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109. Case I. - Given a Side and the Hypotenuse ...
166
110. Case II. — Given an Acute Angle and the Hypotenuse. 167
111. Case III. — Given a Side and an Acute Angle..
168
112. Case IV.- Given the Two Sides....
169
113. When a Side and the Hypotenuse are nearly Equal
114. Four Cases of Oblique Triangles
172
115. Case I. — Given a Side and Two Angles
116. Case II.—Given Two Sides and the Angle opposite One of them, 173
117. Case III. — Given Two Sides and the Included Angle....... 176
118. Case IV. - Given the Three Sides
177
119. Area of a Triangle.
180
120. Heights and Distances – Definitions
181
121. Heights of an Accessible Object....
182
122. Height and Distance of an Inaccessible Object.
123. An Inaccessible Object above a Horizontal Plane
184
124. Object observed from Two Points in Same Vertical Line.... 185
125. Distance between Two Inaccessible Objects
186
126. The Dip of the Horizon
127. Problem of Pothenot or of Snellius
188
Examples ..
189
CHAPTER VIII.
CONSTRUCTION OF LOGARITHMIC AND TRIGONOMETRIC TABLES.
204
205
207
208
209
128. Logarithmic and Trigonometric Tables...
129. Exponential Series..
130. Logarithmic Series.
131. Computation of Logarithms
132. Sin 0 and tan 6 are in Ascending Order of Magnitude
sin e 133. The Limit of is Unity ...
134. Limiting Values of sin 0 and cos 0 .
135. To calculate the Sine and Cosine of 10" and of 1'..
136. To construct a Table of Natural Sines and Cosines.
137. Another Method...
138. The Sines and Cosines from 30° to 600.
139. Sines of Angles Greater than 45o.
140. Tables of Tangents and Secants.
141. Formulæ of Verification.
142. Tables of Logarithmic Trigonometric Functions.
143. The Principle of Proportional Parts...
211
213
214
215
216
217
218
144. To prove the Rule for the Table of Common Logarithms ... 218
145. To prove the Rule for the Table of Natural Sines...
219
146. To prove the Rule for a Table of Natural Cosines.
147. To prove the Rule for a Table of Natural Tangents..
220
148. To prove the Rule for a Table of Logarithmic Sines.
221
149. To prove the Rule for a Table of Logarithmic Cosines. 222
150. To prove the Rule for a Table of Logarithmic Tangents. 222
151. Cases of Inapplicability of Rule of Proportional Parts . 223
152. Three Methods to replace the Rule of Proportional Parts... 224
Examples ...
226
CHAPTER IX.
DE MOIVRE'S THEOREM.
APPLICATIONS.
153. De Moivre's Theorem..
229
р
154. To find all the Values of (cos 0 + V- 1 sin )'.
231
155. To develop cos no and sin no in Powers of sin 0 and cos 0.... 233
156. To develop sin 0 and cos 0 in Series of Powers of 0..
234
157. Convergence of the Series ..
235
158. Expansion of cogn 0 in Terms of Cosines of Multiples of 0... 235
159. Expansion of sinn o in Terms of Cosines of Multiples of 0... 236
160. Expansion of sinn e in Terms of Sines of Multiples of 237
161. Exponential Values of Sine and Cosine..
238
162. Gregory's Series ...
239
163. Euler's Series
240
164. Machin's Series.
241
165. Given sin 0 = x sin (0 + a); expand a Powers of x.
242
166. Given tan X = n tạn 8; expand x in Powers of n.
167. Resolve xn 1 into Factors..
243
168. Resolve an + 1 into Factors ..
244
169. Resolve 2n 2 sin cos 0 + 1 into Factors .
245
170. De Moivre's Property of the Circle ...
247
171. Cote's Properties of the Circle
248
172. Resolve sin 0 into Factors
173. Resolve cos e into Factors
250
174. Sum the Series sin a + sin(a+B) + etc.
251
175. Sum the Series cos O + cos(Q +B) + etc..
252
176. Sum the Series sinm ( + sinn (OL + B)+ etc..
177. Sum the Series sin de sin(Q + B) + etc.
254
178. Sum the Series cosec 0 + cosec 2 0 + cosec 40 + etc..
255
179. Sum the Series tan 8 + { tan +tan + etc..
2
4
180. Sum the Series sin c4 + x sin (Q+B) + etc.
181. Summation of Infinite Series
256
257
PART II.
SPHERICAL TRIGONOMETRY.
CHAPTER X.
FORMULÆ RELATIVE TO SPHERICAL TRIANGLES.
182. Spherical Trigonometry
183. Geometric Principles....
184. Fundamental Definitions and Properties
185. Formulæ for Right Spherical Triangles..
186. Napier's Rules.....
187. The Species of the Parts.
188. Ambiguous Solution
189. Quadrantal Triangles
190. Law of Sines .
191. Law of Cosines..
192. Relation between a Side and the Three Angles
193. To find the Value of cot a sin b, etc.
194. Useful Formula
195. Formulæ for the Half Angles
196. Formulæ for the Half Sides..
197. Napier's Analogies..
198. Delambre's (or Gauss's) Analogies
267
268
270
272
273
274
276
277
278
279
280
281
284
286
287
288
)
CHAPTER XI.
SOLUTION OF SPHERICAL TRIANGLES.
199. Preliminary Observations
200. Solution of Right Spherical Triangles .
297
298
201. Case I. — Given the Hypotenuse and an Angle..
202. Case II. — Given the Hypotenuse and a Side
203. Case III. — Given a Side and the Adjacent Angle....
204. Case IV. Given a Side and the Opposite Angle..
205. Case V. - Given the Two Sides
206. Case VI. - Given the Two Angles.
207. Quadrantal and Isosceles Triangles.
208. Solution of Oblique Spherical Triangles....
209. Case I. – Given Two Sides and the Included Angle
210. Case II. — Given Two Angles and the Included Side..
211. Case III. — Given Two Sides and One Opposite Angle.....
212. Case IV. – Given Two Angles and One Opposite Side.
213. Case V.- Given the Three Sides....
214. Case VI. — Given the Three Angles.
Examples ....
299
300
301
302
303
304
305
307
309
312
313
314
316
CHAPTER XII.
THE IN-CIRCLES AND Ex-CIRCLES. — AREAS.
215. The Inscribed Circle...
324
216. The Escribed Circles..
325
217. The Circumscribed Circle..
326
218. Circumcircles of Colunar Triangles....
328
219. Areas of Triangles. — Given the Three Angles..
220. Areas of Triangles. — Given the Three Sides....
330
221. Areas of Triangles.—Given Two Sides and the Included Angle, 331
332
329
CHAPTER XIII.
APPLICATIONS OF SPHERICAL TRIGONOMETRY.
222. Astronomical Definitions..
223. Spherical Coördinates...
224. Graphic Representation of the Spherical Coördinates..
225. Problems.....
226. The Chordal Triangle...
227. Legendre's Theorem..
228. Roy's Rule......
229. Reduction of an Angle to the Horizon
338
339
341
342
346
348
350
352
230. Small Variations in Parts of a Spherical Triangle
231. Inclination of Adjacent Faces of Polyedrons..
232. Volume of Parallelopiped...
233. Diagonal of a Parallelopiped...
234. Table of Formulæ in Spherical Trigonometry.
Examples ......
353
356
357
358
359
362