Elements of Synthetic Solid Geometry |
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Page 48
... diameters , EG , FH , diameters . and IJ . 51. Theorem . The diameters of a tetrahedron bisect one another . Proof . ABCD is a skew quadrilateral , and BD and AC are its diagonals . But the joins of the middle points of opposite sides ...
... diameters , EG , FH , diameters . and IJ . 51. Theorem . The diameters of a tetrahedron bisect one another . Proof . ABCD is a skew quadrilateral , and BD and AC are its diagonals . But the joins of the middle points of opposite sides ...
Page 49
... diameter . IP is one - third of IA , and P is thus the centroid of the face ABC ( P. Art . 85 ) , and DP is the median to the face ABC . Evidently DP and IJ are com- A planar , and intersect in some point O. Then O is the centre ...
... diameter . IP is one - third of IA , and P is thus the centroid of the face ABC ( P. Art . 85 ) , and DP is the median to the face ABC . Evidently DP and IJ are com- A planar , and intersect in some point O. Then O is the centre ...
Page 60
... diameters and medians of the two tetrahedra coincide , except in length . 4. In the regular tetrahedron the diameters are perpendicular to one another . 5. If the diameters of a tetrahedron terminate in the centres of the faces of a ...
... diameters and medians of the two tetrahedra coincide , except in length . 4. In the regular tetrahedron the diameters are perpendicular to one another . 5. If the diameters of a tetrahedron terminate in the centres of the faces of a ...
Page 66
... diameter . When ADB revolves about AB as an axis , the semicircle generates a sphere of which OD is a radius . Cor . 1. All the radii of a sphere are equal to one another . Therefore , Def . A sphere is a surface every point on which is ...
... diameter . When ADB revolves about AB as an axis , the semicircle generates a sphere of which OD is a radius . Cor . 1. All the radii of a sphere are equal to one another . Therefore , Def . A sphere is a surface every point on which is ...
Page 67
... diameter . A plane which cuts a sphere is a secant plane , and when it passes through the centre it is a diametral plane . 79. Theorem . The join of the centre of a sphere with the middle point of a chord is perpendicular to the chord ...
... diameter . A plane which cuts a sphere is a secant plane , and when it passes through the centre it is a diametral plane . 79. Theorem . The join of the centre of a sphere with the middle point of a chord is perpendicular to the chord ...
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Common terms and phrases
ABē ABCD 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 ruled surfaces segment sheaf of lines sides skew quadrilateral spatial figure sphere spheric geometry spheric line spheric triangle squares surface tangent line Theorem three-faced corner vertex vertices volume
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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.