mid DEF H, the angle ABC [by 9. def. 11] will be equal to the angle DEF, the angle GBC to the angle HEF, and the angle ABG to

will be fimilar to the parallelogram EP. By the fame reafon the parallelogram BN is fimilar to the parallelogram ER, and the parallelogram BK to the parallelogram Ex: Therefore the three parallelograms BM, KB, BN are fimilar to the three parallelograms EP, EX, E R. But [by 24. 11.] the three parallelograms MB, BK, BN are fimilar and equal to the three oppofite parallelograms, and the three parallelograms EP, EX, ER fimilar and equal to the three oppofite ones; wherefore the folids BGML, EHPO are contained under equal numbers of fimilar planes; and fo [by 9. def. 11.] the folid BG ML is fimilar to the folid EHPO. But [by 33. 11.] fimilar folid parallelepipedons are in the triplicate ratio of their homologous fides: Therefore the folid BGMN to the folid EHPO is in the triplicate ratio of the homologous fide BC to the homologous fide EF. But [by 15. 5.] as the folid BGML is to the folid EH PO, fo is the pyramid ABCG to the pyramid DEFH; for the pyramid is a fixth part of that folid, fince the prifm, which is one half of the folid parallelepipedon is thrice the pyramid: Therefore the pyramid ABCG to the pyramid DEFH is in the triplicate ratio of B C to EF. Which was to be demonftrated.

Corollary. From hence it is manifeft, that fimilar pyramids having polygonous bafes, are to one another in the triplicate ratio of the homologous fides: For they being divided into pyramids having triangular bases, because [by 20. 6.] fimilar polygonous bafes are divided into equal numbers of fimilar triangles homologous to the wholes; it will be as one of the pyramids having a triangular base in one pyramid, is to one pyramid having a triangular base in

the

Book XII. the other, fo are all the pyramids having triangular bases in the one pyramid, to all thofe having triangular bafes in the other; that is, fo is one pyramid having the polygonous base to the other pyramid having the polygonous base. But one pyramid with a triangular bafe is to another pyramid with a triangular bafe in the triplicate ratio of their homologous fides: And therefore one pyramid having a polygonous bafe to a fimilar pyramid having fuch a base, is in the triplicate ratio of one of its homologous fides to

the other.

PROP. IX. THEO R.

The bafes of equal pyramids, having triangular bafes, are reciprocally proportional to their altitudes, and thofe triangular pyramids whose bases are reciprocally proportional to their altitudes, are equal to one another.

For let there be equal pyramids having the triangular bafes ABC, DE F, and vertexes the points G, H: I fay the bases of the pyramids ABC G, DEFH are reciprocally proportional to their altitudes; that is, as the base ABC is to the base DEF, fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG.

For complete the folid parallelepipedons BG ML, EHPO. Then because the pyramid AECG is equal to the pyramid DEFH, and the pyramid ABCG is the fixth part of the folid BGM L, and the pyramid DEFH the fixth part of the folid E HPO: The folid BGML [by 15. 5.] will be equal to the folid EHPO. But [by 34. 11.] the bafes of equal folid parallelepipedons are reciprocally proportional to their altitudes: Therefore as the bafe ɓ M is to the bafe fo is the altitude of

H

R

EP,

the folid EHPO to the

altitude of the folid BGML. But as the base BM is to the base E P, fo is the triangle ABC to the triangle DEF: VETherefore as the trian

gle

gle ABC is to the triangle DEF, fo is the altitude of the folid EHPO to the altitude of the folid BGM L. But the altitude of the folid EHPO is the fame as the altitude of the pyramid DE FH, and the altitude of the folid B G ML the fame as the altitude of the pyramid ABCG: There fore as the base ABC is to the base DEF, fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: Wherefore the bafes of the pyramids ABCG, DEFH are reciprocally proportional to their altitudes.

Now let the bafes of the pyramids ABCG, DEFH be reciprocally proportional to their altitudes, and let the bafe ABC be to the base DEF, as the altitude of the pyramid DEFH is to the altitude of the pyramid ABCG: I fay the pyramid ABC G is equal to the pyramid D E F H.

For the fame construction remaining, because as the bafe A B C is to the bafe D E F, fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG, And.. as the bafe ABC is to the bafe DEF, fo is the parallelo gram B M to the parallelogram EP; it will be as the parallelogram B M is to the parallelogram E P, fo is the altitude of the pyramid DEFH to the altitude of the pyramid. ABC G. But the altitude of the pyramid DEFH is the fame as the altitude of the folid parallelepipedon EHPO; and the altitude of the pyramid ABCG the fame as the altitude of the folid parallelepipedon BGML: Therefore as the bafe B M is to the bafe EP, fo is the altitude of the folid parallelepipedon EH PO to the altitude of the folid, parallelepipedon BGML. But thofe folid parallelepipedons. whofe bafes are reciprocally proportional to their altitudes, [by 34. 11.] are equal to one another. Therefore the folid parallelepipedon B G M L is equal to the folid paralle-, lepipedon EH PO. And the pyramid ABCG is a fixth part of the folid BG ML, and the pyramid DE F H is, alfo a fixth part of the folid EHPO; therefore the pyramid ABCG is equal to the pyramid DEF H.

