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Euclid's Fifth, July Fourth

What's math got to do with it?

Seelye Martin 30 June 2010

Inspired: did Lincoln know non-Euclidean theory?

There are two explanations for the phrasing in the Gettysburg Address; they are divided approximately between historians and mathematicians

For both the American Declaration of Independence and Abraham Lincoln’s Gettysburg Address, July 4th is significant; the Continental Congress signed the Declaration on July 4, 1776, while Lincoln gave the Gettysburg Address in November 1863 to memorialize the Union dead resulting from a battle that had its final climatic day on July 4, 1863. Both works use the phrase “all men are created equal”, but their authors seem to mean different things by it. Could mathematics have something to do with it?

But first, let’s look a little more carefully at the context. The phrase written by Thomas Jefferson occurs in the second sentence of the Declaration:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

And the phrase recurs in the first sentence of Lincoln’s Gettysburg address:

Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal.

The question arises, why did Jefferson treat the phrase “all men are created equal” as self-evident, while Lincoln treated it as a proposition? Could the change in phrasing be due to the development of mathematics in the intervening 87 years between the two documents?

For the Declaration, historical documents show that both Jefferson and his co-writer John Adams acknowledged the role of Euclid’s axioms in Jefferson’s statement, which appears to reflect a desire of Jefferson’s to derive a system of government from axioms similar to Euclid’s. At the beginning of the 19th century, Euclid’s was the only geometry. But for Euclid, a problem occurs with the fifth axiom: the parallel postulate. This postulate states that, given a straight line and a point not on this line, there is at most one line that can be drawn through the point that is parallel to the first. In 1826, the Russian mathematician Nikolai Lobachevski and others showed that when space is curved, the fifth axiom could be modified so that more than one parallel line could be drawn through the point, leading to non-Euclidean geometries. Bernhard Riemann continued this work, culminating at Gauss’s request, in an 1854 public lecture at Gottingen on n-dimensional geometries, where Gauss states that he was probably the only person in the audience to understand the work.

What do we know about Lincoln – who had only one year of formal schooling – and the subject of geometry? His law partner Bill Herndon observed that when Lincoln turned 40, he began studying Euclid, because he was concerned that he did not understand the meaning of the word “proof” and “demonstrate”. In Lincoln’s words:

At last I said, - Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.

And, from his 1860 presidential campaign biography: “He studied and nearly mastered the six books of Euclid since he was a member of Congress.”

Given Lincoln’s background in Euclid, there are two explanations for the phrasing in the Address; they are divided approximately between historians and mathematicians. The historians say that Euclid heavily influenced Lincoln, and the phrase “dedicated to the proposition” comes straight from Euclid, so that Lincoln is simply rephrasing Jefferson’s words. The mathematicians, on the other hand, say that the different phrasing shows that Lincoln was aware of non-Euclidean geometries and the violation of the fifth axiom; the change of phrasing from “self-evident” to “dedicated to the proposition”, therefore, shows that Lincoln was aware of other possible propositions as a basis for government. No written evidence exists however, that Lincoln was aware of the work of Lobachevski or Riemann. Unless written evidence appears, we will never know the truth. Finally, even though the writers of the Declaration and the Constitution were politically unable to resolve the conflict between their fine words and slavery, Lincoln spent a lifetime arguing against slavery, aided in part by his study of Euclid.

It seems appropriate to end this rumination with the last sentence from Lincoln’s first inaugural address, given in 1861 just before the war:

I am loath to close. We are not enemies, but friends. We must not be enemies. Though passion may have strained, it must not break our bonds of affection. The mystic chords of memory, stretching from every battle-field, and patriot grave, to every living heart and hearth-stone, all over this broad land, will yet swell the chorus of the Union, when again touched, as surely they will be, by the better angels of our nature.