1the Powers that in BD ſhall ſuſtain the Reſiſtances A and C ſhall
be equal to each other. Let E G be ſuppoſed a Mean-Proporti
onal between E B and F D; therefore as B E is to E G, ſo ſhall
G E be to F D, and A E to C F; and ſo is ſuppoſed the Reſiſtance
of A to the Reſiſtance of C. And becauſe that as E G is to F D,
ſo is A E to C F; by Permutation as G E is to E A, ſo ſhall D F
be to F C: And therefore (in
regard that the two Leavers
69[Figure 69]
D C and G A are divided pro
portionally in the Points F and
E) in caſe the Power that being
placed at D compenſates the
Reſiſtance of C were at G, it
would countervail the ſame Reſiſtance of C placed in A: But by
what hath been granted, the Reſiſtance of A hath the ſame propor
tion to the Reſiſtance of C, that AE hath to C F; that is, B E
hath to E G: Therefore the Power G, or if you will D, placed at
B will ſuſtain the Reſiſtance placed at A: Which was to be de
monſtrated.
be equal to each other. Let E G be ſuppoſed a Mean-Proporti
onal between E B and F D; therefore as B E is to E G, ſo ſhall
G E be to F D, and A E to C F; and ſo is ſuppoſed the Reſiſtance
of A to the Reſiſtance of C. And becauſe that as E G is to F D,
ſo is A E to C F; by Permutation as G E is to E A, ſo ſhall D F
be to F C: And therefore (in
regard that the two Leavers
69[Figure 69]D C and G A are divided pro
portionally in the Points F and
E) in caſe the Power that being
placed at D compenſates the
Reſiſtance of C were at G, it
would countervail the ſame Reſiſtance of C placed in A: But by
what hath been granted, the Reſiſtance of A hath the ſame propor
tion to the Reſiſtance of C, that AE hath to C F; that is, B E
hath to E G: Therefore the Power G, or if you will D, placed at
B will ſuſtain the Reſiſtance placed at A: Which was to be de
monſtrated.
This being underſtood: in the Surface F B of the Priſme D B,
let the Parabolical Line F N B be drawn, whoſe Vertex is B, ac
cording to which let the ſaid Priſme be ſuppoſed to be ſawed, the
Solid compriſed between the Baſe A D, the Rectangular Plane
A G, the Bight Line B G, and the Superficies D G B F being leſt
incurvated according to the Curvity of the Parabolical Line
F N B. I ſay, that
that Solid is through
70[Figure 70]
out of equal Reſi
ſtance. Let it be cut
by the Plane C O pa
rallel to A D; and
imagine two Leavers
divided and ſuppor
ted upon the Fulciments A and C; and let the Diſtances of one
be B A and A F, and of the other B C, and C N. And becauſe in
the Parabola F B A, A B is to B C, as the Square of F A to the
Square of C N, it is manifeſt, that the Diſtance B A of one Leaver,
hath to the Diſtance B C of the other a proportion double to that
which the other Diſtance A F hath to the other C N, And be
cauſe the Reſiſtance that is to be equal by help of the Leaver
B A hath the ſame proportion to the Reſiſtance that is to be
equal by help of the Leaver B C, that the Rectangle D A hath to
the Rectangle O C; which is the ſame that the Line A F hath to
N C, which are the other two Diſtances of the Leavers; it is ma
nifeſt by the fore going Lemma, that the ſame Force that being
let the Parabolical Line F N B be drawn, whoſe Vertex is B, ac
cording to which let the ſaid Priſme be ſuppoſed to be ſawed, the
Solid compriſed between the Baſe A D, the Rectangular Plane
A G, the Bight Line B G, and the Superficies D G B F being leſt
incurvated according to the Curvity of the Parabolical Line
F N B. I ſay, that
that Solid is through
70[Figure 70]out of equal Reſi
ſtance. Let it be cut
by the Plane C O pa
rallel to A D; and
imagine two Leavers
divided and ſuppor
ted upon the Fulciments A and C; and let the Diſtances of one
be B A and A F, and of the other B C, and C N. And becauſe in
the Parabola F B A, A B is to B C, as the Square of F A to the
Square of C N, it is manifeſt, that the Diſtance B A of one Leaver,
hath to the Diſtance B C of the other a proportion double to that
which the other Diſtance A F hath to the other C N, And be
cauſe the Reſiſtance that is to be equal by help of the Leaver
B A hath the ſame proportion to the Reſiſtance that is to be
equal by help of the Leaver B C, that the Rectangle D A hath to
the Rectangle O C; which is the ſame that the Line A F hath to
N C, which are the other two Diſtances of the Leavers; it is ma
nifeſt by the fore going Lemma, that the ſame Force that being