A history of Greek mathematics - Wilbourhall.org

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54 ARCHIMEDES
Now the triangles ADB, BDd, ACd are similar;
DB = BD : Dd = AC : Cd
= AB: Bd, since AD bisects Z BAG,
= (AB + AC):(Bd + Cd)
= (AB + AC):BC.
But AG : GB < 1351 : 780,
therefore AD :
while .RA : jBO = 2 : 1 = 1560 :
Therefore AD :
DB
< 2911 : 780.
780.
Hence AB 2 :BD 2 < (2911 2 + 780 2 ):780 2
< 9082321 :
and, says Archimedes,
AB.BD < 3013|: 780.
608400,
Next, just as a limit is found for AD : DB and AB : BD
from AB : BG and the limit of AG: GB, so we find limits for
AE.EB and AB : BE from the limits of AB : BD and AD : Di?,
and so on, and finally we obtain the limit of AB : BG.
We have therefore in both cases two series of terms a , a lf
(t 2
... a n and b , b l}
b 2
... b n
, for which the rule of formation is
a x
= a + b , a 2
= a x
+ &!,...
where 6 2
= \/ («/ + c 2 ), 6 2
= \/(a 22
+ c 2 ) . . . ;
and in the first case
while in the second case