where m and n may be any numbers, taken at pleasure, provided their assumed values be such as will render the values of x, y, and z, in the above expressions, all positive.

6. It is required to find two square numbers, such that their difference shall be equal to a given number.

Let d = the given difference; which resolve into two factors a, b, making a the greater and b the less.

b

=

x2 = x2 +

Then, putting the side of the less square, and x+ side of the greater, we shall have (x + b)2 2bx + b2 — x2 = d =(ab), or 2bx + b2 = d = Whence, dividing each side of this equation, by b, we shall

(ab):

If, for instance, d = 60, take a × b = 30 × 2, and we shall

= 256 19660, the given difference.

7. As an instance of the great use of resolving formula of this kind into factors, let it be proposed, in addition to what has been before said, to find two numbers, x and y, such that the difference of their squares, x2 — y2, shall be an integral

square.

Here the factors of x2 shall have (x + y) × (x.

૫)

y2, being x + y and x

=

-

y, we

x2 y2. And since this product is to be a square, it will evidently become so, by making each of its factors a square, or the same multiple of a square. Let there be taken, therefore, for this purpose,

Then, by the question, we shall have (x + y) × (x − y) or its equal x2. y2 = m2r2s2; which is evidently a square, whatever may be the value of m, r. s.

--

But by addition and subtraction, the above equations give, when properly reduced,

where, as above said, m, r, and s, may be assumed at pleasure. Thus, if we take m= = 2, we shall have x = r2+s3, and yrs, which expressions will obviously give integral =

values of x and y, if r and s be taken any integral numbers.

8. It is required to find two numbers, such that, if either of them be added to the square of the other, the sums shall be squares.

Let x and y be the numbers sought; and consequently x2+y and y2+x the expressions that are to be transformed into squares. Then, if r x be assumed for the side of the first square, we shall have x2 + y = r2 - 2rx + x2, or y = r2

2rx; and consequently, x= 2r

y

And if sy be taken for the side of the second square, we

shall have y2+

2r

y

=s2+2sy + y2; or, by reducing the

equation, y=4rsy + 2rs2, and consequently, by re

duction, y =

r2-2rs2

and x =

2r3s + s2
;

where r and s may

4rs + 12

4rs + 1

be any numbers, taken at pleasure, provided r be greater than 2s2.

9. It is required to find two numbers, such that their sum and difference shall be both squares.

x be the two numbers sought; then, since their sum is evidently a square, it only remains to make their difference, x2 2x, a square.

For this purpose, therefore, put the root = x shall have x2- 2x x2 - 2rx + p2;

Or, by transposition, and cancelling a on each side of the equation, 2rx

where r may be any number, taken at pleasure, provided it be greater than 2.

10. It is required to find three numbers, such that not only the sum of all three of them, but also the sum of every two, shall be a square number.

Let 4x, x2 4x, and 2x+1, be the three numbers sought; then, 4x+(x2 - 4x) = x2, (x2 — 4x) + (2x+1)= x2 −2x + 1, and 4x+(x2 - 4x) + (2x + 1) = x2 + 2x + 1, being all squares, it only remains to make 4x + (2x + 1), or its equal, 6x+1, a square. For which purpose, let 6x + 1 = n2, and

we shall have, by transposition and division, x=

Where n may be any number, taken at pleasure, provided

it be greater than 5.

QUESTIONS FOR PRACTICE.

1. It is required to find a number x, such that х 1 shall be both squares.

2. It is required to find a number x, such that x+7 shall be both squares.

3. It is required to find a number x, such that 10 - x shall be both squares.

4. It is required to find a number x, such that

x + 1 shall be both squares.

x + 1 and Ans. x= x + 4 and Ans. 57. 10 + x and Ans. x =

6.

x2 + 1 and Ans. 40.

9

5. It is required to find three integral square numbers, such that the sum of every two of them shall be squares.

Ans. (528), (5796)2, and (6325)3. 6. It is required to find two numbers, x and y, such that x2+y and y2 + x shall be both squares.,

2

Ans. x=20 and y = 7%. 7. It is required to find three integral square numbers that shall be in harmonical proportion. Ans. 25, 49, and 1225. 8. It is required to find three integral cube numbers, x3, y3, and z3, whose sum may be equal to a cube.

Ans. x=

= 3, y = 4, z = = 5. 9. It is required to divide a given square number (100) into two such parts, that each of them may be a square number. Ans. 64, and 36.

