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Book XII. GR, RH, HS, SE. therefore the remainder of the cone, viz. the Mpyramid of which the base is the polygon EOFPGRHS, and its

vertex the same with that of the cone, is greater than the solid X. In the circle ABCD describe the polygon ATBYCVDQ_similar 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 6. 1. 12. the square of EG, so is the polygon ATBYCVDQ to the polygon

EOFPGRHS; and as the square of AC to the square of EG, so d. 2. 12. is d the circle A B C D to the circle EFGH; therefore the circe 6. 11. 5. ABCD ise to the circle EFGH, as the polygon ATBYCVDQ_10

L

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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 4. 6. 12. 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, so 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 f. 14. s. the solid X is greater f than the pyramid in the cone EN. but it is

less, as was shown; which is absurd. therefore the circle ABCD is

not

not to the circle EFGH, as the cone AL to any solid which is less than Book XII.
the cone EN. In the same manner it may be demonstrated that the
circle EFGH is not to the circle ABCD, as the cone EN to any fo-
lid 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 possible, 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, so is the solid I to the cone AL. but as the solid I to the
cone AL, fo is the cone EN to fome folid, which must be less f than f. 14. 5.
the cone AL, because the folid I is greater than the cone EN.
therefore as the circle EFGH is to the circle ABCD, so is the cone
EN to a solid less than the cone AL, which was shewn to be im-
possible. 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 demonstrated that neither is the circle ABCD to the circle
EFGH, as the cone AL to any folid less than the cone EN. there-
fore the circle ABCD is to the circle EFGH, as the cone AL to the
cone EN. but as the cone is to the cone, lo s is the cylinder to the g. 15. 5.
cylinder; because the cylinders are triple h of the cones, each of h. 10.12.
each. Therefore as the circle ABCD to the circle EFGH, so are the
cylinders upon them of the fame altitude. Wherefore cones and
cylinders of the fame altitude, are to one another as their bafcs.
Q. E. D.

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SIMILAR
IMILAR cones and cylinders have to one another the Sce N.

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 has to EG, the cone ABCDL shall have the triplicate of that ratio to some solid which is less or greater

than

Book XII. than the cone EFGHN. First, let it have it to a less, viz. to the folid

X. make the same construction as in the preceeding Proposition, and
it may be demonstrated the very fame way as in that Proposition,
that the pyramid of which the base is the polygon EOFPGRHS, and
vertex N is greater than the solid X. Describe also in the circle
ABCD the polygon ATBYCVDQ similar to the polygon
EOFPGRHS, 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

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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, 2.24. Def. AC is a 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, fo is KL to MN; and, b. 15. 5.

alternately, AK to KL, as EM to MN. and the right angles AKL,

EMN are equal; therefore, the sides about these equal angles being c. 6. 6. proportionals, the triangle AKL is similar to the triangle EMN. again, because AK is to KQ_, as EM to MS, and that these sides

are

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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 similar to the triangle EMS. c. 6. 6.
and because it has been shewn that as AK to KL, so is EM to MN,
and that AK is equal to KQ, and EM to MS, as OK to KL, so is
SM to MN; and therefore, the fides 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, so is NE to EM; and by the similarity of the
triangles AKO, EMS, as KA to AQ, so ME to ES; ex aequalid, d. 22. 5.
LA is to AQ, as NE to ES. again, because of the similarity of
the triangles LOK, NSM, as LQ to OK, fo NS to SM; and from
the similarity of the triangles KAQ, MES, as KQ_to QA, fo MS
to SE; ex aequalid, LQ is to QA, as NS to SE, and it was pro-
ved 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,
having the fidcs about all their angles proportionals, are equiangu-
låre and similar to one another. and therefore the pyramid of which c. 5. 6.
the base is the triangle AKQ , and vertex L, is similar to the pyra-
mid the base of which is the triangle EMS, and vertex N, because
their folid angles are equal to one another, and they are contained f.:B. 11.
by the fame number of similar planes. but fimilar pyramids which
have triangular bases have to one another the triplicate 8 ratio of that g.8.11.
which their homologous fides have; therefore the pyramid AKOL
has to the pyramid EMSN the triplicate ratio of that which AK has
to EM. In the fame manner, if straight 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
fame order, the triplicate ratio of that which the fide AK has to the
fide EM; that is, which AC has to EG. but as one antecedent to
its confequent, so are all the antecedents to all the consequents h; h. 12. 5.
therefore as the pyramid AKQL to the pyramid EMSN, fo is the
whole pyramid the base of which is the polygon DOATBYCV, 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 hypothesis, 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 solid X, lo
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 i. 14.5. pyramid contained in it, therefore the solid X is greater i than the

pyramid the base of which is the polygon HSEOFPGR, and vertex
N. but it is also less; which is impossible. therefore the cone of
which the base is the circle ABCD, and vertex L has not to any so-

N

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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 folid 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 solid which is greater than the cone EFGHN,
the triplicate ratio of that which AC has to EG. for, if it be pol-
sible, let it have it to a greater, viz. to the solid Z. therefore, inverse-
ly, the solid Z has to the cone ABCDL the triplicate ratio of that

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