Elements of Synthetic Solid Geometry |
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Page 1
... join of any two arbitrary points in it lies wholly in the surface and coincides with it . Cor . 1. A line cannot lie partly within a plane and partly without it . For the part within the plane must have at least two points in the plane ...
... join of any two arbitrary points in it lies wholly in the surface and coincides with it . Cor . 1. A line cannot lie partly within a plane and partly without it . For the part within the plane must have at least two points in the plane ...
Page 4
... Join A , B by a line . Then , since A and B are two U B V points in U , the join AB lies wholly in U ( Art . 1. Def . ) . 4 For a similar reason the join AB lies wholly in Polyhedra Euler's Theorem C+FE+ 2 Tetra- hedron Parallelepiped ...
... Join A , B by a line . Then , since A and B are two U B V points in U , the join AB lies wholly in U ( Art . 1. Def . ) . 4 For a similar reason the join AB lies wholly in Polyhedra Euler's Theorem C+FE+ 2 Tetra- hedron Parallelepiped ...
Page 5
... join of A and B as a common line . Def . A group of planes having one common line is an axial pencil , and the line is the axis . In contra- distinction to this the pencil of lines in a common plane ( P. Art . 203. Def . ) is called a ...
... join of A and B as a common line . Def . A group of planes having one common line is an axial pencil , and the line is the axis . In contra- distinction to this the pencil of lines in a common plane ( P. Art . 203. Def . ) is called a ...
Page 6
... joins AC and BC respectively . Then D and E and their join lie in U , and in every plane through A , B , and C. Therefore every plane through A , B , and C coincides with U , and forms with U virtually but one plane . Cor . 1. Any three ...
... joins AC and BC respectively . Then D and E and their join lie in U , and in every plane through A , B , and C. Therefore every plane through A , B , and C coincides with U , and forms with U virtually but one plane . Cor . 1. Any three ...
Page 9
... 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 the AAPB = and AOB are each isosceles , and PC and OC are lines from the vertices to the common ...
... 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 the AAPB = and AOB are each isosceles , and PC and OC are lines from the vertices to the common ...
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Common terms and phrases
ABē altitude axis base bisect bisector centroid chord circle circular cone common line complanar congruent conic cosē 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.