We now come to Lobachevskian Geometry by A. S. Smogorzhevsky in the Little Mathematics Library series.As the title of the book suggests the book is about one of the non-Euclidean geometries viz. the one by Lobachevsky. The back cover of the book says:
The author, the late Alexander Smogorzhevsky, D.Sc., was professor of mathematics at Kiev Polytechnical Institute and a specialist in Lobachevskian geometiy. He began his career as a school teacher in the Vinnitsa Region of the Ukraine, and later lectured at Kiev Polytechnical Institute for nearly forty years. He published over a hundred papers, both of original research and of a popularizing character, many of them devoted to non-Euclidean geometry: The Theory of Geometrical Constructions in Lobachevskian Space, On Some Plane Curves in Lobachevskian Geometry, and Lobachevsky’s Basic Ideas, to name a few.
And the Author’s Note before the book begins says:
The aim of this book is to acquaint the reader with the fundamentals of Lobachevsky’s non-Euclidean geometry.
The famous Russian mathematician N. I. Lobachevsky was an outstanding thinker, to whom is credited one of the greatest mathematical discoveries, the construction of an original geometric
system distinct from Euclid’s geometry. The reader will find
a brief biography of N. I. Lobachevsky in Sec. I.
Euclidean and Lobachevskian geometries have much in common,
differing only in their definitions, theorems and formulas as
regards the parallel-postulate. To clarify the reasons for these
differences we must consider how the basic geometric concepts
originated and developed, which is done in Sec. 2.
Apart from a knowledge of school plane geometry and trigonometry reading our pamphlet calls for a knowledge of the transformation known as inversion, the most important features of which are reviewed in Sec. 3. We hope that the reader will be able to grasp its principles with profit to himself and without great difficulty, since it, and Sec. 10, play very important, though ancillary, role in our exposition.
The book was translated from the Russian by V. Kisin and was first published by Mir in 1976 with reprint in 1982.
Contents are as under:
Author’s Note 7
1. A Brief Essay on the Life and Work of N. I. Lobachevsky 9
2. On the Origin of Axioms and Their Role in Geometry 11
3. Inversion 21
4. Map of a Lobachevskian Plane 29
5. A Circle in a Lobachevskian Plane 42
6. Equidistant Curve 46
7. Horocycle 47
8. Selected Theorems of Lobachevskian Geometry 49
9. Supplementary Remarks 52
10. On Natural Logarithms and Hyperbolic Functions 53
11. Measurement of Segments of Hyperbolic Straight Lines 57
12 Basic Formulas of Hyperbolic Trigonometry 60
13. The Lengths of Certain Plane Curves in Lobachevskian Geometry 64