Miscellaneous Papers. 



245 



brecht Diirer in Nuremberg, Bavaria, in 1514, is a well-pre- 

 pared magic square. It appears to be a modification of the 

 square of Moschopulus, by reinversion, or reversing and in- 

 verting, and transposition of the second and third columns. 

 The date, 1514, however, would seem to antedate or be cotem- 

 porary with Moschopulus. The square is perfectly harmonic; 

 and on reinversion is exactly the same as my No. 47, scheme 

 IV, harmonic system, page 59 of volume XIV, Transactions 

 Kansas Academy of Science, 1894. On transposing the third 

 and fourth columns and the third and fourth lines it becomes 

 "perfect" and is No. 21 above. A comparison of Nos. 21 and 

 15 above will show the similarity of those two old magic 

 squares when reduced to a primary condition. They are both 

 probably modifications of some still older magic square. 



As each of the above twenty-four primary squares can be 

 transposed into fifteen other squares, it follows that 24x16, or 

 384, different perfect squares can be constructed from these 

 four schemes. No other schemes for perfect squares of 4 are 

 possible. 



A SIMPLE METHOD. 



A simple method of forming perfect squares of 4 without 

 the aid of a scheme or prearranged plan has been evolved by 

 Mr. D. H. Davison, of Minonk, 111. ; and as still further simpli- 

 fied and modified by the present author is here presented : 



FIRST STEP. 



SECOND STEP. THIRD STEP. 



No. 19. 



COMPLETE SQUARE. 



1. — (a) Place 1 in the upper left-hand corner and (b) its 

 complement, 16, in the alternate or second cell diagonally from 

 it; (c) place 2 in either one of the four cells next to 16, above, 

 below, or on either side, and (d) its complement, 15, in the 

 alternate cell diagonally from it. 



2. — (a) Place 3 in one of the three remaining cells next to 

 16, and (b) its mate, 4, in a cell relatively from 3 as 2 is 

 from 1. For instance, if 2 is in an alternate line and adjacent 

 column from 1, then 4 must be placed in an adjacent column 

 and alternate line from 3. Thus the harmony of the square is 

 preserved, (c and d) Their complements, 14 and 13, are to be 



