170. (Bolyai Farkas – Bolyai János): Tentamen...

juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Cum appendix triplici. Auctore Professore Matheseos et Physices Chemiaeque Publ. Ordinary. Tomus primus. – Appendix...
Maros Vásárhelyini, 1832. Typis Collegii Reformatorum. (4)+XXXII+XLI-LII+XXXIII-XL+LXVII-LXXIV[recte LIV]+502+(2)+26+(2)+LIII-LVIII+5 folding pl(s). (4 plates with copper engravings and 1 large, folding with text); (4)+XVI+402p.+10pl(s). (plate 7 bounded twice)
First edition of Farkas Bolyai's Tentamen with his son’s, János Bolyai's system on non-Euclidean geometry in the first volume. “The most extraordinary two dozen pages in the history of thought" (Halsted) and one of the few absolute rarities in the history of science.
 
The authors
János Bolyai was the most significant figure of Hungarian mathematics. He was born to an old noble family in Transylvania. At the age of 12, he got into the Reformed College in Marosvásárhely (Târgu-Mureș), where his father, Farkas Bolyai taught. The modest teacher’s salary of the father did not allow his son to continue his studies in Göttingen, so in 1818, he went to the military academy in Vienna. He graduated with excellent results in 1823, obtained a qualification as a military engineer, and began to work at the directorate of fortifications in Temesvár. In 1832, he asked his superiors for a temporary exemption from military service to continue his scientific work, but his request was rejected. The following year, at his own request, he was retired. He first moved in with his father, but they did not get along, so he moved to the family’s estate in Domáld, where he started working on his mathematical research. In 1837, he submitted an 8-page paper entitled “Responsio” (Response) to the competition of the Jablonowski Society in Leipzig, in which he also explained ideas ahead of his time and tried to clarify the geometrical role of complex numbers. With this, Bolyai founded the algebraic theory of complex numbers simultaneously with Hamilton. His reviewers were perplexed and did not comprehend his work (an insignificant competitor won the contest). Apart from his main work, the Appendix, no other piece of his was published in his lifetime. The majority of his legacy – left to us in manuscripts of many thousands of pages – is today preserved in the Teleki-Bolyai Library in Marosvásárhely. He lived and created far from the academic world, so the significance of his oeuvre was only recognised after his death. Another reason for his omission was that Bolyai wrote in Latin and German, while at the time, the main task of the Magyar Tudós Társaság (Hungarian Society of Scientists) was the cultivation of the Hungarian language. In the 1860s and 1870s, Arthur Cayley and Felix Klein showed the fundamental connections between Euclidean, non-Euclidean and projective geometry, thus gaining full recognition of Bolyai and Lobachevsky’s theory.
His father, Bolyai Farkas was a professor of mathematics, physics and chemistry at the Reformed College in Marosvásárhely from 1804 until his death and a corresponding member of the Magyar Tudományos Akadémia (Hungarian Academy of Sciences) from 1832. In addition to his scientific work, he is credited with numerous works of fiction. Throughout his life, he corresponded with Carl Friedrich Gauss, with whom he became friends during his university years in Göttingen and who helped him in his work with his criticisms. He began to investigate Euclid’s axiom of parallelism — which his son refuted — and proved that the statement “three points not located on a straight line lie on a circle” is equivalent to it. In the history of mathematics, he was among the first to require the mutual independence of the axioms belonging to a system.
 
The work
In the 26-page appendix to the mathematical work of his father, the young Bolyai established a generalised system of geometry free of Euclidian premises of the parallel postulate. The study contains the foundations of independent, non-Euclidean geometry, which the literature calls Bolyai-Lobachevsky geometry since Bolyai and the Russian Nikolai Ivanovich Lobachevsky made the same discovery independently from each other. The significance of their work was only recognised at the beginning of the 20th century because it provided the mathematical foundations for the development of the general theory of relativity. While Lobachevsky (whose article was published in a Kazan university journal between 1829 and 1830) only proved the possibility of the existence of a geometry in which Euclid’s fifth postulate is false, the absolute geometric studies described by Bolyai are entirely independent of the previously mentioned Euclidean principle, and they can also be used on different types of curved space. With this theory, he reinterpreted parallelism and presented various notable shapes of the hyperbolic plane. He discussed the two geometries together and also drew parallels with spherical geometry. The younger Bolyai worked out his discovery between 1820 and 1824 — mainly during the years he spent at the military academy in Vienna — and from Temesvár, he wrote to his father about his thoughts: “My precondition is that as soon as I get it in order, I will publish a work on parallels. It is not clear yet at this moment, but the road I took almost certainly promises to reach the purpose, if it is possible at all. It is not complete yet, but I found so majestic things that I was astonished myself, and I feel it would be a pity to let it perish. You will also find so yourself, my Dear Father, when you will see it. Now I cannot say more, just that I have created a new world out of nothing: everything I had sent before was only a house of cards in respect to a tower.”
At the beginning of 1825, he presented the theory he had already developed to his father. However, a meeting in 1831 was of decisive importance concerning the publication of his treatise, and as he later admitted “if my father would not have encouraged and, one might say, would not have forced me to write, the Appendix had not seen the light of day either”. Since Farkas Bolyai was unsure about his son’s research results, he sent the manuscript of the “Appendix” to his friend, Gauss. Nevertheless, the mathematician’s answer caused great disappointment to both the father and the son. In his letter, he claimed that he had already obtained these results 30 years ago. He already knew it but did not publish it, and if he praised the discovery, it would be like deifying himself. Gauss’ attitude discouraged János so much that he gave up his mathematical career. His father later published his son, János Bolyai’s essay on absolute geometry – titled as “Appendix” – in his main work, the “Tentamen”.
 
The edition
Printed in 500 copies, the list of subscribers contains 70 names, total of 128 of them were puchased. There are several reasons for the very strange collation. On the one hand, the author kept adding more and more additions to the previously unsold pieces; on the other hand, the printer was unaware of the situation (the page numbering is wrong in many places). Determining completeness thus encounters extremely great difficulties. The study by Samuel Vincent Lemley, in which he attempted to clarify the publication history based on the known copies, is of great help in this regard. He distinguishes 11 parts of the first volume, of which he considers 5 integral parts of the basic work and the other 6 additions. He describes four versions of the second piece, ours is longer than the most complete one, by one blank leaf. Our item includes some of the additions and can be considered complete.
The title page shows two stamps with the inscription "Mathematicai Seminarium". According to the sources, around 1870, the remaining copies were sold.
Despite the relatively high number of copies, the work is among the greatest rarities.
Restored, contemporary hardpaper. Some leaves restored, some stained.
Bolyai bibliography: 353-354. pp.; Horblit: One Hundred Books Famous in Science. 69b.;
Dibner: Heralds of Science. 116.
 
Starting price: 20,000,000 HUF