values of x and y, if r and s be taken numbers. any integral 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 p2 — 2rx + x2, or y = r2 2rx; and consequently, x=- 2r And if s + y be taken for the side of the second square, we shall have y2+ equation, r2- y = 4rsy + 2rs2, and consequently, by re = s2+2sy + y2; or, by reducing the ; where r and s may duction, y p2 - 2rs2 and x 2r2s+s2 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. Let x and x2 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 shall have a2 2x = x2 2rx + r2; r, and we Or, by transposition, and cancelling a2 on each side of the equation, 2rx -2x= 2: whence 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) = 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 we shall have, by transposition and division, x= ານ • n n2 and 1 ; 6 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 1 shall be both squares. 2. It is required to find a number x, such that x +7 shall be both squares. 10 x - 1 and Ans. x= 5. 4 3. It is required to find a number x, such that 10 --- and x shall be both squares. Ans. x = 6. x2 + 1 and Ans. 40 4.0% such 4. It is required to find a number x, such that x + 1 shall be both squares. 5. It is required to find three integral square numbers, that the sum of every two of them shall be squares. 9 Ans. (528), (5796), 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. 3 Ans. x= 3 and 201 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. 3 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 3 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. 2.5 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, 4, and 1296* 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. 629209 629 201 6292 0 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. Ans. 406 518, and 791 9 6 96 " 96 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 Ans. 1, 27 cube numbers. 64 and 125 29. To find three square numbers such, that the difference 27 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 3375 9 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 perfectly equal to it; but yet may approach towards it in such a manner as to differ from it by less than any quantity that can be named. So that we may justly call this the value or sum of the series; not as being a number found by the common method of addition, but such a limitation of the value of the series, taken in all its infinite capacity, that, if it were possible to add all the terms together, one after another, the sum would be equal to that number. In other series, on the contrary, the aggregate, or value of the several terms, taken collectively, has no limitation; which state of it may be expressed by saying, that the sum of the series is infinitely great; or, that it has no determinate or assignable value, but may be carried on to such a length, that its sum shall exceed any given number whatever. Thus, as an illustration of the first of these cases, it may be observed, that if r be the ratio, g the greatest term, and 7 the least, of any decreasing geometric series, the sum, according to the common rule, will be (rg — 1) ÷ (r — 1): and if we suppose the less extreme 7 to be diminished till it becomes = = 0, the sum of the whole series will be rg (r− 1): for it is demonstrable that the sum of no assignable number of terms of the series can ever be equal to that quotient; and yet no number less than it will ever be equal to the value of the series. Whatever consequences, therefore, follow from the supposition of rg (r− 1) being the true and adequate value of the series taken in all its infinite capacity, as if all the parts were actually determined, and added together, no assignable error can possibly arise from them, in any operation or demonstration, where the sum is used in that sense; because, if it should be said that the series exceeds that value, it can be proved, that this excess must be less than any assignable difference; which is, in effect, no difference at all; whence the supposed error cannot exist, and consequently rg ÷ (r — 1 ) may be looked upon as expressing the true value of the series. continued to infinity. 6 6 6 We are, also, farther satisfied of the reasonableness of this doctrine, by finding, in fact, that a finite quantity is frequently convertible into an infinite series, as appears in the case of circulating decimals. Thus, two thirds expressed decimally is 3.66666, &c., +180 + 1000. + 1000 +, &c., continued ad infinitum. But this is a geometric series, the first term of which is, and the ratio ; and therefore the sum of all its terms, continued to infinity, will evidently be equal to, or the number from which 6 ΤΟ TO |