TWJJMY-SIXTn ANNUAL MEETING. 



61 



No. 55. 



Scheme XII (f3g. 46) is the same as scheme IX 

 turned over on its direct central diagonal. The two i ] ^ 

 columns of the trial arrangement govern the outer 1 6 3 

 opposite columns of the squares. 



1 8 



7 2 



6 3 



4 5 



Scheme XIII (fig. 47) is like scheme X turned over on 

 its direct diagonal. It is governed the same as scheme X. 



Scheme XIV, Knight's tour (fig. 48), can be used sepa- 

 rately or in combination with scheme XI. 



1 6 



5 2 



8 3 



4 7 



No. 57. 



The regular transpositions allowed in the squares of all other schemes are not 

 applicable to this or to scheme XI. These two schemes are necessarily governed 

 by rules of their own. 



About 40 other schemes may be constructed, all more or less defective, though 

 many squares may be made from them, sometimes remedying the defect by trans- 

 posing two pairs of contiguous couplets, or by other means. But schemes that 

 are but poorly adapted to squares of four are well adapted to squares of eight; so 

 that perfect squares of 64 places are much easier made than squares of 16 places, 

 and can be made in much greater variety. 



Here are two more schemes. The first is that used in laying 16-page forms in 

 ordinary book i^rinting. 



Scheme XV. 



t7v>. ■^f 



And the one arrangement necessary for 

 proper folding is: 



(18 7 2^ Which gives the primary 

 < 4 5 6 3 ) square: 



No. 58. 

 Primary. 



This primary square shows the relatiffe positions of the pages in the printed 

 sheet before folding. 



The pages are laid in inverted order on the imposing stone. They may also, 

 for various reasons, be laid in three other positions, as shown in the four follow- 

 ing transpositions: 



