VOLUME VII SLICE VIII Cube to Daguerre, Louis (continued 6)

It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable theorem that the five divergent parabolas by their shadows generate and exhibit all the cubic curves.

The five divergent parabolas are curves each of them symmetrical with regard to an axis. There are two non-singular kinds, the one with, the other without, an oval, but each of them has an infinite (as Newton describes it) campaniform branch; this cuts the axis at right angles, being at first concave, but ultimately convex, towards the axis, the two legs continually tending to become at right angles to the axis. The oval may unite itself with the infinite branch, or it may dwindle into a point, and we have the crunodal and the acnodal forms respectively; or if simultaneously the oval dwindles into a point and unites itself to the infinite branch, we have the cuspidal form. (See Parabola.) Drawing a line to cut any one of these curves and projecting the line to infinity, it would not be difficult to show how the line should be drawn in order to obtain a curve of any given species. We have herein a better principle of classification; considering cubic curves, in the first instance, according to singularities, the curves are non-singular, nodal (viz. crunodal or acnodal), or cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex. There is thus a complete division into the five kinds, the complex, simplex, crunodal, acnodal and cuspidal. Each singular kind presents itself as a limit separating two kinds of inferior singularity; the cuspidal separates the crunodal and the acnodal, and these last separate from each other the complex and the simplex.

The whole question is discussed very fully and ably by A. F. Möbius in the memoir “Ueber die Grundformen der Linien dritter Ordnung” (Abh. der K. Sachs. Ges. zu Leipzig, t. i., 1852). The author considers not only plane curves, but also cones, or, what is almost the same thing, the spherical curves which are their sections by a concentric sphere. Stated in regard to the cone, we have there the fundamental theorem that there are two different kinds of sheets; viz., the single sheet, not separated into two parts by the vertex (an instance is afforded by the plane considered as a cone of the first order generated by the motion of a line about a point), and the double or twin-pair sheet, separated into two parts by the vertex (as in the cone of the second order). And it then appears that there are two kinds of non-singular cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones. It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base.

In general a cone consists of one or more single or twin-pair sheets, and if we consider the section of the cone by a plane, the curve consists of one or more complete branches, or say circuits, each of them the section of one sheet of the cone; thus, a cone of the second order is one twin-pair sheet, and any section of it is one circuit composed, it may be, of two branches. But although we thus arrive by projection at the notion of a circuit, it is not necessary to go out of the plane, and we may (with Zeuthen, using the shorter term circuit for his complete branch) define a circuit as any portion (of a curve) capable of description by the continuous motion of a point, it being understood that a passage through infinity is permitted. And we then say that a curve consists of one or more circuits; thus the right line, or curve of the first order, consists of one circuit; a curve of the second order consists of one circuit; a cubic curve consists of one circuit or else of two circuits.

A circuit is met by any right line always in an even number, or always in an odd number, of points, and it is said to be an even circuit or an odd circuit accordingly; the right line is an odd circuit, the conic an even circuit. And we have then the theorem, two odd circuits intersect in an odd number of points; an odd and an even circuit, or two even circuits, in an even number of points. An even circuit not cutting itself divides the plane into two parts, the one called the internal part, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.

We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none,—four if the point is on the odd circuit, none if it is on the even circuit. It at once appears from inspection of the figure of a non-singular cubic curve, which is the odd and which the even circuit. The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode must be regarded as an even circuit.

12. Quartic Curves.—The analogous question of the classification of quartics (in particular non-singular quartics and nodal quartics) is considered in Zeuthen’s memoir “Sur les différentes formes des courbes planes du quatrième ordre” (Math. Ann. t. vii., 1874). A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections. A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is = 4; it follows that the quartic has at most 8 real inflections. The forms of the non-singular quartics are very numerous, but it is not necessary to go further into the question.

We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present themselves always conjointly; thus a circle is a conic passing through the two circular points, and is thereby distinguished from other conics. Similarly a cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, δ = 2, κ = 0); it has not in general, but it may have, a third node or a cusp. Or again, we may have a quartic curve having a cusp at each of the circular points: such a curve is a “Cartesian,” it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m = 4, δ = 0, κ = 2), having a cusp at each of the circular points. The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to their focal properties.

13. Foci.—The points called foci presented themselves in the theory of the conic, and were well known to the Greek geometers, but the general notion of a focus was first established by Plücker (in the memoir “Über solche Puncte die bei Curven einer höheren Ordnung den Brennpuncten der Kegelschnitte entsprechen” (Crelle, t. x., 1833). We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to the two circular points respectively) is a focus. There will be from each circular point λ tangents (λ, a number depending on the class of the curve and its relation to the line infinity and the circular points, = 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular quartic, &c.); the λ tangents from the one circular point and those from the other circular point intersect in λ real foci (viz. each of these is the only real point on each of the tangents through it), and in λ² − λ imaginary foci; each pair of real foci determines a pair of imaginary foci (the so-called antipoints of the two real foci), and the ½λ(λ − 1) pairs of real foci thus determine the λ² − λ imaginary foci. There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.

