In 1833 Hungarian mathematician János Bolyai published "Appendix scientiam spatii absolute veram exhibens: a veritate aut falsitate axiomatis xi Euclidei (a priori haud unquam decidenda) independentem. . . ." appended to a textbook by his mathematician father Farkas Bolyai, entitled Tentamen juventutem studiosam in elementa matheseos purae I pp.  -26  pp. (second series). The two volumes appeared in Maros Vasarhelyini, Hungary (now Romania) printed by Joseph and Simon Kali, at the press of the Reform College.
Although the idea of a non-Euclidean geometry had occurred independently to several nineteenth-century mathematicians, János Bolyai was one of the first to publish an organized, deductive and logically based system that was avowedly non-Euclidean. He was preceded only by Lobachevskii (Lobachevsky), whose "O nachalakh geometrii" (On the Foundations of Geometry) had been published in the obscure periodical, Kazanskii vestnik, izdavaemyi pri Imperatorskom Kazamskom Universitete in Kazan, Russia, in 1829-30, but Bolyai remained unaware of the Russian's work until 1848, when he came across the German translation Lobachevskii's Geometrische Untersuchungen (1840). Bolyai and Lobachevskii are generally given equal credit for the invention of non-Euclidean geometry.
János Bolyai began developing his new geometry in 1820, and completed it five years later. He undertook this task despite the warnings of his father, who discouraged his son in the strongest terms from trying to prove or refute Euclid's parallel axiom; in a letter written in 1820, Farkas told his son not to "tempt the parallels" and to "shy away from it as from lewd intercourse, it can deprive you of all your leisure, your health, your peace of mind and your entire happiness." The elder Bolyai found his son's new geometry of "absolute space" unacceptable, but finally, in the summer of 1831, decided to send János's manuscript to his old friend Carl Friedrich Gauss. Neither of the Bolyais knew that Gauss had been working for thirty years on developing his own non-Euclidean geometry, so János was dreadfully shocked to read in Gauss's reply that he [Gauss] could not praise János's system since to do so would be to praise himself! Despite this blow, János agreed to let his paper be published as an appendix to his father's obscure mathematics textbook printed in a small edition by an equally obscure Hungarian school publisher.
Unsurprisingly, Bolyai's paper failed to attract the attention of contemporary mathematicians, and his new geometry remained almost completely unknown until 1867, when German mathematician Heinrich Richard Baltzer publicized the achievements of Bolyai and Lobachevskii in his Elemente der Mathematik.
The Tentamen was very crudely or amateurishly printed at a school press; copies exhibit the earmarks of non-professional or inexperienced publishing, particularly in the clumsy typography and numerous errata and corrigenda leaves, which must have made the Tentamen extremely difficult to use. These leaves were printed on different paper stocks and were obviously added after the original printing. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) No. 259 included a collation and discussion of tentative issue points. The subscribers' lists in Vol. i (1r+v) and Vol. ii (266v) indicate that 156 copies were subscribed for, and the edition was probably not much larger than this.
In January 2016 antiquarian bookseller William P. Watson of London published preliminary results of his bibliographical researches on Bolyai's work in his Catalogue 21, Science, Medicine, Natural History, item No. 14, from which I quote:
"... Apart from the Appendix, hardly any two copies of the Tentamen agree in collation, and the great variation amongst them, including cancel leaves and gatherings, indicates that the publishing history of this work was confused, and remains confusing.
"Bolyai illustrates his textbook with 14 folding plates, five of which are inventively augmented with numerous small flaps. These plates contain as many as 10 slips, often concealed one behind the other; plate 10 also displays a single volvelle, which has gone unrecorded in most bibliographies to date; although not described in the printed or on-line catalogue entries, it is present in most copies. One point of bibliographic confusion has been clarified: the Horblit/Grolier Catalogue (based on the Smithsonian copy) lists an overslip on plate 6 that is not recorded in any other copy. Upon investigation it appears that an integral part of the plate (the lower portion of the diagram labelled T.144) was inadvertently detached during rebinding and subsequently reattached on a stub, leading to the conclusion that this was a required flap.
"Fewer than 25 copies are known: Stanford University: Haskell Norman collection (sold 29 October 1998 Christie's New York); Yale (Cushing copy, the first volume with Appendix only); Smithsonian Institution (Dibner copy, which was also the copy described in Horblit); Huntington (formerly the Burndy Library; the copy owned by Bolyai's translaor into English, George Bruce Halsted); Boston Public Library; University of Kentrucky (Louisville), and four in private collections. In Europe there are copies recorded at the Royal Society London; University College London; Austrian National Library; Hungarian National Library (Budapest); Leipzig, Göttingen (two, one Gauss's copy) Bordeaux (Jules Hoüel, translator of the Appendix 1867) and Trento (vol 1 only, and that seriously defective, lacking text and all the plates). There are two copies in private collections, one comprising vol. 1 only. There was one in Berlin (lost or destroyed in WWII). The copy sometimes described at Kanazawa Institute of Technology appears to be a ghost.
"There are numerous variations in collation etc. amongst these copies. We are compiling a detailed census and concordance which should be available shortly...."
Kline, Mathematical Thought from Ancient to Modern Times (1972) 873-880.