66 



KANSAS ACADEMY OF SCIENCE. 



Second Group, 



13th Arr. 



No. 28. 



14th Arr. 



No. 29. 



G 3 



loth Arr. 

 I . 6 



7 4 



No. 30. 



( 1.6 3 8 ? 

 5 2 \ 



10th Arr. 

 6 3 8 



17th Arr. 

 (14 5 8? 

 \ 1 6 3 Z S 



No. 32. 



18th Arr. 

 14 5 

 6 7 2 



This makes 288 harmonic squares, of which one-third are perfectly harmonic 

 the other 192 less perfectly so. 



Series 3. — Magic Squares. 



Yet another series, in which the squares add equally in every column and line 

 and regularly in several other directions, but not diagonally, furnishes the means 

 of producing thousands of ordinary magic squares. 



In this series the trial arrangements add 18 in each line; the extremes add 18 



and the means 18. The resulting squares add 34 vertically, horizontally, and 



other regular ways, but not diagonally. The numbered trial arrangements are 



here shown in order : 



First Qroup, 



19th. 

 18 5 4 

 7 3 2 6 



20th. 



18 5 4 



6 2 3 7 



21st. 

 17 6 4 

 8 3 2 5 



22d. 

 17 6 4 

 5 3 2 8 

 27th. 

 15 8 4 

 7 3 2 6 



23d. 

 17 6 4 

 5 2 3 8 



28th. 

 15 8 4 

 7 2 3 6 



24th. 

 16 7 4 

 8 2 3 5 



29th. 

 15 8 4 

 6 2 3 7 



25th. 

 16 7 4 

 5 2 3 8 

 30th. 

 15 8 4 

 6 3 2 7 



26th. 

 16 7 4 

 5 3 2 8 



33d. 

 17 4 6 

 8 5 2 3 



3lRt. 



18 3 6 



7 5 2 4 



34th. 35th 



Second Oroup, 



32d. 

 18 3 6 

 4 2 5 7 

 36th. 37th. 



17 4 6 



3 5 2 8 



39th. 

 13 8 6 

 7 5 2 4 



17 4 6 

 3 2 5 8 



40th. 

 13 8 6 

 7 2 5 4 



14 7 6 

 8 2 5 3 



41st. 

 13 8 6 

 4 2 5 7 



14 7 6 

 3 2 5 8 



42d. 

 13 8 6 



4 5 2 7 



38th. 

 14 7 6 

 3 5 2 8 



