Elements of Synthetic Solid Geometry |
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Page 6
... Take D , E , any points in the 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 ...
... Take D , E , any points in the 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 ...
Page 9
... 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 the AAPB = and AOB are each isosceles , and PC and OC ...
... 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 the AAPB = and AOB are each isosceles , and PC and OC ...
Page 12
... Take OA = ON and join PA and AN . Since 2 PNA = 1 , PA is > PN . And in the triangles POA and PON , PO is common , OA ON , and PA > PN ; = .. Z POA is > < PON . ( P. Art . 67. ) And as L is any planar line not parallel to ON , the Z PON ...
... Take OA = ON and join PA and AN . Since 2 PNA = 1 , PA is > PN . And in the triangles POA and PON , PO is common , OA ON , and PA > PN ; = .. Z POA is > < PON . ( P. Art . 67. ) And as L is any planar line not parallel to ON , the Z PON ...
Page 14
... take any point , A , and through 4 draw the line N parallel to L ( Art . 14. 1 ) . M and N determine a plane , U , which is parallel to L. M C N L A From any point B in L draw BC normal to U ( Ex . 1 ) . Then , as L is parallel to U ...
... take any point , A , and through 4 draw the line N parallel to L ( Art . 14. 1 ) . M and N determine a plane , U , which is parallel to L. M C N L A From any point B in L draw BC normal to U ( Ex . 1 ) . Then , as L is parallel to U ...
Page 32
... take OB OD = any conven- = M ient length , and let A be any point on L , other than O. Let the plane of ABD cut N in C. Then ΔΑΟΒΞΔ AOD . ( P. Art . 52. ) .. AD = AB , and ZADB = Z ABD . . : ≤ CDB is > ≤CBD , and CB is > CD . ( P. Art ...
... take OB OD = any conven- = M ient length , and let A be any point on L , other than O. Let the plane of ABD cut N in C. Then ΔΑΟΒΞΔ AOD . ( P. Art . 52. ) .. AD = AB , and ZADB = Z ABD . . : ≤ CDB is > ≤CBD , and CB is > CD . ( P. Art ...
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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.