14. Distance and Angle. Curves described mechanically.—The notions of distance and of lines at right angles are connected with the circular points; and almost every construction of a curve by means of lines of a determinate length, or at right angles to each other, and (as such) mechanical constructions by means of linkwork, give rise to curves passing the same definite number of times through the two circular points respectively, or say to circular curves, and in which the fixed centres of the construction present themselves as ordinary, or as singular, foci. Thus the general curve of three bar-motion (or locus of the vertex of a triangle, the other two vertices whereof move on fixed circles) is a tricircular sextic, having besides three nodes (m = 6, δ = 3 + 3 + 3 = 9), and having the centres of the fixed circles each for a singular focus; there is a third singular focus, and we have thus the remarkable theorem (due to S. Roberts) of the triple generation of the curve by means of the three several pairs of singular foci.

Again, the normal, qua line at right angles to the tangent, is connected with the circular points, and these accordingly present themselves in the before-mentioned theories of evolutes and parallel curves.

15. Theories of Correspondence.—We have several recent theories which depend on the notion of correspondence: two points whether in the same plane or in different planes, or on the same curve or in different curves, may determine each other in such wise that to any given position of the first point there correspond α′ positions of the second point, and to any given position of the second point a positions of the first point; the two points have then an (α, α) correspondence; and if α, α are each = 1, then the two points have α (1, 1) or rational correspondence. Connecting with each theory the author’s name, the theories in question are G. F. B. Riemann, the rational transformation of a plane curve; Luigi Cremona, the rational transformation of a plane; and Chasles, correspondence of points on the same curve, and united points. The theory first referred to, with the resulting notion of “Geschlecht,” or deficiency, is more than the other two an essential part of the theory of curves, but they will all be considered.

Riemann’s results are contained in the memoirs on “Abelian Integrals,” &c. (Crelle, t. liv., 1857), and we have next R. F. A. Clebsch, “Über die Singularitäten algebraischer Curven” (Crelle, t. lxv., 1865), and Cayley, “On the Transformation of Plane Curves” (Proc. Lond. Math. Soc. t. i., 1865; Collected Works, vol. vi. p. 1). The fundamental notion of the rational transformation is as follows:—

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 of the 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 order deficiency = 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.

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 which does lead 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−1 common intersections, where the α1 points are ordinary points, the α2 points are double points, the α3 points 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−1 may 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 the Comptes 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 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 given point 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.

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 paper Nyt Bydrag, “Contribution to the Theory of Systems of Conics which satisfy four Conditions” (Copenhagen, 1865), translated with an addition in the Nouvelles 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 his Thèse, Recherche des caractéristiques des systèmes élémentaires des courbes planes du troisième ordre (Paris, 1871). Thus, considering the several cases of a cubic

No. of consts.

1. With a given cusp 5

2. ” cusp on a given line 6

3. ” cusp 7

4. ” a given node 6

5. ” node on given line 7

6. ” node 8

7. non-singular 9

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 α1 times, the curve P2 = 0 taken α2 times, &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 δ1 nodes and κ1 cusps, and therefore its class n1 to 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 2α1α2m1m2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P1 = 0, P2 = 0 each line 2α1α2 times; a term α1(n1 + 2δ1 + 3κ1) tangents which are in the 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.)

1 In 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), 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 principal 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, 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, 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 designation fidelissima. 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 include Russia 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).

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 entitled De 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 his Crebratio 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 by De conjecturis 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. 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 his De Quadratura Circuli he professed to have solved the problem; and in his Conjectura de novissimis diebus he 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’s Der 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) and Aufgabe 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 also Mysticism.

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 (see Ethiopia). 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.1 Even in the 5th century A.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 classical Κοσσαῖοι, 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. (see Asa) 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.

On Num. xii. I see Jethro; and consult H. Winckler, Keil. u. das alte Test., 3rd ed., p. 144 sq., and Im Kampfe um den alten Orient, ii. pp. 36 seq., and the literature cited under Mizraim.

(S. A. C.)

1 For Seba, see Sabaeans, and cf. generally the commentaries on Gen. x. 7. In Hab. iii. 7 Cushan (obviously a related form) is parallel to Midian.

