BGML is the fame with the altitude of the pyramid ABCG. there- Bock Xil. fore, as the bafe ABC to the bafe 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 bafes and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz. the bafe 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 conftruction being made, because as the bafe ABC to the base DEF, fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG; and as the bafe ABC to the bafe 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 bafe BM to the bafe EP, fo is the altitude of the folid parallelepiped EHPO to the altitude of the folid parallelepiped BGML. but folid parallelepipeds having their bales and altitudes reciprocally proportional, are equal b to one another. there- b. 34. 11. fore the folid parallelepiped BGML is equal to the folid parallelepi- . ped EHPO. and the pyramid ABCG is the fixth part of the fold BGML, and the pyramid DEFH the fixth part of the folid EHPO. therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bafes, &c. Q. E. D.

PROP. X. THEOR.

EVERY 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 bafe with a cylinder, viz. the circ'e 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 great.r than the triple, or lefs than it. First, Let it be greater than the triple; and defcribe the fquare ABCD in the circle; this fquare is R 3

greater

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 prifm is greater than half of the cylinder; because if a fquare be defcribed about the circle, and a prifm erected upon the fquare, of the fame altitude with the cylinder, the infcribed fquare is half of that circumfcribed; and upon thefe fquare bases are erected folid parallelepipeds, viz. the prifms, of the fame altitude; therefore the prifm upon the fquare ABCD is the half of the prifm upon the fquare defuibed about the circle; becaufe they are to one another as their 2. 32. 11, bafes. and the cylinder is lefs than the prifm upon the fquare deforibed about the circle ABCD. therefore the prifm upon the fquare 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 flands, as was fhewn in Prop. 2. of this Book. Erect prifms upon E cach of thefe triangles of the fame altitude with the cylinder; each of these prifms is greater than half of the fegment of the cylinder in which it is; be

B

caufe if through the points E, F, G, HF
parallels be drawn to AB, BC, CD,

DA, and parallelograms be completed

A

H

D

G

C

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

7.12.

the fegments of the cylinder which are upon the fegments of the circle cut off by AB, BC, CD, DA, are lefs than the folid paralleleFipeds which contain them. therefore the prifms upon the triangles AEB, BFC, CGD, DIIA, 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 straight 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 must at length remain fome 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 fgments of the circle AE, EB, BF, FC, CG, GD, DH, HA. there

As was flown in l'rop. 2. of this Book.

fore

E

A

H

D

G

7.12.

fore the rest of the cylinder, that is the prism of which the bafe is Book XII. the polygon AEBFCGDH, and of which the altitude is the fame with that of the cylinder, is greater than the triple of the cone. but this prifm is triple d of the pyramid upon the fame bafe, of which d. 1. Cor. the vertex is the fame with the vertex of the cone; therefore the pyramid upon the bafe AEBFCGDH, having the fame vertex with the cone, is greater than the cone, of which the bafe is the circle ABCD. but it is alfo lefs, for the pyramid is contained within the cone; which is impoffible. Nor can the cylinder be lefs than the triple of the cone. let it be lefs if poffible. therefore, inversely, the cone is greater than the third part of the cylinder. In the circle ABCD describe a fquare, this fquare is greater than the half of the circle. and upon the square ABCD erect a pyramid having the fame vertex with the cone; this pyramid is greater than the half of the cone; because, as was before demonftrated, if a fquare be defcribed about the circle, the fquare ABCD is the half of it; and if upon thefe fquares there be erected folid parallelepipeds of the fame altitude with the cone, which are alfo prifms, the prifm upon the fquare ABCD fhall be the half of that which is upon the fquare defcribed about the circle; for they are to one another as their bafes; as are alfo the third parts of them. therefore the pyramid the base of which is the square ABCD is half of the pyramid upon the fquare defcribed about the circle. but this laft pyramid is greater than the cone which it contains; therefore the pyramid upon the fquare ABCD having the fame vertex with the cone, is greater than the half of the cone. Bifect the circumferences AB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG, CD, DH, HA. therefore each of the triangles AEB, BFC, CGD, DHA is greater than half of the fcgment of the circle in which it is. upon each of thefe triangics erect pyramids having the fame vertex with the cone. therefore each of thefe pyramids is greater than the half of the fegment of the cone in which it is, as before was demonftrated of the prifms and fogments of the cylinder. and thus dividing each of the circumferences into two equal parts, and joining the points of divifion and their extremities by ftraight lines, and upon the triangles erecting pyramids

R 4

B

F

having

a. 32. 11.

E

A

H

D

G

Book XII. having their vertices the fame with that of the cone, and so on, there muft at length remain fome fegments of the cone which together fhall be less than the excess of the cone above the third part of the cylinder. let thefe be the fegments upon AE, EB, BF, FC,CG, GD, DH, HA. therefore the reft of the cone, that is the pyramid, of which' the bafe is the polygon AEBFCGDH, and of which the vertex is the fame with that of the cone, is greater than the third part of the cylinder. but this pyramid is the third part of the priẩm upon the fame bafe AEBFCGDH, and of the fame altitude with the cylinder. therefore this prifm is greater than the cylinder of which the bafe is the circle ABCD. but it is alfo lefs, for it is contained within the cylinder; which is impoffible. therefore the cylinder is not lefs than the triple of the cone. and it has been demonftrated that neither is it greater than the triple. therefore the cylinder is triple of the cone, or, the cone is the third Fart of the cylinder. Wherefore every cone, &c. Q. E. D.

See N.

B

PROP. XI. THEOR.

F

CONES and cylinders of the fame altitude, are to one

another as their bafes.

Let the cones and cylinders, of which the bafes are the circles ABCD, EFGH, and the axes KL, MN, and AC, EG the diameters of their bafes, be of the fame altitude. as the circle ABCD to the circle EFGH, fo is the cone AL to the cone EN.

If it be not fo, let the circle ABCD be to the circle EFGH, as the cone AL to fome folid either lefs than the cone EN, or greater than it. First, let it be to a folid lefs than EN, viz. to the folid X; and let Z be the folid which is equal to the excess of the cone EN above the folid X; therefore the cone EN is equal to the folids X, Z together. in the circle EFCH defcribe the fquare EFGH, therefore this fquare is greater than the half of the circle. upon the fquare EFGH erect a pyramid of the fame altitude with the cone; this pyramid is greater than half of the cone. for if a fquare be defcribed about the circle, and a pyramid be erected upon it, having the

fame

fame vertex with the cone †, the pyramid inscribed in the cone is half Book XII. of the pyramid circumfcribed about it, because they are to one another as their bases. but the cone is less than the circumfcribed py- a. 6. 12. ramid; therefore the pyramid of which the bafe is the square EFGH, and its vertex the fame with that of the cone, is greater than half of the cone. divide the circumferences EF, FG, GH, HE, each into two equal parts in the points O, P, R, S, and join EO, OF, FP, PG, GR, RH, HS, SE. therefore each of the triangles EOF, FPG, GRH, HSE is greater than half of the fegment of the circle

in which it is. upon each of these triangles erect a pyramid having the fame vertex with the cone; each of thefe pyramids is greater than the half of the fegment of the cone in which it is. and thus dividing each of thefe circumferences into two equal parts, and from the points of divifion drawing ftraight lines to the extremities of the circumferences, and upon each of the triangles thus made erecting pyramids having the fame vertex with the cone, and fo on, there muft at length remain fome fegments of the cone which are together lefsb b. Lemma. than the folid Z. let these be the fegments upon EO, OF, FP, PG,

↑ Vertex is put in place of altitude which is in the Greck, because the pyramid, in what follows, is fuppofed to be circumfcribed about the cone, and fo must have the fame vertex, and the fame change is made in fome places following.

GR,