TWEXTY-SIXTn AN S UAL MEETIXG. 



29 



FIGURATE SERIES. 



By B. B. Smyth, Topeka. 



Between arithmetical and geometrical series are to be found many seriep 

 known as figurate series, based upon the fact that they rejjresent geometrical 

 figures, such as triangles, squares, pentagons, hexagons, octagons, tetrahedrons, 

 hexahedrons, cubes, parallelopipeds, pyramids, cones, cylinders, spheres, and 

 frustums of pyramids and cones. 



Figurate series are produced by addition, to each of the successive terms, of 

 a constantly increasing number, usually the preceding term, the last preceding 

 term but one, the same term of another series, or some other definite term, as 

 the case may require. 



Figurate series are of two kinds — planimetric and volumetric, the former sec- 

 ondary, the latter tertiary. Arithmetical series are primary and linear. 



In most of the following series each term is found by adding to the same 

 term of the preceding series the preceding term of the same series. ' It is equiva- 

 lent to the sum of all the terms of the preceding series up to and including the 

 same term. Thus the seventh term (84) of the tetrahedral series is equal to the 

 sum of the first seven terms of the trigonal series; the eighth term (6i) of the 

 square series is equal to the first eight terms of the alternate series. In the fol- 

 lowing examples linear series are marked {^), planimetric series are marked (2), and 

 volumetric series are marked (^). 



TIUGONAL SERIES. 



Terms: 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th, etc. 

 11111 1 1 1 1 



Numeral Series. 

 55 .... Trigonal Series. 



. Tetrahedral Series. 



(3) 1 5 14 30 55 91 140 204 285 385 



Hexahedral Series. 



This tetrahedron is, of course, a regular equilateral triangular pyramid. The 

 hexahedron is not a cube but a double triangular pyramid. Any term in the 

 hexahedral series is obtained by addition of two terms of the tetrahedral series — 

 a similar term with the preceding term. 



