AN'A-LYT'I-CAL-LY, adj. Performed by analysis, by analytic methods, as opposed to synthetic methods.
AN'A-LY-T1C'I-TY , n. point of analyticity.
A point at which a function of the complex variable z is analytic.
ANCHOR RING or TORUS . A surface in the shape of a doughnut with a hole in it; a surface generated by the rotation, in space, of a circle about an axis in its plane but not cutting the circle. If r is the radius of the circle. A: the distance from the center to the axis of revolution, in this case the z-axis, and the equation of the generating circle is 
then the equation of the anchor ring is 
Its volume is 
and the area of its surface is 
AN'GLE , n. A geometric angle (or simply angle) is a set of points consisting of a point P and two rays extending from P (sometimes it is required that the rays do not lie along the same straight line). The point P is the vertex and the rays are the sides (or rays) of the angle. Two geometric angles arc equal if and only if they are congruent. When the two rays of an angle do not extend along the same line in opposite directions from the vertex, the set of points between the rays is the interior of the angle. The exterior of an angle is the set of all points in the plane that are not in the union of the angle and its interior. A directed angle, then a radian measure of the angle for which one ray is designated as the initial side and the other as the terminal side. There are two commonly used signed measures of directed angles. Jf a circle is drawn with unit radius and center at the vertex of a directed angle, then a radian measure of the angle is the length of an arc that extends counterclockwise along the circle from the initial side to the terminal side of the angle, or the negative of the length of an arc that extends clockwise along the circle from the initial side to the terminal side. The arc may wrap around the circle any number of times. For example, if an angle has radian measure 
it also has radian measure 
etc., or 
Degree measure of an angle is defined so that 
corresponds to radian measure of 
[see sexagesimal —
sexagesimal measure of an angle], A rotation angle consists of a directed angle and a signed measure of the angle. The angle is a positive angle or a negative angle according as the measure is positive or negative. Equal rota tion angles are rotation angles that have the same measure. Usually, angle means rotation angle (e.g., see below, angle of depression, angle of inclination, obtuse angle). A rota such side of the polygon extended through the vertex, and whose measure is equal to the least positive measure that describes a rotation from one side of the angle to the other through the exterior of the polygon. At each vertex of a polygon, there is one interior angle and there are two exterior angles. These definitions suffice for any polygon for which no side contains points of more than two other sides (in other cases, the sides must be ordered in some way so that the angles between them can be defined uniquely).
angle of reflection . See reflection.
angle of refraction . See refraction.
base angles of a triangle . The angles in Ihe triangle having the base of the triangle for their common side.
central angle . See central.
complementary angles . See comple mentary.
conjugate angles . Two angles whose sum is 
Such angles arc sometimes said to be e xplements of each other,
coterminal angles . See cote.rminal.
dihedral angle . See dihedral.
direction angles . See direction —direction angles.
eccentric angle . See ELLIPSE.
Ruler's angles. See euler Euler's angles.
explementary angles . See above, conjugate angles.
face angles . See below, polyhedral angle.
flat angle . Same as straight angle.
hour angle of a celestial point. The angle between the plane of the meridian of the observer and the plane of the hour circle of the star, measured westward from the plane of the meridian. See hour —hour angle and hour circle,
interior angle . See above, angle of a polygon, angles made by a transversal.
measure of an angle . See angle, MIL, radian, SEXAGESIMAL—sexagesimal mea sure of an angle.
obtuse angle . An angle numerically greater than a right angle and less than a straight angle; sometimes used for all angles numeric ally greater than a right angle.
opposite angle . See opposite.
plane angle . See above, angle.
polar angle. See polar —polar coordinates in the plane.
