Page:ScienceAndHypothesis1905.djvu/74

Plane	...	...	...	Sphere cutting orthogonally the fundamental plane.
Line	...	...	...	Circle cutting orthogonally the fundamental plane.
Sphere	...	...	...	Sphere.
Circle	...	...	...	Circle.
Angle	...	...	...	Angle.
Distance between
two points			...	Logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cutting it orthogonally.
Etc.				Etc.

Let us now 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 if produced would 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