Engler — Figurate Numbers. 



55 



Pyrmnidal numbers. These are the numbers in the 

 third row of Table 21 for ^=' successively 1, 2, 3, . . . . 



For^=l, 1 4 10 20 . . (Triangular Pyramid) 

 d=2, 1 5 14 30 . . (Square Pyramid) 

 ^=3, 1 6 18 40 . . (Pentagonal Pyramid) 



They are called pyramidal because the numbers corre- 

 spond to the numbers of points or dots which can be 

 arranged in the form of regular pyramids with the reg- 

 ular polygons of 3, 4, 5 . . . sides for the bases in the 

 respective cases. 



Evidently an indefinite number of sets of series com- 

 parable with the above may be obtained by varying the 

 combination of the columns involving d with the columns 

 of Table 6. Two such sets of series have been exhibited 

 in Tables 13 and 21. 



All such sets of series will satisfy the definition of 

 figurate numbers. The numbers of such series is infinite. 



But it should be noted that the definition of figurate 

 numbers usually given, namely that the n^^ term of any 

 series is equal to the sum of the first n terms of the pre- 

 ceding series, applies also to an infinite number of other 

 sets of series not included among those previously indi- 

 cated and not usually classed as figurate numbers. Such, 

 for example, is the set of series, arbitrarily chosen, 



Table 30. 



which may be extended up and down indefinitely. Evi- 

 dently from any arbitrary series of integers, such a set 



