THE PARALLELOGRAM OF VELOCITIES 349 
reaches the position CD. Let EF be any intermediate 
position of AB, and G the position at the same instant 
of the moving point on it. Then clearly AE\AG—EG\EF\ 
therefore AE-.EG = AG:EF — AC: CD, and it follows that 
G lies on the diagonal AD, which is therefore the actual path 
of the moving point. 
Chaps. 9-19 contain a digression on the construction of 
plane and solid figures similar to given figures but greater or 
less in a given ratio. Heron observes that the case of plane 
figures involves the finding of a mean proportional between 
two straight lines, and the case of solid figures the finding of 
two mean proportionals; in chap. 11 he gives his solution of 
the latter problem, which is preserved in Pappus and Eutocius 
as well, and has already been given above (vol. i, pp. 262-3). 
The end of chap. 19 contains, quite inconsequently, the con 
struction of a toothed wheel to move on an endless screw, 
after which chap. 20 makes a fresh start with some observa 
tions on weights in equilibrium on a horizontal plane but 
tending to fall when the plane is inclined, and on the ready 
mobility of objects of cylindrical form which touch the plane 
in one line only. 
Motion on an inclined plane. 
When a weight is hanging freely by a rope over a pulley, 
no force applied to the other end of the rope less than the 
weight itself will keep it up, but, if the weight is placed on 
inclined plane, and both the plane and the portion of the 
weight in contact with it are smooth, the case is different. 
Suppose, e.g., that a weight in the form of a cylinder is placed 
on an inclined plane so that the line in which they touch is 
horizontal; then the force required to be applied to a rope 
parallel to the line of greatest slope in the plane in order to 
keep the weight in equilibrium is less than the weight. For 
the vertical plane passing through the line of contact between 
the cylinder and the plane divides the cylinder into two 
unequal parts, that on the downward side of the plane being 
the greater, so that the cylinder will tend to roll down; but 
the force required to support the cylinder is the * equivalent 
not of the weight of the whole cylinder, but of the difference