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

xii CONTENTS
CHAPTER VII.
THE ELLIPTIC PLANE TRIGONOMETRY.
PAGE
~~ 80-83. Gerard and Mansion's treatment of the Non-Euclidean
trigonometrical formulae - - - - 136
~ 84. The function ~(x) is continuous. - 143
~ 85. The functional equation q(x + y) + p(x - y)= 20(x)q(y) - 145
~ 86. The function ~(x) and the cosine 145
c a b
~87. The formula cos =cos, cos -146
6c- -
b
~ 88. The formula tan -=cos A tan 149
~ 89. The other trigonometrical formulae - - - 150
~ 90. Further results..- - 152
CHAPTER VIII.
THE CONSISTENCY OF THE NON-EUCLIDEAN GEOMETRIES AND THE IMPOSSIBILITY OF PROVING
THE PARALLEL POSTULATE.
~ 91. A method of proving that the Non-Euclidean Geometries
are consistent -- 153
~~ 92-93. Poincar6's representations of the Non-Euclidean Geometries 154
~~ 94-96. The system of circles passing through a fixed point,
and the Euclidean Geometry - 156
~~ 97-101. The system of circles orthogonal to a fixed circle, and
the hyperbolic geometry 160
~ 102. The impossibility of proving the Parallel Postulate - 170
~ 103. The system of circles cutting a fixed circle diametrally, and
the Elliptic Geometry 171
~ 104. Is the Euclidean Geometry true? 174
INDEX OF NAMES OF AUTHORS -176

SUBJECT INDEX -

177