258[Figure 258]For if it be poſſible, let it fall ſhort of it: and let R T be pro
longed as farre as to A C in V: and then thorow V draw V X pa
rallel to F D. Now, by the thing we have last demonſtrated, A X
ſhall have the ſame proportion unto A R, as A F hath to A E.
But A S hath alſo the ſame proportion to A R: Wherefore (a)
A S is equall to A X, the part to the whole, which is impoſſi
ble. The ſame abſurdity will follow if we ſuppoſe the Toint
T to fall beyond the Line A C: It is therefore neceſſary that
it do fall in the ſaid A C. Which we propounded to be demonstrated.
longed as farre as to A C in V: and then thorow V draw V X pa
rallel to F D. Now, by the thing we have last demonſtrated, A X
ſhall have the ſame proportion unto A R, as A F hath to A E.
But A S hath alſo the ſame proportion to A R: Wherefore (a)
A S is equall to A X, the part to the whole, which is impoſſi
ble. The ſame abſurdity will follow if we ſuppoſe the Toint
T to fall beyond the Line A C: It is therefore neceſſary that
it do fall in the ſaid A C. Which we propounded to be demonstrated.
(a) By 9. of the
fifth.
fifth.
LEMMA III.
Let there be a Parabola, whoſe Diameter
let be A B; and let the Right Lines A C and B D be ^{*} con
tingent to it, A C in the Point C, and B D in B: And two
Lines being drawn thorow C, the one C E, parallel unto
the Diameter; the other C F, parallel to B D; take any
Point in the Diameter, as G; and as F B is to B G, ſo let B
G be to B H: and thorow G and H draw G K L, and H E
M, parallel unto B D; and thorow M draw M N O parallel
to A C, and cutting the Diameter in O: and the Line N P
being drawn thorow N unto the Diameter let it be parallel
to B D. I ſay that H O is double to G B.
let be A B; and let the Right Lines A C and B D be ^{*} con
tingent to it, A C in the Point C, and B D in B: And two
Lines being drawn thorow C, the one C E, parallel unto
the Diameter; the other C F, parallel to B D; take any
Point in the Diameter, as G; and as F B is to B G, ſo let B
G be to B H: and thorow G and H draw G K L, and H E
M, parallel unto B D; and thorow M draw M N O parallel
to A C, and cutting the Diameter in O: and the Line N P
being drawn thorow N unto the Diameter let it be parallel
to B D. I ſay that H O is double to G B.
* Or touch it.
For the Line M N O cutteth the Diameter either in G, or in other Points: and if it do
cut it in G, one and the ſame Point ſhall be noted by the two letters G and O. Therfore F C,
P N, and H E M being Parallels, and A C being Parallels to M N O, they ſhall make the
259[Figure 259]
Triangles A F C, O P N and O H M like to
each other: Wherefore (a) O H ſhall be to
H M, as A F to FC: and ^{*} Permutando,
O H ſhall be to A F, as H M to F C: But
the Square H M is to the Square G L as the Line
H B is to the Line B G, by 20. of our firſt Book
of Conicks; and the Square G L is unto the
Square F C, as the Line G B is to the Line B F:
and the Lines H B, B G and B F are thereupon
Proportionals: Therefore the (b) Squares
H M, G L and F C and there Sides, ſhall alſo be
Proportionals: And, therefore, as the (c)
Square H M is to the Square G L, ſo is the Line
H M to the Line F C: But as H M is to F C, ſo
is O H to A F; and as the Square H M is to
the Square G L, ſo is the Line H B to B G; that
is, B G to B F: From whence it followeth that
O H is to A F, as B G to B F: And Permu
tando, O H is to B G, as A F to F B; But A F is double to F B: Therefore A B and B F
are equall, by 35. of our firſt Book of Conicks: And therefore N O is double to G B:
Which was to be demonſtrated.
cut it in G, one and the ſame Point ſhall be noted by the two letters G and O. Therfore F C,
P N, and H E M being Parallels, and A C being Parallels to M N O, they ſhall make the
259[Figure 259]Triangles A F C, O P N and O H M like to
each other: Wherefore (a) O H ſhall be to
H M, as A F to FC: and ^{*} Permutando,
O H ſhall be to A F, as H M to F C: But
the Square H M is to the Square G L as the Line
H B is to the Line B G, by 20. of our firſt Book
of Conicks; and the Square G L is unto the
Square F C, as the Line G B is to the Line B F:
and the Lines H B, B G and B F are thereupon
Proportionals: Therefore the (b) Squares
H M, G L and F C and there Sides, ſhall alſo be
Proportionals: And, therefore, as the (c)
Square H M is to the Square G L, ſo is the Line
H M to the Line F C: But as H M is to F C, ſo
is O H to A F; and as the Square H M is to
the Square G L, ſo is the Line H B to B G; that
is, B G to B F: From whence it followeth that
O H is to A F, as B G to B F: And Permu
tando, O H is to B G, as A F to F B; But A F is double to F B: Therefore A B and B F
are equall, by 35. of our firſt Book of Conicks: And therefore N O is double to G B:
Which was to be demonſtrated.