ON SPIRALS 
7» 
Lastly, if E be the portion of the sector b'OG bounded by 
b'B, the arc b'zG 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 OC and the arc intercepted between OB and OC of 
the circle with centre 0 and radius OB, it is proved that 
E\F — {OB + %{OG-OB)} : {OB + %{OC-OB)} (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 prin 
ciples. 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 some 
thing 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 dis 
tances 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 parallel- 
trapezium (Prop. 15), and here we have an illustration of the 
extraordinary rigour which he requires in his geometrical