CHORD , n. A chord of any curve (or sur face) is a segment of a Straight line between two designated points of intersection of the line and the curve (surface). See circle, sphere, etc.
chord of contact with reference to a point outside a circle. The chord joining the points of contact of the tangents to the circle from the given point.
focal chords of conics . See focal.
supplemental or supplementary chords in a circle. See supplemental.
CHRISTOFFEL, Elwin Bruno (1 829-1 900). German differential geometer. Invented process of covariant differentiation.
Christoffel symbols . Certain symbols representing particular functions of the coeffi cients, and of the first-order derivatives of the coefficients, of a quadratic differential form. The differential form is usually the first funda mental quadratic differential form of a surface. For the quadratic differential form 
the Christoffel symbols of the first kind are 
where 
separately. For a quadratic form in n variables, we have 
separately. The symbo 
is sometimes replaced by 
The
symbols are symmetric in i and /. For the
quadratic differential form
the Chrisloffel symbols of the second kind are 
where 
separately, 
is the cofactor of 
in the determinant 
divided by 
and 
are the Christoffel symbols of the first kind. The symbol 
is sometimes re placed by 
in keeping with the summation convention, or by 
The symbols are symmetric in i and /, For the quadratic differential form 
in n variables 
{where the summation con vention applies), the Christoffel symbols of the first kind are 
it being assumed that 
for all i and j.
The Christoffel symbols of the second kind are 
where g is the determinant having 
in the 
row and 
column and 
Neither kind of Christoffel symbol is a tensor. The law connecting Christoffel symbols of the second kind in two systems of coordinates 
and 
is 
where and arc the Christoffel symbols 
of Ihe second 
kind in the 
and 
coordinate systems, respectively. See below, Euclidean Christoffel symbols.
Euclidean Christoffel symbols . Christoffel symbols in an Euclidean space (i.e., where rectangular Cartesian coordinates 
exist such that the element of arc length 
is given by 
The Euclidean symbols of the second kind arc all identically zero in rectangular Cartesian coordinates. However, the Euclidean Christoffel symbols arc not all zero in general coordinates and are given by the alternative expression
in terms of the transformation functions and their inverses taking the rettangular Cartesian coordinates 
into the general coordinates 
Since the Euclidean Christoffel symbols of the second kind are all identically zero in rectangular Cartesian coordinates, it follows that covariaiit differentiation is ordinary partial differentiation in rectangular Cartesian co ordinates. Hence successive covariant differ entiation in Euclidean space is a commutative operation even in general coordinates as long as partial differentiation is commutative. Riemann-Christoffel curvature tensor. See RI E.MANN.
Cl'PHER (or CYPHER), n. The symbol 0, denoting zero. Syn. 7ero, naught.
Cl'PHER , v . To compute with numbers; to carry out one or more of the fundamental operations of arithmetic.
CIR'CLE , n. A plane curve consisting of all points at a given distance (called the radius ) from a fixed point in the plane, called the center . The diameter is twice the radius ("diameter" can mean any chord passing through the center; see diameter -diameter of a conic). An arc is one of Ihe Iwo pieces bounded by two points on the circle. The circumference is the length of the circle, which is 
if r is the radius (see 
); sometimes "circumference" is used to mean a circle itself rather than its interior. The area of a circle (i.e., the area of the interior) is 
, or in terms of the diameter 
A unit circle
is a circle with radius 1; it is has circumference 
and area 
A null circle is a circle with radius 
A secant of a circle is any line that is not tangent to the circle and intersects the circle; a chord is Ihe segment of a secant joining the two points of intersection with the circle.
circle of convergence . See convlrgence— circle of convergence of a power series.
circle of curvature . See CURVATURE—curva ture of a curve, curvature of a surface.
circumscribed circle . See CIRCUMSCRIBED .
eccentric circles of an ellipse. See ellipse- parametric equations of an ellipse.
