# Letter from János to Farkas Bolyai (1823)

March 21, 2015

The first time I heard of F. Bolyai I was reading *The Poincaré conjecture*. And here's the story.

**TL;TR** Farkas to Janos Bolyai: «Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.»

Farkas Bolays was a Hungarian mathematician,Photo by Wikipedia mainly known for his work in geometry. He was taught at home by his father until he reached the age of six years when he was sent to the Calvinist school in Nagyszeben. His teachers immediately recognised his talents both in arithmetical calculation and in learning languages. When he was 12y.o. old he left school and was appointed as a tutor to the eight year old Simon Kemény who was the son of Baron Kemény. This meant that Bolyai was now treated as a member of one of the leading families in the country, and he became not only a tutor to Simon but a close friend. In 1790 Bolyai and his pupil both entered the Calvinist College in Kolozsvár where they spent five years.

The time spent in Kolozsvár was important for Bolyai's development. The Enlightenment had spread across Europe and by this time was influencing Transylvania. This meant that Bolyai was presented with its fundamental idea that reason was the route to understanding the universe and to improving the position of man. Knowledge, freedom, and happiness should be the aims of a rational human being. On the other hand nationalist feeling were on the increase in Hungary. The country had been freed from Ottoman rule in 1699 and after an attempt at gaining independence, Hungary had been controlled by the Habsburgs. There was an increasing resentment against the Habsburgs, particularly from the workers, and Bolyai too felt support for Hungarian culture, language, and nationality. There were also conflicting religious pressures as branches of the Christian Church argued against each other and against the ideas of the Enlightenment.

The professor of philosophy at the College in Kolozsvár was an impressive person, and he tried to turn Bolyai against mathematics and towards religious philosophy. Bolyai on the other hand had wide ranging interests, science, mathematics, and literature all interested him and in 1795 after leaving the College he spent a few weeks considering a career as an actor. However, he decided to go abroad with Simon Kemény on an educational trip funded by Baron Kemény and, after a delay caused by an unexpected illness, they set off in the spring of 1796.

First they reached Jena where Bolyai for the first time began to study mathematics systematically. He would go for long walks on his own and think about mathematics as he walked. After six months in Jena Bolyai and Kemény went to Göttingen. There he was taught by Kästner and became a life long friend of Gauss, a fellow student at Göttingen. This was the time when one could say that Bolyai really became a mathematician. He began to think about Euclid's geometrical axioms and in particular the independence of the Fifth Postulate. He discussed these issues with Gauss and his later writing show how important he considered their friendship to be for his mathematical development.

By the autumn of 1798 Bolyai and Kemény had completed their studies but back in Hungary Baron Kemény had hit hard times financially and although he supplied enough money for his son to return, Bolyai was left penniless in Göttingen. He spent a year there relying on charity and borrowed money for food to survive. It was a year of great hardship, yet one where he continued to develop mathematically surrounded by other talented mathematicians. After a year a friend sent him enough money from Hungary to pay the debts he had incurred and he set off on foot to return in July 1799.

Back on the family estate at Bolya he undertook research in mathematics. He went to Kolozsvár where he became a tutor. There he met Zsuzsanna Benkö and they married in 1801. In Zsuzsanna's parents home on 15 December 1802 their son János Bolyai was born. When Farkas Bolyai was offered a job at the Calvinist College in Marosvásárhely he was rather reluctant to accept but his father, wanting his son to have a secure job, pressed him to accept. Bolyai taught mathematics, physics and chemistry at Marosvásárhely for the rest of his life.

Life was not easy for Bolyai in Marosvásárhely. He was paid very little for his teaching at the College and had to take on extra work to bring in extra money. He wrote and published dramas, he ran the College pub, and he designed tiles and cast iron stoves which were produced commercially. Life was not easy at home too, for Bolyai's wife was a difficult person to live with and became increasingly difficult over the years as her health steadily deteriorated. Bolyai taught his son János mathematics, for this was the subject which he hoped that he would follow. Up until János was nine years old students from the College taught János other subjects, and only at this age did he attend school. Mathematics, the subject which Farkas Bolyai loved, was relegated to something to do for relaxation. Certainly he gained little satisfaction from his mathematics teaching at the College for the level of his students was low.

## The letter

János Bolyai wrote a two page letter to his father from Temesvár (now Timisoara, Romania) on the 3rd November 1823.

*My Dearest Father!
I have so many things to write to you about my new discoveries, that I
cannot solve it in another way, than doing nothing else than
writing you this letter; without waiting for your answer to my previous letter;
and perhaps I shouldn't write to you before getting it, but
I will put my letter addressed to the Baroness inPhoto by Mathematics archiveMy Father's envelope
asking you the favour of handing it her personally.
First of all I do answer the Binomial Theorem. For positive integers exponents the
proof, of course is perfect as you have written, but you need to know the form of the series, to be used in the proof,
about these things we will continue to discuss; we will try it
for negative exponents as well. As far as possible.
I have got already the conditions, and once I put them in order, and finish them, I will
publish a paper about the parallels; right at this moment is not finished, but the way
I followed, is almost sure promising to attain the goal, if it were ever possible;
it is not finished but I have found beautiful things, that surprised even me,
and it would be a pity to lose them; my Dearest Father will see and know;
I cannot say more, only that from nothing I have created a new different world;
Everything I sent you before is like a house of cards if compared to a tower.
I am convinced, that it will be in my honour, not less than if I would discover ...
Waiting for your answer, yours for ever indebted son Bolyai.
(P.S. in the lower part of the 1st page)
I am not afraid to justify my inventions in presenting them to My Father, I am convinced and have no fear of misunderstandings,
# (continued on the 1st page left side vertically)
what I wouldn't deserve, and it the sign that in some aspect I feel My Father as being myself.
*

There follows some discussion of the Binomial Theorem for non-integer exponents. He discusses his father's proof and mentions those of Lacroix (1765 - 1843) and Vega (1754 - 1802).