CUSHING, CALEB (1800-1879), American political leader and lawyer, was born in Salisbury, Massachusetts, on the 17th of January 1800. He graduated at Harvard in 1817, was tutor in mathematics there in 1820-1821, was admitted to practice in the court of common pleas in December 1821, and began the practice of law in Newburyport, Mass., in 1824. After serving, as a Democratic-Republican, in the state house of representatives in 1825, in the state senate in 1826, and in the house again in 1828, he spent two years, from 1829 to 1831, in Europe, again served in the state house of representatives in 1833 and 1834, and in the latter year was elected by the Whigs a representative in Congress. He served in this body from 1835 until 1843, and here the marked inconsistency which characterized his public life became manifest; for when John Tyler had become president, had been “read out” of the Whig party, and had vetoed Whig measures (including a tariff bill), for which Cushing had voted, Cushing first defended the vetoes and then voted again for the bills. In 1843 President Tyler nominated him for secretary of the treasury, but the senate refused to confirm him for this office. He was, however, appointed later in the same year commissioner of the United States to China, holding this position until 1845, and in 1844 negotiating the first treaty between China and the United States. In 1847, while again a representative in the state legislature, he introduced a bill appropriating money for the equipment of a regiment to serve in the Mexican War; although the bill was defeated, he raised the necessary funds privately, and served in Mexico first as colonel and afterwards as brigadier-general of volunteers. In 1847 and again in 1848 the Democrats nominated him for governor of Massachusetts, but on each occasion he was defeated at the polls. He was again a representative in the state legislature in 1851, became an associate justice of the supreme court of Massachusetts in 1852, and during the administration (1853-1857) of President Pierce, was attorney-general of the United States. In 1858, 1859, 1862 and 1863 he again served in the state house of representatives. In 1860 he presided over the National Democratic Convention which met first at Charleston and later at Baltimore, until he joined those who seceded from the regular convention; he then presided also over the convention of the seceding delegates, who nominated John C. Breckinridge for the presidency. During the Civil War, however, he supported the National Administration. At the Geneva conference for the settlement of the “Alabama” claims in 1871-1872 he was one of the counsel for the United States. In 1873 President Grant nominated him for chief justice of the United States, but in spite of his great learning and eminence at the bar, his ante-war record and the feeling of distrust experienced by many members of the senate on account of his inconsistency, aroused such vigorous opposition that his nomination was soon withdrawn. From 1874 to 1877 Cushing was United States minister to Spain. He died at Newburyport, Mass., on the 2nd of January 1879. He published History and Present State of the Town of Newburyport, Mass. (1826); Review of the late Revolution in France (1833); Reminiscences of Spain (1833); Oration on the Growth and Territorial Progress of the United States (1839); Life and Public Services of William H. Harrison (1840); and The Treaty of Washington (1873).

CUSHING, WILLIAM BARKER (1842-1874), American naval officer, was born in Delafield, Wisconsin, on the 4th of November 1842. He entered the Naval Academy from New York in 1857, but resigned in March 1861. When, however, the Civil War began, he volunteered into the navy, was rated acting master’s mate, and became a midshipman in October 1861, and a lieutenant in July 1862, serving in the North Atlantic blockading squadron. The work of blockade, and of harassing the Confederates on the coast and the rivers of the Atlantic seaboard, called for much service in boats, and entailed a great deal of exposure. Cushing was distinguished by his readiness to volunteer, his indefatigability, and by his good fortune, the reward of vigilance and intelligence. The feat by which he will be remembered was the destruction of the Confederate ironclad “Albemarle” in the Roanoke river on the 27th of October in 1864. The vessel had done much damage to the Federal naval forces, and her destruction was greatly desired. She was at anchor surrounded by baulks of timber, and a cordon of boats had been stationed to row guard against an expected Federal attack. Lieutenant Cushing undertook the attack on her with a steam launch carrying a spar-torpedo and towing an armed cutter. He eluded the Confederate lookout and reached the “Albemarle” unseen. When close to he was detected, but he had time to drive the steam launch over the baulks and to explode the torpedo against the “Albemarle” with such success that a hole was made in her and she sank. Cushing’s own launch was destroyed. He and the few men with him were compelled to take to the water; one was killed, another was drowned, Cushing and one other escaped, and the rest were captured. Cushing himself swam to the swamps on the river bank, and after wading among them for hours reached a Federal picket boat. For destroying the “Albemarle” he was thanked by Congress and was promoted to be lieutenant-commander. On the 15th of January 1865 he took a conspicuous part in the land attack on the sea-front wall of Fort Fisher. After the war he commanded the “Lancaster” (1866-1867) and the “Maumee” (1868-1869) in the Asiatic Squadron. In 1872 he was promoted commander at what was an exceptionally early age, but he died on the 17th of December 1874 of brain fever. He had suffered extreme pain for years before his death, and in fact broke down altogether under disease contracted in the discharge of his duty.

