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

17] LOBATSCHEWSKY'S GEOMETRY

35

positive line a, and with every obtuse angle A, a negative line
a, such that AF (a)
Further parallels, in both cases, possess the following properties:
" If two lines are parallel, and two planes passing through
them intersect, their intersection is a line parallel to both.
"Two lines parallel to a third are parallel to each other.
" When three planes intersect each other in parallel lines, the
sum of the inner plane angles is equal to t7."
In ~ 9 the circle and sphere of infinite radius are introduced;
the Limiting-Curve and Limiting-Surface* of the Non-Euclidean
Geometry.
In ~~ 11 to 15 he deals with the measurement of triangles
and the solution of the problems of parallels.
At the end of ~ 13 are to be found the fundamental equations
(17) connecting the angles and sides of a plane triangle.
~ 16, and those which follow it, are devoted to the determination, in the Non-Euclidean Geometry, of the lengths of curves,
the areas of surfaces, and the volumes of solids.
After the most important cases have been examined, he adds
a number of pages dealing with definite integrals, which have
only an analytical interest.
From the conclusion I make the following extract, as it is
related to the question already touched upon in the sections
dealing with Bolyai's work-the logical consistency of the
new geometry:: After we obtained the equations (17), which express the
relations between the sides and angles of a triangle, we have
finally given general expressions for the elements of lines,
surfaces, and volumes. After this, all that remains in Geometry
becomes Analysis, where the calculations must necessarily
agree with one another, and where there is at no place the
chance of anything new being revealed which is not contained
in these first equations. From them all the relations of the
geometrical magnitudes to each other must be obtained. If
anyone then asserts that somewhere in the argument a contradiction compels us to give up the fundamental assumption,
which we have adopted in this new geometry, this contradiction
can only be hidden in equations (17) themselves. But we
* See note on p. 80.