János Bolyai | Biography

János Bolyai is remembered as the Hungarian mathematician whose work on non-Euclidean geometry was eclipsed by the work of Nikolai Lobachevsky in Russia and Karl Gauss in Germany. The son of Farkas (Wolfgang) and Susanna Arkos Bolyai, Bolyai was born in Kolosvar, Transylvania, Hungary, on December 15, 1802. Bolyai was education in Marosvasarhely at the Evangelical-Reformed College, where his father was a professor of mathematics, physics, and chemistry.

From childhood, Bolyai showed an aptitude for both mathematics and music; he was an accomplished violinist at an early age. His father--a notable mathematician in his own right who devoted considerable effort to trying to prove the Euclidean theory of parallels--wanted Bolyai to study mathematics in Germany with Gauss; instead, the young man enrolled in the imperial engineering academy in Vienna in 1818 and pursued a military education. However, he still shared his father's passion for proving the Euclidean postulate that there can be only one parallel to a line through a point outside it. The elder Bolyai, in despair about his own lack of success, wrote to his son: "For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it, too, may take all your time, and deprive you of your health, peace of mind, and happiness in life."

While still a student in Vienna, Bolyai began to consider concepts that would alter the focus of studies. Inspired partly by his inability to prove Euclidean parallelismand partly by his exposure to other ideas, Bolyai began to explore the notion of a geometry constructed without the Euclidean axiom. He graduated from military college in 1822 and was commissioned as a sublieutenant. A flamboyant man, Bolyai developed a reputation as a competent swordsman and violinist. He readily accepted challenges to duel; in one account, he crossed swords with 13 consecutive opponents, stipulating only that he be permitted to pause between matches to play a violin selection. According to the story, he defeated all 13 swordsmen and was applauded for his musicianship. Bolyai still found time to pursue his mathematical inquiries, however. In 1823, he wrote to his father that he had made significant progress in his non-Euclidean geometry constructs, stating "from nothing I have created another entirely new world."

Work Eclipsed by Elder Mathematicians

Bolyai's military duties took him to Temesvar, where he was stationed from 1823 to 1826. He managed a visit to his father in 1825 and took with him a manuscript that detailed his theory of absolute space. Although the father rejected the son's concept, he forwarded the manuscript to Gauss in 1831. In 1832, Gauss wrote to the elder Bolyai: "Now something about the work of your son. . . . The whole content of the paper, the paths that your son has taken, and the results to which he has been led, agree almost everywhere with my own meditations, which have occupied me in part already for 30-35 years . . . now I have been saved the trouble [of writing a paper]."

The younger Bolyai was distressed and humiliated; in either 1831 or 1832, Bolyai's work was published as an addendum to a longer work of his father's, entitled Tentamen. Bolyai was dismayed anew on publication--neither father or son had been aware that Lobachevsky had published a paper outlining the concepts of non-Euclidean geometrythree years before their work was issued. Stung by the lack of recognition and his failure to establish his own priority, Bolyai virtually abandoned his scholarly efforts in mathematics. Plagued by poor health--he suffered from chronic fevers--he accepted an invalid's pension and left the military in 1833.

Bolyai returned to Marosvasarhely to live with his father, but the two strong-willed, emotional, and disappointed men were unable to peacefully share the same house. The younger Bolyai retired to a small family estate at Domald; in 1834 he contracted an "irregular marriage" (probably a live-in arrangement with no legal standing) with Rosalie von Orban, with whom he had three children.

In 1837 father and son tried again in vain to build a place for themselves in the world of mathematics. They entered the Jablonow Society prize contest, the subject of which was the geometric construction of imaginary quantities. Several mathematicians, Gauss included, were exploring the subject at the time. Unfortunately, the Bolyais' solutions to the problem were too complicated to win, and in fact, János' solution was similar to William Rowan Hamilton's, who had already published a solution that was easier to understand.

Notes and letters indicate that Bolyai continued to dabble in mathematics. He was interested in absolute geometry, the relationship between absolute trigonometry and spherical trigonometry, and the volume of tetrahedronsin absolute space. Bolyai also dabbled in philosophy, outlining what he termed salvation theory, in which he examined the concepts of individual and universal happiness and the relationship of virtue to knowledge.

Bolyai's father died in 1856, at about the same time Bolyai's arrangement with Rosalie von Oraban ended. Bolyai continued to live at the family estate as a semi-recluse, and his occasional writings from 1856 to 1860 include a memorial to his mother and a lively appreciation of the ballet company at the Vienna Opera House. He died after a lengthy illness and was buried in Marosvasarhely.