THE MAGIC SQUARE. 



gives the sum of 130. This square is the middle of a square of 6 

 times 6 places which so contains the numbers from 15 to 50 that 

 each row gives the sum 195. Finally, this last square is again the 

 middle of an ordinary magic square composed of the numbers from 

 i to 64. 



VII. 

 MAGICAL SQUARES WITH MAGICAL PARTS. 



If we divide a square of 8 times 8 places by means of the two 

 middle lines parallel to its sides into 4 parts containing each 4 times 

 4 spaces, we may propound the problem of so inserting the numbers 

 from i to 64 in these spaces that not only the whole shall form a 

 magic square, but also that each of the 4 parts individually shall be 

 magical, that is to say, give the same sum for each row. This prob- 

 lem also has been successfully solved, as the following diagram will 

 show. 



Fig. 27 



The 4 numbers in each row of any one of the sub-squares here, gives 

 130 ; so that the sum of each one of the rows of the large square 

 will be 260. 



Finally, in further illustration of this idea, we will submit to 

 the consideration of our readers a very remarkable square of the 

 numbers from i to 81. This square, which will be found on the 

 following page (Fig. 28), is divided by parallel lines into 9 parts, of 

 which each contains 9 consecutive numbers that severally make up 

 a magic square by themselves. 



Wonderful as the properties of this square may appear, the law 

 by which the author constructed it is equally simple. We have 



