geometrical progression


From The Collaborative International Dictionary of English v.0.48:

Geometric \Ge`o*met"ric\, Geometrical \Ge`o*met"ric*al\, a. [L.
   geometricus; Gr. ?: cf. F. g['e]om['e]trique.]
   1. Pertaining to, or according to the rules or principles of,
      geometry; determined by geometry; as, a geometrical
      solution of a problem.
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   2. (Art) characterized by simple geometric forms in design
      and decoration; as, a buffalo hide painted with red and
      black geometrical designs.

   Syn: geometric.
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   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.
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   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.
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   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 (Zool.), 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.
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From The Collaborative International Dictionary of English v.0.48:

Progression \Pro*gres"sion\, n. [L. progressio: cf. F.
   progression.]
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   1. The act of moving forward; a proceeding in a course;
      motion onward.
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   2. Course; passage; lapse or process of time.
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            I hope, in a short progression, you will be wholly
            immerged in the delices and joys of religion.
                                                  --Evelyn.
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   3. (Math.) Regular or proportional advance in increase or
      decrease of numbers; continued proportion, arithmetical,
      geometrical, or harmonic.
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   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.
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   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.
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   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.
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   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.
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