of its parallel bent rows of eight numbers, make 260. Also the bent row from 52 descending to 54, and from 43 ascending to 45, and every one of its parallel bent rows of eight numbers, make 260. Also the bent row from 45 to 43, descending to the left, and from 23 to 17, descending to the right, and every one of its parallel bent rows of eight numbers, make 260. Also the bent row from 52 to 54, descending to the right, and from 10 to 16, descending to the left, and every one of its parallel bent rows of eight numbers, make 260. Also the parallel bent rows next to the abovementioned, which are shortened to three numbers ascending and three descending, &c., as from 53 to 4 ascending, and from 29 to 44 descending, make, with the two corner numbers, 260. Also the two numbers, 14, 61, ascending, and 36, 19, descending, with the lower, four numbers situated like them, viz. 50, 1, descending, and 32, 47, ascending, make 260. And, lastly, the four corner numbers, with the four middle numbers, make 260.

So this magical square seems perfect in its kind. But these are not all its properties; there are five other curious ones, which, at some other time, I will explain to you.

Mr. Logan then showed me an old arithmetical book, in quarto, wrote, I think, by one Stifelius, which contained a square of sixteen, that he said he should imagine must have been a work of great labor; but, if I forget not, it had only the common properties of making the same sum, viz. 2056, in every row, horizontal, vertical, and diagonal. Not willing to be outdone by Mr. Stifelius, even in the size of my square, I went home, and made, that evening, the following magical square of sixteen, which, besides having all the properties of the foregoing square of eight, that is, it would make the 2056 in all the same rows and diago

247 281 315 202 182 170 18 188 179 180 87 71

21 285 216 n 22 43

108

|16| 221 228

או

16 241240 209 208 177 176 145

253 1 2.9 36 61 68 93 100 125 132 757

nals, had this added, that a four-square hole being cut in a piece of paper of such a size as to take in and show through it just sixteen of the little squares, when laid on the greater square, the sum of the sixteen numbers, so appearing through the hole, wherever it was placed on the greater square, should likewise make 2056. This I sent to our friend the next morning, who, after some days, sent it back in a letter with these words; "I return to thee thy astonishing or most stupendous piece of the magical square, in which "— but the compliment is too extravagant, and therefore, for his sake, as well as my own, I ought not to repeat it. Nor is it necessary; for I make no question but you will readily allow this square of sixteen to be the most magically magical of any magic square ever made by any magician. (See Plate VII. Fig. 2.)

I did not, however, end with squares, but composed also a magic circle, consisting of eight concentric circles, and eight 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 eight. If you desire it, I will send it; but at present, I believe, you have enough on this subject.

I am, &c.

B. FRANKLIN.

I am glad the perusal of the magical squares afforded you any amusement. I now send you the magical circle. (See Plate VIII.)

Its properties, besides those mentioned in my former, are these.

Half the numbers in any radial row, added with half the central number, make 180, equal to the number of degrees in a semicircle.

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.

There are, moreover, included four other sets of circular spaces, eccentric 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 colored ink, red, blue, green, and yellow.*

These sets of eccentric circular spaces intersect those of the concentric, and each other; and yet the numbers contained in each of the twenty eccentric spaces, taken all around, make, with the central number, the same sum as those in each of the eight concentric, viz. 360. The halves, also of those drawn from the centres

* În the plate they are distinguished by dashed or dotted lines, as different as the engraver could well make them.