54 



KANSAS ACADEMY OF SCIENCE. 



the first couplet occupies the upper alternate and exterior columnar half; the 

 second couplet occupies the lower alternate and interior columnar half. The 

 next quaterniad is just the reverse, the first couplet occupying the lower alter- 

 nate interior half; the second couplet, the upper alternate exterior half. 



Now, after having constructed the scheme — which is the first thing necessary 

 to do in forming any harmonic square — the next step to take is to determine the 

 arrangement of the numbers in the square. To do this, the coupling lines are 

 to be numbered from 1 to 8 at their uppermost end. These numbers will then be 

 in two rows or lines, with number 1 in the upper left-hand corner. 



Numerous arrangements of these eight numbers may be made; and when 

 made under certain laws a harmonic square may always be produced. Such ar- 

 rangements may be divided into three series. 



Series 1. — Harmonic Squares. 



To produce perfectly harmonic squares, so arrange these eight numbers with 1 

 in the upper left-hand corner that they shall add 18 in each line and 9 in each 

 •column. These may be called trial arrangements. 



Under these restrictions, only six arrangements can be made, which, with 

 their corresponding primary squares, are here produced: 



1st AlT. 



17 6 

 8 2 3 



2d Arr. 

 17 4 6 

 8 2 5 3 



3d Arr. 



(16 7 4? 

 ^ 8 3 2 5 J 



The first (see fig. 27) is our square No. 1, from which Nos. 2 to 16 were pro- 

 duced. 



The second arrangement (fig. 28) gives square No. 17, from which 15 other 

 squares are derived in the same manner as the first 16 from No. 1. These are all 

 different from the first 16. 



From the third arrangement (fig. 29) another set of 16 squares can be con- 

 structed by transposition the same as the first 16 were from the first arrange- 

 ment. 



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