BOLYAI, JANOS or JOHANN (1802-1860) and LOBACHEVSKY, N.I(1793-1856)

    Bolyai published his findings in 1832 in an appendix to a mathematical work of his father.  Later it was learned that Lobachevsky had published similar findings as early as 1829-1830, but, because of language barriers and the slowness with which information of new discoveries traveled in those days, Lobachevsky's work did not become known in western Europe for some years..

  There seems little point in discussing here the intricate, and probably unfounded theories explaining how various of these men might have obtained and appropriated information of the findings of some other.  There was considerable suspicion and incrimination of plagiarism at the time.
    Janos(or Johann) Bolyai was a Hungarian officer in the Austrian army, and the son of Farkas(or Wolfgang) Boyai a provincial mathematics teacher and long-time personal friend of Gauss.  The younger Bolyai undoubtedly received considerable stimulus for his study of the parallel postulate from his father, who had earlier shown an interest in the problem.  As early as 1823, janos Bolyai began to understand the real nature of the problem that faced him and a letter written during that year to his father shows the enthusiansm he held for his work.  In this letter hs discloses a resolution to publish a tract on the theory of parallels as soom as he can find the time and opportunity to put the material in order and exclaims, "Out of nothing I have created a strange new universe." The father urged that the proposed tract be published as an appendix to his own large two-volume semiphilosophical work on elementary mathematics.  the expansion and arrangement of ideas proceeded more sloyly than Janos had anticipated, but finally, in 1829, he submitted the finished manuscript to his father and three years later, in 1832, the tract appeared as a twenty-six page appendix to the first volume of his father's work.  Janos Bolyai never published anything further, although he did leave behind a great pile of manuscript pages.  His chief interest was in hwat he called "the absolute science of space," by which he meant the collection of those propositions that are independent of the parallel postulate and that consequently hold in both the Euclidean geometry and the new geometry.

    Nicolai Ivanocitich Lobachevsky spent the greater part of his life at the University of Kasan first as a student, later as a professor of mathematics, and finally as rector.  His earliest paper on non-Euclidean geometry was published in 1829 and 1830 in the Kasan Bullentin, two to three years before Boyai's work appeared in print.  This memoir attracted only slight attention in Russia, and, because it was written in Russian, practically no attention elsewhere.  Lobachevsky followed this initial effort with other presentations.  For example, in the hope of reaching a wider group of readers, he published, in 1840,a little book

written in German entitled Geometrische Untersuchungen zur Theorie der Parallellinien (Geometrical Researches on the Theory of Parallels), and then still later in 1855 a year before his death and after he had become blind, he published in French a final and more condensed treatment entitled Pangeometrie (Pangeometry).  So slowly did information of new discoveries spread in those days that Gauss probably did not hear of Lobachevsky's work until the appearance of the German publication in 1840, and Janos Bolyai was unaware of it until 1848.  Lobachevsky himself did not live to see his work accorded any wide recognition, but the non-Euclidean geometry that he developed is nowadays frequently referred to a Lobachevskian geometry.