A New, Strange World

Janos Bolyai

Birth: 1802 – Romania

Death: 1860 – Romania

Most people think of Count Dracula whenever Transylvania is mentioned. However, I think of another name at first: Janos Bolyai.

Failed dreams of a young mathematician

Janos Bolyai (from now on I will mention him as Bolyai) is the son of Hungarian mathematician Farkas Bolyai. He showed great potential even when he was just 5-6 years old, and mastered calculus* when he was 13.

*Calculus

The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences.

In 1816, Farkas asked his old friend and mathematics teacher Gauss* to take his son as a pupil so that young Bolyai can have the best possible mathematics education. Although, Gauss rejected Farkas’ offer. This wasn’t going to be the only bad new Bolyai gets from Gauss.

*Gauss

A German mathematician who is also known as the prince of mathematics.

For young Bolyai, the best possible choice was to go to Vienna and study military engineering. He was an outstanding student there and finished 7 years of study in just 4 years and joined the army in 1823. He earned a living there until 1834.

A new geometry

Farkas Bolyai spent most of his career for finding a proof (or disproof) for Euclid’s parallel postulate* but failed in the end. It is not a surprise that young Bolyai took the matter in his own hands. He started working on the subject in the early 1820s.

Euclid’s parallel postulate

Also known as the fifth postulate that is given by the Greek mathematician Euclid in the first book of his masterpiece Elements.

Basically, it states that no two infinite parallel lines meet at a point.
L1 and L2 will meet at a point if they are not parallel to each other. Euclid’s fifth states that in essence.

Bolyai spent almost all of his spare time in his army duty for mathematics. In November 3rd, 1823 he wrote a letter to his father, mentioning his findings for the first time with these words:

“… out of nothing, I created a new and strange world.”

A year after this letter Bolyai completed his idea for non-Euclidean geometry. At first, Farkas was distant to his son’s findings, but at 1830 he realized how important his discovery was. This is why he convinced his son to write down his idea. In 1831, Bolyai wrote 24 pages-long appendix in his father’s book.

Farkas sent the appendix to his old friend Gauss to evaluate his son’s work. After reading the appendix, Gauss made two important comments to two separate people…

The idea

Bolyai’s idea at its core: Imagine a new geometry where Euclid’s fifth postulate isn’t true. In other words; a new geometry where parallel lines can meet.

In Euclidean geometry, the shortest path between two points is a straight line. However, according to Bolyai’s non-Euclidean geometry shortest path is a curve. You can read about this subject here.

One could explain Bolyai’s idea like this:

In Euclidean geometry, internal angles of any given triangle add up to 180 degrees. But if we draw a triangle on a sphere (e.g. on Earth), angles will exceed 180 degrees:

Bolyai’s geometry (today is known as hyperbolic or non-Euclidean geometry) was a brand new geometry.

The breakdown

Gauss told one of his friends “I regard this young geometer Bolyai as a genius of the first order.”. But, at the same time he wrote a letter to Farkas and showed a much different attitude:

“… to praise this work would be praising myself, as I’ve had the same ideas some 30-35 years ago.”

Today we know that Gauss, in fact, held similar ideas with Bolyai thanks to a letter of his dated back to 1824. But, this happened much after that he told. It is believed that Gauss wasn’t feeling comfortable about publishing his ideas publicly.

Gauss’ comments on his appendix hit Bolyai hard. His idea stayed unknown for the mathematics community and he was deeply affected after Gauss’ remarks. Soon after, his health had gone bad and he was forced to leave the army in 1834, and he started living an isolated life from his beloved mathematics.

Paranoid

Even after all this, Bolyai kept working on mathematics. In 1848, he received a work that is written by a Russian mathematician named Lobachevsky. Lobachevsky’s work was published in 1829 (before his appendix) and covered almost the same ideas as he had held for non-Euclidean geometry. Gauss knew about this work as well. In fact, he praised Lobachevsky for his work.

After investigating the work deeply, Bolyai believed that Lobachevsky was not a real person, and it was all Gauss behind this work. Unfortunately, Bolyai was slowly losing his mind.

Bolyai stopped his mathematics in his last years and died in poverty in 1860. It is known that he left around 20.000 pages of unpublished mathematical work behind. They can be found in the Bolyai-Teleki library located in Targu Mures.

Today, we honor him by calling non-Euclidean geometry as Bolyai-Lobachevsky geometry.

M. Serkan Kalaycıoğlu

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