"The treatise itself, therefore, contains only twenty-four pages—the most extraordinary two dozen pages in the whole history of thought!""How different with Bolyai János and Lobachévski, who claimed at once, unflinchingly, that their discovery marked an epoch in human thought so momentous as to be unsurpassed by anything recorded in the history of philosophy or of science, demonstrating as had never been proved before the supremacy of pure reason at the very moment of overthrowing what had forever seemed its surest possession, the axioms of geometry."

—George Bruce Halsted, on János Bolyai's treatise on non-Euclidean geometry,

The Science of Absolute Space.

Last week, I gave a talk for undergraduates about hyperbolic geometry, one of my very favorite topics in math. On Twitter, I joked about giving a hyperbolic talk about geometry instead of a talk about hyperbolic geometry.

A talk about hyperbolic geometry is different from hyperbolic talk about geometry: "OMGosh, geometry is the coolest & it will cure cancer!"— Evelyn Lamb (@evelynjlamb) February 5, 2014

But my joke wasn't so far-fetched. While I was preparing for my talk, I encountered several hyperbolic quotes from the early history of hyperbolic geometry. The first two, at the top of this post, are from the lengthy and flowery translator's introduction to János Bolyai's treatise on non-Euclidean geometry. I found it in Roberto Bonola's textbook *Non-Euclidean Geometry, *which is available for free here. Bolyai and Nikolai Lobachevsky (yes, *that* Nikolai Lobachevsky) were the first two mathematicians to publish their work on non-Euclidean geometry.

First, a few basics about non-Euclidean and hyperbolic geometry. Euclid's Elements, one of the most enduring textbooks in history, was the first systematic treatment of geometry. Euclid begins with 23 definitions, 5 axioms, and 5 postulates and derives all sorts of theorems from them. The fifth postulate, also called the parallel postulate, feels very different from the other definitions and postulates. One way of stating it is that if you have a line L and a point P not on L, there is exactly one line that goes through the point P and is parallel to (does not intersect) L.

The other postulates and axioms, statements like "any two points can be connected by a straight line segment," seem much more self-evident. The parallel postulate never sat well with some mathematicians, who felt that there must be a way to derive it from the other axioms and postulates. And for 2000 years, they tried to. The story of non-Euclidean geometry is the story of mathematicians discovering that you can invent geometries that are consistent with the rest of Euclid's geometry but do not follow the parallel postulate. There are two ways of violating the parallel postulate: either there exist no lines through P that do not intersect L, or there exist multiple lines through P that do not intersect L. The former case is called elliptic geometry, and the latter case is called hyperbolic geometry. Non-Euclidean is a general term for either of these cases.

Bolyai's father, Farkas (sometimes called Wolfgang), was also a mathematician. In fact, he had tried to prove the parallel postulate and wrote to his son,

"It is unbelievable that this stubborn darkness, this eternal eclipse, this flaw in geometry, this eternal cloud on virgin truth can be endured."

But Farkas did not want his son to go down the same road he had.

"You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone....I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum [if it's failed to make the grade, even by a smidgeon, it might as well be the worst]. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind.""I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness—aut Caesar aut nihil [either Caesar or nothing]."

"For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life."

Luckily, János did not heed his father's advice:

"I am resolved to publish a work on parallels as soon as I can put it in order, complete it, and the opportunity arises. I have not yet made the discovery but the path which I have followed is almost certain to lead me to my goal, provided this goal is possible. I do not yet have it but I have found things so magnificent that I was astounded. It would be an eternal pity if these things were lost as you, my dear father, are bound to admit when you seen them. All I can say now is that I have created a new and different world out of nothing. All that I have sent you thus far is like a house of cards compared with a tower."

With János's success in reach, Farkas changed his tune and urged him to publish. As János recalls:

“He advised me that, if I was really successful, I should speedily make a public announcement and that for two reasons. One reason is that the idea might easily pass to someone else who would then publish it. Another reason—and one that seems valid enough—is that when the time is ripe for certain things, these things appear in different places in the manner of violets coming to light in early spring. And since scientific striving is like a war of which one does not know when it will be replaced by peace one must, if possible, win; for here preeminence comes to him who is first.”

I found most of the back-and-forth between the Bolyais in Herbert Meschkowski's textbook Noneuclidean Geometry, which begins with this somewhat hyperbolic paragraph:

"The discoverers of noneuclidean geometry fared somewhat like the biblical king Saul. Saul was looking for some donkeys and found a kingdom. The mathematicians wanted merely to pick a hole in old Euclid and show that one of his postulates which he though was not deducible from the others is, in fact, so deducible. In this they failed. But they found a new world, a geometry in which there are infinitely many lines parallel to a given line and passing through a given point; in which the sum of the angles in a triangle is less than two right angles; and which is nevertheless free of contradiction."

These quotes all seem a bit dramatic for geometry, but it's easy not to know how truly revolutionary the discovery of non-Euclidean geometry was. Halsted's description of Bolyai's paper as "the most extraordinary two dozen pages in the whole history of thought" certainly sounds hyperbolic, but can you find 24 other pages that compete with it? This is not a rhetorical question. I'm curious about what else might be in the running. I can't think of anything off the top of my head, but I love hyperbolic geometry so much that I can't possibly be expected to be objective about it!