Book XII. bafe of which is the triangle ABC, and to which DEF is the op

pofite triangle; therefore the pyramid of which the bafe is the triangle ABC, and vertex the point D, is the third part of the prism which has the fame base, viz. the triangle ABC, and DEF is the oppofite triangle. Q. E. D.

COR. 1. From this it is manifeft, that every pyramid is the third part of a prism which has the fame base, and is of an equal altitude with it; for if the bafe of the prifm be any other figure than a triangle, it may be divided into prisms having triangular bases.

COR. 2. Prisms of equal altitudes are to one another as their bafes; because the pyramids upon the fame bases, and of the fame c. 6. 12. altitude, are to one another as their bases.

SIMI

IMILAR pyramids having triangular bases, are one to another in the triplicate ratio of that of their homologous fides.

Let the pyramids having the triangles ABC, DEF for their bases, and the points G, H for their vertices, be fimilar and finfilarly fituated. the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which the fide BC has to the homologous fide EF.

Complete the parallelograms ABCM, GBCN, ABGK, and the folid parallelepiped BGML contained by these planes and those op

pofite to them. and in like manner complete the folid parallelepiped EHPO contained by the three parallelograms DEFP, HEFR,

2. 11. Def. DEHX, and those opposite to them. and because the pyramid ABCG is fimilar to the pyramid DEFH, the angle ABC is equal a

II.

b. 1. Def. 6.

to the angle DEF, and the angle GBC to the angle HEF, and ABG Book XII. to DEH. and AB is b to BC, as DE to EF; that is, the fides about the equal angles are proportionals; wherefore the parallelogram BM is fimilar to EP. for the fame reason, the parallelogram BN is fimilar to ER, and BK to EX. therefore the three parallelograms BM, BN, BK are fimilar to the three EP, ER, EX. but the three BM, BN, BK are equal and similar to the three which are oppofite to them, and the three EP, ER, EX equal and fimilar to the three oppofite to them. wherefore the folids BGML, EHPO are contained by the fame number of fimilar planes; and their folid angles are equal d; and therefore the folid BGML is fimilar to d. B. 11. the folid EHPO. but fimilar folid parallelepipeds have the triplicate

C. 24. II,

e ratio of that which their homologous fides have. therefore the e. 33. II. folid BGML has to the folid EHPO the triplicate ratio of that which the fide BC has to the homologous fide EF. but as the folid BGML is to the folid EHPO, fo is f the pyramid ABCG to the f. 15. 5. pyramid DEFH; because the pyramids are the fixth part of the folids, fince the prifm, which is the half of the folid parallelepiped, g. 28. 11. is triple h of the pyramid. Wherefore likewise the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous fide EF. Q. E. D.

h.

7. 12.

COR. From this it is evident, that fimilar pyramids which have See N. multangular bases, are likewise to one another in the triplicate ratio of their homologous fides. for, they may be divided into fimilar pyramids having triangular bafes, because the fimilar polygons which are their bafes may be divided into the fame number of fimilar triangles homologous to the whole polygons; therefore as one of the triangular pyramids in the first multangular pyramid is to one of the triangular pyramids in the other, fò are all the triangular pyramids in the first to all the triangular pyramids in the other; that is, fo is the firft multangular pyramid to the other. but one triangular pyramid is to its fimilar triangular pyramid, in the triplicate ratio of their homologous fides; and therefore the first multangular pyramid has to the other, the triplicate ratio of that which one of the fides of the firft has to the homologous fide of the other.

R 2

Book XII.

PROP. IX.

THEOR.

THE

HE bafes and altitudes of equal pyramids having triangular bases are reciprocally proportional. and triangular pyramids of which the bases and altitudes are reciprocally proportional, are equal to one another.

Let the pyramids of which the triangles ABC, DEF are the bafes, and which have their vertices in the points G, H be equal to one another. the bafes and altitudes of the pyramids ABCG, DEFH are reciprocally proportional, viz. the base ABC is to the bafe DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG.

