they are contained under the fame Planes; and the Pyramid, whose Base is the Triangle ABD, and Vertex the Point C, has been proved to be a third Part of the Prism, whose Base is the Triangle ABC, and oppofite Base to that the Triangle DEF. Therefore alfo the Pyramid, whofe Bafe is the Triangle ABC, and Vertex the Point D, is a third Part of the Prism having the fame Bafe, viz. the Triangle ABC, and the oppofite Base the Triangle DEF; which was to be demonftrated.

Coroll. 1. It is manifeft from hence, that every Pyramid is a third Part of a Prism, having the fame Base and an equal Altitude; because if the Base of a Prifm, as alfo the oppofite Bafe, be of any other Figure, it may be divided into Prisms having triangular Bafes:

2. Prisms of the fame Altitude are to one another as their Bafes.

PROPOSITION VIII.

THEOREM.

Similar Pyramids, having triangular Bafes, are in a triplicate Proportion of their homologous Sides.

ET there be two Pyramids fimilar and alike fitu

and let their Vertices be the Points G, H. Í fay, the Pyramid ABCG to the Pyramid DEFH has a Proportion triplicate of that which BC has to EF.

*

21

For complete the folid Parallelepipedons BGML, EHPO; then because the Pyramid ABCG is fimilar to the Pyramid DEFH, the Angle ABC fhall be equal to the Angle DEF, the Angle GBC* Def. 9. 11. equal to the Angle HEF, and the Angle ABG equal to the Angle DEH. And AB is to DE as BC is to EF; and fo is BG to EH. Therefore because AB is to DE, as BC is to EF; and the Sides about the equal Angles are proportional, the Parallelogram BM fhall + be fimilar to the Parallelo- † 6. 6. gram EP. For the fame Reason, the Parallelogram BN is fimilar to the Parallelogram ER, and the Pa

rallelogram

33. 11.

† 15.5.

rallelogram BK to the Parallelogram EX. Therefore three Parallelograms B M, KB, BN, are fimilar to three Parallelograms EP, EX, ER; but the three MB, BK, BN, are equal and fimilar to the three opposite ones; as alfo the three EP, EX, ER. Therefore the Solids BGML, EHPO, are contained under equal Numbers of fimilar and equal Planes; and confequently, the Solid BGML is fimilar to the Solid EHPO. But fimilar folid Parallelepipedons are * to each other in a triplicate Proportion of their homologous Sides. Therefore the Solid BGML to the Solid EHPO, has a Proportion triplicate of that which the homologous Side BC has to the homologous Side EF. But as the Solid BGML is to the Solid EHPO, fo is † the Pyramid ABCG to the Pyramid DEFH; for the Pyramid is the one fixth Part of that Solid, fince the Prism, which is the half of the Solid Parallelepipedon is triple of the Pyramid. Wherefore the Pyramid ABCG to the Pyramid DEF H, fhall have a triplicate Proportion to that which BC has to EF; which was to be demonftrated.

Coroll. From hence it is manifeft, that fimilar Pyramids having polygonous Bafes, are to one another in a triplicate Proportion of their homologous Sides. For if they be divided into, Pyramids having triangular Bafes; because their fimilar polygonous Bafes are divided into fimilar Triangles equal in Number, and homologous to the Wholes, it fhall be as one Pyramid having a triangular Base in one of the Pyramids, is to a Pyramid having a triangular Bafe in the other Pyramid, fo are all the Pyramids having triangular Bases in one Pyramid, to all the Pyramids having triangular Bafes in the other Pyramid; that is, fo is one of the Pyramids having the polygonous Base, to the other; bút a Pyramid having a triangular Bafe to a Pyramid having a triangular Base, is in a triplicate Proportion of the homologous Sides. Therefore one Pyramid having a polygonous Bafe to another Pyramid having a fimilar Bafe, is in a triplicate Proportion of their homologous Sides.

PRO

PROPOSITION IX.

THE ORE M..

The Bafes and Altitudes of equal Pyramids, having triangular Bafes, are reciprocally proportional; and thofe Pyramids, baving triangular Bafes, whofe Bafes and Altitudes are reciprocally proportional, are equal.

ET there be equal Pyramids, having the triangu lar Bafes ABC, DEF, and Vertices the Points G, H. I fay, the Bafes and Altitudes of the Pyramids ABCG, DEFH, are reciprocally proportional, that is, as the Bafe ABC is to the Bafe DEF, fo is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG.

