ALGEBRA. 
30 ? 
^-W' a +V^+& c< =a 3 -— a-, therefore 
dividing both Tides by ABC, -L 
Lb 
has been proved equal to ~ — 
quantities 
B A 
&c. have a common difference 
W 
a+V '■>* = x + V - 
2 , or i-j-a-=A- 4 —2tf.V* 
o; from which equa- 
Va-{-y/ b-\- See. —x* — iax- -j-a 
-|-a 2 ; hence x* —2a# 2 -— x-\-a 2 — b. 
tion .* may be found. 
To find the lead common multiple of two quantities ; 
or the lead quantity which is divitible by each of them 
without remainder. The product of the two quantities 
divided by their greated common meafure is their lead 
common multiple. Let a and b be two quantities, .v their 
greated common meafure, m their lead common multi¬ 
ple, and let m contain a, p times, and b, q times, that is, 
a o 
let nv=pa-=.qb ; then -=and iince m is the lead podible, 
p and q are the lead podible, therefore ^ is the fraction - 
in its lowed terms, and q=:-\ hence m—qb=—. 
X X 
Ex. What is the lead common multiple of i 3 and 12 ? 
Their greated common meafure is 6 ; therefore their lead 
. 12X18 
common multiple is —^—=36. 
Every other common multiple of a and b is a multiple 
of m. Let n be any other common multiple Of the two 
quantities; and, if podible, let m be contained in n , r 
times, with a remainder s, which is lefs than m\ then 
n — rm-=zs, and Iince a and b meafure n and rm, they mea¬ 
fure n—rm or s, that is, they, have a common multiple 
lefs than m, which is contrary to the fuppofition. 
To find the lead common multiple of three quantities, 
a, b, andc, take m the lead common multiple of a and b, 
and n the lead common multiple of m and c ; then n is the 
lead common multiple fought. For every common mul¬ 
tiple of a and b is a multiple of m ; therefore, every com¬ 
mon multiple of a, b, and c, is a multiple of m and c ; alfo, 
every multiple of m and c is a multiple of a, b, andc; con- 
fequently, the lead common multiple of m and c is the 
lead common multiple of a, b, and c. 
Three quantities are faid to be in harmoincal proportion, 
when the fird is to the third, as the difference of the fird 
and tfecond is to the difference of the fecond and third. 
Any magnitudes, A, B, C, D, E, &c. are faid to be in 
harmonical progrejjion, if A: C:: A — B : 3 — C ; B : D:: 
B —C : C —jD; C: E:: C—D : D — E, &c. 
The reciprocals of quantities in harmonical progreflion 
are-in arithmetical progreffion. 
Let A, B, C, &c. be in harmonical progreffion, then 
A : C:: A — B-.B —C, therefore AB — AC—AC — BC, and 
1 1 
Again, B:D :B — C:C — D;- therefore BC—BLhzzDB 
=~DC, and, dividing by BCD, -L — -^-=-^--—^-5 and 
Affume 
Then 3 V a —yj —tf — x — y/ - 
By mult. 
-r 
2 yf a 2 -\-b 2 —x 2 A^y^-m, by fubftitution; 
Therefore, y-r^m —x 2 ; alio from the fird equation* 
— b z —x-\-y/ —jy 2 | 3 — -* 3 + 3X 2 \/— y 2 — 3 xy 2 —• 
y 2 y/ — y 2 ; therefore x 3 — %xy 2 —a-, or fubdituting for_y 2 its 
value m —.v 2 , x 3 — 2 mxJ r3 x3=a j that is, 4.x 3 — ynx — a—o, 
a cubic equation whofe roots may be found by approxima¬ 
tion; hence y, and confequently x-j- yf — y 2 , the root re¬ 
quired, may be determined. 
In the fame manner it appears, that the cth root may 
be extracted by the folution of an equation of c dimenlions. 
Let A-\-B be the given binomial ford, in which both 
terms are poilible and A greater than B, its cth root, or 
the cth root of A-\-B.^/Q, a quantity of the fame deferip- 
tion, may fometimes be found in the form a-}- yj b, or y/ a 
b, where a and b are whole numbers. 
Let c V 7 +Bx y/]Q=zx+y, where x and y involve the 
fame fin ds that A y/ 0 and B y/ 0 do, refpedtively. 
Thenfince A-\-Bx\/Q—x-\-y 
V /foii X y/~Q- x—y, 
By mult. V A 2 — B 2 X Q—x‘-—y 2 ; let Q be fo a (Turned 
that A 2 —B 2 XQ may be a perfect cth power —n c , then 
t/ . / /) j n t } ie 
take r— V A-\~Bx yf U 
X 2 - 1)2 y. 
neared integer number; then ■ 
r 
x+y 
X — y — 
n 
r +z 
and *4. 
y—y 
therefore zx—r -\—, 
r 
foppofe s to 
be the ford which A yf 0 and * involve ; alfo let t be the 
rational part of x, and foch that ts does not differ from 
r-f- 
T I 
- by -, then if the root of the binomial be of the 
2 2 
form x-\-y, the value of .*■ is ts-, alfo x 2 
— x 2 — n — t 2 s 2 —• 72; therefore y- 
C V A-{-Bx sj U — ts 4- yf t 2 s 2 — n, or c y/A-\-B — 
ts -j- y/ t 2 s 2 —n 
2 V2~ 
By affuming r, the neared integer to c V A^-B. y/ (J, 
the value of x is accurately determined. 
^2_ 
Let x-^y—r+q exaftly, or x-\-y — q—r\ then--— 
' x+y—i 
therefore the 
111 
~B' e ~D' 
ihat is, they are in arithmetical progreffion. 
On Binomial Surds. 
To extraft the root of a binomial, one or both of whofe 
faftors are quadratic furds. If the index of the root to be 
extracted be an even number, the fquare root may be 
found by the method before given on Surds, when it can 
be expreffed by a binomial of the fame defeription ; and, 
if half the index be an even number, the fquare root may 
again be taken, and fo on, till the root remaining to be 
extrafted is expreffed by an odd number. 
Let the cube root of the binomial V— 1,2 be re¬ 
quired. 
n 1 
==—=*—y-V 
r 
qxx—y 
xAry—q 
differs from zx by q- 
and r- 4 - -r=.ix—q 
r 
qXx- 
which 
qX x —y 
ty—q 
i. e. by the difference of 
x-\~y —<7 
two fractions, each of w hich is lefs than r, when x and y 
are pofitive, x greater than y, and x-\~y greater than 1 
therefore the value of zx does not differ from the truth by 
1, nor the value of x or tsby-; and s is known, confe- 
2 
quently, the neareil integer to t, that is, f, which is an in¬ 
teger, is known. 
Cor. If the ctli root of A■—B be required ; find, by the 
X—i-y x—y 
the cth root of A^-B-, and -—, is the 
rule 
20 VQ 
cth root of A- 
-R. % 
Qb 
