1of reſt, are alwayes determinate, and anſwer in proportion to the

parallels comprehended between two right lines that concur in

an angle, like to the angle B A E, or B A D, or any other

infinitely more acute, alwayes provided it be

But the diminution of the ſpaces thorow which the moveable is

to be conducted along the circumference of the wheel, is

tionate to another kind of diminution, comprehended between

lines that contain an angle infinitely more narrow and acute, than

any rectilineal angle, how acute ſoever, which is that in our

ſent caſe. Let any point be taken in the perpendicular A C, and

making it the centre, deſcribe at the diſtance C A, an arch A M P,

the which ſhall interſect the parallels that determine the degrees of

velocity, though they be very minute, and comprehended within

a moſt acute rectilineal angle; of which parallels the parts that

lie between the arch and the tangent A B, are the quantities of

the ſpaces, and of the returns upon the wheel, alwayes leſſer (and

with greater proportion leſſer, by how much neerer they approach

to the contact) than the ſaid parallels of which they are parts.

The parallels comprehended between the right lines in retiring

wards the angle diminiſh alwayes at the ſame rate, as v.g. A H

ing divided in two equal parts in F, the parallel H I ſhall be

ble to F G, and ſub-dividing F A, in two equal parts, the

lel produced from the point of the diviſion ſhall be the half of

F G; and continuing the ſub-diviſion in infinitum, the ſubſequent

parallels ſhall be alwayes half of the next preceding; but it doth

not ſo fall out in the lines intercepted between the tangent and

the circumference of the circle: For if the ſame ſub-diviſion be

made in F A; and ſuppoſing for example, that the parallel which

cometh from the point H, were double unto that which commeth

from F, this ſhall be more then double to the next following, and

continually the neerer we come towards the contact A, we ſhall

find the precedent lines contein the next following three, four,

ten, an hundred, a thouſand, an hundred thouſand, an hundred

millions of times, and more in infinitum. The brevity therefore of

ſuch lines is ſo reduced, that it far exceeds what is requiſite to make

the project, though never ſo light, return, nay more, continue

unremoveable upon the circumference.

parallels comprehended between two right lines that concur in

an angle, like to the angle B A E, or B A D, or any other

infinitely more acute, alwayes provided it be

But the diminution of the ſpaces thorow which the moveable is

to be conducted along the circumference of the wheel, is

tionate to another kind of diminution, comprehended between

lines that contain an angle infinitely more narrow and acute, than

any rectilineal angle, how acute ſoever, which is that in our

ſent caſe. Let any point be taken in the perpendicular A C, and

making it the centre, deſcribe at the diſtance C A, an arch A M P,

the which ſhall interſect the parallels that determine the degrees of

velocity, though they be very minute, and comprehended within

a moſt acute rectilineal angle; of which parallels the parts that

lie between the arch and the tangent A B, are the quantities of

the ſpaces, and of the returns upon the wheel, alwayes leſſer (and

with greater proportion leſſer, by how much neerer they approach

to the contact) than the ſaid parallels of which they are parts.

The parallels comprehended between the right lines in retiring

wards the angle diminiſh alwayes at the ſame rate, as v.g. A H

ing divided in two equal parts in F, the parallel H I ſhall be

ble to F G, and ſub-dividing F A, in two equal parts, the

lel produced from the point of the diviſion ſhall be the half of

F G; and continuing the ſub-diviſion in infinitum, the ſubſequent

parallels ſhall be alwayes half of the next preceding; but it doth

not ſo fall out in the lines intercepted between the tangent and

the circumference of the circle: For if the ſame ſub-diviſion be

made in F A; and ſuppoſing for example, that the parallel which

cometh from the point H, were double unto that which commeth

from F, this ſhall be more then double to the next following, and

continually the neerer we come towards the contact A, we ſhall

find the precedent lines contein the next following three, four,

ten, an hundred, a thouſand, an hundred thouſand, an hundred

millions of times, and more in infinitum. The brevity therefore of

ſuch lines is ſo reduced, that it far exceeds what is requiſite to make

the project, though never ſo light, return, nay more, continue

unremoveable upon the circumference.

SAGR. I very well comprehend the whole diſcourſe, and upon

what it layeth all its ſtreſſe, yet nevertheleſſe methinks that he

that would take pains to purſue it, might yet ſtart ſome further

queſtions, by ſaying, that of thoſe two cauſes which render the

deſcent of the moveable ſlower and ſlower in infinitum, it is

feſt, that that which dependeth on the vicinity to the firſt term of

the deſcent, increaſeth alwayes in the ſame proportion, like as the

parallels alwayes retain the ſame proportion to each other, &c.

what it layeth all its ſtreſſe, yet nevertheleſſe methinks that he

that would take pains to purſue it, might yet ſtart ſome further

queſtions, by ſaying, that of thoſe two cauſes which render the

deſcent of the moveable ſlower and ſlower in infinitum, it is

feſt, that that which dependeth on the vicinity to the firſt term of

the deſcent, increaſeth alwayes in the ſame proportion, like as the

parallels alwayes retain the ſame proportion to each other, &c.