1to M D, as C Q to Q A: But L B is to B D, by 5 of Archimedes, before recited, as C D
to D A: It is manifeſt therefore, by the precedent Lemma, that C D is to D Q, as L B is to
B M: But as C D is to D Q, ſo is C M to M P: Therefore L B is to B M, as C M to M P:
And it haveing been demonſtrated, that C M is to M P, as C E to E A; L B ſhall be to B M,
as C E to E A. And in like manner it ſhall be demonstrated that ſo is N O to O F; as alſo the
Remainders. And that alſo H K is to K E, as C E to E A, doth plainly appeare by the ſame
5. of Archimedes: Which is that that we propounded to be demonſtrated.
to D A: It is manifeſt therefore, by the precedent Lemma, that C D is to D Q, as L B is to
B M: But as C D is to D Q, ſo is C M to M P: Therefore L B is to B M, as C M to M P:
And it haveing been demonſtrated, that C M is to M P, as C E to E A; L B ſhall be to B M,
as C E to E A. And in like manner it ſhall be demonstrated that ſo is N O to O F; as alſo the
Remainders. And that alſo H K is to K E, as C E to E A, doth plainly appeare by the ſame
5. of Archimedes: Which is that that we propounded to be demonſtrated.
(a) By 4. of the
ſixth.
ſixth.
(b) By 11 of the
fifth,
fifth,
(c) By 14 of the
fifth.
fifth.
By 2. of the ſixth
LEMMA. VI.
And, therefore, let the things ſtand as above; and deſcribe
yet another like Portion, contained betwixt a Right Line, and
the Section of the Rightangled Cone D R C, whoſe Diameter
is R S, that it may cut the Line F G in T; and prolong S R
unto the Line C H in V, which meeteth the Section A B C in
X, and E F C in Y. I ſay, that B M hath to M D, a propor
tion compounded of the proportion that E A hath to A C;
and of that which C D hath to D E.
yet another like Portion, contained betwixt a Right Line, and
the Section of the Rightangled Cone D R C, whoſe Diameter
is R S, that it may cut the Line F G in T; and prolong S R
unto the Line C H in V, which meeteth the Section A B C in
X, and E F C in Y. I ſay, that B M hath to M D, a propor
tion compounded of the proportion that E A hath to A C;
and of that which C D hath to D E.
For, we ſhall firſt demonſtrate, that the Line C H toucheth the Section D R C in the
Point C; and that L M is to M D, as alſo N F to F T, and V Y to Y R, as C D is to E D.
And, becauſe now that L B is to B M, as C E is to E A; therefore, Compounding and Conver
ting, B M ſhall be to L M, as E A to A C: And, as L M is to M D, ſo ſhall C D be to
D E: The proportion, therefore, of B M to M D, is compounded of the proportion that
B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion
of B M to M D, ſhall alſo be compounded of the proportion that E A hath to A C, and of
that which C D hath to D E. In the ſame manner it ſhal be demonſtrated, that O F hath to
F T, and alſo X Y to Y R, a proportion compounded of thoſe ſame proportions; and ſo in
the reſt: Which was to be demonstrated.
Point C; and that L M is to M D, as alſo N F to F T, and V Y to Y R, as C D is to E D.
And, becauſe now that L B is to B M, as C E is to E A; therefore, Compounding and Conver
ting, B M ſhall be to L M, as E A to A C: And, as L M is to M D, ſo ſhall C D be to
D E: The proportion, therefore, of B M to M D, is compounded of the proportion that
B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion
of B M to M D, ſhall alſo be compounded of the proportion that E A hath to A C, and of
that which C D hath to D E. In the ſame manner it ſhal be demonſtrated, that O F hath to
F T, and alſo X Y to Y R, a proportion compounded of thoſe ſame proportions; and ſo in
the reſt: Which was to be demonstrated.
By which it appeareth that the Lines ſo drawn; which fall betwixt
the Sections A B C and D R C, ſhall be divided by the Section E F C
in the ſame Proportion.
the Sections A B C and D R C, ſhall be divided by the Section E F C
in the ſame Proportion.
And C B is to B D, as ſix to fifteen.] For we have ſuppoſed that B K is
double of K D: Wherefore, by Compoſition B D ſhall be to K D as three to one; that is, as
fifteen to five: But B D was to K C as fifteen to four; Therefore B D is to D C as fifteen to nine:
And, by Converſion of proportion and Convert
ing, C B is to B D, as ſix to ſifteen.
double of K D: Wherefore, by Compoſition B D ſhall be to K D as three to one; that is, as
fifteen to five: But B D was to K C as fifteen to four; Therefore B D is to D C as fifteen to nine:
And, by Converſion of proportion and Convert
ing, C B is to B D, as ſix to ſifteen.
N
279[Figure 279]And as C B is to B D, ſo is
E B to B A; and D Z to D A.]
For the Triangles C B E and D B A being
alike; As C B is to B E, ſo ſhall D B be to B A:
And, Permutando, as C B is to B D, ſo ſhall
E B be to B A: Againe, as B C is to C E ſo
ſhall B D be to D A, And, Permutando, as
C B is to B D, ſo ſhall C E, that is, D Z
equall to it, be to D A.
E B to B A; and D Z to D A.]
For the Triangles C B E and D B A being
alike; As C B is to B E, ſo ſhall D B be to B A:
And, Permutando, as C B is to B D, ſo ſhall
E B be to B A: Againe, as B C is to C E ſo
ſhall B D be to D A, And, Permutando, as
C B is to B D, ſo ſhall C E, that is, D Z
equall to it, be to D A.
O
And of D Z and D A, L I and
L A are double.] That the Line L A is
double of D A, is manifeſt, for that B D is the Diameter of the Portion. And that L I is
dovble to D Z ſhall be thus demonſtrated. For as much as ZD is to D A, as two to five:
therefore, Converting and Dividing, A Z, that is, I Z, ſhall be to Z D, as three to two:
L A are double.] That the Line L A is
double of D A, is manifeſt, for that B D is the Diameter of the Portion. And that L I is
dovble to D Z ſhall be thus demonſtrated. For as much as ZD is to D A, as two to five:
therefore, Converting and Dividing, A Z, that is, I Z, ſhall be to Z D, as three to two:
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