A’ Marosvasarhelyt 1829-be nyomtatott Arithemetika Elejének részint röviditett, reszint bovitett, általán jobbitott, 's tisztáltabb kiadása

Marosvasarhely: Kali Simon, 1843.

First edition, an exceptionally fine copy in the original boards, of Bolyai’s work on the foundations of mathematics, and the last of the great works of Farkas Bolyai (1775-1856). It is in part based on the author’s Az arithmethica eleje published in 1830 and on the second volume (the first volume being on geometry) of his magnum opus the Tentamen juventutem studiosam elementa matheseos purae, 1832-33, to which it frequently refers. Farkas Bolyai (1775-1856) was a close friend of Gauss and regarded by the latter as the only man who fully understood Gauss’ metaphysics of mathematics. “He can be taken as a precursor of Gottlob Frege, Pasch, and Georg Cantor; but, as with many pioneers, he did not enjoy the credit that accrued to those that followed him” (DSB). The text of the present work refers frequently to Gauss and Lagrange, along with other mathematicians such as Maclaurin and Taylor. Farkas Bolyai had worked on the parallel postulate and the possibilities of a non-Euclidean geometry from his earliest days as a mathematician in Göttingen, and had corresponded with Gauss on the subject, even sending him a manuscript entitled Theoria parallelarum, but it was his son János who was to achieve the breakthrough, publishing his discovery of non-Euclidean geometry in an Appendix to his father’s Tentamen. In the present work, Farkas writes of his son’s discoveries: “The Appendix is a work worth whole folios. It is to the true geometer a work beautiful, necessary, original, colossal. How many thinkers have even up to the latest time endeavored in vain to make sure the one foundation of the edifice of Euclidean geometry? In the mentioned work is built up an absolute and for all cases true geometry. Few have appreciated its worth, yet it is in one word a true classic” (quoted in Halsted, p. 37). At the time when this work was published, Bolyai was in despair as neither his son Janos nor he himself had received any recognition. He desperately wrote to Gauss (January 18th, 1848): ‘Some years ago I have published another attempt [the present work] in Hungarian but without any results - Mathematics does not grow in this “China”.’ Only one more letter to Gauss followed (February 6th 1853) before Bolyai’s death in 1856. No copies listed on ABPC/RBH. OCLC lists Clark and Michigan in US.

Farkas Bolyai was born near Marosvásárhely in Transylvania, now part of Romania. He was taught at home by his father until he reached the age of six when he was sent to the Calvinist school in Nagyszeben. His teachers immediately recognised his talents both in arithmetic and in languages. When he was 12 he left school and was appointed as a tutor to the eight year old Simon Kemény who was the son of Baron 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 to Simon but a close friend. In 1790 Bolyai and his pupil both entered the Calvinist College in Kolozsvár where they spent five years. There Bolyai was presented with the fundamental idea that reason was the route to understanding the universe and to improving the position of man. Knowledge, freedom, and happiness should be the aims of a rational human being. The professor of philosophy at the College in Kolozsvár tried to turn Bolyai against mathematics and towards religious philosophy, but Bolyai had wide ranging interests, including science, mathematics, and literature.

In 1796, after leaving the College, Bolyai set off on an educational trip with Simon Kemény. First they reached Jena where Bolyai for the first time began to study mathematics systematically. He would go for long walks on his own and think about mathematics as he walked. After six months in Jena, Bolyai and Kemény went to Göttingen. There he was taught by Kästner and became a life long friend of Gauss, a fellow student at Göttingen. He began to think about Euclid’s geometrical axioms and in particular the independence of the Fifth Postulate. He discussed these issues with Gauss and his later writing show how important he considered their friendship to be for his mathematical development.

Bolyai returned to Hungary in July 1799. He went to Kolozsvár, where he became a tutor, and there met Zsuzsanna Benkö, whom he married in 1801. Their son János was born on 15 December 1802. Farkas Bolyai was offered, and somwwhat reluctantly accepted, a job at the Calvinist College in Marosvásárhely, where he taught for the rest of his life. Bolyai taught his son mathematics. János left home in August 1818 to study at the Academy of Engineering at Vienna. Three years later, on 18 September 1821 Bolyai's wife died. Throughout all these difficult years Bolyai was working on the Tentamen.

In 1829 Bolyai finished his principal work, but because of technical and financial problems it was not published until 1832–1833. It appeared in two volumes, with the title Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, method intuitiva, evidentiaque huic propria, introducendi, cum appendice triplici (“An Attempt to Introduce Studious Youth Into the Elements of Pure Mathematics, by an Intuitive Method and Appropriate Evidence, With a Threefold Appendix”). While writing the Tentamen, Bolyai had his first difficulties with his son János. In spite of warnings from his father to avoid any preoccupation with Euclid’s axiom, János not only insisted on studying the theory of parallels, but also developed an entirely unorthodox system of geometry based on the rejection of the parallel axiom, something with which his father could not agree. However, despite misgivings, Bolyai added his son’s paper to the first volume and thus, unwittingly, gave it immortality. In 1834, a Hungarian version of Volume I was published.

“The Tentamen itself, the fundamental ideas of which may date back to Bolyai’s Göttingen days, is an attempt at a rigorous and systematic foundation of geometry (Volume I) and of arithmetic, algebra, and analysis (Volume II). The huge work shows the critical sprit of a man who recognized, as did few of his contemporaries, many weaknesses in the mathematics of his day, but was not able to reach a fully satisfactory solution of them. Nevertheless, when it is remembered that Bolyai worked in almost total isolation, the Tentamen is a most remarkable witness to the sharpness of his mind and to his perseverance” (DSB).

“There are other ideas in the Tentamen which show the quality of Bolyai as a mathematician. For example 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 …

Bolyai retired from his teaching in 1851 and after a number of strokes, died five years later” (MacTutor).

In the present work Bolyai takes further his studies on the foundations of mathematics presented in the Tentamen. He began it in 1829 but only a short introduction and an excerpt were printed (in 1830) because he was under pressure with preparing the Tentamen for publication.

Halsted, ‘Bolyai János,’ American Mathematical Monthly 5 (1898), pp. 35-38.

8vo (207x123 mm), pp. xliv, 386, with two folding lithographed plates on blue paper, one with six folding slips with grey and pink wash colouring; some very occasional and light foxing, a very fine copy, entirely unrestored in original blue boards with paper label to spine. Very rare.

Item #3558

Price: $22,500.00