## 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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**PROP**. I. 49. In any right - angled plane triangle , the hypothe- nuse is to either of the sides , as the radius to the sine of the angle opposite to that side , or to the cosine of the angle adjacent to that side . C G D Let ABC be a ... Page 13

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**Prop**. I , II . If the lengths of AC and CB , or AC and AB , or AB and CB , be known in feet , inches , or any other measure , the an gles may be found by the following proportions . AC : AB :: R :: sine C. Whence the sine of the angle ... Page 14

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**PROP**. IV . 54. The sides of any triangle are to one another as the sines of their opposite angles ; and , conversely , the sines of the angles of any triangle are to one another as the sides which are opposite to the angles . In the ... Page 15

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**Prop**. 3. From any angle A of the triangle ABC draw AD perpendicular to BC . About the centres B and C , with the radii BA and CA , suppose arcs to be described ; then AD will be the sine of the angles B and C. In the right - angled ... Page 16

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**PROP**. V. 59. In any triangle , if a perpendicular from the ver- tex to the base fall within the triangle , the base is to the sum of the other two sides , as their difference is to the difference of the segments of the base made by the ...### 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.