TWENTY-SIXTH ANNUAL MEETING. 



65 



All of the above squares are harmonic, and have all the characteristics of 

 the squares of Scheme I, notwithstanding their varying differences. 



There are four grades of differences in all these squares, namely: 



1st. The differences between adjacent numbers, as between the first and 

 second, the third and fourth, etc. There are eight of these. 



2d. The differences between adjacent pairs, thus coming in the middle of 

 each quaterniad. There are four of these. 



3d. The differences between adjacent quaterniads. There are two of these. 



4th. The difference between the two halves of a series. This is but one. 



The least difference that can be used, without repetition of a number, is 1. 

 Any excess of difference between the extremes of a series more than 1 for each 

 term may be used entirely in the 4th grade, or may be distributed among all 

 the grades. When there is an excess of 1 it must be used in the 4th grade, 

 as in the square "Sums equal 36" above. An excess of 2 or more may be 

 used in the 3d or 4th grade, or both, as in the three squares "sums equal 38, 

 40, and 42" above. An excess of 4 or more may be used in either or all of the 

 last three grades. Ah excess of 8 or more may be distributed anywhere, as in 

 the squares "sums equal 100" above. Excess of difference must be distributed 

 according to the following 



Rule. — So distribute the excess that the differences in any one grade will 

 be exactly the same throughout the varying series. 



This can easily be done thus: 



A difference of 1 in the first grade equals 8 



A difference of 1 in the second grade equals 4 



A difference of 1 in the third grade equals 2 



A difference of 1 in the fourth grade equals 1 



Total 15 



A difference of 1 in the first grade is equal to 2 in the. second, 4 in the 

 third, and S in the fourth, as to results. Thus may any number be easily 

 distributed and series so arranged that squares may be made to sum up any 

 even number. 



Odd Sums. 

 Should it be desired to make a square add aa odd number, it may be ac- 

 complished without fractions by making an extra difference of 1 in the last 

 position of third grade, thus: 



No. 67. 



No. 68. 



Sums = 35. 



Sums = 37. 



In which an extra difference of 1 is placed between the third and fourth sets. 

 These squares, which add 35 and 37, respectively, are arranged according to the 

 third arrangement of scheme I, as shown in square No. 18, that arrangement 

 being best adapted to this purpose. But such squares, though they answer 

 every requirement of a " magic square," are nor harmonic. 



