Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same order) of the co-ordinates (x, y, z), and u′, X′, Y′, Z′ rational and integral functions (X′, Y′, Z′, all of the same order) of the co-ordinates (x′, y′, z′), we transform a given curve u = 0, by the equations of x′ : y′ : z′ = X : Y : Z, thereby obtaining a transformed curve u′ = 0, and a converse set of equations x : y : z = X′ : Y′ : Z′; viz. assuming that this is so, the point (x, y, z) on the curve u = 0 and the point (x′, y′, z′) on the curve u′ = 0 will be points having a (1, 1) correspondence. To show how this is, observe that to a given point (x, y, z) on the curve u = 0 there corresponds a single point (x′, y′, z′) determined by the equations x′ : y′ : z′ = X : Y : Z; from these equations and the equation u = 0 eliminating x, y, z, we obtain the equation u′ = 0 of the transformed curve. To a given point (x′, y′, z′) not on the curve u’ = 0 there corresponds, not a single point, but the system of points (x, y, z) given by the equations x′ : y′ : z′ = X : Y : Z, viz., regarding x′, y′, z′ as constants (and to fix the ideas, assuming that the curves X = 0, Y = 0, Z = 0, have no common intersections), these are the points of intersection of the curves X : Y : Z, = x′ : y′ : z′, but no one of these points is situate on the curve u = 0. If, however, the point (x′, y′, z′) is situate on the curve u′ = 0, then one point of the system of points in question is situate on the curve u = 0, that is, to a given point of the curve u′ = 0 there corresponds a single point of the curve u = 0; and hence also this point must be given by a system of equations such as x : y : z = X′ : Y′ : Z′.
Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same order) of the co-ordinates (x, y, z), and u′, X′, Y′, Z′ rational and integral functions (X′, Y′, Z′, all of the same order) of the co-ordinates (x′, y′, z′), we transform a given curve u = 0, by the equations of x′ : y′ : z′ = X : Y : Z, thereby obtaining a transformed curve u′ = 0, and a converse set of equations x : y : z = X′ : Y′ : Z′; viz. assuming that this is so, the point (x, y, z) on the curve u = 0 and the point (x′, y′, z′) on the curve u′ = 0 will be points having a (1, 1) correspondence. To show how this is, observe that to a given point (x, y, z) on the curve u = 0 there corresponds a single point (x′, y′, z′) determined by the equations x′ : y′ : z′ = X : Y : Z; from these equations and the equation u = 0 eliminating x, y, z, we obtain the equation u′ = 0 of the transformed curve. To a given point (x′, y′, z′) not on the curve u’ = 0 there corresponds, not a single point, but the system of points (x, y, z) given by the equations x′ : y′ : z′ = X : Y : Z, viz., regarding x′, y′, z′ as constants (and to fix the ideas, assuming that the curves X = 0, Y = 0, Z = 0, have no common intersections), these are the points of intersection of the curves X : Y : Z, = x′ : y′ : z′, but no one of these points is situate on the curve u = 0. If, however, the point (x′, y′, z′) is situate on the curve u′ = 0, then one point of the system of points in question is situate on the curve u = 0, that is, to a given point of the curve u′ = 0 there corresponds a single point of the curve u = 0; and hence also this point must be given by a system of equations such as x : y : z = X′ : Y′ : Z′.
It is an old and easily proved theorem that, for a curve ofthe order m, the number δ + κ of nodes and cusps is at most = ½(m − 1)(m − 2); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz. ½(m − 1)(m − 2) − δ − κ, is the “Geschlecht” or “deficiency,” of the curve, say this is = D. When D = 0, the curve is said to be unicursal, when = 1, bicursal, and so on.
The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this rational or (1, 1) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]
A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D; and a curve of any orderdeficiency= or > 3 can be rationally transformed into a curve of the order D + 3, deficiency D.
Taking x′, y′, z′ as co-ordinates of a point of the transformed curve, and in its equation writing x′ : y′ : z′ = 1 : θ : φ we have φ a certain irrational function of θ, and the theorem is that the co-ordinates x, y, z of any point of the given curve can be expressed as proportional to rational and integral functions of θ, φ, that is, of θ and a certain irrational function of θ.In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, φ is a mere linear function of θ, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of θ; it is easy to see that for a given curve of the order m, these functions of θ must be of the same order m.If D = 1, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, φ can be expressed as the square root of a quartic function of θ; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a quartic function of θ.And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. But D = 3, we have no longer the like law, viz. φ is not expressible as the square root of an octic function of θ.
Taking x′, y′, z′ as co-ordinates of a point of the transformed curve, and in its equation writing x′ : y′ : z′ = 1 : θ : φ we have φ a certain irrational function of θ, and the theorem is that the co-ordinates x, y, z of any point of the given curve can be expressed as proportional to rational and integral functions of θ, φ, that is, of θ and a certain irrational function of θ.
