Syllabus of Propositions in Geometry: Intended for Use in Preparing Students for Harvard College and the Lawrence Scientific School |
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Page 18
... four right angles . THEOREM XXIII . If two triedral angles have the three face angles of one respectively equal to the three face angles of the other , the corresponding diedral angles are equal . BOOK VII . PRISMS AND PYRAMIDS ...
... four right angles . THEOREM XXIII . If two triedral angles have the three face angles of one respectively equal to the three face angles of the other , the corresponding diedral angles are equal . BOOK VII . PRISMS AND PYRAMIDS ...
Page 25
... extremities of a diameter , it is the pole of the great circle passing through them . THEOREM III . A sphere may be generated by the revolution of a semicircle about its diameter . THEOREM IV . Through four points not lying in a.
... extremities of a diameter , it is the pole of the great circle passing through them . THEOREM III . A sphere may be generated by the revolution of a semicircle about its diameter . THEOREM IV . Through four points not lying in a.
Page 26
... four points not lying in a plane one sphere , and only one , can be drawn . THEOREM V. A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact . Conversely , a plane perpendicular to a radius of a sphere ...
... four points not lying in a plane one sphere , and only one , can be drawn . THEOREM V. A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact . Conversely , a plane perpendicular to a radius of a sphere ...
Page 28
... four right angles . THEOREM XX . The area of a spherical triangle is equal to its spherical excess . THEOREM XXI . The shortest line that can be drawn on the surface of a sphere between two points is the arc of a great circle , not ...
... four right angles . THEOREM XX . The area of a spherical triangle is equal to its spherical excess . THEOREM XXI . The shortest line that can be drawn on the surface of a sphere between two points is the arc of a great circle , not ...
Page 29
... four great circles . Corollary . The surfaces of two spheres are to each other as the squares of the diameters , or as the squares of the radii . THEOREM XXV . The volume of a sphere is equal to the area of the surface multiplied by one ...
... four great circles . Corollary . The surfaces of two spheres are to each other as the squares of the diameters , or as the squares of the radii . THEOREM XXV . The volume of a sphere is equal to the area of the surface multiplied by one ...
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Syllabus of Propositions in Geometry: Intended for Use in Preparing Students ... Harvard University No preview available - 2015 |
Common terms and phrases
adjacent angles angle formed angles are equal apothem bases and equal bisect bisector BOOK centre circle is equal circular cone circular cylinder circumscribed Corollary Corollary II cubes diameter dicular diedral angles equal altitudes equal bases equal circles equal or sym equally distant given point given straight line hypotenuse included angle intercepted arc interior angles lateral area line a plane line is perpendicular lines are cut lines are parallel mean proportional number of sides opposite angles parallel lines parallelogram passed parallel perpen Plane Geometry plane passing plane perpendicular point of contact prism is equal propositions pyramid is divided pyramid is equal quadrilateral radii radius rectangle rectangular parallelopiped regular polygon respectively equal right angles right triangle secant line segment shortest line slant height Solid Geometry sphere are equal spherical triangle squares straight line drawn surface Syllabus THEOREM VII THEOREM VIII third straight line triangles are equal triangular prism triangular pyramid VIII
Popular passages
Page 26 - The angle of two arcs of great circles is equal to the angle of their planes, and is measured by the arc of a great circle described from its vertex as a pole and included between its sides (produced if necessary).
Page 12 - Prove that the square described on the hypotenuse of a right triangle is equivalent to the sum of the squares described on the other two sides.
Page 10 - If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Page 7 - In the same circle, or in equal circles, equal chords are equally distant from the centre ; and of two unequal chords, the less is at the greater distance from the centre.
Page 4 - If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles.
Page 2 - If two triangles have two sides of one respectively equal to two sides of the other, and the included angles unequal, the triangle which has the greater included angle has the greater third side. If two triangles have two sides of...
Page 19 - An oblique prism is equivalent to a right prism whose base is a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism. Hyp. OM is a right section of oblique prism AD', and OM ' a right prism whose altitude is equal to a lateral edge of AD'. To prove AD' =0= GM' . Proof. The lateral edges of GM
Page 5 - If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.
Page 9 - ... they have an angle of one equal to an angle of the other and the including sides are proportional; (c) their sides are respectively proportional.
Page 20 - The volume of a triangular prism is equal to the product of its base by its altitude. A~ Let V denote the volume, B the base, and H the altitude of the triangular prism CEA-E'.