Vega, Georg Freyherrn von. 1754-1802. Vorlesungen über die Mathematik ... Erster band. Vienna: Wappler & Beck, 1802.
8vo. Contemporary calf-backed marbled boards, wear to joints and corners, rubbing to boards.
Provenance: Károly Szász (1798-1853, ink signature to title page, contemporary marginalia to 2 pages, likely in his autograph); János Bolyai (ink signature to endpaper, dated 1823; ink signature to title page); Farkas Sandorz (later ink signature).

A CRITICAL ARTIFACT OF THE DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY, ONE OF THE MOST PROFOUND DEVELOPMENTS IN SCIENTIFIC HISTORY: THE JANOS BOLYAI-KAROLY SZASZ COPY OF VEGA'S LECTURES, WITH THE AUTOGRAPH OF BOTH.

"I HAVE CREATED A STRANGE NEW WORLD OUT OF NOTHING!" Janos Bolyai proclaimed in his famous 1823 letter announcing the discovery of Non-Euclidean Geometry, an achievement "every bit as significant as the Copernican revolution in astronomy, the Darwinian revolution in biology, or the Newtonian or 20th-century revolution in physics" (Coxeter, p viii). It would be almost 10 years before Bolyai's discovery appeared in print, a short treatise in Latin appended to a larger mathematical work by his father, Farkhas Bolyai, titled Appendix, Scientiam Spatii absolute..., translated to English as "The Science of Absolute Space." Given such an obscure beginning, it is hardly surprising that it would be decades before Janos's treatise gained any real attention and found the recognition it rightfully deserved. In 1891, Janos's short treatise was finally translated into English, and translator and mathematician G.B. Halsted recognized Bolyai's appendix as "the most extraordinary two dozen pages in the whole history of thought!" (Halsted, p XVIII).
As a young student in Vienna, Janos befriended a slightly older student, Károly Szász, who shared his passion for mathematics. The two began to explore the parallel postulate, even as his father warned him in a letter, "I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone ...." It was Szasz, an unsung hero of scientific history, who first conceived the important and central non-Euclidean concept of the "asymptotic parallel". "From the conversations of the two friends were also derived the conception of the line equidistant from a straight line and the other most important idea of the Paracycle" (Bonola, p 97). Armed with such a new generalized conception of lines and curves, together they came to the enormous insight that Euclid's parallel postulate is true only when the paracycle is taken to be a straight line. When Szasz left Vienna in early 1821 to teach Law in Hungary, Bolyai carried their joint speculations further – both formally demonstrating the logical independence of the parallel postulate and further proving the validity and non-contradictory nature of the alternative non-Euclidean hyperbolic geometry.
Bolyai's breakthrough to non-Euclidean geometry in 1823 precedes Lobachevsky's related insight into hyperbolic geometry circa 1826. He clearly outlined the substance of his discovery in an 1825 letter to his father, and in 1826 he also sent a full statement of his new system of thought to one of his former professors, Wolther von Eckwehr. Unfortunately, both of these autograph statements are now lost. Farkhas had conceptual difficulties with Janos' generalization of geometry, and von Eckwehr never responded to the manuscript he received – with the result that Lobachevsky's work on non-Euclidean hyperbolic geometry saw print (in 1829-30) before Bolyai's work was actually published. However, despite the 1832 publication date of Bolyai's work, the manuscript was completed and authorized at the printer by 1829. Farkhas Bolyai himself refers to the Tentamen as "a latin work of 1829" (Kurzer Grundriss eines Versuchs, Târgu Mureş, 1851). Recognizing the ambiguity of the historical situation, historians of mathematics are inclined to acknowledge the respective merit of each man's achievement; and in a spirit of compromise, hyperbolic geometry has come to be called "Bolyai-Lobachevsky geometry."
Vega's Lectures were enormously influential on the younger Bolyai's mathematical development, and indeed they appear to have been the launchpoint for his non-Euclidean investigations. First printed in 1783, Vega's Lectures was the most popular mathematical textbook ever published in Europe (going through more than 90 editions before 1924). It was decidedly the favorite textbook of Bolyai's father Farkhas, and it was in fact the very textbook with which Farkhas trained his own son Janos. Writing to Gauss in 1816, Farkhas boasted that Janos "completely knows the first two volumes of Vega (which I use in my courses), and he is also well versed in the third and fourth volumes." In fact, Bolyai's autograph note on his own copy of the Tentamen, suggests that it was an 1807 edition of Vega's Lectures, specifically von Hoffmann's Critik Der Parallel-Theorie in the second volume, that that initially stimulated Janos' interest in the parallel postulate.
Autograph material relating to the discovery of non-Euclidean geometry is of utmost raity in the marketplace, with no auction sales of such material recorded in the last 40 years. The bulk of Bolyai's manuscripts are preserved at the Teleki library (in Marosvasarhely, Romania), with a few of his books, and another small collection of books at the Hungarian Academy of Sciences in Budapest. This association copy of Vega's "Lectures," a book which had a profound effect on the young mathematician, contains both the autograph of Bolyai, as well as that of his early partner in the exploration of the parallel postulate, Karoly Szasz. A REMARKABLE DISCOVERY FROM BOLYAI'S PERSONAL LIBRARY, AUTOGRAPHED AND DATED IN THE YEAR OF HIS DISCOVERY OF NON-EUCLIDEAN GEOMETRY.

References: Bonola, Roberto. Non-Euclidean Geometry, New York, 1958; Coxeter, H.S.M. "Introduction" to Richard Trudeau's The Non-Euclidean Revolution, Boston, 1987; Halsted, G.B. "Translator's Introduction," The Science of Absolute Space, Austin, 1896; Snygg, John. A New Approach to Differential Geometry, New York, 2012.