Fig: 34. now be made upon EG, AO, Parallelepipeds, whofe Sides are perpendicular to the Bafes in the Height S; thefe will be equal to the oblique ones by 29th or 30th. Wherefore feeing by the firft Part, right Parallelepipeds are equal betwixt themselves, the oblique ones will be equal betwixt themselves likewife. Q. E. D. A PROP. XXXII. Theorem. LL Parallelepipeds whatever of equal Height, are betwixt themselves as their Bases. Let GO and A be the Bafes. Upon CO make the Parallelogram OE equal to A. Upon BC, OE, let Parallelepipeds be understood to be erected in the Altitude K; thefe therefore will be Parts of one Parallelepiped B E K. Therefore the Parallelepiped O EK, is to the Parallelepiped BCK, as the Bafe OE, to the Bafe BC (per 25.7. 11.); that is, by the Conftruction, as the Base A is to the Bafe B C. But because the Bafes OE and A are equal, the Parallelepipeds OEK and AK are equal (by the foregoing). Therefore alfo the Parallelepiped A K is to the Parallelepiped BCK, as the Bafe A is to the Bafe BC. 2. E. D. Fig, 35. Scholium. THAT which hath here been fhew'd of Parallelepi peds, will be demonftrated in the Twelfth Book of Pyramids, Prop. 6. Of all Prifms whatever, in Coroll. 1. after Prop. 9. Of Cones and Cylinders, Prop. 11. PRO P. XXXIII. Theorem. IKE Parallelepipeds (HA and CM) are in a triplicate Proportion of their homologous Sides (AB, BC). Let the Parallelepipeds AH, CM, be like. Therefore all their Planes (by Defin. 9. l. 11.) are like; and confe confequently AE (by Defin. 1. 1.6.) is to BC, EB to BU; and as FB is to BG, fo is EB to BO. Moreover the Angles of the Planes are alfo equal (by the fame). Therefore let the Solids AH, CM, be fo plac'd, that the equal Angles CBO, ABE, may be oppofite, and the Sides A B, CB, may lie fo as to make one ftreight Line; and then EB, OB will also lie fo as to make one ftreight Line. Now let Solids be imagin'd to be conftituted upon the Planes BQ and E C, in fuch fort that the Solids K B, HA, may be one Parallelepi ped, and KB, P O, may make one Parallelepiped, and PO, CM, may make one Parallelepiped likewife. The Solid HA is to the Solid KB (per 25.1.11.) as AE to BR; that is, (per 1. l. 6.) as A B to BC that is, (as I fhew'd above by the Hypothefis) as EB is to BO; that is, (by the fame) as EC is to BQ that is, (per 25. 7. 11.) as the fame Solid KB is to the Solid PO. Therefore the three Solids H A, KB, PO, continue the fame Proportion. But now the Solid KB is to the Solid PO (by the fame) as the Base BR is to the Base BQ; that is, (per 1. 1. 6.) as E B is to BO; that is, as F B is to B G, as it was fhew'd above by the Hypothefis; that is, (by the fame) as the Plane FC is to the Plane BS; that is, (per 25. l. 11.) as the fame Solid PO again is to the Solid CM. Therefore the four Solids, HA, KB, PO, CM, are continually proportional. Therefore (by Defin. 10. l. 5.) the Proportion of the first H A to the fourth CM is triplicate of the Proportion of the first H A to the second KB; that is, triplicate to the Proportion (per 25. l. 11.) of AE to BR; that is, triplicate (per 1. 7. 6.), to the Proportion of the homologous Sides, A B to BC. Q. E.D. [Coroll. (1.) Hence if there be four right Lines continually proportional; as is the first to the fourth, fo is a Parallelepiped defcrib'd upon the first, to a Paral lelepiped like, and in like manner defcrib'd upon the Second. (2.) Upon this also depends that most famous Problem concerning doubling the Cube; of which afterwards, Schol. p. 18. l. 12. (3.) Hence alfo is to be corrected the Error of thofe, who suppose that the Proportion of like Solids is the fame as is that of their Sides. For the Cube of a Line, which is double to another Line, is not only double to Fig. 36. the other, but as 8 to one. And the Cube of aLine, which is triple to another Line, is not only triple to the other Cube, but contains it 27 Times. For 1:2:4:8 and 1:39:27, and the fame Thing is to be faid of all like Bodies whatsoever; as will appear afterwards. (4) Hence the triplicate Proportion of any Quantities whatsoever is the