238 Kansas Academy of Science. 



the series, but may be taken in any order whatsoever. An un- 

 used number of one set may be taken as a member of another 

 set. 



In odd squares all sets must be placed parallel to the first. 

 In even squares compensatory arrangements must be ob- 

 served so as to preserve a rhythmic or balanced effect. The 

 initial members of the several sets in a perfect square need 

 not be equidistant mathematically. It is only necessary in odd 

 squares that the initials of the several sets bear the same re- 

 lation to each other in position as the second does to the first, 

 and in even squares be opposed, so as to balance. 



SECTION 1. — Perfect Square of Four. 



Perfect squares of four cells on each side may be formed, 

 as are harmonic squares, according to certain schemes which 

 are here shown. They may be formed without the aid of a 

 visible scheme; but human ability to see a mental picture of 

 all numbers in position before writing any is not great and the 

 visible scheme is a great help in that direction. 



Before showing any of the schemes a few definitions would 

 seem to be desirable. 



DEFINITIONS. 



An adjacent number, line or column is the one next to it in 

 the same half square, as first and second are adjacent to each 

 other; third and fourth are adjacent. Second and third, though 

 contiguous, are not adjacent. An adjacent quarter is one on 

 the same side, whether vertically or horizontally. 



An altej'nate number, cell, line or column is the second re- 

 moved, or with one intervening, as first and third are alternate ; 

 second and fourth are alternate. 



An opposite line or column is the one in the opposite part of 

 the square that would come against it if we fold the square 

 along its middle line, as first and fourth are opposite; second 

 and third, though contiguous, are opposite. An opposite cell 

 or quarter is the one diagonally or diametrically opposite. 



A couplet is two numbers in succession in a series, con- 

 sisting, in a series whose first term is 1 and whose common 

 difference is 1, of an odd and an even number; and in any 

 other series consists of two numbers side by side when the en- 

 tire series is arranged in pairs from the beginning. The first 

 member of each couplet may be called the antecedent or leader 

 and the second the consequent or follower. 



A pair is two numbers standing side by side in the same 



