Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

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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.
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.