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

65]

INFINITESIMAL GEOMETRY

111

a
become sin A =-,
cot A =-,
a
cot A cot B = 1,
a2 + b2 = 2
cos A = sin B,
a
cos B=-,
C
when we write A, B for X and j.
From the Sine and Cosine Formulae for the Oblique-Angled
Triangle (~ 63) we get at once,
sin A: sin B: sin C = a: b: c,
a2 = b 2 + c - 2b cos A.
Again, a b and c can be made infinitesimals by letting a, b,
and c tend to zero instead of k to infinity. In this case again
the Euclidean relations are obtained.
This result can be stated in other terms:
In the immediate neighbourhood of a point on the Hyperbolic
Plane, the formulae of the Euclidean Geometry hold true.
Or, again:
The Euclidean Formulae hold true in Infinitesimal Geometry
on the Hyperbolic Plane.
These theorems have an important bearing upon the question
as to whether the Hyperbolic Geometry can actually represent
the external relations of the space in which we live. The
experimental fact that, within the limits of error to which all
actual observations are subject, the sum of the angles of a
triangle is two right angles does not prove that the geometry
of our space is the Euclidean Geometry. It might be a Hyperbolic Geometry in which the parameter k was very great.
The Geometry of Bolyai and Lobatschewsky can be made
to fit in with the facts of experience by taking k large enough.
The Postulate of Euclid reaches the same end by another
means. It is a better means, for it gives a simpler geometry.