Almost from the time Euclid stated his five postulates, mathematicians felt that there was something different about the fifth one. Intuitively, they felt that the first four postulates were somehow more fundamental; perhaps the fifth postulate could be derived from the others, and therefore was not a true axiom. Thus began a 2000-year search for a proof of the fifth postulate, one pursued by such luminaries as the astronomer (and mathematician) Ptolemy (90–168), the poet (and mathematician) Omar Khayyam (1050–1153), and the Italian priest (and mathematician) Giovanni Girolamo Saccheri, S.J. (1667–1733). Saccheri wrote a book called Euclidus Vindicatus (“Euclid Vindicated”) in which he constructed a whole geometrical system based on the the
assumption that the fifth postulate is false, then claimed that the consequences would be so bizarre that the postulate must be true.
While most 18th-century mathematicians didn’t care about axioms, the mood shifted in the 19th century. Mathematicians started to focus on the foundations of their work. They revisited geometry, no longer taking Euclid for granted, but examining his assumptions.
Around 1824, Russian mathematician Nikolai Lobachevsky was working on the problem. At some point, he realized that the parallel postulate was just one possible assumption, and that the contrary assumption is equally valid. Instead of saying “there is at most one line through a point parallel to a given line,” Lobachevsky essentially explored the idea that “there are many lines....” Unlike Saccheri, Lobachevsky realized that the resulting system of geometry was entirely consistent. In other words, he invented an entirely new non- Euclidean geometry, sometimes called hyperbolic geometry.
In Lobachevsky’s geometry, there are no similar triangles except for congruent ones. By way of analogy, think of triangles on the surface of a sphere. For small triangles, the surface is almost planar, so the sum of the angles is close to 180°. But as the triangles get bigger, the angles need to get bigger because of the curvature of the surface. Lobachevsky’s model was similar, but with space curved in the opposite way, so that bigger triangles corresponded to smaller angles.
Lobachevsky’s results, first published in 1826, were met with dismissal and scorn from the Russian mathematical community, and Lobachevsky himself was marginalized. One person who did recognize the
mathematical community, and Lobachevsky himself was marginalized. One person who did recognize the
validity of Lobachevsky’s work was Gauss, who learned Russian to read Lobachevsky’s book. But in general, it would take many years before his work became an accepted part of mathematics. Today, Lobachevsky’s discovery is considered to be a monumental turning point in the history of mathematics.
Nikolai Ivanovich Lobachevsky (1792–1856)
In the early 19th century, Russia was not a major center for mathematics (despite Euler spending much of his career in St. Petersburg). There were no great Russian mathematicians. Yet by the middle of the 20th century, Russia was a mathematical superpower. This transformation began with the first great Russian mathematician, Nikolai Ivanovich Lobachevsky.
Lobachevsky did not come from a major city, nor did he attend one of the two great universities (Moscow and St. Petersburg); he was not sent abroad to learn from the leading thinkers of Europe. He did not come from the aristocracy or even the upper middle class; he and his brother were charity students at their local school. He grew up in Kazan, a provincial city on the Volga river that did not even have a university until 1805.
Lobachevsky entered the recently founded university in 1807. (Interestingly, Tolstoy and Lenin attended the same school decades later.)
When Lobachevsky started at the University of Kazan, there was no one to teach mathematics—students studied on their own. Fortunately, Martin Bartels, one of Gauss’s former professors, soon joined the faculty. After receiving a master’s degree and continuing to study privately with Bartels, Lobachevsky was appointed as an adjunct professor in 1814. He would go on to spend most of his career at the university, eventually being elected its rector (similar to president) in 1827.
Despite his humble origins, Lobachevsky was never afraid to challenge conventional opinions. His
groundbreaking work on non-Euclidean geometry was submitted in 1826, but was not widely known until it was published as a book in 1832. The book was publicly ridiculed in a review by Ostrogradsky, an important Russian mathematician who studied with Cauchy.
Lobachevsky continued his work on non-Euclidean geometry for the rest of his life, refining it and publishing books about it in various languages. By the 1840s, Gauss recognized the importance of the work, even reading some of Lobachevsky’s books in the original Russian. Gauss nominated him for membership in the Göttingen Academy of Sciences, a great honor at the time. Yet Lobachevsky was still ostracized by the Russian
mathematical establishment until the end of his career.
In his later years, Lobachevsky’s life took a tragic turn. He lost his job at the university, his house, and most of his property, suffered the deaths of two of his children, and then became blind. Even under these
circumstances, he persisted in his work, dictating a major new book, Pangeometry, just before his death in 1856.
Often when a new idea emerges in math or science, it is discovered independently by multiple people at roughly the same time. This was the case with non-Euclidean geometry. At about the same time Lobachevsky was working in Kazan, a young Hungarian mathematician named János Bolyai made a similar discovery. A few years later, Bolyai’s father Farkas Bolyai, a well-known math professor and friend of Gauss, included the son’s results as an appendix to one of his own books. Farkas sent Gauss the book. Although Gauss privately remarked that young Bolyai was a genius, the letter he sent Farkas had a discouraging message:
If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years.
The letter is typical of Gauss, both in his refusal to give credit to others and in his insistence that his own unpublished thoughts gave him priority. (We now know that Gauss had indeed discovered many of the same ideas, but had decided not to publish them because he was afraid of the reaction.) Why he acknowledged Lobachevsky’s work but dismissed Bolyai’s we will never know. But whatever the reason, the results were tragic. Bolyai was devastated by Gauss’s response and never attempted to publish in mathematics again. Even sadder, he became mentally unstable. When he came across Lobachevsky’s book sometime later, he was convinced that “Lobachevsky” was actually a pseudonym for Gauss, whom he believed had stolen his ideas. * * *
Once non-Euclidean geometry was discovered, many mathematicians wrestled with what they considered to be an important question: which geometry is actually correct, Euclid’s or Lobachevsky’s? Gauss took the question quite seriously, and proposed an ingenious experiment to test the theory.
First, find three mountains forming a triangle that are some distance apart, but close enough so that a person standing on top of each with a telescope can see the others. Then set up surveying equipment on each peak to accurately measure the angles of the triangle. If the angles add up to 180°, then Euclid is right; if their sum is less than 180°, then Lobachevsky is.
The actual experiment was never conducted. But over time, the question became moot. Other mathematicians would ultimately prove the independence of the fifth postulate, showing that if Euclidean geometry is
consistent, then so is Lobachevskian geometry. Meanwhile, mathematicians began to treat questions of reality as irrelevant. While math was originally invented to understand aspects of the world we live in, by the end of the 19th century, it began to be seen as a purely formal exercise.