To Peter Collinson
I. MS not found; printed in Benjamin Franklin, Experiments and Observations on Electricity (London, 1769), pp. 354–6. II. MS not found; facsimile: Royal Society;2 also draft: American Philosophical Society.
Of Franklin’s magic square James Ferguson wrote that it went “far beyond any thing of the kind I ever saw before; and the magic circle (which is the first of the kind I ever heard of, or perhaps any one besides) is still more surprising.”3 There are two descriptions of the circle, basically similar, but different in important respects: one is Franklin’s letter to Collinson of about 1752, printed as Letter XXVII in Experiments and Observations, 1769; the other survives as a draft in the American Philosophical Society, and, revised and expanded, in a letter to John Canton, May 29, 1765. Both descriptions are printed here, the second in the revised version of the letter to Canton.
I am glad the perusal of the magical squares afforded you any amusement. I now send you the magical circle.
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 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, excentric with respect to the first, each of these sets containing five spaces. The centers 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 colour’d ink, red, blue, green, and yellow.*
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 centers A and C, taken above or below the double horizontal line, and of those drawn from centers 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.
These interwoven circles make so perplexed an appearance, that it is not easy for the eye to trace every circle of numbers one would examine, through all the maze of circles intersected by it; but if you fix one foot of the compasses in either of the centers, and extend the other to any number in the circle you would examine belonging to that center, the moving foot will point the others out, by passing round over all the numbers of that circle successively. I am, &c.
[Extract of a letter to John Canton, May 29, 1765]
It is compos’d of a Series of Numbers from 12 to 75 inclusive, divided in 8 concentric Circles of Numbers, and rang’d in 8 Radii of Numbers, with the Number 12 in the Center, which Number, like the Center, is common to all the Circles and to all the Radii.
The Numbers are so dispos’d, as that all the Numbers in any one of the Circles, added together, make, with the central Number, just 360, the Number of Degrees in a Circle.
The Numbers in each Radius also, with the central Number, make just 360.
Also Half of any of the said 8 Circles, taken above or under the horizontal double Line with Half the Central Number, make 180, or half the Degrees in a Circle. So likewise do the Numbers in each Half Radius, with half the Central Number.
There are moreover included 4 other Sets of concentric Circles, 5 in each Set, the several Sets distinguish’d by Green, Yellow, Red, and Blue Ink, and each Set drawn round a Center of the same Colour. These Sets of Circles intersect the first 8 and each other; and the Numbers contain’d in each of these 20 Circles, do also, with the Central Number, make 360. Their Halves also, taken above or under the horizontal Line, do, with half the central Number, make 180.
Observe, That there is no one of the Numbers but what belongs to at least two different Circles, some to three, some to four, and some to five; and yet all so plac’d (with the central Number which belongs to all) as never to break the requir’d Number 360 in any one of the 28 Circles.
2. Facsimile of BF’s letter to John Canton, May 29, 1765. The original MS was offered for sale in Parke-Bernet Galleries, Catalogue 63 (Nov. 16–17, 1938), item 73, and by the Rosenbach Galleries in 1943.
3. Tables and Tracts, relative to Several Arts and Sciences (London, 1767), p. 309. Ferguson’s description of the magic circle, pp. 312–17, is essentially the same BF gave Canton.
4. The date is assigned by the editors: the letter was written after the preceding one.