Lines. Therefore the Parallelepip. ABCDEF PROP. XXXI. Solid Parallelepipedons ALEKGMBI, CPO HQDN ftanding upon equal Bafes ALEK, CPOO, and having the fame Altitude, are equal to one another. * (1)Let the Parallelepipedons AB,CD have their Sides perpendicular to the Planes of their Bafes, and at the Side CD continued out, make a the a 18. 6. Pgr. PRTS equal and fimilar to the Pgr. KELA; b and confequently the Ppp. PRTSQVYX is 27.11. equal and fimilar to the Ppp. AB. Produce OoE, & 10 def. NDd, oPZ, DQF, ERB, dVg, TSZ, YXF; 11. and draw Ed, Bg, ZF. are 30 def. The Planes OEN, CRVH, STYF are. parallel; and the Pgrs. ALEK, CPO, PRTS, 11. *The Altitude is a Perpendicular drawn from the Plane of the Bafe to the oppofite Plane. N 4 PRBZ a b PRBZ are equal; therefore fince the Ppp. Ppp. be с = AB. Q. E. D. d PRVOSTYX = But if the Ppps. AB, CD have their Sides oblique to their Bafes; fet Parallelepipedons upon thofe Bafes, that have the fame Altitude, and have their Sides parallel to their Bases: f 29. 11. thefe fhall be equal f to one another, and to the oblique ones; and confequently the oblique ones AB, CD, are equal to one another. QE. D. & I ax. I. g PROP. XXXII. Solid Parallelepipedons ABCD, EFGL, having the fame Altitude, are to one another as their Bases. h n 45. I. 31. 1. k A E H h I Continue out EHI, and make the Pgr. EF AB, and compleat 31.11. is manifeft that the Ppp. 125.11. EFGL::'FI: (AB) EF. the Ppp. FINM; it FINM (ABCD): PROP. XXXIII. Similar folid Parallelepipedons ABCD, EFGH, are to one another in the triplicate proportion of their Homologous Sides AI, EK. Con b a C a 1. 6. Continue out the right Lines AIL, DIO, BIN, and make IL, IO, IN equal to EK, KH, KF, and fo the Ppp. IXMT is equal and fimilar to the Ppp. EFGH. and compleat the 31. 1. Ppps. IXPB, DLYQ. then fhall AI: IL (EK)" byp. :: DI: IO (HK) :: BI: IN (KF); that is, the Pgr.AD: DL:: DL: IX:: BO: IT; that is, the Ppp. ABCD: DLQY:: DLQY: IXBP :: IXBP: IXMT (EFGH). Therefore the Ra- f 10 def. 5. tio of ABCD to EFGH is triplicate of the Ratio of ABCD, to DLQY, or of AI to EK. 81. 6 Q: E. D. CORO L. f Hence if there be four right Lines continually Proportional, as the first is to the fourth, fo is a Parallelepipedon defcrib'd upon the first to a fimilar Parallelepipedon fimilarly describ'd upon the fecond. PROP. XXXIV. The Bafes and Altitudes of equal folid Parallelepipedons ADCB, EHGF, are reciprocally propor tional; .d e 32. 11. conft. tional; viz. AD: EH:: EG: AC. and thofe Solid Parallelepipedons ADCB, EHGF whofe BaJes and Altitudes are reciprocally proportional, are equal. (1.) Let the Sides CA, GE be perpend. to the Bafes; now if the Altitudes of the Solids be equal, their Bafes fhall be alfo equal; which is manifeft enough. But if the Altitudes be unequal, from the greater EG take away EI AC, and draw the Plane IK through I, parallel to the Base EH. с a ΕΙ d e 1 Hyp. Then AD: EH :: Ppp. ADCB : EHIK:: Ppp. EHGF: EHIK:: GL: IL :: GE: IE (AC). Therefore it is manifeft that AD: EH::GE: AC. Q. E. D. g h : 2 Hyp. ADCB: EHIK :: AD: EH " EG:EI :: GL: IL *:: Ppp. EHGF : EHIK. whence the Ppp. ADCB1EHGF. Q. E. D. Again, let the Sides be oblique to the Bases, upon these Bases erect Parallelepip. of the fame height, whofe Sides are at right Angles to the Bases; then fhall the oblique Parallelepipedons be equal to thefe. Whence fince thefe (by Part 1.) have their Bases and Altitudes reciprocally proportional, therefore thofe will be reciprocal. Q. ED. 4 CO CORO L. What has been demonftrated in Prop. 29, 30, 31, 32, 33, 34, may likewife be apply'd to triangular Prifms, which are the halves of Parallelepipedons, as is manifeft from Prop. 28. Therefore, 1. Triangular Prifms of equal Altitudes are to one another as their Bases. 2. If they have the fame or equal Bases, and the fame Altitudes, they are equal. 3. If they be fimilar, their Proportion is triplicate of the Proportion of their Homologous Sides. 4. If they be equal, their Bafes and Altitudes are reciprocally proportional; and if the Bafes and Altitudes be reciprocally proportional, they are equal. PROP. XXXV. If there be two equal plain Angles BAC, EDF, and from the Vertices A, D of those Angles, two right Lines AG, DH be elevated above the Planes, in which the Angles are, containing equal Angles with the Lines first given, each to its CorreSpondent one, viz. the Ang. GAB= =HDE, and GAC HDF; and if in thefe elevated Lines AG, DH, any Points G, H, be taken, from which the Lines GI, HK be drawn perpendicular to the Planes BAC, |