CUSHION (from O. Fr. coisson, coussin; according to the New English Dict., from Lat. coxa, a hip; others say from Lat. culcita, a quilt), a soft bag of some ornamental material, stuffed with wool, hair, feathers, or even paper torn into fragments. It may be used for sitting or kneeling upon, or to soften the hardness or angularity of a chair or couch. It is a very ancient article of furniture, the inventories of the contents of palaces and great houses in the early middle ages constantly making mention of it. It was then often of great size, covered with leather, and firm enough to serve as a seat, but the steady tendency of all furniture has been to grow smaller. It was, indeed, used as a seat, at all events in France and Spain, at a very much later period, and in Saint-Simon’s time we find that at the Spanish court it was still regarded as a peculiarly honourable substitute for a chair. In France the right to kneel upon a cushion in church behind the king was jealously guarded and strictly regulated, as we may learn again from Saint-Simon. This type of cushion was called a carreau or square. When seats were rude and hard the cushion may have been a necessity; it is now one of the minor luxuries of life.

The term “cushion” is given in architecture to the sides of the Ionic capital. It is also applied to an early and simple form of the Romanesque capitals of Germany and England, which consist of cubical masses, square at the top and rounded off at the four corners, so as to reduce the lower diameter to a circle of the same size as the shaft.

CUSHMAN, CHARLOTTE SAUNDERS (1816-1876), American actress, was born in Boston, Massachusetts, on the 23rd of July 1816. Her father, a West India merchant, left his family in straitened circumstances, and Charlotte, who had a fine contralto voice, went on the operatic stage. In 1835 she successfully appeared at the Tremont theatre as the countess Almaviva in The Marriage of Figaro. But her singing voice failing her she entered the drama, and played Lady Macbeth in the same year. She then engaged herself as a stock actress, but was soon given leading parts. In 1842 she managed and played in the Walnut Street theatre in Philadelphia. She accompanied Macready on an American tour, winning a great reputation in tragedy, and in 1845 and in 1854-1855 she fulfilled successful engagements in London. She was a keen student, and acquired a large range of classic rôles. Her best parts were perhaps Lady Macbeth and Queen Katherine, her most popular Meg Merrilies, in a dramatization of Scott’s Guy Mannering. Her figure was commanding and her face expressive, and she was animated by a temperament full of vigour and fire. These qualities enabled her to play with success such male parts as Romeo and Cardinal Wolsey. During her later years Miss Cushman worked hard as a dramatic reader, in which capacity she was much appreciated. Her last appearance on the stage took place on the 15th of May 1875, at the Globe theatre, Boston, in which city she died on the 18th of February 1876.

See Emma Stebbins’s Charlotte Cushman, her Letters and Memories of her Life (Boston, 1878); H. A. Clapp’s Reminiscences of a Dramatic Critic (Boston, 1902); and W. T. Price, A Life of Charlotte Cushman (New York, 1894).

CUSP (Lat. cuspis, a spear, point), a projecting point, or pointed end. In architecture (Fr. feuille, Ital. cuspide, Ger. Knöpfe), a cusp is the point where the foliations of tracery intersect. The earliest example of a plain cusp is probably that at Pythagoras school, at Cambridge,—of an ornamented cusp at Ely cathedral, where a small roll, with a rosette at the end, is formed at the termination of a cusp. In the later styles the terminations of the cusps were more richly decorated; they also sometimes terminate not only in leaves or foliages, but in rosettes, heads and other fanciful ornaments. The term “feathering” is used of the junction of the foliated cusps in window tracery, but is usually restricted to those cases where it is ornamented with foliage, &c.

CUSTARD1 APPLE, a name applied to the fruit of various species of the genus Anona, natural order Anonaceae. The members of this genus are shrubs or small trees having alternate, exstipulate leaves, and flowers with three small sepals, six petals arranged in a double row and numerous stamens. The fruit of A. reticulata, the common custard apple, or “bullock’s heart” of the West Indies, is dark brown in colour, and marked with depressions, which give it a quilted appearance; its pulp is reddish-yellow, sweetish and very soft (whence the name); the kernels of the seeds are said to be poisonous. The sour-sop is the fruit of A. muricata, native of the West Indies. The plant, which is a small tree, has become naturalized in some parts of India where it is extensively cultivated, as elsewhere in the tropics. It is covered with soft prickles, is of a light-greenish hue, and has a peculiar but agreeable sour taste, and a scent resembling that of black currants. The sweet-sop is produced by A. squamosa, also a native of the West Indies and widely cultivated in the tropics. It is known as the custard apple by Europeans in India. It is an egg-shaped fruit, with a thick rind and luscious pulp. An acrid principle, fatal to insects, is contained in its seeds, leaves and unripe fruits, which, powdered and mixed with the flour of gram (Cicer arietinum), are used to destroy vermin. A. Cherimolia yield the Peruvian cherimoyer, which is held to be a fruit of very superior flavour, and is much esteemed by the creoles. A. palustris, alligator apple, or cork-wood, a native of South America and the West Indies, is valued for its wood, which serves the same purposes as cork; the fruit, commonly known as the alligator-apple, is not eaten, being reputed to contain a dangerous narcotic principle.

