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S. Hunter Christie Erof.
PLANE AND SPHERICAL
RULES, EXAMPLES, & PROBLEMS.
BY H. W. JEANS, F. R. A. S.
Royal Naval College, Portsmouth; formerly Mathematical
THE following pages contain the principal Rules in Plane and Spherical Trigonometry. The investigations of these rules requiring some knowledge of Mathematics, will be given in the second part of this treatise. The Author has been induced to make the present volume consist entirely of the rules and their applications; and to place the demonstrations in a separate part, in order that the student may be enabled to proceed through the practical part of plane and spherical trigonometry as soon as he is acquainted with vulgar and decimal fractions, and has acquired a sufficient knowledge of algebra to work an easy equation.
The collection of problems at the end of the book have been chiefly selected for Naval Students;
most of them can be solved by means of the rules contained in the book. The problems marked with the letter (a) have been added for the use of those who have already made some progress in mathematics; they will present little difficulty to the student who is acquainted with analytical trigonometry.
The use of the table of log haversines* now becoming generally known, reduces considerably the labour of working out some of the problems in navigation it may also be applied, with equal
This table under the name of logarithmic versed sines may be found in MENDOZA Rios, calculated to five places of decimals : the table in NORIE called log. rising, may be formed from it by the addition of the constant log 5.30103. DR. INMAN re-calculated the above table of MENDOZA Rios, and carried it to six places of decimals, and arranged it in a much more convenient form for use: he has inserted it in the last edition of his tables under the name of log. haversines. Lastly, in the recent work on Navigation by LIEUT. RAPER, the student will find a similar table called LOG. SINE SQUARE. The reason for adopting the two last mentioned names will appear, from considering the formula, by means of which this table may be constructed from the common table of log. sines A A -namely, sin.22
ver. 4, or log. sine square == log. half
versine A. RAPER derived his name from the first side of this equation: INMAN from the last, contracting half versine into haversine.
advantage, to the principal cases in Plane and Spherical Trigonometry. Rules have accordingly been now for the first time adapted to this table. Other rules are also given for the same cases suited to the Logarithmic Tables in more general use, such as HUTTON's, or those in MR. RIDDLE'S excellent work on Navigation.
The young student who may use this volume as an Introduction to Navigation will not find it necessary to read, at first, more than a certain portion of it. The Author has marked in the table of contents with the letter (n) the articles which may be sufficient for this purpose. All the examples under each of the rules thus selected should be worked out; with the exception of a few of the more difficult. Such student may also solve a few of the problems at the end of the book, as far perhaps, as the 20th.
In finding the answers to the examples and problems, the Author has generally taken them from the Tables by inspection; that is, if INMAN'S tables have been used, to the nearest 15": if RAPER'S or RIDDLE's, to the nearest minute or