Elements of Synthetic Solid Geometry |
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Page 6
... complanar or coplanar . Thus all the parts of a figure in plane geom- etry are complanar . Cor . 2. Two intersecting lines are complanar and determine one plane . For , taking a point in each line , and the point of inter- section , we ...
... complanar or coplanar . Thus all the parts of a figure in plane geom- etry are complanar . Cor . 2. Two intersecting lines are complanar and determine one plane . For , taking a point in each line , and the point of inter- section , we ...
Page 7
... at OF . Then AB , CD , EF are complanar . Similarly and ZAOE ZAOP = 1 ; = .. Z DOE is < a ] . BOF BOP = 1 ; = .. Z DOF is > al . B D ( hyp . ) Therefore , in revolving OP from the position OE to THE LINE AND THE PLANE . 7 PART II PAGE.
... at OF . Then AB , CD , EF are complanar . Similarly and ZAOE ZAOP = 1 ; = .. Z DOE is < a ] . BOF BOP = 1 ; = .. Z DOF is > al . B D ( hyp . ) Therefore , in revolving OP from the position OE to THE LINE AND THE PLANE . 7 PART II PAGE.
Page 9
... complanar with OA and OB . Then OP is to OC . Proof . Take OA = 0B = any con- venient length . Join AB , cutting OC in C , and join PA , PB , PC . The right - angled triangles POA and POB are congruent , and there- fore PA PB . Hence ...
... complanar with OA and OB . Then OP is to OC . Proof . Take OA = 0B = any con- venient length . Join AB , cutting OC in C , and join PA , PB , PC . The right - angled triangles POA and POB are congruent , and there- fore PA PB . Hence ...
Page 11
... plane . Def . 2. The angle between a given line and its pro- jection upon a plane is taken to be the angle between the given line and the plane . Def . 3. The angle between two non - complanar THE LINE AND THE PLANE . 11.
... plane . Def . 2. The angle between a given line and its pro- jection upon a plane is taken to be the angle between the given line and the plane . Def . 3. The angle between two non - complanar THE LINE AND THE PLANE . 11.
Page 12
Nathan Fellowes Dupuis. Def . 3. The angle between two non - complanar lines is the angle between two intersecting lines respectively parallel to the given lines . 12. Theorem . The angle between a line and its pro- jection on a plane is ...
Nathan Fellowes Dupuis. Def . 3. The angle between two non - complanar lines is the angle between two intersecting lines respectively parallel to the given lines . 12. Theorem . The angle between a line and its pro- jection on a plane is ...
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Common terms and phrases
AB² altitude axis base bisect bisector centroid chord circle circular cone common line complanar congruent conic cos² cube cuboid curve cylinder cylindroid denote diagonals diameter dihedral angles draw ellipse equal face angles form a sheaf four frustum given line given point Hence hyperbola infinity intersection isoclinal line join line-segment locus mean centre median meet middle point mon line non-complanar lines normal number of faces octahedron opposite parabola parallel lines parallelepiped parallelogram pass perpendicular planar line plane figure plane geometry point equidistant polygon polyhedra polyhedron prism prismatoid projection Proof pyramid radius rectangle regular tetrahedron right angle right section right-bisector plane secant line segment sheaf of lines sides skew quadrilateral spatial figure spheric geometry spheric line spheric triangle squares surface tangent line Theorem three-faced corner vertex vertices volume
Popular passages
Page 236 - To the many of my fellow-teachers in America who have questioned me in regard to the Non-Euclidean Geometry, I would now wish to say publicly that Dr. Smith's conception of that profound advance in pure science is entirely sound. . . . Dr. Smith has given us a book of which our country can be proud. I think it the duty of every teacher of geometry to examine it carefully."— From Prof.
Page 237 - OF EUCLID'S ELEMENTS. Including Alternative Proofs, together with additional Theorems and Exercises, classified and arranged. By HS HALL, MA, and FH STEVENS, MA, Masters of the Military and Engineering Side, Clifton College. Gl.
Page 67 - The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the other two sides.
Page 1 - The projection of a line on a plane is the locus of the projections of all its points.
Page 234 - ... University of Ohio, of Pennsylvania, of Michigan, of Wisconsin, of Kansas, of California, of Missouri, Stanford University, etc., etc. "Those acquainted with Mr. Smith's text-books on conic sections and solid geometry will form a high expectation of this work, and we do not think they will be disappointed. Its style is clear and neat, it gives alternative proofs of most of the fundamental theorems, and abounds in practical hints, among which we may notice those on the resolution of expressions...
Page 238 - AND BESSEL'S FUNCTIONS. Crown 8vo. IQJ. 6d. WILSON (JM)— ELEMENTARY GEOMETRY. Books I. to V. Containing the Subjects of Euclid's first Six Books. Following the Syllabus of the Geometrical Association. By JM WILSON, MA, Head Master of Clifton College. New Edition. Extra fcap. 8vo. 4*.
Page 237 - RICHARDSON.— THE PROGRESSIVE EUCLID. Books I. and II. With Notes, Exercises, and Deductions. Edited by AT RICHARDSON, MA, Senior Mathematical Master at the Isle of Wight College.
Page 97 - S'-A'B'C' be two triangular pyramids having equivalent bases situated in the same plane, and equal altitudes. To prove that S-ABC =c= S'-A'B'C'. Proof. Divide the altitude into n equal parts, and through the points of division pass planes parallel to the plane of the bases, forming the sections DEF, GHI, etc., D'E'F', G'H'I', etc. In the pyramids S-ABC and S'-A'B'C' inscribe prisms whose upper bases are the sections DEF, GHI, etc., D'E'F', G'H'I', etc.
Page 234 - GEOMETRY. 12mo. $2.60. WORKS BY ISAAC TODHUNTER, FRS Late Principal Lecturer on Mathematics in St. John's College. PLANE CO-ORDINATE GEOMETRY, As Applied to the Straight Line and the Conic Sections.