In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, φ is a mere linear function of θ, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of θ; it is easy to see that for a given curve of the order m, these functions of θ must be of the same order m.
If D = 1, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, φ can be expressed as the square root of a quartic function of θ; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a quartic function of θ.
And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. But D = 3, we have no longer the like law, viz. φ is not expressible as the square root of an octic function of θ.
Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.
It is a form of the theorem for the case D = 1, that the co-ordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be √1 − θ²·1 − k²θ², and writing θ = snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.
It will be observed that the equations x′ : y′ : z′ = X : Y : Z before mentioned do not of themselves lead to the other system of equations x : y : z = X′ : Y′ : Z′, and thus that the theory does not in anywise establish a (1, 1) correspondence between the points (x, y, z) and (x′, y′, z′) of two planes or of the same plane; this is the correspondence of Cremona’s theory.
In this theory, given in the memoirs “Sulle trasformazioni geometriche delle figure piani,”Mem. di Bologna, t. ii. (1863) and t. v. (1865), we have a system of equations x′ : y′ : z′ = X : Y : Z whichdoeslead to a system x : y : z = X′ : Y′ : Z′, where, as before, X, Y, Z denote rational and integral functions, all of the same order, of the co-ordinates x, y, z, and X′, Y′, Z′ rational and integral functions, all of the same order, of the co-ordinates x′, y′, z′, and there is thus a (1, 1) correspondence given by these equations between the two points (x, y, z) and (x′, y′, z′). To explain this, observe that starting from the equations of x′ : y′ : z′ = X : Y : Z, to a given point (x, y, z) there corresponds one point (x′, y′, z′), but that if n be the order of the functions X, Y, Z, then to a given point x′, y′, z′ there would, if the curves X = 0, Y = 0, Z = 0 had no common intersections, correspond n² points (x, y, z). If, however, the functions are such that the curves X = 0, Y = 0, Z = 0 have k common intersections, then among the n² points are included these k points, which are fixed points independent of the point (x′, y′, z′); so that, disregarding these fixed points, the number of points (x, y, z) corresponding to the given point (x′, y′, z′) is = n² − k; and in particular if k = n² − 1, then we have one corresponding point; and hence the original system of equations x′ : y′ : z′ = X : Y : Z must lead to the equivalent system x : y : z = X′ : Y′ : Z′; and in this system by the like reasoning the functions must be such that the curves X′ = 0, Y′ = 0, Z′ = 0 have n′² − 1 common intersections. The most simple example is in the two systems of equations x′ : y′ : z′ = yz : zx : xy and x : y : z = y′z′ : z′x′ : x′y′; where yz = 0, zx = 0, xy = 0 are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system. In the case where X, Y, Z are of an order exceeding 2 the required number n² − 1 of common intersections can only occur by reason of common multiple points on the three curves; and assuming that the curves X = 0, Y = 0, Z = 0 have α1+ α2+ α3... + αn−1common intersections, where the α1points are ordinary points, the α2points are double points, the α3points are triple points, &c., on each curve, we have the conditionα1+ 4α2+ 9α3+ ... (n − 1)² αn−1= n² − 1;but to this must be joined the conditionα1+ 3α2+ 6α3... + ½n(n − 1) αn−1= ½n (n + 3) − 2(without which the transformation would be illusory); and the conclusion is that α1, α2, ... αn−1may be any numbers satisfying these two equations. It may be added that the two equations together giveα2+ 3α3... + ½ (n − 1) (n − 2) αn−1= ½ (n − 1) (n − 2),which expresses that the curves X = 0, Y = 0, Z = 0 are unicursal. The transformation may be applied to any curve u = 0, which is thus rationally transformed into a curve u′ = 0, by a rational transformation such as is considered in Riemann’s theory: hence the two curves have the same deficiency.
