A history of Greek mathematics - Wilbourhall.org

,
44 ARCHIMEDES
values for each of the ratios A'H' : H'M and H'H: A'H' which
are alike independent of H, H' and then, secondly, to equate
the ratio compounded of these two to the known value of
H'M.
ratio HH' :
the
(ex.) We have, from (2),
A'H : H'M = OA :
(/?) From (1) and (2), separando,
(OA + AM). (3)
AH:A3I= OA'iA'M, (4)
A'H' :
Equating the values of the ratio A'M :
we have 6A' AH : = A'H' : OA
whence HH :
OH
A'M
= OH' :
=0A: AM. (5)
AM
-
= OH' : OH,
or HH'.A'H' = OH 2 ,
so that HH' : A'H' = OH' 2 : ^i?'
But, by (5), OA' : A'H' = AM: A'M,
and, componendo, OH : A'H' — AA' : A'M.
given by (4). (5),
A'H', (since OA = OA')
2
.
(6)
By substitution in (6),
HH' : A'H = A A' 2 : A'M
2 .
(7)
Compounding with (3), we obtain
HH :
H'M = (A A' 2 : A 'M 2 ) (OA .
:
OA
+ AM). (8)
[The algebraical equivalent of this is
m + n 4 r 3
n "
(2r—h) 2 i
(r-{-h)
+ n 4r 3
... , m .
which reduces to = —=-= =-z
on 3/rr — h 6
or h — z 3h 2 r-\ r = 3 0, as above.]
m + n
Archimedes expresses the result (8) more simply by producing
OA to D so that
OA = AD, and then dividing AD at