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

THE QUADRATURE OF THE PARABOLA 91
the triangle PQq is half of the parallelogram and therefore
more than half the segment. And so on (Prop. 20).
We now have to sum n terms of the above geometrical
series. Archimedes enunciates the problem in the form, Given
a series of areas A, B, C, D . . . Z, of which A is the greatest, and
each is equal to four times the next in order, then (Prop. 23)
A+B + C+... + Z+iZ = §A.
The algebraical equivalent of this is of course
i+H(i) 2 +...+(ir l =f-iar i =
:!f
1^-
To find the area of the segment, Archimedes, instead of
taking the limit, as we should, uses the method of reductio ad
absurdum.
Suppose K — f
. A
PQq.
(1) If possible, let the area of the segment be greater than K. ,
We then inscribe a figure in the recognized manner ' ' such
that the segment exceeds it by an area less than the excess of
the segment over K. Therefore the inscribed figure must be
greater than K, which is impossible since
A + B + C+...+Z< §4,
where A = APQq (Prop. 23).
(2)
If
If possible, let the area of the segment be less than K.
then APQq = A, B = \A, G = \B, and so on, until we
arrive at an area X less than the excess of K over the area of
the segment, we have
A + B + C+ ... +X + iX = %A = K.
Thus K exceeds A + B + C+ ... + X by an area less than X,
and exceeds the segment by an area greater than X.
It follows that A +B + C+ ... +X> (the segment) ; which
is impossible (Prop. 22).
Therefore the area of the segment, being neither greater nor
less than K, is equal to K or f APQq.
On Floating Bodies, I, II.
In Book I of this treatise Archimedes lays down the fundamental
principles of the science of hydrostatics. These are
1
4
92 ARCHIMEDES
deduced from Postulates which are only two in number.
first which begins Book I is this
1
The
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 medias res with the theorem
(Prop. 2) that ' 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 tvprjKa, evprjKa, namely the problem of