TWEXTY-SIXTH ANNUAL MEETING. 



69 



No. 6. 



No. 7, 



No. 8. 



Any number in this square except 7, 9, 10, 12, 16, 28, 30, and 36, may be 

 brought into the upper left-hand corner. These eight numbers immediately 

 surround the central square of four numbers in square No. 9. These numbers, 

 too, may be put into the same corner; but in that case the sums of the 

 diagonal lines will not be 111. 



Squares like the above may be produced almost without limit. They are, 

 however, not harmonic. In explanation of this it may be said that a per- 

 fectly harmonic square of 6 places on a side is impossible; since, in every at- 

 tempt at it there will always be found left four numbers that can only be put 

 in according to the square of two, and since this adds equally only twice, 

 while six additions are needed, a choice must be made. 



I now present a square on the old-fashioned plan— that of transposing 

 numbers from their regular order vertically and horizontally. There is no 

 special merit in this; and I only present it to show that, in addition to the 

 lines, columns, and diagonals adding regularly 111, the six numbers on either 

 side of the center, and some curious basket-shaped figures (to be pointed out) 

 near the center also add the same. For instance, the following numbers add 

 together 111: 16, 23, 21, 27, 14, 10; 15, 10, 22, 20, 27, 17; 22, 14, 15, 9, 23, 28; 21, 28, 

 16. 17, 9, 20; 21, 13, 8, 4. 30, 35; 28, 9, 23, 15, 16, 20; 14, 21, 22, 17, 10, 27; 23, 

 14, 28, 15, 21, 10; and, finally, 9, 16, 22, 27, 20, 17. Almost any number will add 

 one-third of 111 [37] with the number diametrically opposite; and any four 

 numbers at the corners of a rectangle, rhomb, rhomboid, or square, having for 