1 The term “custard,” now given to a dish made with eggs beaten up with milk, &c., and either served in liquid form or baked to a stiff consistency, originally denoted a kind of open pie. It represents the older form “crustade,” Fr. croustade, Ital. crostata, from crostare, to encrust.

CUSTER, GEORGE ARMSTRONG (1839-1876), American cavalry soldier, was born in New Rumley, Harrison county, Ohio, on the 5th of December 1839. He graduated from West Point in 1861, and was at once sent to the theatre of war in Virginia, joining his regiment on the battlefield of Bull Run. Afterwards he served on the staff of General Kearny, and on that of General W. F. Smith in the Peninsular Campaign. His daring and energy, and in particular a spirited reconnaissance on the Chickahominy river, brought him to the notice of General McClellan, who made him an aide-de-camp on his own staff, with the rank of captain. A few hours afterwards Custer attacked a Confederate picket post and drove back the enemy. He continued to serve with McClellan until the general was relieved of his command, when Custer returned to duty with his regiment as a lieutenant. Early in 1863 General Pleasonton selected him as his aide-de-camp, and in June 1863 Custer was promoted to the rank of brigadier-general of volunteers. He distinguished himself at the head of the Michigan cavalry brigade in the battle of Gettysburg, and frequently did good service in the remaining operations of the campaign of 1863. When the cavalry corps of the Army of the Potomac was reorganized under Sheridan in 1864, Custer retained his command, and took part in the various actions of the cavalry in the Wilderness and Shenandoah campaigns. At the end of September 1864, he was appointed to command a division, and on the 9th of October fought, along with General Merritt, the brilliant cavalry action called the battle of Woodstock. Soon afterwards he was made brevet-major-general, U.S.V., having already won the brevets of major, lieutenant-colonel and colonel U.S.A., for his services at Gettysburg, Yellow Tavern and Winchester. His part in the decisive battle of Cedar Creek (q.v.) was most conspicuous. He served with Sheridan in the last great cavalry raid, won the action of Waynesboro, and in the final campaign added to his laurels by his conduct at Dinwiddie and Five Forks, and in other operations. At the close of the war he received the brevets of brigadier and major-general in the regular army, and was promoted major-general of volunteers. In 1866 Custer was made lieutenant-colonel of the 7th U.S. Cavalry, and took part under General Hancock in the expedition against the Cheyenne Indians, upon whom he inflicted a crushing defeat at Washita river on the 27th of November 1868. In 1873 he was sent to Dakota Territory to serve against the Sioux.

In 1876 an expedition, of which Custer and his regiment formed part, was made against the Sioux and their allies. As the advanced guard of the troops under General Terry, Custer’s force arrived at the junction of Big Horn and Little Big Horn rivers, in what is now the state of Montana, on the night of June 24; the main body was due to join him on the 26th. Unfortunately, the presence of what was judged to be a small isolated force of Indians was reported to the general. On the 25th, dividing his regiment into three parties, he moved forward to surround this force. But instead of meeting only a small force of Indians, the 7th were promptly attacked by the full forces of the enemy. The flanking columns maintained themselves with difficulty until Terry came up. Custer and 264 men of the centre column rode into the midst of the enemy and were slaughtered to a man.

The general’s wife, Elizabeth Bacon Custer, who accompanied him in many of his frontier expeditions, wrote Boots and Saddles, Life with General Custer in Dakota (1885), Tenting on the Plains (1887) and Following the Guidon (1891). General Custer himself wrote My Life on the Plains (1874).

See F. Whittaker, Life of General George A. Custer (1876).

His brother Thomas Ward Custer (1845-1876), in spite of his youth, fought in the early campaigns of the Civil War. Becoming aide-de-camp to General Custer, he accompanied him throughout the latter part of the war, distinguishing himself by his daring on all occasions, and winning successively the brevets of captain, major and lieutenant-colonel, though he was barely twenty years of age when the war ended. He was first lieutenant in the 7th cavalry when he fell with his brother at the Little Big Horn.

