Johnson Publishing Company, 1928

Preface

There seems to be a real need for a short yet rigorous text in spherical trigonometry which shall contain little more theory than is needed for the solution of spherical triangles, with a few applications to add interest to the subject. This book is the result of an effort to meet that demand. As there are often but few class hours to be devoted to the subject, the aim of the writer has been to present matters in such a way as to enable the student to master the elements with little help from outside. The average student finds it hard to do this when he must pick out the essentials for himself.

A knowledge of spherical geometry is not presupposed, but a brief presentation of the principal concepts and theorems is given in Chapter I. Most of the theoretical examples bear directly on the development of the theory in the text. The data for the numerical solution of triangles are such as to require a minimum of labor in computation while offering a wider variety of applications of the theory than is usual in problem sets. Especial attention is called to the summarization of the tests for the number of solutions in the ambiguous cases in four short and complete tables, material usually covering several pages and often incomplete. The student is encouraged to check all solutions, and methods of checking are given in every case.

The time required for the discussion and solution of the general spherical triangle may be reduced by half by considering only Cases 1, 2, and 3 in Chapter III, explaining how the other cases may be solved by means of the polar triangle.

The terminology and symbols used follow the recommendations in the *Report of the National Committee on Mathematical Requirements* under the auspices of the Mathematical Association of America (1923).

**Contents**

I. Concerning Solid and Spherical Geometry

I. Planes and Lines in SpaceII. Certain Properties of the Sphere

- Lines perpendicular to a plane
- Parallel planes
- Perpendicular planes

- Definitions
- Circles on a sphere
- Length of arc of a circle
- Poles of a circle
- Spherical angles
- The spherical triangle
- Relations of sides and angles in spherical triangles
- Polar triangles
- Applications of polar triangle theory
- Lunes

II. The Right Spherical Triangle

- Classification according to the number of right angles
- Formulas for the solution of right spherical triangles
- About species
- Napier's Rules
- The six cases of right spherical triangles
- Figures solvable by means of right spherical triangles

III. The Oblique Spherical Triangle

I. Certain Relations between the Trigonometric Functions of the Sides and Angles of a General Spherical TriangleII. The Solution of the General Spherical Triangle

- Law of sines
- Law of cosines for sides
- Law of cosines for angles
- The trigonometric functions of the half angles in terms of the sides
- The trigonometric functions of the half sides in terms of the angles
- Napier's Analogies
- Delambre's Analogies (Gauss's Formulas)

- The six cases of the oblique spherical triangle
- Case I. Given the three sides

case I'. Given the three angles- Case 2. Given two sides and the included angle

Case 2'. Given two angles and the included side- Case 3. Given two sides and the angle opposite one of them

Case 3'. Given two angles and the side opposite one of them- Tables for determining the number of solutions in the ambiguous cases
- Suggestions for checking solutions

IV. The Area of a Spherical Triangle

- Introductory theorems concerning areas
- Spherical degrees
- The area of the spherical triangle in spherical degrees and in square units of radius
- L'Huillier's formula for the area in terms of the sides

V. Practical Applications of Spherical Trigonometry

I. Applications to GeographyII. Applications to Astronomy

- Latitude and longitude
- Great circle sailing

- The celestial sphere
- The horizon system
- First equatorial system
- The ecliptic
- A second equatorial system
- Comparison of the three systems of coordinates
- Projection of the three systems of coordinates
- Relation of altitude of pole to latitude of observer
- The astronomical triangle

Historical Sketch

Appendix

Index