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

15, 16, 17]

LOBATSCHEWSKY

33

advance; for he says regarding the Parallel Postulate, "a
rigorous proof of this truth has not hitherto been discovered;
those which have been given can only be called explanations,
and do not deserve to be considered as mathematical proofs
in the full sense." *
Between 1823 and 1826 Lobatschewsky had entered upon
the path which finally led him to his great discovery. It is
known that in 1826 he read a paper to the Physical-Mathematical Society of Kasan, entitled, Exposition succincte des
principes de la geometric, avec une d6monstration rigoureuse du
the'or'me des paralleles. The MSS. of this work does not survive,
and the last clause in the title is ominous, for it suggests that
he had not yet reached his goal. But in 1829-30 he published
a memoir in Russian, On the Principles of Geometry,t and in
a footnote to the first page he explains that the work is an
extract from the Exposition succincte.
This memoir and many other works of Lobatschewsky have
come down to us, for, unlike Bolyai, he was a prolific writer.
He published book after book, hoping to gain for the NonEuclidean Geometry the recognition it deserved-a recognition
which in his lifetime it wholly failed to receive. But his first
published work contains all that is essential to the treatment
of the subject; and fully establishes the truth and value of
his discovery. Thus, if the year 1826 cannot, with absolute
certainty, be taken as the date at which Lobatschewsky had
solved the problem, there is not the least doubt that his discovery of the Non-Euclidean Geometry was an accomplished
fact in the year 1829.
~ 17. This memoir consists of nearly seventy pages. The
earlier sections, ~~ 1 to 7, deal with the ordinary geometrical
notions of surface, line, point, distance, etc. In ~ 8 he introduces his theory of parallels.
This section reads as follows: *
* 1 am indebted to Dr. D. M. Y. Sommerville for a rendering of the
Appendix I. by Vasiliev to the Russian translation of Bonola's La
geometria non-euclidea. From this Appendix the sentence in the text
is taken.
+ When Lobatschewsky's works appeared in Russian. We give the
titles in English. This work is available in German in Engel's
translation. See Engel u. St;ckel's Uricunlden zur Geschichte der nichteuklidischen Geometrie, I. (Leipzig, 1898).
+ Cf. Engel, loc. cit. p. 10.
N.-E.G. c