CUSTINE, ADAM PHILIPPE, Comte de (1740-1793), French general, began his military career in the Seven Years’ War. He next served with distinction against the English in the War of American Independence. In 1789 he was elected to the states-general by the bailliage of Metz. In October 1791 he again joined the army, with the rank of lieutenant-general and became popular with the soldiers, amongst whom he was known as “général moustache.” General-in-chief of the army of the Vosges, he took Spires, Worms, Mainz and Frankfort in September and October 1792. He carried on the revolutionary propaganda by proclamations, and levied heavy taxes on the nobility and clergy. During the winter a Prussian army forced him to evacuate Frankfort, re-cross the Rhine and fall back upon Landau. He was accused of treason, defended by Robespierre, and sent back to the army of the north. But he dared not take the offensive, and did nothing to save Condé, which the Austrians were besieging. Sent to Paris to justify himself, he was found guilty by the Revolutionary Tribunal of having intrigued with the enemies of the republic, and guillotined on the 28th of August 1793. (See French Revolutionary Wars.)

See A. Rambaud, Les Français sur le Rhin (Paris, 1880); A. Chuquet, Les Guerres de la Révolution (1886-1895; vol. vi. “L’Expédition de Custine”).

CUSTOM (from O. Fr. custume, costume or coustume; Low Lat. costuma, a shortened form of consuetudo), in general, a habit or practice. Thus a tradesman calls those who deal with him his “customers,” and the trade resulting as their “custom.” The word is also used for a toll or tax levied upon goods; there was at one time a distinction between the tax on goods exported or imported, termed magna custuma (the great custom), and that on goods taken to market within the realm, termed parva custuma (the little custom), but the word is now used in this sense only in the plural, to signify the duties levied upon imported goods. It is also used as a name for that department of the public service which is employed in levying the duty.

In law, such long-continued usage as has by common consent become a rule of conduct is termed custom. Jessel, M. R. (Hammerton v. Honey, 24 W. R. 603), has defined it as “local common law. It is common law because it is not statute law; it is local law because it is the law of a particular place, as distinguished from the general common law. Local common law is the law of the country (i.e. particular place) as it existed before the time of legal memory.” There has been much discussion among jurists as to whether custom can properly be reckoned a source of law (see Jurisprudence). As to the distinction between prescription (which is a personal claim) and custom, see Prescription. The adoption of local customs by the judiciary has undoubtedly been the origin of a great portion of the English common law. Blackstone divides custom into (1) general, which is the common law properly so called, and (2) particular, which affects only the inhabitants of particular districts. The requisites necessary to make a particular custom good are: (1) it must have been used so long that the memory of man runneth not to the contrary; (2) it must have been continued, and (3) enjoyed peaceably; (4) it must be reasonable, and (5) certain; (6) it must be compulsory, and not left to the option of every man whether he will use it or no; (7) it must be consistent with other customs, for one custom cannot be set up in opposition to another. Customs may be of various kinds, for example, customs of merchants, customs of a certain district (such as gavelkind and borough English), customs of a particular manor, &c. The word custom is also generally employed for the usage of a particular trade or market; for a trade custom to be established to the satisfaction of the law it must be a uniform and universal practice so well defined and recognized that contracting parties must be assumed to have had it in their minds when they contracted (Russell, C. J., Fox-Bourne v. Vernon, 10 Times Rep. 649).

In the history of France the term “custom” was given to those special usages of different districts which had grown up into a body of local law, as the “custom of Paris,” the “custom of Normandy” (see France: Law and Institutions).

CUSTOMARY FREEHOLD, in English law, a species of tenure which may be described as a variety of copyhold. It is also termed privileged copyhold or copyhold of frank tenure. It is a tenure by copy of court roll, but not expressed to be at the will of the lord. It is, in fact, only a superior kind of copyhold, and the freehold is in the lord. It is subject to the general law of copyholds, except where the law may be varied by the custom of the particular manor. (See Copyhold.)

CUSTOM-HOUSE, the house or office appointed by a government where the taxes or duties (if any) are collected upon the importation and exportation of commodities; where duties, bounties or drawbacks payable or receivable upon exportation or importation are paid or received, and where vessels are entered and cleared. In the United Kingdom there is usually a custom-house established at every port or harbour to which any considerable amount of shipping resorts, the officer in charge being called “collector of customs”; in the minor ports the officer is usually termed “superintendent of customs” or “principal coast officer.”

CUSTOM DUTIES, the name given to taxes on the import and export of commodities. They rank among the most ancient, as they continue to prevail as one of the most common modes, in all countries, of levying revenue for public purposes. In an insular country like the United Kingdom customs duties came in process of time to be levied only or chiefly in the seaports, and thus applied only to the foreign commerce, where they may be brought under the control of fair and reasonable principles of taxation. But this simplification of customs duties was only reached by degrees; and during a long period special customs were levied on goods passing between England and Scotland; and the trade of Ireland with Great Britain and with foreign countries was subjected to fiscal regulations which could not now stand in the light of public reason. The taxes levied, on warrant of some ancient grant or privilege, upon cattle or goods at a bridge or a ferry or other point of passage from one county or province to another, of which there are some lingering remains even in the United Kingdom, and those levied at the gates of cities on the produce of the immediate country—a not uncommon form of municipal taxation on the European continent—are all of the nature of customs dues. It is from the universality of this practice that the English term “customs” appears to have been derived.

