Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . Ik Let the st. line AB be± to each of the planea CD, EF. Then must CD be parallel to EF. If not, let them meet, and let the st. line GH be their com-mon section. In GH take any pt. K, and join AK, BK. Then •.• AB is± to the plane EF, .-. AB isxto BK, a st. line in that plane, XI. Def. 2. and .-. z A BK is a rt. z . So also, L BAK is a rt. z . Hence two z s of the A ABK are together = two rt. z s :which is irapo^.sibl? I. 17. .•. the planes CD, EF do not ibeet when produced. and .-. Ci) is II to EF. XI. D
Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . Ik Let the st. line AB be± to each of the planea CD, EF. Then must CD be parallel to EF. If not, let them meet, and let the st. line GH be their com-mon section. In GH take any pt. K, and join AK, BK. Then •.• AB is± to the plane EF, .-. AB isxto BK, a st. line in that plane, XI. Def. 2. and .-. z A BK is a rt. z . So also, L BAK is a rt. z . Hence two z s of the A ABK are together = two rt. z s :which is irapo^.sibl? I. 17. .•. the planes CD, EF do not ibeet when produced. and .-. Ci) is II to EF. XI. Def. 7. 326 EUCLIDS ELEMENTS. [Book XI. Proposition XV. Theorem. [f two stroAght lines, meeting one another, be jMrallel to twoother straight lines, which meet one another, but are not in thesame plane with the first two ; the plane, which passes throughthese, must be parallel to the plane passing through the Let AB, BG, two st. lines meeting one another, be || to DE,EF, which meet one another, but are not in the same planewith AB, BC. Then must the plane AC be \\ to the plane DF. From B draw BG ± to the plane DF, meeting it in G. XL G draw GH || to ED, and GK \\ to EF. I. 31. Then •.• BG is x to the plane DF, .: BG is X to GH and GK, lines in that plane, XI. Def. .•. each of the z s BGH, BGK is a rt. z .Again •.• BA and GR are both || to ED,.: BA is II to GH,and .•. IS GBA, BGH together = two ri. I L BGH is a rt. ^ ... I GBA is a rt. z .Hence GB is ± to BA ; and GB is ± to BC, for the same reason ; .-. GB is ± to the plane AC. XL 4. Also, GB is ± to the plane DF; Constr. .-. the plane AC is || to the plane DF. XL 14. Q. K D. XT. 9. L 29. Book XI.] PROPOSITION XVI. 327 Proposition XVI. Thkorem. If two parallel planes be cut by another plane, their commonuctions vnih it are paraJld. K / I / \/ / / / n / Ji I
Size: 2048px × 1220px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1870, bookidelementsofge, bookyear1879
