SCIENCE. 



569 



. 

 i 



in. 



10. 



wards to the top of the square, and leaving out a num- exterior row of cells in Fig. E is now to be lettered, Mnt*. 



b,-r wlicn- the cell UalriMily filled up, us oak-red for the and the numbers trarwlerred to Fif. G, by the c : ' 



vquari- ( t 1. rules already given for nquares oddly even. The ^ ' 



The square will now be magic, and it to be placed magic square of 10 will thus be completed, as shown 



in tiie interior ol another tquure G of 100 cells. The below. 

 Natural Square of 1U. 



Magic 

 squares in 

 compart 

 ments. 



Magic Square of 10. 



e 



p 



M,.- .. 

 . Ml : 

 10. 



This naturally leads us to another property which 

 may be given to thes,e squares. 



Magic Squares in Compartments. 

 Example in the square of 8. 



Take the first 8 numbers and the last 8, and arrange 

 them in a square of 16 cells ; do the same with the 

 numbers 9 to 16, and 49 to 56 for the second square ; 

 then 17 to 24, and 41 to 48, for the third square. Last- 

 ly, 25 to 40 for the fourth square, as seen in the Figures 

 annexi d. 



Make each of these magical by the rules given for 

 the square of 4, then putting them together in a square 

 of 64 cells, they will appear as below, and possess of 

 course not only the common properties of these squares, 

 but also that of being composed of four similar magic 

 squares. 



Masjic Square of 8. 



Wo have now given methods of constructing magic 

 squares, possess-ing all the properties tliey were thought 

 capable of containing previous to the appearance of Dr. 

 Franklin's square of 16', which was published, we be- 

 lieve, for the first time in Ferguson's Tablet and Tracts, 

 under the title of the Manic Sqmre of Squares. 



The principal property of this square was, that the 

 amount of each half diagonal ascending, added to its 

 adjacent diagonal descending taken from any of the 

 four sides, and likewise all the parallels to their half 

 diagonals, was the same as that of the vertical or hori- 

 zontal columns. 



This was effected by an arrangement of the num. 

 bers, by which a property unknown or unnoticed be- 

 fore was given to these squares, viz. the equality of the 

 sum of each small square or clu>ter of four cells, taken 

 In the second edition of Mutton's Montucla, there w a magic square of 16 by Mr. Dalby, professor in ih Koj-al .Military College, giten 

 as an improvement of Dr. Franklin's hquare. liy means of two varieties of a particular arrangement, he has to far succeeded a* to get the 

 diagonals of the large square, and of each of its four squares of 8, equal to their other columns ; but in gaming this property be has lost an 

 essential one of Franklin's square, viz, the equality of the sums of every square of four adjacent cell* through the whole square. The fault 

 lies not in the arrangement, but in the distribution of the numbers. It is a curious circumstance, tha- the ammgrment be baa made use of is 

 of that class which is necessary for forming Franklin's magic circles ; and the profesaor was probably not aware of the property inherent in 

 tbc square of forming a magnificent circle of this sort, 



VOL. XVII. PARU 11. j. C 