In this theory, given in the memoirs “Sulle trasformazioni geometriche delle figure piani,”Mem. di Bologna, t. ii. (1863) and t. v. (1865), we have a system of equations x′ : y′ : z′ = X : Y : Z whichdoeslead to a system x : y : z = X′ : Y′ : Z′, where, as before, X, Y, Z denote rational and integral functions, all of the same order, of the co-ordinates x, y, z, and X′, Y′, Z′ rational and integral functions, all of the same order, of the co-ordinates x′, y′, z′, and there is thus a (1, 1) correspondence given by these equations between the two points (x, y, z) and (x′, y′, z′). To explain this, observe that starting from the equations of x′ : y′ : z′ = X : Y : Z, to a given point (x, y, z) there corresponds one point (x′, y′, z′), but that if n be the order of the functions X, Y, Z, then to a given point x′, y′, z′ there would, if the curves X = 0, Y = 0, Z = 0 had no common intersections, correspond n² points (x, y, z). If, however, the functions are such that the curves X = 0, Y = 0, Z = 0 have k common intersections, then among the n² points are included these k points, which are fixed points independent of the point (x′, y′, z′); so that, disregarding these fixed points, the number of points (x, y, z) corresponding to the given point (x′, y′, z′) is = n² − k; and in particular if k = n² − 1, then we have one corresponding point; and hence the original system of equations x′ : y′ : z′ = X : Y : Z must lead to the equivalent system x : y : z = X′ : Y′ : Z′; and in this system by the like reasoning the functions must be such that the curves X′ = 0, Y′ = 0, Z′ = 0 have n′² − 1 common intersections. The most simple example is in the two systems of equations x′ : y′ : z′ = yz : zx : xy and x : y : z = y′z′ : z′x′ : x′y′; where yz = 0, zx = 0, xy = 0 are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system. In the case where X, Y, Z are of an order exceeding 2 the required number n² − 1 of common intersections can only occur by reason of common multiple points on the three curves; and assuming that the curves X = 0, Y = 0, Z = 0 have α1+ α2+ α3... + αn−1common intersections, where the α1points are ordinary points, the α2points are double points, the α3points are triple points, &c., on each curve, we have the condition
α1+ 4α2+ 9α3+ ... (n − 1)² αn−1= n² − 1;
but to this must be joined the condition
α1+ 3α2+ 6α3... + ½n(n − 1) αn−1= ½n (n + 3) − 2
(without which the transformation would be illusory); and the conclusion is that α1, α2, ... αn−1may be any numbers satisfying these two equations. It may be added that the two equations together give
α2+ 3α3... + ½ (n − 1) (n − 2) αn−1= ½ (n − 1) (n − 2),
which expresses that the curves X = 0, Y = 0, Z = 0 are unicursal. The transformation may be applied to any curve u = 0, which is thus rationally transformed into a curve u′ = 0, by a rational transformation such as is considered in Riemann’s theory: hence the two curves have the same deficiency.
Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the conics which satisfy given conditions, and to other geometrical questions, contained in theComptes rendus, t. lviii. (1864) et seq. The theorem of united points in regard to points in a right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), “Sur les courbes planes ou à double courbure dont les points peuvent se déterminer individuellement—application du principe de correspondance dans la théorie de ces courbes.”
The theorem is as follows: if in a unicursal curve two points have an (α, β) correspondence, then the number of united points (or points each corresponding to itself) is = α + β. In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (α, β) correspondence between the two points is given by an equation of the form (*θ, 1)α(φ, 1)β= 0 between their parameters θ and φ; at a united point φ = θ, and the value of θ is given by an equation of the order α + β. The extension to curves of any given deficiency D was made in the memoir of Cayley, “On the correspondence of two points on a curve,”—Proc. Lond. Math. Soc.t. i. (1866;Collected Works, vol. vi. p. 9),—viz. taking P, P′ as the corresponding points in an (α, α′) correspondence on a curve of deficiency D, and supposing that when P is given the corresponding points P′ are found as the intersections of the curve by a curve Θ containing the co-ordinates of P as parameters, and having with the given curve k intersections at the point P, then the number of united points is a = α + α′ + 2kD; and more generally, if the curve Θ intersect the given curve in a set of points P′ each p times, a set of points Q′ each g times, &c., in such manner that the points (P, P′) the points (P, Q′) &c., are pairs of points corresponding to each other according to distinct laws; then if (P, P′) are points having an (α, α′) correspondence with a number = a of united points, (P, Q′) points having a (β, β′) correspondence with a number = b of united points, and so on, the theorem is that we havep (a − α − α′) + q (b − β − β′) + ... = 2kD.
The theorem is as follows: if in a unicursal curve two points have an (α, β) correspondence, then the number of united points (or points each corresponding to itself) is = α + β. In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (α, β) correspondence between the two points is given by an equation of the form (*θ, 1)α(φ, 1)β= 0 between their parameters θ and φ; at a united point φ = θ, and the value of θ is given by an equation of the order α + β. The extension to curves of any given deficiency D was made in the memoir of Cayley, “On the correspondence of two points on a curve,”—Proc. Lond. Math. Soc.t. i. (1866;Collected Works, vol. vi. p. 9),—viz. taking P, P′ as the corresponding points in an (α, α′) correspondence on a curve of deficiency D, and supposing that when P is given the corresponding points P′ are found as the intersections of the curve by a curve Θ containing the co-ordinates of P as parameters, and having with the given curve k intersections at the point P, then the number of united points is a = α + α′ + 2kD; and more generally, if the curve Θ intersect the given curve in a set of points P′ each p times, a set of points Q′ each g times, &c., in such manner that the points (P, P′) the points (P, Q′) &c., are pairs of points corresponding to each other according to distinct laws; then if (P, P′) are points having an (α, α′) correspondence with a number = a of united points, (P, Q′) points having a (β, β′) correspondence with a number = b of united points, and so on, the theorem is that we have
p (a − α − α′) + q (b − β − β′) + ... = 2kD.