equation of a circle in the plane . In rect angular Cartesian coordinates, 
where r is the radius of the circle and the center is at the point (/;, k). When the center is at the origin, this becomes 
(See DISTANCE—distance be tween two points.) In polar coordinates, the equation is
where 
is the radius vector, 
the vectorial angle, 
the polar coordinates of the
center, and r the radius. When the center of the circle is at the origin, this equation becomes 
The parametric equations
of a circle are 
where 
is the angle between the positive x-axis and the radius from the origin to the given point, and a is the radius of Ihe circle.
equations of a circle in space . The equa tions of any two surfaces whose intersection is the circle; a sphere and a plane, each containing the circle, would suffice.
escribed circle . See escribed.
family of circles . All the circles whose equations can be obtained by assigning particular values to an essential constant in the equation of a circle. E.g., 
describes the family of circles with their centers at the origin, r being the essential constant in this case. See pfncil -pencil of circles.
great circle . A section of a sphere by a plane which passes through its center; a circle (on a sphere) which has its diameter equal lo that of the sphere.
hour circle of a celcslial point . The great circle on the celestial sphere that passes through the poinl and the north and south celestial poles. Sec hour —hour angle and hour circle.
imaginary circle. The name given to the set of points which satisfy the equation 
or 
Both coordinates of such a point can not be real. Although no points in the real plane have such coordinates, this termin ology is desirable because of the algebraic properties common to these imaginary co ordinates and the real coordinates of points on real circles.
inscribed circle. See INSCRIBED.
nine-point circle. The circle through the midpoints of the sides of a triangle, the feet of Ihe perpendiculars from the vertices upon the sides, and the midpoints of the line segments between the vertices and the point of inter section of the altitudes.
osculating circle . Sec curvature curva ture of a curve.
parallel circle . See surface —surface of revolution.
small circle . A section of a sphere by a plane thai docs not pass through Ihe center of the sphere.
squaring the circle . See squaring.
CIR'CUIT, n. flip-flop circuit . In a com puting machine, any bistable circuit that remains in either of its two stable slates until receiving a signal changing it to the other slate. Such a circuit often involves a charac teristic configuration of vacuum tubes,
CIR'CU-LANT , n. A determinant in which Ihe elements of each row are the elements of the previous row slid one place to the right (with the last element put first). The elements of the main diagonal are all identical.
CIR'CU-LAR, adj. circular argument or reasoning. An argument which is incorrect because it makes use of the theorem to be proved, or makes use of a theorem that is a consequence of the theorem to be proved but is not known to be true. One flagrant type of circular reasoning is to use Ihe theorem to be proved as a reason for a step in its "proof."
circular cone and cylinder . See CONE, CYLINDER.
circular functions. The trigonometric func tions.
circular (or cyclic) pernutation . See per mutation.
circular point of a surface . An elliptic point of the surface at which 
See surface funda mental coefficients of a surface, and llliptic — elliptic point of a surface. For a circular point, the principal radii of normal curvature are equal, and the Dupin indicatrix is a circle. A surface is a sphere if and only if all its points are circular points. The points where an ellipsoid of revolution cuts its axis of revolu tion are circular points. See i'LANAR - planar point of a surface, UMBILICAL— umbilical point of a surface.
circular region. See region.
uniform circular motion . See uniform.
CIR'CUM-CEN'TER, n. circumcenter of a triangle. The center of the circumscribed circle (the circle passing through the three vertices of the triangle); the point of inter section of the perpendicular bisectors of the sides; point 
in the figure.
CIR'CUM-CIR'CLE , n. Same as circum scribed CIRCLE.
CIR-CUM'FER-ENCE , n. (I) See circle. (2) The boundary of any region whose bound ary is a simple closed curve (e.g., a polygon). Syn, periphery, perimeter.
circumference of a sphere . The circum ference of any great circle on the sphere.
CIR'CUM-SCRIBED' , udj. A configuration composed of lines, curves, or surfaces, is said to be circumscribed about a polygon (or polyhedron) if every vertex of the latter is incident upon the former and the polygon (or poly hedron) is contained in the configuration. A polygon (or polyhedron) is circumscribed about a configuration if every side of the polygon (or face of the polyhedron) is tangent 10 the configuration and the configuration is contained in the polygon (or polyhedron). If one figure is circumscribed about another, the latter is said to be inscribed in the former.
