If two planes, which cut one another, be each of them perpendicular to a third plane. their common section must be perpendicular to the same plane. B E F Let the two planes AB, BC be each to a third plane, and let BD be the common section of AB and BC. Then must BD be 1 to the third plane. If it be not, draw, in the plane AB, the st. line DE 1 to AD, the common section of AB with the third plane; I. 11. and draw, in the plane BC, the st. line DF 1 to DC, the common section of BC with the third plane. I. 11. Then the plane AB is to the third plane, and DE is drawn in the plane ABI to the common section, .. DE is to the third plane. So also, DF is to the third plane. XI. Def. 3. Hence, from the pt. D, two st. lines are drawn to the third plane, and on the same side of it; which is impossible. XI. 13. .. no other line but BD can be to the third plane at D; .. BD is to the third plane. Q. E. D. If a solid angle be contained by three plane angles, any two of them must be together greater than the third. Let the solid at A be contained by the three plane s BAC, CAD, DAB. Any two of these must be together greater than the third. If the 4s BAC, CAD, DAB, be all equal, any two of them are together greater than the third. If they are not equal, let BAC be that 4, which is not less than either of the other two, and is greater than one of them, DAB. At A, in the plane passing through AB, AC, make LBAE = L DAB, I. 23. and make AE=AD, and through E draw the st. line BEC, cutting AB, AC, in the pts. B, C ; and join DB, DC. Then in AS ABD, ABE, :: AD= AE, and AB is common, and ▲ BAD Then... DB, DC together are greater than BC, and DB=BE, a part of BC, .. DC is greater than EC. Then in As ADC, AEC, = L BAE, I. 4. I. 20. · AD=AE, and AC' is common, and DC greater than EC, .. DAC is greater than EAC. Also, by construction, ▲ DAB= ▲ BAE, I. 25. .. 48 DAC, DAB together are greater than 4 s BAE, EAC together; that is, s DAC, DAB together are greater than 4 BAC. L Again, 4 BAC is not less than either of the 8 DAC, DAB, and ... 4 BAC with either of them is greater than the other. Q. E. D, PROPOSITION XXI. THEOREM. Every solid angle is contained by plane angles, which are together less than four right angles. First, let the solid at A be contained by three plane s BAC, CAD, DAB. These shall be together less than four right angles. Take, in each of the st. lines AB, AC, AD, any points B, C, D, and join BC, CD, DB. Then the solid at B is contained by the three plane ▲ 8 CBA, ABD, DBC, :: ≤ 8 CBA, ABD are together greater than ▲ DBC. XI.20. So also, 4 s BCA, ACD are together greater than ▲ BCD, and 4 s CDA, ADB are together greater than CDB. .. the six 48 CBA, ABD, BCA, ACD, CDA, ADB are together greater than the three ▲s DBC, BCD, CDB, and are.. together greater than two rt. 4 s. Again, the three s of each of the As ABC, ACD, ADB are together equal to two rt. 4s, I. 32. .. the nine s CBA, BAC, ACB, ACD, CDA, DAC, ADB, DBA, BAD are together equal to six rt. 4s; and of these the six s CBA, ACB, ACD, CDA, ADB, DBA, have been proved to be together greater than two rt. 4s, and the three 4s BAC, CAD, DAB, which contain the solid at A, are together less than four rt. 4 s. NEXT, let the solid at A be contained by any number of planes BAC, CAD, DAE, EAF, FAB. These must be together less than four rt. 4 s. B Let the planes, in which the s are, be cut by a plane, and let the common sections of it with those planes be BC, CD, DE, EF, FB. Then the solid at B is contained by the three plane ▲ § CBA, ABF, FBC, of which any two are together greater than the third, .. ▲ 8 CBA, ABF are together greater than ▲ FBC. L XI. 20. So also, the two planes at each of the pts. C, D, E, F, which are at the bases of the As having the common vertex A, are together greater than the third at the same pt., which is one of the 4s of the polygon BCDEF. ..all the 4 s at the bases of the As are together greater than all the 4s of the polygon. Now all the 4s of the As together twice as many rt. 4 s as there are ▲s, that is, as there are sides in the polygon BCDEF: I. 32. and all the 4s of the polygon, together with four rt. 4s, together twice as many rt. 4s as there are sides in the polygon. I. 32. Cor. 1 ... all the 4s of the As together all the s of the polygon together with four rt. 4 s. But all the s at the bases of the As have been proved to be together greater than all the ▲ s of the polygon; .. all the s at the vertex A are together less than four rt. 4 s. Q. E. D. Miscellaneous Exercises on Book XI. 1. If two straight lines in one plane, be equally inclined to another plane, they will be equally inclined to the common section of the two planes. 2. Two planes intersect at right angles in the line AB; from a point C in this line are drawn CE and CF in one of the planes, so that the angle ACE is equal to ACF. Shew that CE and CF will make equal angles with any line through Cin the other plane. 3. ABC is a triangle; the perpendiculars from A, B, on the opposite sides, meet in D, and through D is drawn a straight line, perpendicular to the plane of the triangle; if E be any point in this line, shew that EA, BC; EB, CA; and EC, AB; are respectively perpendicular to each other. 4. A number of planes have a common line of intersection: what is the locus of the feet of perpendiculars on them from a given point? 5. Two perpendiculars are let fall from any point on two given planes: shew that the angle between the perpendiculars will be equal to the angle of inclination of the planes to one another. 6. If perpendiculars AF, A'F', be drawn to a plane from two points A, A', above it, and a plane be drawn through A perpendicular to AA', its line of intersection with the given plane is perpendicular to FF". 7. Prove that equal straight lines drawn from a given point to a plane are equally inclined to the plane. 8. Prove that the inclination of a plane to a plane is equal to the angle between the perpendiculars to the two planes. 9. From a point above a plane two straight lines are drawn, the one at right angles to the plane, the other at right angles |