OF THE MAGIC CYCI.OVDI.lTi:. 



21 



singular that this extension should have hitherto escaped notice, and thai the magic circle 

 in its present form should yet admit of improvement. To establish this point let as form 



the two elementary squares (d) and (//) thus: 



('0 



• ('/') 



and also similar squares, (e), (V) and (/), (/') by means of the binary combinations 3, G . 

 (4, 5) and (3,4), (6, 3). The repetition of these will give/our fundamental square ■ /' • 

 (E) and (/)'), (E), the second and fourth of which arc imperfecta magic; and from thi -• 

 by combination, we shall obtain the eight imperfect magic squares (DE), (I) £?), &c, the 

 first of which is here subjoined. The manner iu which these 

 eight are increased to forty-eight, we have already explained, 

 and taking the three combinations 1, 2, 3, 4 || 5, 6, 7, 8, and 

 1, 4, 7, 6 || 5, 8 3, 2, and 1, 2, 7, 8 || 5, 6, 3, 1, we shall 

 immediately find by their changes the number IS . 32 = 1536 

 magic circles with our additional properties; IS . 1G0 = 7G80 

 of Dr. Franklin's limited form; and the total number without 

 distinction of properties will result as before G . 9G 2 = 55296. 

 The principles by which we have been guided to the various 

 results in this paper, we judge of considerable moment in re- 

 gard to the magical combination of numbers. They may bo 



readily applied to the extensive series of 25G numbers 1, 2, :! . . . 256, first consid red in 

 a magic form by Dr. Franklin, and left combined by him in an imperfect magic square, 

 similar in properties to the base of his magic circle. This he no doubt intended as n 

 generalized base of an enlarged magic circle, but owing to an oversight in two or three 

 of the numbers, it would not be applicable. In Dr. Hutton's Mathematical Dictionary, 

 which I have lately consulted, this imperfection is said to have been noticed bv Mr. Dal by, 

 "first Professor at the Royal Military College;" and to this ingenious mathematician is 

 ascribed the formation of a remarkable magic square of sixteen perfect magic Bquan -. in- 

 cluding the above series; and which is given in the work just cited. Without the Bligl 

 intimation of Mr. Dalby's square, or that of any other person, I had been led to several 

 analogous arrangements. My method is that employed in case of the perfect magic 

 square (AB). I formed from the binary combination-; (l, 16), (2, 15), (3, 14), (4, 13 

 and (5, 12), (6, 11), (7, 10), (S, 9), the elementary Bquares 



(g) 



vol. x. — G 



