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

#### List of thumbnails

< >
< >
page |< < of 701 > >|
<archimedes>
<text>
<body>
<chap>
<p type="main">
<s>
of reſt, are alwayes determinate, and anſwer in proportion to the
<lb/>
parallels comprehended between two right lines that concur in
<lb/>
an angle, like to the angle B A E, or B A D, or any other
<lb/>
infinitely more acute, alwayes provided it be
<lb/>
But the diminution of the ſpaces thorow which the moveable is
<lb/>
to be conducted along the circumference of the wheel, is
<lb/>
tionate to another kind of diminution, comprehended between
<lb/>
lines that contain an angle infinitely more narrow and acute, than
<lb/>
any rectilineal angle, how acute ſoever, which is that in our
<lb/>
ſent caſe. </s>
<s>Let any point be taken in the perpendicular A C, and
<lb/>
making it the centre, deſcribe at the diſtance C A, an arch A M P,
<lb/>
the which ſhall interſect the parallels that determine the degrees of
<lb/>
velocity, though they be very minute, and comprehended within
<lb/>
a moſt acute rectilineal angle; of which parallels the parts that
<lb/>
lie between the arch and the tangent A B, are the quantities of
<lb/>
the ſpaces, and of the returns upon the wheel, alwayes leſſer (and
<lb/>
with greater proportion leſſer, by how much neerer they approach
<lb/>
to the contact) than the ſaid parallels of which they are parts.
<lb/>
</s>
<s>The parallels comprehended between the right lines in retiring
<lb/>
wards the angle diminiſh alwayes at the ſame rate, as
<emph type="italics"/>
v.g.
<emph.end type="italics"/>
A H
<lb/>
ing divided in two equal parts in F, the parallel H I ſhall be
<lb/>
ble to F G, and ſub-dividing F A, in two equal parts, the
<lb/>
lel produced from the point of the diviſion ſhall be the half of
<lb/>
F G; and continuing the ſub-diviſion
<emph type="italics"/>
in infinitum,
<emph.end type="italics"/>
the ſubſequent
<lb/>
parallels ſhall be alwayes half of the next preceding; but it doth
<lb/>
not ſo fall out in the lines intercepted between the tangent and
<lb/>
the circumference of the circle: For if the ſame ſub-diviſion be
<lb/>
made in F A; and ſuppoſing for example, that the parallel which
<lb/>
cometh from the point H, were double unto that which commeth
<lb/>
from F, this ſhall be more then double to the next following, and
<lb/>
continually the neerer we come towards the contact A, we ſhall
<lb/>
find the precedent lines contein the next following three, four,
<lb/>
ten, an hundred, a thouſand, an hundred thouſand, an hundred
<lb/>
millions of times, and more
<emph type="italics"/>
in infinitum.
<emph.end type="italics"/>
The brevity therefore of
<lb/>
ſuch lines is ſo reduced, that it far exceeds what is requiſite to make
<lb/>
the project, though never ſo light, return, nay more, continue
<lb/>
unremoveable upon the circumference.</s>
</p>
<p type="main">
<s>SAGR. </s>
<s>I very well comprehend the whole diſcourſe, and upon
<lb/>
what it layeth all its ſtreſſe, yet nevertheleſſe methinks that he
<lb/>
that would take pains to purſue it, might yet ſtart ſome further
<lb/>
queſtions, by ſaying, that of thoſe two cauſes which render the
<lb/>
deſcent of the moveable ſlower and ſlower
<emph type="italics"/>
in infinitum,
<emph.end type="italics"/>
it is
<lb/>
feſt, that that which dependeth on the vicinity to the firſt term of
<lb/>
the deſcent, increaſeth alwayes in the ſame proportion, like as the
<lb/>
parallels alwayes retain the ſame proportion to each other, &c. </s>
</p>
</chap>
</body>
</text>
</archimedes>