Nikolai Ivanovich Lobachevsky (1792 - 1856)

Lobachevsky was born to a poor family in Kazan in Russia. He attended the local school on a scholarship, and later studied at the recently founded Kazan University. Like Daniel Bernoulli before him, Lobachevsky originally intended to study medicine, but fortunately for mathematics he soon switched to mathematics and physics.

He quickly made excellent progress and within a few years was a professor, teaching a wide range of subjects including maths, physics and astronomy.

This period in his life was not entirely happy. Lobachevsky clashed frequently with the head of the University administration - curator M L Magnitski - over its conservative approach to modern scientific and philosophic developments.

However, things changed with the succession of Tsar Nicholas I to the Russian throne. A new curator was appointed, bringing in a far more tolerant system, and Lobachevsky was appointed rector.

Lobachevsky was rector for almost 20 years, during which time the University went from strength to strength, both in its faculties and its student numbers. Lobachevsky can be given much of the credit for this success.

He continued to teach an ever increasing selection of mathematical subjects, from mechanics to differential equations, from the calculus of variations to hydrodynamics.

But it is in the field of geometry that Lobachevsky left his mark. Lobachevsky’s work on geometry had really important implications for modern geometry - he along with Gauss can be said to be one of the discoverers of non-Euclidean geometry.

Bolyai and Lambert both had the same opportunities as Lobachevsky but could not go the final mile and embrace the utterly new territory that lay at their feet. Lobachevsky could as did Gauss, though Gauss did not publish his work. Riemann developed their work even further.

In 1837 Lobachevsky published his article Géométrie imaginaire, and a summary of his new geometry Geometrische Untersuchungen zur Theorie der Parellellinien was published in Berlin in 1840.

His major work, Geometriya completed in 1823, was not published in its original form until 1909.

He retired from the University in 1846, and fell ill after the death of his eldest son, eventually losing his sight as a result of the great stress. Sadly, Lobachevsky’s mathematical contributions were not recognised during his lifetime, and he died a poor man, not knowing the importance of his work.

He is, however, immortalised in a song by the American mathematician, Tom Lehrer: as the song goes ‘Nikolai Ivanovich Lobachevsky was his name’.

Lobachevsky’s mathematics

The background to Non-Euclidean geometry is easy to describe.

Euclidean geometry is the geometry of flat surfaces, unlike say, the geometry on the surface of a sphere, which not unaturally is called Spherical geometry. Euclidean geometry starts off with a set of assumptions and basic ideas - called axioms and postulates. These are taken as given. Using them, Theorems are progressivley discovered and proved. Each Theorem when proved, may be used in the proof of a new Theorem. Here is an example of this at work.

Theorem already proved: alternate angles of a pair of parallel lines are equal; corresponding angles of a pair of parallel lines are equal;

Postulates - angles on a straight line sum to 180^\circ; lines may be extended indefinitely; through a given point may be drawn a line parallel to any other line;

THEOREM: The angles of a triangle add up to 180^\circ

PROOF: Take any triangle. Extend a side of the triangle, and draw a line parallel to another side as shown. Mark equal alternate and corresponding angles as shown:

Because of the properties of parallel lines (alternate angles, the angles marked \angle B; corresponding angles, the angles marked ) there are two sets of equal angles as marked. The angles A, B and C now make a straight line and so

\angle A+\angle B+\angle C=180^\circ \mbox{as required. The proof is complete.}

The fifth postulate is one of the ones we have just made use of: though a given point may be drawn a line paralell to a given line.

Notice that it says ‘a’ line, not lines - there is just one such parallel. On this simple point much would emerge.

Lobachevsky did not try to prove this fifth postulate as a theorem nor to disprove alternative versions of it. Instead he took the alternative forms and simply pursued the geometry that resulted. There were only two alternatives - there was no parallel; or there were more than one such parallel.

The geometry was strange - although he didn’t know it at first, he was looking at the geometry on curved surfaces. On such surfaces triangles did not have angle sums of 180º. In the first case they were less and in the second more. Lobachevsky categorised euclidean geometry - in which there was exactly one parallel line through a point, parallel to another line - as a special case of this more general geometry.

He published his work on non-euclidean geometry, the first account of the subject to appear in print, in 1829. It was published in the Kazan Messenger but rejected for publication when it was submitted to the St Petersburg Academy of Sciences.