## 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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**circles**be divided into the same number of equal parts , whatever number of those parts is contained in any arc AC of one cir- cle , subtending a given**angle ABC**, the same number of parts will be contained in the arc GH of the other ... Page 7

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**angle**less than a right an- gle , are respectively equal to the sine , tangent , and secant of the complement of that arc or ...**ABC**, BAC being equal ( 5. 1 ) , each of them is 60 degrees =**angle**ACB ; therefore the**triangle**ACB being ... Page 12

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**Trigonometry**to the Mensuration of the Sides and**Angles**of**Triangles**described on a Plane , or Plane**Trigonometry**properly so called . 48. In any right - angled**triangle ABC**( fig . page 21 ) , if the hypothenuse AC be made radius , the ... Page 13

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**angle**, and therefore is equal to the**angle ABC**. From the equian- gular**triangles ABC**, AEG , 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 AB , or AB and CB , be ... Page 14

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**triangle ABC**into two right- angled**triangles**CAD , BAD . About the centre A , with the ra- dius AD , suppose arcs to be de- scribed on each side of the point D , then DC , DB will be the tan- B gents of the opposite**angles**CAD , BAD ...### 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.