Posted by: Dave Richeson | February 18, 2011

## Lincoln and squaring the circle

I’d heard a long time ago that Abraham Lincoln was a largely self-taught man and that he read Euclid’s Elements on his own. Right now I’m reading Doris Kearns Goodwin’s Team of Rivals: The Political Genius of Abraham Lincoln, and from it I learned that not only did he read Euclid, he spent some time trying to square the circle. Today we think of circle squarers as mathematical cranks. But remember that this was in the 1850′s, more than two decades before Lindemann’s proof that $\pi$ is transcendental—the result which proved conclusively that it is impossible to square the circle.

Here’s the relevant passage:

During his nights and weekends on the circuit, in the absence of domestic interruptions, [Lincoln] taught himself geometry, carefully working out propositions and theorems until he could proudly claim that he had “nearly mastered the Six-books of Euclid.” His first law partner, John Stuart, recalled that “he read hard works—was philosophical—logical—mathematical—never read generally.”

Herndon describes finding him one day “so deeply absorbed in study he scarcely looked up when I entered.” Surrounded by “a quantity of blank paper, large heavy sheets, a compass, a rule, numerous pencils, several bottles of ink of various colors, and a profusion of stationery,” Lincoln was apparently “struggling with a calculation of some magnitude, for scattered about were sheet after sheet of paper covered with an unusual array of figures.” When Herndon inquired what he was doing, he announced “that he was trying to solve the difficult problem of squaring the circle.” To this insoluble task posed by the ancients over four thousand years earlier, he devoted “the better part of the succeeding two days… almost to the point of exhaustion.”

Pretty cool!

Aside: There are two mathematical oddities here. First of all, it is strange that they mention the six books of Euclid, rather than the thirteen books. [Update: Now that I think about it, the first six books are the ones covering plane geometry. Book 7 is where the number theory begins. Then the end of Elements covers solid geometry.] Second, I’m curious to know where the author got the figure “over four thousand years ago” for the origin of the circle squaring problem. If the origin of the problem is marked by the first approximation of $\pi$, then that’s not a terrible exageration (as far as I am aware, the earliest known approximation is found in the Egyptian Rhind papyrus, which dates back to roughly 1650 BCE). But if we mean the classical problem (Is it possible to create a square with the same area as a given circle using only a straightedge and compass?), then it is a much younger problem than she asserts.

Second aside: Lincoln was not the only mathematically-inclined president. For example, James Garfield discovered a new proof of the Pythagorean theorem.

## Responses

1. Squaring the circle may be an apt metaphor for Lincoln’s presidency, struggling to reconcile the irreconcilable.

2. Delightful post. It reminded me of many wonderful hours, days and weeks spent trying to trisect an angle with compass and straightedge as a child!

Today, I still think the time was well spent.

3. [...] Szemerédi-Trotter Theorem via the polynomial ham sandwich theorem. But you could also learn from Division by Zero that Abraham Lincoln had tried to square the circle (and, of course, wasn’t a crank). This week could not go by without many posts on Watson [...]

4. Oh come on. Everybody is talking about Garfield’s “new” proof of PT, just because he was a US president (with a cool name;). However, that proof is exactly half of the standard proof (http://www.matematikaria.com/geometri/pythagorean-theorem-proof.html), with nothing special except a reference to trapezoids (instead of squares). Wow. :-P