## Elements of Geometry: Plane and Solid |

### From inside the book

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Page 269

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**parallelopiped**, ABcd - c ' , will be formed , of the same altitude as the given figure , and hav- ing an equivalent base ( Const . , 328 ) . Now ABcd - c ' ABCD - C ' ( 519 ) ; for taking AD ' as the base of the latter , then ABcd - c ... Page 270

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**parallelopipeds**with equal bases are to each other as their altitudes . L F : F L K F N E E D D D A B Α ' A B Given : Two ...**parallelopiped**AK . Since AN and A'L ' are commensurable , ( Const 270 SOLID GEOMETRY . - BOOK VIII . Page 272

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**parallelopipeds**, P , P ' , with dimensions a , b , c , and a ' , b ' , c ' , respectively ; To Prove : P : Paxbxc : a ' x b'x c ' . Let be a third rectangular**parallelopiped**whose dimen- sions are 272 SOLID GEOMETRY . -BOOK VIII . Page 273

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**parallelopiped**whose faces are all squares . COR . The edges of a cube are all equal . 532. DEFINITION . The unit of volume is the cube whose edge is the linear unit , and whose base is , consequently , the unit of area . 533. COR . 1 ... Page 274

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**parallelopiped**is equivalent to a rectangular**parallelopiped**having the same altitude and an equivalent base ( 525 ) ; and the volume of the latter is measured by the product of its three dimensions , that is , of its base and ...### Contents

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### Common terms and phrases

ABē ABCD ACē altitude angle formed apothem arc AC bisector bisects chord circumf circumference circumscribed coincide cone of revolution Const construct cylinder DEFINITION diagonals diagram for Prop diameter dihedral angles divided draw edges equiangular equiangular polygon equidistant equilateral triangle equivalent EXERCISE find a point frustum given circle given line given point given straight line greater homologous hypotenuse inscribed intercept interior angles intersecting isosceles triangle line drawn line joining locus meet mid point number of sides numerical measures parallel parallelogram parallelopiped pass perimeter perpendicular plane MN polyhedral angle polyhedron prism produced PROPOSITION Prove pyramid quadrilateral radii radius ratio rect rectangle regular polygon regular polyhedrons right angle right triangle SCHOLIUM secant segment similar slant height sphere spherical polygon spherical triangle square straight angle tangent THEOREM triangle ABC trihedral vertex vertical angle volume

### Popular passages

Page 308 - A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The bounding arcs are the sides of the polygon ; the...

Page 298 - Sphere is a body bounded by a uniformly curved surface, all the points of which are equally distant from a point within called the center.

Page 283 - 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.

Page 113 - To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus : A : B : : C : D; and read, A is to B as C to D.

Page 373 - The object of these primers is to convey information in such a manner as to make it both intelligible and interesting to very young pupils, and so to discipline their minds as to incline them to more systematic after-studies. They are not only an aid to the pupil, but to the teacher, lightening the task of each by an agreeable, easy, and natural method of instruction.

Page 178 - ... the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle.

Page 123 - In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Page 118 - If the product of two numbers is equal to the product of two others, either two may be made the extremes of a proportion and the other two the means.

Page 179 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle, is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle.

Page 272 - Two prisms are to each other as the products of their bases by their altitudes ; prisms having equivalent bases are to each other as their altitudes; prisms having equal altitudes are to each other as their bases ; prisms having equivalent bases and equal altitudes are equivalent.