Background
János Bolyai was born December 15, 1802, in Koloszvär (German, Klausenburg), Transylvania, Hungary (now Cluj, Rumania), to Farkas (Wolfgang) Bolyai and Susanna von Arkos Bolyai.
Cluj-Napoca, Romania
Plaque on Bolyai's birthplace at Kolozsvár (Cluj)
János Bolyai
Matzleinsdorf Evangelical Protestant cemetery, Vienna, Austria
János Bolyai's original grave.
Cluj-Napoca, Romania
Statue von János Bolyai in Klausenburg.
János Bolyai, Hungarian mathematician.
János Bolyai, Hungarian mathematician.
Theresian Military Academy, Wiener Neustadt, Lower Austria, Austria
János Bolyai studied at the Imperial and Royal Military Academy (TherMilAk) in Vienna from 1818 to 1822.
János Bolyai was born December 15, 1802, in Koloszvär (German, Klausenburg), Transylvania, Hungary (now Cluj, Rumania), to Farkas (Wolfgang) Bolyai and Susanna von Arkos Bolyai.
Janos Bolyai received his early education in Marosväsärhely, where his father was a professor of mathematics, physics, and chemistry at Evangelical-Reformed College. From 1815 to 1818, he studied at the college where his father taught. By the age of 13, he had mastered calculus and other forms of analytical mechanics, receiving instruction from his father.
He studied at the Imperial and Royal Military Academy (TherMilAk) in Vienna from 1818 to 1822.
In 1823 Janos had finished his courses at the academy and had entered upon a military career, beginning as a sublieutenant. His duties took him first to Temesvar (now Timisoara, Rumania), in 1823-1826, then to Arad (Rumania), in 1826-1830, and finally to Lemberg (now Lvov, W. Ukraine), where in 1832 he was promoted to the lieutenant second class. During his military service, he was often plagued with intermittent fever, but he built up a reputation as a dashing officer who dueled readily. In 1833 he was pensioned off as a semi-invalid, and he returned to his father’s home in Marosvasarhely.
While visiting his father in February 1825, Janos had shown him a manuscript that contained his theory of absolute space, that is, a space in which, in a plane through a point P and a line / not through P there exists a pencil of lines through P which does not intersect l. When this pencil reduces to one line, space satisfies the Euclidean axiom. Farkas Bolyai could not accept this geometry, mainly because it depended on an arbitrary constant, but he finally decided to send his son’s manuscript to Gauss, who could not accept this theory and explicitly stated it in his letters.
It is now known from Gauss’s diaries and from some of his letters that he was not exaggerating; but for Janos the letter was a terrible blow, since it robbed him of the priority. Even after he became convinced that Gauss spoke the truth, he felt that Gauss had done wrong in remaining silent about his discovery. Nevertheless, he allowed his father to publish his manuscript, which appeared as an appendix to the elder Bolyai’s Tentamen (1832), under the title “Appendix scientiam spatii absolute veram exhibens” (“Appendix Explaining the Absolutely True Science of Space”). This classic essay of twenty-four pages, which contains Janos’ system of non-Euclidean geometry, is the only work of his published in his lifetime. Gauss’s letters had such a discouraging influence on him that he withdrew into himself more and more, and for long periods he did hardly any mathematics. Disappointment grew when his essay evoked no response from other mathematicians.
In an attempt to re-establish themselves in mathematics, both father and son participated in the Jablonow Society prize contest in 1837. The subject was the rigorous geometric construction of imaginary quantities, at that time a subject to which many mathematicians (for example, Augustin Cauchy, W. R. Hamilton, and Gauss) were paying attention. The Bolyais’ solutions were too involved to gain a prize, but Janos’ solution resembled that of Hamilton, which was published about the same time, although in simpler terms, and which considered complex numbers as ordered pairs of real numbers. Again the Bolyais had failed to obtain due recognition.
Janos continued to do mathematical work, however, some of it strong and some, because of his isolation, very weak. His best work was that on his absolute geometry, on the relation between absolute trigonometry and spherical trigonometry, and on the volume of the tetrahedron in absolute space. On the last subject, there are notes written as late as 1856. Nikolai Lobachevski’s Geometrische Untersuchungen zur Theorieder Parallellinien (1840), which reached him through his father in 1848, worked as a powerful challenge, for it established independently the same type of geometry that he had discovered.
In his later days, he occasionally worried about the possibility of contradictions in his absolute geometry - a real difficulty that was not overcome until Beltrami did so later in the nineteenth century. Janos also worked on a salvation theory, which stressed that no individual happiness can exist without universal happiness and that no virtue is possible without knowledge.
After his death in 1860, the “Appendix” was practically forgotten until Richard Baltzer discussed the work of Bolyai and Lobachevski in the second edition of his Elemente der Mathematik (1867). Jules Houel, a correspondent of Baltzer’s, then translated Lobachevski’s book into French (1867) and did the same with Bolyai’s “Appendix” (1868). Full recognition came with the work of Eugenio Beltrami (1868) and Felix Klein (1871).
In his religious affiliation, János Bolyai was an Evangelist, and after his death, he was buried in the Evangelical-Reformed Cemetery in Marosvasdrhely.
From his father, Janos had inherited an interest in the theory of parallels; but in 1820 his father warned him against trying to prove the Euclidean axiom that there can be only one parallel to a line through a point outside of it. However, in the same year, Janos began to think in a direction that led him ultimately to non-Euclidean geometry. In 1823, after vain attempts to prove the Euclidean axiom, he found his way by assuming that geometry can be constructed without the parallel axiom; and he began to construct such a geometry. “From nothing I have created another entirely new world,” he jubilantly wrote his father in a letter of 3 November 1823.
The precocious lad was first taught by Bolyai's father and showed early proficiency not only in mathematics but also in other fields, such as music. He mastered the violin at an early age. He was also an accomplished polyglot speaking nine foreign languages, including Chinese and Tibetan.
After his retirement from the army, Janos lived with his father, who was then a widower. This arrangement lasted only a short time, however. Tension grew between father and son, who were both disappointed at the poor reception given their work, and Janos withdrew to the small family estate at Donald, visiting Marosvasarhely only occasionally. In 1834 he contracted an irregular marriage with Rosalie von Orban. The couple had three children, the first was born in 1837.
Janos’ father died in 1856 and his relationship with Rosalie ended at about the same time, thus depriving him of two of his few intimate contacts. However, in the four years left to him, he did have his good moments. He could write enthusiastically about the ballet performances of the Vienna Opera and compose some beautiful lines to the memory of his mother.