In 1865, the Reverend J.P. Gulliver asked Abraham Lincoln how he came to acquire his famous rhetorical skill. The President gave an unusual response:

“In the course of my law-reading I constantly came upon the word ‘demonstrate.’ I thought, at first, that I understood its meaning, but soon became satisfied that I did not…. At last I said, ‘Lincoln, you can never 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 propositions in the six books of Euclid at sight.”

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Geometry is the cilantro of math: Few people have neutral feelings about it.

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Not the Constitution, not scripture, but geometry—that’s where Lincoln went when he needed to learn to persuade. Euclid was a mathematician in Greek North Africa in the 4th century B.C., who gathered and systematized the geometric knowledge of his day. His “Elements,” in Lincoln’s time and to a lesser extent our own, is the standard model of mathematical proof or “demonstration.” Starting with axioms that the reader can hardly doubt, Euclid builds up a rich body of knowledge about angles, line segments, circles and figures, step by careful step.

There’s something special about geometry, as any high-school student or former high-school student can tell you. Geometry is the cilantro of math: Few people have neutral feelings about it. There are those who hate it, who say geometry was the moment when math stopped making sense to them. Others say it was the *only* part of math that made sense to them.

The young Lincoln was one of those who couldn’t get enough. His law partner William Herndon, who often had to share a bed with Lincoln at small country inns during their sojourns around Illinois, recalled that the future president would stay up late into the night with a candle lit, deep in Euclid, his long legs hanging over the edge of the bed. Herndon once found Lincoln in a haggard state, having spent two full days trying to solve the old conundrum of squaring the circle. “His attempt to establish the proposition having ended in failure,” Herndon remembers, “we, in the office, suspected that he was more or less sensitive about it and were therefore discreet enough to avoid referring to it.”

Still, for most high-school students “Lincoln thought geometry was important” isn’t a good enough reason to study it. Will today’s teen, somewhere down the boulevard of life, be called upon to square a circle (hopefully not, since that was proved to be impossible in 1882), or be asked to demonstrate that the sum of the exterior angles of a polygon is 360 degrees? I keep waiting for that to happen to me and it never has, even though I’m a professor of mathematics.

Some would say that the point of geometry is its beauty. That was the poet Edna St. Vincent Millay’s take: “Euclid alone has looked on beauty bare,” she wrote in a sonnet. Millay’s Euclid is an unearthly figure, struck by a shaft of insight on a “holy, terrible day.” Not like the rest of us, who might get to hear Beauty’s footsteps hurrying off down a faraway hallway, Millay says—if we’re lucky.

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As a mathematician, I get a lot of benefit from geometry’s prestige. It feels good when people think that the work you do is mysterious, eternal, elevated above the common plane. “How was your day?” “Oh, holy and terrible, the usual.” But the harder you push that point of view, the more you incline people to see the study of geometry as an obligation. It acquires the slightly musty smell of something that’s good for you.

Anyway, geometry isn’t just Euclid, and it hasn’t been for a long time. Even in Lincoln’s time, geometry had begun to outgrow its Euclidean baby clothes. Mathematicians like Bolyai, Lobachevsky and Riemann were developing new notions of distance, space and curvature that their forebears couldn’t have imagined. By the beginning of the 20th century, it was more accurate to speak of geometries than geometry.

Today the subject is expanding faster than ever before. Mathematicians study the geometry of social networks and analyze the spread of a pandemic in terms of motion through those networks. We can model personality and political ideology as points in higher-dimensional space. Startling new developments in artificial intelligence are nothing but applied geometry, analyzing paths in the infinite-dimensional space of all possible algorithms.

But Abraham Lincoln wasn’t an AI developer or a paradigm-busting pure mathematician. His interest in Euclid arose, as he told Rev. Gulliver, because he needed to know what “proof” was. What distinguished Lincoln as a thinker, his friend and fellow lawyer Henry Clay Whitney recalled, wasn’t his brilliance; lots of people in public life are smart, and among them one finds both the good and the bad. What made Lincoln special was integrity, his belief that you should not say something unless you have demonstrated that it is right. Whitney writes: “It was morally impossible for Lincoln to argue dishonestly; he could no more do it than he could steal; it was the same thing to him in essence, to despoil a man of his property by larceny, or by illogical or flagitious reasoning.”

In Euclid, Lincoln found a language in which it’s very hard to dissimulate, cheat or dodge the question. Geometry is a form of honesty.

The ultimate reason for young people to learn how to write a proof is that the world is full of bad logic, and we need to know the difference. Geometry teaches us to be skeptical when someone says they’re “just being logical.” If they are talking about an economic policy, or a cultural figure whose behavior they deplore, or a concession they want you to make in a relationship, rather than a congruence of triangles, they aren’t just being logical. They want you to mistake an assertively expressed chain of opinions for proof of a theorem.

Knowing geometry protects you: Once you’ve experienced the sharp click of an honest-to-goodness proof, you’ll never fall for this trick again. Tell your “logical” opponent to go square a circle.

—Mr. Ellenberg is a professor of mathematics at the University of Wisconsin. This essay is adapted from his new book “Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else,” which will be published May 25 by Penguin Press.

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