54 
ARCHIMEDES 
Now the triangles ADB, BDd, AGd are similar ; 
therefore AD : DB = BD : Dd = AC : Cd 
= AB : Bd, since AD bisects Z BAG, 
= {AB + AG) : (Bd + Gd) 
= (AB + AG) : BG. 
But AG: GB < 1351 : 780, 
while BA :BG = 2:1 = 1560 : 780. 
Therefore AD : DB < 2911:780. 
Hence AB 2 : BD 2 < (2911 2 + 780 2 ): 780 2 
< 9082321 :608400, 
and, says Archimedes, 
AB : BD < 3013|: 780. 
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 
A E: EB and AB : BE from the limits of AB: BD and AD: DB, 
and so on, and finally we obtain the limit of AB: BG. 
We have therefore in both cases two series of terms a 0 , a x , 
a 2 ... a n and h 0 , h x , h 2 ... h n , for which the rule of formation is 
a x = a 0 + h 0 , a 2 = a 1 + b 1 , 
where h x — V(a 2 + c 2 ), h 2 = V(a 2 + c 2 ) ... ; 
and in the first case 
a 0 = 265, b 0 = 306, c = 153, 
while in the second case