rl^P:- V*; 
and 
{DG) = (DB). 
v m + n 
DIVISIONS OF 
Since {DG) : (FB) = m : n, 
(DB); (DG) = (m + n): m. 
Now 
(DB) = Y2?rh {a 2 + aaf + a' 2 ) 
m 
FIGURES 343 
(DG): {FB) = 4:1, 
(DB): {DG) = 5:4. 
(DR) = 5698, 
(DG) = 4558f. 
Let y be the height (CM) of the 
cone CFG. 
Then DH: AH = CK : KA, 
or h:^{a — a') = {x + h):^a, 
whence x is known. 
Cone CDE=Y- n of 2 x, 
cone CFG = (CDE) + ~ m —(DB), 
v m + n 
cone GAB= (CDE) + (DD). 
Now, says Heron, 
(CAB) + (CDE) _(x + h) 3 + x 3 
{CFG) ” y 3 
[He might have said simply 
{CDE): {CFG) = x 3 : f.] 
This gives y or Dili, 
whence Xdf is known. 
Now AD 2 = AH 2 + DH 2 
= {-|(a —a/)} 2 + A 2 , 
so that HD is known. 
Therefore DF = ^ .HD is 
• h 
known. 
t 14.12 
# + h = 3l - = 48, 
and x = 48 — 12 = 36. 
(cone CDE) = 4158, 
(cone CFG) = 4158 + 4558f = 8716f, 
(cone GAB) =4158 + 5698 = 9856. 
8 71 
y 3 - 5 _ . (48 3 + 36 3 ) 
u 9856 + 4158 v ; 
= 8716§ • -WoT 4 4- — 97805, 
whence y — 46 approximately. 
Therefore LM = y — x = 10. 
HD 2 = (3i) 2 + i2 2 
= 156|, 
and HD = 12-|. 
Therefore DF = . 12-| 
= 10 
T2‘