figurate numbers

Figurate \Fig"ur*ate\, a. [L. figuratus, p. p. of figurare. See
   {Figure}.]
   1. Of a definite form or figure.
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            Plants are all figurate and determinate, which
            inanimate bodies are not.             --Bacon.
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   2. Figurative; metaphorical. [Obs.] --Bale.
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   3. (Mus.) Florid; figurative; involving passing discords by
      the freer melodic movement of one or more parts or voices
      in the harmony; as, figurate counterpoint or descant.
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   {Figurate counterpoint} or {Figurate descant} (Mus.), that
      which is not simple, or in which the parts do not move
      together tone for tone, but in which freer movement of one
      or more parts mingles passing discords with the harmony;
      -- called also {figural}, {figurative}, and {figured
      counterpoint} or {descant} (although the term figured is
      more commonly applied to a bass with numerals written
      above or below to indicate the other notes of the
      harmony).

   {Figurate numbers} (Math.), numbers, or series of numbers,
      formed from any arithmetical progression in which the
      first term is a unit, and the difference a whole number,
      by taking the first term, and the sums of the first two,
      first three, first four, etc., as the successive terms of
      a new series, from which another may be formed in the same
      manner, and so on, the numbers in the resulting series
      being such that points representing them are capable of
      symmetrical arrangement in different geometrical figures,
      as triangles, squares, pentagons, etc.

   Note: In the following example, the two lower lines are
         composed of figurate numbers, those in the second line
         being triangular, and represented thus:
         . 1, 2, 3, 4, etc. . . . 1, 3, 6, 10, etc. . . . . . .
         . etc. 1, 4, 10, 20, etc . . . . . . . . . . . .
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