A history of Greek mathematics - Wilbourhall.org

348 HERON OF ALEXANDRIA
case suppose the circle AG to roll along AE till ODG takes
the position 0'FE\ then D will be at F where AE = BF.
And similarly if a whole revolution is performed and OBA is
again perpendicular to AE. Contrary, therefore, to the principle
that the greater circle moves quicker than the smaller on
the same axis, it would appear that the movement of the
smaller in this case is as quick as that of the greater, since
BH = AG, and BF = AE. Heron's explanation is that, e.g.
in the case where the larger circle rolls on AE, the lesser
circle maintains the same speed as the greater because it has
favo motions ; for if we regard the smaller circle as merely
fastened to the larger, and not rolling at all, its centre
will
move to 0' traversing a distance 00' equal to AE and BF;
hence the greater circle will take the lesser with it over an
equal distance, the rolling of the lesser circle having no effect
upon this.
The parallelogram of velocities.
Heron next proves the parallelogram of velocities (chap. 8);
he takes the case of a rectangle, but the proof is applicable
generally.
The way it is put is this. A
point moves with uniform velocity
along a straight line AB, from A
to B, while at the same time AB
moves with uniform velocity always
parallel to itself with its extremity
A describing the straight line AG.
Suppose that, when the point arrives at B, the straight line