48 
ARCHIMEDES 
(1) 
Draw QMN through M parallel to AK. Produce K'M to 
meet KG produced in F. 
By similar triangles, 
FA : AM = K'A' : A'M, or FA : h = a : h', 
whence FA = AH (k, suppose). 
Similarly A'F = A'H' (1c, suppose). 
Again, by ¿similar triangles, 
(FA + AM) : (A'K' + A'M) = AM : A'M 
= (AK + AM): (EA' + A'M), 
or (k + h) : (cl + It') — (ct + h) : (k + h'), 
i. e. (Jc + h) (k' + h') = (a + h) (a + h'). 
Now, by hypothesis, 
m:n = (k + h): (k' + h') 
= (k + h) (k' + h'):(k' + h'f 
= (a + h) (a + h') : (k' + h') 2 [by (1)]. 
Measure AR, A'R' on AA' produced both ways equal to a. 
Draw RP, R'P' at right angles to RR' as shown in the figure. 
Measure along MN the length MV equal to MA' or h', and 
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, 
2m:n = 2 (a + h) (a + h') : (k' + h') 2 
= 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, O GQNK is equal to D AA'K'K, whence 
GQ. QN = A A'. A'K' =(h + h') a = 2 m, (4) 
and therefore Q is on the rectangular hyperbola with KF, 
KK' as asymptotes and passing through A'. 
(2) 
H 
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1823.2