The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every geometrical problem. We can by means of it investigate the class of a curve, number of inflections, &c.—in fact, Plücker’s equations; but it is necessary to take account of special solutions: thus, in one of the most simple instances, in finding the class of a curve, the cusps present themselves as special solutions.
Imagine a curve of order m, deficiency D, and let the corresponding points P, P′ be such that the line joining them passes through a givenpoint O; this is an (m − 1, m − 1) correspondence, and the value of k is = 1, hence the number of united points is = 2m − 2 + 2D; the united points are the points of contact of the tangents from O and (as special solutions) the cusps, and we have thus the relation n + κ = 2m − 2 + 2D; or, writing D = ½(m − 1)(m − 2) − δ − κ, this is n = m(m − 1) − 2δ − 3κ, which is right.
Imagine a curve of order m, deficiency D, and let the corresponding points P, P′ be such that the line joining them passes through a givenpoint O; this is an (m − 1, m − 1) correspondence, and the value of k is = 1, hence the number of united points is = 2m − 2 + 2D; the united points are the points of contact of the tangents from O and (as special solutions) the cusps, and we have thus the relation n + κ = 2m − 2 + 2D; or, writing D = ½(m − 1)(m − 2) − δ − κ, this is n = m(m − 1) − 2δ − 3κ, which is right.
The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other geometrical problems.
There is one application of the theory of the (α, α′) correspondence between two planes which it is proper to notice.
Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u′; then, writing u = x + iy, u′ = x′ + iy′, the equation determines real values of (x, y), and of (x′, y′), corresponding to any given real values of (x′, y′) and (x, y) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x′, y′); for example in the imaginary circle u² + u′² = (a + bi)², the correspondence is given by the two equations x² − y² + x′² − y′² = a² − b², xy + x′y′ = ab. We have thus a means of geometrical representation for the portions, as well imaginary as real, of any real or imaginary curve. Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.
Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u′; then, writing u = x + iy, u′ = x′ + iy′, the equation determines real values of (x, y), and of (x′, y′), corresponding to any given real values of (x′, y′) and (x, y) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x′, y′); for example in the imaginary circle u² + u′² = (a + bi)², the correspondence is given by the two equations x² − y² + x′² − y′² = a² − b², xy + x′y′ = ab. We have thus a means of geometrical representation for the portions, as well imaginary as real, of any real or imaginary curve. Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.
16.Systems of Curves satisfying Conditions.—The researches of Chasles (Comptes Rendus, t. lviii., 1864, et seq.) refer to the conics which satisfy given conditions. There is an earlier paper by J. P. E. Fauque de Jonquières, “Théorèmes généraux concernant les courbes géométriques planes d’un ordre quelconque,”Liouv.t. vi. (1861), which establishes the notion of a system of curves (of any order) of the index N, viz. considering the curves of the order n which satisfy ½n(n + 3) − 1 conditions, then the index N is the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers (February 1864), considering the conics which satisfy four conditions, establishes the notion of the two characteristics (μ, ν) of such a system of conics, viz. μ is the number of the conics which pass through a given arbitrary point, and ν is the number of the conics which touch a given arbitrary line. And he gives the theorem, a system of conics satisfying four conditions, and having the characteristics (μ, ν) contains 2ν − μ line-pairs (that is, conics, each of them a pair of lines), and 2μ − ν point-pairs (that is, conics, each of them a pair of points,—coniques infiniment aplaties), which is a fundamental one in the theory. The characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate conics in the system, and how often each is to be counted. It was thus that Zeuthen (in the paperNyt Bydrag, “Contribution to the Theory of Systems of Conics which satisfy four Conditions” (Copenhagen, 1865), translated with an addition in theNouvelles Annales) solved the question of finding the characteristics of the systems of conics which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conics which satisfy five conditions of contact with a given curve or curves (Cayley,Comptes Rendus, t. lxiii., 1866;Collected Works, vol. v. p. 542), and “On the Curves which satisfy given Conditions” (Phil. Trans.t. clviii., 1868;Collected Works, vol. vi. p. 191).
It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of conics satisfying 4 conditions, yet what is really determined is in every case the number of the conics which satisfy 5 conditions; the characteristics of the system (4p) of the conics which pass through 4p points are (5p), (4p, 1l), the number of the conics which pass through 5 points, and which pass through 4 points and touch 1 line: and so in other cases. Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the number of the cubics (9p), (8p, 1l), &c., which pass through 9 points, pass through 8 points and touch 1 line, &c.; but it is in the investigation convenient to seek for the characteristics of the systems of cubics (8p), &c., which satisfy 8 instead of 9 conditions.
