Benjamin Franklin Papers

From Benjamin Franklin to John Canton, 29 May 1765

To John Canton

MS not available:9 facsimile of ALS: The Royal Society

Franklin’s interest in magic squares and circles, which began in the 1730s when he was clerk of the Pennsylvania Assembly, has been demonstrated in earlier volumes; see above, III, 458–9; IV, 392–403. In 1752 he sent Peter Collinson, among other examples of his constructions, the first version of his complicated magic circle, for which he used inks of four colors in addition to black. The letter printed below makes clear that he had revised and improved this circle by 1765, when he sent a new drawing to John Canton, again using four extra colors of ink. To illustrate Franklin’s descriptive letter of 1752 to Collinson (above, IV, 399–400) the editors reproduced (p. 401) the revised diagram of 1765 as it had been adapted for printing in a single ink in Experiments and Observations on Electricity, 1769 edition, facing p. 355.1 With this reproduction was printed (IV, 400, 403) the text of Franklin’s explanation to Canton, which appears on the sheet with the diagram as an essential part of the present letter. Only the main text of this letter is printed below, together with illustrations of two additional magic squares mentioned in the postscript and included on another sheet of the letter. For the illustration and detailed explanation of the magic circle the reader is referred to the indicated pages in volume IV.

Cravenstreet, May 29. 1765

Dear Sir,

As you seem’d desirous of seeing the magic Circle I mention’d to you, I have revis’d the one I made many Years since, and with some Improvements, send it you.

I have made it as distinct as I could, by using Inks of different Colours for the several Sets of interwoven Circles;2 and yet the whole makes so perplext an Appearance, that I doubted whether the Eye could in all Cases easily trace the Circle of Numbers one would examine, through all the Maze of Circles intersected by it: I have therefore, in the middle Circle, mark’d the Centers of the Green, Red, Yellow, and Blue Sets;3 so that when you would cast up the Numbers in any Circle of either of those Colours, if you fix one Foot of the Compasses in the Center of the same Colour, and extend the other to any Number in that Circle, it will pass round over all the rest successively.

This magic Circle has more Properties than are mention’d in the Description of it, some of them curious and even surprizing;4 but I could not mark them all without occasioning more Confusion in the Figure, nor easily describe them without too much Writing. When I have next the Pleasure of seeing you, I will point them out. I am, Dear Sir, Your most obedient humble Servant

B Franklin

P.S. You have my curious Square of 8, and the great perfect one of 16;5 I enclose one of 6, and one of 4, which I assure you I found more difficult to make, (particularly that of 6) tho nothing near so good.

Mr Canton

By B:F.

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.

The Diagonals are to be reckon’d by Halves, not crossing but turning at right Angles from the Center, by which 4 Varieties are made instead of two.

By Doctor Benj. Franklin

[Note numbering follows the Franklin Papers source.]

9The original MS has been listed in several sales catalogues during the last thirty years. Its most recent appearance was at the Charles Hamilton Sale of Sept. 28, 1967, lot 279, when it was bought by Charles Sessler, Inc., for an anonymous client.

1As indicated in the present letter, BF used the colored inks to identify four sets of circles, concentric within each set but eccentric to the principle series of concentric circles in black ink. The centers of the eccentric sets all lie near the rim of the smallest circle of the principle series and are identified respectively by the letters A, B, C, and D. In the black-and-white reproductions the circles of series A are printed with light solid lines, those of series B with fine dotted lines, those of series C with short dashes, and those of series D with long dashes. This substitute for color in the eccentric circles originated with James Ferguson, who first printed Franklin’s magic circle in his Tables and Tracts, relative to Several Arts and Sciences (London, 1767), plate III, in this manner and suggested in his explanatory text (pp. 312–17) that in redrawing the figure by hand the reader should restore the colors.

2That is, the sets of eccentric circles.

3Marked, A, B, C, and D.

4In addition to the properties BF mentioned in his description (above, IV, 400, 403), Ferguson pointed out (Tables and Tracts, p. 317) that adding any four adjacent numbers that form a “square” plus half the central number (12), always produces a total of 180, the number of degrees in a semicircle. For example, taking the two outer numbers in the upper right-hand segment (14 and 72) and adding the two outer numbers in the next clockwise segment (25 and 63) and one half the central number (6), one arrives at the total of 180.

5Above, IV, 394, 397.

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