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altitude of the pyramid ABCG: therefore, as the base ABC to Book XII.
the base DEF, so is the altitude of the pyramid DEFH to the
altitude of the pyramid ABCG: wherefore the bases and alti-
tudes of the pyramids ABCG, DEFH are reciprocally pro-
portional.

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 alti-
tude 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 parallelo. gram 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 same with the altitude of the solid parallelopiped EHPO; and the altitude of the pyramid ABCG is the same with the altitude of the solid parallelopiped BGML:as, therefore, the base BM to the base EP, so is the altitude of the solid parallelopiped EHPO to the altitude of the solid parallelopiped BGML. But solid parallelopipeds having their bases and altitudes recipro.' cally proportional, are equal to one another. Therefore the b 34. 11. solid parallelopiped BGML is equal to the solid parallelopiped EHPO. And the pyramid ABCG is the sixth part of the solid BGML, and the pyramid DEFH is the sixth part of the solid EHPO. Therefore the pyramid ABCG is equal to the pyramtid DEFH. Therefore the bases, &c. Q. E. D.

i

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 same base with a cylinder, viz. the circle ABCD, and the såme 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 must either be greater than 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 greater than the half of the circle ABCD*:

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As was shown in prop. 2. of this book

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Book XI1 Upon the square ABCD erect a prism of the same altitude

with the cylindçr; this prism is greater than half of the cy-
linder; because if a square be described about the circle, and
å prism erected upon the square, of the same altitude with the
cylinder, the inscribed square is half of that circumscribed;
and
upon
these square

bases are erected solid parallelopipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is the half of the prism upon the square

described about the circle: because they are to one another 32.11.

as their basesa : and the cylinder is less than the prisın upon the

square described 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 cir-
cumferences 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
stands, as was sliown in prop. 2. of

A
this book. Erect prisms upon each
of these triangles of the same altitude

H н
with the cylinder; each of these
prisms is greater than half of the seg.
ment of the cylinder in which it is; B

D because if, tlırough the points E, F, G, H, parallels be drawn to AB, BC, CD, DA, and parallelograms be T completed upon the same AB, BC, CD, DA, and solid parallelopípeds

с be erected upon the parallelograms; the prisms upon the

triangles AEB, BFC, CGD, DHA are the halves of the solid B2 Cor. 7. parallelopipedsb. And the segments of the cylinder which 12.

are upon the segments of the circle cut off by AB, BC,CD,DA, are less than the sofid parallelopipeds which contain them. Therefore the prisms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the segments 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 division to the extremities of the circumferences, and upon the triangles thus made, prisms be erected of the same

altitude with the cylinder, and so on, there must at length rec Lem.

main some segments of the cylinder which together are less than the excess of the cylinder above the triple of the cone. Iet them be those upon the segments of the circle AE, EB, BF,

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D

FC, CG, GD, DH, HA. Therefore the rest of the cylin. Book XII. der, that is, the prism of which the base is the polygon AEBFCGDH, and of which the altitude is the same with that of the cylinder, is greater than the triple of the cone; but this prism is tripled of the pyramid upon the same base, of which d 1. Cor. the vertex is the same with the vertex of the cone; therefore 12. the 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 come 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

H square ABCD is the half of it; and if, upon these squares, there be erect

es

A А ed solid parallelopipeds of the same altitude with the cone, which are also prisms, the prism upon the square ABCD shall be the half of that whichE

G is upon the square described about the circle ; for they are to one another as their basese; as are also the

e 32. 11.

B В third parts of them: therefore the pyramid, the base of which is the

F 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 same vertex with the cone, is greater than the half of the cone. Bisect the circumferences AB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA: therefore each of the triangles AEB, BFC) CGD, DHA is greater than half of the segment of the circle in which it is : upon each of these triangles erect pyramids having the same vertex with the cone. Therefore each of these pyramids is greater than the half of the segment of the cone 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 their vertices the same with that of the cone, and so on, there must at-length remain some segments of the cone, which together shall be less than the excess of the cone above

H

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"Book XII. the third part of the cylinder. Let these be the segments opend upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the

rest of the cone, that is, the pyramid,
'of which the base is the polygon
AEBFCGDH, and of which the ver- A A

D
tex is the same with that of the cone,
is greater than the third part of the
cylindor. But this pyramid is the

E third part of the prism’upon the same

G base AEBFCGDH, and of the same altitude with the cylinder. Therefore this prism is greater than the cylin.

B der of which the base is the circle

F ABCD. But it is also less; for it is contained within the cylinder; which is impossible. Therefore the cylinder is not less than the triple of the cone. And it has been demonstrated that neither is it greater than the triple. Therefore the cylinder is triple of the cone, or the cone is the third part of the cylinder. Wherefore every cone, &c. Q. E. D.

PROP. XI. THEOR.

See Note,

CONES and cylinders of the same altitude, are to one another as their bases.

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Let the cones and cylinders, of which the bases are the circles ABCD, EFGH, and the axes KL, MN, and AC, EG the diameters of their bases, be of the same altitude. As the circle ABCD to the circle EFGH, so is the cone AL to the conc EN.

If it be not so, 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 solid less than EN, viz. to the solid X; and let Z be the solid which is equal to the excess of the cone EN above the solid X; therefore the cone EN is equal to the solids X, Z together. In the circle EFGH describe the square EFGH, therefore this square is greatex than the half of the circle: upon the square EFGH erećt a Pyramid of the same altitude with the cone; this pyramid is greater than half &f the cone. For, if a square be described about the circle, and a pyramid be crected upon it, having the same rer. tex with the cone", the pyramid inscribed in the cone is half Book xft. of the pyramid circumscribed about it, because they are to one another as their bases a : but the cone is less than the circum-a 6. 12 scribed pyramid; therefore the pyramid of which the base is the square EFGH, and its vertex the same with that of the cone, is greater than halfof 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

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the segment of the circle 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 same vertex with the cone, and so on, there must at length reinain some segments of the cone which are together less b than the solid Z: let these be the segments upon EO, OF, 6 Lem.

* Vertex is put in place of altitude which is in the Greek, 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.

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