polyhedral angle . The configuration formed by the lateral faces of a polyhedron which have a common vertex (A-BCDEF in the figure); the positional relation of a set of planes determined by a point and the sides of some polygon whose plane does not contain the point. The planes (ABC, etc.) are faces of
the angle; the lines of intersection of the planes are edges of the polyhedral angle. Their point of intersection (A) is the vertex. The angles (BAC, CAD, etc.) between two successive edges are face angles. A section of a polyhedral angle is the polygon formed by cutting all the edges of the angle by a plane not passing through the vertex.
quadrant angles . See quadrant.
quadrantal angles . See quadrantal.
reflex angle. An angle greater than a straight angle and less than two straight angles; an angle between 
and 
related angle . See related.
right angle . Half of a straight angle; an angle of 
or 
radians.
solid angle . See solid.
spherical angle . The figure formed at the intersection of two great circles on a sphere; the difference in direction of the arcs of two great circles at their point of intersection. In the figure, the spherical angle is APB. It is equal to the plane angles A'PB' and AOB. See spherical —spherical degree.
straight angle . An angle whose sides lie on the same straight line, but extend in opposite directions from the vertex; an angle of 
or 
radians. Syn. flat angle. supplementary angles. See supplementary.
tetrahedral angle . A polyhedral angle having four faces.
trihedral angle . A polyhedral angle having three faces.
trisection of an angle. See trisectiqn.
vertex angle . The angle opposite the base of a triangle.
vertical angles . Two angles such that each side of one is a prolongation, through the vertex, of a side of the other.
zero angle . The figure formed by two rays drawn from the same point in the same direction (so as to coincide); an angle whose measure in degrees is 0.
AN'GU-LAR , adj. Pertaining to an angle; circular; around a circle.
angular acceleration . See acceleration — angular acceleration.
angular distance . See distance- angular distance between two points.
angular momentum . See momentum — moment of momentum.
angular speed . See SPEED.
angular velocity . See VELOCITY.
AN'HAR-MON'IC RATIO . See RATIO - anharmonic ratio.
AN-NI'HI-LA'TOR, n. The annihilator of a set S is the class of all functions of a certain type which annihilate S in the sense of being zero at each point of S. E.g., if the Functions are continuous linear functional and S is a subset of a normed linear space N , then the annihilator of S is the linear subset S' of the first conjugate space N* which consists of all continuous linear functionals which are zero at each point of S. Analogously, the annihi lator of a linear subset S of Hilbert space is the orthogonal complement of S.
AN'NU-AL, adj. Yearly.
annual premium (net annual premium). See PREMIUM.
annual rent. Rent, when the payment period is a year. See rent.
AN-NU'I-TANT , n. The life (person) upon whose existence each payment of a life annuity is contingent, i.e., the beneficiary of an annuity.
AN-NU'I-TY, n. A series of payments at regular intervals. An annuity contract is a written agreement setting forth the amount of the annuity, its cost, and the conditions under which it is to be paid (sometimes called an annuity policy, when the annuity is a temporary annuity). The payment interval of an annuity is the time between successive payment dates; the term is the time from the beginning of the first payment interval to the end of the last one. An annuity is a simple annuity, or a general annuity, according as the payment intervals do, or do not, coincide with the interest conversion periods. A deferred annuity (or intercepted annuity) is an annuity in which the first payment period begins after a certain length of time has lapsed; it is an immediate annuity if the term begins immediately. An annuity due is an annuity in which the payments are made at the beginning of each period. If the payments are made at the end of the periods, the annuity is called an ordinary annuity. An annuity certain is an annuity that provides for a definite number of payments, as contrasted to a life annuity, which is a series of payments at regular intervals during the life of an individual (a single-life annuity) or of a group of individ uals (a joint-life annuity). A last survivor annuity is an annuity payable until the last of two (or more) lives end. An annuity whose payments continue forever is called a perpe- tuity. A temporary annuity is an annuity extending over a given period of years, provided the recipient continues to live throughout that period, otherwise terminating at his death. A reversionary annuity is an annuity to be paid during the life of one person, beginning with the death of another. An annuity whose payments depend upon certain conditions, such as some person (not neces sarily the beneficiary) being alive, is called a contingent annuity. A forborne annuity is a life annuity whose term began sometime in the past; i.e., the payments have been allowed to accumulate with the insurance company for a stated period. In case a group contributes to a fund over a stated period and at the end of the period the accumulated fund is con verted into annuities for each of the survivors, the annuity is also called a forborne annuity. A life annuity is curtate, or complete, according as a proportionate amount of a payment is not made, or is made, for the partial period from the last payment before death of the beneficiary to the time of death. A complete annuity is also called an apportionate annuity and a whole-life annuity. An annuity is increasing if each payment after the first is larger than the preceding payment; it is decreasing if each payment except the last is larger than the next payment. Also see tontine —tontine annuity.
