Modern Pure Solid Geometry

The scope of this book is more limited than its title indicates. The nine chapters deal respectively with preliminary ideas, trihedral angles, skew quadrilaterals, tetrahedra, transversals, oblique cones, spheres, inversion, and recent geometry of the tetrahedron. The treatment is synthetic; but excluding anharmonic ratios, involution, conies, and even the complete quadrilateral. On the other hand, the author uses the concept of the imaginary sphere. The book may be criticised for its limited scope, but it is a useful collection of theorems on elementary topics. In particular, the chapters on tetrahedra bring together information not readily accessible elsewhere.Modern Pure Solid Geometry. -- From review. PREFACE ............................................... v CHAPTER I. PRELIMINARY ................................ 1 1. Introductory Propositions and Problems ........ 1 2. The Hyperbolic Group of Lines ................. 9 3. Harmonic Forms ................................ 12 4. Homothetic Figures ............................ 16 5. Perspective Tetrahedrons ...................... 21 CHAPTER II. THE TRIHEDRAL ANGLE ....................... 27 1. The Orthocentric Line ......................... 27 2. The Supplementary Trihedral Angle ............. 30 3. Isoclinal Lines and Planes .................... 32 4. Axes .......................................... 37 5. Centroidal Lines .............................. 40 CHAPTER III. THE SKEW QUADRILATERAL ................... 42 CHAPTER IV. THE TETRAHEDRON ........................... 48 1. The Centroid .................................. 48 a. Bimedians and Bialtitudes .................. 48 b. Medians .................................... 51 c. The Circumscribed Parallelepiped ........... 58 2. Altitudes ..................................... 61 a. Special Cases .............................. 61 b. The General Case ........................... 66 c. The Monge Point ............................ 68 3. The Bisecting Planes of the Dihedral Angles ... 71 4. The Spheres Touching the Four Faces of a Tetrahedron ................................... 72 a. The Inscribed Sphere ....................... 72 b. The Existence and Distribution of the Escribed Spheres .......................... 74 c. Relative Position of the Centers to Each Other ................................ 78 d. The Points of Contact with the Faces ....... 80 e. Relations Between the Radii ................ 82 5. Volume of Tetrahedron ......................... 86 6. Special Tetrahedrons .......................... 91 a. The Trirectangular Tetrahedron ............. 91 b. The Isosceles Tetrahedron .................. 94 7. Miscellaneous Propositions .................... 102 CHAPTER V. TRANSVERSALS ............................... 111 1. The Skew Quadrilateral ........................ 111 2. The Tetrahedron ............................... 115 CHAPTER VI. THE OBLIQUE CONE WITH A CIRCULAR BASE...... 123 CHAPTER VII. SPHERES .................................. 133 1. Preliminaries ................................. 133 2. Inverse Points ................................ 135 3. Orthogonal Spheres ............................ 136 4. Poles and Polar Planes with Respect to a Sphere ........................................ 140 5. The Imaginary Sphere .......................... 149 6. Centers, Axes, and Planes of Similitude ....... 151 7. The Power of a Point with Respect to a Sphere . 161 8. The Radical Plane of Two Spheres .............. 167 9. Coaxal Pencils of Spheres ..................... 177 10. Coaxal Nets of Spheres ........................ 191 11. Four Spheres .................................. 201 CHAPTER VIII. INVERSION ............................... 214 CHAPTER IX. RECENT GEOMETRY OF THE TETRAHEDRON ........ 230 A. THE GENERAL TETRAHEDRON .......................... 230 1. Harmonic Points and Planes .................... 230 a. Tetrahedral Poles and Polar Planes ......... 230 b. Desmic Systems of Tetrahedrons ............. 235 2. Isogonal Points ............................... 240 3. Antiparallel Sections ......................... 247 4. Spheres Related to the Tetrahedron ............ 250 a. Analogues of the Nine-Point Circle ......... 250 b. The Apollonian Spheres of the Tetrahedron... 252 B. SPECIAL TETRAHEDRONS ............................. 256 1. The Circumscriptible Tetrahedron .............. 256 2. The Orthocentric Tetrahedron .................. 261 3. The Isodynamic Tetrahedron .................... 276 a. The Lemoine Point and the Lemoine Plane .... 276 b. The Lemoine Tetrahedron .................... 282 c. The Newberg Spheres ........................ 286 4. The Isogonic Tetrahedron ...................... 290 BIBLIOGRAPHICAL NOTES ................................. 297 INDEX ................................................. 305