- Let a plane pass through BA, AC, and let DAE be the common section of it with the given plane : :. AB, AC, DAE, are in one plane. Then :: ca is I to the plane, and meets DAE, :: ZCAE is. a right Z; for the same reason BAE is a right 2. :: Z CAE = _BAE, the less to the greater, which is impossible. Also, from the same point above a plane there can be drawn but one perpendicular to that plane; for if there could be two they would be ll, which is absurd. D E A PROPOSITION XV. Planes to which the same straight line is perpendicular are parallel to one another. Let as be to each of the planes CD, EF; these planes are || to each other. If not, they will meet if produced: let them meet; their common section is a straight line ga, in which take any point K, and join AK, BK. Then : AB is 1 EF it is I to BK .: ZABK is a right 2 ; for the same reason KAB is a right Z; :. two Zs of the A A B K are = two right 2$, which is impossible; :: the planes CD, EF, though produced, do not meet; therefore they are 11. PROPOSITION XVI. If two straight lines meeting one another be parallel to two other straight lines which meet one another, but are not in the same plane with the first two; the plane which passes through these is parallel to the plane through the others. Let A B, BC, meeting in B, be || to DE, EF, meeting in E, but which are not in the same plane with A B, BC: the planes through A B, BC, and DE, EF, are ll. From B, draw BG + to plane through ED, DF, through a, where BG meets the plane; draw gh || to ED, and GK || to EF. And :: BG is 1 to plane DF, But BGH= a right a i. Z For the same reason GB is 1 BC, and .. GB is 1 to the plane through AB, BC; and it is to plane through DE, EF. :: BG is at once + to the plane through AB, BC, and the plane through DE, EF; hence these planes are parallel. BA. FH or EG. D PROPOSITION XVII. sections with it are parallels. If not, EF, gh, will meet if produced either on the side of First let them be pro- Hence, :: EFK is in the plane plane, .. K is a point in the plane For the same reason, K is a point in the plane cd; :. the planes AB, CD, meet if produced, which is contrary to the hypothesis. :. EF and gu do not meet when produced on the side of FH. In the same manner it may be shown that they do not meet when produced on the side EF: wherefore EF and hg are parallel. 1 A B. PROPOSITION XVIII. If two straight lines be cut by parallel planes, they shall be cut in the same ratio. Let A B, CD, be cut by the parallel planes GH, KL, Mn, in the points A, E, B; C, F, D; then AE H Н : EB ::CF : FD. G E K Join AC, BD, AD; let ad meet Kl in x; join Ex, XF. Then :: the parallel planes KL, MN, are cut by EXDB, :: Ex is parallel to BD; and :: GH, kl, are cut by the plane AXFC, the common sections Ac, XF, are parallel. Then :: Ex is parallel to BD in A ABD, and : xp is parallel to Ac in A ADC, .. AE : EB :: AX : XD. And AX : XD :: CF : FD. N B M .. AE : EB :: CF : FD. PROPOSITION XIX. If a straight line be perpendicular lo a plane, every plane pass ing through it shall be perpendicular to that plane. Let all be + to plane ck; every plane through A B shall be I to CK. Let plane de pass through AB; and let ce be the common section of D E and cK. Take any point F in ce, draw ra in DE, I to CE; then : AB is 1 to ck, < ABF is a right Z, but GFB is a right L, :. AB and fg are ll, and AB is 1 to the plane CK, :: FG is = to the same plane. But one plane is to another plane, when the straight lines drawn in one of the planes, + to their G H B common section, are also I to the other plane. And any line FG, in the plane de, drawn to the line ce, is also I to CK. .. the plane DE is I to CK. And the same may be proved of any other plane passing through A B. PROPOSITION XX. If two planes which intersect be each perpendicular to a third plane, their common section shall be perpendicular to the same plane. Let the two planes AB, BC, be each 1 to the third plane, and let bd be the common section of AB, BC; BD shall be + to the third plane. If it be not, from D in AB draw DE + AD, the common section of AB with the third plane; in bc draw DF + CD, the common section of BC with the same plane. Then : AB is I to plane ADC, and DE is I to AD, :. DE is 1 to ADC; similarly, DF is + to ADC; :. from the same point d, two 19, DE, DF, to the same plane Adc, have been C drawn, which is impossible; .. the only line that can be drawn from D I to adc is BD, the common section of AB and BC. PROPOSITION XXI. If a solid angle be contained by three plane angles, any two of them are greater than the third. Let the solid Z A, be contained by the three plane 29, BAC, CAD, DAB; any two of them shall be greater than the third. If the three angles be all the proposition is evident. If not, let BAC be not less than either of the other two, and > DAB. At the point a, in the plane through Ba, Ac, make Z BAE = _ DA B. Make AE = AD; through E draw bec, cutting AB, AC, in B and c, and join DB, DC. And :: DA = AE, and AB is common, and DAB= ZEAB, :: DB= EB; but :: BD, DC, are > BC and BD=BE, :. Dc is > EC; and :: DA= AE and Ac is common, also DC> EC, :: Z DAC is > EAC; and Z DAB= _ BAE, :: ZDAC+ DAB is > Z BAC. Also, 2 BAC is not less than either of the other two; 1:2 BAC + either of the other two, must be > than the remaining angle. PROPOSITION XXII. Every solid angle is contained by plane angles, which together are less than four right angles. ABD First, let the solid < A be contained by the three plane 25, BAC, CAD, DAB; take in each of the lines AB, AC, AD, any points B, C, D; join BC, CD, DB. Then .: the <S CBA, ABD, DBC, make the solid < B, :: Z CBA + L are > < DBC; similarly 2 BCA + 2 ACD, are '> < DCB; and Z CDA + ZADB, are > BDC. i. Z CBA, ABD, BCA, ACD, CDA, ADB, are > ZS DBC, BCD, BDC; but Z DBC + 2 DCB + XBDC = two right 2$. i. Z CBA + 2 ABD + Z BCA + 2 ACD + 2 CDA + L ADB are > two right ¿ $; and : the three xs of each of the A $ ABC, ACD, ADB, are = two right 2$, . the nine _s of these As, viz., the <SCBA, BAC, ACB, ACD, ADC, DAC, ADB, ABD, BAD, are = six right_$; and of these the ZS CBA, ABD, BCA, ACD, ADC, ADB, are > two right Zs. ... the three remaining 2$, BAC, BAD, DAC, are < four right Zs. Next, let the solid < at a be contained by any number of |