The Works of Benjamin Franklin

or stops the perspiration, by clogging the perspiratory ducts, or, perhaps, by not admitting the perspirable parts to separate. Paper wet with size and water will not dry so soon as if wet with water only.

A vessel of hot water, if the vapor can freely pass from it, soon cools. If there be just fire enough under it to add continually the heat it loses, it retains the same degree. If the vessel be closed, so that the vapor may be retained, there will from the same fire be a continual accession of heat to the water, till it rises to a great degree. Or, if no fire be under it, it will retain the heat it first had for a long time. I have experienced, that a bottle of hot water stopped, and put in my bed at night, has retained so much heat seven or eight hours, that I could not in the morning bear my foot against it, without some of the bedclothes intervening.

During the cold fit, then, perspiration being stopped, great part of the heat of the blood, that used to be dissipated, is confined and retained in the body; the heart continues its motion, and creates a constant accession to that heat; the inward parts grow very hot, and, by contact with the extremities, communicate that heat to them. The glue of the blood is by this heat dissolved, and the blood afterwards flows freely, as before the disorder.

SIR,

TO PETER COLLINSON.

Magical Square of Squares.*

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

* The dates of the letters, in which the account of Magical Squares and Magical Circles was communicated to Mr. Collinson, are not known; but in a letter from James Logan to Mr. Collinson, dated February 14th, 1750, the following mention is made of them. "Our Benjamin Franklin," says Mr. Logan, "is certainly an extraordinary man, one of a singular good judgment, but of equal modesty. He is clerk of our Assembly, and there, for want of other employment, while he sat idle, he took it into his head to think of magical squares, in which he outdid Frenicle himself, who published above eighty pages in folio on that subject alone."

In reply to a letter from Mr. Logan on this subject, Franklin wrote (January 20th, 1749-50,) "The magical squares, how wonderful soever they may seem, are what I cannot value myself upon, but am rather ashamed to have it known I have spent any part of my time in an employment that cannot possibly be of any use to myself or others." EDITOR.

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 VII. Fig. 1.)

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 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