the cone EFGHIN to fome folid, which must be less than the cone Book XII. ABCDL, because the folid Z is greater than the cone EFGHN. therefore the cone EFGHN has to a folid which is less than the cone i. 14. 5. ABCDL, the triplicate ratio of that which EG has to AC, which was demonftrated to be impoffible. therefore the cone ABCDL has not to any folid greater than the cone EFGHN, the triplicate ratio of that which AC has to EG; and it was demonftrated that it could not have that ratio to any folid lefs than the cone EFGHN. therefore the cone ABCDL has to the cone EFGHN, the triplicate ratio of that which AC has to EG. but as the cone is to the cone, fok k. 15. 5. the cylinder to the cylinder, for every cone is the third part of the cylinder upon the fame bafe, and of the fame altitude. therefore alfo the cylinder has to the cylinder, the triplicate ratio of that which AC has to EG. Wherefore fimilar cones, &c. Q. ED.

IF a

Fa cylinder be cut by a plane parallel to its oppofité See N. planes, or bafes; it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other.

R

Let the cylinder AD be cut by the plane GH parallel to the oppofite planes O AB, CD, meeting the axis EF in the point K, and let the line GH be the common fection of the plane GH and the furface of the cylinder AD. let AEFC be the parallelogram, in any position of it, by the revolution of which about the straight line EF the cylinder AD is defcribed; and let GK be the common fection of the plane GH, and the plane AEFC. and because the parallel planes AB, GH are cut by the plane AEKG, AE, KG, their common fections with it, are parallel; wherefore AK is a parallelogram, T and GK equal to EA the ftraight line from the center of the circle AB. for the fame reafon, each of the ftraight lines

G

Book XII. drawn from the point K to the line GH may be proved to be equal to thofe which are drawn from the center of the circle AB to its circumference, and are therefore all equal to one anob.15.Def.1, ther. therefore the line GH is the circumference of a circle of which the center is the point K. therefore the plane GH divides the cylinder AD into the cylinders AH, GD; for they are the fame which would be defcribed by the revolution of the parallelograms AK, GF about the ftraight lines EK, KF. and it is to be fhewn that the cylinder AH is to the cylinder HC, as the axis EK to the axis KF.

R

L

N

S

Produce the axis EF both ways; and take any number of ftraight lines EN, NL, each equal to EK; and any number FX, XM, each equal to FK; and let planes parallel to AB, CD pafs through the points L, N, X, M. therefore the common fections of thefe planes with the cylinder produced are circles the centers of which are the points L, N, X, M, as was proved of the plane GH; and thefe planes cut off the cylinders PR, RB, DT, TQ and becaufe the axes LN, NE, EK are all equal, therefore c. 11. 12. the cylinders PR, RB, BG are to one another as their bafes. but their bafes are equal, and therefore the cylinders PR, RB, BG are equal. and becaufe

A

E

B

G

K

H

F

D

X

M

the axes LN, NE, EK are equal to one T
another, as alfo the cylinders PR, RB,
BG, and that there are as many axes as V
cylinders; therefore whatever multiple

the axis KL is of the axis KE, the fame multiple is the cylinder PG
of the cylinder GB. for the fame reafon, whatever multiple the axis
MK is of the axis KF, the fame multiple is the cylinder QG of the
cylinder GD. and if the axis KL be equal to the axis KM, the cy-
linder PG is equal to the cylinder GQ; and if the axis KL be
greater than the axis KM, the cylinder PG is greater than the cy-
linder GQ; and if lefs, lefs. fince therefore there are four magni-
tudes, viz. the axes EK, KF, and the cylinders BG, GD, and that
of the axis EK and cylinder BG there has been taken any equimul-
tiples whatever, viz. the axis KL and cylinder PG; and of the axis

KF and cylinder GD, any equimultiples whatever, viz. the axis Book XII. KM and cylinder GQ; and it has been demonftrated if the axis

KL be greater than the axis KM, the cylinder PG is greater than. the cylinder GQ; and if equal, equal; and if lefs, lefs. therefore d d. 5. Def.g. the axis EK is to the axis KF, as the cylinder BG to the cylinder GD. Wherefore if a cylinder, &c. Q. E. D.

PROP. XIV. THEOR.

CONES and cylinders upon equal bafes are to one

another as their altitudes.

Let the cylinders EB, FD be upon the equal bafes AB, CD. as the cylinder EB to the cylinder FD, fo is the axis GH to the axis KL.

