inverse. See {Invert}.]

1. Opposite in order, relation, or effect; reversed;

inverted; reciprocal; -- opposed to {direct}.

2. (Bot.) Inverted; having a position or mode of attachment

the reverse of that which is usual.

3. (Math.) Opposite in nature and effect; -- said with

reference to any two operations, which, when both are

performed in succession upon any quantity, reproduce that

quantity; as, multiplication is the inverse operation to

division. The symbol of an inverse operation is the symbol

of the direct operation with -1 as an index. Thus sin-1 x

means the arc whose sine is x.

{Inverse figures} (Geom.), two figures, such that each point

of either figure is inverse to a corresponding point in

the order figure.

{Inverse points} (Geom.), two points lying on a line drawn

from the center of a fixed circle or sphere, and so

related that the product of their distances from the

center of the circle or sphere is equal to the square of

the radius.

{Inverse}, or {Reciprocal}, {ratio} (Math.), the ratio of the

reciprocals of two quantities.

{Inverse}, or {Reciprocal, {proportion}, an equality between

a direct ratio and a reciprocal ratio; thus, 4 : 2 : : 1/3

: 1/6, or 4 : 2 : : 3 : 6, inversely.

think, judge. See {Reason}.]

1. (Math.) The relation which one quantity or magnitude has

to another of the same kind. It is expressed by the

quotient of the division of the first by the second; thus,

the ratio of 3 to 6 is expressed by 3/6 or 1/2; of a to b

by a/b; or (less commonly) the second is made the

dividend; as, a:b = b/a.

Note: Some writers consider ratio as the quotient itself,

making ratio equivalent to a number. The term ratio is

also sometimes applied to the difference of two

quantities as well as to their quotient, in which case

the former is called arithmetical ratio, the latter,

geometrical ratio. The name ratio is sometimes given to

the rule of three in arithmetic. See under {Rule}.

2. Hence, fixed relation of number, quantity, or degree;

rate; proportion; as, the ratio of representation in

Congress.

{Compound ratio}, {Duplicate ratio}, {Inverse ratio}, etc.

See under {Compound}, {Duplicate}, etc.

{Ratio of a geometrical progression}, the constant quantity

by which each term is multiplied to produce the succeeding

one.

priv. + ? harmonic.] (Math.)

Not harmonic.

{The anharmonic function} or {ratio} of four points abcd on a

straight line is the quantity (ac/ad):(bc/bd), where the

segments are to be regarded as plus or minus, according to

the order of the letters.

to double, fr. duplex double, twofold. See {Duplex}.]

Double; twofold.

{Duplicate proportion} or {ratio} (Math.), the proportion or

ratio of squares. Thus, in geometrical proportion, the

first term to the third is said to be in a duplicate ratio

of the first to the second, or as its square is to the

square of the second. Thus, in 2, 4, 8, 16, the ratio of 2

to 8 is a duplicate of that of 2 to 4, or as the square of

2 is to the square of 4.

geometricus; Gr. ?: cf. F. g['e]om['e]trique.]

Pertaining to, or according to the rules or principles of,

geometry; determined by geometry; as, a geometrical solution

of a problem.

Note: Geometric is often used, as opposed to algebraic, to

include processes or solutions in which the

propositions or principles of geometry are made use of

rather than those of algebra.

Note: Geometrical is often used in a limited or strictly

technical sense, as opposed to mechanical; thus, a

construction or solution is geometrical which can be

made by ruler and compasses, i. e., by means of right

lines and circles. Every construction or solution which

requires any other curve, or such motion of a line or

circle as would generate any other curve, is not

geometrical, but mechanical. By another distinction, a

geometrical solution is one obtained by the rules of

geometry, or processes of analysis, and hence is exact;

while a mechanical solution is one obtained by trial,

by actual measurements, with instruments, etc., and is

only approximate and empirical.

{Geometrical curve}. Same as {Algebraic curve}; -- so called

because their different points may be constructed by the

operations of elementary geometry.

{Geometric lathe}, an instrument for engraving bank notes,

etc., with complicated patterns of interlacing lines; --

called also {cycloidal engine}.

{Geometrical pace}, a measure of five feet.

{Geometric pen}, an instrument for drawing geometric curves,

in which the movements of a pen or pencil attached to a

revolving arm of adjustable length may be indefinitely

varied by changing the toothed wheels which give motion to

the arm.

{Geometrical plane} (Persp.), the same as {Ground plane} .

{Geometrical progression}, {proportion}, {ratio}. See under

{Progression}, {Proportion} and {Ratio}.

{Geometrical radius}, in gearing, the radius of the pitch

circle of a cogwheel. --Knight.

{Geometric spider} (Zo["o]l.), one of many species of

spiders, which spin a geometrical web. They mostly belong

to {Epeira} and allied genera, as the garden spider. See

{Garden spider}.

{Geometric square}, a portable instrument in the form of a

square frame for ascertaining distances and heights by

measuring angles.

{Geometrical staircase}, one in which the stairs are

supported by the wall at one end only.

{Geometrical tracery}, in architecture and decoration,

tracery arranged in geometrical figures.

n : the relative magnitudes of two quantities (usually expressed

as a quotient)

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