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

91, 92, 93] POINCARE'S REPRESENTATIONS

155

Plane. - Sphere cutting orthogonally the
fundamental plane.
Line. - Circle cutting orthogonally the fundamental plane.
Sphere. - - Sphere.
Circle. - - Circle.
Angle. - - Angle.
Distance between The logarithm of the anharmonic
two points. ratio of these two points and of
the intersections of the fundamental
plane with the circle passing through
these points and cutting it orthogonally.
Etc. Etc.
"Let us take Lobatschewsky's theorems and translate them
by the aid of this dictionary, as we would translate a German
text with the aid of a German-French dictionary. We shall
then obtain the theorems of ordinary geometry. For instance,
Lobatschewsky's theorem: 'The sum of the angles of a
triangle is less than two right angles ' may be translated thus:
' If a curvilinear triangle has for its sides arcs of circles which
cut orthogonally the fundamental plane, the sum of the angles
of this curvilinear triangle will be less than two right angles.'
Thus, however far the consequences of Lobatschewsky's
hypotheses are carried, they will never lead to a contradiction;
in fact, if two of Lobatschewsky's theorems were contradictory, the translation of these two theorems made by the
aid of our dictionary would be contradictory also. But
these translations are theorems of ordinary geometry, and no
one doubts that ordinary geometry is exempt from contradiction." *
~ 93. To Poincare is also due another representation of the
Hyperbolic Geometry, which includes that given in the preceding section as a special case. We shall discuss this representation at some length, as also a corresponding one for the
Elliptic Geometry, since from these we can obtain in a simple
and elementary manner the proof of the impossibility of
* Poincar6, La Science et P'Hypothese. English translation by Greenstreet, p. 41 et seq.