Produce the axis KL to the point N, and make LN equal to the axis GH, and let CM be a cylinder of which the bafe is CD, and axis LN. and because the cylinders EB, CM have the fame altitude,, they are to one another as their bafes. but their bafes a. 11. are equal, therefore alfo the cylin

ders EB, CM are equal. and becaufe the cylinder FM is cut by the plane CD parallel to its oppofite planes, as the cylinder CM to E the cylinder FD, fo is the axis LN to the axis KL. but the cylinder CM is equal to the cylin der EB, and the axis LN to the

F

K

G

L

b. 13. 73.

axis GH. therefore as the cylin-A

der EB to the cylinder FD, fo is

H

M

N

the axis GH to the axis KL. and as the cylinder EB to the cylinder FD, fo is the cone ABG to the cone CDK, because the c. 15. s cylinders are triple 4 of the cones. therefore alfo the axis GH is d. 10. 12. to the axis KL, as the cone ABG to the cone CDK, and the cylinder EB to the cylinder FD. Wherefore cones, &c. Q. E. D

274

Book XII.

Sea N.

2. 11. 12.

b. A. S.

THE

PROP. XV. THE OR.

HE bafes and altitudes of equal cones and cylinders are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the cones and cylinders are equal to one another.

Let the circles ABCD, EFGH, the diameters of which are AC, EG, be the bafes, and KL, MN the axes, as also the altitudes, of equal cones and cylinders; and let ALC, ENG be the cones, and AX, EO the cylinders. the bafes and altitudes of the cylinders AX, EO are reciprocally proportional; that is, as the bafe ABCD to the bafe EFGH, fo is the altitude MN to the altitude KL.

Either the altitude MN is equal to the altitude KL, or these altitudes are not equal. First, let them be equal; and the cylinders AX, EO being alfo equal, and cones and cylinders of the fame altitude being to one another as their bases. therefore the bafe ABCD is equal to the base EFGH; and as the base ABCD is to the base EFGH, fo is the altitude MN to the altitude KL. but let the

altitudes KL, MN be

N

R

0

C. 7. S..

planes of the circles EFGH, RO; therefore the common fection of the plane TYS and the cylinder EO is a circle, and confequently ES is a cylinder, the base of which is the circle EFGH, and altitude MP. and because the cylinder AX is equal to the cylinder EO, as AX is to the cylinder ES, fo is the cylinder EO to the faine ES. but as the cylinder AX to the cylinder ES, fo is the bafe ABCD to the bafe EFGH; for the cylinders AX, ES are of the fame altitude; and as the cylinder EO to the cylinder ES, d. 13. 12. fod is the altitude MN to the altitude MP, because the cylinder

EO is cut by the plane TYS parallel to its oppofite planes. there- Book XII. fore as the bafe ABCD to the bafe EFGH, fo is the altitude MN to the altitude MP. but MP is equal to the altitude KL; wherefore as the bafe ABCD to the bafe EFGH, fo is the altitude MN to the altitude KL; that is, the bafes and altitudes of the equal cylinders AX, EO are reciprocally proportional.

But let the bafes and altitudes of the cylinders AX, EO, be reciprocally proportional, viz. the bafe ABCD to the bafe EFGH, as the altitude MN to the altitude KL. the cylinder AX is equal to the cylinder EO.

Firft, let the bafe ABCD be equal to the bafe EFGH, then becaufe as the base ABCD is to the base EFGH, fo is the altitude MN to the altitude KL; MN is equal to KL, and therefore the b. A. 5. cylinder AX is equal to the cylinder EO.

b

But let the bafes ABCD, EFGH be unequal, and let ABCD be the greater; and because as ABCD is to the bafe EFGH, fo is the altitude MN to the altitude KL, therefore MN is greater than KL; then, the fame conftruction being made as before, because as the base ABCD to the bafe EFGH, fo is the altitude MN to the altitude KL; and because the altitude KL is equal to the altitude MP; therefore the bafe ABCD is to the bafe EFGH, as the cylinder AX to the cylinder ES; and as the altitude MN to the altitude MP or KL, fo is the cylinder EO to the cylinder ES. therefore the cylinder AX is to the cylinder ES, as the cylinder EO is to the fame ES. whence the cylinder AX is equal to the cylinder EO. and the fanie reasoning holds in cones. Q. E. D.

2. II. IZA

T

PROP. XVI. PRO B.

O defcribe in the greater of two circles that have the fame center, a polygon of an even number of equal fides, that fhall not meet the leffer circle.

Let ABCD, EFGH be two given circles having the fame center K. it is required to infcribe in the greater circle ABCD a polygon of an even number of equal fides, that shall not meet the leffer circle.

Thro' the center K draw the straight line BD, and from the point G, where it meets the circumference of the lefter circle, draw