ARCHIMEDES’S SYSTEM (BY OCTADS) 
41 
= 22780912 
powers of 
rst myriad 
ad (Sevrepa 
l pvpidSes, 
in full or 
)CtL pvpLaSf.y 
6560 (Dio 
inserted to 
b myriads 
r Apollonius 
3f which is, 
is, Book II: 
tetrads (sets 
&c., ‘ simple 
100, 10000 2 , 
) successive 
(3 Kcd y\yx 
a less con- 
;rs of 10000 
mtury A.D.) 
be ordinary 
;he ‘ double 
ve the other, 
o on. Thus 
ind so on. 
s purpose of 
Archimedes’s 
is: 
first order; ■ 
as the unit 
ambers from 
10 8 , or 100000000, to 10 16 , or 100000000 2 ; similarly 10 16 is 
taken as the unit of the third order, which consists of all 
numbers from 10 1G to 10 24 , and so on, the 100000000th order 
consisting of all the numbers from (10 0 0 0 0 0 0 0) 99999999 to 
(lOOOOOOOO) 100000000 , i.e. from Ю 8 ^ 10 *- 1 ) to lO 810 *. The aggre 
gate of all the orders up to the 100000000th form the first 
period ; that is, if P = (ЮООООООО) 10 *, the numbers of the 
first period go from 1 to P. Next, P is the unit of the first 
order of the second period; the first order of the second 
period then consists of all numbers from P up to 100000000 P 
or P. 10 s ; P.10 8 is the unit of the second order (of the 
second period) which ends with (100000000) 2 P or P.10 16 ; 
P.10 1G begins the third order of the second period, and so 
on; the 100000000th order of the second period consists of 
the numbers from (lOOOOOOOO) 99999999 P or P. l0 8 -h° s -i) 
(lOOOOOOOO) 100000000 P or P. IO 8 - 10 ", i.e. P 2 . Again, P 2 is the 
unit of the first order of the third period, and so on. The 
first order of the 100000000th period consists of the numbers 
from P 10 _1 to P 10 * -1 . 10 8 , the second order of the same 
period of the numbers from P 10 - 1 л 0 s to P 10 * -1 . 10 1G , and so 
on, the (10 8 )th order of the (10 8 )th period, or the period 
itself, ending with P 10 * -1 .10 8Л ° 8 , i.e. P x0 \ The last number 
is described by Archimedes as a ‘ myriad-myriad units of the 
myriad-myriadth order of the myriad-myriadth period (at 
pvpLaKuryvpLoa-rdу neptodov р.орсаккгр.орюа-тооп арсбршп pvpiaL 
pvpLciSes) ’. This system was, however, a tour de force, and has 
nothing to do with the ordinary Greek numerical notation. 
Fractions. 
(a) The Egyptian system 
We now come to the methods of expressing fractions. A 
fraction may be either a submultiple (an ‘ aliquot part ’, i. e. 
a fraction with numerator unity) or an ordinary proper 
fraction with a number not unity for numerator and a 
greater number for denominator. The Greeks had a pre 
ference for expressing ordinary proper fractions as the sum 
of two or more submultiples ; in this they followed the 
Egyptians, who always expressed fractions in this way, with 
the exception that they had a single sign for §, whereas we