Miscellaneous Papers. 255 



1X4= 4 



13X3 = 39 



8X2 = 16 



5X1= 5 



27 64 



For example, take the numbers 1, 14, 22, 27, as in the first 

 line of square No. 25. The first and lowest number in the line 

 is 1 ; four times that is 4. The next number is 14 ; the differ- 

 ence between that and 1 is 13 ; three times 13 is 39. The next 

 difference is 8 ; twice that is 16. Finally, the difference? be- 

 tween 22 and 27 is 5 ; once that is S. The sum of these differ- 

 ences equals 27, the last number taken; the sum of the pro- 

 ducts equals 64, equal to the sum of the numbers. 



SQUARE. 10X4= 40 



10 36 30 24 ^fivl-_II 



31 23 11 35 _fiOi- _R 



20 26 40 14 _OAi- D 



39 15 19 27 24 lOO 



Again, take the numbers 10, 36, 30, 24, as in the above 

 square, in the order in which they occur in the line. The first 

 is 10, which, multiplied by 4 equals 40 ; the difference between 

 10 and 36 is 26, which multiplied by 3 equals 78; the next 

 difference is 6 minus, which multiplied by 2 gives 12 to be sub- 

 tracted; the final difference is 6, also to be subtracted. The 

 sum of the differences is 24,^ the last number taken ; the sum of 

 the products is 100, equal to the sum of the numbers in the line. 



Several other examples are here given of the same problem 

 performed in various ways, but always ending in the same re- 

 sult, namely : The sum of the differences is always equal to the 

 last number taken in the operation ; the sum of the products is 

 always equal to the total of all the numbers taken. The ex- 

 amples above given can undoubtedly be understood by in- 

 spection without further elucidation. 



10 100 30 100 



If, after arranging a series of numbers in four sets pre- 

 paratory to constructing a magic square, we take the successive 

 differences in the first set and multiply them in order by 4, 3, 

 2, and 1, respectively, and to the products add the differences 

 between the initials of the sets multiplied in consecutive order 

 by 3, 2, and 1, we obtain as sum of the differences the highest 



