötter, “Die Entwickelung der synthetischen Geometrie” (Jahresb. der deutschen math. ver., 1898–1901); E. Pascal, Repertorio di matematiche superiori, ii. “Geometrìa” (Milan, 1900); H. Wieleitner, Bibliographie der höheren algebraischen Kurven für den Zeitabschnitt von 1890–1894 (Leipzig, 1905).
Text-books:—G. Salmon, A Treatise on the Higher Plane Curves (Dublin, 1852, 3rd ed., 1879); translated into German by O. W. Fiedler, Analytische Geometrie der höheren ebenen Kurven (Leipzig, 2te Aufl., 1882); L. Cremona, Introduzione ad una teoria geometrica delle curve piane (Bologna, 1861); J. H. K. Durège, Die ebenen Kurven dritter Ordnung (Leipzig, 1871); R. F. A. Clebsch and C. L. F. Lindemann, Vorlesungen über Geometrie, Band i. and i2 (Leipzig, 1875–1876); H. Schroeter, Die Theorie der ebenen Kurven dritter Ordnung (Leipzig, 1888); H. Andoyer, Leçons sur la théorie des formes et la géométrie analytique supérieure (Paris, 1900); Wieleitner, Theorie der ebenen algebraischen Kurven höherer Ordnung (Leipzig, 1905). (A. Ca.; E. B. El.)
CURVILINEAR, in architecture, that which is formed by curved or flowing lines; the roofs over the domes and vaults of the Byzantine churches were generally curvilinear. The term is also given to the flowing tracery of the Decorated and the Flamboyant styles.
CURWEN, HUGH (d. 1568), English ecclesiastic and statesman, was a native of Westmorland, and was educated at Cambridge, afterwards taking orders in the church. In May 1533 he expressed approval of Henry VIII.’s marriage with Anne Boleyn in a sermon preached before the king. In 1541 he became dean of Hereford, and in 1555 Queen Mary nominated him to the archbishopric of Dublin, and in the same year he was appointed lord chancellor of Ireland. He acted as one of the lords justices during the absence from Ireland of the lord deputy, the earl of Sussex, in 1557. On the accession of Elizabeth, Curwen at once accommodated himself to the new conditions by declaring himself a Protestant, and was continued in the office of lord chancellor. He was accused by the archbishop of Armagh of serious moral delinquency, and his recall was demanded both by the primate and the bishop of Meath. In 1567 Curwen resigned the see of Dublin and the office of lord chancellor, and was appointed bishop of Oxford. He died on the 1st of November 1568.
See John Strype, Life and Acts of Archbishop Parker (3 vols., Oxford, 1824), and Memorials of Thomas Cranmer (2 vols., Oxford, 1840); John D’Alton. Memoirs of the Archbishops of Dublin (Dublin, 1838).
CURWEN, JOHN (1816–1880), English Nonconformist minister and founder of the Tonic Sol-Fa system of musical teaching, was born at Heckmondwike, Yorkshire, of an old Cumberland family. His father was a Nonconformist minister, and he himself adopted this profession, which he practised till 1864, when he gave it up in order to devote himself to his new method of musical nomenclature, designed to avoid the use of the stave with its lines and spaces. He adapted it from that of Miss Sarah Ann Glover (1785–1867) of Norwich, whose Sol-Fa system was based on the ancient gamut; but she omitted the constant recital of the alphabetical names of each note and the arbitrary syllable indicating key relationship, and also the recital of two or more such syllables when the same note was common to as many keys (e.g. “C, Fa, Ut,” meaning that C is the subdominant of G and the tonic of C). The notes were represented by the initials of the seven syllables, still in use in Italy and France as their names but in the “Tonic Sol-Fa” the seven letters refer to key relationship and not to pitch. Curwen was led to feel the importance of a simple way of teaching how to sing by note by his experiences among Sunday-school teachers. Apart from Miss Glover, the same idea had been elaborated in France since J. J. Rousseau’s time, by Pierre Galin (1786–1821), Aimé Paris (1798–1866) and Emile Chevé (1804–1864), whose method of teaching how to read at sight also depended on the principle of “tonic relationship” being inculcated by the reference of every sound to its tonic, by the use of a numeral notation. Curwen brought out his Grammar of Vocal Music in 1843, and in 1853 started the Tonic Sol-Fa Association; and in 1879, after some difficulties with the education department, the Tonic Sol-Fa College was opened. Curwen also took to publishing, and brought out a periodical called the Tonic Sol-fa Reporter, and in his later life was occupied in directing the spreading organization of his system. He died at Manchester on the 26th of May 1880. His son John Spencer Curwen (b. 1847), who became principal of the Tonic Sol-Fa College, published Memorials of J. Curwen in 1882. The Sol-Fa system has been widely adopted for use in education, as an easily teachable method in the reading of music at sight, but its more ambitious aims, which are strenuously pushed, for providing a superior method of musical notation generally, have not recommended themselves to musicians at large.
CURZOLA (Serbo-Croatian Korčula or Karkar), an island in the Adriatic Sea, forming part of Dalmatia, Austria; and lying west of the Sabionicello promontory, from which it is divided by a strait less than 2 m. wide. Its length is about 25 m.; its average breadth, 4 m. Curzola (Korčula), the capital and
