m 
N U M B E R. 
and (hilling's. The ftrae authors, however, employed a 
Hill more fun pie notation, by dropping the Mv, and fup- 
plyingits place with a point; thus, inftead of ¥ro@.M 
they wrote ^ro/3.5]^; and for they wrote 
043 ) 0 . 00 . 
/ / 
This laft number, with the addition of unity, becomes 
io,ooo 2 =ioo,000,000,. which was the greateft extent of the 
Greek arithmetic ; and for common purpofes it was quite 
fufiicient, becaufe their units of weight and meafure, jfuch 
as the.talent and Hade, were greater than our pound and 
foot. But fomething more was wanted, to embrace the 
objects of fcience. With that view, Archimedes com- 
pofed a curious tract, entitled Yup.f/.lhi;, or Arenarius, in 
which he endeavoured to (how that, following the eftima- 
tion of the aftronomers of his time, it was pofiible to ex- 
prefs the number of particles of fand which would be re¬ 
quired to fill thefpliere of the univerfe. He a (Turned the 
limit of the received arithmetical fyftem, or the fquare of 
a myriad, as the root of a new (bale of progrefiion, which 
therefore advanced eight times fafter than the common 
decimal arrangement. Succeftive periods, which he termed 
oclads, were thus formed, riling above each other, by the 
continued multiplication of a hundred millions. Ar¬ 
chimedes propofed to carry this comprehenfive fyftem as 
far as-eight periods, which would therefore correfpond 
' to a number expreffed on our (bale by fixty-four digits. 
From the nature of a geometrical progrefiion, he demon¬ 
itrated, that proportional numbers mud range at equal 
diftances in the fyftem ; and confequentiy, that the pro¬ 
duct of two numbers will have its place determined by 
the fum of their feparate ranks ; a principle with which 
the theory of logarithms has fince rendered us familiar. 
The fine fpeculation of the Sicilian philofopher does not 
appear, however, to have been carried into effebf ; and 
Archimedes, without actually performing his calcula¬ 
tions, contents himfelf with merely pointing out the pro- 
cefs, and with (fating the approximate refults. But Apol¬ 
lonius, who .certainly holds among the ancients the next 
rank as a geometer, refumed that fch'eme of numeration, 
fimplified the conftruftion of the fcale, and reduced it to 
commodious practice. For the fake of convenience, he 
preferred the fimple myriad as the root of the fyftem, 
which therefore proceeded by fuccefiive tetrads, or periods 
that correfpond to four of our digits. As an example of 
this improved notation, the number exprefling the cir¬ 
cumference of a circle which has unit for its diameter, 
would be thus reprefented : 
3. 1415 9265 3589 793 2 38+6 264-3 3832 795 ° 2824. 
7. atuE 7<p7r0 C/jiA /3 yup-r @xi “7 7^(3 
As that very important office which, in our fyftem of 
notation, the cipher performs, by marking the rank of 
the digits, was unknown to the Greeks, they were obliged, 
when the lower periods failed, to repeat the letters 
Mt>. or the contraction of Mvgix. Thus, to fignify 
37,0000,0000,0000, they wrote kQMv.Mv.Mv. When 
units were to be feparately expreffed, Diophantus and 
Eutocius-prefixed the contrablion M°, as an abbreviation 
for 'Monad, or unit. • 
Having thus given an idea of the Grecian notation for 
integer numbers, it remains to (ay a few words on their 
method of representing fractions. A fmall daflt fet on 
the right of a number made of that number the denomi¬ 
nator of a fraftion, of which unity was the numerator : 
thus, 7' fignified A, ¥ %, py.x &c. but the frac¬ 
tion { had a particular character, as C, or When the 
numerator was not unity, the denominator was placed as 
we fet our exponents. Thus, uft, reprefented 15 64 , or 
and reprefented 7 121 , or T | T ; alfo eZ.y.yfpptf' 7 '^ 0 ' 
=2633 544? 3 1 7 76 = rrfi • 
•We (hall now, from the Commentaries of Eutocius and 
Theon, give a few examples, which may elucidate the 
arithmetical operations of the Greeks, and convey fome 
idea of the labour and addrefs by which that ingenious 
people, with a fyftem of notation fa decidedly inferior to 
our own, were yet enabled to perform calculations of 
very confiderable intricacy. 
The Example in Addition. —From Eutocius, theorem 4. 
of the meafure of the circle. 
. t y &) Kcc . 847 3921 
| . riv 60 8400. 
3 )ij . (Srxoi 908 2321 
In this example the method of proceeding is fo obvious, 
that it heeds no explanation, being performed exactly as 
we do our compound addition of feet and inches; or 
pounds, (hillings, and pence; but it is more fimple, on 
account of the conftant ratio of 10 between any character 
and the lucceeding one. 
Example in Subtraction. —Eutocius, theorem 3 on the 
meafure of the circle. 
6- /7%V 9 3 6 36 
(3 . 2 3409 
£ 7 0327 
This example alfo is fo fimple, that the reader will find no 
difficulty in following the operation, by proceeding from 
right to left, as in our fubtrablion, which method feems 
fo obvioufly advantageous and fimple, that one can hardly 
conceive why the Greeks (hould ever proceed in the con¬ 
trary way ; though there are many inftances which makes 
it evident that they did, both in addition and fubtrablion, 
work from left to right. 
In multiplication, the Greeks appear to have followed 
the fame method as that which was formerly pradii fed 
with the crofs multiplication of duodecimals, and nearly 
correfponding to the ordinary treatment of compound 
quantities in algebra. They proceeded, as in their writing, 
from left to right. The produ 61 of each numeral of the 
multiplier into every numeral of the multiplicand, was 
let down feparately ; and thefe diftinbi elements were af¬ 
terwards collebted together into one total amount. For 
the fake of compacrnefs, the partial products were often 
grouped or.interfperfed, though foinetimes apparently fet 
down at random ; but lfili they were always noted; nor 
was any contrivance employed limilar to that mental pro- 
cefs of carrying lucceflively tens to the higher places, 
which abridges and liinplifies fo much the operations- of 
modern arithmetic. Thefe remarks will be confirmed by 
the following inftance, where the Greek numerals are like- 
wile expreffed in our figures: 
c £ s 
S ce. [3 a. 
MM 7 * 
a @ 7 x r 
M ' * 
a. r ■/. £ 
£ a » £ 
M 
This operation will be eafily underftood. In the firft line, 
er multiplied into o-, gives 0, or '200X200=40,000; a into 
M 
| gives a. or 300X60=12,000; and a- into 1 gives a, or 
200 x 5=1000. In the fecond line, | into < 7 , again, makes 
3 "ft 
265 
265 
4 
12.. . 
1 . . 1 
12.. .; 
36.. 
3 • • 
1 . . . 
3 • • 
2 5 
