A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE MECHANICS 347
the first chapter or chapters of the real Mechanics which had
been lost. The treatise would doubtless begin with generalities
introductory to mechanics such as we find in the (much
interpolated) beginning of Pappus, Book VIII. It must then
apparently have dealt with the properties of circles, cylinders,
and spheres with reference to their importance in mechanics ;
for in Book II. 21 Heron says that the circle is of all figures
the most movable and most easily moved, the same thing
applying also to the cylinder and sphere, and he adds in
support of this a reference to a proof in the preceding Book '
'.
This reference may be to I.
21, but at the end of that chapter
he says that 'cylinders, even when heavy, if placed on the
ground so that they touch it in one line only, are easily
moved, and the same is true of spheres also, a matter which
we have already discussed ' ; the discussion may have come
earlier in the Book, in a chapter now lost.
The treatise, beginning with chap. 2 after the passage
interpolated from the BapovXitos, is curiously disconnected.
Chaps. 2-7 discuss the motion of circles or wheels, equal or
unequal, moving on different axes (e.g. interacting toothed
wheels), or fixed on the same axis, much after the fashion of
the Aristotelian Mechanical problems.
Aristotle s
Wheel.
In particular (chap. 7) Heron attempts to explain the puzzle
of the Wheel of Aristotle ' ', which remained a puzzle up to quite
modern times, and gave rise to the proverb, rotam Aristotelis
'
magis torquere, quo magis torqueretur \ l '
The question is ', says
the Aristotelian problem 24, '
why does the greater circle roll an
equal distance with the lesser circle when they are placed about
the same centre, whereas, when they roll separately, as the
size of one is to the size of the other, so are the straight lines
traversed by them to one another V 2 Let AC, BD be quadrants
of circles with centre bounded by the same radii, and draw
tangents AE, BF at A and B. In the first case suppose the
circle BD to roll along BF till D takes the position H\ then
the radius ODC will be at right angles to AE, and C will be
at G, a point such that AG is equal to BH. In the second
1
2
See Van Capelle, Aristotelis quaestiones mechanicae, 1812, p. 263 sq.
Avist. Mechanica, 855 a 28.
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