92 
ARCHIMEDES 
deduced from Postulates which are only two in number. The 
first which begins Book I is this: 
‘ let it be assumed that a fluid is of such a nature that, of the 
parts of it which lie evenly and are continuous, that which is 
pressed the less is driven along by that which is pressed the 
more; and each of its parts is pressed by the fluid which is 
perpendicularly above it except when the fluid is shut up in 
anything and pressed by something else ’; 
the second, placed after Prop. 7, says 
‘ let it be assumed that, of bodies which are borne upwards in 
a fluid, each is borne upwards along the perpendicular drawn 
through its centre of gravity 
Prop. 1 is a preliminary proposition about a sphere, and 
then Archimedes plunges in mediae res witli the theorem 
(Prop. 2) that f the surface of any fluid at rest is a sphere the 
centre of which is the same as that of the earth ’, and in the 
whole of Book I the surface of the fluid is always shown in 
the diagrams as spherical. The method of proof is similar to 
what we should expect in a modern elementary textbook, the 
main propositions established being the following. A solid 
which, size for size, is of equal weight with a fluid will, if let 
down into the fluid, sink till it is just covered but not lower 
(Prop. 3); a solid lighter than a fluid will, if let down into it, 
be only partly immersed, in fact just so far that the weight 
of the solid is equal to the weight of the fluid displaced 
(Props. 4, 5), and, if it is forcibly immersed, it will be driven 
upwards by a force equal to the difference between its weight 
and the weight of the fluid displaced (Prop. 6). 
The important proposition follows (Prop, 7) that a solid 
heavier than a fluid will, if placed in it, sink to the bottom of 
the fluid, and the solid will, when weighed in the fluid, be 
lighter than its true weight by the weight of the fluid 
displaced. 
The problem of the Crown. 
This proposition gives a method of solving the famous 
problem the discovery of which in his bath sent Archimedes 
home naked crying evprjKa, evprjKa, namely the problem of 
determin 
crown. 
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