Lincoln, Gettysburg, and Geometry

381px-Abraham_Lincoln_O-77_by_Gardner,_1863In my last blog, I showed a connection between Lincoln’s Cooper Union speech and Euclid’s Elements. There is also a connection between the fundamental postulates in a mathematical geometry and the fundamental postulates in a system of government or political geometry.

All mathematical geometries have a core set of postulates from which all statements or theorems about the relationships between the objects in the geometry are derived. Different sets of postulates result in different sets of geometric theorems. For example: in the geometry of a flat surface, Euclidean geometry, there is a parallel line postulate from which we can deduce that the sum of three angles in any flat surface triangle equals 180 degrees. In the geometry of a sphere, all lines intersect and therefore there is no parallel line postulate. As a result, the sum of the three angles in any spherical triangle is greater than 180 degrees and less than 540 degrees.

All political geometries, likewise, have a core set of postulates from which the laws of the political geometry are derived. The opening lines of the second paragraph of the Declaration of Independence stated the fundamental postulate of the new American democracy, “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.” This new American political geometry was radically different from previous political geometries.

The meaning of the word “all” in the fundamental postulate of American democracy eventually led to the Civil War. Initially, Lincoln viewed the Civil War as a struggle to preserve the Union. Only later did he see it as a struggle to make the fundamental postulate of American democracy a reality for all Americans. Students of mathematics come to fully appreciate and understand terms like “all,” “each,” and “every” only after they have acquired a sufficient level of mathematical maturity. Lincoln had acquired sufficient political maturity by the time he gave his Gettysburg Address on November 19, 1863.

Lincoln began his Gettysburg Address by reminding his audience of the fundamental postulate of American democracy. The Civil War was about making this postulate a reality for all Americans. He then tells Americans that we, the living, have the responsibility to finish the work of the living and dead Americans who so nobly fought to advance the fundamental postulate of American democracy. The struggle goes on today.

All human enterprises are connected. Whether the enterprise involves art, music, literature, dance, theater, mathematics, physics, chemistry, biology, political science, computer science, history, psychology, religion, sports, etc., there is always some connection to be found. When I find a connection that I have not realized before, I experience one of life’s special moments, a moment of epiphany.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have you experienced?

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Photo credit: “Abraham Lincoln O-77 by Gardner, 1863” by Alexander Gardner – Library of Congress. Taken on November 8, 1863, just 11 days before Lincoln’s address at Gettysburg.



What Links Abraham Lincoln to Euclid’s Elements?

Lincolnx400On February 27, 1860 Abraham Lincoln gave a speech at the Cooper Union Institute in New York City. The vast majority of the audience of approximately 1,300 were members of the Republican party. Lincoln, like many in the audience, was well aware that this speech was his one chance to show the party’s movers and shakers that the prairie lawyer from Illinois was cut from presidential timber. Lincoln’s physical appearance, poorly tailored suit, awkward gait, and frontier twang caused many in the audience to form the initial opinion that he would not be a suitable Republican candidate for President.

That evening, Lincoln’s speech was sensational, evaporating any doubts about his suitability as a Republican candidate for President. The audience was awed and dazzled by his command of historical facts and the airtight logic of his speech. Three months later Lincoln received the Republican nomination for President, and the Cooper Union speech effectively became the Republican Party platform for the 1860 presidential election.

In his book, Lincoln at Cooper Union, Harold Holzer brilliantly captures that magical evening and gives the reader a wonderful description of the events leading up to Cooper Union and the events afterward that made Lincoln President.

After reading and studying Holzer’s book several times, I realized that Lincoln’s Cooper Union speech was, astonishingly, modeled after a classic Euclidean geometric proof. This made sense. Both Lincoln and Thomas Jefferson admired and read Euclid’s Elements as a way to improve the mind and promote an individual’s ability to think and reason logically. For me, a former high school geometry teacher and Civil War buff, this was a thrilling realization, a moment of epiphany.


Lincoln organized his Cooper Union speech into three sections. The objective of the first section was to demonstrate or prove that Senator Steven Douglas’ principle of popular sovereignty was a false doctrine. The principle of popular sovereignty claimed that only the states had the power to decide whether or not to allow slavery into the territory of a new state, and the Federal government had no control over the spread of slavery. Lincoln believed that the Federal government had the right and duty to control the spread of slavery.

He began his Euclidian proof by stating a postulate that all parties in the popular sovereignty debate could agree with:  “Our fathers, when they framed the Government under which we live, understood this question just as well, and even better, than we do now.” “Question” refers, of course, to the spread of slavery, and the postulate was a direct quote from a speech given by Senator Douglas.

The manner in which Lincoln constructs and develops the statements in his proof is brilliant. It’s like watching a creative master teacher presenting a proof of a theorem to a class of geometry students. Each statement in his proof is a logical consequence of previous statements, all supported by facts in the historical record. During the course of his speech, Lincoln referred to the postulate no less than 15 times.

In section three, Lincoln’s goal is to rally the Republican Party around the principles that slavery is wrong and should not be allowed to spread beyond where it already exists. Republicans have a duty to publicly declare these principles and should not compromise them in order to appease Southern slaveholders. To prove his point, Lincoln used proof by contradiction, a sophisticated technique commonly used by mathematicians to prove a theorem.  The proof begins with the assumption that the negation of the statement is true. At Cooper Union, the negation of the statement Lincoln was attempting to prove was, “Slavery is right.” From the assumed statement, one draws logically correct conclusions that eventually lead to a statement that’s false. Therefore the negation of the statement must be false and the original statement must be true.

To fully appreciate and understand the Cooper Union speech, read Holzer’s book. Even if you normally don’t enjoy history books, I think you’ll find Lincoln at Cooper Union a great read. If Lincoln had failed at Cooper Union, he would not have become President, and the course of world history would have taken a far different path.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have your experienced?

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