## Die Theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung zur vorgeschichte der nichteuklidischen Geometrie ...

Leipzig: Teubner, 1895.

First edition, Engel’s own annotated copy, of this important collection of texts in the pre-history of non-Euclidean geometry, covering the period 1482-1837, and including works by Euclid, Wallis, Saccheri, Lambert, Gauss, Schweikart and Taurinus, many of which are virtually unobtainable in their original editions. Also included are extracts from letters between Gauss, Wolfang Bolyai, Bessel and Schumacher. Engel’s notes were perhaps made in preparation for a second edition, which never appeared.

Engel & Stäckel include a facsimile of the historic letter from Gauss to Taurinus, written at Göttingen on 8 November, 1814, in which Gauss describes his own prior discovery of non-Euclidean geometry: “The assumption that the sum of the three angles [of a triangle] is less than 180° leads to a curious geometry, quite different from ours (the Euclidean), but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it with the exception of the determination of a constant, which cannot be designated *a priori*. The greater one takes this constant, the nearer one comes to Euclidean Geometry, and when it is chosen infinitely large the two coincide. The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible. For example, the three angles of a triangle become as small as one wishes, if only the sides are taken large enough; yet the area of the triangle can never exceed a definite limit, regardless of how great the sides are taken, nor indeed can it ever reach it. All my efforts to discover a contradiction, an inconsistency, in this Non-Euclidean Geometry have been without success, and the one thing in it which is opposed to our conceptions is that, if it were true, there must exist in space a linear magnitude, determined for itself (but unknown to us). But it seems to me that we know, despite the say-nothing word-wisdom of the metaphysicians, too little, or too nearly nothing at all, about the true nature of space, to consider as absolutely impossible that which appears to us unnatural.”

The editors Engel & Stäckel also published (separately) other historico-critical editions of works on non-Euclidean geometry. In particular, Engel published a translation of Lobachevsky’s *Elements of Geometry* (1829) and of his *New elements of geometry with a complete theory of parallel lines*. Stäckel published a critical edition of the correspondence between Gauss and Wolfgang Bolyai as well as works on the lives and works of Wolgang and János Bolyai.

Friedrich Engel (1861-1941) received his doctorate at Leipzig in 1883, and in the following year travelled to Oslo to assist Sophus Lie in writing up his researches on transformation groups, eventually published in three volumes as *Theorie der Transformationsgruppen* (1888-93). Engel published Lie’s collected works in six volumes and prepared a seventh (not published until 1960). He also edited the works of Hermann Grassmann, which did much to bring this author’s obscure works to the attention of the mathematical world. Engel received many honours for his work including the Lobachevsky Gold Medal and the Norwegian Order of St Olaf. He was elected to the Saxon Academy of Sciences, the Russian Academy of Sciences, the Norweigian Academy of Science and Letters, and the Prussian Academy of Sciences.

Sommerville, p. 126.

8vo (236 x 156 mm), pp. x, 325, with 145 figures in the text, and a facsimile of a letter by Gauss over 4 pages at the end. Interleaved with blanks throughout. Signed by Engel on front-free endpaper and with pencil notes in his hand in the margins and on several of the interleaved blanks. Contemporary half-calf, capitals with some light wear, block shaken a bit, ex-library and withdrawal stamps to the half-title and titel, book sellers ticket to front pastedwon.

Item #3522

**
Price:
$950.00
**