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

Supports '.
ON THE CENTRE OF GRAVITY 351
As, however, the principles are the same whether
the body is supported or hung up, it does not follow that
this was a different work from that known as wept {vy&v.
Chaps. 32-3, which are on the principles of the lever or of
weighing, end with an explanation amounting to the fact
that ' greater circles overpower smaller when their movement
is about the same centre', a proposition which Pappus says
that Archimedes proved in his work ire pi {vyoov. 1 In chap. 32,
too, Heron gives as his authority a proof given by Archimedes
in the same work. With I. 33 may be compared II. 7,
where Heron returns to the same subject of the greater and
lesser circles moving about the same centre and states the
fact
that weights reciprocally proportional to their radii are
in equilibrium when suspended from opposite ends of the
horizontal diameters, observing that Archimedes proved the
'
proposition in his work On the equalization of inclination
(presumably la-oppoiviai).
Book II. The five mechanical powers.
Heron deals with the wheel and axle, the lever, the pulley,
the wedge and the screw, and with combinations of these
powers. The description of the powers comes first, chaps. 1-6,
and then, after II. 7, the proposition above referred to, and the
theory of the several powers based upon it (chaps. 8-20).
Applications to specific cases follow. Thus it is shown how
to move a weight of 1000 talents by means of a force of
5 talents, first by the system of wheels described in the
BapovXKo?, next by a system of pulleys, and thirdly by a
combination of levers (chaps. 21-5). It is possible to combine
the different powers (other than the wedge) to produce the
same result (chap. 29). The wedge and screw are discussed
with reference to their angles (chaps. 30-1). and chap. 32 refers
to the effect of friction.
Mechanics in daily life;
queries and answers.
After a prefatory chapter (33), a number of queries resembling
the Aristotelian problems are stated and answered
(chap. 34), e.g. 'Why do waggons with two wheels carry
a weight more easily than those with four wheels?', 'Why
1
Pappus, viii, p. 1068. 20-3.
352 HERON OF ALEXANDRIA
do great weights fall to the ground in a shorter time than
lighter ones V, Why does a stick break sooner when one
'
puts one's knee against it in the middle V, 'Why do people
use pincers rather than the hand to draw a tooth ? ',
'
Why
is it easy to move weights which are suspended ? ', and
1
Why is it the more difficult to move such weights the farther
the hand is away from them, right up to the point of suspension
or a point. near it ? ', Why are great ships turned by a rudder
'
although it is so
1
small ?', 'Why do arrows penetrate armour
or metal plates but fail to penetrate cloth spread out ?
Problems on the centre of gravity, &c.
II. 35, 36, 37 show how to find the centre of gravity of
a triangle, a quadrilateral and a pentagon respectively. Then,
assuming that a triangle of uniform thickness is supported by
a prop at each angle, Heron finds what weight is supported
by each prop, (a) when the props support the triangle only,
(b) when they support the triangle plus a given weight placed
at any point on it (chaps. 38, 39). Lastly, if known weights
are put on the triangle at each angle, he finds the centre of
gravity of the system (chap. 40) ;
the problem is then extended
to the case of any polygon (chap. 41).
Book III deals with the practical construction of engines
for all sorts of purposes, machines employing pulleys with
one, two, or more supports for lifting weights, oil-presses, &c.
The Catoplrica.
This work need not detain us long.
Several of the theoretical
propositions which it contains are the same as propositions
in the so-called Catoptrica of Euclid, which, as we have
seen, was in all probability the work of Theon of Alexandria
and therefore much later in date. In addition to theoretical
propositions, it contains problems the purpose of which is to
construct mirrors or combinations of mirrors of such shape
as will reflect objects in a particular way, e.g. to make the
right side appear as the right in the picture (instead of the
reverse), to enable a person to see his back or to appear in
the mirror head downwards, with face distorted, with three
eyes or two noses, and so forth. Concave and convex