The elementary problems in regard to cubics are solved very completely by S. Maillard in hisThèse,Recherche des caractéristiques des systèmes élémentaires des courbes planes du troisième ordre(Paris, 1871). Thus, considering the several cases of a cubic
he determines in every case the characteristics (μ, ν) of the corresponding systems of cubics (4p), (3p, 1l), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir “Almindelige Egenskaber, &c.,”Danish Academy, t. x. (1873), considers the problem in reference to curves of any order, and applies his results to cubic and quartic curves.
The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics (μ, ν) of a system of curves by consideration of the special or degenerate forms of the curves included in the system. The quantities which have to be considered are very numerous. Zeuthen in the case of curves of any given order establishes between the characteristics μ, ν, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23 independent equations), involving (besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves. It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.
17.Degeneration of Curves.—The remarks which follow have reference to the analytical theory of the degenerate curves which present themselves in the foregoing problem of the curves which satisfy given conditions.
A curve represented by an equation in point-co-ordinates may break up: thus if P1, P2, ... be rational and integral functions of the co-ordinates (x, y, z) of the orders m1, m2... respectively, we have the curve P1α1P2α2... = 0, of the order m, = α1m1+ α2m2+ ..., composed of the curve P1= 0 taken α1times, the curve P2= 0 taken α2times, &c.Instead of the equation P1α1P2α2... = 0, we may start with an equation u = 0, where u is a function of the order m containing a parameter θ, and for a particular value say θ = 0, of the parameter reducing itself to P1α1P2α2.... Supposing θ indefinitely small, we have what may be called the penultimate curve, and when θ = 0 the ultimate curve. Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called “summits” on the component curves P1= 0, P2= 0 respectively; a summit Σ is a point such that, drawing from an arbitrary point O the tangents to the penultimate curve, we have OΣ as the limit of one of these tangents. The ultimate curve together with its summits may be regarded as a degenerate form of the curve u = 0. Observe that the positions of the summits depend on the penultimate curve u = 0, viz. on the values of the coefficients in the terms multiplied by θ, θ², ...; they are thus in some measure arbitrary points as regards the ultimate curve P1α1P2α2... = 0.It may be added that we have summits only on the component curves P1= 0, of a multiplicity α1> 1; the number of summits on such a curve is in general = (α1² − α1)m1². Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = m² − m, = (α1m1+ α2m2+ ...)² − (α1m1+ α2m2+ ...). Taking P1= 0 to have δ1nodes and κ1cusps, and therefore its class n1to be = m1² − m1− 2δ1− 3κ1, &c., the expression for the number of tangents to the penultimate curve is= (α1² − α1) m1² + (α2² − α2) m2² + ... + 2α1α2m1m2++ α1(n1+ 2δ1+ 3κ1) + α2(n2+ 2δ2+ 3κ2) + ...where a term 2α1α2m1m2indicates tangents which are in the limit the lines drawn to the intersections of the curves P1= 0, P2= 0 each line 2α1α2times; a term α1(n1+ 2δ1+ 3κ1) tangents which are in thelimit the proper tangents to P1= 0 each α1times, the lines to its nodes each 2α1times, 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 be1Λδ1... 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.
A curve represented by an equation in point-co-ordinates may break up: thus if P1, P2, ... be rational and integral functions of the co-ordinates (x, y, z) of the orders m1, m2... respectively, we have the curve P1α1P2α2... = 0, of the order m, = α1m1+ α2m2+ ..., composed of the curve P1= 0 taken α1times, the curve P2= 0 taken α2times, &c.
Instead of the equation P1α1P2α2... = 0, we may start with an equation u = 0, where u is a function of the order m containing a parameter θ, and for a particular value say θ = 0, of the parameter reducing itself to P1α1P2α2.... Supposing θ indefinitely small, we have what may be called the penultimate curve, and when θ = 0 the ultimate curve. Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called “summits” on the component curves P1= 0, P2= 0 respectively; a summit Σ is a point such that, drawing from an arbitrary point O the tangents to the penultimate curve, we have OΣ as the limit of one of these tangents. The ultimate curve together with its summits may be regarded as a degenerate form of the curve u = 0. Observe that the positions of the summits depend on the penultimate curve u = 0, viz. on the values of the coefficients in the terms multiplied by θ, θ², ...; they are thus in some measure arbitrary points as regards the ultimate curve P1α1P2α2... = 0.
It may be added that we have summits only on the component curves P1= 0, of a multiplicity α1> 1; the number of summits on such a curve is in general = (α1² − α1)m1². Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = m² − m, = (α1m1+ α2m2+ ...)² − (α1m1+ α2m2+ ...). Taking P1= 0 to have δ1nodes and κ1cusps, and therefore its class n1to be = m1² − m1− 2δ1− 3κ1, &c., the expression for the number of tangents to the penultimate curve is
= (α1² − α1) m1² + (α2² − α2) m2² + ... + 2α1α2m1m2++ α1(n1+ 2δ1+ 3κ1) + α2(n2+ 2δ2+ 3κ2) + ...
where a term 2α1α2m1m2indicates tangents which are in the limit the lines drawn to the intersections of the curves P1= 0, P2= 0 each line 2α1α2times; a term α1(n1+ 2δ1+ 3κ1) tangents which are in thelimit the proper tangents to P1= 0 each α1times, the lines to its nodes each 2α1times, 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 be1Λδ1... 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 theEncy. 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).