In particular, the circumscribed circle of a polygon is a circle which passes through the vertices of the polygon. The polygon is then an inscribed polygon of the circle , if the poly gon is regular and s is the length of a side and n the number of sides, the radius of the circle is 
If the polygon is a triangle with sides a, b, c, and 
then
If the polygon is regular and has n sides, its area is 
and its perimeter is
where r is the radius of the circumscribed circle. A circumscribed poly gon of a circle is a polygon which has its sides tangent to the circle. The circle is then an inscribed circle of the polygon (see INSCRIBED— inscribed circle of a triangle). If the polygon is regular, its area is
and its perimeter is
where r is the radius of the inscribed circle and n the number of sides of the polygon. If s is the length of a side of the polygon and n the number of its sides, the radius is 
A circumscribed sphere of a poly hedron is a sphere which passes through all the vertices of the polyhedron (the polyhedron is then said It) be inscribed in the sphere). An inscribed sphere of a polyhedron is a sphere which is tangent to all the faces of the polyhedron (the polyhedron is then said to be circumscribed about the sphere). A circum scribed pyramid of a cone is a pyramid having its base circumscribed about the base of the cone and its vertex coincident with that of the cone. The cone is then an inscribed cone of the pyramid.
A circumscribed cone of a pyramid is a cone whose base is circumscribed about the base of the pyramid and whose vertex coincides with the vertex of the tone. The pyramid is then an inscribed pyramid of the cone.
A circumscribed prism of a cylinder is a. prism whose bases are coplanar with, and circumscribed, about, the bases of the cylinder. The lateral faces of the prism arc then tangent t o the cylindrical surface and the cylinder is an inscribed cylinder of the prism . A cir-cumscribed cylinder of a prism is a cylinder whose bases are coplanar with, and cir cumscribed about, the bases of the cylinder. The lateral edges of the prism arc then elements of the cylinder and the prism is an inscribed prism of the cylinder.
CIS'SOID (cissoid of Diocles) , n. The plane locus of a variable point on a variable line passing through a fixed point on a circle, where the distance of the variable point from the fixed point is equal to the distance from the line's intersection with the circle to its intersection with a fixed tangent to the circle at the extremity of the diameter through the fixed point; the locus of the foot of the per pendicular from the vertex of a parabola to a variable tangent. If a is taken as the radius
of the circle in the first definition, the polar equation of the cissoid is 
its Cartesian equation is 
The curve has a cusp of the first kind at the origin, the x-axis being the double tangent. The cissoid was first studied by Diodes about 200 B.C, who gave it the name "Cissoid" (meaning like ivy).
CIV'IL , adj. civil year . See year.
CLA1RAUT (or CLAIRAULT) , Alexis Claude (1713-1765). French analyst, dif ferential geometer, and astronomer.
Clairaut's differential equation. A dif ferential equation which is of the form 
for some function /. The general solution is 
and a singular solution is given by the parametric equations 
and 
CLASS , n. (Statistics) Often the set of all observations of a random variable are grouped into classes by divisions of the range of the variable. E.g., a variable whose range is the interval |0, 100] may be grouped into class intervals ten units wide with 
for x in the first interval,
the second, etc. The class limits or class bounds are the upper and lower bounds of the values in a class inter val. The class frequency is the frequency with which the random variable assumes a value in a given class interval.
class of a plane algebraic curve . The greatest number of tangents that can be drawn to it from any point in the plane not on the curve,
equivalence class. See EQUIVALENCE.
subclass . Same as SUBSET. See SET, NUMBER.
CLAS'SI-FI-CA'TION , n. one-way classifi cation (Statistics). A classification of values of a random variable according to several classes of a single factor, e.g., the factor might be sex and the classes female and male.
