3 <j 6 
A L G E B 
rule which is ufually given for difcovering when the mul¬ 
tiplication of two numbers is not juft. 
Let g a-^f-x be the multiplicand 
9 b-\-y the multiplier 
Then 3 1 ab-\-e)bx-\-r)ay-\-xy is the product; 
and, if the film of the digits in the multiplicand be divided 
by 9, the remainder is x ; if the fum of the digits in the 
multiplier be divided by 9, the remainder isjy; and, if the 
fum of the digits in the product be divided by 9, the re¬ 
mainder is the fame as when the fum of the digits in xy is 
divided by 9, unlefs the work is wrong. 
No general rule can be laid down for finding integral, 
correfponding values when the final equation is not a fim- 
ple one, aln.ofi: every particular cafe requiring a different 
procefs. 
On Continued FraBions. 
a . 
To reprefent - in a continued fraction. 
Let b be contained in a, p 
times, with a remainder c; 
again, let c be contained in b, 
q times, with a remainder d, 
and l'o on, then we have, 
a—pb-\-c 
bzaqc-\-d 
c—rd-\-e 
See. 
b) a(p 
c) H<? 
e See. 
Or 
a . r 
--p+- = p + 
q cJ r<£ 
~p-Y 
, d 
?+ ~ 
tszp- 
< 7 + 
t=P+ 
rd-\-e 
1 
f+ 
That is, -£=P+ 
e 
T +d 
Sc c. 
1 
J+&C. 
Cor. 1. An approximation may thus be made to the 
value of a fraction whofe numerator and denominator are 
dn too high terms, and, the farther the divifion is conti¬ 
nued, the nearer will the approximation be to the true 
value. 
Cor. 2. This approximation is alternately lefs and 
greater than the true value. 
Thus p is lefs than -r, and p 
0 
— is greater, becatife a part of the denominator of the 
frattion is omitted; alfo 5+^ is too great for the denomi- 
aator, therefore p-\- 
— is lefs than 8 c c. 
1 o 
qAr~ 
Ex. To find a fraction which fball nearly exprefs 
x00000’ 
R A. 
100000)314159(3 
300000 
14159)100000(7 
99113 
887)14159(15 
887 
4435 
854)887(1 
854 
33 & c. 
Here p= 3 , q— 7, r~is, s—i 
= 3 +--- 
, 8 cc. therefore 
100000 
7+- 
15+ See. 
The firft approximation is 3, which is too little; the 
—= 3 + 
next is 3too great; the next is 3-J- 
7 + 
15 
If ? 1 
—-=—7, too little; and fo on. This fraClionexprefies, 
106 106 
nearly, the circumference of a circle in terms of its dia¬ 
meter ; therefore the circumference is greater than 3 dia¬ 
meters, lefs than — diameters, and greater than 211, & c . 
7 106 
To find the value of a continued fraction when the de¬ 
nominators q , r, s, Sec. recur in any certain order. 
1 .,1 
' i 
Ex. 1. Let- 
-==.*; then 
r + 
r + # 
ex, or 
r-\-x 
qr+qx+i 
r- J- &c. 
hence r-\-x—qrx-\-qx*Jr*x, and 
:o, from the folution of which quadratic the 
* s +r*-= 
value of .*• may be obtained. 
r 
Ex. 2. Let I a -f- ■ 
7 
f 
zx ; by fquaring 
a- j- Sc c. 
both fides, 
J*' 
zx f xza-\- -; and x 3 —ax 
\Ja-\- Sec. 
>—b— 0 ; whence the value of .*• may be found. 
In the fame manner may the values of other quantities, 
which run on in infinitum , be found, if the factors recur. 
fl _ 
Ex. 1. To find the value of 
infinitum. 
^'a\JaSe c. in 
r . _ 
Let \ a\/ a\f a 8 cc.=x ; by fquaring both fides, 
Wa^a&cc. thatis, ax—x 2 , or x=a. 
,r 
Ex. 2. Required the value of 
Sec. in infinitum. 
Let J a+^+x/'a+^b+ 8 ec.=zx, by fquaring both 
fides, c-L 
yJb+Sec. =* z , and 
