THEOR. XX. PROP. XXXI.
If a Right-Line ſhall be in any manner inclined
upon the Horizontal Line, the Plane produced
from a given point in the Horizon untill it
meet with the Inclined Plane, along which
the Deſcent is made in the ſhorteſt of all
Times, is that which ſhall divide the Angle
contained between the two Perpendiculars
drawn from the given Point, the one unto the
Horizontal Line, the other to the Inclined
Line, into two equal parts.
upon the Horizontal Line, the Plane produced
from a given point in the Horizon untill it
meet with the Inclined Plane, along which
the Deſcent is made in the ſhorteſt of all
Times, is that which ſhall divide the Angle
contained between the two Perpendiculars
drawn from the given Point, the one unto the
Horizontal Line, the other to the Inclined
Line, into two equal parts.
Let C D be a Line inclined in any manner upon the Hori
zontal Line A B, and let any point A be given in the Hori
zon, and from it let A C be drawn Perpendicular to A B,
and A E Perpendicular to C D, and let the Line F A divide the
Angle C A E into two equal parts. I ſay, that of all Planes incli
ned out of any point of the Line C D to the point A that ſame pro
duced along F A is it along
129[Figure 129]
which the Deſcent is made in
the ſhorteſt of all Times. Let
F G be drawn Parallel to AE;
the alternate Angles G F A
and F A E ſhall be equal: But
E A F is equal to that other
F A G: Therefore of the Tri
angle the Sides F G and G A
ſhall be equal. If therefore
about the Center G, at the di
ſtance G A, a Circle be deſcri
bed it ſhall paſſe by F, and ſhall
touch the Horizontal, and the Inclined Lines in the points A and F:
For the Angle G F C is a Right Angle, and likewiſe G F is equidiſtant
to A E: Whence it is manifeſt that all Lines produced from the point
A unto the inclined Plane do extend beyond the Circumference, and,
which followeth of conſequence, that the Motions along the ſame do
take up more Time than along F A. Which was to be demonſtrated.
zontal Line A B, and let any point A be given in the Hori
zon, and from it let A C be drawn Perpendicular to A B,
and A E Perpendicular to C D, and let the Line F A divide the
Angle C A E into two equal parts. I ſay, that of all Planes incli
ned out of any point of the Line C D to the point A that ſame pro
duced along F A is it along
129[Figure 129]which the Deſcent is made in
the ſhorteſt of all Times. Let
F G be drawn Parallel to AE;
the alternate Angles G F A
and F A E ſhall be equal: But
E A F is equal to that other
F A G: Therefore of the Tri
angle the Sides F G and G A
ſhall be equal. If therefore
about the Center G, at the di
ſtance G A, a Circle be deſcri
bed it ſhall paſſe by F, and ſhall
touch the Horizontal, and the Inclined Lines in the points A and F:
For the Angle G F C is a Right Angle, and likewiſe G F is equidiſtant
to A E: Whence it is manifeſt that all Lines produced from the point
A unto the inclined Plane do extend beyond the Circumference, and,
which followeth of conſequence, that the Motions along the ſame do
take up more Time than along F A. Which was to be demonſtrated.