NUMBER. 
£ 6 / 3 , or £ex ioo=«=ii,ooo; £ into f makes y%, cr 60X60 
297 
=3600 ; and £ into t makes r, or 60X 5 = 300. Ladly, in 
the third line, £ into o, again, gives the product a, or 
5X200=1000 ; s into £ likewife gives r, or 5X60=300 ; 
and e into e gives xe, or 5 x 5=15. 
We (hall next extrad a more intricate example of mul¬ 
tiplication, and one in which fractions are concerned : 
(Olfll) 
9 ‘“ 
183 8 ^j- 
cc cj y v) 
9 ‘* 
1 8 3 ? TT 
r 
3 . 
M M M 
8 . 
3 . . . . 
8 . . . 
let, 
r 
8 1 8 ^- 
MM// 
8 . 
64 ... . 
y (3 0 fjc x S 
u& 
r 
24 . . . 
M M / 
64 . . 
6 5 4 rr 
v r v 0 £ Sr 
r‘“ 
3 • • • • 
24 . . . 
9 • • 
2 4 . 
(0 l Y) 
2 4 TT 
8 . . . 
% » * 
r‘* 
64 . . 
2 4 
xS 
r“* 
6 4 
6 T®f 
r 
r‘“ 
6 5 4 tt 
2 4 VT 
6 tY 
rAi) a. cr v a 
e-r, T^t) a. cr v (3 
M 
sra 
xa 
3 3 812 5 1 itT T 2*i 
or, 3,381,252^ 
judge corredly, mud have drawn their aftronomy from 
the fame fource, exprefs the radius in parts of the circum¬ 
ference, making it equal to 57 0 18', or 3438 minutes, and 
not to 3600', which would refult from its own fubdivifion. 
The operations with fexagefimal fradions were performed 
in the defcending fcale on a principle quite fimilar to that 
which Archimedes had before laid down. Each period 
of the multiplier, (fill proceeding from the left hand, was 
multiplied in a period of the multiplicand ; and this pro¬ 
duct was then thrown to a rank deprefled as much as the 
conjoined defcents of both its fadors. Thus, minutes 
multiplied into feconds produced thirds, and feconds 
multiplied into thirds produced fifths. Theon propoles, 
as an example of the proceft, to find the fquare of the 
fide of a regular decagon infcribed in a circle, and which, 
according to the calculation of Ptolemy, meafures, in 
fexagefimal parts of the radius, 37°4'55". The multi¬ 
plication was thus effeded : 
l ,e 37 ° 4' 55 " 
6 n 37 4 55 
ax£0 
/ 
>r 
/ 3 At 
era 
OK 
1369° 148' 3035" 
148 16 
2035 
220 
220 
yy.c 
3° 2 * 4 5 
1375 
14 ' 
It may fuffice to remark the fractional produds 
merely. Thus, a. multiplied into r‘ tt gives uin f 3 ‘“, or 
loooX-^^ 8 i 8 t 2 7 ; a into r‘“ gives r‘“» or 8 ooX-jV 
= 6 5 4 T 6 t ; X into r‘ tt gives yj r‘“, or 3oX T Y=4 T 6 T ; 5nto 
r‘“ gives $-r‘ tt , or 8 X^=6^| and r‘ s into r‘“ gives ra.^ a > 
« r tVXtt—T2 1 !- 
As the management of fuch complex fractions proved 
mod laborious, they were gradually laid afide for the ufe 
of fexagefimals, which the aftronomers had introduced. 
The divifion of the circumference of the circle into 360 
equal parts, or degrees, was no doubt originally founded 
on the fuppofed length of the year, which, exprelfed in 
round numbers, confifts of twelve months, each compofed 
of thirty., days. The radius, approaching to the fixth part 
of the circumference, would contain nearly 60 of thofe 
degrees ; and, after its ratio to the circumference was 
more accurately determined, the radius (till continued to 
be diftinguifhed into the fame number of divisions, and 
which likewife bore the fame name. As calculation now 
aimed at greater accuracy, each of thefe 60 divilions of 
the radius was, following the uniform progreffion, again 
lubdivided into 60 equal portions, called minutes ; and, re¬ 
peating the procefs of fexagefimal fubdivifion, feconds and 
thirds were fucceflively formed. The fame plan of divi¬ 
fion, and the fame names, were transferred to the circum¬ 
ference of the circle, though the degrees and minutes 
employed to meafure arcs were fenfibly different from 
thofe contained in the radius. It is curious, however, to 
remark, that the Hindoos, who, fo far as we are able to 
Vol. XVII. No. 1178. 
In the firlt line, a£ multiplied into a£ gives «r£ 9 , or 37 0 
X 37°=! 369° ; ^multiplied into S gives or 3 7 °X4' 
= 148'; and a£ multiplied into »s gives /?Ae, or 37 0 X55 ,, 
/ 
=2035". In the fecond line, S into f\£, again, gives g/2.5), or 
4 , X37°— 1 48 / ; ^ into S gives ir, or 4'X = i6 " ; and S into 
ve gives ok , or 4'X 55"=220'". And ladly, in the third 
line, ve into a£ gives, as before, £?Ae, or 55"x 37°=2035"; 
vt into S gives likewife ox, or 55"=4'x 22 o'"; and ve into 
»s gives ^xe, or 55 "X 55"=3o25"". Thefefeveral products 
being now cclleded and reduced, formed the total refult 
a.TotS' iSt xe, or I375°4'_I4" io""2 5""; but all the termsbe- 
yond the feconds were in pradice omitted as infignificant. 
It is one of the mod beautiful theorems in geometry, 
that the fide of an infcribed decagon is the greater feg- 
ment of the radius divided into extreme and mean ratio. 
The fquare now found ought, therefore, to equal the pro- 
dud of the radius into its (mailer fegment, or 22 0 55' 5". 
But 6o° (22 0 55' 5")= 1 375° 5', differing only by 14 fe¬ 
conds from the lad refult ; a remarkable indance of the 
accuracy which the fexagefimal fydem of computation 
was capable of attaining within a very moderate compafs. 
The divifion of the Greeks was dill more intricate than 
their multiplication : for which reafon itfeems they gene¬ 
rally preferred the fexagefimal divifion; and no example 
is left at length by any of thofe writers, except in the lat¬ 
ter forin. But thebe are fufficient to throw fome light on 
the procefs they followed in the divifion of common 
numbers; and Delambre has accordingly fuppofed the 
following example : 
a su 7) T ?i ^ 4 T rj (a # » 7 1823)3323329(1823 
f n @ ■ y 1823 
pv . 
Pf*‘ 
S . O . X 
7 • r v £ 
15003 
J4584 
4192 
3646 
v £ 9 
V £ G 
5469 
5469 
4 G 
This 
