Now, let the line a b be projected on to the plane of the base, by drawing lines from a and b at right angles to the base, and meeting it in a′ b′; the line a′ b′, produced, will meet A B produced in m. If the lines b c and a c be projected in the same way on to the base, to the points b′ c′ and a′ c′; then B C and b′ c′ produced, will meet in n, and A C and a′ c′ produced, will meet in o. The two triangles A B C and a′ b′ c′ are such, that the lines joining A to a′, B to b′, and C to c′, will, if produced, meet in a point, namely, the point on the base A B C which is the projection of D. Any two triangles which fulfil this condition are the possible base and projection of the section of a pyramid; therefore the sides of such triangles, if produced in pairs, will meet (if they are not parallel) in three points which lie in one straight line. A four-dimensional pyramid may be defined as a figure bounded by a polyhedron of any number of sides, and the same number of pyramids whose bases are the sides of the polyhedron, and whose apices meet in a point not in the space of the base. If a four-dimensional pyramid on a tetrahedral base be cut by a space which passes through the four sides of the pyramid in such a way that the sides of the sectional figure be not parallel to the sides of the base; then the sides of these two tetrahedra, if produced in pairs, will meet in lines which all lie in one plane, namely, the plane of intersection of the space of the base and the space of the section. If now the sectional tetrahedron be projected on to the base (by drawing lines from each point of the section to the base at right angles to it), there will be two tetrahedra fulfilling the condition that the line joining the angles of the one to the angles of the other will, if produced, meet in a point, which point is the projection of the apex of the four-dimensional pyramid. Any two tetrahedra which fulfil this condition, are the possible base and projection of a section of a four-dimensional pyramid. Therefore, in any two such tetrahedra, where the sides of the one are not parallel to the sides of the other, the sides, if produced in pairs (one side of the one with one side of the other), will meet in four straight lines which are all in one plane. APPENDIX F. EXERCISES ON SHAPES OF THREE DIMENSIONS. The names used are those given in Appendix B. Find the shapes from the following projections: 1. Syce projections: Ratis, Caput, Castrum, Plagua. Alvus projections: Merum, Oculus, Fulmen, Pruinus. Moena projections: Miles, Ventus, Navis. 2. Syce: Dies, Tuba, Lituus, Frons. Alvus: Sagitta, Regnum, Tellus, Fulmen, Pruinus. Moena: Tibia, Tunica, Robur, Finis. 3. Syce: Nemus, Sidus, Vertex, Nix, Cerva. Alvus: Lignum, Haedus, Vultus, Nemus, Humerus. Moena: Dexter, Princeps, Equus, Dux, Urbs, Pullis, Gens, Monstrum, Miles. 4. Syce: Amphora, Castrum, Myrtus, Rota, Palma, Meta, Trabs, Ratis.