Spherical Trigonometry - pocketboek

SPHERICAL TRIGONOMETRY AFTER THE CESARO METHOD J. D. H. DONNAY. E. M., Ph. D. 1945 INTERSCIENCE PUBLISHERS. INC. - NEW YORK. N. Y. To THE MEMORY OF GIUSEPPE CESARO 1849 - 1939 CRT8TALLOGRAPHER AND MINERALOGIST PROFESSOR AT THE UNIVERSITY OF LIEGE PREFACE From a practical standpoint spherical trigonometry is useful to engineers and geologists, who have to deal with surveying, geodesy, and astronomy to physicists, chemists, mineralogists, and metallurgists, in their common study of crystallography to Navy and Aviation officers, in the solu tion of navigation problems. For some reason, however, spherical trigonometry is not recognized as a regular subject in many American college curricula. As a consequence, the teacher of a science for which a working knowledge of spher ical triangles is desirable usually finds he has to impart it to the students himself, or to use ready-made formulae that his listeners have never seen, let alone derived, before. This book has been written in an attempt to meet this situation. It can be covered in about ten to twelve lectures and could well serve as a text in a one-unit course for a quarter or a semester. It aims at giving the strict minimum, as briefly as possible. The straightforward and time-saving Cesaro method seems particularly suitable for this purpose, tying together as it does, from the outset, the concepts of spherical and plane trigonometry. This method has with stood the test of experience. For years Belgian students have thrived on it. Personally I have taught it for eight years to crystallography students at the Johns Hopkins University, and for two years to freshmen classes at Laval University, with gratifying results. The order in whichthe subject matter is arranged may appear unorthodox. It has proved satisfactory, however, from the teaching point of view. Through the use of the stereographic projection Ch. I, the concept of spherical excess somewhat of a stumbling block is mastered from vii viii PREFACE the start. A working knowledge of Cesaros key-triangles is acquired, as soon as they are established Ch. II, through the derivation of Napiers and Delambres formulae and several expressions of the spherical excess Ch. III. This much insures the understanding of the method. The treat ment of the oblique-angled triangle Ch. IV is thereby so simplified that one can dispense with that of the right-angled triangle. The latter properly follows as a particular case Ch. V. Examples of computations are given Ch. VI, both with logarithms and with the calculating machine. They are followed by a selection of problems, completely worked out Ch. VII, and a number of exercises with answers Ch. VIII. Cross-references are made in the theoretical chapters to appropriate applications. Proofs believed to be new are marked by asterisks. In conclusion may I be permitted to say that this booklet was planned jointly by Cesaro and myself, several years ago. He who was to have been the senior author passed away shortly afterwards. In introducing Cesaros elegant method to the English-speaking public, I would like to think that my writing will reflect his influence and, to some degree at least, his reverence for simplicity and rigor. J. D. H. D. Hercules Powder Company Experiment Station Wilmington, Delaware April 1945 CONTENTS PAGE Dedication v Preface . vn Introduction 3 1. Purpose of Spherical Trigonometry 3 2. The Spherical Triangle 4 3.Comparison between Plane and Spherical Trigo nometry 5 4. Fields of Usefulness of Spherical Trigonometry. 5 I. The Stereographic Projection 7 1. Definition 7 2. First Property of the Stereographic Projection. 7 3. Second Property of the Stereographic Projection. 10 II. Cesaros Key-Triangles 13 1. Cesaros Triangle of Elements 13 2. Cesaros Derived Triangle 16 3. Polar Triangles 18 4. Derivation of the Formulae of Spherical Trigo nometry 20 Exercises 21 III. How the Key-Triangles are Put to Work 22 1. Napiers Analogies 22 2. Delambres Formulae 22 3...