CLOCK'WISE , adj. In the same direction of rotation as that in which the hands of a clock move around the dial.
counterclockwise . In the direction of rota tion opposite to that in which the hands of a clock move around the dial.
CLOSED , adj. closed curve . A curve which has no end points. Tech. A set of points which is the image of a circle under a con tinuous transformation. See CONTINUOUS— continuous function, curve, SIMPLE- simple closed curve.
closed interval . See interval.
closed mapping or transformation. (1) See OPEN—open mapping. (2) A linear trans formation T is said lo be closed if it has the property that, if 
and 
exist, where 
is in the domain D of T for each n , then 
is in D and 
This is equivalent to stating that the set of points of type 
is closed in the Cartesian product 
oi' the closure of the domain D and the closure of R, the runge of T. See open —open mapping theorem.
closed region. See region.
closed set . A set of points 
such that every accumulation point of 
is a point of 
; the complement of an open set. The set of points on and within a. circle is a closed set.
closed surface . A surface with no bound ary curves; a space such that each point has a neighborhood iopulogically equivalent to the interior of a circle. See surface.
closed with respect to a binary operation. See binary —binary operation.
CLO'SURE , n. closure of a set of points .
The set which contains the given set and all accumulation points of the given set. The closure of a closed set is the set itself, while the closure of any set is closed. The set of all accumulation points of a given set is the derived set. The closure of a set 
is usually denoted by 
and the derived set by 
CLUS'TER , adj. cluster point. Same as ACCUMULATION POINT.
CO'A-LITION , n. In an n -pcrson game, a set of players, more than one in number, who coordinate their strategies, presumably for mutual benefit. See game— cooperative game.
CO-AL'Tl-TUDE , n. coaltitude of a celestialpoint. Same as zinith distance .
CO-AX'I-AL, adj. coaxial circles . Circles such that all pairs of the circles have the same radical axis.
coaxial planes. Planes which pass through the same straight line. The line is called the axis.
CO-BOUND'A-RY , n. See cohomology — cohomology group.
COCHRAN, William Gemmell {1909- ). Scottish statistician, who has spent many years in the U.S.
Cochran's theorem . If 
are independently and normally distrib uted random variables with zero mean and unit variance, and if 
are k quadratic forms in the variables 
with
ranks 
and 
then a necessary and sufficient condition that 
be each independently distributed with tne 
distribution with 
degrees
of freedom is that 
On the basis of this theorem it follows that if 
is a random sample from a normal distribution with mean u and variance 
, then 
is distributed as 
with 
degrees of freedom, where 
is the mean of the sample. This theorem is useful, for example, in establishing the indepen dence of the mean and sum of squares of deviations around the mean in random samples from a normal population.
CO-CY'CLE , n. See cohomologY - cohomology group.
CODAZZI, Delfino (1824-1873). Italian differential geometer.
Codazzi equations . The equations
and
involving the fundamental coefficients of Ihe first and second orders of a surface. In tensor notation: 
There arc no relations between these funda mental coefficients and their derivatives which cannot be derived from the Gauss equation and the Codazzi equations, for these three equations uniquely determine a surface to within its position in space. See CHRISTOF- fel —Chrisioffel symbols.
CO-DEC'LI-NA'TION , n. codeclination of a celestial point. Ninety degrees minus the declination; the complementary angle of the declination. See hour hour angle and hour circle. Syn. polar distance.
COD'ING , n. In machine computation, the detailed preparation from the programmer's instructions or flow charts, of machine com mands that will lead to the solution of the problem at hand. See problem —problemformulation, PROGRAMMING -programming for a computing machine.
