261

the folid BGML is the fame with the altitude of the pyramid Book XII. ABCG. therefore, 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 bafes 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 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 conftruction being made, because as the base ABC to the bafe 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 bases and altitudes reciprocally proportional, are equal to one another. therefore the b. 34. xx. folid parallelepiped BGML is equal to the folid parallelepiped EHPO. and the pyramid ABCG is the fixth part of the folid 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.

EV

b

PROP. X. THEOR.

VERY 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 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 must either be greater
than the triple, or lefs than it.
triple; and describe the fquare

First, Let it be greater than the
ABCD in the circle; this fquare is

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

2. 32. II.

erect a prism of the fame altitude with the cylinder; this prifm is
greater than half of the cylinder; because if a fquare be described
about the circle, and a prifm erected upon the fquare, of the fame
altitude with the cylinder, the infcribed fquare is half of that cir-
cumfcribed; and upon thefe fquare bafes are erected folid parallele-
pipeds, viz. the prifins, of the fame altitude; therefore the prifm
upon the fquare ABCD is the half of the prifm upon the fquare de-
fcribed about the circle; becaufe they are to one another as their
baits. and the cylinder is lefs than the prifin upon the fquare de-
fcribed 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. Bife&t the circumferences AB, BC, CD, DA in the
points E, F, G, II; and join AE, EB, RF, 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 hewn in Prop.
2. of this Book. Ere&t prifins upon each E
of thefe triangles of the fame altitude
with the cylinder; each of thefe prifins
is greater than half of the fegment of
the cylinder in which it is; becaule if
thro' the points E, F, G, HI parallels be F
drawn to AB, PC, CD, DA, and pa-
rallelograms be compicted upon the fame

BK

A

H

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

7.12.

ments of the cylinder with are upon the figments 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 piifins be erected of the fame altitude with the cylinder, and fo on, there muft at length remain fome c. Lemma. fgments of the cylinder which together are lefs than the excess of the cylinder above the triple of the cone. let them be thofe upon the fegments of the circle AE, EB, BF, FC, CG, GD, DH, HA. thereAs was fhewn in Prop. 2. of this Book,

c

fore the rest of the cylinder, that is the prifin 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

7.12.

this prifm is tripled 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 deferibe a fquare, this fquare is greater than the half of the circle. and upon the fquare ABCD erect a pyramid having the fame vertex with the cone; this pyramid is greater than the half of the cone; becaufe, as was before demonftrated, if a fquare be defcribed about the circle, the fquare ABCD is the haif of it; and if upon thefe fquares there be created folid parallelepipeds of the fame altitude with the cone, which are alfo prifins, the prifm upon the square ABCD shall be the half of that E 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 pyra

H

mid the bafe of which is the fquare ABCD is half of the pyramid
upon the fquare defcribed about the circle. but this last pyramid
is greater than the cone which it contains; therefore the pyramid
upon the fquare ABCD having the fame vertex with the cone, is
Bifect the circumferences AB,
greater than the half of the cone.
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 fegment of the
circle in which it is. upon each of thefe triangles ere&t pyramids
having the fame vertex with the cone. therefore each of thefe py-
ramids is greater than the half of the fegment of the cone in which
it is, as before was demonftrated of the piilms and fegments of the
cylinder. and thus dividing each of the cicrumferences into two
equal parts, and joining the points of divifion and their extremi
ties by ftraight lines, and upon the triangles erecting pyramids hav,

3. 32. 11.

Bock XII. ing their vertices the fame with that of the cone, and fo on, there

See N.

A

H

G

muft at length remain fome fegments of the cone which together
fhall be less than the excefs of the cone above the third part of the
cylinder. let these be the fegments upon AE, EB, BF, FC, CG,
GD, DH, HA. therefore the rest 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 py-E
ramid is the third part of the prifm
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. there-
fore the cylinder is not less than the triple of the cone. and it has
been demonftrated that neither is it greater than the triple. there-
fore the cylinder is triple of the cone, or, the cone is the third part
of the cylinder. Wherefore every cone, &c. Q. E. D.

CON

PROP. XI. THEO R.

F

ONES 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 solid X; and let Z be the folid which is equal to the excefs of the cone EN above the folid X; therefore the cone EN is equal to the folids X, Z together. in the circle EFGH defcribe the fquare EFGH, therefore this fquare is greater than the half of the circle. upon the Square EFGH erect 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 pyramid be erected upon it, having the

fame vertex with the cone, the pyramid infcribed in the cone is half Book XII. of the pyramid circumfcribed about it, because they are to one another as their bafes. but the cone is lefs than the circumfcribed py- a. 6. 12. ramid; therefore the pyramid of which the bafe is the fquare 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 thefe triangles erect a pyramid having the fame vertex with the cone; each of these pyramids is greater than the half of the fegment of the cone in which it is. and thus dividing each of these 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 must at length remain fome fegments of the cone which are together lefs b. Lemma. than the folid Z. let thefe be the fegments upon EO, OF, FP, PG,

+ Vertex is put in place of altitude which is in the Greek, because the pyramid, in what follows, is fuppofed to be circumferibed about the cone, and so must have the fame vertex. and the fame change is made in some places following.