572 



Science, 



Curiosities 



in. 



Properties 

 of these 

 squares. 

 Square of 



Additional 

 properties 

 'of the 

 square of 

 16. 



17 



18 



19 



SCIENCE. 



20 21 



22 



23 



24 



The preceding varieties of arrangements for the petty 

 squares are in three rows, each row containing a diffe- 

 rent kind of them. Some of the distributions will be 

 found .not suitable for the yarieties..in all the three rows, 

 being calculated only for those in one or two of these 

 rows. 



The first six varieties in each roxv are to have their 

 majors added, as shown in No*. 1, 9, and 17- The two 

 last of each row must have the majors placed, as shown 

 in Nos. 7> 15, and 23, 



We annex examples of magic squares constructed 

 from the preceding Tables, and as they possess all the 

 properties of Dr. Franklin's square of squares, joined 

 to most of those which were previously known, we 

 have given them the name of Union Magic Squares. 

 The squares of 8 are constructed, 



No. 1. from distribution No. 1. arrangement No. 10. 



2 ...... 2 



3 ...... 3 



Of 12 are constructed, 



No. 1 ...... 



2 2 



Of 16 is constructed, No. 3 



8 



18 



No. 10 



4 

 No. 20 



Properties of these Squares. Square ofl6. 



1. The amount of each column vertical or hori- 

 zontal, is ... . . 



2. Each half column vertical or horizontal, is 



3. Haifa diagonal ascending, added to half a 

 diagonal descending, taken from any of 

 four sides, is the same with all the parallels 

 these diagonals, 



4. The amount of the 4 corner numbers in theT 

 large square, and also in any square of 12, 8, > 

 or 4, taken throughout the square, is, J 



5. The sum of the numbers of any square or 



'J2056 



half a") 

 >f the ( 

 lels to f 



1028 



514. 



cluster of 4 cells taken throughout the square, > 514 

 is '..... . . J 



(These are all the principal properties of Dr. Frank- 

 lin's square.) 



Additional Properties of the Square of 1 6. 



2056 



514 



6. The amount of each of the two diagonals is 



7. The square is composed of l6 squares of 4, 

 each of which is a complete magic square, the 

 amount of their vertical and horizontal co- i 

 lumns, and of each diagonal, is . } 



8. The square of 8 may be taken 9 times, and the 

 square 1 2 four times in the large square, each be- 

 ing a complete magic square. 



9. The amount of the diagonals M a, a O, ascending 

 and descending on each side of the vertical division 

 NB, is ... 1028 



The same with all their parallels to the top of the 



square. 

 The same with the diagonals and their parallels on 



each side of the other vertical divisions OC and 



PD. 

 The same with all the diagonals and their parallels, 



reckoning from any of the other sides, making in 



all 156 equal sums. 



10. If the square is cat into four columns, through the 

 vertical divisions BN, CO, and DP, the parts may 

 be exchanged at pleasure, and all the properties re- 

 main the same. 



11. If it is in like manner cut into four, through the 

 horizontal divisions FG, HI, ami KL, these may 

 likewise be exchanged, and the properties remain 

 unaltered. By having the option of these mutual 

 interchanges, the square may be arranged in 576. 

 different ways, without having occasion to alter a 

 single figure in the petty squares. 



The squares of 12 and of 8 possess similar properties 

 as far as their dimensions go. The numbers in the co- 

 lumns of the square of 12 amount to 870. The sums 

 of their diagonals on each vertical or horizontal divi- ' 

 sion, ascending and descending, amount to 580. The 

 number of these equal sums with their parallels is 72. 

 By cutting it through the two vertical or horizontal di- 

 visions, it may be arranged in 36 different ways, with- 

 out altering its properties. 



The square of 8 may be arranged in four different PLATE 

 ways as in Plate CCCCLXXX1V. tccci~\xxrv. 



2. Magic Circles. 



The only magic circle that has hitherto appeared is Magic 

 that of Dr. Franklin, which was published fifty years circles. 

 ago by Mr. Ferguson in his Tables and Tracts. 



The arrangement of the minors in the petty squares, 

 is totally different from any of those given" in the pre- 

 ceding pages, we therefore subjoin a few varieties of it. 



This circle contains 64 num- 

 bers, proceeding regularly from 

 12 to 75, and, as may readily be 

 supposed, is constructed from a 

 magic square of 8, a copy of which 

 is annexed, the numbers being 

 placed in the same order as they 

 are found in the circle. The sum 

 of the first and last number is 87- 

 Four times this is 348, which is 

 the amount of each column. This 

 added to 12, a number placed in 

 the centre of the circle, makes 360. 



6273l4'253o'414S57 



7 164 231 639 j.32 



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 09 66*121 



1920,6768515213536 



548 



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