9. The angles of elevation of two distant objects being given, together with the oblique angle contained be- 11. Of quadrantal or rectilateral spherical triangles 12. GENERAL FORMULE for the solutions of the dif- ferent cases of oblique-angled spherical triangles 345 to 353 VIII. A Table of the right ascensions and declinations d xxvi BOOKS, For the Use of Schools, published by the same Author. I The COMPLETE PRACTICAL ARITHMETICIAN: Containing several new and useful Improvements. The Fourth Edition, corrected. II. A KEY TO THE COMPLETE PRACTICAL ARITHMETICIAN: Containing answers to all the questions in that Work; several useful Notes and Observations on Arithmetic, a Synopsis of Logarithmical Arithmetic, &c. Also, general Demonstrations of the Rules in the Complete Practical Arithmetician. The Second Edition. III. A SHORT AND EASY INTRODUCTION TO THE SCIENCE OF GEOGRAPHY: Containing an accurate Description of the Situation, Extent, Boundaries, Divisions, chief Cities, &c. of the several Empires, Kingdoms, States, and Countries, in the known World: With the Nature, Use, and Construction of Maps. The Sixth Edition, corrected and improved. IV. A NEW TREATISE ON THE USE OF THE GLOBES, or, A PHILOSOPHICAL VIEW OF THE EARTH AND HEAVENS: Comprehending an Account of The Figure, Magnitude, and Motion of the Earth; with the natural Changes of its Surface, caused by Floods, Earthquakes, &c. Together with the elementary Principles of Meteorology and Astronomy; The Theory of the Tides, &c The Third Edition, corrected and improved. V. HAWNEY'S COMPLETE MEASURER; or, the whole Art of Measuring. A New Edition, corrected and greatly improved. VI. THE ELEMENTS OF PLANE GEOMETRY: Comprehending the first SIX BOOKS of EUCLID, from the Text of DR. SIMSON, with Notes Critical and Explanatory. To which is added, Book VII. containing several important propositions, not to be found in EUCLID, and Book VIII. consisting of PRACTICAL GEOMETRY. The whole explained in an easy and familiar manner, for the Instruction of Youth. EXPLANATION OF THE CHARACTERS or MARKS USED IN THE FOLLOWING WORK. +, Plus or more, the sign of addition; as AD+DC, signifies that the line AD is to be increased by the line DC; and 4+3 signifies that the num ber 4 is to be increased by the number 3. -, Minus or less, the sign of subtraction, and shews that the second quantity is to be taken from the first; as CB-GB shews that the line CB is to be diminished by the line GB. x, Into or by, the sign of multiplication; as EDX DC signifies the rectangle formed by the lines ED and DC, and axb expresses the product of the quantity a by the quantity b. Also ab, or ab signifies the same thing. PB +, Divided by, as PB÷Cs, or signifies that PB is to be divided by cs. CS AB2, AB3, signify the square and cube of AB; also I 14 signifies that 14 is to be involved to the third power, and then the fourth root is to be extracted. √or A, Va or a3, express the square and cube root A A A of A. =, Equal to, as ABCD, shews that AB is equal to CD. Avinculum or parenthesis, serves to link two or more quan tities together, as A+B xm, or (A+B). M, signifies that A and B are first to be added together, together, and then to be multiplied by the quantity m. Proportion, A : B :: c`: D signifies that a has A to в the same ratio which c has to D, and is usually read A is to B as C is to D. Angle, as A, signifies the angle A. -Greater than, as AB, shews that A is greater than B. Less than, as AB, shews A to be less than B. The other characters are explained among the definitions' in the work. N. B. The letters within the parentheses, at the beginning of the different paragraphs of the work, are references. Thus, (C. 2.) refers to the article marked (£) at page 2; (R. 27.) refers to the article marked (R) at page 27, and so on. ERRATA. Preface, page vii, line 13, for have, read has. 99. line 21, for vers 2A-sine 2A, read vers 2A+sine A. 216. line 16, for (C sine B. sine c), read (sine B. sine C). M. AN INTRODUCTION TO PLANE AND SPHERICAL TRIGONOMETRY. BOOK I. CHAPTER I. THE NATURE AND PROPERTIES OF LOGARITHMS. (A) Definition. LOGARITHMS are a series of numbers contrived to facilitate arithmetical calculations; so that by them the work of multiplication is performed by addition, division by subtraction, involution by multiplication, and the extraction of roots by division. They may therefore be considered as indices to a series of numbers in geometrical progression, where the first term is an unit, Let 1 . go2 . y2 . g3 . g4.75. 76. &c. be such a series, increasing &c. decreasing from 1; which last series, agreeably to the established notation in algebra, may be thus expressed, 1 p-1, p-2.r-3.p-4. p−5.p−6, &c. Here the common ratio is r, and the indices 1.2.3, &c, or -1.-2.-3, &c. are logarithms. Hence it is obvious, that if a series of numbers be in geometrical progression, their logarithms will constitute a series in arithmetical progression. And, where the series is increasing, the terms of the geometrical progression are obtained by multiplication, and those of the arithmetical progression, or logarithms, by addition; on the contrary, if the series be decreasing, the terms |