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

### From inside the book

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Page 12

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**tive**; that is , a must , by ( 496 ) , be greater than 90 ° . We may express this result as follows : An angle and its opposite leg in a spherical right ( 517 ) triangle must be both less or both greater than 90 ° , or by ( 522 ) both ... Page 46

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**tive**. Demonstration . Let the excess of the angle above 180 ° be M , which must be less than 90 ° ; and the ( 706 ) angle is 180 ° + M. Now , if we change - N into M in ( 189-194 ) , they become , by ( 196-201 ) , ( 707 ) sin . ( 180 ... Page 47

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**tive**; and , therefore , the numerator must be likewise positive . But if s were greater than 180 ° , sin s would by ( 705 ) be negative , since s must be less than 270 ° , as each side is less than 180 ° , and consequently the sum ... Page 48

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**tive**or ( sa ) is positive , that is , ( 727 ) ( 728 ) ( 729 ) or by ( 679 ) or s > a , or 2 s > 2 a , a + b + c > 2 a , b + c > a ; that is , each side of a spherical triangle is less than ( 730 ) the sum of the other two . 55. Theorem ... Page 54

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**tive**at the same time . But as ( A— C ) and ( 771 ) ( a - c ) are both less than 90 ° , these terms can be negative only when A is less than C and a less than c which agrees with ( 768 ) . - 64. Theorem . Of two sides of a spherical ...### 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 ABC+ the surface AC the perpendicular adjacent angles angles are given angles respectively equal AP and PC ar.co B'OC Corollary cosec cosine of half cotan Demonstration differs from 90 equal to 90 fall on AC given angle given leg given sides given value greater than 90 h tang half the sum Hence hypothenuse included angle Lemma less from 90 less than 90 Let ABC fig let fall logarithm lunary surface means of 496 middle negative obtuse opposite angle opposite side perpendicular BP perpendicular to OA planes BOC Problem quotient right angle right triangle fig right triangle PBC Scholium second member Secondly side BC sides and angles sides equal Solution of Spherical solve a spherical solve the triangle spherical right triangle spherical triangle ABC SPHERICAL TRIGONOMETRY substituted supplements surface ABC tangent of half Thirdly tive trian triangle ABC figs

### 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.