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Geometrical pacePace Pace, n. [OE. pas, F. pas, from L. passus a step, pace,
orig., a stretching out of the feet in walking; cf. pandere,
passum, to spread, stretch; perh. akin to E. patent. Cf.
Pas, Pass.]
1. A single movement from one foot to the other in walking; a
step.
2. The length of a step in walking or marching, reckoned from
the heel of one foot to the heel of the other; -- used as
a unit in measuring distances; as, he advanced fifty
paces. ``The heigh of sixty pace .' --Chaucer.
Note: Ordinarily the pace is estimated at two and one half
linear feet; but in measuring distances be stepping,
the pace is extended to three feet (one yard) or to
three and three tenths feet (one fifth of a rod). The
regulation marching pace in the English and United
States armies is thirty inches for quick time, and
thirty-six inches for double time. The Roman pace
(passus) was from the heel of one foot to the heel of
the same foot when it next touched the ground, five
Roman feet.
3. Manner of stepping or moving; gait; walk; as, the walk,
trot, canter, gallop, and amble are paces of the horse; a
swaggering pace; a quick pace. --Chaucer.
To-morrow, and to-morrow, and to-morrow, Creeps in
this petty pace from day to day. --Shak.
In the military schools of riding a variety of paces
are taught. --Walsh.
4. A slow gait; a footpace. [Obs.] --Chucer.
5. Specifically, a kind of fast amble; a rack.
6. Any single movement, step, or procedure. [R.]
The first pace necessary for his majesty to make is
to fall into confidence with Spain. --Sir W.
Temple.
7. (Arch.) A broad step or platform; any part of a floor
slightly raised above the rest, as around an altar, or at
the upper end of a hall.
8. (Weaving) A device in a loom, to maintain tension on the
warp in pacing the web.
Geometrical pace, the space from heel to heel between the
spot where one foot is set down and that where the same
foot is again set down, loosely estimated at five feet, or
by some at four feet and two fifths. See Roman pace in
the Note under def. 2. [Obs.]
To keep, or hold, pace with, to keep up with; to go as
fast as. ``In intellect and attainments he kept pace with
his age.' --Southey. Geometrical progressionProgression Pro*gres"sion, n. [L. progressio: cf. F.
progression.]
1. The act of moving forward; a proceeding in a course;
motion onward.
2. Course; passage; lapse or process of time.
I hope, in a short progression, you will be wholly
immerged in the delices and joys of religion.
--Evelyn.
3. (Math.) Regular or proportional advance in increase or
decrease of numbers; continued proportion, arithmetical,
geometrical, or harmonic.
4. (Mus.) A regular succession of tones or chords; the
movement of the parts in harmony; the order of the
modulations in a piece from key to key.
Arithmetical progression, a progression in which the terms
increase or decrease by equal differences, as the numbers
[lbrace2]2, 4, 6, 8, 1010, 8, 6, 4, 2[rbrace2] by the
difference 2.
Geometrical progression, a progression in which the terms
increase or decrease by equal ratios, as the numbers
[lbrace2]2, 4, 8, 16, 32, 6464, 32, 16, 8, 4, 2[rbrace2]
by a continual multiplication or division by 2.
Harmonic progression, a progression in which the terms are
the reciprocals of quantities in arithmetical progression,
as 1/2, 1/4, 1/6, 1/8, 1/10. geometrical proportionProportion Pro*por"tion, n. [F., fr. L. proportio; pro before
+ portio part or share. See Portion.]
1. The relation or adaptation of one portion to another, or
to the whole, as respect magnitude, quantity, or degree;
comparative relation; ratio; as, the proportion of the
parts of a building, or of the body.
The image of Christ, made after his own proportion.
--Ridley.
Formed in the best proportions of her sex. --Sir W.
Scott.
Documents are authentic and facts are true precisely
in proportion to the support which they afford to
his theory. --Macaulay.
