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SALV. You ſee then, that the Propoſition which ſaith, that

the motion by the perpendicular is more ſwift than by the incli

nation, holds not true univerſally, but onely of the motions,

which begin from the extremity, namely from the point of reſt:

without which reſtriction, the Propoſition would be ſo deficient,

that its very direct contrary might be true; namely, that the mo

tion in the inclining plane is ſwifter than in the perpendicular:

for it is certain, that in the ſaid inclination, we may take a ſpace

paſt by the moveable in leſs time, than the like ſpace paſt in the

perpendicular. Now becauſe the motion in the inclination is in

ſome places more, in ſome leſs, than in the perpendicular; there

fore in ſome places of the inclination, the time of motion of the

moveable, ſhall have a greater proportion to the time of the motion

of the moveable, by ſome places of the perpendicular, than the

ſpace paſſed, to the ſpace paſſed: and in other places, the pro

portion of the time to the time, ſhall be leſs than that of the

ſpace to the ſpace. As for example: two moveables departing

from their quieſcence, namely, from the point C, one by the per

pendicular C B, [in Fig. 4.] and the other by the inclination C A,

in the time that, in the perpendicular, the moveable ſhall have

paſt all C B, the other ſhall have paſt C T leſſer. And therefore

the time by C T, to the time by C B (which is equal) ſhall have

a greater proportion than the line C T to C B, being that the

ſame to the leſs, hath a greater proportion than to the greater.

And on the contrary, if in C A, prolonged as much as is requi

ſite, one ſhould take a part equal to C B, but paſt in a ſhorter

time; the time in the inclination ſhall have a leſs proportion to

the time in the perpendicular, than the ſpace to the ſpace. If

therefore in the inclination and perpendicular, we may ſuppoſe

ſuch ſpaces and velocities, that the proportion between the ſaid

ſpaces be greater and leſs than the proportion of the times; we

may eaſily grant, that there are alſo ſpaces, by which the times

of the motions retain the ſame proportion as the ſpaces.

the motion by the perpendicular is more ſwift than by the incli

nation, holds not true univerſally, but onely of the motions,

which begin from the extremity, namely from the point of reſt:

without which reſtriction, the Propoſition would be ſo deficient,

that its very direct contrary might be true; namely, that the mo

tion in the inclining plane is ſwifter than in the perpendicular:

for it is certain, that in the ſaid inclination, we may take a ſpace

paſt by the moveable in leſs time, than the like ſpace paſt in the

perpendicular. Now becauſe the motion in the inclination is in

ſome places more, in ſome leſs, than in the perpendicular; there

fore in ſome places of the inclination, the time of motion of the

moveable, ſhall have a greater proportion to the time of the motion

of the moveable, by ſome places of the perpendicular, than the

ſpace paſſed, to the ſpace paſſed: and in other places, the pro

portion of the time to the time, ſhall be leſs than that of the

ſpace to the ſpace. As for example: two moveables departing

from their quieſcence, namely, from the point C, one by the per

pendicular C B, [in Fig. 4.] and the other by the inclination C A,

in the time that, in the perpendicular, the moveable ſhall have

paſt all C B, the other ſhall have paſt C T leſſer. And therefore

the time by C T, to the time by C B (which is equal) ſhall have

a greater proportion than the line C T to C B, being that the

ſame to the leſs, hath a greater proportion than to the greater.

And on the contrary, if in C A, prolonged as much as is requi

ſite, one ſhould take a part equal to C B, but paſt in a ſhorter

time; the time in the inclination ſhall have a leſs proportion to

the time in the perpendicular, than the ſpace to the ſpace. If

therefore in the inclination and perpendicular, we may ſuppoſe

ſuch ſpaces and velocities, that the proportion between the ſaid

ſpaces be greater and leſs than the proportion of the times; we

may eaſily grant, that there are alſo ſpaces, by which the times

of the motions retain the ſame proportion as the ſpaces.

SAGR. I am already freed from my greateſt doubt, and con

ceive that to be not onely poſſible, but neceſſary, which I but

now thought a contradiction: but nevertheleſs I underſtand not

as yet, that this whereof we now are ſpeaking, is one of theſe

poſſible or neceſſary caſes; ſo as that it ſhould be true, that the

time of deſcent by C A, to the time of the fall by C B, hath the

ſame proportion that the line C A hath to C B; whence it may

without contradiction be affirmed, that the velocity by the incli

nation C A, and by the perpendicular C B, are equal.

ceive that to be not onely poſſible, but neceſſary, which I but

now thought a contradiction: but nevertheleſs I underſtand not

as yet, that this whereof we now are ſpeaking, is one of theſe

poſſible or neceſſary caſes; ſo as that it ſhould be true, that the

time of deſcent by C A, to the time of the fall by C B, hath the

ſame proportion that the line C A hath to C B; whence it may

without contradiction be affirmed, that the velocity by the incli

nation C A, and by the perpendicular C B, are equal.