1is to E F, as F E is to E C, that F C determineth the Time along C O:
And if a part of the Horizontal Line T C double to C A be divided in
two equal parts in V, the extenſion towards X ſhall be prolonged in in
finitum, whilſt it ſeeks to meet with the prolonged Line A E: And the
proportion of the Infinite Line T X to the Infinite Line V X, ſhall be
no other than the proportion of the Infinite Line V X to the Infinite
Line X C.
And if a part of the Horizontal Line T C double to C A be divided in
two equal parts in V, the extenſion towards X ſhall be prolonged in in
finitum, whilſt it ſeeks to meet with the prolonged Line A E: And the
proportion of the Infinite Line T X to the Infinite Line V X, ſhall be
no other than the proportion of the Infinite Line V X to the Infinite
Line X C.
We may conclude the ſelf-ſame thing another way by reaſſuming the
ſame Reaſoning that we uſed in the Demonſtration of the firſt Propoſi
tion. For reſuming the Triangle A B C, repreſenting to us by its Pa
rallels to the Baſe B C the Degrees of Velocity continually encreaſed ac
cording to the encreaſes of the Time; from which, ſince they are infi
nite, like as the Points are infinite in the Line A C, and the Inſtants
in any Time, ſhall reſult the Superficies of that ſame Triangle, if we
underſtand the Motion to continue for ſuch another Time, but no far
ther with an Accelerate, but with an Equable Motion, according to the
greateſt degree of Velocity acquired, which degree is repreſented
by the Line B C. Of ſuch degrees ſhall be made up an Aggregate like to
a Parallelogram A D B C, which is the double of
119[Figure 119]
the Triangle A B C. Wherefore the Space which
with degrees like to thoſe ſhall be paſſed in the ſame
Time, ſhall be double to the Space paſt with the de
grees of Velocity repreſented by the Triangle A B C:
But along the Horizontal Plane the Motion is Equa
ble, for that there is no cauſe of Acceleration, or Re
tardation: Therefore it may be concluded that the
Space C D, paſſed in a Time equall to the Time A C is double to the
Space A C: For this Motion is made ex quiete Accelerate according
to the Parallels of the Triangle; and that according to the Parallels
of the Parallelogram, which, becauſe they are infinite, are donble to
the infinite Parallels of the Triangle.
ſame Reaſoning that we uſed in the Demonſtration of the firſt Propoſi
tion. For reſuming the Triangle A B C, repreſenting to us by its Pa
rallels to the Baſe B C the Degrees of Velocity continually encreaſed ac
cording to the encreaſes of the Time; from which, ſince they are infi
nite, like as the Points are infinite in the Line A C, and the Inſtants
in any Time, ſhall reſult the Superficies of that ſame Triangle, if we
underſtand the Motion to continue for ſuch another Time, but no far
ther with an Accelerate, but with an Equable Motion, according to the
greateſt degree of Velocity acquired, which degree is repreſented
by the Line B C. Of ſuch degrees ſhall be made up an Aggregate like to
a Parallelogram A D B C, which is the double of
119[Figure 119]the Triangle A B C. Wherefore the Space which
with degrees like to thoſe ſhall be paſſed in the ſame
Time, ſhall be double to the Space paſt with the de
grees of Velocity repreſented by the Triangle A B C:
But along the Horizontal Plane the Motion is Equa
ble, for that there is no cauſe of Acceleration, or Re
tardation: Therefore it may be concluded that the
Space C D, paſſed in a Time equall to the Time A C is double to the
Space A C: For this Motion is made ex quiete Accelerate according
to the Parallels of the Triangle; and that according to the Parallels
of the Parallelogram, which, becauſe they are infinite, are donble to
the infinite Parallels of the Triangle.
Moreover it may farther be obſerved, that what ever degree of
ſwiftneſs is to be found in the Moveable, is indelibly impreſſed upon it
of its own nature, all external cauſes of Acceleration or Retardation
being removed; which hapneth only in Horizontal Planes: for in de
clining Planes there is cauſe of greater Acceleration, and in the riſing
Planes of greater Retardation. From whence in like manner it fol
loweth that the Motion along the Horizontal Plane is alſo Perpetual:
for if it be Equable, it can neither be weakned nor retarded, nor much
leſſe deſtroyed. Farthermore, the degree of Celerity acquired by the
Moveable in a Natural Deſcent, being of its own Nature Indelible and
Penpetual, it is worthy conſideration, that if after the Deſcent along a
declining Plane a Reflexion be made along another Plane that is riſing,
in this latter there is cauſe of Retardation, for in theſe kind of Planes
ſwiftneſs is to be found in the Moveable, is indelibly impreſſed upon it
of its own nature, all external cauſes of Acceleration or Retardation
being removed; which hapneth only in Horizontal Planes: for in de
clining Planes there is cauſe of greater Acceleration, and in the riſing
Planes of greater Retardation. From whence in like manner it fol
loweth that the Motion along the Horizontal Plane is alſo Perpetual:
for if it be Equable, it can neither be weakned nor retarded, nor much
leſſe deſtroyed. Farthermore, the degree of Celerity acquired by the
Moveable in a Natural Deſcent, being of its own Nature Indelible and
Penpetual, it is worthy conſideration, that if after the Deſcent along a
declining Plane a Reflexion be made along another Plane that is riſing,
in this latter there is cauſe of Retardation, for in theſe kind of Planes
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