24 
ARCHIMEDES 
“ works on the lever ” ’d Simplicius refers to problems on the 
centre of gravity, KtvrpofiapLKd, such as the many elegant 
problems solved by Archimedes and others, the object of which 
is to show how to find the centre of gravity, that is, the point 
in a body such that if the body is hung up from it, the body 
will remain at rest in any position. 1 2 This recalls the assump 
tion in the Quadrature of the Parabola (6) that, if a body hangs 
at rest from a point, the centre of gravity of the body and the 
point of suspension are in the same vertical line. Pappus has 
a similar remark with reference to a point of support, adding 
that the centre of gravity is determined as the intersection of 
two straight lines in the body, through two points of support, 
which straight lines are vertical when the body is in equilibrium 
so supported. Pappus also gives the characteristic of the centre 
of gravity mentioned by Simplicius, observing that this is 
the most fundamental principle of the theory of the centre of 
gravity, the elementary propositions of which are found in 
Archimedes’s On Equilibriums (irepi iaoppomdov) and Heron’s 
Mechanics. Archimedes himself cites propositions which must 
have been proved elsewhere, e. g. that the centre of gravity 
of a cone divides the axis in the ratio 3:1, the longer segment 
being that adjacent to the vertex 3 ; he also says that ‘ it is 
proved in the Equilibriums ’ that the centre of gravity of any 
segment of a right-angled conoid (i. e. paraboloid of revolution) 
divides the axis in such a way that the portion towards the 
vertex is double of the remainder. 4 It is possible that there 
was originally ^larger work by Archimedes On Equilibriums 
of which the surviving books On Plane Equilibriums formed 
only a part; in that case irepl £vyS>v and KevTpo/SapiKd may 
only be alternative titles. Finally, Heron says that Archi 
medes laid down a certain procedure in a book bearing the 
title ‘ Book on Supports ’. 5 
4. Theon of Alexandria quotes a proposition from a work 
of Archimedes called Catoptrica (properties of mirrors) to the 
effect that things thrown into water look larger and still 
larger the farther they sink. 0 Olympiodorus, too, mentions 
1 Heron, Mechanics, i. 32. 
2 Simpl. on Arist. De caelo, ii, p. 508 a 80, Brandis; p. 543. 24, Heib. 
3 Method, Lemma 10. 4 On Floating Bodies, ii. 2, 
5 Heron, Mechanics, i. 25. 
6 Theon on Ptolemy’s Syntaxis, i, p. 29, Halma. 
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