fion, and there is only ore equation expressing 
-their relation ; but it is required also that they 
may be rational, which circumstance cannot be 
expressed by an equation: another condition 
therefore must be assumed, in such a manner as 
to obtain a solution in rational numbers. 
Let the given square be a 1 ; let one of the 
squares sought be .v 2 , the other is a 2 — a 2 . Let 
rx — a also be a side of the last square, therefore 
r 2 \‘ 2 — 2rxa — a 2 ' Ter- a 1 — v 2 
Bv transp. 
Divide by x 
Therefore 
-j- x 2 — 2rxa 
-j- X —Sra 
2ra 
And rx — a = 
r 2 +l 
lr 2 a r 2 — I 
+T * = + 1 
Let r therefore be assumed at pleasure, and 
2 ra r l — 1 
-~ — - — , — —a, which must always be ra- 
■ r 1 -f- 1 r -j- 1 
.tional, will be the sides of the two squares re- 
quired. 
Thus, if a 2 — 100; then, if r — 3, the sides 
of the two squares are 6 and 8, for 36 -j- 64 zz: 
100. 
Also, let a 2 — 64. Then, if r rz: 3, the sides 
32 
and 
24 1024 
— - ; and 
5 25 
576 
: 64. 
of the squares are 
_ 1600 
~ 25~ 
The reason of the assumption of rx — u as 
a side of the square a 1 — a- 2 , is, that being 
squared and put equal to this last, the equation 
manifestly will be simple, and the root of such 
an equation is always rational. 
Ex. 2. To find two square numbers whose 
difference is given. 
Let a 2 and y 2 be the square numbers, and a 
their difference. 
Put 
+ 
r, and - 
zv -j- v 2 
4 
. If x and 
y are required only to be rational, 
then take <o at pleasure, and x— ----- , whence x 
v 
and y are known. 
But if .v and y are required to be whole num- 
bers, take for z and v any two factors that pro- 
duce a, and are both even or both odd numbers. 
And this is ssible only where a is either an 
odd number greater than 1, or a number divi- 
sible by 4. 
~~ J — - — are the numbers sought. 
Then 
For the product of two odd numbers is odd, 
and that of two. even numbers is divisible by 4. 
Also if z and v are both odd or both 'ever., 
z -j- V Z V , 
— - — ana — - — must oe integers. 
Ex. 1. If a — 27, take v ~ 1, then z = 27 ; 
and the squares are 196 and 169. Or z may be 
6, and v zz: 3, and then the squares are 36 and 9. 
2, If a — 12, take v zz: 2, and z — 6 ; and 
the squares, are 16 and 4. 
Of the Origin and Composition of Equa- 
tions; AND OF THE SlGNS AND Co-EFFI- 
CIENTS OF THEIR TERMS. 
The higher orders of equations, and their ge- 
neral affections, are best investigated by con- 
sidering their origin from the combination of 
inferior equations. 
In this general method, all the terms of any 
equation are brought to one side, and the equa- 
Vol. I. 
A L G K E R A. 
t?©n is expressed by making them equal to 0. 
Therefore, if a root of the equation be inserted 
instead of (a) the unknown quantity, the posi- 
tive terms will be equal to the negative, and the 
whole must be equal to 0. 
Def When any equation is put into this form, 
the term in which ( v) the unknown quantity is 
of the highest power is made the first, that in 
which the index of „v is less by 1 is the second, 
and so on, till the last into which the unknown 
quantity does not enter, and which is called the 
absolute term. 
Prop. I. If any number of equations be mul- 
tiplied together, an equation will be produced, 
of which the dimension is equal to the sum of 
the dimensions of the equations multiplied. 
If any number of simple equations be mul- 
tiplied together, as a — a — 0, a- — b ~ 0, 
x — c ~ 0, &e. the product will be an equation 
of a dimension, containing as many units as 
there are simple equations. In like manner, if 
higher equations are multiplied together, as a 
cubic and a quadratic, one of the fifth order is 
produced, and so on. 
Conversely. An equation of any dimension is 
considered as compounded either of simple 
equations, or of other such that the sum of their 
dimensions is equal to the dimension of the 
given one. By the resolution of equations these 
inferior equations are discovered, and by inves- 
tigating the component simple equations, the 
roots of any higher equation are found. 
Cor. I. An equation admit f as many solu- 
tions, or has as many roots, a 3 there are "simple 
equations which compose it. 
Cor. 2. And conversely no equation can have 
more roots than it has dimensions. 
Cor. 3. Imaginary or impossible roots must 
enter an equation by pairs ; for they arise from 
quadratics, in which both the roots are such. 