accumulated value of an annuity . The accumulated value (or amount) of an annuity at a given date is the sum of the compound amounts of the annuity payments to that date. The amount of an annuity is ihe accumulated value at the end of the term of the annuity.
annuity bond . See B ond. cash equivalent (or present value) of an annuity. See value —present value.
consolidated annuities (consols). See CONSOLIDATED.
ANN'U-LUS , n. [pl. annuli]. The portion of a plane bounded by two concentric circles in the plane. The area of an annulus is the difference between the areas of the two circles, namely 
where R is the radius of the larger circle and 
is the radius of the smaller.
A-NOM'A-LY , n. anomaly of a point. Sec polar— polar coordinates in the plane,
AN-TE-CED'ENT, n. (1) The first term (or numerator) of a ratio; that term of a ratio which is compared with the other term. In the ratio 2/3, 2 is the antecedent and 3 is the consequent, (2) See implication,
AN'TI-DE-RIV'A-TIVE , n, antiderivative of a function. Same as the primitive or INDEFINITE integral of the function. See integral — indefinite integral.
AN'TI-HY-PER-BOL'IC functions. Same as I NVERSE hyperbolic funclions. See hyper- bolic —inverse hyperbolic functions.
AN-TI-LOG'A-RITHM , n. antilogarithm of a given number. The number whose logarithm is the given number; e.g., 
2 = 100,
Syn. inverse logarithm. To find an anti- logarithm corresponding to a given logarithm that is not in the tables, subtract the next smaller mantissa from the given one and from the next larger one, divide the former differ ence by the latter and annex the quotient to the number corresponding to the smaller mantissa. See characteristic.
AN'TI-PAR'AL-LEL, adj. anliparailel lines. Two lines which make, with two given lines,
angles that are equal in opposite order. In the figure, the lines AC and AD are anliparallel with respect to the lines EB and EC, since 
and 
Two parallel lines have a similar property. The parallel lines A D and GH also make equal angles wilh the lines EB and EC, but in the 
same order; i.e., 
and antiparallel vectors. See parallel —parallel vectors.
AN-TiP'O-DAL, adj. antipodal points. Points on a sphere at opposite ends of a diameter,
AN'TI-SYM-MET'RIC, adj. antisymmetric dyadic. See dyad.
AN'TI-TRIG'O-NO-MET'RIC , adj. antitrigonometric function. Same as inverse
TRIGONOMETRIC FUNCTION. See TRIGONOMET RIC—inverse trigonometric funetion.
A'PEX , n. [pl. apexes or apices]. A highest point relative to some line or plane. The apex of a triangle is the vertex opposite the side which is considered as the base; the apex of a cone is its vertex.
APOLLONIUS of Perga (c. 255-170 B.C.), A great Greek geometer.
Problem of Apollonius. To construct a circle tangent to three given circles.
A POS-TE'RI-O'RI, adj. a posteriori knowl edge. Knowledge from experience. Syn. Empirical knowledge.
a posteriori probability. See probability — empirical or a posteriori probability,
A-POTH'E-CAR-Y , n. apothecaries' weight.
The system of weights used by druggists. The pound and the ounce arc the same as in troy weight, but the subdivisions are different. See denominate numbers in the appendix.
