## 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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... ABC be an angle at the centre of the circle ACF , standing on the arc AC ; the

... ABC be an angle at the centre of the circle ACF , standing on the arc AC ; the

**angle ABC**: four right an- gles :: arc AC : whole circumference ACF . D H K B G F Produce AB till it meet the cir- E cumference in E , and through B draw DBF ... Page 2

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**angle ABC**, the same number of parts will be contained in the arc GH of the other circle , subtending the same**angle ABC**. 5. COR . 4. Hence , if a circle of any radius whatever be di- vided into 360 equal parts , called degrees , each ... Page 3

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**angle ABC**: 4 right angles :: the arc AC circumfe- rence ADEFA ( 1. ) , therefore the**angle ABC**= ACX4 right angles . But four right angles are a constant circum . ADEFA quantity , and the circumferences of circles are as their diame ... 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 ...**ABC**, BAC being equal ( 5. 1 ) , each of them is 60 degrees =**angle**ACB ; therefore the**triangle**ACB being equiangular ... Page 12

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**triangle ABC**( fig . page 21 ) , if the hypothenuse AC be made radius , the perpendicular BC be- comes the sine , and the base AB the cosine of the angle at the base CAB . But if the base AB be made radius , the per- pendicular BC ...### 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.