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 theEncy. 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.)
1In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane.
1In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane.
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), andMemorials of Thomas Cranmer(2 vols., Oxford, 1840); John D’Alton.Memoirs of the Archbishops of Dublin(Dublin, 1838).
See John Strype,Life and Acts of Archbishop Parker(3 vols., Oxford, 1824), andMemorials 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 anumeralnotation. Curwen brought out hisGrammar of Vocal Musicin 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 theTonic 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, publishedMemorials of J. Curwenin 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-CroatianKorčulaorKarkar), 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 andprincipal port, is a fortified town on the east coast, and occupies a rocky foreland almost surrounded by the sea. Besides the interesting church (formerly a cathedral), dating from the 12th or 13th century, theloggiaor 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, together with the massive walls and towers, erected in 1420, and the 15th-century Franciscan monastery, with its beautiful Venetian Gothic cloister. The main resources of the islanders are boat-building (for which they are celebrated throughout the Adriatic), fishing and sea-faring, the cultivation of the vine, corn and olives, and breeding of mules. Pop. (1900) of island, 17,377; of capital (town and commune), 6486. Prehistoric grave-mounds are common on the hills of the interior, and in later times Curzola may have been a Phoenician settlement. Its early history is very obscure, but it was certainly colonized by Greeks from Cnidus. The present name is a corruption of the Gr.Κέρκυρα Μέλαινα, or Lat.Corcyra Nigra, “Black Corcyra”; and is perhaps due to the dark pines which still partly cover the island. In 998 Curzola first came under Venetian suzerainty. During the 12th century it was ruled by Hungary and Genoa in turn, and enjoyed a brief period of independence; but after 1255 its hereditary counts again submitted to Venice. The Roman Catholic see of Curzola, created in 1301, was only suppressed in 1806. Curzola surrendered to the Hungarians in 1358, was purchased by Ragusa (1413-1417), and finally declared itself subject to Venice in 1420. In 1571 it defended itself so gallantly against the Turks that it obtained the designationfidelissima. From 1776 to 1797 it succeeded Lesina as the main Venetian arsenal in this region. During the Napoleonic wars it was ruled successively by Russians, French and British, ultimately passing to Austria in 1815.
CURZON OF KEDLESTON, GEORGE NATHANIEL,1st Baron(1859- ), English statesman, eldest son of the 4th baron Scarsdale, rector of Kedleston, Derbyshire, was born on the 11th of January 1859, and was educated at Eton and Balliol College, Oxford. At Oxford he was president of the Union, and after a brilliant university career was elected a fellow of All Souls College in 1883. He became assistant private secretary to Lord Salisbury in 1885, and in 1886 entered parliament as member for the Southport division of S.W. Lancashire. He was appointed under secretary for India in 1891-1892 and for foreign affairs in 1895-1898. In the meantime he had travelled in Central Asia, Persia, Afghanistan, the Pamirs, Siam, Indo-China and Korea, and published several books describing central and eastern Asia and the political problems connected with those regions. In 1895 he married Mary Victoria Leiter (d. 1906), daughter of a Chicago millionaire. In January 1899 he was appointed governor-general of India, where his extensive knowledge of Asiatic affairs showed itself in the inception of a strong foreign policy, while he took in hand the reform of every department of Indian administration. He was created an Irish peer on his appointment, the creation taking this form, it was understood, in order that he might remain free during his father’s lifetime to re-enter the House of Commons. Reaching India shortly after the suppression of the frontier risings of 1897-98, he paid special attention to the independent tribes of the north-west frontier, inaugurated a new province called the North West Frontier Province, and carried out a policy of conciliation mingled with firmness of control. The only trouble on this frontier during the period of his administration was the Mahsud Waziri campaign of 1901. Being mistrustful of Russian methods he exerted himself to encourage British trade in Persia, paying a visit to the Persian Gulf in 1903; while on the north-east frontier he anticipated a possible Russian advance by the Tibet Mission of 1903, which rendered necessary the employment of military force for the protection of the British envoys. The mission, which had the ostensible support of China as suzerain of Tibet, penetrated to Lhasa, where a treaty was signed in September 1904. In pursuance of his reforming policy Lord Curzon appointed a number of commissions to inquire into Indian education, irrigation, police and other branches of administration, on whose reports legislation was based during his second term of office as viceroy. With a view to improving British relations with the native chiefs and raising the character of their rule, he established