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

ON SPIRALS 75
Lastly, it' E be the portion of the sector b'OC bounded by
b'B, the arc b'zC of the circle and the arc BC of the spiral, and
F the portion cut off between the arc BC of the spiral, the
radius 00 and the arc intercepted between OB and 00 of
the circle with centre
and radius OB, it is proved that
E:F= {0B + %(0C-0B)}:{0B + i(0C-0B)} (Prop. 28).
On Plane Equilibriums, I, II.
In this treatise we have the fundamental principles of
mechanics established by the methods of geometry in its
strictest sense. There were doubtless earlier treatises on
mechanics, but it may be assumed that none of them had
been worked out with such geometrical rigour. Archimedes
begins with seven Postulates including the following principles.
Equal weights at equal distances balance ; if unequal
weights operate at equal distances, the larger weighs down
the smaller. If when equal weights are in equilibrium something
be added to, or subtracted from, one of them, equilibrium
is not maintained but the weight which is increased or is not
diminished prevails. When equal and similar plane figures
coincide if applied to one another, their centres of gravity
similarly coincide ; and in figures which are unequal but
similar the centres of gravity will be ' similarly situated '.
In any figure the contour of which is concave in one and the
same direction the centre of gravity must be within the figure.
Simple propositions (1-5) follow, deduced by reductio ad
absurdum; these lead to the fundamental theorem, proved
first for commensurable and then by reductio ad absurdum
for incommensurable magnitudes, that Two magnitudes,
whether commensurable or incommensurable, balance at distances
reciprocally proportional to the magnitudes (Props.
6, 7). Prop. 8 shows how to find the centre of gravity of
a part of a magnitude when the centres of gravity of the
other part and of the whole magnitude are given. Archimedes
then addresses himself to the main problems of Book I, namely
to find the centres of gravity of (1) a parallelogram (Props.
9, 10), (2) a triangle (Props. 13, 14), and (3) a paralleltrapezium
(Prop. 15), and here we have an illustration of the
extraordinary rigour which he requires in his geometrical
76 ARCHIMEDES
proofs.
We do not find him here assuming, as in The Method,
that, if all the lines that can be drawn in a figure parallel
(and including) one side have their middle points in a straight
line, the centre of gravity must lie somewhere on that straight
line ; he is not content to regard the figure as made up of an
infinity of such parallel lines ;
pure geometry realizes that
the parallelogram is made up of elementary parallelograms,
indefinitely narrow if you please, but still parallelograms, and
the triangle of elementary trapezia, not straight lines, so
that to assume directly that the centre of gravity lies on the
straight line bisecting the parallelograms would really be
a i^etitio principii. Accordingly the result, no doubt discovered
in the informal way, is clinched by a proof by reductio
ad absardum in each case. In the case of the parallelogram
ABCD (Prop. 9), if the centre of gravity is not on the straight
line EF bisecting two opposite sides, let it be at H. Draw
HK parallel to AD. Then it is possible by bisecting AE, ED,
then bisecting the halves, and so on, ultimately to reach
a length less than KH. Let this be done, and through the
points of division of AD draw parallels to AB or DC making
a number of equal and similar parallelograms as in the figure.
The centre of gravity of each of these parallelograms is
similarly situated with regard to it.
to
Hence we have a number
of equal magnitudes with their centres of gravity at equal
distances along a straight line. Therefore the centre of
gravity of the whole is on the line joining the centres of gravity
of the two middle parallelograms (Prop. 5, Cor. 2). But this
is impossible, because H is outside those parallelograms.
Therefore the centre of gravity cannot but lie on EF.
Similarly the centre of gravity lies on the straight line
bisecting the other opposite sides AB, CD; therefore it lies at
the intersection of this line with EF, i.e. at the point of
intersection of the diagonals.