1half of O D, would paſſe in the Times E A and D A. If it be proved
therefore that theſe Spaces M H and L H are in duplicate proportion to
the Times E A and D A; We ſhall have done that which was intended.
But in the fourth Propoſition of the Firſt Book we have demonſtrated:
That the Spaces paſt by two Moveables with an Equable Motion are
to each other in a proportion compounded of the proportion of the Velo
cities and of the proportion of the Times: But in this caſe the propor
tion of the Velocities and the proportion of the Times is the ſame (for
as the half of P E is to the half of O D, or the whole P E to the whole
O D, ſo is A E to A D: Therefore the proportion of the Spaces paſ
ſed is double to the proportion of the Times. Which was to be demon
ſtrated.
therefore that theſe Spaces M H and L H are in duplicate proportion to
the Times E A and D A; We ſhall have done that which was intended.
But in the fourth Propoſition of the Firſt Book we have demonſtrated:
That the Spaces paſt by two Moveables with an Equable Motion are
to each other in a proportion compounded of the proportion of the Velo
cities and of the proportion of the Times: But in this caſe the propor
tion of the Velocities and the proportion of the Times is the ſame (for
as the half of P E is to the half of O D, or the whole P E to the whole
O D, ſo is A E to A D: Therefore the proportion of the Spaces paſ
ſed is double to the proportion of the Times. Which was to be demon
ſtrated.
Hence likewiſe it is manifeſt, that the proportion of the ſame Spaces
is double to the proportions of the greateſt degrees of Velocity: that is,
of the Lines P E and O D: becauſe P E is to O D, as E A to D A.
is double to the proportions of the greateſt degrees of Velocity: that is,
of the Lines P E and O D: becauſe P E is to O D, as E A to D A.
COROLARY I.
Hence it is manifeſt, that if there were many equal Times taken in or
der from the firſt Inſtant or beginniug of the Motion, as ſuppoſe
A D, D E, E F, F G, in which the Spaces H L, L M, M N, N I
are paſſed, thoſe Spaces ſhall be to one another as the odd numbers
from an Vnite: ſcilicet, as 1, 3, 5, 7. For this is the Rate or pro
portion of the exceſſes of the Squares of Lines that equally exceed
one another, and the exceſſe of which is equal to the least of them,
or, if you will, of Squares that follow one another, beginning ab
Unitate. Whilſt therefore the degree of Velocity is encreaſed ac
cording to the ſimple Series of Numbers in equal Times, the Spaces
paſt in thoſe Times make their encreaſe according to the Series of
odd Numbers from an Vnite.
der from the firſt Inſtant or beginniug of the Motion, as ſuppoſe
A D, D E, E F, F G, in which the Spaces H L, L M, M N, N I
are paſſed, thoſe Spaces ſhall be to one another as the odd numbers
from an Vnite: ſcilicet, as 1, 3, 5, 7. For this is the Rate or pro
portion of the exceſſes of the Squares of Lines that equally exceed
one another, and the exceſſe of which is equal to the least of them,
or, if you will, of Squares that follow one another, beginning ab
Unitate. Whilſt therefore the degree of Velocity is encreaſed ac
cording to the ſimple Series of Numbers in equal Times, the Spaces
paſt in thoſe Times make their encreaſe according to the Series of
odd Numbers from an Vnite.
SAGR. Be pleaſed to ſtay your Reading, whilſt I do paraphraſe
touching a certain Conjecture that came into my mind
but even now; for the explanation of which, unto your under
ſtanding and my own, I will deſcribe a ſhort Scheme: in which I
fanſie by the Line A I the continuation of the Time after the firſt
Inſtant, applying the Right Line A F unto A according to any
Angle: and joyning together the Terms I F, I divide the Time A I
in half at C, and then draw C B parallel to I F. And then conſide
ring B C, as the greateſt degree of Velocity which beginning from
Reſt in the firſt Inſtant of the Time A goeth augmenting accord
ing to the encreaſe of the Parallels to B C, drawn in the Triangle
A B C, (which is all one as to encreaſe according to the encreaſe
of the Time) I admit without diſpute, upon what hath been ſaid
already, That the Space paſt by the falling Moveable with the
touching a certain Conjecture that came into my mind
but even now; for the explanation of which, unto your under
ſtanding and my own, I will deſcribe a ſhort Scheme: in which I
fanſie by the Line A I the continuation of the Time after the firſt
Inſtant, applying the Right Line A F unto A according to any
Angle: and joyning together the Terms I F, I divide the Time A I
in half at C, and then draw C B parallel to I F. And then conſide
ring B C, as the greateſt degree of Velocity which beginning from
Reſt in the firſt Inſtant of the Time A goeth augmenting accord
ing to the encreaſe of the Parallels to B C, drawn in the Triangle
A B C, (which is all one as to encreaſe according to the encreaſe
of the Time) I admit without diſpute, upon what hath been ſaid
already, That the Space paſt by the falling Moveable with the