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LXVII: TO PETER COLLINSON 1 - Benjamin Franklin, The Works of Benjamin Franklin, Vol. II Letters and Misc. Writings 1735-1753 [1904]

Edition used:

The Works of Benjamin Franklin, including the Private as well as the Official and Scientific Correspondence, together with the Unmutilated and Correct Version of the Autobiography, compiled and edited by John Bigelow (New York: G.P. Putnam’s Sons, 1904). The Federal Edition in 12 volumes. Vol. II (Letters and Misc. Writings 1735-1753).

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LXVII

TO PETER COLLINSON1

Sir:—

According to your request, I now send you the arithmetical curiosity, of which this is the history.

Being one day in the country, at the house of our common friend, the late learned Mr. Logan, he showed me a folio French book filled with magic squares, wrote, if I forget not, by one M. Frenicle, in which, he said, the author had discovered great ingenuity and dexterity in the management of numbers; and though several other foreigners had distinguished themselves in the same way, he did not recollect that any one Englishman had done any thing of the kind remarkable.

I said it was, perhaps, a mark of the good sense of our English mathematicians, that they would not spend their time in things that were merely difficiles nugæ, incapable of any useful application. He answered, that many of the arithmetical or mathematical questions publicly proposed and answered in England were equally trifling and useless. “Perhaps the considering and answering such questions,” I replied, “may not be altogether useless, if it produces by practice an habitual readiness and exactness in mathematical disquisitions, which readiness may, on many occasions, be of real use.” “In the same way,” says he, “may the making of these squares be of use.” I then confessed to him that in my younger days, having once some leisure (which I still think I might have employed more usefully), I had amused myself in making this kind of magic squares, and at length had acquired such a knack at it that I could fill the cells of any magic square of reasonable size with a series of numbers as fast as I could write them, disposed in such a manner as that the sums of every row, horizontal, perpendicular, or diagonal, should be equal; but not being satisfied with these, which I looked on as common and easy things, I had imposed on myself more difficult tasks, and succeeded in making other magic squares, with a variety of properties, and much more curious. He then showed me several in the same book of an uncommon and more curious kind; but as I thought none of them equal to some I remembered to have made, he desired me to let him see them; and, accordingly, the next time I visited him I carried him a square of eight, which I found among my old papers, and which I will now give you, with an account of its properties. (See Plate IV., Fig. 1.)

lf1438-02_figure_008

MAGICAL SQUARES.

The properties are:

1. That every straight row (horizontal or vertical) of eight numbers added together makes 260, and half each row half 260.

2. That the bent row of eight numbers, ascending and descending diagonally, viz., from 16 ascending to 10, and from 23 descending to 17; and every one 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 above-mentioned, 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 diagonals, 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 IV., Fig. 2.)

lf1438-02_figure_009

MAGIC CIRCLE OF CIRCLES.

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.