The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

7, 8, 9]

GAUSS AND BOLYAI

19

substitute for it the assumption that the sum of the angles of a
triangle is two right angles-or that the locus of the points
equidistant from a straight line is another straight linedifferent geometries can be created. One of these is the
Euclidean Geometry, in which only one parallel can be drawn
to a straight line from a point outside it. Another is what
Dehn calls the Semi-Euclidean Geometry, in which an infinite
number of parallels can be drawn.*
~ 9. The Work of Gauss (1777-1855).
Though Bolyai and Lobatschewsky were the first to
publicly announce the discovery of the possibility of a NonEuclidean Geometry and to explain its content, the great
German mathematician Gauss had also independently, and
some years earlier, come to the same conclusion. His results
had not been published, when he received from Wolfgang
Bolyai, early in 1832, a copy of the famous Appendix, the
work of his son John.
This little book reached Gauss on February 14, 1832. On
the same day he wrote to Gerling, with whom he had been
frequently in correspondence on mathematical subjects: t
".. Further, let me add that I have received this day a
little book from Hungary on the Non-Eitlidean Geometry. In
it I find all my own ideas and RESULTS, developed with
remarkable elegance, although in a form so concise as to offer
considerable difficulty to anyone not familiar with the subject.
The author is a very young Austrian officer, the son of a friend
of my youth, with whom, in 1798, I have often discussed these
matters. However at that time my ideas were still only
slightly developed and far from the completeness which they
have now received, through the independent investigation of
this young man. I regard this young geometer v. Bolyai as a
genius of the highest order..."
The letter in which Gauss replied to Wolfgang Bolyai three
weeks later is better known, but deserves quotation from the
light it throws upon his own work: $
". o. If I commenced by saying that I am unable to praise
this work (by John), you would certainly be surprised for a
moment. But I cannot say otherwise. To praise it would be to
* Cf. Halsted, Science, N.S. vol. xiv. pp. 705-717 (1901).
tCf. Gauss, Werke, vol. viii. p. 220.
+ Gauss, Werke, vol. viii. p. 220.