Geometry of Four Dimensions |
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24-hedroids ABCD absolutely perpendicular planes altitude axiom cells centre circle common perpendicular planes congruent contains convex polygon corresponding cube dihedral angles directing-polygon distance dodekahedron double prism droid edges elements Elliptic Geometry equal regular Euclidean face angles figure four dimensions frustum given plane given point half-hyperplanes half-line drawn half-plane hedroid hedrons hyper Hyperbolic Geometry hypercone hypercube hyperparallelopiped hyperplane angle hyperplane Art hyperprism hyperpyramid hypersolid hyperspace hypersphere hypersurface hypervolume ikosahedron infinity interior isocline isocline planes Jouffret line perpendicular lines at infinity lying Non-Euclidean Geometry number of vertices pair parallel planes parallelopiped pendicular pentahedroid plane absolutely perpendicular plane angle plane intersects plano-polyhedral angle Point Geometry points collinear polyhe polyhedral angles polyhedroidal angle prism cylinder projection PROOF prove radii reciprocal nets regular polyhedroid rotation Schoute segment sphere spherical surface symmetrically situated symmetry tetrahedroidal angle tetrahedron THEOREM triangle triangular prism trihedral Veronese vertex vertex-edge volume
Popular passages
Page 194 - THEOREM 559. // two trihedral angles have the three face angles of one equal respectively to the three face angles of the other, the corresponding dihedral angles are equal and the trihedral angles are either equal or symmetrical. Fig. I Given in trihedral angles V-ABC and V'-A'B'C', /.AVB = /.A'V'B', /.BVC = /.B'V'C', /CVA = /C'V'A'.
Page 221 - Theorem. —A line perpendicular to one of two parallel lines is perpendicular to the other.
Page 1 - Distance well proved that there are no more than three dimensions, because of the necessity that distances should be defined, and that the distances defined should be taken along perpendicular lines, and because it is possible to take only three lines that are mutually perpendicular, two by which the plane is defined and a third measuring depth; so that if there were any other distance after the third it would be entirely without measure and without definition. Thus Aristotle seemed to conclude from...
Page 128 - Similar figures are those which are alike in form. As in the case of triangles, which have been considered, two figures, to be similar, must have their corresponding sides in proportion, and the angles of one equal to the corresponding angles of the other. Any two circles are similar.
Page 110 - If two triangles have two sides and the included angle of one equal respectively to two sides and the included angle of the other, the triangles are equal.
Page 10 - Donkin, regard the alleged notion of generalised space as only a disguised form of algebraical formulisation ; but the same might be said with equal truth of our notion of infinity in algebra, or of impossible lines, or lines making a zero angle in geometry, the utility of dealing with which as positive substantiated notions no one will be found to dispute. Dr Salmon, in his extension of Chasles...
Page 4 - ... the accidents of perception, and there being but one straight [line between two points would not be a necessity, but only something taught in each case by experience. Whatever is derived from experience possesses a relative generality only, based on induction. We should therefore not be able to say more than that, so far as hitherto observed, no space has yet been found having more than three dimensions.] 4.
Page 203 - CONES. 343 705. DEF. A circular cone is a cone whose base is a circle. The straight line joining the vertex and the centre of the base is called the axis of the cone. If the axis is perpendicular to the base, the cone is called a right cone. If the axis is oblique to the base, the cone is called an oblique cone.
Page 13 - Manning points out that a study of the fourth-dimensional geometry enables one to prove the theorems in geometry of three dimensions, just as a consideration of the latter enables us to prove theorems in plane geometry. The book is a very scholarly and searching piece of work. (New York: The Macmillan Company. 1914. 348 p. $2.00.) In his Elementary geometrical optics, Mr. AS Ramsey of Magdalene College, Cambridge, has provided a textbook with a rather specific and local use in mind. Nevertheless...
Page 124 - Two trihedral angles are equal when two face angles and the included dihedral angle of the one are equal, respectively, to two face angles and the included dihedral angle of the other, and similarly placed.