## 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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**supplement**of AB , and the angle DCB is the**supplement**of ACB . 15. DEF . 3. The chord of any arc is a right line drawn from one extremity of the arc to the other . Let the points B and E be in the circle , and join B , E ; then the ... Page 6

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**supplement**of AB ; draw the radius Cb , and produce it till it meet the circle in E , and the line TAt in t ; demit bf and EF perpendicular to the dia- meter AD ; then bf is the sine of the arc ABHb , or of the an- gle ACb , and EF is ... Page 8

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**supplements**to each other , namely , Ab greater , and AB less than a quadrant ; then will their sines bf and BF be equal . Because Ab + AB = 180 ° = Ab + bD , the arc AB = 6D , or the angle_bCD = BCA . But the radii BC , ¿ C are equal ... Page 9

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**supplement**of the proposed arc to a whole circle . Thus , the sine of the arc AHDLË , is EF , and is equal to BF , the sine of the arc AB = AE , the**supplement**of AHDLE to a whole circle . And so of the other lines . 39. The versed sine ... Page 15

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**supplement**of A ( 32. 1 ) , 180 ° ( < C + < B ) = ZA . Whence , sine C : AB :: s . A : BC . Again , if the angles A and C , and the side AB , be given , to find the rest , then Sine C AB :: s . A : BC , : and 180 ° - ( < A + < C ) = 2B ...### 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.