2. Harmonic relation between parts, or between different
things of the same kind; symmetrical arrangement or
adjustment; symmetry; as, to be out of proportion. ``Let
us prophesy according to the proportion of faith.' --Rom.
xii. 6.
3. The portion one receives when a whole is distributed by a
rule or principle; equal or proper share; lot.
Let the women . . . do the same things in their
proportions and capacities. --Jer. Taylor.
4. A part considered comparatively; a share.
5. (Math.)
(a) The equality or similarity of ratios, especially of
geometrical ratios; or a relation among quantities
such that the quotient of the first divided by the
second is equal to that of the third divided by the
fourth; -- called also geometrical proportion, in
distinction from arithmetical proportion, or that in
which the difference of the first and second is equal
to the difference of the third and fourth.
Note: Proportion in the mathematical sense differs from
ratio. Ratio is the relation of two quantities of the
same kind, as the ratio of 5 to 10, or the ratio of 8
to 16. Proportion is the sameness or likeness of two
such relations. Thus, 5 to 10 as 8 to 16; that is, 5
bears the same relation to 10 as 8 does to 16. Hence,
such numbers are said to be in proportion. Proportion
is expressed by symbols thus: a:b::c:d, or a:b = c:d,
or a/b = c/d.
(b) The rule of three, in arithmetic, in which the three
given terms, together with the one sought, are
proportional.
Continued proportion, Inverse proportion, etc. See under
Continued, Inverse, etc.
Harmonical, or Musical, proportion, a relation of three
or four quantities, such that the first is to the last as
the difference between the first two is to the difference
between the last two; thus, 2, 3, 6, are in harmonical
proportion; for 2 is to 6 as 1 to 3. Thus, 24, 16, 12, 9,
are harmonical, for 24:9::8:3.
In proportion, according as; to the degree that. ``In
proportion as they are metaphysically true, they are
morally and politically false.' --Burke. geometrical squareQuadrat Quad"rat, n. [F. quadrat, cadrat. See Quadrate.]
1. (Print.) A block of type metal lower than the letters, --
used in spacing and in blank lines. [Abbrev. quad.]
2. An old instrument used for taking altitudes; -- called
also geometrical square, and line of shadows. Geometrically
Geometrically Ge`o*met"ric*al*ly, adv.
According to the rules or laws of geometry.
Ratio of a geometrical progressionRatio Ra"ti*o, n. [L., fr. reri, ratus, to reckon, believe,
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.
Meaning of Geometrical from wikipedia
- to
ratios of
geometrical quantities, and
contributed to the
development of
analytic geometry. Omar
Khayyam (1048–1131)
found geometric solutions to cubic...
-
Geometrical optics, or ray optics, is a
model of
optics that
describes light propagation in
terms of rays. The ray in
geometrical optics is an abstraction...
- A
geometric progression, also
known as a
geometric sequence, is a
mathematical sequence of non-zero
numbers where each term
after the
first is
found by...
- P. Moran,
Geometrical Probability.
Eugene Seneta,
Karen Hunger Parshall, François Jongmans, "Nineteenth-Century
Developments in
Geometric Probability:...
-
geometric algebra (also
known as a
Clifford algebra) is an
algebra that can
represent and mani****te
geometrical objects such as vectors.
Geometric algebra...
-
geometric analysis includes both the use of
geometrical methods in the
study of
partial differential equations (when it is also
known as "
geometric PDE")...
- In astronomy, the
geometric albedo of a
celestial body is the
ratio of its
actual brightness as seen from the
light source (i.e. at zero
phase angle)...
- In mathematics, a
geometric transformation is any
bijection of a set to
itself (or to
another such set) with some
salient geometrical underpinning, such...
- In mathematics, a
geometric series is a
series summing the
terms of an
infinite geometric sequence, in
which the
ratio of
consecutive terms is constant...
- In mathematics, the
geometric mean is a mean or
average which indicates a
central tendency of a
finite collection of
positive real
numbers by
using the...
