## A Treatise on Spherics: Comprising the Elements of Spherical Geometry, and of Plane and Spherical Trigonometry, Together with a Series of Trigonometrical Tables |

### From inside the book

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Page vii

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**meet**the sphere's surface in two points : lastly , that straight line , so terminated , is to be bisected . Now this process is plainly impracticable , without wholly destroying the form of the given solid . But neither is it necessary ... Page 27

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**meet**together , but do not belong to the same circle : If , therefore , two straight lines be drawn touching any two circular arches , on a sphere's surface , which include a spherical angle , the one straight line touching the one arch ... Page 28

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**meet**in P , and Px be a straight line touching PF in P , and Py a straight line that touches PH in P , the plane rectilineal angle x Py is equal to the spherical angle FPH . ( 42. ) COR . 1. If two arches of circles , in a sphere's ... Page 29

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**meet**the sphere , in any other point , than in that point of intersection : a plane , therefore , which**meets**a sphere in any point , so as to be at right angles to the sphere's radius , at that point , is said to touch the sphere in ... Page 30

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**meet**, are ( Art . 45. ) in the same plane , all the angles made by them , at their point of concourse , are ( E. Cor . 2. 15. 1. ) together equal to four right angles : wherefore ( Art . 41. ) all the spherical angles made by the ...### Other editions - View all

### Common terms and phrases

angle opposite base bisect circle EFH circumference co-tangent common section cosine describe a circle describe Art diameter drawn equal and parallel equal arches equal circles equal spheres equal to FE equations Euclid's Elements Find Art fore four right angles given angle given arch given circle given great circle given point given sphere given triangle greater hypotenuse Introd join Art less measure meet oblique angles opposite angle parallel circles perpendicular plane triangle polar distance polar triangle pole Problem PROP proposition quadrantal triangle radius rical triangle right angles right-angled spherical triangle SCHOLIUM semi-circumference shewn side BC sin S sin sine sphe sphere's center sphere's surface spherical angle spherical distance Spherical Geometry spherical polygon spherical tri Spherical Trigonometry straight line tangent Theorem three angles three sides touch the circle triangle ABC trigonometrical functions wherefore

### Popular passages

Page 53 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

Page 46 - BC shall be equal to the base EF ; and the triangle ABC to the triangle DEF ; and the other angles, to which the equal sides are opposite, shall be equal, each to each, viz.

Page iii - A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.

Page 43 - Theorem. If two spherical triangles on the same sphere, or on equal spheres, are equilateral with respect to each other, they are also equiangular with respect to each other.

Page 53 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 53 - BC common to the two triangles, which is adjacent to their equal angles ; therefore their other sides shall be equal, each to each, and the third angle of the one to the third angle of the other, (26.

Page iii - A diameter of a circle is a straight line drawn through the center and terminated both ways by the circumference, as AC in Fig.

Page 132 - If two triangles have two sides and the included angle in the one equal to two sides and the included angle in the other, each to each, the two triangles will be equal.

Page 38 - THEOREM. The sum of the sides of a spherical polygon is less than the circumference of a great circle.

Page 50 - If two angles of a triangle be equal to one another, the sides also which subtend the equal angles shall be equal to one another.