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.
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.
53[Figure 53]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.
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: