A history of Greek mathematics - Wilbourhall.org

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 :AC=EG: EF;
therefore AE:EG = AG: EF = AG: 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. 1 1 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 construction
of a toothed wheel to move on an endless screw,
after which chap. 20 makes a fresh start with some observations
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 an
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