b b с 16.11. Pianes AG, DB, makes the Sections AH, DC parallel. By the fame Reason AD, HC are parallel. Therefore ADCH is a Parallelogram. And in like manner, the other Planes of the 35 def. 1. Parallelepipedon are Parallelograms. Therefore fince AF is parallel to HG, and AD to 10. 11. HC, the Ang. FAD fhall be = CHG; whence becaufe AF = HG, and AD HC, and confequently AF: AD:: HG: HC; therefore the Triangles FAD, GAH are Similar and Equal; and fo the Parallelograms AE, HB are 6 ax. 1. Similar and Equal. And the fame may be prov'd of the other parallef Planes. Therefore, d 34. I. 7.5. f 6.6. 4. I. &c. d e h PROP. XXV. = If a folid Parallelepipedon ABCD be cut by a Plane EF parallel to oppofite Planes AD, BC; it fhall be as the Bafe AH is to the Bafe BH, fo is the Solid AHD to the Solid BHC. Conceive the Parallelepip. ABCD to be continued out both ways, and take AI = AE, and BK EB, and put the Planes IQ, KP, parallel to the Planes AD, BC. The Parallelograms IM, 336. 1.& AH, and * DL, DG, and IQ, AD, EF, &c. def. 6. are Similar and Equal. Whence the Ppp. AQ=AF; and for the fame Reason the Ppp. BP BF. Therefore the Solids IF and EP are 1. K 1 11. 24. 11. 10. def. k F Equimultiples of the Solids AF, EC, b and the than KH, in like man All this may be apply'd to any Prifm. CORO L. or Q. E. D. Whence v6 def. 5. If any Prifm be cut by a plane Parallel to oppofite Planes, the Section fhall be a Figure Equal and Similar to the oppofite Planes. PROP. XXVI. Probl. To make a folid Angle AHIL equal to a given folid Angle CDEF, at a Point A, in a given Right Line AB. From any point F, let fall FG perpend. to 11. 11. the Plane; and draw the right Lines DF, FE, EG, GD, CG. Make AHCD, and the Ang. HAIDCE, and AI = CE; and in the Plane HAI make the Ang. HAK = DCG, and AK = CG; then raise KL perpend. to the Plane HAI, and let KL be = GF, and draw AL. Then fhall the folid Angle AHIL be equal to the folid Angle CDEF given. PROP: N 2 PROP. XXVII. From a right Line given AB to defcribe a Parallelepipedon AK Similar, and in like manner fituate to a folid Parallelepipedon given CD. a 26. II. b d e 12. 6. 22. 5. a Make a folid Angle A of the plane Angles BAH, HAI, BAI, which are equal to FCE, ECG, FCG; alfo make FC: CE :: BA: AH. And CE: CG :: AH: AI, (whence by Equality it fhall be FC: CG :: BA : AÏ) and compleat the Ppp. AK, which fhall be fimilar to the given one. 1 def. 6. For by Conftr. the Pgrs. BH, FEd; and HI, d 24. 11. EG; and EG; and BI, FG, are fimilar; and fo the oppofite Planes of thefe to the oppofite Planes of thofe: therefore the fix Planes of the Solid AK are fimilar to the fix Planes of the f9. def.11 Solid CD, and confequently AK, CD are fimilar. Q. E. F. PROP D A E PROP. XXVIII. H B b a G If a folid Parallelepipedon AB be cut by a Plane FGCD, passing a b 24. II. 34. 1. thro' the Diagonals DE, CG, of two oppofite Planes AE, HB; the Solid AB will be cut in half by the Plane FGCD. For because DC, FG are equal and parallel, the Plane FGCD is a Pgr. and fo the Pgrs. AE, HB are a fimilar and equal: alfo the Triangles AFD,HGC, CGB, DFE, are 2 equal and fimilar. But the Pgrs. AC, AG, are a equal and fimilar to FB, FD; therefore all the Planes of the Prism FGCDAH are alfo equal and fimilar to all the Planes of the Prifm FGCDEB; def. 9.11. and therefore this Prifm is equal to that. Q. E. D. а a PROP. XXIX. Solid Parallelepipedons AGHEFBCD, AGHE MLKI, ftanding upon the fame Bafe AGHE, and having the fame Altitude; that is, ftanding between the parallel Planes AGHE, FLKD, whofe infiftent N 3 Lines Lines AF, AM, are in the fame right Lines AG, For if AFMEDI, GBLHCK, the Solid PROP. XXX. Solid Parallelepipedons ABCDEFGH and ABCDKIML, that stand upon the fame Bafe ACBD, and have the fame Altitude, whofe infiftent Lines CF, CI, DE, DM, BG, BK, AH, AL, do not fall in the fame ftreight Lines, are equal to For let ML, GH, and IK, FE continued out, meet in N and P, and FE, IK in O, Q. and join CN, DO, BQ, AP. с = Now the Parallelepip. ABCDEFGH © ABCDLKNO, they being both upon the fame Base AC, and their infiftent Lines CF, CN, DE, DO, BG, BQ fall in the fame right Lines FO, GP. And for the fame Reason, the Parallelepip. ABCDONPQ Parallelepip. ABC DIMLK; for they have the fame Base AC, and the infiftent Lines CI, CN, BK, BQ, DM, DỌ, AL, AP, are in the fame ftreight = |