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

13, 14]

THE APPENDIX

29

In a footnote he adds " pronounced BN asymptotic to AM."
Bolyai always used the word parallel and the symbol II
in the sense of equidistant, while he reserved the word
asymptotic and this symbol III for the new parallels, in the
sense in which we shall see Lobatschewsky used the term.
The properties of the new parallels are then established.
N
M D C A E
FIG. 12.
(ii) The properties of the circle and sphere of infinite radius
are obtained. It is shown that the geometry on the sphere of
infinite radius is identical with ordinary plane geometry.
(iii) Spherical Geometry is proved to be independent of the
Parallel Postulate.
(iv) The formulae of the Non-Euclidean Plane Trigonometry
are obtained with the help of the sphere of infinite radius.
(v) Various geometrical problems are solved for the NonEuclidean Geometry; e.g. the construction of a " square "
whose area shall be the same as that of a given circle.*
Bolyai laid particular stress upon the demonstration of the
theorems which can be established without any hypothesis as
to parallels. He speaks of such results as absolutely true, and
they form part of Absolute Geometry or the Absolute Science of
Space. As the title of the Appendix shows, one of his chief
objects was to build up this science.
In the Appendix he says little about the question of the
impossibility of proving the truth of the Euclidean Parallel
* Of course the Non-Euclidean "square " is not a quadrilateral with
equal sides and all its angles right angles. A rectangle is impossible
in the Non-Euclidean plane. The square of Bolyai is simply a regular
quadrilateral. The angles are equal, but their size depends on the
sides.