H

2

D

12.

FC, CG, GD, DH, HA. Therefore the reft of the cylin- Book XII. der, that is, the prifm of which the base is 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. 7. 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 less than the triple of the cone. Let it be lefs, if poffible: Therefore, inverfely, the cone is greater than the third part of the cylinder. In the circle ABCD describe a square; 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; 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 also 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 e; as are also the third parts of them: Therefore the pyramid, the bafe of which is the fquare ABCD, is half of the pyramid upon the fquare defcribed about the circle: But this laff 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, GD, 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 these triangles erect pyramids having the fame vertex with the cone. Therefore each of these pyramids is greater than the half of the fegment of the cone in which it is, as before was demonftrated of the prifms and fegments of the cylinder; and thus dividing each of the circumferences into two equal parts, and joining the

E

B

F

G

points

e 32. II,

H

Book XII. points of divifion and their extremities by ftraight lines, and upon the triangles erecting pyramids having their vertices the fame with that of the cone, and fo on, there must at length remain some segments of the cone, which together shall be less than the excess 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 E of the cylinder. But this pyramid is the third part of the prifm upon the fame base AEBFCGDH, and

See N.

A

B

F

G

of the fame altitude with the cylin-
der. Therefore this prifm is great-
er than the cylinder of which the
bafe is the circle ABCD. But it is also lefs, for it is contained
within the cylinder; which is impoffible. Therefore the cy-
linder 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 part
of the cylinder. Wherefore every cone, &c. Q.E. D.

ONES and cylinders of the fame altitude, are to one another as their bases.

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 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 less 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 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 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, ha

ving

a 6. 12,

ving the fame vertex with the cone, the pyramid infcribed Book XII. in the cone is half of the pyramid circumfèribed about it, because they are to one another as their bafesa: But the cone is less than the circumfcribed pyramid; therefore the pyramid of which the base 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

1

circle in which it is: Upon each of these 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 division 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 some segments of the cone which are together lefs b b Lem. 1. than the folid Z: Let these be the fegments upon EO, OF, FP,

S

PG,

Vertex is put in place of altitude which is in the Greek, because the pyra mid, in what follow, 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.

Book XII. PG, GR, RH, HS, SE: Therefore the remainder of the cone, viz.the pyramid of which the base is the polygon EOFPGRHS, and its vertex the fame with that of the cone, is greater than the folid X: In the circle ABCD defcribe the polygon ATBYCVDQ fimilar to the polygon EOFPGRHS, and upon it erect a pyramid having the fame vertex with the cone AL: a 1. 12. And because as the fquare of AC is to the fquare of EG, so a is the polygon ATBYCVDQ to the polygon EOFPGRHS; and as the fquare of AC to the fquare of EG, fo is the circle ABCD to the circle EFGH; therefore the circle ABCD c is to the circle EFGH, as the polygon ATBYCVDQ to the poly

b 2. 12. 11. 5.

d 6. 12.

e 14. 5.

Z

gon EOFPGRHS: But as the circle ABCD to the circle EFGH, fo is the cone AL to the folid X; and as the polygon ATBYCVDQ to the polygon EOFPGRHS, fo is d the pyramid of which the bafe is the first of these polygons, and vertex L, to the pyramid of which the bafe is the other polygon, and its vertex N: Therefore, as the cone AL to the folid X, fo is the pyramid of which the base is the polygon-ATBYCVDQ, and vertex L, to the pyramid the base of which is the polygon EOFPGRHS, and vertex N: But the cone AL is greater than the pyramid contained in it; therefore the folid X is greater e than the pyramid in the cone EN. But it is lefs, as was shown,

which

which is abfurd: Therefore the circle ABCD is not to the circle Book XII. EFGH, as the cone AL to any folid which is less than the cone EN. In the fame manner it may be demonstrated that the circle EFGH is not to the circle ABCD, as the cone EN to any folid less than the cone AL. Nor can the circle ABCD be to the circle EFGH, as the cone AL to any folid greater than the cone EN: For, if it be poffible, let it be so to the folid I, which is greater than the cone EN: Therefore, by inverfion, as the circle EFGH to the circle ABCD, fo is the folid I to the cone AL: But as the folid I to the cone AL, fo is the cone EN to some solid, which must be less a than the cone a 14. 5. AL, because the folid I is greater than the cone EN: Therefore, as the circle EFGH is to the circle ABCD, fo is the cone EN to a folid less than the cone AL, which was fhewn to be impoffible: Therefore the circle ABCD is not to the circle EFGH, as the cone AL is to any folid greater than the cone EN: And it has been demonftrated that neither is the circle ABCD to the circle EFGH, as the cone AL to any folid lefs than the cone EN: Therefore the circle ABCD is to the circle EFGH, as the cone AL to the cone EN: But as the cone is to the cone, fob is the cylinder to the cylinder, because the cy- b 15. 5. linders are triple c of the cone each to each. Therefore, as C IO. 12. the circle ABCD to the circle EFGH, fo are the cylinders upon them of the fame altitude. Wherefore cones and cylinders of the fame altitude are to one another as their bases. Q. E. D.

PROP. XII. THEOR.

IMILAR cones and cylinders have to one another the triplicate ratio of that which the diameters of their bales have.

Let the cones and cylinders of which the bases are the circles ABCD, EFGH, and the diameters of the bases AC, EG, and KL, MN, the axis of the cones or cylinders, be fimilar: The cone, of which the base is the circle ABCD, and vertex the point L, has to the cone of which the base is the circle EFGH, and vertex N, the triplicate ratio of that which AC has to EG. For if the cone ABCDL has not to the cone EFGHN the triplicate ratio of that which AC has to EG, the cone ABCDL fhall have the triplicate of that ratio to fome folid which is lefs

See N.