THEOR. XXI. PROP. XXXII.
If two points be taken in the Horizon, and any
Line ſhould be inclined from one of them to
wards the other, out of which a Right-Line is
drawn unto the Inclined Line, cutting off a
part thereof equal to that which is included
between the points of the Horizon, the De
ſcent along this laſt drawn ſhall be ſooner per
formed, than along any other Right Lines pro
duced from the ſame point unto the ſaid Incli
ned Line. And along other Lines which are
on each hand of this by equal Angles a De
ſcent ſhall be made in equal Times.
Line ſhould be inclined from one of them to
wards the other, out of which a Right-Line is
drawn unto the Inclined Line, cutting off a
part thereof equal to that which is included
between the points of the Horizon, the De
ſcent along this laſt drawn ſhall be ſooner per
formed, than along any other Right Lines pro
duced from the ſame point unto the ſaid Incli
ned Line. And along other Lines which are
on each hand of this by equal Angles a De
ſcent ſhall be made in equal Times.
In the Horizon let there be two points A and B, and from B incline
the Right Line B C, in which from the Term B take B D equal to
the ſaid B A, and draw a Line from A to D. I ſay, that the De
ſcent along A D is more ſwiftly made, than along any other whatſoever
drawn from the point A unto the inclined Line B C. For out of the
points A and D unto B A and
131[Figure 131]
B D draw the Perpendiculars
A E and D E, interſecting one
another in E: and foraſmuch as
in the equicrural Triangle A B D
the Angles B A D and B D A
are equal, the remainders to the
Right-Angles D A E and E D A
ſhall be equal. Therefore a Circle
deſcribed about the Center E at
the diſtance A E ſhall alſo paſſe
by D; and the Lines B A and
B D will touch it in the points A
and D. And ſince A is the end of the Perpendicular A E, the Deſcent
along A D ſhall be ſooner performed, than along any other produced from
the ſame Term A unto the Line B C beyond the Circumference of the
Circle: Which was firſt to be proved.
the Right Line B C, in which from the Term B take B D equal to
the ſaid B A, and draw a Line from A to D. I ſay, that the De
ſcent along A D is more ſwiftly made, than along any other whatſoever
drawn from the point A unto the inclined Line B C. For out of the
points A and D unto B A and
131[Figure 131]B D draw the Perpendiculars
A E and D E, interſecting one
another in E: and foraſmuch as
in the equicrural Triangle A B D
the Angles B A D and B D A
are equal, the remainders to the
Right-Angles D A E and E D A
ſhall be equal. Therefore a Circle
deſcribed about the Center E at
the diſtance A E ſhall alſo paſſe
by D; and the Lines B A and
B D will touch it in the points A
and D. And ſince A is the end of the Perpendicular A E, the Deſcent
along A D ſhall be ſooner performed, than along any other produced from
the ſame Term A unto the Line B C beyond the Circumference of the
Circle: Which was firſt to be proved.
But if in the Perpendicular A E being prolonged any Center be taken as
F, and at the diſtance F A the Circle A G C be deſcribed cutting the
Tangent Line in the points G and C; drawing A G and A C they ſhall
make equal Angles with the middle Line A D by what hath been afore
F, and at the diſtance F A the Circle A G C be deſcribed cutting the
Tangent Line in the points G and C; drawing A G and A C they ſhall
make equal Angles with the middle Line A D by what hath been afore
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