[ 2°9 ] 
When in the afcending and defending parts of the fcale, the gradation 
proceeds by a tenfold value from the right hand towards the left, the rank 
of numbers, thus generated, is called the decimal fcale. 
As every place in this lcale is ten times the value of its next right hand 
place ; therefore the firft place in thefraCtional part, is _y ofthe place of units ; 
and the fecond, third, fourth, &c. defcending places in the fractional part, 
is t 4 tt, rm» -ro 4 ro-> P ar f the P 5ace of unitS - 
Therefore every decimal fraction is equal to a feries arifing from m ulti— 
plying the firft, fecond, third, fourth, &c. terms of the decrealing geome- 
trical progreffion -L- + — '-o + toW 4* by the firft, fecond, 
third, and fourth, &c. terms in the given fraction refpeCtively. 
Thus. Let the given fraction be 0,3587 or _yJ* o 7 ._. 
Then 0,3587=^ x 3+-^ X 5 + Ww X 8 4. T ^ 4 - TV x 7 
4- ~L_ 4* • _ 4- 
I I O O » I o \ 
3 _ 
1 o 
- 3 0 0 0 \ 
~~T~o"~o~o TT *» T 
1 o 
500 
"o ~o o"o* 
4- -• 
3 o 
o* o ' c3‘ o 
1 '3"o' o o 
7 
10000 
3 S B r 
i "o"6 "o' o’ 
2. Every decimal fraction arifes from divifton, when the dividend is lefs 
than the divifor. 
For, divifor : dividend : : io : firft; term of the fraction ; 
: : ioo : fum of the firft and fecond terms 
:: 1000 : fum of the firft, fecond, and third, terms, 
&c. 
And according to the ratio of the divifor to the dividend, the quotient, 
or decimal fraction, will be finite or infinite. 
3. Among thofe decimal fractions which are infinite, or do not end, fome 
of them recur , or circulate j that is, the fame figure or -figures run over 
again and again ad infinitum . 
As 0,333 j 0,2323 &c . ; 0,758758 &c.; 0,999 &c. 
Here 0,33 3 &c. z=JL _ 4 -_ * -I- 1 4 - L , &c. 
' J J J 1 o ^ 100 1 i^Too I 10000 * 
"^3^3 — ts'4' tIo 4" ■nroo-'i" T oTTo-) &C, 
°> 7 ^ 57 ^ 5 & c ' “4o4'x4-o4"--^^4'T-o4--o-o-4'-r 
0,999, &C. =A+TW+T^+iTrm $V- 
Vox.. LVJII. E e 
00000 
+ TToVVoV) 
The 
