PHILOSOPHICAL SOCIETY OF WASHINGTON. 



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lines or parallelograms, for which the sum is equal to 34, and the 

 sum of two figures on the straight line equidistant from the center 

 ^17. 



Figure 4, which has the 6 squares on a side, will readily be 

 perceived to be symmetrical. The sum of two symmetrical num- 

 bers always being equal to 26, we have of course a great num- 

 ber of parallelograms, for which the sum of the numbers at the 

 angular points is always equal to 52, and a number of hexagons 

 having the sum of the numbers at the angular points equal to 78, 

 all which have this property that the diagonals joining opposite, 

 joints will always be mutually bisected at the centre, &c. 



The magic square of six on a side cannot be constructed, at least 

 not symmetrical like these others, this may be shown as follows : 



Fig. 5. 



If in figure 5 we attempt to fill up the diagonals so that the sum 

 of the extremes shall be equal to 37, their difierence must be an 

 odd multiple of 5 ; the only numbers that can fill these conditions 

 are: 



1 8 15 22 29 36 



6 11 16 21 26 31 



11 14 17 20 23 26 



16 17 18 19 20 21 



The 2d and 3d, 2d and 4th, and 3d and 4th cannot co-exist, 



