First Part of an Elementary Treatise on Spherical Trigonometry |
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Page 36
... signs of the several terms must be carefully attended to by means of ( 496 ) . The two values of A , given by ( 643 ) , correspond respectively to the two triangles which satisfy the problem . And the one , which belongs to each trian ...
... signs of the several terms must be carefully attended to by means of ( 496 ) . The two values of A , given by ( 643 ) , correspond respectively to the two triangles which satisfy the problem . And the one , which belongs to each trian ...
Page 39
... signs of the several terms must be attended to by means of ( 496 ) . ( 655 ) ( 656 ) ( 657 ) Either value of AP , given by ( 653 ) , may be used , and there will be two different triangles solving the problem , except when AP + PC ( fig ...
... signs of the several terms must be attended to by means of ( 496 ) . ( 655 ) ( 656 ) ( 657 ) Either value of AP , given by ( 653 ) , may be used , and there will be two different triangles solving the problem , except when AP + PC ( fig ...
Page 47
... sign in ( 687 ) is posi- ( 715 ) 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 ...
... sign in ( 687 ) is posi- ( 715 ) 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 ...
Page 52
... as ( 759 ) . 62. Scholium . In using ( 749 ) and ( 758 ) , the signs of the terms must be attended to by means of ( 496 ) . EXAMPLES . 1. Given in the spherical triangle ABC ( 52 SPHERICAL TRIGONOMETRY . [ CH . III . Ý II .
... as ( 759 ) . 62. Scholium . In using ( 749 ) and ( 758 ) , the signs of the terms must be attended to by means of ( 496 ) . EXAMPLES . 1. Given in the spherical triangle ABC ( 52 SPHERICAL TRIGONOMETRY . [ CH . III . Ý II .
Page 56
... signs of the second and fourth terms . Divide the two terms of the second ratio of ( 785 ) by sin . PBC and reduce , by ( 11 ) , ( 787 ) cos . C : cos . A : 1 sin . B cotan . PBC cos . B. Make the product of the means equal that of the ...
... signs of the second and fourth terms . Divide the two terms of the second ratio of ( 785 ) by sin . PBC and reduce , by ( 11 ) , ( 787 ) cos . C : cos . A : 1 sin . B cotan . PBC cos . B. Make the product of the means equal that of the ...
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.