## 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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**90**, 60 , 9 , & c . , by which means they could accurately express several parts of the periphery of the circle in whole numbers , which they considered far more clear , and**less**intricate**than**frac- tions . LEMMA II . 7. ALL angles are ... Page 7

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**under 90**degrees , or of an angle less than a right an- gle , are respectively equal to the sine , tangent , and secant of the complement of that arc or angle . Draw BI perpendicular to the diameter HL , then CF - BI . But BI , HK , CK ... Page 8

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**90**degrees , or a quadrant , or a right angle , is equal to radius . This is manifest from an inspection of the figure . 28. The tangent of 45 degrees is radius . For**then the**angle ACT being half a right angle , the other acute angle ... Page 39

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**less than 90**; therefore this exam- ple is not ambiguous ( 75 ) . Hence the angle B = 180 ( 39 53+ 107 40 ' ) = 32 ° 27 ′ . Sine C sine B :: AB : AC . Sine 107 40 ′ : sine 32 27 ′ :: 532 : AC . Log . sine 32 27 ' + log . 532 · Log ... Page 60

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**than**when it approaches near the end of it , at H ; and , on the contrary , that it increases faster in the first ...**90**degrees may be found , whose tangent shall exceed that line ; consequently the tangent has no limit to its increase ...### 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.