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 section, 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 section will also be perpendicular to that plane. E А Let the two planes AB, CB be each of them perpendicular to the plane ACD; then will their common section BD be also 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 section DC (I. 11.) Then because the plane AB is perpendicular to the plane ACD (by Hyp.), the line De will also be perpendi. cular to that plane (VII. 3.) And since the plane co is perpendicular to the plane ACD (by Hyp.), the line di will also be perpendicular to that plane (VII. 3.) But lines which are perpendicular to the same plane are parallel to each other (VII. 4.); whence the lines DE, DF meet, and are parallel at the same time, which is absurd. These lines, therefore, are not perpendicular to the plane ACD; and the same may be thewn of any other line but dB; whence DB is perpendicular to ACD, as was to be thewn. • 1. A folid angle is that which is made by three or more plane angles, which meet each other in the same point. 2. Similar folids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes. 3. A prism is a folid whose ends are parallel, equal, and like plane figures, and its sides parallelograms. 4. A parallelepipedon is a prism contained by fix ра. rallelograms, every opposite two of which are equal, alike, and parallel. 5. A rectangular parallelepipedon is that whose bounding planes are all rectangles, which are perpendicular to each other. 6. A cube is a prism, contained by fix equal square fides, or faces. 7. A pyramid is a solid whose base is any right lined plane figure, and its sides triangles, which meet each other in a point above the base, called the vertex. 8. A cylinder is a solid 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. 10. A cone is a solid 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 right 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 such as have their altitudes and the diameters of their bases proportional. 13. A sphere is a solid generated by the revolution of a semi-circle about its diameter, which remains fixed. 14 The axis of a sphere is the right line about which the semi-circle revolves; and the centre is the same as that of the semi-circle. 15. The diameter of a sphere is any right line passing through the centre, and terminated both ways by the surface. PRO P. 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 so on : there will at length remain a magnitude less than the least of the proposed magnitudes. D AE G Let AB and c be any two magnitudes, of which ab is the greater ; then, if from AB there be taken more than its half; and from the remainder more than its half; and so on : there will at length remain a magnitude less than c. For fince AB and c are each finite magnitudes, it is evident that c may be taken fuch a number of times as at length to become greater than AB. Let, therefore, de be such a multiple of c as is greater than AB, and divide it into the parts DF, FG, GE, each equal to c. Also from AB take bh greater than its half; and from the remainder Ah, take hk greater than its half, and so on, till there be as many divisions in AB' as there are in DE. Then because de is greater than AB, and BH, taken from AB, is greater than its half, but eg, taken from de, is not greater than its half; the remainder GD will be greater than the remainder ha. And, again, because op is greater than HA, and HK, taken from ha, is greater than its half, but GF, taken from GD, is not greater than its half; the remainder FD will be greater than the remainder AK. But fp is equal to c by construction, whence c is greater than AK; or, which is the same thing, ak is less than c, as was to be shewn. |