The lines EF, GH, therefore, do not meet on the fide FH; and, in the fame manner, it may be proved, that they do not meet on the fide EG; confequently they are parallel to each other. Q.E.D. PRO P. XIII. THEOREM. If a right line be perpendicular to a plane, every plane which paffes through it will also be perpendicular to that plane. D G A H Let the right line AB be perpendicular to the plane CK; then will every plane which paffes through that line be also perpendicular to CK. For let ED be any plane which paffes by the line AB; and in this plane draw any right line GF perpendicular to the common fection CE (I. 11.) Then, because the line AB is perpendicular to the plane CK (by Hyp.), it will also be perpendicular to the line CE; and the angle ABF will be a right angle (VII. Def. 2.) And fince the angles ABF, GFB are each of them a right angle, and the lines AB, GF are in the fame plane, they will be parallel to each other (VII. 4.) Since, therefore, thefe lines are parallel to each other, and one of them, AB, is perpendicular to the plane CK, the other, GF, will also be perpendicular to that plane (VII. 5.) But one plane is perpendicular to another, when any right line that can be drawn in it, at right angles to the common fection, is also at right angles to the other plane (VII. Def. 3.); whence the plane ED is perpendicular to the plane CK, as was to be shewn. PROP. XIV. THEOREM. If two planes which cut each other, be each of them perpendicular to a third plane, their common fection will also be perpendicular to that plane. Let the two planes AB, CB be each of them perpendicular to the plane ACD; then will their common section BD be alfo perpendicular to ACD. For if not, let DE be drawn in the plane AB, at right angles to the common section AD; and DF in the plane CB at right angles to the common fection DC (I. 11.) Then because the plane AB is perpendicular to the plane ACD (by Hyp.), the line DE will also be perpendicular to that plane (VII. 3.) And fince the plane CB is perpendicular to the piane ACD (by Hyp.), the line DF will also be perpendicular to that plane (VII. 3.)' But lines which are perpendicular to the fame plane are parallel to each other (VII. 4.); whence the lines DE, DF meet, and are parallel at the fame time, which is abfurd. These lines, therefore, are not perpendicular to the plane ACD; and the fame may be shewn of any other line but DB; whence DB is perpendicular to ACD, as was to be fhewn. C: 2. воок VIII. ་་་ ་ DEFINITIONS. 1. A folid angle is that which is made by three or more plane angles, which meet each other in the fame point: 2. Similar folids, contained by plane figures, are fuch as have all their solid angles equal, each to each, and are bounded by the fame number of similar planes. 3. A prism is a folid whofe ends are parallel, equal, and like plane figures, and its fides parallelograms. 4. A parallelepiped is a prifm contained by fix parallelograms, every oppofite two of which are equal, alike, and parallel. 5. A rectangular parallelepiped is that whofe bounding planes are all rectangles, which are perpendicular to each other. 6. A cube is a prism, contained by fix equal fquare fides, or faces. 7. A pyramid is a folid whose base is any right lined plane figure, and its fide triangles, which meet each other in a point above the base, called the vertex. 8. A cylinder is a folid generated by the revolution of a right line about the circumferences of two equal and parallel circles, which remain fixed. 9. The axis of a cylinder is the right line joining the centres of the two parallel circles, about which the figure is described. I 10. A cone is a folid generated by the revolution of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle. 11. The axis of a cone is the rights line joining the vertex, or fixed point, and the centre of the circle about which the figure is described. 12. Similar cones and cylinders are fuch as have their altitudes and the diameters of their bafes proportional. 13. A fphere is a folid generated by the revolution of a femi-circle about its diameter, which remains fixed. 14. The axis of a fphere is the right line about which the femi-circle revolves; and the centre is the fame as that of the femi-circle. 15. The diameter of a fphere is any right line paffing through the centre, and terminated both ways by the furface. PROP. I. LEMMA. If from the greater of two magnitudes, there be taken more than its half; and from the remainder, more than its half; and fo on there will at length remain a magnitude lefs than the leaft of the propofed magnitudes, Let AB and C be any two magnitudes, of which AB is the greater; then, if from AB there be taken more than/ |