78 



KAXSAS ACADEMY OF SCIENCE. 



tion, adds as many times the central number as the number of places in the 

 row. This is not harmonic. 



c5^. 60. 



PENTAGONS. 



A magic pentagon is presented in figure 61, in which the rows of numbers 



tLSon.^ 



.»?i_3. 6/, 



on either side add as many times the central number as the number of 

 pentagons in the row. The five pentagons in any corner added to the middle 

 pentagon at the opposite side, adds six times the central number. 



The harmony of arrangement of the numbers may readily be seen on in- 

 spection. 



OCTAGONS. 

 A single octagon (a modified circle) is presented in figure 62, in which the 

 alternate series is used instead of the numerical series. In this octagon 



