THEOR. I. PROP. I.
The time in which a Space is paſſed by a Movea
ble with a Motion Vniformly Accelerate, out of
Reſt, is equal to the Time in which the ſame
Space would be paſt by the ſame Moveable
with an Equable Motion, the degree of whoſe
Velocity is ſubduple to the greateſt and ulti
mate degree of the Velocity of the former Vni
formly Accelerate Motion.
ble with a Motion Vniformly Accelerate, out of
Reſt, is equal to the Time in which the ſame
Space would be paſt by the ſame Moveable
with an Equable Motion, the degree of whoſe
Velocity is ſubduple to the greateſt and ulti
mate degree of the Velocity of the former Vni
formly Accelerate Motion.
Let us by the extenſion A B repreſent the Time, in which the
Space C D is paſſed by a Moveable with a Motion Vniformly
Accelerate, out of Reſt in C: and let the greateſt and laſt de-
83[Figure 83]
gree of Velocity acquired in the Inſtants of the Time
A B be repreſented by E B; and conſtitute at plea
ſure upon A B any number of parts, and thorow the
points of diviſion draw as many Lines, continued
out unto the Line A E, and equidiſtant to B E,
which will repreſent the encreaſe of the degrees of
Velocity after the firſt Inſtant A. Then divide B E
into two equall parts in F, and draw F G and A G
parallel to B A and B F: The Parallelogram A G
F B ſhall be equall to the Triangle A E B, its Side
G F dividing A E into two equall parts in I: For
if the Parallels of the Triangle A E B be continued
out unto I G F, we ſhall have the Aggregate of all
the Parallels contained in the Quadrilatural Figure
equal to the Aggregate of all the Parallels compre
hended in the Triangle A E B; For thoſe in the Triangle I E F are equal
to thoſe contained in the Triangle G I A, and thoſe that are in the Tra
pezium are in common. Now ſince all and ſingular the Inſtants of Time
do anſwer to all and ſingular the Points of the Line A B; and ſince the
Parallels contained in the Triangle A E B do repreſent the degrees of Ac
celeration or encreaſing Velocity, and the Parallels contained in the Pa
rallelogram do likewiſe repreſent as many degrees of Equable Motion or
unencreaſing Velocity: It appeareth, that as many Moments of Velocity
paſſed in the Accelerate Motion according to the encreaſing Parallels of the
Triangle A E B, as in the Equable Motion according to the Parallels of
the Parallelogram G B: Becauſe what is wanting in the firſt half of the
Space C D is paſſed by a Moveable with a Motion Vniformly
Accelerate, out of Reſt in C: and let the greateſt and laſt de-
83[Figure 83]gree of Velocity acquired in the Inſtants of the Time
A B be repreſented by E B; and conſtitute at plea
ſure upon A B any number of parts, and thorow the
points of diviſion draw as many Lines, continued
out unto the Line A E, and equidiſtant to B E,
which will repreſent the encreaſe of the degrees of
Velocity after the firſt Inſtant A. Then divide B E
into two equall parts in F, and draw F G and A G
parallel to B A and B F: The Parallelogram A G
F B ſhall be equall to the Triangle A E B, its Side
G F dividing A E into two equall parts in I: For
if the Parallels of the Triangle A E B be continued
out unto I G F, we ſhall have the Aggregate of all
the Parallels contained in the Quadrilatural Figure
equal to the Aggregate of all the Parallels compre
hended in the Triangle A E B; For thoſe in the Triangle I E F are equal
to thoſe contained in the Triangle G I A, and thoſe that are in the Tra
pezium are in common. Now ſince all and ſingular the Inſtants of Time
do anſwer to all and ſingular the Points of the Line A B; and ſince the
Parallels contained in the Triangle A E B do repreſent the degrees of Ac
celeration or encreaſing Velocity, and the Parallels contained in the Pa
rallelogram do likewiſe repreſent as many degrees of Equable Motion or
unencreaſing Velocity: It appeareth, that as many Moments of Velocity
paſſed in the Accelerate Motion according to the encreaſing Parallels of the
Triangle A E B, as in the Equable Motion according to the Parallels of
the Parallelogram G B: Becauſe what is wanting in the firſt half of the