But

For complete the folid Parallelepipedons BGML, EHPO. Then because the Pyramid ABCG is equal to the Pyramid DEFH, and the Solid BGML is fextuple, the Pyramid ABCG, and the Solid EHPO fextuple of the Solid DEF H, the Solid BGML fhall be equal to the Solid EHPO. * But the Bafes* 15. 5. and Altitudes of equal folid Parallelepipedons are reciprocally proportional. Therefore, as the Bafe B M is to the Bafe EP, fo is + the Altitude of the Solid † 34. 11. EHPO to the Altitude of the Solid B GML. as the Bafe BM is to the Base EP, fo is † the Triangle ABC to the Triangle DEF. Therefore, as the Triangle ABC is to the Triangle DEF, fo is the Altitude of the Solid EHPO to the Altitude of the Solid BGML. But the Altitude of the Solid EHPO is the fame as the Altitude of the Pyramid DEFH; and the Altitude of the Solid B GML the fame as the Altitude of the Pyramid ABCG. Therefore, as the Base ABC is to the Bafe DEF, fo is the Altitude of the Pyramid DEFH to the Altitude of the Pyramid A BCG. Wherefore the Bafes and Altitudes of the equal Pyramids ABCG, DEFH, are reciprocally proportional; and if the Bafes and Altitudes of the Pyramids ABCG, DEF H, are reciprocally proportional, that is, if the Bafe ABC to the Bafe DEF, be as the Altitude of the Pyramid DEFH

† 34. II.

DEFH to the Altitude of the Pyramid ABCG. I fay, the Pyramid ABCG is equal to the Pyramid DEFH: For, the fame Conftruction remaining, because the Base ABC to the Base DEF, is as the Altitude of the Pyramid DEFH to the Altitude of the Pyramid ABCG, and as the Bafe ABC is to the Base DEF, fo is the Parallelogram BM to the Parallelogram EP; the Parallelogram BM to the Parallelogram EP fhall be alfo as the Altitude of the Pyramid DEFH is to the Altitude of the Pyramid ABCG. But as 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 the Base BM to the Base EP will be as the Altitude of the folid Parallelepipedon EHPO to the Altitude of the folid Parallelepipedon BGML. But thofe folid Parallelepipedons, whofe Bafes and Altitudes are reciprocally proportional, are equal to each other. Therefore the folid Parallelepipedon BGML is equal to the folid Parallelepipedon EHPO; and the Pyramid ABCG is a fixth Part of the Solid BG ML. And in like manner the Pyramid DEFH is a fixth Part of the Solid EHPO. Therefore the Pyramid ABCG is equal to the Pyramid DEFH. Wherefore the Bafes and Altitudes of équal Pyramids, having triangular Bafes, are reciprocally proportional; and thofe Pyramids, having triangular Bafes, whofe Bafes and Altitudes are reciprocally proportional, are equal; which was to be demonftrated.

PROPOSITION X.

THEOREM.

Every Cone is a third Part of a Cylinder, having the fame Bafe, and an equal Altitude.

L

ET a Cone have the fame Bafe as a Cylinder, viz. the Circle ABCD, and an Altitude equal to it. I fay the Cone is a third Part of the Cylinder; that is, the Cylinder is triple to the Cone.

For

"

For if the Cylinder be not triple to the Cone, it fhall be greater or lefs than triple thereof. First, let it be greater than triple to the Cone, and let the Square ABCD be described in the Circle ABCD, then the Square ABCD is greater than one half of the Circle ABCD. Now let a Prifm be erected upon the Square ABCD, having the fame Altitude as the Cylinder, and this Prifm will be greater than one half of the Cylinder; because, if a Square be circumfcribed about the Circle ABCD, 'the infcrib'd Square will be one half of the circumfcribed Square; and if a Prism be erected upon the circumfcrib'd Square of the fame Altitude as the Cylinder, fince Prisms are * to one * 2 Cor. another as their Bafes, the Prifm erected upon the of this. Square ABCD is one half of the Prism erected upon the Square described about the Circle ABCD. But the Cylinder is leffer than the Prifm erected on the Square defcribed about the Circle ABCD. Therefore the Prism erected on the Square ABCD, having the fame Height as the Cylinder, is greater than one half of the Cylinder. Let the Circumferences A B, BC, CD, DA, be bifected in the Points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA. Then each of the Triangles AEB, BF C, CGD, DHA, is greater than the half of each of the Segments in which they ftand. Let Prisms be erected from each of the Triangles AEB, BF C, CGD, DHA, of the fame Altitude as the Cylinder, then every one of these Prisms erected is greater than its correfpondent Segment of the Cylinder. For because, if Parallels be drawn through the Points E, F, G, H, to AB, BC, CD, DA, and Parallelograms be compleated on the faid AB, BC, CD, DA, on which are erected folid Parallelepipedons of the fame Altitude as the Cylinder; then each of those Prisms that are on the Triangles AE B, BF C, CGD, DHA, are Halves + of each of the folid Parallelepipedons; and the Segments of the Cylinder are lefs than the erected folid Parallelepipedons; and confequently the Prisms that are on the Triangles AEB, BFC, CGD, DHA, are greater than the Halves of the Segments of the Cylinder; and fo bifecting the other Circumferences, joining Right Lines, and on every one of the Triangles erecting Prifms of the fame Height as the Cylinder;

7.

+ This follows from 2

of this.