7. 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 y+r the expressions that are to be transformed to squares.

Then, if r-x be assumed for the side of the first square, we shall have x2+y= r2 — 2rx + x2, or, by

reduction, x= ; and if s+y be taken for the

2r

numbers required;

where r and s may be any numbers, taken at pleasure, provided r be greater than 2s2.

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

Let x and x-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 its root =x-r, and we shall have x-2x= x2 - 2rx+r, or, by

transposition, 2rx-2x= r2; whence x=

xo — 2x = 4 (‚—'—° 7)2 —‚—¡ ; where r may be any num

r-1

problems. The excellent old Kersey, after amplifying and illustrating it in a variety of ways, concludes his chapter thus: "For a farther account of this rare speculation, see Andersonus, theorem 2, of Vieta's mysterious doctrine of Angular Sections; and likewise Herigonius, at the latter end of the first tome of his Cursus Mathematicus."

ber taken at pleasure provided it be greater

than 1.

9. 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, & 4x + (x2 − 4x) + (2x + 1) = x2 + 2x + 1, being all evidently squares, it only remains to make the quantity 4x + (2x+1), or its equal, 6x + 1, a square; for which purpose, let 6x+1=a, and we

be any number taken at pleasure, provided it be greater than 5.

QUESTIONS FOR PRACTICE.

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

5

Ans. ✰=2

4

2. It is required to find a number a such, that x+4 and x+7 shall both be squares.

3. It is required to find two numbers such, that, if their product be added to the sum of their squares, the result shall be a square.

Ans. 3 and 5 4. Find two numbers such, that, if the square of each be added to their product, the sums shall be both Ans. 9 and 16

squares.

5. It is required 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 6. To find two whole numbers such, that, if unity be added to each of them, and also to their halves, the sums, in both cases, shall be squares. Ans. 48 and 1680

7. It is required to find three square numbers, that shall be in arithmetical progression.

Ans. 1, 25, and 49 8. It is required to find three square numbers, that shall be in harmonical proportion.

Ans. 1225, 49, and 25 9. To find three whole numbers such, that, if to the square of each the product of the other two be added, the sums shall be squares.

Ans. 9, 73, and 328 10. To find three numbers in geometrical progression such, that the difference of every two of them shall be a square number.

Ans. 567, 1008, and 1792 11. To find three numbers such, that, if each of them be added to the product of the other two, the sums shall be all squares. Ans. 1, 7, and 9 12. It is required to resolve 4225, which is the square of 65, into two other integral squares. Ans. 2704 and 1521 13. It is required to resolve 9+ 2o, or 85, into

two other integral squares.

Ans. 72+6

14. It is required to find three square numbers such, that their sum shall be a square.

144

Ans. 9, 16, and 25

15. To find three numbers such, that their sum and their three differences shall be all squares.

Ans. 24, 168, and 249 16. To find three numbers in geometrical progression such, that, if the mean be added to each of the extremes, the sum, in both cases, shall be squares. Ans. 5, 20, and so 17. To find three square numbers such, that the sum of every two of them shall be

squares.

Ans. 44, 1179, and 240°

18. To find two integral numbers such, that if unity be added to each, as also to their sum and difference, the four results shall be squares.

Ans. 1368, 840, or 2208 and 528 19. To find three square numbers such, that the differences of every two of them shall be squares. Ans. 153, 185, and 6972

20. To find three numbers such, that the sum, or difference, of any two of them shall be a square number. Ans. 434657, 420968, and 150568 21. It is required to find three square numbers such, that the difference between every two of them and the third shall be a square number.

Ans. 149, 241, and 269°

22. To find two numbers such, that their sum shall be equal to the sum of their cubes.

23. To find three numbers such, that, if each of them be added to the cube of their sum, the three results shall be all cubes.

24. To find three numbers such, that, if each of

them be subtracted from the cube of their sum, the three remainders shall be all cubes.

OF CONTINUED FRACTIONS.

(H) A CONTINUED FRACTION is that which has for its denominator a whole number and a fraction; and which latter fraction has, also, for its denominator, a whole number and a fraction; and so on, continually, or till the series terminates, by being broken off, after a certain number of terms.

(i) The two last questions, here given, are reckoned among the most difficult of any of those that have been proposed by Diophantus; and, if solved by his method, the operation will be found extremely intricate and laborious.

Vieta was of opinion, that a number, composed of two known cubes, could not be resolved into any other two cubes; but Fermat, in his observations on the questions of Diophantus, has pointed out a method by which such cubes may be determined; though the calculation, indeed, extends to numbers that are exceedingly complex. Thus, in the simple case of dividing 9, which is the sum of the two cubes 8 and 1, into two other cubes, Pére de Billy, in his Diophantus Redevivus, has found, by following the rule laid down by Fermat, that the sides of the two new cubes, auswering the conditions required, are

12436177733990097836481

and

60962383566137297449

487267171714352336560 60962383566137297449

but few, it is believed, will be inclined to ascertain whether the answer, here given, be true or false.