
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a
Hungarian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, who developed
absolute geometry—a geometry that includes both
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
and
hyperbolic geometry. The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.
Early life

Bolyai was born in the Hungarian town of Kolozsvár,
Grand Principality of Transylvania (now
Cluj-Napoca in
Romania
Romania ( ; ro, România ) is a country located at the crossroads of Central, Eastern, and Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, Serbia to the southwest, Moldova to the east, a ...
), the son of Zsuzsanna Benkő and the well-known mathematician
Farkas Bolyai.
By the age of 13, he had mastered
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and other forms of
analytical mechanics, receiving instruction from his father. He studied at the
Imperial and Royal Military Academy (TherMilAk) in
Vienna
en, Viennese
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, postal_code_type = Postal code
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, timezone = CET
, utc_offset = +1
, timezone_DST ...
from 1818 to 1822.
Career
Bolyai became so obsessed with
Euclid's
parallel postulate that his father, who had pursued the same subject for many years, wrote to him in 1820: "You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone...Learn from my example."
János, however, persisted in his quest and eventually came to the conclusion that the postulate is independent of the other axioms of geometry and that different consistent geometries can be constructed on its negation. In 1823, he wrote to his father: "I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe."
Between 1820 and 1823 he had prepared a treatise on
parallel lines that he called
absolute geometry. Bolyai's work was published in 1832 as an appendix to a mathematics textbook by his father.
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
, on reading the Appendix, wrote to a friend saying "I regard this young
geometer Bolyai as a genius of the first order."
To Bolyai, however, Gauss wrote: "To praise it would amount to praising myself. For the entire content of the work...coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."
In 1848 Bolyai learned that
Nikolai Ivanovich Lobachevsky had published a similar piece of work in 1829. Though Lobachevsky published his work a few years earlier than Bolyai, it contained only
hyperbolic geometry. Working independently, Bolyai and Lobachevsky pioneered the investigation of
non-Euclidean geometry.
In addition to his work in geometry, Bolyai developed a rigorous geometric concept of
complex numbers as ordered pairs of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. Although he never published more than the 24 pages of the Appendix, he left more than 20,000 pages of mathematical manuscripts when he died. These can now be found in the
Teleki-Bolyai Library
The Teleki Library ( hu, Teleki Téka, ro, Biblioteca Teleki-Bolyai), also known as Teleki-Bolyai Library and Bibliotheca Telekiana, is a historic public library and current museum in Târgu-Mureş, Romania. One of the richest Transylvanian col ...
in
Târgu Mureș, where Bolyai died. His grave lies in the Lutheran Cemetery in Târgu Mureș.
Personal life
He was an accomplished linguist speaking several foreign
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...
s:
German,
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
,
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
,
Italian,
Romanian. He learned the violin and performed in Vienna.
No original portrait of Bolyai survives. An unauthentic picture appears in some encyclopedias and on a Hungarian postage stamp.
Legacy
The
Babeș-Bolyai University in
Cluj-Napoca, that was established in 1959, bears his name, as does the
János Bolyai Mathematical Institute
Bolyai Institute is the mathematics institute of the Faculty of Sciences of the University of Szeged, named after the Hungarian mathematicians, Farkas Bolyai, and his son János Bolyai, the co-discoverer of non-Euclidean geometry. Its director is L ...
at the
University of Szeged. The crater
Bolyai on the
Moon and
1441 Bolyai
1441 Bolyai, provisional designation , is a dark asteroid from the central region of the asteroid belt, approximately 13 kilometers in diameter. It was discovered on 26 November 1937, by Hungarian astronomer György Kulin at Konkoly Observator ...
, a
minor planet
According to the International Astronomical Union (IAU), a minor planet is an astronomical object in direct orbit around the Sun that is exclusively classified as neither a planet nor a comet. Before 2006, the IAU officially used the term '' ...
discovered in 1937, are also named after him.
Several primary and secondary schools in the
Carpathian Basin
The Pannonian Basin, or Carpathian Basin, is a large Sedimentary basin, basin situated in south-east Central Europe. The Geomorphology, geomorphological term Pannonian Plain is more widely used for roughly the same region though with a somewh ...
bear his name; for instance, Bolyai János Műszaki Szakközépiskola in
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population ...
, Bolyai János Gyakorló Általános Iskola és Gimnázium in
Szombathely, and the Bolyai János Általános Iskola in
Debrecen
Debrecen ( , is Hungary's second-largest city, after Budapest, the regional centre of the Northern Great Plain region and the seat of Hajdú-Bihar County. A city with county rights, it was the largest Hungarian city in the 18th century and i ...
. Streets in Budapest, Cluj-Napoca, and
Timișoara are named after Bolyai.
The professional society of Hungarian mathematicians bears his name. There is also a mathematical award given out every five years, named the
Bolyai Prize. Bolyai is a minor character in the 1969 science-fiction/fantasy story
"Operation Changeling", where his unique abilities allow the protagonists to navigate the non-Euclidean geometry of
Hell
In religion and folklore, hell is a location in the afterlife in which evil souls are subjected to punitive suffering, most often through torture, as eternal punishment after death. Religions with a linear divine history often depic ...
.
Works
* "Appendix scientiam spatii absolute veram exhibens; a veritate aut falsitate axiomatis XI Euclidei, a priori haud unquam", appendix to
Farkas Bolyai, ''Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi (An Attempt to Introduce Studious Youths to the Elements of Pure Mathematics)'', 1832.
** English translation
"The Science Absolute of Space: Independent of the Truth or Falsity of Euclid's Axiom XI (Which Can Never Be Decided A Priori)" The Neomon, Austin, 1896.
File:Bolyai-1.jpg, 1896 copy of Bolyai's "''The science absolute of space, independent of the truth of falsity of Euclid's axiom XI (which can never be decided a priori)''"
File:Bolyai-2.jpg, Appendix to "''The science absolute of space''"
File:Bolyai-3.jpg, First page to "''The science absolute of space''"
Gallery
References
Sources
*
Martin Gardner (2001) ''Non-Euclidean Geometry'', Chapter 4 of ''The Colossal Book of Mathematics'',
W. W. Norton & Company
*
Marvin Greenberg
Marvin Jay Greenberg (December 22, 1935 – December 12, 2017) was an American mathematician.
Education
Greenberg studied at Columbia University where he received his bachelor's degree in 1955 (he was a Ford Scholar as an undergraduate) and ...
(1994) ''Euclidean and Non-Euclidean Geometries: Development and History'', 3rd edition,
W. H. Freeman
* Elemér Kiss (1999) ''Mathematical Gems from the Bolyai Chests. János Bolyai's discoveries in number theory and algebra as recently deciphered from his manuscripts''. Translated by Anikó Csirmaz and Gábor Oláh.
Akadémiai Kiadó, Budapest; TypoTeX, Budapest, ;
* Tibor Weszely (2013) ''János Bolyai. Die ersten 200 Jahre'',
Birkhäuser, (translated from Hungarian by Manfred Stern),
* Ana Todea, Maria Fűllop, Monica Avram (2004) ''Oameni de știință mureșeni - Dicționar biobibliografic'', CJ Mureș Biblioteca Județeană Mureș, tipografia Mediaprint SRL
* Silva Oliva (2018) ''Janos Bolyai. Uno sguardo psicoanalitico su genio matematico e follia'', ed Mimesis.
External links
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{{DEFAULTSORT:Bolyai, Janos
1802 births
1860 deaths
Scientists from Cluj-Napoca
Austrian Empire people
Transylvanian Hungarians
Theresian Military Academy alumni
19th-century Hungarian mathematicians
Geometers