A history of Greek mathematics - Wilbourhall.org

102 ARCHIMEDES
of the two smaller semicircles adjoin, and NP be drawn at
right angles to AB meeting the external semicircle in P, the
area of the apfi-qXos (included between the three semicircular
arcs) is equal to the circle on PN as diameter (Prop. 4). In
Prop. 5 it is shown that, if a circle be described in the space
between the arcs AP, AN and the straight line PN touching
all three, and if a circle be similarly described in the space
between the arcs PB, NB and the straight line PN touching
all three, the two circles are equal. If one circle be described
in the dpftrjXos touching all three semicircles, Prop. 6 shows
that, if the ratio of AN to NB be given, we can find the
relation between the diameter of the circle inscribed to the
dpftrjXos and the straight line AB ; the proof is for the particular
case AN = §BN, and shows that the diameter of the
inscribed circle = -f^AB.
Prop. 8 is of interest in connexion with the problem of
trisecting any angle. If AB be any chord of a circle with
centre 0, and BC on AB produced be made equal to the radius,
draw CO meeting the circle in D, E
;
then will the arc BD be
one-third of the arc AE (or BF, if EF be the chord through E
parallel to AB). The problem is by this theorem reduced to
a v ever is (cf. vol. i, p. 241).