AP'O-THEM, n. The perpendicular distance from the center of a regular polygon to a side. Syn. short radius.
AP-PAR'ENT , adj. apparent distance. See
distance —angular distance between two points.
apparent time. Same as apparent solar time. See time.
APPLIED MATHEMATICS. See mathematics.
AP-POR'TION-A-BLE, adj. apportionable annuity. See annuity.
AP-PROACH' , p. approach a limit. See LIMIT.
AP-PROX'I-MATE , adj., v. To calculate nearer and nearer to a correct value; used mostly for numerical calculations. E.g., one approximates the square root of 2 when he finds, in succession, the numbers 1.4, 1.41, 1.414, whose successive squares are nearer and nearer to 2. Finding any one of these decimals is also called approximating the root; that is, to approximate may mean either to secure one result near a desired result, or to secure a succession of results approaching a desired result.
approximate result, value, answer, root, etc. One that is nearly but not exactly correct. Sometimes used of results either nearly or exactly correct. See root —root of an equation.
AP-PROX'I-MA'TION , n (!) A result that is not exact, but is accurate enough for some specific purpose. (2) The process of obtaining such a result.
approximation by differentials . See differential .
successive approximations . The successive steps taken in working toward a desired result or calculation. Sec approximate.
A PRI-O'RI , adj. a priori fact. Used in about the same sense as axiomatic or self- evident fact.
a priori knowledge . Knowledge obtained from pure reasoning from cause to effect, as contrasted to empirical knowledge (knowledge obtained from experience); knowledge which has its origin in the mind and is (supposed to be) quite independent of experience.
a priori reasoning . Reasoning which arrives at conclusions from definitions and assumed axioms or principles; deductive reasoning.
AR'A-BIC, adj. Arabic numerals: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0: introduced into Europe from Arabia, probably originating in India. Also
Called HINDU-ARABIC NUMERALS.
AR'BI-TRAR'Y , adj. arbitrary assumption.
An assumption constructed at the pleasure of the individual without regard to its being con sistent either with the laws of nature or (some times) with accepted mathematical principles.
arbitrary constant. See constant.
arbitrary 
. A statement is true for arbitrary 
if it is true for any numerical value (usually restricted to be positive) which may be assigned to 
. This idiom usually occurs in situations where small values of 
are of the most interest.
arbitrary function in the solution of partial differential equations. A symbol that stands for an unspecified function in an expression that satisfies the differential equation whatever function (of some specified type) may be substituted for the symbol. E.g., 
is a solution of 
if 
is any differentiable function.
arbitrary parameter. Same as parameter in its most commonly used sense. The addition of the attribute arbitrary places emphasis upon the fact that this particular parameter is not determined, but may be any member of some set (e.g., any real number).
ARC' , n. A segment, or piece, of a curve. Tech. (1) The image of a closed interval [a, b ] under a one-to-one continuous trans formation; i.e., a simple curve that is not closed. (2) A curve that is not a closed curve. If a curve is the continuous image of the interval [a, b ], then an arc of the curve is any arc that is the image of an interval lc,d] contained in [a,b|.
arc length. The length of an arc. See length —length of a curve.
degree of arc. An arc of a circle is an arc of one degree if it subtends an angle of one degree at the center of the circle. The measure of an arc in degrees is the measure of the angle subtended at the center of the circle.
differential (or element) of are. See element — element of integration.
limit of the ratio of an arc to i ts chord. See limit— limit of the ratio of an arc to its chord.
minor arc of a circle. Sec sector —sector of a circle. Syn, short arc.
ARC CO-SE'CANT, n. The arc cosecant of a number x is an angle (or number) whose cosecant is x, written 
x or arc csc x .
E.g., 
2 is 
or in general,
See trigonometric —in-
verse trigonometric function. Syn. inverse cosecant, anticosccanl. (In the above figure, >? is in radians.)