Archimedes Texts

122
would have you consider three Places, which
I call Points; the two Ends, that is the Steel
and the Peathers, and the third is the Loop in
the Middle for throwing the Dart by; and the
two Spaces between the two Ends and the
Loop, I shall call the Radii. I shall not dis
pute about the Reasons of these Names, which
will appear better from the Consideration of
the Thing itself. If the Loop be placed ex
actly in the Middle of the Dart, and the Fea
ther End be just equal in Weight to the Steel,
both Ends of the Dart will certainly hang even
and be equally poised; if the steel End be the
Heaviest, the Feather will be thrown up, but
yet there will be a certain Point in the Dart
further towards the heavy End, to which if
you slip the Loop, the Weight will be imme
diately brought to an equal Poise again; and
this will be the Point by which the larger Ra
dius exceeds the smaller just as much as the
smaller Weight is exceeded by the larger. For
those who apply themselves to the Study of
these Matters, tell us, that unequal Radii may
be made equal to unequal Weights, provided
the Number of the Parts of the Radius and
Weight of the right Side, multiplied together,
be equal to the Number of those Parts on the
opposite left Side: Thus if the Steel be three
Parts, and the Feather two, the Radius be
tween the Loop and the Steel must be two, and
the other Radius between the Loop and the
Feather must be three. By which Means, as
this Number five will answer to the five on the
opposite Side, the Radii and the Weights an
swering equally to one another, they will hang
even and be equally poised. If the Number
on each Side do not answer to one another,
that Side will overcome on which that Inequa
lity of Numbers lies. I will not omit one Ob
servation, namely, that if equal Radii run out
from both Sides of the Loop, and you give the
Ends a twirl round in the Air they will de
seribe equal Circles; but if the Radii be un
equal, the Circles which they describe, will be
unequal also. We have already said that a
Wheel is made up of a Number of Circles:
Whence it is evident, that if two Wheels let
into the same Axis be turned by one and the
same Motion, so as when one moves the
other cannot stand still, or when one stands
still the other cannot move; from the Length
of the Radii or Spokes in each Wheel we may
come at the Knowledge of the Force which is
in that Wheel, remembring always to take the
Length of the Radius srom the very Center of
the Axis. If these Principles are sufficiently
understood, the whole Secret of all these En
gines of which we are here treating, will be
manisest; especially with Relation to Wheels
and Leavers. In Pullies indeed we may con
sider some surther Particulars: For both the
Rope which runs in the Pully and the little
Wheel in the Pully are as the Plain, whereon
the Weight is to be carried with the middle
Motion, which we observed in the last Chapter
was between the most Easy and the most Dif
ficult, inasmuch as it is neither to be raised up
nor let down, but to be drawn along upon the
Plain keeping always to one Center. But that
you may understand the Reason of the Thing
more clearly, take a Statue of a thousand
Weight; if you hang this to the Trunk of a
Tree by one single Rope, it is evident this Rope
must bear the whole thousand Weight. Fasten
a Pully to the Statue, and into this Pully let
the Rope by which the Statute hangs, and bring
this Rope up again to the Trunk of the Tree,
so as the Statue may hang upon the double
Rope, it is plain the Weight of the Statue is
then divided between two Ropes, and that the
Pully in the Middle divides the Weight equal
ly between them. Let us go on yet further,
and to the Trunk of the Tree fasten another
Pully and bring the Rope up through this
likewise. I ask you what Weight this Part of
the Rope thus brought up and put through
the Pully will take upon itself: You will say
five hundred; do you not perceive from hence
that no greater Weight can be thrown upon
this second Pully by the Rope, than what the
Rope has itself; and that is five hundred. I
shall therefore go no farther, having, I think,
demonstrated that a Weight is divided by Pul
lies, by which means a greater Weight may be
moved by a smaller; and the more Pullies
there are, the more still the Weight is divided;
from whence it follows that the more Wheels
there are in them, so many more Parts the
Weight is split into and may so much the more
easily be managed.