DIVISION 
ô9 
remainder ( = 474201) has now to be divided by t arva (1351), 
and it would be seen that the latter would go into the former 
r (300) times, but not v (400) times. Multiplying arva by r, 
IX . fx£ 
we obtain M^r (405300), which, when subtracted from M y 8<ra 
(474201), leaves (68901). This has again to be divided 
by y arva and goes v (50) times; multiplying t arva by v, we 
have M y £(f>v (67550), which, subtracted from M rj~^)a (68901), 
leaves y arva (1351). The last quotient is therefore a (1), and 
the whole quotient is y arva (1351). 
An actual case of long division where both dividend and 
divisor contain sexagesimal fractions is described by Theon. 
The problem is to divide 1515 20' 15" by 25 12' 10", and 
Theon’s account of the process amounts to the following: 
Divisor. Dividend. 
25 12' 10" 1515 20' 
25.60 = 1500 
15' 
Quotient. 
First term 60 
Remainder 15 = 900' 
Sum 
920' 
12'.60 = 
720' 
Remainder 
200' 
II 
o 
CO 
o 
t-H 
V 
10' 
Remainder 
190' 
to 
Ox 
*<! 
II 
175' 
15' = 900" 
Second term 7' 
Sum 915" 
12'. 7' = 84" 
Remainder 831" 
10". 7'= 1" 10'" 
Remainder 829" 50'" Third 
25.33"= 825" . term 33'" 
Remainder 4" 50'" = 290'" 
12'. 33"= 396'" 
(too great by) 106" 
Thus the quotient is something less than 60 7' 33". It will 
be observed that the difference between this operation of