1tion of the Squares of C B, D B, E B, or, if you will, in the doubled
proportion of the Lines. And if unto the Moveable moved beyond B
towards C with the Equable Lation we ſuppoſe the Perpendicular
Deſcent to be ſuperadded according to the quantity C I, in the Time
B C it ſhall be found conſtituted in the Term I. And proceeding farther,
146[Figure 146]
in the Time D B, namely,
in the double of B C, the
Space of the Deſcent down
wards ſhall be quadruple to
the firſt Space C I: For
it hath beendemonſtrated in
the firſt Trastate, that the
Spaces paſſed by GraveBo
dies with a Motion Natu
rally Accelerate are in du
plicate proportion of their Times. And it likewiſe followeth, that the
Space E H paſſed in the Time B E, ſhall be as G. So that it is manifeſtly
proved, that the Spaces E H, D F, C I, are to one another as the Squares
of the Lines E B, D B, C B. Now from the points I, F, and H draw
the Right Lines I O, F G, H L, Parallel to the ſaid E B; and each of
the Lines H L, F G, and I O ſhall be equal to each of the other Lines
E B, D B, and C B; as alſo each of thoſe B O, B G, and B L, ſhall be
equal to each of thoſe C I, D F, and E H: And the Square H L ſhall
be to the Square F G, as the Line L B to B G: And the Square F G
ſhall be to the Square I O, as G B to B O: Therefore the Points I, F,
and H are in one and the ſame Parabolical Line. And in like manner
it ſhall be demonſtrated, any equalparticles of Time of whatſoever Mag
nitude being taken, that the place of the Moveable whoſe Motion is
compounded of the like Lations, is in the ſame Times to be found in the
ſame Parabolick Line: Therefore the Propoſition is manifeſt.
proportion of the Lines. And if unto the Moveable moved beyond B
towards C with the Equable Lation we ſuppoſe the Perpendicular
Deſcent to be ſuperadded according to the quantity C I, in the Time
B C it ſhall be found conſtituted in the Term I. And proceeding farther,
146[Figure 146]in the Time D B, namely,
in the double of B C, the
Space of the Deſcent down
wards ſhall be quadruple to
the firſt Space C I: For
it hath beendemonſtrated in
the firſt Trastate, that the
Spaces paſſed by GraveBo
dies with a Motion Natu
rally Accelerate are in du
plicate proportion of their Times. And it likewiſe followeth, that the
Space E H paſſed in the Time B E, ſhall be as G. So that it is manifeſtly
proved, that the Spaces E H, D F, C I, are to one another as the Squares
of the Lines E B, D B, C B. Now from the points I, F, and H draw
the Right Lines I O, F G, H L, Parallel to the ſaid E B; and each of
the Lines H L, F G, and I O ſhall be equal to each of the other Lines
E B, D B, and C B; as alſo each of thoſe B O, B G, and B L, ſhall be
equal to each of thoſe C I, D F, and E H: And the Square H L ſhall
be to the Square F G, as the Line L B to B G: And the Square F G
ſhall be to the Square I O, as G B to B O: Therefore the Points I, F,
and H are in one and the ſame Parabolical Line. And in like manner
it ſhall be demonſtrated, any equalparticles of Time of whatſoever Mag
nitude being taken, that the place of the Moveable whoſe Motion is
compounded of the like Lations, is in the ſame Times to be found in the
ſame Parabolick Line: Therefore the Propoſition is manifeſt.
SALV. This Concluſion is gathered from the Converſion of the
firſt of thoſe two Propoſitions that went before, for the Parabola
being, for example, deſcribed by the points B H, if either of the
two F or I were not in the deſcribed Parabolick Line, it would be
within, or without; and by conſequence the Line F G would be
either greater or leſſer than that which ſhould determine in the Pa
rabolick Line; Wherefore the Square of HL would have, not to
the Square of F G, but to another greater or leſſer, the ſame pro
portion that the Line L B hath to BG, but it hath the ſame propor
tion to the Square of F G: Therefore the point F is in the Parabo
lick Line: And ſo all the reſt, &c.
firſt of thoſe two Propoſitions that went before, for the Parabola
being, for example, deſcribed by the points B H, if either of the
two F or I were not in the deſcribed Parabolick Line, it would be
within, or without; and by conſequence the Line F G would be
either greater or leſſer than that which ſhould determine in the Pa
rabolick Line; Wherefore the Square of HL would have, not to
the Square of F G, but to another greater or leſſer, the ſame pro
portion that the Line L B hath to BG, but it hath the ſame propor
tion to the Square of F G: Therefore the point F is in the Parabo
lick Line: And ſo all the reſt, &c.
SAGR. It cannot be denied but that the Diſcourſe is new, in
genious and concludent, arguing ex ſuppoſitione, that is, ſuppoſing
that the Tranſverſe Motion doth continue alwaies Equable, and
genious and concludent, arguing ex ſuppoſitione, that is, ſuppoſing
that the Tranſverſe Motion doth continue alwaies Equable, and