the Imperial Cadet corps, settled the question of Berar with the nizam of Hyderabad, reduced the salt tax, and gave relief to the smaller income-tax payers. Lord Curzon exhibited much interest in the art and antiquities of India, and during his viceroyalty took steps for the preservation and restoration of many important monuments and buildings of historic interest. In January 1903 he presided at the durbar held at Delhi in honour of the coronation of King Edward VII. It was attended by all the leading native princes and by large numbers of visitors from Europe and America; and the magnificence of the spectacle surpassed anything that had previously been witnessed even in the gorgeous ceremonial of the East. On the expiration of his first term of office, Lord Curzon was reappointed governor-general. His second term of office was marked by the passing of several acts founded on the recommendations of his previous commissions, and by the partition of Bengal (1905), which roused bitter opposition amongst the natives of that province. A difference of opinion with the commander-in-chief, Lord Kitchener, regarding the position of the military member of council in India, led to a controversy in which Lord Curzon failed to obtain support from the home government. He resigned (1904) and returned to England. In 1904 he was appointed lord warden of the Cinque Ports; in the same year he was given the honorary degree of D.C.L. by Oxford University, and in 1908 he was elected chancellor of the university. In the latter year he was elected a representative peer for Ireland, and thus relinquished any idea of returning to the House of Commons. In 1909-1910 he took an active part in defending the House of Lords against the Liberals. Lord Curzon’s publications includeRussia in Central Asia(1889);Persia and the Persian Question(1892);Problems of the Far East(1894; new ed., 1896).
See Caldwell Lipsett,Lord Curzon in India, 1898-1903 (1906); and C. J. O’Donnell,The Failure of Lord Curzon(1903).
See Caldwell Lipsett,Lord Curzon in India, 1898-1903 (1906); and C. J. O’Donnell,The Failure of Lord Curzon(1903).
CUSANUS, NICOLAUS(Nicholas of Cusa) (1401-1464), cardinal, theologian and scholar, 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, Kues or Cusa, on the Moselle, in the archbishopric of Trier (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 university of Padua, where he took his doctor’s degree in law in his twenty-third year. Failing in his first case he abandoned the legal profession, and resolved to take holy orders. After filling several subordinate offices he became archdeacon of Liége. He was a member of the council of Basel, and dedicated to the assembled fathers a work entitledDe concordantia Catholica, in which he maintained the superiority of councils over popes, and assailed the genuineness of the False Decretals and the Donation of Constantine. A few years later, however, he had reversed his position, and zealously defended the supremacy of the pope. He was entrusted with various missions in the interests of Catholic unity, the most important being to Constantinople, to endeavour to bring about a union of the Eastern and Western churches. From 1440 to 1447 he was in Germany, acting as papal legate at the diets of 1441, 1442, 1445 and 1446. In 1448, in recognition of his services, Nicholas V. raised him to the cardinalate; and in 1450 he was appointed bishop of Brixen against the wish of Sigismund, archduke of Austria, 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 ecclesiastical abuses and bring back the monastic life to the original rule of poverty, chastity and obedience—a mission which he 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 11th of 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 hisCrebratio Alcorani, a treatise against Mahommedanism, in support of the expedition against the Turks proposed at that assembly. Some time before his 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.
Although one of the great leaders in the reform movement of the 15th century, Nicholas of Cusa’s interest for later times lies in his philosophical much more than in his political or ecclesiastical activity. As in religion he is entitled to be called one of the “Reformers 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 byDe conjecturis libriduo 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. God is the absolute maximum and also the absolute minimum, who can be neither greater nor less than He is, and who comprehends all that is or that can be (“deum esse omnia, ut non possit esse aliud quam est”). Cusanus thus laid himself open to the charge of pantheism, which did not fail to be brought against him in his own day. His chief philosophical doctrine was taken up and developed more than a hundred years later by Giordano Bruno, who calls him the divine Cusanus. In mathematical and physical science Cusanus was much in advance of his age. In a tract,Reparatio Calendarii, presented to the council of Basel, he proposed the reform of the calendar after a method resembling that adopted by Gregory. In hisDe Quadratura Circulihe professed to have solved the problem; and in hisConjectura de novissimis diebushe prophesied that the world would come to an end in 1734. Most noteworthy, however, in this connexion is the fact that he anticipated Copernicus by maintaining the theory of the rotation of the earth.
