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SCIENCE.. 



Example in the Square of 12. 



Scienc; 



NATURAL SQUARE. 



MAGIC SQUARE. 



These squares are capable of being greatly varied, as 

 it is not necessary to insert the numbers in the natural 

 square in a regular order, from the first to the last of 

 the series, but we may commence with any number. 



Of Mngic Squarjes of 4, and its Multiples. 



Of magic The lowest root from which a magic square of even 



squares of numbers can be constructed, is that of 4. This is rea- 



4, and its djjy done as follows : 



multiples. A 



In the square annexed A, divided into 

 16 cells, insert the numbers 1 to 16 in their 

 natural order. 



In a similar square B, insert the num- 

 bers in the diagonal cells as they stand in 

 the diagonals of the natural square A ; 

 then beginning at the right hand corner at 

 the bottom, (wherein No. 16. in A is,) in- 

 sert the numbers 1 to 16 regularly in the 

 reverse order, leaving out a number where- 

 the cell is already filled up. The square 

 B will now be magic, the . amount of the 

 vertical, horizontal, and diagonal columns 

 being each of the same, or 34s. , 



This method is simple, and may be made 

 ue of to produce a great variety of these 

 squares ; for the order of the numbers in 

 toe natural square may be altered in a 



13151416 

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great many ways ; as for instance in Fig. C 

 annexed, which is made magic in this 

 way, as in Fig. D. 



Instead of beginning at the bottom, and 

 inserting the numbers in the reverse order, 

 after the diagonal cells are filled up, it will 

 be found readier in these small squares to 

 proceed thus : 



Reverse the order of numbers 14 and 15 at the bot- 

 tom of A, and place them at the top of B. Do the 

 same with 2 and 3 at the top of A, and place them at 

 the bottom of B. Reverse numbers 5 and 9 on tne 

 left of A, and place them on the right of B, as also 8' 

 and 12 on the right of A, and place them on the left 

 of B. 



In the same way the squares of a, 12, 16, &c. may 

 be made magic. 



Example in the square of 8. 



Instead of giving here the square of 8 alone, we shall. Example h 

 give it as it stands in the interior of 10, by which we the square 

 can illustrate both this method, and the preceding of 8< 

 rules for squares odd by even. 



In a square of 100 cells, insert the numbers 1 to 100 

 in their natural order, draw strong lines round the in- 

 terior square of 8, and divide it into four smaller squares, 

 as shown .in Fig. E. In another square of 64 cells, in- 

 sert the diagonals as they are found in each of the small 

 squares in the interior squares of E, as shown in Fig. 

 F. Beginning, then, at the bottom of the right, where 

 8^ is placed, insert the numbers as they stand in the in- 

 terior square of E, in a reverse order, proceeding up- 



