80 



KANSAS ACADEMY OF SCIENCE. 



A new hexagonal star is presented in figure 66, in which each of the ten 

 full lines of five places adds five times the central number, and each of the 





corners except two, with the adjacent three places, adds four times the central 

 number. This star is not harmonic. 



A magic pentagonal star is shown in figure 67, in which each of the full 

 lines except two adds five times the central number. 



ijLCth}' too. 



In the octsgonal star (figure G8) each of the full lines except two adds five 

 times the central number. The square in the center adds equally in every 

 direction. It is not harmonic, though many arrangements may be made. 



In the small circle (fig. 69) the circumference equals three times the 

 diameter in both inner and outer circles, adding diameters of circles sep- 

 arately. 



In figure 70, v/hich is the square of 4 in a new form, the diameter and the 

 quarters are equal, as in the case of the octagon (fig. 62), which is a modifica- 

 tion of this; and the circumference, owing to the series, is slightly less than 