See Taxation; Protection; Tariff.

CUSTOS ROTULORUM, the keeper of the English county records, and by virtue of that office the highest civil officer in the county. The appointment until 1545 lay with the lord chancellor, but is now exercised by the crown under the royal sign-manual, and is usually held by a person of rank, most frequently the lord-lieutenant of the county. He is one of the justices of the peace. In practice the records are in the custody of the clerk of the peace. This latter official was, until 1888, appointed by the custos rotulorum, but since the passing of the Local Government Act of that year, the appointment is made by the standing joint-committee of the county council. Lambarde described the custos rotulorum as a “man for the most part especially picked out either for wisdom, countenance or credit.”

CUSTOZZA, a village of Italy, in the province of Verona, 11 m. S.W. of Verona, famous as the scene of two battles between the Austrians and the Italians in the struggle for Italian unity. The first battle of Custozza was fought on the 23rd-25th of July 1848, the Austrians commanded by Field-Marshal Radetzky being victorious over the Piedmontese army under King Charles Albert. The second battle was fought on the 24th of June 1866, and resulted in the complete victory of the Austrians under the archduke Albert, over the Italian army of King Victor Emmanuel I. (See Italian Wars, 1848-1870.)

CÜSTRIN, or Küstrin, a town of Germany, in the kingdom of Prussia, a fortress of the first rank, at the confluence of the Oder and Warthe, 18 m. N.E. from Frankfort-on-Oder and 51 m. N.E. of Berlin by rail. Pop. (1900) 16,473 (including the garrison). It consists of the town proper within the strong fortifications, a suburb on the left bank of the Oder, and one on the right bank of the Warthe. There are three Evangelical churches and one Roman Catholic, and a handsome town hall. There are bridges over both rivers. Cüstrin has some manufactories of potato-meal, machinery, pianos, furniture, cigars, &c., and there is a considerable river trade.

About 1250 a town was erected on the site of Cüstrin, where a fishing village originally stood. From 1535 till 1571 it was the residence of John, margrave of Brandenburg-Cüstrin, who died without male heirs in 1571. Cüstrin was the prison of Frederick the Great when crown-prince, and the scene of the execution of his friend Hans Hermann von Katte on the 6th of November 1730.

CUTCH, or Kach, a native state of India within the Gujarat division of Bombay, with an area of 7616 sq. m. It is a peninsular tract of land, enclosed towards the W. by the eastern branch of the Indus, on the S. by the Indian Ocean and the Gulf of Cutch, and on the N. and E. towards the interior, by the great northern Runn, a salt morass or lake. The interior of Cutch is studded with hills of considerable elevation, and a range of mountains runs through it from east to west, many of them of the most fantastic shapes, with large isolated masses of rock scattered in all directions. The general appearance of Cutch is barren and uninteresting. The greater part is a rock destitute of soil, and presenting the wildest aspect; the ground is cold, poor and sterile; and the whole face of the country bears marks of volcanic action. From the violence of tyranny, and the rapine of a disorderly banditti, by which this district long suffered, as well as from shocks of earthquakes, the villages have a ruinous and dilapidated appearance; and, with the exception of a few fields in their neighbourhood, the country presents a rocky and sandy waste, with in many places scarcely a show of vegetation. Water is scarce and brackish, and is chiefly found at the bottom of low ranges of hills, which abound in some parts; and the inhabitants of the extensive sandy tracts suffer greatly from the want of it. Owing to the uncertainty of the periodical rains in Cutch, the country is liable to severe famines, and it has suffered greatly from plague.

The temperature of Cutch during the hot season is high, the thermometer frequently rising to 100° or 105° F.; and in the months of April and May clouds of dust and sand, blown about by hurricanes, envelop the houses, the glass windows scarcely affording any protection. The influence of the monsoon is greatly moderated before it reaches this region, and the rains sometimes fail altogether. Bhuj, the capital of the state, is situated inland, and is surrounded by an amphitheatre of hills, some of which approach within 3 or 4 m. of the city. The hill of Bhuja, on which the fort is situated, rises to the height of 500 ft. in the middle of the plain, and is detached from other high ground. The residency is 4 m. distant in a westerly direction. There are many mountain streams, but no navigable rivers. They contain scarcely any water except in the rainy season, when they are very full and rapid, and discharge themselves into the Runn, all along the coast of which the wells and springs are more or less impregnated with common salt and other saline ingredients.

