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IT

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was formerly observed, (Chap. III.)

that if there are more unknown quantities in a question, than equations by which their relations are expressed, it is indetermined; or it may admit of an infinite number of answers. Other circumstances, however, may limit the number in a certain manner; and these are various, according to the nature of the problem. The contrivances by which such problems are resolved are so very different in different cases, that they cannot be comprehended in general rules.

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EXAMPLE I.

To divide a given Square number. into two. parts, each of which mall be a square number.

There are two quantities fought in this question, and there is only one equation expressing their relation; but it is required also, that they may be rational, which circumstance cannot be expressed by añi equation; another condition therefore must be assumed in such a manner as to obtain a folution in rational numbers.

Let the given square be a’, let one of the squares sought be x3, the other is a _*?. Let rxa also be a fide of the last fquare, therefore gr?x?—2rxata=a?—** By transp.

groziom? + x2=2rxa Divide by *

g2x+x=2ra

2ra Therefore

g2+ 2ra

And rx

a

-a

I
a.

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十1

Let r, therefore, be assumed at pleasure,

and

ära
and
g2 + 1 m2 +1

a, which must always be rational, will be the sides of the two squares required.

1

Thus, if a:=100. Then, if r=3, the sides of the two squares are 6 and 8, for 36+64=100.

Also, let a’=64. Then, if r=2, the sides

32 24 1024
of the squares are

and
5 5

25 25 1600

; and

**576

=64.

25

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The reason of the assumption of rxas a side of the square a?—x, 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 AMPLE II.

To find two squate numbers whqfe difference

is given.

Let

Let x2 and ya be the square numbers, and a their difference.

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? +2zvtv2

4 있 4

zv=x2-y2=a. If x and y are required only to be rational, then take v at pleasure, and z: = whence x and y are known.

But, if x and y are required to be whole numbers,' Take for z and v any two factors that produce a, and are both even or both odd numbers, and this is posible only where a is either an odd number greater than 1, or a number divifible by 4. Then and are the numbers fought.

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2

2

For the product of two odd numbers is odd, and that of two even numbers is dis visible by 4. Also, if x and v are both

odd,

i

z tu odd, or both even,

and

must be in

tegers.
Ex. I.

If a=27 take v=1, then x= 27; and the squares are 196 and 169: or ở may

be
9
and v=3

and then the squares
9.
2. If a=12, take v=2, and x=6; and
the squares are 16 and 4.

are 36 and

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To find a sum of money in pounds and fil

lings, whose half is just its reverse. Note. The reverse of a sum of money, as 81. 12 s. is 12 l. 8 s.

Let x be the pounds and y the shillings.
The sum required is 20x+y
Its reverse is 2oy+x

20x+y
Therefore, =20y+x

20x+y=40y+2x

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2

18x=39

x:y:: (39 : 18 :: ) 13 : 6.

In

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