()K rill. M \i.u CYC] OVOL1 l l . 



I" 



The square (AB) will directlj bring to view the arrangement adopted in the drawing. 



\\ e have merely to increase each of its numbers by 11, in order t<> adapt il i" the amount 

 360, employed by the author of the magic circle, instead of 260 above-mentioni d; and to 

 place the same horizontal rows of the corresponding magic Bquare in succession round 

 the principal rings of the cyclovolute, above and below the principal diameter .1 I . 



Let us now determine the number of magic cyclovolutes that ma) be constructed l>\ 

 arranging differently the terms of the series 1,2,3. ..64. With tin- view, let us return to 

 the first squares («), (a), and let them be combined with similar squares formed from the 

 reverse binary combinations (5, -1) and (6, 3) and constituting (c) and (/■') lure given : 



1 " / \ 





We shall then obtain, as before, the two new fundamental squares (.1 ) and /.' 



("*') 



B 



which arc precisely similar to (A) and (/J); and may be employed for like purpOSt -. It 

 we use (A) as a primitive in the manner of (.1) already treated, we shall find from il and 

 the derivative {Ii') the compound square {A B)< in which and an) simil ir arrangi m< at, 

 attention must be paid to the order of the letters J. />' . a- indicative of the particular 

 mode of combination. Besides the two compound Bquares (AB), (A II ). thus obtained, 

 we may in a similar manner obtain the Bquares [BA), [B I . and also the four additional 

 squares {AB') {AB), and (BA) (B'A); so that the elemental) forms (a) and (a') consi- 

 dered as invariable, will thus lead to eight com pom ul magic Bquares, having all the same 

 properties. It is also evident that the binary combination- 1,8), 3, 6), and 1,8), 5,4), 

 will also furnish squares analogous to (//), (</'), by which and the remaining combinations 

 (2, 7), (4, ')), and (2, 7), (G, 3), we shall be enabled to form sixteen new compound m 

 squares, which, with the preceding eight, amount to twenty-four different arrangi mi nts, 

 as given at the end of this note. 



We may immediately derive other form- from these, -nnpl\ bj mi' rchanging the first 

 and second vertical rows, the third and fourth, &c. I!\ virtue of the forma .1. 1 . the 

 resulting squares will he similar to {AB) ; and, in tin- way, the number of arrangi mi 

 becomes forty-eight. These arc the only changes that i an be made m the \- rtii al ro* -. 

 so as to give essentially different magic cyclovolutes. To obtain other arrangemi ati m 

 must change the order of the horizontal rows, which, commencing with the upper, »■■ 

 shall briefly denote b) the number 1, J, 3, I | 5, 6, 7, 8. [f we interchange ever) two 

 of these, the resulting form will he 2, I. 1,3 ' 6,5,8,7; and taking also th. 

 direct form, 1, 8, 3. 6 || 5, I. 7. 2. which is equallj applicabli , th< n will, in like mani 



