over the fucceeding terms of the 
n —r-j-l . r-\-i 
equation, beginning with thefecond; alfo, place the fign 
-j- under the firft and laft terms, and under every other 
term 4* or —according as the fquare of that term multi¬ 
plied by the fradtion which hands over it is greater or lefs 
than the product of the adjacent terms; then there will 
be as many impoffible roots in the equation as there are 
changes of thefe figns from 4- to — and from —■ to -f-. 
Ex. Let the propofed equation be x 3 —4** + 4 ^—-6=o. 
In the feries of fractions -, -, -, if each terra be di- 
1 2 3 
vided by that which precedes it, we obtain -, to b© 
placed over the terms of the equation 
ALGEBRA 305 
be repeated till the coefficient of one of the unknown 
quantities is unity, and the coefficient of the other a whole 
number; then an integral value of the former may be ob¬ 
tained by fubftituting nothing, or any whole number, for 
the other; and, from the preceding equations, integral 
values of the quantities propofed may be found. 
Ex. 1. Let 5x4-77=129; to find the correfponding in¬ 
tegral values of x and y. Dividing the whole equation 
by 5, the lefs coefficient, 
3 3 
v2 
PART III. 
ON UNLIMITED PROBLEMS . 
WHEN there are more unknown quantities than inde¬ 
pendent equations, the number of correfponding .values 
which thefe quantities admit is indefinite. This number 
may be leffened by rejecting all the values which are not 
integers; it may be farther leffened by rejecting all the 
negative values; and Hill farther, by rejecting all values 
which are not fquare or cube numbers, See. By reduc¬ 
tions of this kind the number of anfwcrs may he confined 
within definite limits. 
If a fimple equation exprefs the relation of tvyo un¬ 
known quantities, and their correfponding integral values 
be required, divide the whole equation by the coefficient 
which is the lefs of the two, and fuppofe that part of the 
quotient, which is in a fraffional form, equal to fame 
whole number; tints a new fimple equation is obtained, 
with which we may proceed as before; let the operation 
VUL. I. No. 20. 
. , 2 V ,4 
Or A': 
Affitme 
= 5-74 
-*y 
4—2 y 
5 
■pi ex 4- 
- 2 y—sP> 
x ’ —4** 4-4 x —6=0 
__ 4 4 — 4 
and. fince ——, or —, is greater than 4X 1, die fign. -fc- 
3 3 
4' 16 . 
muff be placed under the fecond term ; but—, or —, is 
lefs than —4X—6, or 24, therefore the figp—muff he 
placed under the third term; and 4- being placed under 
the firft and laft terms, there are two changes of figns; 
therefore the equation contains two impoffible roots. 
Scholium. The difeovery of the number of impoffi¬ 
ble roots in an equation has given great trouble to algebra- 
ifts, and their researches, hitherto, have-not been attended 
with any great fuccefs. In a cubic equation a 3 — qx-\-r 
yZ 
—o two roots are impoffible or not, according as-— 
4 fl 7 
ispofitive or negative. A biquadratic, x’-j-ip^-l-rx-pi 
—o, has two impoffible roots when two roots of the equa¬ 
tion y 3 -\-2i]y ' 2 4-tf 2 —4x.j V —r 2 —o, are impoffible; and all its 
roots aje impoffible, when the roots of this cubic are all 
poffible and two of them negative. 
Dr. Waring has given rules for determining the number 
of impoffible roots in equations of five and fix dimenfions, 
but the inveftigationof thefe rules cannot.ppffibly he intro r 
duced within our limits. 
Sir If. Newton’s rule, fpecified as above, is general and 
eafdy applied, but as it is deduced from the nature of the 
inferior limits, it will not always dote6l impoffible roots. 
The proof alfo is defedlive, as it does not extend to that 
part of the rule which refpecls the number of impoffible 
roots. Thus far, however, it may be depended upon, 
that it never fhews impoffible roots, but when there are 
fuch in the propofed equation. 
Then 2— -y—2p-\- - 
2 
P 
y— 2—2 p - 
2 
Let p—2s, thenjy— 2—5?, and —/ 4 #= 
3 + 5 X 4 -25—34-73. 
If 5—0, then'41=3 and y—2, the only pofitive whole 
numbers which anfwer the condition of the equation ; for 
if s—i; then x=io andy—— 3; and if 5——1, thenx;^; 
—4 andy—7. ■ • 
If -the-iimple equation contain more unknown quanti¬ 
ties, their correfponding integral, values may be found in 
the fame manner. 
Ex. 2. Let 4 a4 -374 io =3 X/; to find the correfpond¬ 
ing integral values of a, y, and v. Dividing the -whole, 
equation by 3, the leaft coefficient, 
, 1 , •*' 4*1 ■ . 2P ’ 
* 47434——=^4 — 
jz=.v- 
Affume 
-■=.p, or an— a —1 —ip 
Then x—2v —3 p—1 
And >—n~ ? n 4 - 3 /)' 4 -i—34-/)=4ifr— v —2, 
and fubftituting for p and v nothing, or any whole num¬ 
bers, integral values ■©£-x and-y are known. If n—3, and 
p— 1, then a— 2 and y~—i 3 if 0—4 and p=z o, then 
a= 7 andy——6, Sec. 
Ex. 3. To find what numbers are divifible by 3 without 
remainders. • ^ 
Let <2, b, c, d, See. be the digits, or figures in the unit’s, 
ten’s, hundred's, thoufand’s, &c. place of any number, 
th.en the number is a 4 - io ^4 ioo<: 4 iooo ^4 &c. this divid¬ 
ed by 3 is - + 3 J+- 4-33C 4-f +3330-4-^ 4. See. or 
•>3 3 3 
a+b-\-c.-\-d-\- , 8 c c. 
----- 43 " 433 c 4333^4 &c * which is a 
11 , , a-\-b+c-\-d-\- Sec. . , 
whole,pumber when ——— -—1-is a whole num- 
3 
her, that is, any number is divifible by 3 if the fum of its 
•digits be divifible by 3. Thus 111,252, 7851, &c. are di¬ 
vifible by 3. 
In the fame manner, any number is divifible by 9 if the 
fum of its digits be divifible by 9. 
For 
a4-io44ioor4-ioooa4- <1'C. 
9 .9 
rt 4 ^ 4 c 44 }- & c > 
4-r 1 
9 y 
See. which is a whole number when 
- b —p ~ I T c *4— - 
9 ’ 9 
1 e 4-1 x 
o. 4 ^4 ( '4 ( ^ ,&C. 
IS a 
whole number. Thus 684, 6588, See. are divifible by 9,' 
Cor. i. Hence, iff any number and the fum of its digits 
be each divided by 9, the remainders are equal. 
Cor. 2. From this property of 9 may be deduced the 
3 1 rufe 
