SCIENCE. 



/5G5 



.3, f>, to fill up the fifth and last column. 

 TJjjg S q U aie will he one of the com- 

 ponent pirts of tin? n quired square, 

 and will ! ina^ii-, lor the sum of each 

 column, \\hrtlu-r hori/i'iit il, vertical or 

 diagonal, is the same, as the five figures 

 of the progression are contained in each 

 without the same figure being repeat- 

 eO. 



Now in another square of 25 cells, 

 Fig. 2. inscribe in the first column, the 

 root 5 and its multiples, beginning with 

 a -cypher, viz. 0, 5, 10, 15, 20, and in 

 any order at pleasure, such for example, 

 5, 0, 15, 10, 20. Th.a fill up the 

 square according t > the same principles 

 as before, taking care not to assume 

 the same number in the series always 

 to begin with. Thus for example, as 

 in the former square, the third figure in the series was 

 taken, in the present one the fourth must be assumed, 

 and thus we shall have a square of multiples as seen 

 in Fig. 2. This is the second component of the re- 

 quired magic square, and is itself magic, since the 

 sum of each column is always the same. 



Now to obtain the magic square required, nothing 

 more is necessary but to inscribe in a third square of 

 25 cells, Fig. 8. the sum of the numbers found in the 

 corresponding cells of the preceding two. For ex- 

 ample 5 and 1, or 6, on the first of the left at the top 

 of the required square, and 3, or 3 in the second, 

 ami so on. By these means we shall have the square 

 Fig. 3. which will necessarily be magic. 



Jtematks. 



By this method any of the num- 

 bers may be made to fall on any of 

 the cells at pleasure ; for example, 1 

 on the central cell. We have only to 

 fill up the middle band with the series 

 of numbers in such a manner that 

 1 may be in the centre, as seen in 

 Fig. 4. and then to fill up the rest of 

 the square, according to the above prin- 

 ciples, beginning at the highest column 

 when the lowest has been filled up. 

 To form the second square, place a 

 cypher in the centre as seen in Fig. 5. 

 and fill up the remaining cells in the 

 same manner as before, taking care 

 as in the former, not to assume the 

 same quantities for beginning the co- 

 lumns. 



In the last place, form a third square, 

 by adding together the numbers in the 

 similar cells, and you will have the 

 annexed square Fig. 6. where 1 will ne- 

 cessarily occupy the centre. 



Remarks. 



1st, We must here observe, that 

 when the number of the root is not 

 prime, that is, if it be 9, 15, 21, &c. it 

 is impossible to avoid a repetition of 

 some of the numbers, at least in one 



Fig. 4. 



FIR. 5. 



Fig. 6. 



I-.,' I 



of the diagonals ; but in that case it 

 mutt be arranged in uch a manner, 

 that the number repent' d in that diag- 

 onal, ihall be the middle one of the 

 progression ; for example 5, if the root 

 of the square by y ; 8, if it be 1.0 : 

 and as the square formed by the mul- 

 tiples will be liable to the tame acci- 

 dent, care must be taken in filling them 

 up, that the opposite diagonal shall 

 contain the mean multiple between 

 and the greatest ; for example 36, 

 if the root be <J, 105, if it be 15, 

 &c. 



2d, The same thing may be done 

 in squares which have a prime 

 number for their root. By way of ex- 

 ample, we shall form a magic square 

 of the two first of the annexed ones, 

 in the first of which, Fig. 7. the num- 

 ber 3 is repeated in the diagonal de- 

 scending from right to left, and in the 

 second, Fig. 8. 10 is repeated in the 

 diagonal from left to right. This, how- 

 ever, does not prevent the third square, 

 Fig. 9. formed by their addition from 

 being magic. 



Magic Squares of Odd Numlert tvi/A Border*. 



There is an additional property which it has been Magic 

 found can be given to these squares, viz. that whatever iquare*f 

 may be the dimensions, any one or two or more of the AA 

 exterior rows may be removed all round the square, 

 and the remaining square still continue magic. They 

 are constructed by the following rules.; 



Preliminary Remarks on the Natural Square. 



1. In the middle there is a cell, which we shall call Prelimin- 

 the centre. arymnadu 



'2. One half of all the other numbers in the square 00 ***. 

 are le>s, arid the other half greater than the centre. The ni 

 former we shall call Minors, the latter Majors. 



The cell in the centre is now to have a strong line 

 drawn round it, the cells next to this are likewise to 

 be bounded by a strong line, and so on with each sur. 

 rounding row to the extremity of the square. These 

 lines will appear as so many eccentric squares, and the 

 spaces bounded by them containing the numbers we 

 shall call belts. 



3. The belt next the centre we shall call the 1st belt, 

 and continue numbering them outwards 2d, 3d, 4th 

 belt, &c. 



4. Those belts having the odd numbers we shall call 

 the odd bells, those having the even numbers we shall 

 call the even belts. 



5. Supposing now that the square is divided diagon- 

 ally into tour parts, we shall distinguish them by the 

 names of the upper, the loner, the left, and the right 

 quarters ; and we may here observe, that the minor* 

 occupy all the upper quarters, the left quarter from the 

 top to the cells opposite the centre inclusive, and the 

 right quarter from the top to the cells opposite the cen- 

 tre exclusive. 



