DICT.TW Dictionary Taiwan

 Tri·an·gu·lar a.
 1. Having three angles; having the form of a triangle.
 2. Bot. Oblong or elongated, and having three lateral angles; as, a triangular seed, leaf, or stem.
 Triangular compasses, compasses with three legs for taking off the angular points of a triangle, or any three points at the same time.
 Triangular crab Zool., any maioid crab; -- so called because the carapace is usually triangular.
 Triangular numbers Math., the series of numbers formed by the successive sums of the terms of an arithmetical progression, of which the first term and the common difference are 1. See Figurate numbers, under Figurate.
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 Fig·ur·al a.
 1. Represented by figure or delineation; consisting of figures; as, figural ornaments.
 2. Mus. Figurate. See Figurate.
 Figural numbers. See Figurate numbers, under Figurate.
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 Fig·ur·ate a.
 1. Of a definite form or figure.
    Plants are all figurate and determinate, which inanimate bodies are not.   --Bacon.
 2. Figurative; metaphorical. [Obs.]
 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.
 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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