20 ON THE CONSTRUCTION AND DIFFERENT FORMS 



result 8, 1, G, 3 || 4, 5, 2, 7. These may have each four of their rows inverted, relatively 

 to their partial squares, in the order 4, 3, 2, 1 || 8, 7, 6, 5, &c. Every magic square (C) 

 of the forty-eight above considered will thus give seven others, (C,), (CJ, . . . (C ); and 

 accordingly, the total number of different arrangements similar to (C) will become 

 48 . 8 = 384. We may obviously invert each of these arrangements in the circular dis- 

 tribution of their rows, round the principal rings of the drawing, and we shall thus have it 

 in our power to construct 7G8 corresponding magic cyclovolutes. 



In the formation of all the arrangements hitherto determined, we have kept in view the 

 various properties ascribed to the magic cyclovolute, based on the square (AB). By 

 merely excluding the consideration of that particular property, connected with the num- 

 bers taken in pairs along the radii of the drawing, as already mentioned, an additional 

 group of magic cyclovolutes may be constructed, which with the 768 preceding, will 

 amount to 6044; and other groups would result in case the semi-volutes, or the semi-radii, 

 or both, were left unnoticed as minor properties. It would be somewhat tedious, and of 

 no particular interest, to ascertain the precise number of forms corresponding to each of 

 these limited hypotheses. We shall therefore omit any results of this kind, and proceed 

 directly to investigate the total number of unclassified magic cyclovolutes, including the 

 6044 above determined, and all the general properties originally enumerated. 



Let us then return to the form 1, 2, 3, 4 || 5, 6, 7, 8, adopted for any magic square (C), 

 and let us observe that no two odd, or two even numbers, indicative of the horizontal rows, 

 can be disposed and arranged successively in any derivative form, considered as the basis 

 of a magic cyclovolute. The reason of this will appear from the square (AB), to which 

 all additional forms must preserve a like structure. Attending to this essential condition, 

 the number of combinations of the four odd rows, 1, 3, 5, 7, or of the four even rows, 

 2, 4,6, 8, taken in pairs, is k (4-3) = 6; and these combined in fours, of which two are odd 

 and two even, will give 6 . 6 = 36 forms, and consequently eighteen different arrange- 

 ments analogous to 1, 2, 3, 4 || 5, 6, 7, 8. But any of these forms so constructed, as 

 for instance, the preceding, admits of four changes with respect to the rows 1, 2, 3, 4, 

 and also four for the rows 5, 6, 7, 8. These rows may therefore be combined 4 .4 = 16 

 ways; and taking the inverse form 4, 3, 2, 1 || 8, 7, 6, 5, as equally admissible, we shall 

 have thirty-two arrangements, instead of the single primitive form 1,2, 3, 4 || 5,6,7,8; 

 and the eighteen different arrangements previously determined will generate the number 

 18 . 32 = 576. By viewing these combinations in a less specific manner, the number 

 just found will result as the square of (4'3-2*l), the permutations of either the odd or 

 even rows. We shall thus increase the forty-eight magic squares first considered to 

 48 . 576 = 3 . 96 2 or 27648, which may be all obtained from the series of sixty-four num- 

 bers, 1, 2, 3 ... 64, and converted into magic cyclovolutes. The total number of these 

 remarkable arrangements, with all the leading properties in my paper in the Transactions 

 of the Society, therefore, amount to 6 . 96^ = 55296. 



In connexion with our subject, we shall here bring to notice a new imperfect magic 

 square, analogous to that adopted by Dr. Franklin in the construction of his magic circle; 

 but which so far generalizes it, as to include the particular property of the numbers taken 

 in pairs along the several radii, as already mentioned in case of the cyclovolute. It is 



