A history of Greek mathematics - Wilbourhall.org

48 ARCHIMEDES
Draw QMN through M parallel
meet KG produced in F.
to AK. Produce K'M to
By similar triangles,
FA:AM = K'A':A'M, or FA:h = a:h' y
whence FA = AH (k, suppose).
Similarly # A'E — A'H' (Jc, suppose).
Again, by^similar triangles,
(FA + AM) :
(A'K' + A'M) = AM: A'M
= (AK + AM):(EA' + A'M),
or (k + h):(a + h') = (a + h):(¥ + h') }
i. e. (k + h) (k' + h') = (a + h)(a + h f ). (1)
Now, by hypothesis,
m:n = (k + h):(k' + hf)
= (k + h)(k' + h'):(k' + h') 2
= (a + h) (a + hf) :
{¥ + h!f [by (I)]. (2)
Measure AR, A f R! on AA f produced both ways equal to a.
Draw RP, R'P' at right angles to RR /
Measure along MJS T the length MV equal to
draw PP' through V, A' to meet RP, R'P'.
Then QV=k' + h', P'V= V2 (a + h'),
PV= V2(a + h),
whence PV.P'V = 2 (a + h) {a + h!) ;
and, from (2) above,
as shown in the figure.
MA' or h\ and
2m:n=2(a + h) (a + h') :
(¥ + h'f
= PV.P'V: QV 2 .
(3)
Therefore Q is on an ellipse in which PP' is a diameter, and
Q V is an ordinate to it.
Again, GQNK is equal to AA'K'K, whence
GQ.QN= AA'. A'K' =(h + h!) a = 2ra, (4)
and therefore Q is on the rectangular hyperbola with KF,
KK' as asymptotes and passing through A'.