1Velocity encreaſed in the manner aforeſaid would be equal to the
Space that the ſaid Moveable would paſſe, in caſe it were in the
ſame Time A C, moved with an Uniform Motion, whoſe degree of
Velocity ſhould be equal to E C, the half of B C. I now proceed
farther, and imagine the Moveable; having deſcended with an
Accelerate Motion, to have in the Inſtant
C the degree of Velocity B C: It is ma
85[Figure 85]
nifeſt, that if it did continue to move
with the ſame degree of Velocity B C,
without farther Acceleration, it would
paſſe in the following Time C I, a Space
double to that which it paſſed in the equal
Time A C, with the degree of Uniform
Velocity E C, the half of the Degree B C.
But becauſe the Moveable deſcendeth
with a Velocity encreaſed alwaies Uni
formly in all equal Times; it will add to
the degree C B in the following Time
C I, thoſe Tame Moments of Velocity
that encreaſe according to the Parallels of
the Triangle B F G, equal to the Triangle
A B C. So that adding to the degree of
Velocity G I, the half of the degree F G, the greateſt of thoſe ac
quired in the Accelerate Motion, and regulated by the Parallels of
the Triangle B F G, we ſhall have the degree of Velocity I N, with
which, with an Uniform Motion, it would have moved in the
Time C I: Which degree I N, being triple the degree E C, pro
veth that the Space paſſed in the ſecond Time C I ought to be tri
ple to that of the firſt Time C A. And if we ſhould ſuppoſe to be
added to A I another equal part of Time I O, and the Triangle to
be enlarged unto A P O; it is manifeſt, that if the Motion ſhould
continue for all the Time I O with the degree of Velocity I F,
acquired in the Accelerate Motion in the Time A I, that degree
I F being Quadruple to E C, the Space paſſed would be Quadruple
to that paſſed in the equal firſt Time A C: But continuing the
encreaſe of the Uniform Acceleration in the Triangle F P Q like
to that of the Triangle A B C, which being reduced to equable
Motion addeth the degree equal to E C, Q R being added, equal
to E C, we ſhall have the whole Equable Velocity exerciſed in the
Time I O, quintuple to the Equable Velocity of the firſt Time A C,
and therefore the Space paſſed quintuple to that paſt in the firſt
Time A C. We ſee therefore, even by this familiar computation,
That the Spaces paſſed in equal Times by a Moveable which
departing from Reſt goeth acquiring Velocity, according to the
encreaſe of the Time, are to one another as the odd Numbers ab
Space that the ſaid Moveable would paſſe, in caſe it were in the
ſame Time A C, moved with an Uniform Motion, whoſe degree of
Velocity ſhould be equal to E C, the half of B C. I now proceed
farther, and imagine the Moveable; having deſcended with an
Accelerate Motion, to have in the Inſtant
C the degree of Velocity B C: It is ma
85[Figure 85]nifeſt, that if it did continue to move
with the ſame degree of Velocity B C,
without farther Acceleration, it would
paſſe in the following Time C I, a Space
double to that which it paſſed in the equal
Time A C, with the degree of Uniform
Velocity E C, the half of the Degree B C.
But becauſe the Moveable deſcendeth
with a Velocity encreaſed alwaies Uni
formly in all equal Times; it will add to
the degree C B in the following Time
C I, thoſe Tame Moments of Velocity
that encreaſe according to the Parallels of
the Triangle B F G, equal to the Triangle
A B C. So that adding to the degree of
Velocity G I, the half of the degree F G, the greateſt of thoſe ac
quired in the Accelerate Motion, and regulated by the Parallels of
the Triangle B F G, we ſhall have the degree of Velocity I N, with
which, with an Uniform Motion, it would have moved in the
Time C I: Which degree I N, being triple the degree E C, pro
veth that the Space paſſed in the ſecond Time C I ought to be tri
ple to that of the firſt Time C A. And if we ſhould ſuppoſe to be
added to A I another equal part of Time I O, and the Triangle to
be enlarged unto A P O; it is manifeſt, that if the Motion ſhould
continue for all the Time I O with the degree of Velocity I F,
acquired in the Accelerate Motion in the Time A I, that degree
I F being Quadruple to E C, the Space paſſed would be Quadruple
to that paſſed in the equal firſt Time A C: But continuing the
encreaſe of the Uniform Acceleration in the Triangle F P Q like
to that of the Triangle A B C, which being reduced to equable
Motion addeth the degree equal to E C, Q R being added, equal
to E C, we ſhall have the whole Equable Velocity exerciſed in the
Time I O, quintuple to the Equable Velocity of the firſt Time A C,
and therefore the Space paſſed quintuple to that paſt in the firſt
Time A C. We ſee therefore, even by this familiar computation,
That the Spaces paſſed in equal Times by a Moveable which
departing from Reſt goeth acquiring Velocity, according to the
encreaſe of the Time, are to one another as the odd Numbers ab