CO'EF-FI'CIENT , n. In elementary algebra, the numerical part of a term, usually written before the literal part, as 2 in 
or 
(See parenthesis.) Tn general, the product of all the factors of a term except a certain one (or a certain set), of which the product is said to be the coefficient. E.g., in 
is the coefficient of z, 
the coefficient
of v, 
the coefficient of yz, etc. Most
commonly used in algebra for the constant factors, as distinguished from the variables.
binomial coefficients. See binomial.
coefficient of alienation. See coirrelation — normal correlation.
coefficient of correlation . See correla tion -correlation coefficient.
coefficient of friction. See FRICTION.
coefficient of linear expansion . (1) The quotient of the change in length of a rod, due to one degree change of temperature, and the original length (not the same at all tem peratures). (2) The change in length of a unit rod when the temperature changes one degree centigrade beginning at 
coefficient of strain . See one —one-dimen sional strains.
confident of thermal expansion . A term used to designate both the coefficient of linear expansion and the coefficient of volume expansion.
coefficient of variation . See variation.
coefficient of volume (or cubical) expansion. (1) The change in volume of a unit cube when the temperature changes one degree. (The coefficient thus defined is different at different temperatures.) (2) The change in unit volume due to a change of 
beginning at 
coefficients in an equation . (1) The coeffi cients of Ihe variables. (2) The constant term and the coefficients of all the terms containing variables. If the constant term is not in cluded, the phrase coefficients of the variables in the equation is often used.
confidence coefficient. See confidence.
correlation coefficient. See roiiRLLATiON,
detached coefficients, multiplication and di vision by means of . Abbreviations of the ordinary multiplication and division processes used in algebra. The coefficients alone (with their signs) are used, the powers of the variable occurring in the various terms being understood from the order in which the coefficients arc written, missing powers being assumed to be present with zero coefficients. E.g., 
is multiplied by 
by using the expressions 
and 
Sec synthetic- synthetic division.
determinant of the coefficients. See deter minant —determinant of the coefficients.
differential coefficient . Same as derivative.
leading coefficient. See leading.
Legcndre's coefficients. See legendre— Legendre's polynomials.
matrix of the coefficients. See matrix.
phi coefficient (Statistics). A coefficient obtained from a four-fold table, in which the two variables are essentially dichoto-mous. The phi coefficient 
is defined by
where 
is computed from the cell entries and the marginal totals, which arc the basis of the expected values. See chi-square.
regression coefficient . See regression.
relation between the roots and coefficients of a polynomial equation. See R oot —root of an equation.
undetermined coefficients . See undetermined .
CO-FAC'TOR , n. cofactor of an element of a determinant . See minor- minor of an element of a determinant.
cofactor of an element of a matrix . This is defined only for square matrices, and is the same as the cofactor of the same element in the determinant of Ihe matrix.
CO-FlN'AL , adj . cofinal subset . See MOORE (E. H.)—Moore-Smith convergence.
CO-FUNC'TION , n. Sec trigonometric— trigonometric cofunctions.
COHEN, Paul Joseph (1934- ). Ameri can analyst, topological group theorist, and logician. Fields Medal recipient (1966). Proved the independence in set theory of the axiom of choice, and the independence of the continuum hypothesis from the axiom of choice, thus completing the nega tive solution of the continuum problem (Hilbert's first problem), the first part of the solution having been given by Kurt Godel in 1938-40.
CO-HER'ENT-LY , adv. coherently oriented. See MANIFOLD, SIMPLEX.
CO'HO-MOL'O-GY , adj. cohomology group . Let A' be an n -dimensional simplicial complex and let 
be the boundary operator, so that the boundary of a p -chain 
is 
Then 
and the boundary operator maps the group of 
into the group of 
In particular, 
for each p -simplex 
where 
are (p-l)-simplexes and 
arc elements of the group G whose elements are used as coeffi cients in forming chains. If the matrix 
its transpose is 
and can be used to define a mapping 
the chain 
being called the coboundary of 
This operator can be extended to all (p — l)-chains by the definition 
A chain is called a cocycle if its coboundary is zero. The r-dimeridional cohomology group is the quotient group, 
, where 
is the group of all i--dimensional cocyclcs of K and 
is the group of all cycles which arc 0 or are coboundaries of an (r-l)-chain of K. The concepts of homology and cohomology can be defined for certain generalizations of simplicial complexes (called complexes) for which each complex has a dual complex such that the homology group of one is the cohomology group of the other.