Farkas Bolyai
Encyclopedia
Farkas Bolyai was a Hungarian mathematician
, mainly known for his work in geometry
, and of his son János Bolyai
.
), a village near Hermannstadt
, Grand Principality of Transylvania, Habsburg Empire (now Sibiu
, Romania
). His father was Gáspár Bolyai and his mother Krisztina Vajna. Farkas was taught at home by his father until the age of six years when he was sent to the Calvinist school in Nagyszeben. His teachers recognized his talents in arithmetics and in learning languages. He learned Latin, Greek, Romanian
, Hebrew and later also French, Italian and English. With 12 years he left school and was appointed as a tutor to the eight year old son of the count
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 but a real friend to the count's son. In 1790 Bolyai and his pupil both entered the Calvinist College in Klausenburg (today Cluj-Napoca
) where they spent five years.
The professor of philosophy at the College in Klausenburg tried to turn Bolyai against mathematics and towards religious philosophy. Bolyai, however, decided to go abroad with Simon Kemény on an educational trip in 1796 and began to study mathematics systematically at German
universities first in Jena
and then in Göttingen
. In these times Bolyai became a close friend of Carl Friedrich Gauss
.
He returned home to Klausenburg in 1799. It was there he met and married Zsuzsanna Benkő and where their son János Bolyai
– later an even more famous mathematician than his father – was born in 1802. Soon thereafter he accepted a teaching position for mathematics and sciences at the Calvinist College in Marosvásárhely (today Târgu-Mureş
), where he spent the rest of his life.
and the parallel axiom.
His main work, the Tentamen (Tentamen iuventutem studiosam in elementa matheosos introducendi), was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis. In this work, he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above. His study of the convergence of series includes a test equivalent to Raabe's test, which he discovered independently and at about the same time as Raabe
. Other important ideas in the work include a general definition of a function and a definition of an equality between two plane
figures if they can both be divided into a finite equal number of pairwise congruent
pieces. See also Bolyai-Gerwien theorem.
He first dissuaded his son from the study of non-Euclidean geometry
, but by 1830 he became enthusiastic enough to persuade his son to publish his way-breaking thoughts.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, mainly known for his work in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, and of his son János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
.
Biography
Bolyai was born in Bolya (BuiaSeica Mare
Şeica Mare is a commune located in Sibiu County, Romania. It is composed of six villages: Boarta, Buia, Mighindoala, Petiş, Şeica Mare and Ştenea...
), a village near Hermannstadt
Sibiu
Sibiu is a city in Transylvania, Romania with a population of 154,548. Located some 282 km north-west of Bucharest, the city straddles the Cibin River, a tributary of the river Olt...
, Grand Principality of Transylvania, Habsburg Empire (now Sibiu
Sibiu
Sibiu is a city in Transylvania, Romania with a population of 154,548. Located some 282 km north-west of Bucharest, the city straddles the Cibin River, a tributary of the river Olt...
, Romania
Romania
Romania is a country located at the crossroads of Central and Southeastern Europe, on the Lower Danube, within and outside the Carpathian arch, bordering on the Black Sea...
). His father was Gáspár Bolyai and his mother Krisztina Vajna. Farkas was taught at home by his father until the age of six years when he was sent to the Calvinist school in Nagyszeben. His teachers recognized his talents in arithmetics and in learning languages. He learned Latin, Greek, Romanian
Romanian language
Romanian Romanian Romanian (or Daco-Romanian; obsolete spellings Rumanian, Roumanian; self-designation: română, limba română ("the Romanian language") or românește (lit. "in Romanian") is a Romance language spoken by around 24 to 28 million people, primarily in Romania and Moldova...
, Hebrew and later also French, Italian and English. With 12 years he left school and was appointed as a tutor to the eight year old son of the count
Count
A count or countess is an aristocratic nobleman in European countries. The word count came into English from the French comte, itself from Latin comes—in its accusative comitem—meaning "companion", and later "companion of the emperor, delegate of the emperor". The adjective form of the word is...
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 but a real friend to the count's son. In 1790 Bolyai and his pupil both entered the Calvinist College in Klausenburg (today Cluj-Napoca
Cluj-Napoca
Cluj-Napoca , commonly known as Cluj, is the fourth most populous city in Romania and the seat of Cluj County in the northwestern part of the country. Geographically, it is roughly equidistant from Bucharest , Budapest and Belgrade...
) where they spent five years.
The professor of philosophy at the College in Klausenburg tried to turn Bolyai against mathematics and towards religious philosophy. Bolyai, however, decided to go abroad with Simon Kemény on an educational trip in 1796 and began to study mathematics systematically at German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
universities first in Jena
Jena
Jena is a university city in central Germany on the river Saale. It has a population of approx. 103,000 and is the second largest city in the federal state of Thuringia, after Erfurt.-History:Jena was first mentioned in an 1182 document...
and then in Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
. In these times Bolyai became a close friend of Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
.
He returned home to Klausenburg in 1799. It was there he met and married Zsuzsanna Benkő and where their son János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...
– later an even more famous mathematician than his father – was born in 1802. Soon thereafter he accepted a teaching position for mathematics and sciences at the Calvinist College in Marosvásárhely (today Târgu-Mureş
Târgu-Mures
Târgu Mureș is the seat of Mureș County in the north-central part of Romania. As of January 1, 2009 the city had a population of 145,151 inhabitants, making it the 16th most-populated city in Romania.-Names and etymology:...
), where he spent the rest of his life.
Mathematical work
Bolyai's main interests were the foundations of geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and the parallel axiom.
His main work, the Tentamen (Tentamen iuventutem studiosam in elementa matheosos introducendi), was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis. In this work, he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above. His study of the convergence of series includes a test equivalent to Raabe's test, which he discovered independently and at about the same time as Raabe
Joseph Ludwig Raabe
Joseph Ludwig Raabe was a Swiss mathematician.-Life:As his parents were quite poor, Raabe was forced to earn his living from a very early age by giving private lessons. He began to study mathematics in 1820 at the Polytechnicum in Vienna, Austria...
. Other important ideas in the work include a general definition of a function and a definition of an equality between two plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
figures if they can both be divided into a finite equal number of pairwise congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
pieces. See also Bolyai-Gerwien theorem.
He first dissuaded his son from the study of non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
, but by 1830 he became enthusiastic enough to persuade his son to publish his way-breaking thoughts.