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.
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.
{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.
{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.
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.