## Geometrical Researches on the Theory of Parallels. Translated from the Original by George Bruce Halstead.

Austin: University of Texas, 1891.

First obtainable edition in English (see below) of Lobachevsky’s revolutionary work on non-Euclidean geometry, the rare offprint issue from the Bulletin of the University of Texas, Austin.

“The researches that culminated in the discovery of non-Euclidean geometry arose from unsuccessful attempts to ‘prove’ the axiom of parallels in Euclidean geometry. This postulate asserts that through any point there can be drawn one and only one straight line parallel to a given straight line. Although this statement was not regarded as self-evident and its derivation from the other axioms of geometry was repeatedly sought, no one openly challenged it as an accepted truth of the universe until Lobatchewsky published the first non-Euclidean geometry... In Lobatchewsky’s geometry an infinity of parallels can be drawn through a given point that never intersect a given straight line.

“Nicolai Ivanovich Lobatchewsky was born in Novgorod, Russia, and studied at the University of Kazan, where in 1827 he was appointed professor. His fundamental paper was read to his colleagues in Kazan in 1826 but he did not publish the results until 1829-30 when a series of five papers appeared in the Kazan University Courier... He amplified his findings (still in Russian) in 1836-8 under the title ‘New Elements of Geometry, with a Complete Theory of Parallels’. In 1840 he published a brief summary in Berlin under the title *Geometrische Untersuchungen zur Theorie der Parallellinien*” (PMM).

“*Geometrische Untersuchungen zur Theorie der Parallellinien*, which he published in Berlin in 1840, is the best exposition of his new geometry; following its publication, in 1842, Lobachevsky was, on the recommendation of Gauss, elected to the Göttingen Gesellschaft der Wissenschaften” (DSB).

The present work is Halsted’s translation of *Geometrische Untersuchungen*. George Bruce Halsted (1853-1922) was the first student of J. J. Sylvester at Johns Hopkins University and held chairs at several universities, including the University of Texas (1894-1903) where the mathematicians L. E. Dickson and R. L. Moore were among his students. “In a period when American mathematics had few distinguished names, the eccentric and sometimes spectacular Halsted established himself as an internationally known scholar, creative teacher, and promoter and popularizer of mathematics. He was a member of and active participant in the major mathematical societies of the United States, England, Italy, Spain, France, Germany and Russia. His activities penetrated deeply in three main fields: translations and commentaries on the works of Nikolai Lobachevski, János Bolyai, Girolamo Saccheri, and Henri Poincaré; studies in the foundations of geometry; and criticisms of the slipshod presentations of the mathematical textbooks of the day” (DSB).

The first English translation of Lobachevsky’s classic work appeared in February 1891 in *Scientiae Baccalaureus* (Vol. 1, No. 3), a short-lived and obscure journal published under the auspices of the Missouri School of Mines; it was then reprinted by the University of Texas, Austin, in May 1891 with a new preface by the translator Halsted. Nevertheless, most bibliographies cite the Austin printing as the first. Halsted himself writes in the preface to the May printing, “Of the immortal essay now first appearing in English...” and even Cajori in his 1922 obituary notice in the *American Mathematical Monthly* writes: “His most important work was the translation of writings on non–Euclidean geometry. Lobachevski’s Researches on the Theory of Parallels and Bolyai’s Science Absolute of Space were first published at Austin, Texas, in 1891, as parts of ‘Neomonic Series’.” We have been unable to find any copy of the *Scientiae Baccalaureus* printing in auction records, and there seem to be fewer than 10 institutional holdings of this journal. The present University of Texas printing is also very rare.

Horblit 69a; cf. PMM 293.

8vo (231 x 155 mm), pp 50, original printed wrappers, preserved in a half-leather box (embossed stamp of Bowdoin College on upper wrapper, three punch holes to inner margin (not affecting text); slight toning and crease to inner edge of wrapper; neat ink numbers at wrapper top edge, internally clean).

Item #2861

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Price:
$9,500.00
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