And an equation of an even dimension may 
have all its roots, or any even number of them, 
impossible ; but an equation of an odd dimen- 
sion must at least have one possible root. 
Cor. 4. The roots are either positive or nega- 
tive, according as the roots of the simple equa- 
tions, from which they are produced, are posi- 
tive or negative. 
Cor. 5. When one root of an equation is dis- 
covered, one of the simple equations is found, 
from which the given one is compounded. The 
given equation, therefore, being divided by this 
simple equation, will give an equation of a di- 
mension lower by 1. 
Prop. II. To explain the general properties 
of the signs and co-efficients of the terms of an 
equation. 
Let .v — a — _ 0, x — b — O, x — c — 0, 
x — e/—0, &c. be simple equations, of which 
the roots are any positive quantities -f - L, 
r, -{— d , 8< c. and let x — |— tn zz: 0, x — 0, 
&c. be simple equations, of which the roots are 
any negative quantities — z», — », and let any 
number of these equations be multiplied toge- 
ther, as in the following table : 
x — a — 0 
X x — b — 0 
X * 
bx -}- ah 
~c— 0 
0, a Quadratic. 
n 
ah 
ac 
-j- be 
X x 2 -i- ac x -v — aA = 0, a Cubic. 
X x — m — 0 
x 1 — a ") ob \ — a be 1 , 
-fw ) -am f X S ) < l Uadratit - 
— b/n I 
+ 
&c. 
H 
From this table it is plain, 
1. That in a complete equation the number 
of terms is always greater by unit than the di- 
mension of the equation. 
2. The co-efficient of the first term is 1. 
The co-efficient of the second term is the sunt 
of all the roots (a, b, c, tn, See.) with their signs 
changed. 
The co-efficient of the third term is the sum 
of all the products that can be made by multi- 
plying any two of the roots together. 
The co-efficient of the- fourth term is the sum 
of all the products which can be made by mul- 
tiplying together any three of the roots with 
their signs changed ; and so of others. 
The last term is the product of all the roots, 
with their signs changed. 
3. From induction it appears, that in any 
equation (the terms being regularly arranged as 
in the preceding example) there are as many 
positive roots as there are changes in the signs 
of the terms from -j- to — , and from — to -j-; 
and the remaining roots are negative. The rule 
also may be demonstrated. 
Note. The impossible roots in this rule are 
supposed to be either positive or negative. 
Cor. If a term of an equation is wanting, the 
positive and negative parts of its co-efficient 
must then be equal. If there is no absolute term, 
some of the roots — 0, and the equation may 
be depressed by dividing all the terms by the 
lowest power of the unknown quantity in any* 
of them. In this case also, .v — 0 — 0, ,v — (j 
— O, &c. may be considered as so many of the 
component simple equations* by which the 
given equation being divided, it will be de- 
pressed so many degrees. 
Of the Transformation of Equations. 
Prop. I. The affirmative roots of an equation 
become negative, and the negative become af- 
firmative, by changing the signs of the alternate 
terms, beginning with the second. 
Thus the roots of the equation a 4 — x 3 
1 f)x 2 -j- 4qx — 30 zz 0, are — J— 1 , -j- 2, -}- 3, 
— o, whereas the roots of the equation a 4 4- 
a 5 — 19a 2 — 49a — 30 nz 0, are — 1, _ 9 
~ 5> 
The reason of this is derived from the com- 
position of the co-efficients of these terms, which 
consist of combinations of odd numbers of the 
roots, as explained in the preceding head. 
Prop. II. An equation may he transformed 
into another that shall have its roots greater or 
less than the roots of the given equation bv 
some given difference. 
Let e be the given difference ; then y — x -f 
and and if for .v and its powenTia 
the given equation, y -f e and its powers be in- 
serted, a new equation will arise, in which the 
unknown quantity is y, and its value will be 
A L <?. 
Let the equation proposed be a’ — fix 2 -f- q X 
— - r = 0, of which the roots must be- diminish- 
ed by e. By inserting for a and its powers y e 
and ita powers, the equation required is, 
y 3 -j- ?> y 2 -f~ 3e z y -j~ e 1 ) 
- pf - - p* 2 
~i~ 77 + 7“ 
= 0 . 
Cor. 1 . T he use of this transformation is to 
take away the second, or any other intermediate 
term ; for as the co-efficients of all the terms" of 
the transformed equation, except the first, in- 
volve the powers of e, and known quantities 
only, by putting the co-efficient of any term 
equal to 0. and resolving that equation, a value 
of e may be determined, which being substituted, 
will make that term to vanish. 
Thus let the co-effident 3r — p = 0, and 
e ~ jp, which being substituted for ?, the new 
equation will want the second term. And uni- 
versally the co-efficient of the first term of an 
