T^VENTY-SIXTH ANXUAL MEETING. 



71 



In No. 1 the sixty-four numbers are taken as two series, the odd numbers 

 in one series, the even numbers in another; then four consecutive odd num- 

 bers are taken in one group to represent an odd number, four even ones to 

 represent an even number, then filling as a quadruple square of 4. 



In No. 2 the sixty-four numbers are divided into four series, and the first 

 number of each series is taken to fill the first quadruple member of the 

 square, the second number to fill the second member, and so on. 



These two are not the only ways that numbers may be taken to fill the 

 squares. The sixty-four numbers may be divided into series of 32, 16, 8, 

 4, 2 or 1 places; the series may be regular, alternate, or half complementary; 

 they may be direct or half reversed; each group of four may be taken in reg- 

 ular order, either direct, reversed, or crosswise of the series, or one-half 

 may be complementary to the other half. 



Any of the schemes and arrangements in Square of Four is applicable to 

 this square; and many of them are better adapted to this square than to 

 the Square of Four. 



The harmony of the square, aside from the scheme and arrangement, is 

 owing largely to 



(a) The preparation of the series; 



(b) The mode of selection of the groups to fill the quadruple squares; and 



(c) The mode of laying the several groups. 



Plan. 



In laying any of these squares, in order to have them perfectly harmonic, 

 it will be necessary to observe sevei'al precautions: 



1st. The several groups of four numbers must be so commenced in the dif- 

 ferent quadruple squares that no two in the same double line or double col- 

 umn shall begin in the same relative position, and not more than two in 

 each diagonal. 



2d. Increase in the first square may be made in either of three directions; 

 but no two of the quadruple squares in any column or line should increase in 

 the same direction, and no more than two in the diagonals. 



This may be accomplished by commencing the different quadruple squares 

 according to a prepared plan, as either of the following, the numbers indi- 

 cating the relative square in each quadruple square: 



And increase in each of the quadruple squares may be made thus: 



These plans are offered merely as suggestions. It will readily be seen that 

 many different plans and modes may be adopted, including four diagonal direc- 

 tions. 



