C. 1. 12.

Book xır: 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 describe the polygon ATBYCVDQ_fimilar to the polygon EOFPGRHS, and upon it erect a pyramid of the fame altitude with the cone AL. and because as the square of AC is to the square of EG, foo is the polygon ATBYCVDQ to the polygon

EOFPGRHS; and as the square of AC to the square of EG, fo d. 2. 12. is d the circle ABCD to the circle EFGH; therefore the circle ABCD is e to the circle EFGH, as the polygon ATBYCVDQ_to L

NI
D

H
Q

S

R

C. II. S.

a. 6. 12.

the polygon EOFPGRHS. but as the circle ABCD to the circle EFGH, so is the cone AL to the solid X ; and as the polygon ATBYCVDQ to the polygon EOFPGRHS, so is a the pyramid of which the base is the first of those polygons, and vertex L, to the pyramid of which the base is the other polygon, and its vertex N. therefore as the cone AL to the solid 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 f than the pyramid in the code EN. but it is less, as was shewn; which is absurd. therefore the circle ABCD is

f. 14. 5.

not to the circle EFGH, as the cone AL to any folid which is less Book XII. than the cone EN. In the fame manner 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 posible, let it be fo to the solid I which is greater than the cone EN. therefore, by inversion, as the circle EFGH to the circle ABCD, fo is the folid I to the cone AL. but as the folid I to the coae AL, fo is the cone EN to some solid, which must be less than the cone AL, because the folid I is greater f. 14.5. 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 shewn to be impossible. therefore the circle ABCD is not to the circle EFGH, as the conc AL is 10 any solid greater than the cone EN. and it has been demonstrated that neither is the circle ABCD to the circle EFGH, as the cone AL to any folid less 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, so

8 is the cylinder to the cylinder ; because the cylinders are triple g. 15. 5. 'h of the cones, cach of each. Therefore as the circle ALCD to h. 10. 12. the circle EFGH, 1o 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.

SIMILAR cones and cylinders have to one another See N.

the triplicate ratio of that which the diameters of

their bases 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 axes of the cones or cylinders, be similar. 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 bas to EC, the cone ABCDL shall have the triplicate of that ratio to some folid which is less or gtcater

Book XII. than the cone EFGHN. First, let it have it to a less, viz. to the folid W X. make the same construction as in the preceding Proposition, and

it may be demonstrated the very fame way as in that Propofition,
that the pyramid of which the base is the polygon EOFPGRHS,
and vertex N is greater than the folid X. Describe also in the
circle ABCD the polygon ATBYCVDQ similar to the polygon
FOFPGRHS, upon which erect a pyramid having the fame vertex
with the cone; and let LAQ be one of the triangles containing the
pyramid upon the polygon ATBYCVDQ_ the vertex of which is

N
L

L; and let NES be one of the triangles containing the pyramid upon the polygon EOFPGRHS of which the vertex is N; and join KQ,

MS. because then the cone ABCDL is similar to the cone EFGHN, a. 24. Def. AC is ' to EG, as the axis KL to the axis MN; and as AC to EG,

so bis AK to EM; therefore as AK to EM, so is KL to MN; and, b. 15. S. alternately, AK to KL, as EM to MN. and the right angles AKL,

EMN are equal; therefore, the Gdes about these equal angles being proportionals, the triangle AKL is similar to the triangle EMN. again, because AK is to KQ, as EM to MS, and that these sides

11.

c. 6. 6.

are about equal angles AKQ , EMS, because these angles are, Book XII.
each of them, the fame part of four right angles at the centers K, m
M; therefore the triangle AKQ is fimilar to the triangle EMS. c. 6. 6.
and because it has been shewa that as AK to KL, fo is EM to MN,
and that AK is equal to KQ, and EM to MS, as QK to KL, so is
SM to MN; and therefore, the sides about the right angles OKL,
SMN being proportionals, the triangle LKQ is similar to the tri-
angle NMS. and because of the similarity of the triangles AKL,
EMN, as LA is to AK, fo is NE to EM; and by the similarity of the
triangles AK, EMS, as KA to AQ, fo ME to ES; ex aequalid, d. 24. 5.
LA is to AQ, as NE to ES. again, because of the similarity of
the triangles LOK, NSM, as IQ to QK, fo NS to SM; and from
the similarity of the triangles KAQ, MES, as KQ to QA, fo MS
to SE; ex aequali 4, LQ is to QA, as NS to SE. and it was proved
that QA is to AL, as SE to EN; therefore, again, ex aequali, as
QL to LA, fo is SN to NE. wherefore the triangles LQA, NSE,
baving the sides about all their angles proportionals, are equiange-
lare and similar to one another. and therefore the pyramid of which e. 5.6.
the bale is the triangle AKQ , and vertex L, 15 similar to the pyra-
mid the base of which is the triangle EMS, and vertex N, because
their folid angles are equalf to one another, and they are contained f. 8. 11.
by the same number of similar planes. but fimilar pyramids which
have triangular bafes have to one another the triplicate 6 ratio of that s. 8.12.
which their homologous fides have; therefore the pyramid AKQL
has to the pyramid EMSN the triplicate ratio of that which AK has
to EM. In the same manner, if Atraight lines be drawn from the
points D, V, C, Y, B, T to K, and from the points H, R, G, P,
F, O to M, and pyramids be erected upon the triangles having the
same vertices with the cones, it may be demonstrated that each py-
ramid in the first cone has to each in the other, taking them in the
same order, the triplicate ratio of that which the fide AK has to the
side EM; that is, which AC has to EG. but as one antecedent to
its consequent, so are all the antecedents to all the consequents h; h. 12. So
therefore as the pyramid AKOL to the pyramid EMSN, fo is the
whole pyramid the base of which is the polygon DQATBYCV, and
vertex L, to the whole pyramid of which the base is the polygon
HSEOFPGR, and vertex N. wherefore also the first of these two
last named pyramids has to the other the triplicate ratio of that which
AC has to EG. but, by the Hypothefis, the cone of which the base
is the circle ABCD, and vertex L has to the folid X, the triplicate

Book XII. ratio of that which AC has to EG; therefore as the cone of which

the base is the circle ABCD, and vertex L, is to the folid X, so is the pyramid the base of which is the polygon DQATBYCV, and vertex L to the pyramid the base of which is the polygon HSEOFPGR and vertex N. but the said cone is greater than the pyramid contained in it. therefore the folid X is greater i than the Pyramid the base of which is the polygon HISEOFPGR, and vertex N. but it is also less; which is imposible. therefore the cone of which the base is the circle ABCD, and vertex L has pot to any fo

i. 14. 5.

lid which is less than the cone of which the base is the circle EFGH and vertex N, the triplicate ratio of that which AC has to EG. In the same manner it may be demonstrated that neither has the cone EFGHN to any solid which is less than the cone ABCDL, the triplicate ratio of that which EG has to AC. Nor can the cone ABCDL have to any folid which is greater than the cone EFGHN, the triplicate ratio of that which AC has to EG. for, if it be possible, let it have it to a greater, viz. to the folid Z. therefore, inversely, the solid Z has to the cone ABCDL the triplicate ratio of that which EG has to AC. but as the solid Z is to the cone ABCDL, fc'is