570 



SCIENCE, 



Science, 

 Curiosities 



any where throughout the whole square,.that sum being 

 in all cases -.double the amount' of the first and last num- 

 bers of the progression. 



In accomplishing this, Dr. Franklin. gave up the pro* 

 perty which these squares were generally made to pos- 

 sess, that of the two entire .diagonals being each the 

 same sum as the other columns. As by a different ma- 

 nagement this property may be retained, we shall not 

 enter particularly into his method; but proceed directly 

 to show how magic squares may ba constructed, pos- 

 sessing all the properties of Dr. Franklin's square, con- 

 joined with most of those which were formerly known. 



As we shall frequently have occasion to mention, the 

 square of 4, we shall give to squares of that size the 

 name of petty squares, and, as was observed in a preced- 

 ing section, we shall call the first half of the numbers 

 in a natural square minors, and the other half majors. 



In the petty square B, page 568, the mi- 

 nors are disposed as annexed. Adding 

 them vertically, each column amounts to 9. 

 Adding them horizontally, the amount is 

 5, 13, 13, 5, two similar numbers being the 

 sum of the extreme, and two others the 

 sum of the middle columns. The majors, 

 it will be seen, are so placed, that the sum 

 of each, and its adjacent minor, is alter- 

 nately 16 and 18 in the horizontal columns. 



The position of the minors in the petty squares, 

 of which the magic square of 8 is composed, is simi- 



the same 



square are 

 Those in 



H 



17 



2223 



19 



18 



20 



21 



lar. The minors in the first 

 as those in B, mentioned above, 

 the other squares, although composed of 

 higher numbers, are arranged in the same 

 order. Thus the minors in the third square 

 are as in H annexed, where it may bs ob- 

 served, that the numbers as they ascend 

 from 17 to 24 occupy cells similarly situ- 

 ated as the numbers ascending from 1 to 8 

 in the square B. 



This disposition of the lower numbers in the petty 

 squares, which may be varied in a great number of 

 ways, we shall in future distinguish under the general 

 appellation of the Arrangement of the Minors. 



By the arrangement of the minors in the preceding 

 squares, .they possess the property of the sum of the di- 

 agonals being each equal to the sum of the vertical or ho- 

 rizontal columns, but they are deficient of that property 

 possessed by Dr. Franklin's square, of the sums of any four 

 contiguous cells being the same through 

 the square, and no possible alteration of 

 the position of the majors can give them 

 this property. The arrangement of the 

 minors must be altered. 



One of the numerous arrangements suit- 

 able for this purpose is shown in I annex- 

 ed, where it is observed, that the sums of 

 the horizontal columns are alternately 5, 

 13 r 5, 13, instead of 5, 13, 13, 5, as in the 

 preceding squares. 



equal to l() and 18 alternately, the proper 

 t-y wanted will be found, as in K annexed. 



Arranging the minors of the square of 

 8, in the above order, they will stand 

 as in the square L, and adding the ma- 

 jors so as to make the amount of each 

 with its minor in the horizontal line, tlter- 

 nately 64 and 66, the square will be completed as seen 

 in M... 



L M 



In this square, it may be observed, that each of its 

 petty squares is possessed of the desired properties ; 

 but the centre square is- not, as at the junction of-the 

 petty squares some of the clusters of 4 cells contain 

 numbers, whose amount is not exactly 130, as it. ought 

 to be. Thus the numbers in the four cells in the cen- 

 tre, are 59, 46, 25, and 16, amounting to 146. Here 

 they exceed the proper sum, in other places they fall 

 below it: 



This defect cannot be remedied, by altering the or- 

 rangement of the minors in the petty squares. It is to 

 be effected, by distributing the minors methodically 

 through the whole square. Thus in square M, the mi- 

 nors 1 to S are placed in the first square, 9 to 16 in the 

 second, and so on. 



This disposition, it is evident, will not do. Num 

 bers 1 to 8 must therefore be separated,, and thrown 

 into different petty squares. The minors in the. other 

 petty squares must likewise be separated and otherwise 

 disposed of. 



To distinguish this thorough arrangement from th 

 preceding one, we shall grve it the title of the Distri- 

 bution ofthe Minors. 



A considerable number of distributions may be 

 found suitable for these squares of 8, of 12, and of 

 16. 



In these Tables, the first horizontal column is di- 

 vided into as many parts as there are petty squares in 

 that square to which each Table refers. 



Under each of tbese divisions are two vertical co- 

 lumns, containing eight numbers, which are to be 

 placed in each -of the-petty squares, and arranged in 

 eight cells, in the same order as the numbers 1 to 8 are 

 in some one of the varieties of arrangements which fol- 

 low the Tables. 



Distribu- 

 tion of the 

 minors for 

 the square 

 of 12. 



Adding now the majors, so as to mak the sum of 

 each, with its adjoining- minor in the horizontal column, 



Distribution of the Minors for the Square ofVZ. 



No. 1. for arrangements A and B. No. 2. for arrangements A and B". 



