## A Treatise of Plane and Spherical Trigonometry: In Theory and Practice ; Adapted to the Use of Students ; Extracted Mostly from Similar Works of Ludlam, Playfair, Vince, and Bonnycastle |

### From inside the book

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**AB**, and the angle HCB is the complement of the angle ACB . Therefore the cosine , cotangent , and cosecant of any ...**side**is radius . 25. Hence , if the radius of any circle be divided into 10,000,000 equal parts , and the length of ... Page 8

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**Ab**increases , it is evident that the sine bf , the tangent At , and the secant Ct must decrease till the point b ...**side**of the centre C , whence the cosines have their origin . 36. Again , At the tangent , and Ct the secant of**Ab**... Page 12

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**side**, or to the cosine of the angle adjacent to that**side**. C G D Let ABC be a right - angled triangle , of which the hypothenuse is AC . From the centre A , with any radius AD , describe the arc DE ; from D draw DF at right angles to**AB**... Page 13

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**AB**: BC : AE :: EG , or**AB**: BC : R tan . A. 51. Scholium to Prop . I , II . If the lengths of AC and CB , or AC and ...**side**, the segments of that**side**will be to each other as the tangents of the parts into which the vertical angle is ... Page 14

... sides which are opposite to the angles . In the triangle ABC , the

... sides which are opposite to the angles . In the triangle ABC , the

**side AB**: side AC :: sine ZACB , opposite to the former**side AB**: sine ZABC , opposite to the latter side AC ; and conversely . B E F A C D On the side BA ( produced if ...### Common terms and phrases

90 degrees adjacent angle AHDL algebra analogy angle ABC angle ACB Answer arc or angle base centre chord circle comp complement cosecant cosine cotangent Euclid's Elements find the angles find the rest geometry Given the side greater than 90 half the sum half their difference height Hence hypothenuse AC included angle less than 90 logarithmic sines mathematics measured mechanical philosophy negative opposite angle perp perpendicular plane triangle plane trigonometry PROP propositions quadrant AH quantity right-angled spherical triangle right-angled triangle Scholium secant side AB side AC sides and angles sine a sine sine and cosine sine² sines and tangents solution spherical angle spherical triangle ABC spherical trigonometry supplement tables tangent of half theorems third side three angles three sides triangle are given trigono versed sine yards

### Popular passages

Page 12 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.

Page ix - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.

Page 23 - Then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term, or answer.

Page 13 - In any triangle, twice the rectangle contained by any two sides is to the difference between the sum of the squares of those sides, and the square of the base, as the radius to the cosine of the angle included by the two sides. Let ABC be any triangle, 2AB.BC is to the difference between AB2+BC2 and AC2 as radius to cos.

Page 87 - The cosine of half the sum of two sides of a spherical triangle is to the cosine of half their difference as the cotangent of half the included angle is to the tangent of half the sum of the other two angles. The sine of half the sum of two sides of a spherical...

Page 74 - The sum of any two sides is greater than the third side, and their difference is less than the third side.