I did not, however, end with squares, but composed also a magic circle, consisting of 8 concentric circles, and 8 radial rows, filled with a series of numbers from 12 to 75 inclusive, so disposed as that the numbers of each circle, or each radial row, being added to the central number 12, they make exactly 360, the number of degrees in a circle; and this circle had, moreover, all the properties of the square of 8. If you desire it, I will send it; but at present, I believe, you have enough on this subject. I am, &c.
I am glad the perusal of the magical squares afforded you any amusement. I now send you the magical circle. (_See Plate_ XII.)
Its properties, besides those mentioned in my former, are these.
Half the number in any radial row, added with half the central number, make 180, equal to the number of degrees in a semi-circle.
Also half the numbers in any one of the concentric circles, taken either above or below the horizontal double line, with half the central number, make 180.
And if any four adjoining numbers, standing nearly in a square, be taken from any part, and added with half the central number, they make 180.
_Published as the Act directs, April 1, 1806, by Longman, Hurst, Rees & Orme, Paternoster Row._]
There are, moreover, included four other sets of circular spaces, excentric with respect to the first, each of these sets containing five spaces. The centres of the circles that bound them, are at A, B, C, and D. Each set, for the more easy distinguishing them from the first, are drawn with a different coloured ink, red, blue, green, and yellow.[62]
These sets of excentric circular spaces intersect those of the concentric, and each other; and yet the numbers contained in each of the twenty excentric spaces, taken all around, make, with the central number, the same sum as those in each of the 8 concentric, viz. 360. The halves, also of those drawn from the centres A and C, taken above or below the double horizontal line, and of those drawn from centres B and D, taken to the right or left of the vertical line, do, with half the central number, make just 180.
It may be observed, that there is not one of the numbers but what belongs at least to two of the different circular spaces; some to three, some to four, some to five; and yet they are all so placed as never to break the required number 360, in any of the 28 circular spaces within the primitive circle.