Proportion of the Cubes of the fame Quantities. Let there be any two Quantities in the triplicate Proportion of the Quantities AB, BC; they fhall also be as AB Cube is to BC Cube.] Scholium THAT which hath here been fhew'd of Parallelepi PROP. XXXIV. Theorem. F the Parallelepipeds (BM, CK) be equal, their Bafes and Altitudes are reciprocally proportional; (that is, the Bafe AM is to the Bafe FK, as reciprocally the Height FC is to the Height AB). ་་་ And if they be reciprocally proportional, their Bafes and Altitudes are equal. Part I. Firft let the Sides be perpendicular to the Bafes. If now the Altitudes of the Solids BM, CK be equal, the thing is manifest. If the Altitudes be unequal, from the greater FC cut off FE equal to BA and thro' E draw the Plane EL parallel to FK. The Bafe A M is to the Base FK, (per 25. 1.11.) as the Solid BM is to the Solid EK; that is, (becaufe by the Hypothefis the Solids BM, CK are equal) as the Solid CK is to the Solid EK; that is, (by the fame) as CG is to EG; that is, (per 1. l. 6.) as CF is to EF; that is, by the Construction, as CF to BA. Q. E.D. Then let the Sides be oblique to the Bafes. Let right Parallelepipeds be erected upon the fame Bases in the fame fame Height. The oblique Parallelepipeds will per 29. and 30. 1. 11.) be equal to thefe: Wherefore feeing thefe, by the first Part, have their Bafes and Altitudes reciprocal, thofe alfo fhall be fo likewife. Q. E.D. Part. II. Let the Altitudes be unequal, and the Sides perpendicular to the Bafes; and from the greater CF take E F equal to AB. The Solid B M is to the Solid EK, (per 32. l. 11.) as A M is to FK, that is, by the Hypothefis, as CF is to AB; that is, by the Conftruction, as CF is to EF; that is, as CG is to (a) EG; that is, (b) as the Solid CK is to the fame Solid E K. Therefore the Solids B M and CK have the fame Proportion to EK: Therefore they are equal. 2. E. D. Corollaries. (a) Per 1. 1.6. (b) Per 25. WHAT Affections have been demonftrated of Pa- 11. rallelepipeds, Prop. 29, 30, 31, 32, 33, 34, do alfo agree to triangular Prifms, which are the Halves of Parallelepipeds. As is manifeft from Prop. 28. Therefore, 1. Triangular Prifms, which are of equal Height, are as their Bafes, A, B. 2. If they be like, their Proportion is triplicate to the Proportion of the Sides, oppofite to the Angles. 3. If they be equal, they reciprocate their Bases and Altitudes; and if they reciprocate their Bafes and Altitudes, they are equal. Scholium. WHAT bath here in Prop. 34. been fhew'd of Pa- Fig. 37. Fig. 38. PROP. XXXV. S very long, and fubfervient to the following Propofition, which we will demonftrate without it. PROP. XXXVI. Theorem.... A Parallelepiped (DH) made of three proportional Right Lines (A, B, C) is equal to a Parallelepiped (IN), which is made of the Mean (B), and is equiangular to the former. Let the Bafe FD of the Parallelepiped DH have the Side EF equal to A, and the other Side ED equal to C: And the Side E G which stands upon the Base equal to B. Thus the Parallelepiped DH will be made of the three right Lines, A, B, C. Then let the Three Sides, LX, IX, XM, (and confequently all the reft) of the Parallelepiped IN be equal to the middle Line B: And the folid Angle X equal to the folid Angle E the Parallelepiped IN will be made of the Mean B; and be equiangular to the former. I fay alfo that it is equal. For feeing by the Hypothefis and the Conftruction, as FE is to LX, fo reciprocally I X is to DE, the Bafes Per 14.1.6.* DF, IL will be equal. Now becaufe the folid Angles at E and X are equal; if they be put within Per defin. one another, † they will coincide; and because of the Equality of the right Lines, EG, XM, the Points M and G, will coincide. Wherefore both the Solids will have one perpendicular Altitude; to wit, the right Line which is let fall from, the Points M, G, (now become one) unto the Plane of the Bafe. The Solids therefore DH, IN are equal. 2. E.D. Per 32. 1. II. Scholium. WE will further obferve what is of great Ufe, that of three Lines drawn into or multiplied one by ano ther |