Euclid's diagram accompanying his proof of Pythagoras' right-triangle theorem: In any right triangle, the sum of the squares of the two shorter sides equals the square of the longest side.
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Hey hey hi Mumfacolyte --
Here's the story I remember about Lincoln and Euclid, and I suspect I got it from Sandburg's bio (super-recommended),
When Lincoln first began dabbling in politics, he helped a winning candidate, and was rewarded with an Illinois state job as a surveyor -- a job for which he had no previous experience. He was given a theodolite and a crude book of instructions, and went about surveying farms and roads.
In his crude surveying primer, he kept encountering footnotes that said "qv. Euclid," was curious to know who this Euclid guy was, inquired, and was loaned a copy of The Elements. Lincoln and Euclid clicked instantly, and he dove into it like a starving man at a banquet.
Late in his life Lincoln claimed to have proven every demonstration in Euclid, and described it as the singlemost influential event of his intellectual life. As so many other geniuses before and after, he was astonished at the fundamental concept -- that things about the World could actually be PROVEN, known with absolute certainty; and that this huge volume of Proofs were interconnected, each new proof resting with perfect certainty on the proofs that had come before.
We don't normally think of Lincoln as math-minded, but in fact, if his memoir is true, he could blow just about anyone out of the water with his adoring mastery of Euclid.
The utterances from the mouths of politicians are such cotton-candy, ill-defined, evasive crap (e.g., what, exactly, do American politicians mean by "freedom" as they goose applause from unsophisticated audiences?) ... I'm convinced Euclid's demands of clear, precise definitions and ironclad logical step-by-step demonstrations evolved into the core of Lincoln's political and debating skills. More than any of his political rivals and contemporaries, he could instantly spot the flaws and lies in any political argument, and could illuminate the holes and gaffs so that audiences quickly saw which speaker was spewing fluff, and which speaker was on solid "Euclidean" ground.
The Lincoln-Douglas Debates are particularly interesting to read in detail from this perspective. Though Lincoln lost the Senate race to Douglas, the debates, widely published, launched Lincoln from backwoods obscurity to national attention. He was making sense about and illuminating very important national matters -- and Douglas, the far more florid and celebrated orator, wasn't.
I don't want to overplay the importance of Euclid to Lincoln's political success. He also had a personality that can only be described as Magic, people couldn't help but love and admire him, he was a wildly funny raconteur, and these skills made juries his adoring slaves and made Lincoln one America's most successful trial lawyers.
Two memoirs about first encountering Euclid as children (these are paraphrases from memory):
Bertrand Russell: "I never imagined anything could be so delicious."
John Stuart Mill read the first few pages of The Elements and screamed: "This is preposterous!" (That, of course, wasn't his final verdict, he kept reading every preposterous word.)
It's all aesthetics, but my absolute favorite is Euclid's proof (Book IX, Proposition 20) that there is an infinity of prime numbers; there is no greatest prime number.
And he does it all with unmarked straightedge, dividers, and line segments. (Today we'd go straight to the algebra and arithmetic of the argument.)
I think it's enough for every intelligent, civilized person just to be "touched" -- well, maybe punched and clobbered a few times -- by Euclid, to have thoroughly mastered just a half-dozen of the proofs. Mastering Euclid's proof of Pythagoras' right-triangle theorem -- that earns every woman, man and child the right to say forevermore that they truly understand the World, and how Reality is truly constructed.
Two particularly fascinating works based entirely on Euclid's example:
Spinoza's "Ethics" -- right, wrong, good, evil, in human behavior, all defined and proven precisely in the Euclid way.
Newton's "Principia" -- though he'd already invented the calculus secretly to quickly grasp the System of the World, Newton published his results entirely in the traditional Euclidean method of "The Elements," without the slightest hint that any other method could match the precision and perfection of Euclid.
My reading of the extensive Dover notes regarding the Parallel Postulate seems to make it clear that even Euclid thought there was something very fishy -- and "un-Euclidean" -- about this particular foundation of his geometry. I'm certain Euclid would have embraced the modern discoveries of non-Euclidean geometry with instant understanding and great pleasure.