79 
ingle 
cannot 
medians ; so 
'Prop. 14). 
ct character, 
iding on the 
figures are 
10 for the 
, 13 for the 
f gravity of 
, if AD, BC 
i EF is the 
e of gravity 
finding the 
ips. 1-8) and 
(Props. 9, 10). 
that, if P, F' 
V their centres 
en together is 
ON PLANE EQUILIBRIUMS, I, II 
This is merely preliminary. Then begins the real argument, 
the course of which is characteristic and deserves to be set out. 
Archimedes uses a series of figures inscribed to the segment, 
as he says, ‘in the recognized manner’ {yi/copifim)- The rule 
is as follows. Inscribe in the segment the triangle ABB' with 
the same base and height; the vertex A is then the point 
of contact of the tangent parallel to BB'. Do the same with 
the remaining segments cut off by AB, AB', then with the 
segments remaining, and so on. If BRQPAP'Q'R'B' is such 
a figure, the diameters through Q, Q', P, P', R, R' bisect the 
straight lines AB, AB', AQ, AQ', QB, Q'B' respectively, and 
BB' is divided by the diameters into parts which are all 
equal. It is easy to prove also that PP', QQ', RR' are all 
parallel to BB', and that AL: LM: MN: NO = 1: 3 : 5 : 7, the 
same relation holding if the number of sides of the polygon 
is increased; i.e. the segments of AO are always in the ratio 
of the successive odd numbers (Lemmas to Prop. 2). The 
centre of gravity of the inscribed figure lies on AO (Prop. 2). 
If there be two parabolic segments, and two figures inscribed 
in them ‘ in the recognized manner ’ with an equal number of 
sides, the centres of gravity divide the respective axes in the 
same proportion, for the ratio depends on the same ratio of odd 
numbers 1: 3 : 5 : 7 ... (Prop. 3). The centre of gravity of the 
parabolic segment itself lies on the diameter AO (this is proved 
in Prop. 4 by reductio ad absurdum in exactly the same way 
as for the triangle in I. 13). It is next proved (Prop. 5) that 
the centre of gravity of the segment is nearer to the vertex A 
than the centre of gravity of the inscribed figure is; but that 
it is possible to inscribe in the segment in the recognized 
manner a figure such that the distance between the centres of 
gravity of the segment and of the inscribed figure is less than 
any assigned length, for we have only to increase the number 
of sides sufficiently (Prop. 6). Incidentally, it is observed in 
Prop. 4 that, if in an^ segment the triangle with the same 
base and equal height is inscribed, the triangle is greater than 
half the segment, whence it follows that, each time we increase 
the number of sides in the inscribed figure, we take away 
more than half of the segments remaining over; and in Prop. 5 
that corresponding segments on opposite sides of the axis, e. g. 
QRB, Q'R'B' have their axes equal and therefore are equal in