## 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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**perp**. falls within the triangle , and BAC = BAD - DAC , when the**perp**. falls without . Hence C = 180 ° -B- BAC . Lastly , s . C : s . B :: AB : AC . Remark . When two sides and the included angle of a plane triangle are given , the ... Page 65

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**perp**. to the radius AE , and they will be the sines of the arcs AB , AC , * This subject is treated with perspicuity and elegance by Legendre , Elements de Geometrie , p . 338 . I AD . Draw BD cutting CE in I. Draw IK PLANE ... Page 66

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**perp**. to AE , and IM , BL**perp**. to DH . Because the arc BD is bisected in C , EC will be**perp**. to the chord BD , and therefore will bisect it in I ( 3.3 ) ; therefore BI will be the sine , and EI the cosine of BC or CD . Since BD ... Page 82

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**perp**. to the lines OFB and Fm , is**perp**. to the plane OBA ( 4. 2. Sup . ) . Draw FE and BD**perp**. to OA , and join CE ; then CE will be**perp**. to AO . For CFE , CFO being right angles , CE2 - EF2 = CF2 = CO2 - OF2 ( Spherical ... Page 83

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**perp**. to AO , the angle CEF is the inclination of the planes AOC , AOB ( Def . 4. 2. Sup . ) , therefore the angle CEF = CАВ ( 23 ) . By construction CE is the sine , and OE is the cosine of the arc AC , CF is the sine , and OF is the ...### Common terms and phrases

90 degrees adjacent angle AHDL algebra analogy angle ABC angle ACB Answer arc AC arc or angle base centre chord circle comp complement cosecant cosine cotangent Euclid's Elements find the angles find the rest formulæ 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 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.