72 



KANSAS ACADEMY OF SCIENCE. 



Second Method, 



Here are presented the same squares as Nos. 1 and 2, constructed according 

 to the second method : 



No. 3. 



These squares are sufficiently harmonic to add 130 in all the different ways 

 indicated in Square of Four. 



No, 3 is obtained from No. 1 by transposition ; No. 4 from No. 2. 



Third Method. 



Here, now, are two squares written by the third method indicated. They are 

 derived by transposition from Nos. 3 and 4, respectively: 



No. 6. 



These squares, and all other squares built upon some definite plan, are per- 

 fectly harmonic, both as wholes and as parts. 



Knight's Tour Squares. 



These squares are constructed according to Schemes XI and XIV of the 

 Square of Pour. They solve the problem of passing the knight by his pecu- 

 liar leaps over the sixty-four squares of the chess-board without doubling or 

 missing a square. 



These squares are not harmonic. Of the thousands of different solutions 

 of the problem very few are harmonic to any degree. 



There are three methods of constructing squares according to these 

 schemes in order to have them at ail harmonic. 



1st. Reversing the direction after passing once around with each series of 

 sixteen numbers, thus making three reversals. 



