1equal. Draw Horizontal Parallels to the Line of Elevation E F and
B D, which cutteth A E in G. And be-
97[Figure 97]
cauſe the proportion of F A to A D, is
double the proportion of E A to A B; and
as F A to A D, ſo is E A to A G: There
fore the proportion of E A to A G, is dou
ble the proportion of E A to A B: There
fore A B is a Mean-Proportional between
E A and A G: And becauſe the Time of the
Deſcent along A B, is to the Time of the De
ſcent along A G, as A B to A G; and the
Time of the Deſcent along AG, is to the Time of the Deſcent along A E, as
A G is to the Mean-proportional between A G and A E, which is A B:
Therefore ex equali, the Time along A B is to the Time along A E, as A B
unto it ſelf: Therefore the Times are equal: Which was to be demonſtrated.
B D, which cutteth A E in G. And be-
97[Figure 97]cauſe the proportion of F A to A D, is
double the proportion of E A to A B; and
as F A to A D, ſo is E A to A G: There
fore the proportion of E A to A G, is dou
ble the proportion of E A to A B: There
fore A B is a Mean-Proportional between
E A and A G: And becauſe the Time of the
Deſcent along A B, is to the Time of the De
ſcent along A G, as A B to A G; and the
Time of the Deſcent along AG, is to the Time of the Deſcent along A E, as
A G is to the Mean-proportional between A G and A E, which is A B:
Therefore ex equali, the Time along A B is to the Time along A E, as A B
unto it ſelf: Therefore the Times are equal: Which was to be demonſtrated.
THEOR. VIII. PROP. VIII.
In Planes cut by the ſame Circle, erect to the
Horizon, in thoſe which meet with the end of
the erect Diameter, whether upper or lower,
the Times of the Motions are equal to the
Time of the Fall along the Diameter: and in
thoſe which fall ſhort of the Diameter, the
Times are ſhorter; and in thoſe which inter
ſect the Diameter, they are longer.
Horizon, in thoſe which meet with the end of
the erect Diameter, whether upper or lower,
the Times of the Motions are equal to the
Time of the Fall along the Diameter: and in
thoſe which fall ſhort of the Diameter, the
Times are ſhorter; and in thoſe which inter
ſect the Diameter, they are longer.
Let A B be the Perpendicular Diameter of the Circle erect to the
Horizon. That the Times of the Motions along the Planes pro
duced out of the Terms A and B unto the Circumference are equal,
hath already been demonſtrated: That the Time of the Deſcent along
the Plane D F, not reaching to the
98[Figure 98]
Diameter is ſborter, is demonſtrated
by drawing the Plane D B, which
ſhall be both longer and leſſe decli
ning than D F. Therefore the Time
along D F is ſhorter than the Time
along D B, that is, along A B. And
that the Time of the Deſcent along
the Plane that interſecteth the Dia
meter, as C O is longer, doth in the
ſame manner appear, for that it is
longer and leſſe declining than C B: Therefore the Propoſition is de
monſtrated.
Horizon. That the Times of the Motions along the Planes pro
duced out of the Terms A and B unto the Circumference are equal,
hath already been demonſtrated: That the Time of the Deſcent along
the Plane D F, not reaching to the
98[Figure 98]Diameter is ſborter, is demonſtrated
by drawing the Plane D B, which
ſhall be both longer and leſſe decli
ning than D F. Therefore the Time
along D F is ſhorter than the Time
along D B, that is, along A B. And
that the Time of the Deſcent along
the Plane that interſecteth the Dia
meter, as C O is longer, doth in the
ſame manner appear, for that it is
longer and leſſe declining than C B: Therefore the Propoſition is de
monſtrated.