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 mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
. 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 town of Kolozsvár,
Grand Principality of Transylvania (now
Cluj-Napoca
Cluj-Napoca ( ; ), or simply Cluj ( , ), is a city in northwestern Romania. It is the second-most populous city in the country and the seat of Cluj County. Geographically, it is roughly equidistant from Bucharest (), Budapest () and Belgrade ( ...
in
Romania
Romania is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern and Southeast Europe. It borders Ukraine to the north and east, Hungary to the west, Serbia to the southwest, Bulgaria to the south, Moldova to ...
), the son of Zsuzsanna Benkő and the well-known mathematician
Farkas Bolyai.
By the age of 13, he had mastered
calculus
Calculus 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 generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
and other forms of
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
, receiving instruction from his father. He studied at the
Imperial and Royal Military Academy (TherMilAk) in
Vienna
Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...
from 1818 to 1822, and Bolyai received his commission as sub-lieutenant. At the age of 21, he was already a lieutenant, at the age of 22, a first lieutenant and at the age of 24, a captain.
Career
Bolyai became so obsessed with
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's
parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
If a line segment intersects two straight lines forming two interior ...
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 (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
, on reading the Appendix, wrote in a letter, "I regard this young
geometer Bolyai as a genius of the first order."
To his old friend Farkas 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."
Indeed, Gauss had disclosed his discovery of a consistent non-Euclidean geometry in a letter in 1827, and in 1829 wrote that he feared backlash if he published about it. János suspected that Gauss had been secretly informed about his discoveries by his father, causing a rift between him and his father. He later bitterly complained about Gauss's attitude.
In 1848 Bolyai learned that
Nikolai Ivanovich Lobachevsky had published a piece of work similar to his appendix in 1829. Though Lobachevsky published his work a few years earlier than Bolyai, it contained only
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
. 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
as ordered pairs of
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
. 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 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 that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
s beside his native Hungarian, these being
German,
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
,
French,
Italian and
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
Cluj-Napoca ( ; ), or simply Cluj ( , ), is a city in northwestern Romania. It is the second-most populous city in the country and the seat of Cluj County. Geographically, it is roughly equidistant from Bucharest (), Budapest () and Belgrade ( ...
, that was established in 1959, bears his name, as does the professional society of Hungarian mathematicians, the
János Bolyai Mathematical Institute, at the
University of Szeged
The University of Szeged () is a Public university, public research university in Szeged, Hungary. Established as the Jesuit Academy of Kolozsvár in present-day Cluj-Napoca in 1581, the institution was re-established as a university in 1872 by ...
. 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 or state in the afterlife in which souls are subjected to punishment after death. Religions with a linear divine history sometimes depict hells as eternal destinations, such as Christianity and I ...
.
The crater
Bolyai on the
Moon
The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and
1441 Bolyai, 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 ''minor ...
discovered in 1937, are also named after him.
Several primary and secondary schools in the
Carpathian Basin also bear Bolyai's name; for instance, Bolyai János Műszaki Szakközépiskola in
Budapest
Budapest is the Capital city, capital and List of cities and towns of Hungary, most populous city of Hungary. It is the List of cities in the European Union by population within city limits, tenth-largest city in the European Union by popul ...
, Bolyai János Gyakorló Általános Iskola és Gimnázium in
Szombathely, and the Bolyai János Általános Iskola in
Debrecen
Debrecen ( ; ; ; ) is Hungary's cities of Hungary, second-largest city, after Budapest, the regional centre of the Northern Great Plain Regions of Hungary, region and the seat of Hajdú-Bihar County. A city with county rights, it was the large ...
. Streets in Budapest, Cluj-Napoca, and
Timișoara
Timișoara (, , ; , also or ; ; ; see #Etymology, other names) is the capital city of Timiș County, Banat, and the main economic, social and cultural center in Western Romania. Located on the Bega (Tisza), Bega River, Timișoara is consider ...
are named after Bolyai.
Gallery
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 or 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''"
References
Sources
*
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...
(2001) ''Non-Euclidean Geometry'', Chapter 14 of ''The Colossal Book of Mathematics'',
W. W. Norton & Company
*
Jeremy Gray (2004) ''Janos Bolyai, Non-Euclidean Geometry, and the Nature of Space,'' MIT Press
*
Marvin Greenberg (2008) ''Euclidean and Non-Euclidean Geometries: Development and History'', 4th 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
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields:
* Springer continues to publish science (parti ...
, (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
Hyperbolic geometers
Scientists from Cluj-Napoca
Mathematicians from the Austrian Empire
Transylvanian Hungarians
Theresian Military Academy alumni
19th-century Hungarian mathematicians