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