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In school geometry the angles of every triangle add up to 180°, and through a point outside a line you can draw exactly one parallel. Both facts rest on one assumption, Euclid's fifth postulate. For two thousand years mathematicians tried to prove it from Euclid's other axioms. Nikolai Lobachevsky took a different road: he asked what happens if it is false, and in 1829 he published the answer. This post covers who he was, how he defined parallel lines, numbers worked from his own formula, and the misreadings to avoid.
Who was Nikolai Lobachevsky?
His MacTutor biography gives his birth in Nizhny Novgorod on 1 December 1792 and his death in Kazan on 24 February 1856. His father died when he was seven, and his mother moved her three sons to Kazan, where they studied on government scholarships. He entered Kazan University in 1807, where his teacher Martin Bartels taught a history of mathematics course that discussed Euclid's theory of parallels.
He became a full professor in 1822 and rector of the university in 1827, a post he held for 19 years, steering it through a cholera epidemic in 1830 and a fire in 1842. After he retired in 1846 his health declined and he went blind. MacTutor notes that his work was not recognised in his lifetime.
What did he do differently?
On 11 February 1826 he asked his department at Kazan to hear a paper on a new geometry; that text has not survived. His first publication on the subject appeared in the Kazan Messenger in 1829, the first account of non-Euclidean geometry to appear in print. In 1840 he published a summary in Berlin, translated into English in 1891 by George Bruce Halsted as Geometrical Researches on the Theory of Parallels. In its preface he says Legendre's attempts to prove the postulate had done nothing for the theory.
He was not alone. János Bolyai found the same geometry independently, and Gauss had worked on it privately. MacTutor states that claims that Gauss steered Lobachevsky or Bolyai to their discoveries have been refuted. Our post on Carl Friedrich Gauss covers his wider work.
How did Lobachevsky define parallel lines?
In proposition 16 he takes a line and a point A off it, with AD the perpendicular from A of length p. Every line through A either cuts the given line or does not. The boundary lines between the two groups he calls parallel, and the angle between a parallel and AD he calls the angle of parallelism, written Π(p).
If Π(p) is a right angle, there is one parallel and you are back in Euclid. If Π(p) is less than a right angle, there are two parallels, one on each side, and every line between them misses the given line too. Propositions 19 and 20 then show that a triangle's angle sum can never exceed two right angles, and that if one triangle has a sum of exactly two right angles, every triangle does. So either all triangles have 180°, as in Euclid, or all have less.
Worked example: his formula for the angle of parallelism
In proposition 36 he derives tan(½Π(x)) = e^(−x), where x is measured in a fixed unit that belongs to the geometry itself. Here is what it gives.
x = 0. tan(½Π) = e⁰ = 1, so ½Π = 45° and Π = 90°. At zero distance the parallel is perpendicular to AD, as in Euclid.
x = 0.1. e^(−0.1) ≈ 0.9048, and the angle whose tangent is 0.9048 is about 42.14°, so Π ≈ 84.3°.
x = 1. e^(−1) ≈ 0.3679, giving ½Π ≈ 20.20° and Π ≈ 40.4°.
x = 2. e^(−2) ≈ 0.1353, giving ½Π ≈ 7.71° and Π ≈ 15.4°.
Close to the line the parallel sits nearly at a right angle to AD, so the geometry is hard to tell from Euclid's. Far from it, the parallels lean over sharply and a wide fan of lines through A never meets the given line.
What do students often get wrong?
"Non-Euclidean means Euclid was wrong." Lobachevsky showed that a consistent geometry exists without the fifth postulate. Euclid's geometry is still correct as one possible system, and it is the one your board exams use.
Mixing up the two non-Euclidean geometries. In Lobachevsky's geometry angle sums are less than 180°. Sums greater than 180°, as for triangles drawn on a sphere, belong to a different geometry, which our post on Bernhard Riemann touches on.
Confusing Π(p) with π. Π(p) is an angle that depends on a distance; π is a constant.
Forgetting the unit. In tan(½Π(x)) = e^(−x), x is not in metres. It is measured in the geometry's own unit of length.
How did the idea finally win acceptance?
MacTutor traces the slow road. In 1866, ten years after his death, Hoüel published a French translation of the 1840 summary. Beltrami gave a concrete model of the geometry in 1868, Klein fitted it into his view of geometry as the study of properties kept under groups of transformations, and Henri Poincaré made further contributions in 1882 and 1887. These ideas later helped give relativity a natural mathematical foundation.
Postulates and the proofs built on them are the subject of our post on Euclid's Elements. If you want to revisit those foundations first, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
What can a student take from Lobachevsky's way of working?
Change the question. Instead of one more failed proof of the postulate, he asked what follows without it.
Trust the logic past your intuition. His results looked strange, but each followed from the axioms he kept.
Keep going without applause. From 1829 on he kept developing a geometry few colleagues accepted.
Next step: use tan(½Π(x)) = e^(−x) to find Π(0.5), and check that you get about 62.5°.
