While re-reading the wonderful parallel biography Team of Rivals: The Political Genius of Abraham Lincoln by Doris Kearns Goodwin, I was reminded of this passage from Lincoln’s time on the Illinois traveling law circuit in the 1850s, the interlude between his term in the House of Representatives and his ascent to the presidency:

Life on the circuit provided Lincoln the time and space he needed to remedy the “want of education” he regretted all his life. During his nights and weekends on the circuit, in the absence of domestic interruptions, he 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.”

[Law partner William] 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.”

Doris Kearns Goodwin, Team of Rivals: The Political Genius of Abraham Lincoln, pages 152-153

I have two thoughts on this: one mathematical, and one political (albeit the politics of the 19th century).

I must admit that I’m charmed by the mental image of Lincoln, like so many amateur (and professional) mathematicians before and after him, deeply engrossed after a hard day’s work by the classical problem of squaring the circle, described by Wikipedia as “the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge.”

A subtle historical detail was left out of the above account, one that I would not expect a popular history book to include. While it’s known today that squaring the circle is impossible, this was not a settled question during Lincoln’s lifetime. Indeed, the impossibility of squaring the circle was settled in 1882, seventeen years after Lincoln’s death, when Ferdinand von Lindemann proved the transcendence of — that is not a root of any polynomial with integer coefficients. All this to say, when Lincoln spent two days attempting to square a circle, he was actually working on a celebrated open problem in mathematics that was easily understood by amateur mathematicians of the day… in much the same way that the Twin Prime Conjecture attracts attention today.

(As a personal aside: I still remember the triumph I felt a student many, many years ago when I read through this proof in Field Theory and Its Classical Problems and understood it well enough to stand at the chalkboard for the better part of an hour to present it to my teacher.)

Politically, I was reminded of the wonderful book Abraham Lincoln and The Structure of Reason by David Hirsch and Dan Van Haften. Hirsch and Van Haften argue that Lincoln’s studies of geometry were not merely for idle leisure or personal satisfaction, in the same way that people recreationally solve crossword puzzles today. Instead, they argue that Lincoln’s penchant for persuasive rhetoric was shaped (pardon the pun) by his study of geometry, and that Lincoln’s speeches tended to follow the same six-part outline that Euclid employed when writing geometric proofs in The Elements.