Wherefore the bafes of equal pyramids having triangu→ lar bafes, are reciprocally proportional to their, altitudes; and thofe triangular pyramids, whofe bafes are reciprocally proportional to their altitudes, are equal to one another. Which was to be demonftrated.

Every cone is a third part of a cylinder which has the fame bafe and an equal altitude.

B

H

For let a cone have the fame bafe as a cylinder, viz. the circle A B C D, and an equal altitude: I fay the cone is a third part of the cylinder; that is, the cylinder is thrice the cone. For if the cylinder be not thrice the cone; it will be either greater than thrice the cone, or lefs. Firft let it be greater than thrice the cone; and describe a fquare A B C D in the circle ABCD: Then the square A B C D is greater than one half the circle A B C D. And upon the fquare A B C D erect a prifm of the fame altitude as the cylinder, which prifm will be greater than one half the cylinder; because if a square be described about the circle A B C D, the infcribed fquare will be one half the circumfcribed fquare; and there are erected upon those bafes folid parallelepipedons of the fame altitude, viz. the prisms themfelves; and fo the prisms are to one another as their bases; therefore the prifm erected upon the fquare A B C D is one half the prifm erected upon the fquare defcribed about the circle ABCD; and the cylinder is lefs than the prism erected upon the fquare described about the circle A BCD: Therefore the prifm defcribed upon the fquare ABCD of the fame altitude as the cylinder, is greater than one half that cylinder. Bifect the parts A B, BC, CD, DA of the circumference in the points, E, F, G, H, and join A E, E B, BF, FC, CG, GD, DH, HA. Then each of the triangles AEB, BFC, CG D, DHA is greater than one half the fegment of the circle A B C D wherein it ftands, as has been already proved [fee 2. 12.] Upon each of the triangles AEB, BFC, CG D, DHA erect a prifm of the same altitude as the cylinder: Then will each of these prisms be greater than one half the respective segment of the cylin der; because if thro' the points E, F, G, H parallels be drawn to A B, BC, CD, DA, and parallelograms at them

be

be completed, upon which folid parallelepipedons of the fame altitude as the cylinder are erected; each of thefe erected parallelepipedons will be double the prifms which are in the triangles A E B, BF C, CGD, DHA; and the fegments of the cylinder are lefs than the erected folid parallelepipedons; therefore the prifms in the triangles AEB, BFC, CG D, DHA are greater than one half the refpective fegments of the cylinder; and fo let the remaining parts of the circumference be bifected, right lines be joined, and upon each of the triangles erect prifms of the fame altitude as the cylinder, and do this always, till at laft [by 1. 10.] fome fegments of the cylinder remain being less than the excess of the cylinder above thrice the cône. Let fuch fegments be left, and let them be AE, FB, BF, FC, CG, GD, OH, HA. Then the remaining prifm, whose base is the polygon A E B FCGDH, and altitude the fame as that of the cylinder, is greater than thrice the cone. But [by cor. 7. 12.] the prifm whose bafe is the polygon AEBFCGDH, and altitude the fame as that of the cylinder, is thrice the pyramid whose base is the polygon A E BFCGDH, and vertex the fame as that of the cone; and therefore the pyramid whose base is the polygon AEBFCG DH, and vertex the fame as that of the cone, is greater than the cone whose base is the circle ABCD: but it is lefs too, because it is contained in it, which is impoffible: Therefore the cylinder is not greater than thrice the cone. I lay moreover, that the cylinder is not less than thrice the cone. For if poffible let the cy linder be less than thrice the cone; then inversely the cone will be greater than a third part of the cylinder. Defcribe the fquare ABCD in the circle ABCD; this fquare will be greater than one half the circle ABC D, and upon the fquare A B C D erect a pyramid, having the fame vertex as that of the cone; then the erected pyramid will be greater than one half the cone; because, as we have already demonftrated, if a square be defcribed about the circle, the fquare ABCD will be one half of it. And if folid parallelepipedons be erected upon thofe fquares, having the fame altitude with that of the cone, which are alfo called prifms; that erected upon the fquare ABCD will be one half of that erected upon the fquare defcribed about the

CG 2

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