The works of Cusanus were published in a complete form by Henri Petrie (1 vol. fol., Basel, 1565). See F. A. Scharpff’sDer Kardinal und Bischof Nikolaus von Cusa als Reformator in Kirche, Reich, und Philos. des 15. Jahrhund. (Tübingen, 1871); J. M. Düx,Der deutsche Kard. Nicolaus von Cusa und die Kirche seiner Zeit(Regensburg, 1848); R. Falckenberg,Grundzüge d. Philos. d. Nikolaus Cusanus(Breslau, 1880) andAufgabe und Wesen d. Erkenntniss bei Nikolaus von Kues(Breslau, 1880); T. Stumpf,Die politischen Ideen des Nikolaus von Cues(Cologne, 1865); M. Glossner,Nikolaus von Cusa und Marius Nizolius als Vorläufer der neueren Philosophie(Münster, 1891); F. Fiorentino,Il Risorgimento filosofico nel quattro cento(Naples, 1885); Axel Herrlin,Studier i Nicolaus af Cues’ Filosofi(Lund, 1892); H. Höffding,Hist. of Mod. Phil.(Eng. trans., 1900), bk. i. chap. x.; F. J. Clemens,Giordano Bruno und Nikolaus Cusanus(Bonn, 1847); R. Zimmermann,Der Card. Nikolaus Cusanus als Vorläufer Leibnitzens(Vienna, 1852); J. Übinger, Philosophie des Nikolaus Cusanus (Würzburg, 1881); art. by R. Schmid in Herzog-Hauck,Realencyk.s.v. “Cusanus”; see alsoMysticism.
The works of Cusanus were published in a complete form by Henri Petrie (1 vol. fol., Basel, 1565). See F. A. Scharpff’sDer Kardinal und Bischof Nikolaus von Cusa als Reformator in Kirche, Reich, und Philos. des 15. Jahrhund. (Tübingen, 1871); J. M. Düx,Der deutsche Kard. Nicolaus von Cusa und die Kirche seiner Zeit(Regensburg, 1848); R. Falckenberg,Grundzüge d. Philos. d. Nikolaus Cusanus(Breslau, 1880) andAufgabe und Wesen d. Erkenntniss bei Nikolaus von Kues(Breslau, 1880); T. Stumpf,Die politischen Ideen des Nikolaus von Cues(Cologne, 1865); M. Glossner,Nikolaus von Cusa und Marius Nizolius als Vorläufer der neueren Philosophie(Münster, 1891); F. Fiorentino,Il Risorgimento filosofico nel quattro cento(Naples, 1885); Axel Herrlin,Studier i Nicolaus af Cues’ Filosofi(Lund, 1892); H. Höffding,Hist. of Mod. Phil.(Eng. trans., 1900), bk. i. chap. x.; F. J. Clemens,Giordano Bruno und Nikolaus Cusanus(Bonn, 1847); R. Zimmermann,Der Card. Nikolaus Cusanus als Vorläufer Leibnitzens(Vienna, 1852); J. Übinger, Philosophie des Nikolaus Cusanus (Würzburg, 1881); art. by R. Schmid in Herzog-Hauck,Realencyk.s.v. “Cusanus”; see alsoMysticism.
CUSH,the eldest son of Ham, in the Bible, from whom seems to have been derived the name of the “Land of Cush,” commonly rendered “Ethiopia” by the Septuagint and by the Vulgate. The locality of the land of Cush has long been a much-vexed question. Bochart maintained that it was exclusively in Arabia; Schulthess and Gesenius held that it should be sought for nowhere but in Africa (seeEthiopia). Others again, like Michaelis and Rosenmüller, have supposed that the name Cush was applied to tracts of country both in Arabia and in Africa, but the defective condition of the ancient knowledge of countries and peoples, as also the probability of early migrations of “Cushite” tribes (carrying with them their name), will account for the main facts. The existence of an African Cush cannot reasonably be questioned, though the term is employed in the Old Testament with some latitude. The African Cush covers Upper Egypt, and extends southwards from the first cataract (Syene, Ezek. xxix. 10). That the term was also applied to parts of Arabia is evident from Gen. x. 7, where Cush is the “father” of certain tribal and ethnical designations, all of which point very clearly to Arabia, with the very doubtful exception of Seba, which Josephus (Ant.ii. 10. 2) identifies with Meroë.1Even in the 5th centuryA.D.the Himyarites, in the south of Arabia, were styled by Syrian writers Cushaeans and Ethiopians. Moreover, the Babylonian inscriptions mention the Kashshi, an Elamite race, whose name has been equated with the classicalΚοσσαῖοι,Κίσσιοι, and it has been held that this affords a more appropriate explanation of Cush (perhaps rather Kash), the ancestor of (the Babylonian) Nimrod in Gen. x. 8. Although decisive evidence is lacking, it seems extremely probable that several references to Cush in the Old Testament cannot refer to Ethiopia, despite the likelihood that considerable confusion existed in the minds of early writers. The Cushite invasion in 2 Chron. xiv. (seeAsa) is intelligible if the historical foundation for the story be a raid by Arabians, but in xvi. 8 the inclusion of Libyans shows that the enemy was subsequently supposed to be African. In several passages the interpretation is bound up with that of Mizraim (q.v.), and depends in general upon the question whether Ethiopia at a given time enjoyed the prominence given to it.