harmonical proportion

Proportion \Pro*por"tion\, n. [F., fr. L. proportio; pro before
   + portio part or share. See {Portion}.]
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   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.
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            The image of Christ, made after his own proportion.
                                                  --Ridley.
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            Formed in the best proportions of her sex. --Sir W.
                                                  Scott.
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            Documents are authentic and facts are true precisely
            in proportion to the support which they afford to
            his theory.                           --Macaulay.
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   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.
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   3. The portion one receives when a whole is distributed by a
      rule or principle; equal or proper share; lot.
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            Let the women . . . do the same things in their
            proportions and capacities.           --Jer. Taylor.
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   4. A part considered comparatively; a share.
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   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.
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   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:
         [1913 Webster] a:b::c:d, or a:b = c:d, or a/b = c/d.
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      (b) The rule of three, in arithmetic, in which the three
          given terms, together with the one sought, are
          proportional.
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   {Continued proportion}, {Inverse proportion}, etc. See under
      {Continued}, {Inverse}, etc.

   {Harmonical proportion} 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.
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