1of A, ſhall be made in the ſame Time, as along A B from Reſt in A. Let
C D be equal to C B, and drawing B D, let B E be applied equal to both
B D and D C. I ſay B E is the Plane required. Continue out E B to
meet the Horizontal Line A G in G;
105[Figure 105]
and let G F be a Mean-Proportional be
tween the ſaid E G and G B. E F ſhall
be to F B, as E G is to G F; and the
Square E F ſhall be to the Square F B, as
the Square E G is to the Square G F;
that is as the Line E G to G B: But
E G is double to G B: Therefore the
Square of E F is double to the Square of F B: But alſo the Square of
D B is double to the Square of B C: Therefore, as the Line E F is to
F B, ſo is D B to B C: And by Compoſition and Permutation, as E B is
to the two D B and B C, ſo is B F to B C: But B E is equal to the two
D B and B C: Therefore B F is equal to the ſaid B C, or B A. If there
fore A B be underſtood to be the Time of the Fall along A B, G B ſhall
be the Time along G B, and G F the Time along the whole G E: There
fore B F ſhall be the Time along the remainder B E, after the Fall from
G, or from A, which was the Propoſition.
C D be equal to C B, and drawing B D, let B E be applied equal to both
B D and D C. I ſay B E is the Plane required. Continue out E B to
meet the Horizontal Line A G in G;
105[Figure 105]and let G F be a Mean-Proportional be
tween the ſaid E G and G B. E F ſhall
be to F B, as E G is to G F; and the
Square E F ſhall be to the Square F B, as
the Square E G is to the Square G F;
that is as the Line E G to G B: But
E G is double to G B: Therefore the
Square of E F is double to the Square of F B: But alſo the Square of
D B is double to the Square of B C: Therefore, as the Line E F is to
F B, ſo is D B to B C: And by Compoſition and Permutation, as E B is
to the two D B and B C, ſo is B F to B C: But B E is equal to the two
D B and B C: Therefore B F is equal to the ſaid B C, or B A. If there
fore A B be underſtood to be the Time of the Fall along A B, G B ſhall
be the Time along G B, and G F the Time along the whole G E: There
fore B F ſhall be the Time along the remainder B E, after the Fall from
G, or from A, which was the Propoſition.
PROBL. II. PROP. XIV.
A Perpendicular and a Plane inclined to it being
given, to find a part in the upper Perpendicu
lar which ſhall be paſt ex quiete in a Time
equal to that in which the inclined Plane is
paſt after the Fall along the part found in the
Perpendicular.
given, to find a part in the upper Perpendicu
lar which ſhall be paſt ex quiete in a Time
equal to that in which the inclined Plane is
paſt after the Fall along the part found in the
Perpendicular.
Let the Perpendicular be D B, and the Plane inclined to it A C. It is
required in the Perpendicular A D to find a part which ſhall be
paſt ex quiete in a Time equal to that in which the Plane A C is
paſt after the Fall along the ſaid part. Draw the Horizontal Line C B;
and as B A more twice A C is to A C, ſo let E A be to A R; And from
R let fall the Perpendicular R X unto D B. I ſay X is the point requi
red. And becauſe as B A more twice A C is to A C, ſo is C A to A E,
by Diviſion it ſhall be that as B A more A C is to A C, ſo is C E to E A:
And becauſe as B A is to A C, ſo is E A to A R, by Compoſition it ſhall
be that as B A more A C is to A C, ſo is E R to R A: But as B A more
A C is to A C, ſo is C E to E A: Therefore, as C E is to E A, ſo is E R,
to R A, and both the Antecedents to both the Conſequents, that is, C R
required in the Perpendicular A D to find a part which ſhall be
paſt ex quiete in a Time equal to that in which the Plane A C is
paſt after the Fall along the ſaid part. Draw the Horizontal Line C B;
and as B A more twice A C is to A C, ſo let E A be to A R; And from
R let fall the Perpendicular R X unto D B. I ſay X is the point requi
red. And becauſe as B A more twice A C is to A C, ſo is C A to A E,
by Diviſion it ſhall be that as B A more A C is to A C, ſo is C E to E A:
And becauſe as B A is to A C, ſo is E A to A R, by Compoſition it ſhall
be that as B A more A C is to A C, ſo is E R to R A: But as B A more
A C is to A C, ſo is C E to E A: Therefore, as C E is to E A, ſo is E R,
to R A, and both the Antecedents to both the Conſequents, that is, C R