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

X

CONTENTS

CHAPTER III.
THE HYPERBOLIC PLANE GEOMETRY.
PAGE
~ 21. Lobatschewsky's treatment of parallels 40
~ 22. Hilbert's Axiom of Parallels - - 42
~~ 23-25. Some theorems on parallels - - - - 43
~ 26. Properties of the figure formed by two parallel rays through
two given points and the segment joining these points - 47
~ 27. The angle of parallelism - - 50
~ 28. Saccheri's Quadrilateral - - - - - - - 51
~ 29. The quadrilateral with two right angles - - - - 52
~ 30. The quadrilateral with three right angles - - - - 52
~ 31. The sum of the angles of a triangle - - - 53
~ 32. Not-intersecting lines have a common perpendicular - - 54
~ 33. Parallel lines are asymptotic - -- 56
~ 34. The shortest distance between two not-intersecting lines is
their common perpendicular, and on each side of this the
lines continually diverge -- 58
~ 35. The correspondence between a right-angled triangle and a
quadrilateral with three right angles - - - 59
~ 36. The series of associated right-angled triangles - - 63
~~ 37-38. Proper and Improper Points - - - 66
~ 39. The perpendiculars to the sides of a triangle at their middle
points are concurrent - - -- 68
~ 40. The Parallel Constructions 71
~~41-43. Given p, to find II(p) - -- 71
~ 44. Construction of a common parallel to two given straight
lines in one plane -- 74
~ 45. Given II(p), to find p -- 76
~~ 46-47. Corresponding points - - -77
~ 48. The Limiting-Curve or Horocycle 80
~ 49. The Equidistant-Curve - - -- 82
~ 50. The Measurement of Area. Equivalent polygons - - 84
~ 51. Equivalent triangles - 85
~~ 52-53. The areas of triangles and polygons - - - 88
CHAPTER IV.
THE HYPERBOLIC PLANE TRIGONOMETRY.
H~ 54-56. Some theorems on concentric limiting-curves - 91
~ 57. The equation of the limiting-curve - - - 97