10. It is required to find two numbers, such that their difference may be equal to the difference of their squares, and that the sum of their squares shall be a square number.

Ans. 4 and 3. 11. To find two numbers, such that if each of them be added to their product, the same shall be both squares.

Ans. and 5.

12. To find three square numbers in arithmetical progression. Ans. 1, 25, and 49. 13. To find three numbers in arithmetical progression, such that the sum of every two of them shall be a square number. Ans. 120, 8402, and 15601. 14. To find three numbers, such that if to the square of

each, the sum of the other two be added, the three sums shall be all squares. Ans. 1,, and 16. 15. To find two numbers in proportion as 8 is to 15, and such that the sum of their squares shall be a square number. Ans. 576 and 1080. 16. To find two numbers, such that if the square of each be added to their product, the sums shall be both squares.

Ans. 9 and 16. 17. To find two whole numbers such, that the sum or difference of their squares, when diminished by unity, shall be a square. Ans. 8 and 9.

25

1296

18. It is required to resolve 4225, which is the square of 65, into two other integral squares. Ans. 2704 and 1521. 19. To find three numbers in geometrical proportion, such that each of them, when increased by a given number (19), shall be square numbers. Ans. 81, 5, and 20. To find two numbers, such that if their product be added to the sum of their squares, the result shall be a square number. Ans. 5 and 3, 8 and 7, 16 and 5, &c. 21. To find three whole numbers such, that if to the square of each the product of the other two be added, the three sums shall be all squares. Ans. 9, 73, and 328.

62920 629209

62920

22. To find three square numbers, such that their sum, when added to each of their three sides, shall be all square numbers. Ans. 4418 13254, and 19818 = roots required. 23. To find three numbers in geometrical progression, such, that if the mean be added to each of the extremes, the sums, in both cases, shall be squares. Ans. 5, 20, and 80.

24. To find two numbers such, that not only each of them, but also their sum and their difference, when increased by unity, shall be all square numbers. Ans. 3024 and 5624.

25. To find three numbers such, that whether their sum be added to, or subtracted from, the square of each of them, the numbers thence arising shall be all squares.

969 969

96

Ans. 406, 518, and 121. 26. To find three square numbers such, that the sum of their squares shall also be a square number.

25

Ans. 9, 16, and 144. 27. To find three square numbers such, that the difference of every two of them shall be a square number.

Ans. 485809, 34225, and 23409.

28. To divide any given cube number (8), into three other cube numbers. Ans. 1, 27 64, and 127. 29. To find three square numbers such, that the difference

between every two of them and the third shall be a square number. Ans. 1492, 2412, and 2692. 30. To find three cube numbers such, that if from each of them a given number (1) be subtracted, the sum of the remainders shall be a square number.

Ans. 4913 21952 and 8.

3375) 33759

OF THE SUMMATION AND INTERPOLATION OF

INFINITE SERIES.

THE doctrine of Infinite Series is a subject which has engaged the attention of the greatest mathematicians, both of ancient and modern times; and when taken in its whole extent, is, perhaps, one of the most abstruse and difficult branches of abstract mathematics.

To find the sum of a series, the number of the terms of which is inexhaustible, or infinite, has been regarded by some as a paradox, or a thing impossible to be done; but this difficulty will be easily removed, by considering that every finite magnitude whatever is divisible in infinitum, or consists of an indefinite number of parts, the aggregate, or sum of which, is equal to the quantity first proposed.

A number actually infinite, is, indeed, a plain contradiction to all our ideas; for any number that we can possibly conceive, or of which we have any notion, must always be determinate and finite; so that a greater may still be assigned, and a greater after this; and so on, without a possibility of ever coming to an end of the increase or addition.

:

This inexhaustibility, therefore, in the nature of numbers, is all that we can distinctly comprehend by their infinity for though we can easily conceive that a finite quantity may become greater and greater without end, yet we are not, by that means, enabled to form any notion of the ultimatum, or last magnitude, which is incapable of farther augmentation.

Hence we cannot apply to an infinite series the common notion of a sum, or of a collection of several particular numbers, which are joined and added together, one after another; as this supposes that each of the numbers composing that sum, is known and determined. But as every series generally observes some regular law, and continually approaches towards a term, or limit, we can easily conceive it to be a whole of its own kind, and that it must have a certain real value, whether that value be determinable or not.

Thus, in many series, a number is assignable, beyond which no number of its terms can ever reach, or, indeed, be ever