64 



KANSAS ACADEMY OF SCIENCE. 



The above schedule does not include schemes that do not fulfil the require- 

 ments — to provide for "two odd numbers, and no more, in each column, line, 

 and diagonal." Even the schemes in the above schedule differ very much as to 

 their capacity. Those selected and shown by diagram (Schemes I to IV et seq.) 

 are the most perfect. 



Special Sums and Varying Series. 



Suppose it be desired to construct a square that will add a certain number in 

 each column or line, it is only necessary to take as extremes of the series two 

 numbers ivhose sum equals half of the number desired; then select the other 

 fourteen numbers of the series between those two. 



Nor is it necessary that the several terms of the series shall be equidistant; it 

 is only necessary that the common differences shall be harmonic, rhythmic, or 

 concordant. For instance, the series 1, 3, 6, 8, 12, 14, 17, 19, 23, 25, 28, 30, 34, 36, 

 39, 41 is harmonic; because the differences rise and fall with a certain rhythm 

 that fully satisfies the requirement. In other words, the differences 2, 3, 4, recur 

 regularly. 



The following examples are given to show some of the possibilities of these 

 harmonic squares. These squares are made according to square No. 1, 1st ar- 

 rangement, Scheme I : 



No. TO. 

 Common differ- 

 ences 1, 1, 1, "2. 



No. 60. 

 Common differ- 

 ences 1, 1, 2, 1. 



No. Gl. 

 Common differ- 

 ences 1, 1, 2, 2. 



No. 02. 

 Common differ- 

 ences 1, 1, 2, 'i. 



Sums = 36. 



Sums = 38. 



Sums = 40. 



Sums = 42. 



Or suppose, for instance, it be desired to make the sums equal 100, it can be 

 done in many different ways. The following examples from varying series will 

 show some of the modes of accomplishing this result: 



No. 63. 

 Common differ- 

 ences 3, 3, 4, 4. 



No. 64. 

 Common differ- 

 ences 1, 4, 8, 8. 



No. 65. 

 Common differ- 

 ences 2, 4, 5, 6. 



Sums = 100. 



Sums = 100. 



Sums = 100. 



Sums = 100. 



