“The Copernicus of Geometry”

A young Nikolai Lobachevksy. Image: Lev Kriukov (father), via Wikimedia Commons.

On December 1st, 1792 one man, who would create a revolution in geometry, was born. Actually, a lot of people were born on December 1st, 1792. I can’t name any others, but I’m 99% sure that more than one person was born on that day. I’m not a betting man, but if I was I’d even gamble that more than three were born that day. But I don’t really care about them (no offense to them of course, I’m sure they were fine people). I only care about Nikolai Lobachevsky, the man who would take geometry from the ideas of Euclid, throw those ideas away (he didn’t do that), and change the rules and our perception of shapes, angles, and all things geometric.

Since Euclid’s Elements, circa 300 BC, geometry had been looked at in Euclidean way. Euclid’s axioms and postulates were how it was, and mathematicians had to work within those confines. One similarity is how humanity thought that everything revolved around the Earth, the very human, egotistic geocentric model of the cosmos. In Elements, Euclid’s mathematical magnum opus, which may be the most influential treatise of all time, Euclid creates axioms, propositions, and proofs giving an overview of Euclid’s ideas on number theory and geometry. One of the most important axioms within Elements is Euclid’s parallel postulate.

The parallel postulate states that if a line segments intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. For many, many years, around two thousand, various mathematicians attempted to prove this postulate. With no success, again and again mathematicians would never see the day the postulate was proven, nor were their names engraved in history as the person to prove such an important postulate. However, ideas of negating the postulate altogether came into fashion in the early 19th century. Lobachevsky took this task to hand and worked his way towards an entirely new view of geometry.

Lobachevsky decided to abandon the idea of the parallel postulate, negating its meaning trying to see if there was a possibility for geometry that did not follow the rules Euclid put into place. Lobachevsky worked around the idea that there exist two lines parallel to a given line through a given point not on the line. In 1829, Lobachevsky published a paper in the Kazan Messenger on his new, non-Euclidean geometry, doing this before anyone else had. Unfortunately due to the small nature of the paper, as well as the fact that it was Russian, his work went largely unnoticed. While others like the famous János Bolyai later discovered this new non-Euclidean geometry completely separate from Lobachevsky, they discovered it years after Lobachevsky did. This new geometry became known as hyperbolic geometry, a geometry that Pringles would sponsor if people didn’t hate math and for some reason math had sponsors. A new form of geometry was born, and Lobachevsky discovered his own personal heliocentric cosmos.

Lobachevsky had many other findings. He discovered the angle of parallelism in hyperbolic geometry, the computation for the roots of a polynomial, and the “Lobachevsky criterion for convergence of an infinite series.” When it comes to his life, it unfortunately wasn’t as great as his discoveries. The combination of his radical new theories, findings that were found same time others discovered them (this can be seen with the Graeffe’s method, which is the computation of the roots of a polynomial that I previously mentioned, and Peter Dirichlet’s definition of a function), and being Russian led him to quite the sad ending. Left without the ability to walk and blind with no job due to his quickly deteriorating health, his life ended in poverty. He had lost his son he loved the most to tuberculosis, came from a poor family, died a poor man, and worked hard all his life without much humor or relaxation. He is quite the Russian stereotype. If I were to make a movie about Russia, he would be the person who symbolizes the Russian winter.

Luckily for Lobachevsky, and moreover mathematics as a whole, his legacy and ideas in his works have lived on. Much work has been done in hyperbolic geometry since his time, as well as the extension of non-Euclidean geometry to Riemannian geometry. Taking what we consider as fact and not only negating it but also proving there is more to it, in this sense going from Euclidean to non-Euclidean geometry, is a revolutionary task that not many people in the history of, well, the universe, have done. It’s like that one Arcade Fire song, they just tell us lies.

On February 24th, 1856, a lot of people died. Like, a lot of people. I don’t know how many people, but I assume there were quite a few. When you think of how many people die each day, it’s slightly horrifying. On that day Nikolai Lobachevsky died, a poor man with no vision and not much left to live for. However on February 24th, 1856, many people were born. And even today, even more people were born. And who knows, maybe the next Nikolai Lobachevsky was born today.

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