Complete the parallelograms AC, AG, GC, DF, DH, HF; and the folid parallelepipeds BGML, EHPO contained by these planes

and those oppofite to them. and because the pyramid ABCG is equal to the pyramid DEFH, and that the folid BGML is fextuple of the pyramid ABCG, and the solid EHPO sextuple of the pya. 1. Ax. 5. ramid DEFH; therefore the folid BGML is equal a to the folid EHPO. but the bafes and altitudes of equal folid parallelepipeds are b. 34. 11. reciprocally proportional; therefore as the bafe BM to the base

EP, fo is the altitude of the folid EHPO to the altitude of the folid c. 15. 5. BGML. but as the base BM to the base EP, fo is the triangle ABC to the triangle DEF; therefore as the triangle ABC to the triangle DEF, so is the altitude of the folid EHPO to the altitude of the folid BGML. but the altitude of the folid EHPO is the fame with the altitude of the pyramid DEFH; and the altitude of

the folid BGML is the fame with the altitude of the pyramid Book XII. ABCG. therefore, as the base ABC to the base DEF, fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG. wherefore the bases and altitudes of the pyramids ABCG, DEFH are reciprocally proportional.

Again, Let the bases and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz. the base ABC to the bafe DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG. the pyramid ABCG is equal to the pyramid DEFH.

The fame construction being made, because as the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG; and as the base ABC to the base DEF, fo is the parallelogram BM to the parallelogram EP; therefore the parallelogram BM is to EP, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG. but the altitude of the pyramid DEFH is the fame with the altitude of the folid parallelepiped EHPO; and the altitude of the pyramid ABCG is the fame with the altitude of the folid parallelepiped BGML. as, therefore, the base BM to the base EP, fo is the altitude of the solid parallelepiped EHPO to the altitude of the folid parallelepiped BGML. but folid parallelepipeds having their bases and altitudes reciprocally proportional, are equal to one another. therefore the b. 34. 11. folid parallelepiped BGML is equal to the folid parallelepiped EHPO. and the pyramid ABCG is the fixth part of the solid BGML, and the pyramid DEFH the fixth part of the folid EHPO. therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bases, &c. Q. E. D.

E

b

PROP. X. THEOR.

OVERY cone is the third part of a cylinder which has the fame bafe, and is of an equal altitude with it.

Let a cone have the fame base with a cylinder, viz. the circle ABCD, and the fame altitude. the cone is the third part of the cylinder; that is, the cylinder is triple of the cone.

If the cylinder be not triple of the cone, it muft either be greaterthan the triple, or less than it. First, Let it be greater than the triple; and describe the square ABCD in the circle; this square is

Book XII. greater than the half of the circle ABCD +. upon the fquare ABCD erect a prifm of the fame altitude with the cylinder; this prism is greater than half of the cylinder; because if a square be described about the circle, and a prism erected upon the square, of the same altitude with the cylinder, the inscribed fquare is half of that cir-. cumfcribed; and upon these fquare bases are erected folid parallelepipeds, viz. the prisms, of the fame altitude; therefore the prism upon the fquare ABCD is the half of the prism upon the fquare described about the circle; because they are to one another as their 32. 11. bafes . and the cylinder is lefs than the prifm upon the fquare described about the circle ABCD. therefore the prism upon the square ABCD of the fame altitude with the cylinder, is greater than half of the cylinder. Bifect the circumferences AB, BC, CD, DA in the points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA, then, each of the triangles AEB, BFC, CGD, DHA is greater than the half of the fegment of the circle in which it ftands, as was fhewn in Prop. 2. of this Book. Erect prifms upon each of these triangles of the fame altitude with the cylinder; each of these prisms. is greater than half of the fegment of B the cylinder in which it is; because if thro' the points E, F, G, H parallels be F drawn to AB, BC, CD, DA, and parallelograms be completed upon the same

E

A

H

AB, BC, CD, DA, and folid parallelepipeds be erected upon the parallelograms; the prifms upon the triangles AEB, BFC, CGD, b. 2. Cor. DHA are the halves of the folid parallelepipeds . and the feg

7. 12.

ments of the cylinder which are upon the fegments of the circle cut off by AB, BC, CD, DA, are lefs than the folid parallelepipeds which contain them. therefore the prifms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the fegments of the cylinder in which they are. therefore if each of the circumferences be divided into two equal parts, and ftraight lines be drawn from the points of divifion to the extremities of the circumferences, and upon the triangles thus made prifms be erected of the fame altitude with the cylinder, and fo on, there muft at length remain fome c. Lemma. fegments of the cylinder which together are lefs than the excess of the cylinder above the triple of the cone. let them be those upon the As was fhewn in Prop. 2. of this Book.