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Benjamin Franklin and His Magic Squares

Recently, a 2-part, 4-hour documentary by Ken Burns was aired on PBS on the life of Benjamin Franklin and this rekindled my interest in the Founder. While browsing the Benjamin Franklin Papers, held here at the Library in the Manuscript Division, I came across a letter that Franklin wrote to his sister, Jane Mecom, on September 20, 1787. In it he references the costs of war in terms of ammunition, provisions, fair pay for troops, etc., writing:

It seems to me, that, if statesmen had a little more arithmetic, or were more accustomed to calculation, wars would be much less frequent.

Page from Practica arithmetice showing various magic squares.

Page from Practica Arithmetice, showing various magic squares. 1539.

I was well acquainted with Franklin’s experiments and inventions, but this casual mention of arithmetic spurred me to find out what I could about his relationship with mathematics. Franklin didn’t have a thorough education and was largely self-taught, including in mathematics. On page 98 of his autobiography Franklin writes:

And now it was that, being on some occasion made asham’d of my ignorance in figures, which I had twice failed in learning while at school, I took Cocker’s book of arithmetick, and went through the whole by myself with great ease.

This surprised me given his adeptness at creating magic squares, which require skill with arithmetic and quite a bit of creativity. In these matrices each row, column, and major diagonal equal the same number; similar to, but not quite the same as, sudoku.

These squares have been around for thousands of years and can be found in many cultural traditions such as Chinese, Indian, and Islamic. Some cultures attached mystical meanings to them and over time magic triangles, circles, hexagons, and other shapes have also appeared.

In his autobiography, Franklin writes that his time as clerk of the Pennsylvania Assembly in the 1730s and 1740s was so excruciatingly dull that he passed the time by “…making magic squares, or circles, or any thing to avoid weariness.” In addition to the typical features of a magic square, Franklin included bent rows. For example, picture lines that begin in the lower left and bottom corners of the square and draw nearer as they approach the center diagonally until they meet, forming an arrow (^). One of his most famous squares, which appeared in The Gentleman’s Magazine (v. 38, 1768), can be seen here with an explanation alongside:

Benjamin Franklin. A Magic Square of Squares. The Gentleman’s Magazine, v.38. 1768: 313.

If after reading this post you too would like to learn more about founding father’s interest in science and business, see the research guide Benjamin Franklin: Man of Business and Science, featuring books, manuscript collections, and more.

Further explore the fun of puzzles using these subject headings in the Library of Congress online catalog:

For even more diversions see our 5-part series Mathematical Games of Martin Gardner!

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Mental Arithmetic

However, practical arithmetic standardized professionalization by treating all students as potential workers in fields that required numeracy skills–from banking and farming to masonry and carpentry. The business application of arithmetic education became so ingrained–apparently to Davidson’s dismay–that by 1895, the West Virginia State Series curriculum defined arithmetic as: “the solving of problems from its various applications to business life.”