1the ſaid K ω in H, and A S is parallel unto the Line that toucheth
in P; It is neceſſary that P I hath unto P H either the ſame propor
tion that N ω hath to ω O, or greater; for this hath already been
demonſtrated.] Where this is demonſtrated either by Archimedes himſelf, or by
any other, doth not appear; touching which we will here inſert a Demonſtration, after that
we have explained ſome things that pertaine thereto.
in P; It is neceſſary that P I hath unto P H either the ſame propor
tion that N ω hath to ω O, or greater; for this hath already been
demonſtrated.] Where this is demonſtrated either by Archimedes himſelf, or by
any other, doth not appear; touching which we will here inſert a Demonſtration, after that
we have explained ſome things that pertaine thereto.
C
LEMMA I.
Let the Lines A B and A C contain the Angle B A C; and from
the point D, taken in the Line A C, draw D E and D F at
pleaſure unto A B: and in the ſame Line any Points G and L
being taken, draw G H & L M parallel to D E, & G K and
L N parallel unto F D: Then from the Points D & G as farre
as to the Line M L draw D O P, cutting G H in O, and G Q
parallel unto B A. I ſay that the Lines that lye betwixt the Pa
rallels unto F D have unto thoſe that lye betwixt the Par
allels unto D E (namely K N to G Q or to O P; F K to D O;
and F N to D P) the ſame mutuall proportion: that is to ſay,
the ſame that A F hath to A E.
the point D, taken in the Line A C, draw D E and D F at
pleaſure unto A B: and in the ſame Line any Points G and L
being taken, draw G H & L M parallel to D E, & G K and
L N parallel unto F D: Then from the Points D & G as farre
as to the Line M L draw D O P, cutting G H in O, and G Q
parallel unto B A. I ſay that the Lines that lye betwixt the Pa
rallels unto F D have unto thoſe that lye betwixt the Par
allels unto D E (namely K N to G Q or to O P; F K to D O;
and F N to D P) the ſame mutuall proportion: that is to ſay,
the ſame that A F hath to A E.
For in regard that the Triangles A F D, A K G, and A N L
257[Figure 257]
are alike, and E F D, H K G, and M N L are alſo alike: There-
fore, (a) as A F is to F D, ſo ſhall A K be to K G; and as F D is to
F E, ſo ſhall K G be to K H: Wherefore, ex equali, as A F is to F
E, ſo ſhall A K be to K H: And, by Converſion of proportion, as
A F is to A E, ſo ſhall A K be to K H. It is in the ſame manner
proved that, as A F is to A E, ſo ſhall A N be to A M. Now A
N being to A M, as A K is to A H; The (b) Remainder K N ſhall
be unto the Remainder H M, that is unto G Q, or unto O P, as
A N is to A M; that is, as A F is to A E: Again, A K is to
A H, as A F is to A E; Therefore the Remainder F K ſhall be to
the Remainder E H, namely to D O, as A F is to A E. We might in
like manner demonstrate that ſo is F N to D P: Which is that that
was required to be demonstrated.
257[Figure 257]are alike, and E F D, H K G, and M N L are alſo alike: There-
fore, (a) as A F is to F D, ſo ſhall A K be to K G; and as F D is to
F E, ſo ſhall K G be to K H: Wherefore, ex equali, as A F is to F
E, ſo ſhall A K be to K H: And, by Converſion of proportion, as
A F is to A E, ſo ſhall A K be to K H. It is in the ſame manner
proved that, as A F is to A E, ſo ſhall A N be to A M. Now A
N being to A M, as A K is to A H; The (b) Remainder K N ſhall
be unto the Remainder H M, that is unto G Q, or unto O P, as
A N is to A M; that is, as A F is to A E: Again, A K is to
A H, as A F is to A E; Therefore the Remainder F K ſhall be to
the Remainder E H, namely to D O, as A F is to A E. We might in
like manner demonstrate that ſo is F N to D P: Which is that that
was required to be demonstrated.
(a) By 4. of the
ſixth.
ſixth.
(b) By 5. of the
fifth.
fifth.
LEMMA II.
In the ſame Line A B let there be two Points R and S, ſo diſpo
ſed, that A S may have the ſame Proportion to A R that
A F hath to A E; and thorow R draw R T parallel to E D,
and thorow S draw S T parallel to F D, ſo, as that it may
meet with R T in the Point T. I ſay that the Point T fall
eth in the Line A C.
ſed, that A S may have the ſame Proportion to A R that
A F hath to A E; and thorow R draw R T parallel to E D,
and thorow S draw S T parallel to F D, ſo, as that it may
meet with R T in the Point T. I ſay that the Point T fall
eth in the Line A C.