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BGML is the fame with the altitude of the pyramid ABCG, there- Back Xil. fore, 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 base 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 same 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, so is the parallelogram BM to the parallelogram EP; there

fore 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 same with the altitude of the folid parallelepiped EHP'O; and the altitude of the pyramid ABCG is the same with the altitude of the soäid parallelepiped BGML. as, therefore, the base BM to the base EP, fo is the altitude of the folid parallelepiped EHPO to the altitude of the solid parallclepiped BGML. but folid parallelepipeds having their baies and altitudes reciprocally proportional, are equal b to one another. there- b. 34.11. fore the solid parallelepiped BGML is equal to the folid parallelepiped EHPO. and the pyramid ARCG is the fixth part of the socji! BGML, and the pyrami] DEFH the sixth part of the foli EHPO. therefore the pyramid ABCG is equal to the pyramid DEFH, Therefore the bases, &c. Q. E. D.

PROP. X. THEOR.

EVERY cone is the third part of a cylinder which has

the same base, and is of an equal altitude with it.

Let a cone have the fame base with a cylinder, viz. the circ'e ABCD, and the same al:itude. the cone is the third part of the cy. linder; that is, the cylinder is triple of the cone.

If the cylinder be not triple of the cone, it must either be great than the triple, or less than it. First, Let it be greater than tie triple; and describe the square ABCD in the circle; this faraleis

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Book XII greater than the half of the circle ABCD +. upon the square ABCD

erect a prism of the same 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 square is half of that circunscribed; and upon these square bases are erected solid parallelepipeds, viz. the prisms, of the same altitude; therefore the prism upon the square ABCD is the half of the prism upon the square de

fuibed about the circle; because they are to one another as their 2. 32. 11, balese. and the cylinder is less than the prism upon the square de

fcribed about the circle ABCD, therefore the prism upon the square
ABCD of the same altitude with the cylinder, is greater than half
of the cylinder. Bisect 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 segment of the circle in
which it Nands, as was shewn in Prop.

A.
2. of this Book. Erect prisms upon E

H cach of these triangles of the fame altitude with the cylinder ; each of these prisms is greater than half of the feg

B

D
ment of the cylinder in which it is; be-
cause if through the points E, F, G, H

F
parallels be drawn to AB, BC, CD,
DA, and parallelograms be completed

С
upon the fame AB, BC, CD, DA, and solid parallelepipeds be e-

rected upon the parallelograms; the prisms upon the triangles AEB, b. 2. Cor. BFC, CGD, DHA are the halves of the solid parallelepipeds b. ar.d

the segments of the cylinder which are upon the segments of the circle cut off by AB, BC, CD, DA, are less than the solid parallele. fipeds which contain them. therefore the prisms upon the triangles AER, BFC, CGD, DIA, are greater than half of the segments of the cylinder in which they are. therefore if each of the circumfcrcnces be divided into two cqual parts, and straight lines be drawn from the points of division to the extremities of the circumferences, and upon the triangles thus made prisms be erected of the fame alti

tiide with the cylinder, and so on, there must at length remain some Lomma. segments of the cylinder which together are less than the excess of

the cylinder above the triple of the cone. let them be those upon sigments of the circle AE, EB, BF, FC, CG, GD, DH, HA. there+ As itas slicwn in l'rop. 2.

fore

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of this Book.

fore the rest of the cylinder, that is the prism of which the base is Book XII. the polygon AEBFCGDH, and of which the altitude is the same m with that of the cylinder, is greater than the triple of the cone. but this prism is triple d of the pyramid upon the same base, of which d.s. Cor. the vertex is the same with the vertex of the cone; therefore the

7.1%. pyramid upon the base AEBFCGDH, having the same vertex with the cone, is greater than the cone, of which the base is the circle ABCD. but it is also less, for the pyramid is contained within the cone; which is impossible. Nor can the cylinder be less than the triple of the cone. let it be less if possible. therefore, inversely, the cone is greater than the third part of the cylinder. In the circle ABCD describe a square, this square is greater than the half of the circle. and upon the square ABCD erect a pyramid having the same vertex with the cone; this pyramid is greater than the half of the cone; because, as was before demonstrated, if a square be described about the circle, the square ABCD is

H the half of it; and if upon these squares there be erected folid paralle

А lepipeds of the fame altitude with the cone, which are also prisins, the

G prism upon the square ABCD shall be the half of that which is upon the square described about the circle; for B they are to one another as their ba

F fes*; as are also the third parts of thein, therefore the pyramid the base of which is the square ABCD is half of the pyramid upon the square described about the circle. but this last pyramid is greater than the cone which it contains ; therefore the pyramid upon the square ABCD having the fame vertex with the cone, is greater than the half of the cone. Bilect the circumferences AB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA. thereforç each of the triangles AEB, BFC, CGD, DHA is greater than half of the fogment of the circle in which it is. upon each of these triangics erect pyramids having the same vertex with the cone. therefore each of these pyramids is greater than the half of the segment of the cane in which it is, as before was demonstrated of the prisms and segments of the cylinder, and thus dividing each of the circumferences into two equal parts, and joining the points of division and their extremities by straight lines, and upon the triangles erecting pyramids

having

E

a. 32. 11.

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Book XII. having their vertices the same with that of the cone, and so on,

there must at length remain some segonents of the cone which toge-
ther shall be less than the excess of the cone above the third part of the
cylinder. let these be the segments upon AE, EB, BF, FC,CG, GD,
DH, HA. therefore the rest of the

H
cone, that is the pyramid, of which
the base is the polygon AEBFCGDH,

D
and of which the vertex is the same
with that of the cone, is greater than

E

G the third part of the cylinder, but this pyramid is the third part of the pri in upon the fame base AEBFCGDH, B and of the fame altitude with the cy

F linder, therefore this prism is greater than the cylinder of which the base is the circle ABCD, but it is also less, for it is contained within the cylinder ; which is imposfible. therefore the cylinder is not less 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.

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See N.

CONES

ONES and cylinders of the same altitude, are to one

another as their bases.

Lct the cones and cylinders, of which the bases are the circles ABCD, EFGII, and the axes KL, MN, and AC, EG the diameters of their bales, be of the same altitude. as the circle ABCD to the circle EFGHI, 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 some solid either less than the cone EN, or greater than it. First, let it be to a folid less 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 solids X, Z together. in the circle EFCH describe the square EFGH, therefore this square is greater than the half of the circle. upon the square EFGl crcit a pyramid of the fame altitude with the cone ; this pyramid is greater than half of the cone. for if a square be defcribed about the circle, and a prramid be erected upon it, having the

fame

fame vertex with the cone t, the pyramid inscribed in the cone is half Book XII. of the pyramid circumscribed about it, because they are to one ano ther as their bases, but the cone is less than the circumscribed py- a. 6. 12. ramid ; therefore the pyramid of which the base is the square EFGH, and its vertex the same 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. thercore each of the triangles EOF, FPG, GRH, HSE is greater than half of the fegment of the circle L

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D

H
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S

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in which it is. upon each of these triangles erect a pyramid having the same vertex with the cone; each of these pyramids is greater than the half of the segment of the cone in which it is. and thus dividing each of these circumferences into two equal parts, and from the points of division drawing straight 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 so on, there must at length remain some fegments of the cone which are together lessb b. Lemma. than the solid Z. let these be the segments upon EO, OF, FP, PG,

+ Vertex is put in place of altitude which is in the Greck, because the pyramid, in what follows, is supposed to be circumscribed about the cone, and so must have the same vertex, and the same change is made in some places following.

GR,

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