THEOR. XIX. PROP. XXX.
If a Perpendicular be let fall from any point of the
Horizontal Line, and out of another point in
the ſame Horizontal Line a Plane be drawn
forth untill it meet the Perpendicular, along
which a Moveable deſcendeth in the ſhorteſt
time unto the ſaid Perpendicular, this Plane
ſhall be that which cutteth off a part equall to
the diſtance of the aſſigned point from the end
of the Perpendicular.
Horizontal Line, and out of another point in
the ſame Horizontal Line a Plane be drawn
forth untill it meet the Perpendicular, along
which a Moveable deſcendeth in the ſhorteſt
time unto the ſaid Perpendicular, this Plane
ſhall be that which cutteth off a part equall to
the diſtance of the aſſigned point from the end
of the Perpendicular.
Let the Perpendicular B D be let fall from the point B of the Ho
rizontal Line A C, in which let there be any point C, and in the
Perpendicular let the Diſtance B E be ſuppoſed equal to the Di
ſtance B C, and draw C E. I ſay, that of all Planes inclined out of
the point C till they meet the Perpendicular C E is that, along which
in the ſhorteſt of all Times the Deſcent
128[Figure 128]
is made unto the Perpendicular. For
let the Planes C F and C G be inclined
above and below, and draw I K a Tan
gent unto the Semidiameter B C of the
deſcribed Circle in C, which ſhall be
equidiſtant from the Perpendicular;
and unto the ſaid C F let E K be Paral
lel cutting the Circumference of the Cir
cle in L: It is manifeſt that the Time of
the Deſcent along L E is equal to the
Time of the Deſcent along C E: But
the Time along K E is longer than along
L E: Therefore the Time along K E is
longer than that along C E: But the
Time along K E is equal to the Time a
long C F, they being equal, and drawn
according to the ſame Inclination: Likewiſe ſince C G, and I E are
equal, and inclined according to the ſame Inclination, the Times of the
Motions along them ſhall be equal: But H E being ſhorter than I E, the
Time along it is alſo ſhorter than I E: Therefore the Time alſo along
C E, (which is equal to the Time along H E) ſhall be ſhorter than the
Time along I E: The Propoſition, therefore, is manifeſt.
rizontal Line A C, in which let there be any point C, and in the
Perpendicular let the Diſtance B E be ſuppoſed equal to the Di
ſtance B C, and draw C E. I ſay, that of all Planes inclined out of
the point C till they meet the Perpendicular C E is that, along which
in the ſhorteſt of all Times the Deſcent
128[Figure 128]is made unto the Perpendicular. For
let the Planes C F and C G be inclined
above and below, and draw I K a Tan
gent unto the Semidiameter B C of the
deſcribed Circle in C, which ſhall be
equidiſtant from the Perpendicular;
and unto the ſaid C F let E K be Paral
lel cutting the Circumference of the Cir
cle in L: It is manifeſt that the Time of
the Deſcent along L E is equal to the
Time of the Deſcent along C E: But
the Time along K E is longer than along
L E: Therefore the Time along K E is
longer than that along C E: But the
Time along K E is equal to the Time a
long C F, they being equal, and drawn
according to the ſame Inclination: Likewiſe ſince C G, and I E are
equal, and inclined according to the ſame Inclination, the Times of the
Motions along them ſhall be equal: But H E being ſhorter than I E, the
Time along it is alſo ſhorter than I E: Therefore the Time alſo along
C E, (which is equal to the Time along H E) ſhall be ſhorter than the
Time along I E: The Propoſition, therefore, is manifeſt.