Various causes have contributed to thin the population of this country. In 1813 it was ravaged by a famine and pestilence, which destroyed a great proportion of its inhabitants,—according to some accounts, nearly one-half. This, joined to the tyranny and violence of the government until the year 1819, and subsequently to a succession of unfavourable seasons, forced many of the cultivators to remove to Sind and other countries. The inhabitants numbered 488,022 in 1901, being a decrease of 13% during the decade, due to the famines of 1899-1900. One-third are Mahommedans and the remainder Hindus of various castes. The Jareja Rajputs form a particular class, being the aristocracy of the country; and all are more or less connected with the family of the rao or prince. There are in Cutch about 200 of these Jareja chiefs, who all claim their descent from a prince who reigned in Sind about 1000 years ago. From him also the reigning sovereign is lineally descended, and he is the liege lord of whom all the chiefs or nobles hold their lands in feu, for services which they or their ancestors had performed, or in virtue of their relationship to the family. They are all termed the brotherhood of the rao or Bhayad, and supposed to be his hereditary advisers, and their possessions are divided among their male children. To prevent the breaking down of their properties, the necessary consequence of this law of inheritance, there is no doubt that infanticide was common among them, and that it extended to the male as well as the female progeny, but it has been put down by the Infanticide Rules, which provide for the registration of Jareja children. The Jarejas have a tradition that when they entered Cutch they were Mahommedans, but that they afterward adopted the customs and religion of the Hindus. It is certain, indeed, that they still retain many Mahommedan customs. They take oaths equally on the Koran or on the Shastras; they employ Mussulman books; they eat from their hands; the rao, when he appears in public, alternately worships God in a Hindu pagoda and a Mahommedan mosque; and he fits out annually at Mandvi a ship for the conveyance of pilgrims to Mecca, who are maintained during the voyage chiefly by the liberality of the prince. The Mahommedans in Cutch are of the same degenerate class as those usually found in the western parts of India. The natives are in general of a stronger and stouter make, and even handsomer, than those of western India; and the women of the higher classes are also handsome. The peasants are described as intelligent, and the artizans are justly celebrated for their ingenuity and mechanical skill. The palace at Mandvi, and a tomb of one of their princes at Bhuj, are fair specimens of their architectural skill. The estimated gross revenue is £126,322. There are special manufactures of silver filigree-work and embroidery. The maritime population supplies the best sailors in India. There are cotton presses and ginning factories.

The country of Cutch was invaded about the 13th century by a body of Mahommedans of the Summa tribe, who under the guidance of five brothers emigrated from Sind, and who gradually subdued or expelled the original inhabitants, consisting of three distinct races. Cutch continued tranquil under their sway for many years, until some family quarrel arose, in which the chief of an elder branch of the tribe was murdered by a rival brother. His son Khengayi fled to Ahmedabad to seek the assistance of the viceroy, who reinstated him in the sovereignty of Cutch, and Morvi in Káthiáwár, and in the title of rao, about the year 1540. The succession continued in the same line from the time of this prince until 1697, when a younger brother, Pragji, murdered his elder brother and usurped the sovereignty. This line of princes continued till 1760 without any remarkable event, when, in the reign of Rao Ghodji, the country was invaded four times by the Sinds, who wasted it with fire and sword. The reign of this prince, as well as that of his son Rao Rayadan, by whom he was succeeded in 1778, was marked by cruelty and blood. The latter prince was dethroned, and, being in a state of mental derangement, was during his lifetime confined by Fateh Mahommed, a native of Sind, who continued, with a short interval (in which the party of the legal heir, Bhaiji Bawa, gained the ascendancy), to rule the country until his death in 1813. It was in the reign of Fateh Mahommed that a communication first took place with the British government. During the contests for the sovereignty between the usurper and the legal heir, the leader of the royal party, Hansraj, the governor of Mandvi, sought the aid of the British. But no closer connexion followed at that time than an agreement for the suppression of piracy, or of inroads of troops to the eastward of the Runn or Gulf of Cutch. But the gulf continued notwithstanding to swarm with pirates, who were openly encouraged or connived at by the son of Hansraj, who had succeeded his father, as well as by Fateh Mahommed. The latter left several sons by different wives, who were competitors for the vacant throne. Husain Miyan succeeded to a considerable portion of his father’s property and power. Jugjevan, a Brahman, the late minister of Fateh Mahommed, also received a considerable share of influence; and the hatred of these two factions was embittered by religious animosities, the one being Hindu and the other Mahommedan. The deceased rao had declared himself a Mahommedan, and his adherents were preparing to inter his body in a magnificent tomb, when the Jarejas and other Hindus seized the corpse and consigned it to the flames, according to Hindu custom.