## First Part of an Elementary Treatise on Spherical Trigonometry |

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Page 8

**Demonstration**. To demonstrate the preceding rules , it is only necessary to compare all the equations which can be deduced from them , with those previously obtained ( 472 ) . Let there be the spherical right triangle ABC ( fig . 2. ) ... Page 10

8. Lemmas The sine and cosecant of an angle , ( 496 ) which is greater than 90 ° and less than 180 ° , are positive ; but its cosine , tangent , cotangent , and secant are negative .

8. Lemmas The sine and cosecant of an angle , ( 496 ) which is greater than 90 ° and less than 180 ° , are positive ; but its cosine , tangent , cotangent , and secant are negative .

**Demonstrations**Let the given angle be 90 ° + N ... Page 11

**Demonstration**. The factors cos . a and cos . 6 of the second member of the above equation are ...**Demonstrations**Let the quantity by which an angle differs from 90 ° be N ; and the angle is either 90 ° ( 508 ) + N or 90 ° — N. But ... Page 12

12. Corollary . The equation ( 470 ) ( 518 ) COS . A = = cos . a sin . B , leads also to the result that an angle dissers less from 90 ° than its opposite leg , the case of either side , equal to 90 ° , being excepted .

12. Corollary . The equation ( 470 ) ( 518 ) COS . A = = cos . a sin . B , leads also to the result that an angle dissers less from 90 ° than its opposite leg , the case of either side , equal to 90 ° , being excepted .

**Demonstration**. Page 13

**Demonstration**. Since the second member of ( 518 ) is the product of the two fractions cos . a and sin . B , ( 520 ) the first member must be less than either of them . Thus , neglecting the sines , cos .### What people are saying - Write a review

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### Other editions - View all

First Part of an Elementary Treatise on Spherical Trigonometry (Classic Reprint) Benjamin Peirce No preview available - 2017 |

First Part of an Elementary Treatise on Spherical Trigonometry Benjamin Peirce No preview available - 2016 |

### Common terms and phrases

A'BC acute adjacent angles angles are given becomes calculated called Corollary corresponding cosec cosine cotan deduced Demonstration denote determined differs divided equal to 90 equation EXAMPLES expressions factor fall on AC Fourthly fractions given angle gives greater than 90 half the sum hemisphere Hence hypothenuse impossible included angle known leads legs Lemma less than 90 Let ABC fig let fall logarithm lunary surface measured middle negative numerator obtained obtuse opposite angle opposite side perpendicular perpendicular BP positive preceding Problem proportion proved quantity quotient reduce result right angle Rules satisfy Scholium second member Secondly sides and angles sides equal signs sine Solution solve a spherical solve the triangle spherical right triangle spherical triangle ABC substituted supplements surface ABC tang tangent of half Theorem Thirdly tive trian triangle ABC figs whence

### Popular passages

Page 69 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.

Page 1 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Page 69 - THEOREM. The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles, multiplied by the tri-rectangular triangle.

Page 8 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.

Page 8 - II. The sine of the middle part is equal to the product of the cosines of the opposite parts.

Page 30 - Any angle is greater than the difference between 180° and the sum of the other two angles.

Page 51 - 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.

Page 51 - The cosine of half the sum of two angles of a spherical triangle is to the cosine of half their difference as the tangent of half the included side is to the tangent of half the sum of the other two sides.

Page 71 - ... and the sum of the angles in all the triangles is evidently the same as that of all the angles of the polygon ; hence, the surface of the polygon is equal to the sum of all its angles, diminished by twice as many right angles as it has sides less two, into the tri-rectangular triangle.