Boundless History: 1832 – Publication of the principal work of mathematician and Academy member Farkas Bolyai, Tentamen, as well as the work of his son, János Bolyai, entitled Appendix

One of the most significant events in the history of 19th-century Hungarian science was the publication of Farkas Bolyai’s (1775–1856) magnum opus, Tentamen, which included as an appendix the 26-page study Appendix by his son, János Bolyai (1802–1860). These two works played a decisive role in the history of mathematics, particularly in the fields of geometry and axiomatic thinking.

Farkas Bolyai was born in 1775 in Bólya, near Nagyszeben (today Sibiu, Romania), and spent most of his life in Transylvania. He studied first at the colleges of Nagyenyed and Kolozsvár, and subsequently continued his mathematical studies in Jena and Göttingen, where he established a relationship with Carl Friedrich Gauss. It was in Göttingen that he began to explore the question of Euclid’s parallel postulate, which ultimately led to results, via the work of his son János, that fundamentally transformed geometric thinking.

In 1832, Farkas Bolyai became a corresponding member of the Hungarian Learned Society in the Section of Natural Sciences. His principal work, Tentamen, written as a textbook but containing numerous original scientific results, was published in two volumes in 1832–1833. In this work, he sought to establish an axiomatic foundation for mathematics, paying particular attention to the systematic organisation of arithmetic and geometry.

János Bolyai was born in Kolozsvár (today Cluj, Romania) in 1802. Recognising his talent at an early age, his father urged him to continue his studies at the University of Göttingen; however, due to a lack of financial resources and professional support, this plan did not materialise. Instead, Bolyai enrolled at the military academy in Vienna and later pursued a career as a military engineer. Encouraged by his father, he devoted himself to mathematics from an early age, particularly the question of the parallel postulate.

Between 1820 and 1823, he developed the fundamental principles of non-Euclidean geometry. His approach radically transformed the science of geometry by demonstrating that Euclid’s fifth postulate ‒ the famous parallel postulate ‒ could not be derived from the other axioms, thus allowing for the existence of alternative geometric systems. He summarised his discoveries in this regard in Appendix, published in 1832. The non-Euclidean geometry he established later provided an important starting point for numerous theories in 20th-century physics, particularly the theory of relativity.

Farkas Bolyai’s Tentamen did not become widely known during his lifetime. Although it contained significant mathematical results ‒ such as the principles regarding the mutual independence of axioms within a single system and the principle of permanence ‒ it attracted little significant attention within academic circles. János Bolyai’s Appendix did not receive immediate recognition either. Upon the volume’s publication, his father sent a copy to Gauss, who spoke of it with appreciation but also restraint, claiming that he himself had arrived at similar ideas. Although Gauss never formally published these thoughts, the Russian mathematician Nikolai Ivanovich Lobachevsky arrived at similar discoveries independently of Bolyai, which is why the scholarly literature refers to their theories collectively as Bolyai–Lobachevsky geometry.

After János Bolyai’s death in 1860, his manuscripts remained unexamined for a considerable period. In response to international pressure, at the end of the 19th century, the Hungarian Academy of Sciences established a committee to review his manuscript legacy, which spanned some 14,000 pages. However, this effort proved fruitless, and the material was returned to Marosvásárhely in 1894. Recognition of his scientific achievements began to emerge only in the early 20th century, and today he is regarded as one of the most important figures in the history of mathematics. To honour his legacy, on the centenary of his birth the Hungarian Academy of Sciences established the Bolyai Prize, an international award presented every five years.