that is, 4/ b is partly rational and partly irrational, 
which is impotliblej therefore x±za, and confequently 
\'y— yi. 
If two furds, ^/xand-^/y, cannot be reduced to others 
which have the fame irrational part, their produdt is ir¬ 
rational. 
If poffible, let 4/ xy— a-}-;?, then xy—x 2 -\~znx-\- n 2 , and 
2 » a - 4 - h * , „ , znxdrri 1 
and, fince x and y are integers,-- 
ALGEBRA. 
And d ! ^s.x t -\-y t 
By addition, c-^aJ zsf 1 
By fubtradlion, a — \Ja — b— 24/q artd the rpot' 
y 
By involution, 
a -J- &—x : -\-cx c 'y -J-c.- x c 
tional; therefore, a—x c -\-c. - x c 
2 
cy c ‘+r- 
3 
■y-c- 
4-c.—w c 
2 _, - •> 
quently, a — b\ —x — y. 
If c be an odd number, a and b, one or both, quadratic 
furds, and * and^ involve the fame furds that a and b do, 
__L 
refpedtively, when a-\-b\ =:,r-J-y, then 
By involution, a-\-b=.x c -f-c* c 
‘y-f-r.- -x c 
2 
x c 2 y 2 -\- &c. and bezzex 1 'y- J-r. 
hence a — b—x c — cx c 'y-j-c .~—-* c 
, r 1 c —2_ 
y 
f- 
-f- a/ a 2 —b 
+ 
-a/ a 2 —b 
x - x 
is an integer, or n is a multiple of x; let n—rx, thenj— 
x-|-2rx-}-Ax2=i-^r-^-rhA-, and \J y—\ -\-r aJ x, i. e. \J y 
■and aJ x may be reduced to the fame irrational part, which 
is contrary to the fuppolition. 
One furd, aJ x cannot be made up of two others, 4/ m 
and aJ it, which have not the fame irrational part. 
If poffible, let x== a] n ; then, by fquaring 
both (ides, x—m-\~n-\-z\/mn, and x — m — n—2\/mn,ara- 
tional quantity equal to an irrational one, which is abfurd. 
Let a-\-b 1 =zx-\-y, where s is an even number, a a ra¬ 
tional quantity, b a quadratic furd, x and^< one or both of 
___l_ 
them quadratic furds, then a — bf—x — -y. 
c. -.- x c V- 4 - &c. and, fince c is even, the odd 
2 ^ . 
terms of the feries are rational, and the even terms irra- 
■y-f- &c. and b— 
~ 3 /+ &c. hence a — b—x c — cx c 'y 
-I c _2 
-— -x c ~ 3 y 3 -{- &c. and, confe- 
2 
'^ 3 -}-&c. 
-I C -2 
2 2 
From this conclufion it appears that the fquare root of 
a+Ajbcaw only be expreffed by a binomial of the form 
w+jr, one or both of which are quadratic furds, whew 
a 1 —b is a perfedl fquare. 
-aiiS 
The values of x and y are 
there are therefore four different va-. 
r 
- aJ a 2 —b 
lues of x-\-y. 
Ex. 1. Required the fquare root of 3 \f 2. 
In this cafe, (72=3, aJI—zaJz, and a 1 — b= 9—S=i; 
r» _l. j — 
hence, x=z aJ TJ—— Qz, andjy: 
2 
-.aJ - -=15 therefore, 
x-\-y—Aj 2-\- 1. 
Ex. 2. Required the fquare root of 7—2 \J 10. 
°— b—a. and x— 
■y-f Here 7, a/ b=z\/ 10, 
Vb 
•^5; aUoy=-Q - -—aJz ; therefore, x — y=yb 5- 
the root required. 
Ex. 3. Required the fqua re roo t of 4 aJ —5—1. 
Here a =—1, v / ^ = +V / — 5 > a% —^=8i, and 
rT+9__ 3 , a if 0j y—J~-2Z—A/—b i 
2 2 
root required is 2 -\-\f —5. 
therefore, the 
a—bV zzex—y. 
y+<- 
-,- x c 3 y 3 4 - &c. where the odd terms involve the 
23 
fame furd that x does, becaufe c is an odd number, and 
the even terms the fame furd that y does; and, fince no 
part of a can confdt of y, or its parts, a~x c -\-c. C * 
x c 3 y 3 -\- &c. therefore, by evolution, a — b} L —x — y. 
The fquare root of a binomial, one of whofe fadfors is 
a quadratic furd, and the other rational, may fometimes 
be expreifed by a binomial, one or both of whofe fadtors 
are quadratic furds. 
Let a-\- aJ b be the given binomial, and fuppofe 
''Ja-\-\J~b—x-\-y, where * and y are one or both qua¬ 
dratic furds, 
Then V a—\J b— x —y 
By multiplication, \f a 2 — b—x* — y 2 
Alfo, by fquaring both fides of the firft equation, 
a \/-b=:x 2 -j- 2 xy -\-y 2 
PART II. 
ON THE NATURE OF EQUATIONS. 
ANY equation, involving the powers of one unknown, 
quantity, may be reduced to the form x n — •px n ~'-\-qx n ' *■—• 
&c. 2=0; where the whole expreffion is made equal to 
nothing, the terms are arranged according to the dimen^ 
lions of the unknown quantity, the coefficient of the high- 
eft dimenfion is unity, and the coefficients, p, q, r, See. are 
affedfed with their proper figns. 
An equation, where the index of the higheft power of 
the unknown quantity is n, is faid to be of n dimenfions; 
and, in fpeaking fimply of an equation of n dimenfions. 
We understand one reduced to the above form, unlefs the 
contrary be expreffed. 
Any quantity, x 11 — px n '-\-qx 7: 2 ....-\-Px — Q, may be 
fuppofed toarifefromthe multiplication of x — a.x — b.x — c. 
&c. continued to n fadtors. For, by adtually multiplying 
the fadtors together, we obtain a quantity of n dimen¬ 
fions, fimilar to the propofed quantity, x n — px n ‘-f* 
qx n —*— See. and, if a, b, c, d, Sec. can be fo affumed that 
the coefficients of the correfponding terms in the tvvo'quan* 
tities become equal, the whole expreffions coincide. And 
thefe coefficients may be made equal, becaufe we ffiall have 
n equations to. determine the n quantities, c, b, c, d, Sec. 
If then the quantities, a, b, c, d. Sec. be properly affumed, 
the equation x n — px l '-\-qx'‘~~ 2 —&c. 22:0, is the fame, 
with ^— a.x—b.x — c. See. —o. 
This proof, however, though ufually given, is imper- 
fe 61 ; for, if the n equations be reduced to cue, containing; 
only 
