Nikolai Ivanovich Lobachevsky

 

 

 

BIOGRAPHY:

Nikolai Ivanovich Lobachevsky was born Dec. 1, 1792 to a poor family in the town of Nizhny Novgorod, Russia. His father was a railroad clerk assigned to a surveying office. When he was seven, his father died and, out of this bad situation came some good fortune as his mother relocated the family to the city of Kazan. In the city was a good school, The Kazan Gymnasium, that Lobachevsky first attended in 1802. He graduated from the Gymnasium in 1807 and enrolled in Kazan University.

Again fortune smiled on Lobachevsky as the University had opened just two years earlier as part of reforms begun by Tzar Alexander I. Many important professors had been chosen to teach at the new University including Martin Bartels from Germany. Bartels was to be a major influence on Nikolai’s life indirectly, as he was friends with Gauss and corresponded regularly with him.

Bartels soon introduced Lobachevsky to mathematics. It is also known that Bartels taught a course on the history of mathematics and used a text by Montucla, another major influence on Lobachevsky.

Lobachevsky received a Master’s in Physics and Mathematics in 1811. He was appointed a lecturer at Kazan University and thus began a career there that involved several positions of increasing importance. Around 1822 things took a turn for the worse for a short time as Tzar Alexander I, in his later life, grew distrustful of modern science and philosophy. He thought these were products of the French Revolution and a threat to Orthodox Religion. (Maybe he also thought these were a threat to people who held titles such as King, Emporer and Tzar. After all Tzar is the Russian translation of Caesar.)

In 1826 Tzar Nicholas I became ruler and his regime was much more tolerant. Lobachevsky was recognized as someone who would help bring important changes to the school. During this period of time the University thrived. Lobachevsky made it a personal task to be sure all the science labs were properly equipped. He was given positions of increasing importance until he became rector in 1827, a position he held for 19 years. The University survived two potential disasters while he held this position: a major fire and a cholera epidemic.

In his personal life Lobachevsky was less successful. He married a very young woman when he was 40 years old. It would be kind to say this was not a marriage made in heaven. The union did produce 7 children about the only source of joy Lobachevsky derived from the marriage. Due to his large family and, some say, poor financial management, some claim Lobachevsky was almost destitute as he neared death. What is certain is that his heavy academic and administrative load, coupled with the death of his favorite son led to his premature death in 1856. Like many other now-famous people, he died without anyone recognizing his important contributions to the field of mathematics.

CONTRIBUTIONS:

Lobachevsky did not waste long periods of time trying to prove Euclid’s Fifth Postulate. LeGendre, for example, devoted 29 years to this effort. Instead Lobachevsky studied geometry where the Fifth Postulate did not necessarily hold. He considered Euclidean geometry to be a special case of this more general geometry.

Lobachevsky’s Postulate

Lobachevsky assumed that parallel lines meet each other at infinity.

Here’s how he developed this thinking. The diagram below illustrates Euclid’s Fifth Postulate.

 

Here u, L and M are segments. If a + b < Π, L and M must intersect each other on the same side of u if they are extended enough. In the special case where a = b = Π/2, Euclid never said whether the two lines meet or not.

The non-Euclidean Parallel Postulate is that parallel lines are two straight lines that meet each other at infinity. Lobachevsky said nothing at all about “straight” or “infinity”.

 

The next diagram shows key points in Lobachevsky’s idea.

 

Draw a straight line L and a point P not on L. draw radiating straight lines A, B and C from point P. Lines under line C such as A and B cut across line L. Lines above C such as E and D do not cut across L on the right side of line PM. Line C is special. It may or may not cut across L, we don’t know. Assuming the existence of straight line C, we say that C is parallel to straight line L and angle φ is that Parallel Angle.

 

Look at Figure 2 above. Place a point on the circumference of Circle O. Then push the center of the circle out to infinity. The circumference will stretch toward straight and the radii will have infinite length as shown in Figure 3. Lobachevsky named this the Limit Circle.

Because of the nature of infinite it is hard to picture what is going on at the center of the circle now out at infinity. This is because we can’t draw actual pictures that resemble what’s happening out there. Any pictures we draw are, by definition, finite. Even if we draw the lines from Earth to the Moon, they don’t approximate infinity.

Now look at the figure below.

 

This is a portion of the smaller circle above in Figure 2 with two other concentric circles drawn in. Arcs a, b and c are portions of the concentric circles and represent length in Euclidean geometry. Now we push the center out to infinity again.

 

This is a small piece of the larger circle shown if figure 3 above.

To keep things simple, let a = 1   k=1  and   e=base of natural logarithm.

b and c change according to a. k is constant and depends on the unit length.

Now rewrite b to variable y. And

1/y = f(x).

Finally:

y = exp(-x) --- (*).   Where x = distance ps and y = distance st.

This (*) is called Lobachevsky’s revolutionary starting point.

INFLUENCES:

As stated previously, Lobachevsky was greatly influenced by Bartels, Gauss and Montucla. Bartels was the first instructor who interested Lobachevsky in mathematics and, because Bartel’s friendship with Gauss, was the initial connection between the two. Gauss apparently had worked with non-Euclidean geometry but kept his work to himself. He may have influenced Lobachevsky as to what direction to take in his research. Montucla was the author of the book used during Bartel’s course. In the book Montucla discusses Euclid’s Elements and his theories of parallel lines in detail, so it seems likely Lobachevsky’s interest in the Fifth Postulate were stimulated by these lectures.