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common speed, yet such only of them as lie in the straight 
lines aia and bib would at any instant be moving vertically 
upward and such only as lie in am and j3i/3 vertically down- 
ward, and of these again only the two that lie in G would 
be moving vertically upward in one straight line and only 
the two that lie in A vertically downward in another. 
Lay a straight edge in and along the straight line C. 
Shift this straight edge both backward and forward so as 
never to be taken off either drum and so as always to touch 
each of the drum faces and be at right angles to the higher 
axis. Let ciCc be the line 
marked out by the points 
where the lower drum face 
is touched, and of this line 
let GCi be the portion which 
lies on the same side of 
the plane through G per- 
pendicular to the lower 
axis as Ca x and therefore 
Gc the portion which lies 
on the same side as Ca. 
If then through any point 
X in c-JJc two planes be 
drawn perpendicular to 
the axes, and therefore 
cutting one another in a 
vertical straight line and 
the drums in a lower and 
a higher vertical circle, 
the straight edge when it 
passes through X touches 
the higher circle in some 
point Y ; and a string may 
be stretched, first under 
the lower drum a x a along 
