A R I T H 
wh-th will begin after as many places of figures as there 
were 2’s or 5’s divided by. If the whole denominator va- 
nifhes in dividing by 2 or 5, the decimal will be finite, and 
will confifl of as many places as you perform divifions. 
Ex. 1. Required to find whether the decimal equal to 
.gi-jL. be finite or infinite ; if infinite, how many places the 
repe-end will confifl: of, whether there will be a finite deci- 
srrul to the left of the repetend, and how many ? 
Fit ft, 9768—2=4884-7-2—24^2-^-2=1221; 
Then, 1221)999999(719; here are fix nines made life of 
before nothing remain, and the denominator has been di¬ 
vided by 2 three times, and cannot be abridged any more ; 
therefore the decimal will be infinite, and will confifl of 
three finite decimals and fix pure repetends. Thus 
— •008497133. 
Ex. 2. Whether is the decimal equal to ^ finite or in¬ 
finite ? Firft, 16-2-2—S-f-2=4-^2—2=2=1 ; hence the 
decimal is finite, and conlifts of four places. Thus ^= 
^1875- 
ADDITION of CIRCULATING DECIMALS. 
Make the repetends fimilar and conterminous, by the 
rule before laid down, and find their film as in common 
Addition. Divide this fum by as many nines as there are 
places in the repetend, and the remainder is the repetend 
of the fum ; which rttuft be fet under the figures added, 
with cyphers to the left hand when it has not fo many pla¬ 
ces as the repetends. Carry the quotient of this divifion 
to the next column, and proceed with the reft as in finite 
decimals. 
Ex. 1. 
Similar & confer. 
Ex. 2. 
Similar & confer. 
39' 6 545 
= 
39 ' 6 545 ° 
3'6 
= 
3 6666 
81 -046 
= 
81 -04666 
8-3476 
= 
8-3476 
42-35 
= 
42'35555 
9'3 
9-3333 
9 v8 37 
= 
9'33777 
o -375 
0-3750 
8 7"534 
= 
8 7 ' 534 °o 
0’27 
= 
0-2777 
1-9 
= 
1-99999 
7-4 
rx 
7-4444 
Sum 
262.42840 
Sum 
29-4348 
SUBTRACTION of CIRCULATING DECIMALS. 
Make the repetends fimilar and conterminous, as in Ad¬ 
dition; then fubtraft as ufual: only, if the repetend of 
the fubducend exceed the repetend of the minuend, make 
the laft figure of the remainder 1 lefs than it would be if 
the expreffions were finite. 
Ex. 1. Ex. 2. 
From 76-32 — 76-322* From 89*576 = 89.5760 
Take 54*7617 = 54-7617 Take 12-5846 — 12-5846 
Remainder 21 -5604 
Remainder 76-9913 
MULTIPLICATION of CIRCULATING DECIMALS. 
When a repetend is to be multiplied by a finite number, 
multiply as in common numbers, only obferve what mult 
be carried from the beginning of the repetend to the end 
of it. And make all the lines fimilar and conterminous 
when they are to be added. In multiplying a finite deci¬ 
mal by a (ingle repetend, multiply by the repetend and 
divide by -9 or -l 5 ‘ In more complex cafes, reduce the 
repetends to vulgar fractions, then find the produft of 
thefe fraftions as ufual, and reduce the fra6tion exprefting 
the produdl to a decimal, if neceflary. 
Ex. 1. 716-2935 Ex. 2. 2-104 Ex. 3. 3-028 
•27 1-2 17 
55140548 
143258711 
4208 
21044 
21202 
30288 
193-399260 
2 ' 5 2 53 
51 1 49 i 
M £ T I C, 
Ex. 4. 27-1241 
3'6 
9 )162-7446 
1808273 
813723 
iSgi 
Or 3-6 
Then 
- - 5._» * __ ! 
-a a—•> 3— 3 
27 - i 241 x 11 
298-3651 
I X 3 
99'455°3 pvodudt. 
99-45503 
DIVISION of CIRCULATING DECIMALS. 
If the dividend only be a repetend, divide as in coni', 
mon numbers, bringing down always the recurring fi¬ 
gures, till the quotient become as exafl as requilite. It 
either the dividend or divifor, or both, be repetends, ret 
duce them to their finite values, and then apply the rule 
in Divifion of Vulgar Fruitions. 
Ex. 1. i-2)2 - 5253(2-104416 Ex. 2. 17)51-491(3-028 
Ex. 3. Divide 22-690183 by 52-7678. Being reduced to 
their finite values, they are- 2 f4fuu§ 3 > and ; 
Then 2 |S 4 « 7 mH 3 xi f ! ?"r —°‘43 quotient. 
If the divifor be a circulate, whatever the dividend is, 
take the finite value of the divifor, and by its denomina¬ 
tor multiply the dividend, then divide the product by the 
numerator, and you will have the quotient fought. 
Ex. Divide 5-7648 by 8-47— Firft multiply 5-7648 
by 90, the product is 51-884, which divide by 763, the quo¬ 
tient is -68, 
INVOLUTION. 
Involution is the railing of powers from any proposed 
number called its firft power or root. The higher powers 
are frequently diftinguifhed by an index annexed to the 
root tifing according to the height of the power. Thus, 
the index of the fquare or fecond power is 2, of the cube 
or third power is 3, and fo on. 
Rule. Multiply the given root continually into itfelf 
till the number of multiplications be one lefs than the in¬ 
dex of the power fought, and the laft product will be the' 
power required. 
Thus fuppofe 5 the root or firft power; 
Then 5x5=25, the fquare or fecond power, or 5°. 
5X5X5=125, the cube or third power, or 5 s . 
5X 5X 5 X 5=625, the biquadrate or fourth power, or 
6x6x6x6x6=46656, the furfclidor fifth power,or6*. 
And here obferve, if any number end with 5 or 6, all the 
powers of that number will end with 5 or 6. Alfo, the 
fum of any two or more indices of powers will be the in¬ 
dex of a power refulting from the multiplication of thele 
leveral powers into each other. As 5*'r 3 = 5 !! x 5 3 = 2 5 X 
125=3135, the fifth power of 5. 
Ex. 1. What is the fquare or fecond power of 37 f An- 
fwer, 1369. 
Ex. 2. What is the fquare of ? Thus -£xf=ff-, ar >f. 
Ex. 3. What is the cube of 2-07 ? Anlwer, 8-869743. 
Ex.4. What is the fourth power of -022. Anfwer, 
•000000234256. 
Ex. 5. What is the fquare or fecond power of 6^. An¬ 
fwer, 40-96. 
EVOLUTION or EXTRACTION of ROOTS. 
This is the reverfe of Involution; for, as that is finding 
the power of a given root, fo this is finding the root or 
firft power of any given higher power, either exadly, or 
■by approximation fufliciently near, in decimals. The roots 
which approximate are called Curd roots, and tliofe which 
are perfectly accurate are called rational roots. 
To ExtraEl the Square Root. Set a point over every other 
figure of the given number, beginning at the place of 
units in integers, and proceeding to the left; and begin¬ 
ning at the place of hundreds in decimals, and proceeding 
to the right; and this will mark it out into as many pe¬ 
riods as there will be figures in the root. Find the great- 
eft fquare contained in the firft period towards the left 
2 £ hand. 
1 IOX.JI. No, 65. 
