v 
ММ 
9000000 
30000 
9000 
1500 
750 
МрЛб/ЗЬ' 
30000 
100 
30 
5 
ol 
¿2 
,0Л 6al'W8' 
9000 
30 
9 
H 
1 1 
2 4 
у афе aWS'rj' 
1500 
5 
H 
i 
4 
1 
8 
y\rvfi\s'l' r 8'т)' t<5-' 
750 
oi 
^2 
1 i 
2 4 
1 
8 
1 
1 
^V 
ofiov М'/З^тгв i<5 _/ together 9082689^. 
The following is one among many instances in which Heron 
works out a multiplication of two numbers involving fractions. 
He has to multiply 4f| by 7~f, which he effects as follows 
(Geom. 12, 68): 
4.7 — 28, 
4 62 248 
* • 6 4 — “6 4 > 
3 3 7 2 3 1 
64•' 64 
33, 6 2 _ 2046 1_ _ 3 1 I 6 2 JL_ ■ 
64'64 — 64 ‘ 6 4 64 ' 64 1 64 > 
the result is therefore 
The multiplication of 37° 4' 55" (in the sexagesimal system) 
by itself is performed by Theon of Alexandria in his com 
mentary on Ptolemy’s Syntaxis in an exactly similar manner. 
(8) Division. 
The operation of division depends on those of multiplication 
and subtraction, and was performed by the Greeks, mutatis 
mutandis, in the same way as we perform it to-day. Suppose, 
for example, that the process in the first of the above multi- 
P 7 r0 
plications had to be reversed and M¿aa (1825201) had to be 
divided by y aTva (1351). The terms involving the successive 
powers of 10 would be mentally kept separate, as in addition 
and subtraction, and the first question would be, hew many 
times does one thousand go into one million, allowing for the 
fact that the one thousand has 351 behind it, while the one 
million has 825 thousands behind it. The answer is one 
thousand or y a, and this multiplied by the divisor t atvcl gives 
p\e M<T 
M,a w’hich, subtracted from M y eaa, leaves Macros. This 
remai 
and il 
т (30C 
we ob 
(4742» 
by 
have 
leaves 
the wl 
An 
divisor 
The pr 
Theon 
Divisc 
25 12' 
Thus th 
be obse]