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FROM THE LIBRARY OF FRANK MORLEY, 1860-1937 Professor of Mathematics in this University, 1900-1929; Emeritus, 1929-1937 THE GIFT OF MRS. MORLEY

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P K E F A O E . T N preparing this work the aim has been to furnish just so much of Trigonometry as is actually taught in our best schools and colleges. Consequently, all investigations that are important only for the special atudent have been omitted, except the development of functions in series. The principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and inter- esting problems have been selected with a view to awaken a real love for the study. Much time and labor have been spent in devising the simplest proofs for the propositions, and in exhibiting the best methods of arranging the logarithmic work. The object of the work on Surveying is to present this subject in a clear and intelligible way, according to the best methods in actual use; and also to present it in so small a compass that students in general may find the time to acquire a competent knowledge of this very interesting and important study. The author is under particular obligation for assistance to G. A. Hill, A.M., of Cambridge, Mass., to Prof. James L. Patterson, of Schenectady, N.Y., to Dr. F. N. Cole, of A n n Arbor, Mich., and to Prof. S. F. Norris, of Baltimore, Md. G. A. WENTWORTH. Exeter, N.H., July, 1895

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C O N T E N T S . PLANE TRIGONOMETRY. CHAPTEE I. Eunctions of Acute Angles : Angular measure, page 1; trigonometric functions, 3; representation of functions by lines, 7 ; changes in the functions as the angle changes, 10; functions of complementary angles, 11; relations of the functions of an angle, 12; formulas forf indinga ll the other functions of an angle, when one function of the angle is given, 15; functions of 45°, 30°, 60°, 17. CHAPTEE II. The Eight Triangle: Given parts of a triangle, 19. Solutions without logarithms, 19 Case I., when an acute angle and the hypotenuse are given, 19 Case II., when an acute angle and the opposite leg are given, 20 Case III., when an acute angle and an adjacent leg are given, 20 Case IV., when the hypotenuse and a leg are given, 21 Case V., when the two legs are given, 21. General method of solving a right triangle, 22 ; solutions by logarithms, 24 ; area of the right triangle, 26; the isosceles triangle, 31; the regular polygon, 33. CHAPTEE III. Goniometrt: Definition of goniometry, 36 ; angles of any magnitude, 36; general definitions of the functions of angles, 37 ; algebraic signs of the func- tions, 39 ; functions of a variable angle, 40 ; functions of angles greater than 360°, 42; formulas for acute angles extended tp all angles, 43; reduction of the function of all angles to the functions of angles in the first quadrant, 46 ; functions of angles that differ by 90°, 48 ; functions of a negative angle, 49; functions of the sum of two angles, 51 ; func- tions of the difference of two angles, 53 ; functions of twice an angle, 55; functions of half an angle, 55 ; sums and differences of functions, 56. CHAPTEE IV. The Oblique Triangle : Law of sines, 60 ; law of cosines, 62 ; law of tangents, 64. Solu- tions: Case I., when one side and two angles are given, 64; Case II.,

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Tl TRIGONOMETRY. when two sides and the angle opposite to one of them are given, 66; Case III., when two sidea and the included angle are given, 71; Case IV., when the three sides are given, 74; area of a triangle, 78-79. CHAPTER V. Miscellaneous Examples : Plane Trigonometry, 82-99 ; goniometry, 99-105. Examination Papers, 106-116. CHAPTER VI. Construction of Tables : Logarithms, 117 ; exponential and logarithmic series, 120 ; trigo- nometric functions of amall angles, 125; Simpson's method of con- structing a trigonometric table, 127; De Moivre's theorem, 128; expansion of sinx, cosx, and tanx, in infinite aeries, 132. SPHERICAL TRIGONOMETRY. CHAPTER VII. The Right Spherical Triangle : Introduction, 135 ; formulas relating to right spherical triangles, 137 ; Napier's rulea, 141. Solutions: Case I., when the two legs are given, 142 ; Case II., when the hypotenuse and a leg are given, 142 ; Case III., when a leg and the opposite angle are given, 143 ; Case IV., when a leg and an adjacent angle are given, 143 ; Caae V., when the hypotenuse and an oblique angle are given, 144; Case VI., when the two oblique angles are given, 144. The isosceles spherical triangle, 149. CHAPTER VIII. The Oblique Spherical Triangle : Fundamental formulas, 150; formulas for half angles and sides, 152 ; Gauss's equations and Napier's analogies, 154. Solutions: Case I., when two sidea and the included angle are given, 156; Caae II., when two anglea and the included aide are given, 158; Case III., when two sides and an angle opposite to one of them are given, 160; Case IV., when two anglea and a side opposite to one of them are given, 162 ; Case V., when the three sides are given, 163 ; Case VI., when the three angles are given, 164. Area of a spherical triangle, 166. CHAPTER IX. Applications op Spherical Trigonometry: To reduce an angle measured in space to the horizon, 170 ; to find the distance between two places on the earth's surface, when the latitudes of the placea and the difference in their longitudea are known, 171; the celestial sphere, 171; spherical co-ordinates, 174; the astro- nomical triangle, 176 ; astronomical problems, 177-185.

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CONTENTS. Vl SURVEYING. CHAPTER I. Definitions. Instruments and Their Uses: Definitions, 135 ; instruments for measuring lines, 136 ; chaining, 136 ; obstacles to chaining, 138 ; the surveyor's compass, 141 ; uses of the compass, 143 ; verniers, 145 ; the surveyor's transit, 149 ; uses of the transit, 150 ; the theodolite, 150 ; the railroad compass, 150; plotting, 153. CHAPTEE II. Land Surveying: Determination of areas, 155 ; rectangular surveying, 159; field notes, computation, and plotting, 160; supplying omissions, 164; irregular boundaries, 164 ; obstructions, 164; modification of the rectangular method, 167 ; variation of the needle, 168 ; methods of establishing a true meridian, 170 ; dividing land, 173 ; United States public lands, 176; Burt's solar compass, 177 ; laying out the public lands, 179 ; Plane-table surveying, 181; the three-point problem, 186. CHAPTEE III. Triangulation : Introductory remarks, 187 ; the measurement of base lines, 188 ; the measurement of angles, 189. CHAPTEE TV. Levelling: Definitions, 190 ; the X level, 191 ; the levelling-rod, 191; differ- ence of level, 192 ; levelling for section, 195; subatitutea for the X level, 198 ; topographical levelling, 200. CHAPTEE V. Eailroad Surveying: General remarks, 202 ; cross-section work, 202 ; railroad curves, 203.

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