18 



ON THE CONSTRUCTION AND DIFFERENT FORMS 



If we repeat these partial squares, the first and third, (a) and (6), vertically; and the 

 second and fourth, (a 1 ) and (6'), horizontally, we shall obtain by their junction the follow- 

 ing complete magic squares, (A) and (B), which as respects their diagonal and leading 

 rows, are constituted in a manner similar to their components, (a), (a!) and (b), (b 1 ): 



{A) 



(*) 



(•**) 



1 16 



63 50 



57 56 

 7 10 



8 9 

 58 55 



64 49 

 2 15 



17 32 

 47 34 



24 25 

 42 39 



41 40 

 23 26 



48 33 

 1831 



3 14 



61 52 



611 

 60 53 



59 54 

 5 12 



62 51 

 4 13 



19 30143 38 

 45 36 2128 



22 27146 35 

 44 37 20 29 



Of these fundamental squares, which we prefer in their present forms, cither may be 

 considered as a primitive, the other its derivative. To suit our present purpose we shall 

 take (A) as primitive, and we shall mentally diminish its several numbers by unity in 

 accordance with the series 0, 1, 2... 7. By multiplying each of the remainders thus 

 obtained by eight, and adding their products in successive order to the corresponding 

 numbers, similarly posited in the derivative (B), we shall readily construct the perfect 

 magic square (AB), of which all the rows, whether vertical, 

 horizontal, or diagonal, amount to 260; and which is obvious- 

 ly composed of four perfect magic squares, having their diago- 

 nals and other rows each equal to 130, the common amount 

 of every four adjacent numbers in the entire square. 



The arrangement here given is remarkable. It will remain 

 perfectly magic when any number of its vertical rows are 

 removed in order, and placed in succession after the last ; 

 and also when any number of its upper or lower horizontal 

 rows undergo a similar displacement. Such permanency in the magic structure of (AB) 

 follows from the peculiar mode in which the binary combinations taken from the series 

 1, 2, 3 .. .8, are disposed in the fundamental forms (A) and (B); and hence we are imme- 

 diately led to the constant result of the several principal and secondary rings of the magic 

 cyclovolute. 



As to the general result common to all the volutes and the particular property, inserted 

 in the Bulletin of the society, (No. 13, 1840), in reference to the sixteen semi-volutes, 

 reckoned from the extremities of the principal diameters A A', BB', and the corresponding 

 points a, a', and b, b', they arc a consequence of (AB) rendered perfectly magic, as well 

 as the four squares which compose it. With these semi-volutes, we may also notice the 

 sixteen semi-radii in the drawing. The four numbers in each of them with half the aux- 

 iliary 12 amount to ISO; and a like result will always hold good for every four numbers 

 taken in adjacent pairs from the centre, or, from the remote extremities of any radius. 

 These supplemental properties render peculiar the form of the drawing as now considered, 

 and give it every possible generality. 



