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

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
350 HERON OF ALEXANDRIA
between the two portions into which the vertical plane cuts it
(chap. 23).
On the centre of gravity.
This brings Heron to the centre of gravity (chap. 24).
Here
a definition by Posidonius, a Stoic, of the ' centre of gravity
or ' centre of inclination ' is given, namely ' a point such that,
if the body is hung up at it, the body is divided into two
equal parts ' (he should obviously have said ' divided by any
vertical plane through the "point of suspension into two equal
parts'). But, Heron says, Archimedes distinguished between
the ' centre of gravity ' and the ' point of suspension ', defining
the latter as a point on the body such that, if the body is
hung up at it, all the parts of the body remain in equilibrium
and do not oscillate or incline in any direction.
'
" Bodies", said
Archimedes, " may rest (without inclining one way or another)
with either a line, or only one point, in the body fixed ".' The
1
centre of inclination ', says Heron, ' is one single point in any
particular body to which all the vertical lines through the
points of suspension converge.' Comparing Simplicius's quotation
of a definition by Archimedes in his KevrpofiapiKa, to
the effect that the centre of gravity is a certain point in the
body such that, if the body is hung up by a string attached to
that point, it will remain in its position without inclining in
any direction, 1 we see that Heron directly used a certain
treatise of Archimedes. So evidently did Pappus, who has
a similar definition. Pappus also speaks of a body supported
at a point by a vertical stick : if, he says, the body is in
equilibrium, the line of the stick produced upwards must pass
through the centre of gravity. 2 Similarly Heron says that
the same principles apply when the body is supported as when
it is suspended. Taking up next (chaps. 25-31) the question
of ' supports ', he considers cases of a heavy beam or a wall
supported on a number of pillars, equidistant or not, even
or not even in number, and projecting or not projecting
beyond one or both of the extreme pillars, and finds how
much of the weight is supported on each pillar. He says
that Archimedes laid down the principles in his Book on
'
1
Simplicius on Be caelo, p. 543. 31-4, Heib.
2
Pappus, viii, p. 1032. 5-24.