SUMMARY OF CHAPTER VII 1. DEFINITIONS. (1) Plane Section of a Surface· the points common to a surface and an intersecting plane. § 468. (2) Closed Surface· -a surface such that every plane section of it consists of one or more closed lines. § 469. a surface made up of intersecting planes. § 470. (4) Convex Polyhedron· one which lies wholly on one side or the other of each of its faces. § 470. (5) Tetrahedron, Pentahedron, Hexahedron, Octahedron, Dodecahedron, Icosahedron. See § 470. (6) Prism a polyhedron of which two faces are convex polygons lying in parallel planes and identically equal, while the remaining faces are parallelograms. § 471. (7) Right Prism -one whose lateral edges are perpendicular to its bases. § 472. (8) Right Section of a Prism-a section made by a plane perpendicular to the lateral edges. § 473. (9) Lateral Area of a Prism—the sum of the areas of the lateral faces. § 474. (10) Altitude of a Prism· bases. § 475. (11) Truncated Prism a polyhedron having parallel lateral edges like a prism, but bases which are neither parallel nor equal. § 483. (12) Volume of a Polyhedron· the space enclosed by the polyhedron, or the measure of that space. §§ 485, 502. (13) Parallelepiped · a prism whose bases are parallelograms. § 489. (14) Right Parallelepiped — one in which a set of lateral edges is perpendicular to the bases. § 490. (15) Rectangular Parallelepiped · a right parallelepiped whose bases are rectangles. § 491. (16) Cube a parallelepiped whose faces are all squares. § 492. (17) Pyramid--a polyhedron one of whose faces is a polygon, and the remaining faces are triangles having a common vertex. § 509. (18) Altitude of a Pyramid- the perpendicular distance from the vertex to the base. § 509. (19) Regular Pyramid- one whose base is a regular polygon and whose vertex lies on the perpendicular to the base drawn from its centre. § 510. (20) Slant Height of a Regular Pyramid - the altitude of any of its lateral faces. § 510. (21) Truncated Pyramid — the figure determined by the base and any section of a pyramid and the portions of the lateral faces intercepted between them. § 511. (22) Frustum of a Pyramid· parallel. § 511. a truncated pyramid whose bases are (23) Similar Polyhedrons · two polyhedrons which have the same number of faces similar, each to each, and similarly placed, and have their corresponding polyhedral angles equal. § 530. (24) Regular Polyhedron - one whose faces are equal regular polygons, and whose polyhedral angles are all equal. § 537. 2. AXIOMS. (1) If two figures are identically equal, their volumes are equal (Axiom 13). § 486. (2) If equal volumes be added to equal volumes, or to the same volume, their sums are equal (Axiom 14). § 487. 3. THEOREMS ON THE EQUALITY OF PRISMS. (1) Two prisms, or truncated prisms, are identically equal if the three faces forming a trihedral angle in one are identically equal to the three faces forming a trihedral angle in the other, and are similarly placed. §§ 481, 484. (2) Two right prisms are identically equal if their bases are identically equal and they have equal altitudes. § 482. 4. THEOREMS ON THE SECTIONS OF PRISMS AND PYRAMIDS. (1) The sections of a prism made by parallel planes are polygons which are identically equal. § 476. (2) If a pyramid is cut by a plane parallel to its base, the edges and the altitude are divided in the same ratio, the section made by the plane is similar to the base, and the area of the section is to the area of the base as the squares of their distances from the vertex. § 513. (3) If two pyramids have equal altitudes and bases of equal areas, the areas of sections made by planes equidistant from their vertices are equal. § 516. 5. THEOREMS ON THE LATERAL AREAS OF PRISMS AND PYRAMIDS. (1) The lateral area of a prism is equal to the product of a lateral edge and the perimeter of a right section of the prism. § 479. (2) The lateral area of a right prism is equal to the product of its altitude and the perimeter of its base. § 480. (3) The lateral area of a regular pyramid is equal to one-half the product of the perimeter of the base and the slant height. § 512. 6. THEOREMS ON THE VOLUME OF A PRISM. (1) An oblique prism is equal in volume to a right prism whose base (2) The volumes of two rectangular parallelepipeds having identically (4) The volume of any rectangular parallelepiped is the product of (5) The volume of any parallelepiped is equal to the product of its altitude and the area of its base. § 505. (6) The volume of a prism is equal to the product of its altitude and the area of its base. §§ 506, 507. (7) The volume of a truncated triangular prism is equal to the sum of the volumes of three pyramids whose common base is one base of the prism and whose vertices are the three vertices of the other base. § 527. (8) The volume of a truncated right triangular prism is equal to the product of the area of that face to which the edges are perpendicular and one-third the sum of the lateral edges. § 528. (9) The volume of any truncated triangular prism is equal to the product of the area of a right section and one-third the sum of the lateral edges. § 529. 7. THEOREMS ON THE VOLUME OF A PYRAMID. (1) If two triangular pyramids have equal altitudes and equal bases, they have equal volumes. § 517. (2) The volume of a triangular pyramid is one-third the volume of a triangular prism of the same base and altitude. § 521. (3) The volume of a triangular pyramid is equal to one-third the product of its altitude and the area of its base. § 522. (4) The volume of any pyramid is equal to one-third the product of its altitude and the area of its base. § 523. (5) The volumes of any two pyramids are in the same ratio as the products of their altitudes and the areas of their bases. § 524. (6) The volumes of two pyramids having equal altitudes are in the same ratio as the areas of their bases, or having equal bases are in the same ratio as their altitudes. § 525. (7) The volumes of two tetrahedrons having a trihedral angle of the one equal to a trihedral angle of the other are in the same ratio as the products of the edges which meet in the vertices of these angles. § 526. 8. THEOREMS ON SIMILAR POLYHEDRONS. (1) Any two edges or diagonals of a polyhedron are in the same ratio (2) The surfaces of two similar polyhedrons (i.e. the sums of the § 533. (4) The volumes of two similar tetrahedrons are in the same ratio as the cubes of their homologous edges. § 535. (5) The volumes of any two similar polyhedrons are in the same ratio as the cubes of two homologous edges. §.536. 9. MISCELLANEOUS THEOREMS. (1) The opposite lateral faces of a parallelepiped are parallel and identically equal. § 493. (2) Any two opposite faces of a parallelepiped may be taken as the bases. § 494. (3) The plane passed through two diagonally opposite edges of a parallelepiped divides it into two triangular prisms which are equal in volume. § 495. (4) In any polyhedron the number of edges increased by two is equal to the number of faces together with the number of vertices (Euler's Theorem). § 541. (5) The sum of the face angles of any polyhedron together with eight right angles is equal to four times as many right angles as the polyhedron has vertices. § 542. CHAPTER VIII CYLINDERS AND CONES SECTION I CYLINDERS 543. DEFINITIONS. If a straight line moves so as always to remain parallel to a fixed straight line, while some point of it traverses a fixed curve not in a plane with the fixed line, it is said to describe a cylindrical surface. Each position of the moving line is parallel to its former position, hence the line is said to move parallel to itself. The moving line is called the generator of the cylindrical surface, and the guiding curve, the director. |