11. To find two numbers, x and y, such that x+y, x2+y, and y2+x may be all squares. Ans. x, and y= 12. To find three square numbers in arithmetical progression. Ans. 1, 25, and 49. 13. To find three square numbers in harmonical proportion. Ans. 1225, 49, and 25. 14. To find three numbers in arithmetical progression, such that the sum of every two of them may be a Square number. Ans. 1201, 8401, and 15601. 15. To find three numbers, such that, if to the square of each of them the sum of the other two be added, the three sums shall be all squares. Ans. 3, 6, and 1. 16. To find two numbers in proportion as 8 is to 15, and such that the sum of their squares shall make a square number. Ans. 576 and 1080, 17. To find four numbers such that, if a square number (100) be added to the product of every two of them, the sums shall be all squares. Ans. 12, 32, 88, and 168. 18. 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 4. 19. To find three numbers in geometrical proportion, such that every one of them, being increased by a given number (19), shall make square numbers. 1296 Ans. 81,, and 25 20. To find two numbers such that, if their product be added to the sum of their squares, it shall make a square number. Ans. 5 and 3, 8 and 7, 16 and 5, &c. 21. To divide a given number (10) into four such parts, that the sum of every three of them may be a square number. 186 Ans. 1, 6, 289' 289 22. To find three square numbers, such that their sum, being severally added to their three sides, shall make square numbers. Ans. 44181, 13254, and 19881 62920 62920' 62920-roots required. 23. To find two numbers, such that their sum, being increased and lessened, either by their difference, or the difference of their squares, the sums and remainders shall be all squares. Ans. 43 and 24. To find two numbers' such, that not only each number, but also their sums and their difference, being 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 particular number, the numbers thence arising shall be all squares. Ans. 406, 518, and 701. 26. To find three square numbers, such that the sum of their squares shall also be a square number. 961 91 96 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. 24, 27' 125, and 1. 29. Two cube numbers (8 and 1) being given, to find two other cube numbers, whose difference shall be equal to the sum of the given cubes. Ans. 600 and 4913. 343 343 30. To divide a given number (28) composed of two cube numbers (27 and 1) into two other cube numbers. Ans. 63284705 and 28340511, the roots. 11446828 31. To find three cube numbers, such that, if from every one of them a given number (1) be subtracted, the sum of the remainders shall be a square. Ans. 4913 21952, and 8. 3375' * The answers to many of these questions cannot be given in whole numbers. 32. To find three numbers such that, if they be severally added to the cube of their sum, the three sums thence arising shall be all cubes. 157464' 157464' 157464 Ans. 1538 18577 and 23025 33. To find three numbers in arithmetical proportion, such that the sum of their cubes shall be a cube. Ans. 3, 4, 5, or 149, 256, 363, &c. 34. To find three cube numbers, such that their sum shall be a cube number. Ans. 33, 43, and 53, or 213, 193, 183, &c. 35. To find two numbers such that their sum shall be equal to the sum of their cubes. Ans. & and 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 in all ages, and is, perhaps, one of the most abstruse and difficult branches of abstract mathematics. To find the sum of a series, the number of whose terms is inexhaustible, or infinite, has been considered 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 infinite number of parts, whose aggregate, or sum, is equal to the quantity first proposed. A number actually infinite is, indeed, a plain contradiction to all our ideas; for any number which we can possibly conceive, or of which we have any notion, must always be determinate and finite; so that a greater may be still 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, in the nature of numbers, is, therefore, 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 from thence enabled to form any notion of the ultimatum, or last magnitude, which is incapable of further augmentation. We cannot, therefore, apply to an infinite series the common notion of a sum, or a collection of several particular numbers, which are joined and added together, one after another; for this supposes that those particulars are all 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 equal to it: but yet may approach to it in such a manner as to want less than any given difference. And this we may call 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. Again, in other series, the value has no limitation; and this may be expressed by saying, that the sum of the series is infinitely great; or, which is the same thing, 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. According to the common rule for summing up a finite progression of a geometric decreasing series, where r is the ratio, the greatest term, and a the least, the sum is (rl-a)÷(r−i); and if we suppose a, the less extreme, to be actually decreased to 0, then the sum of the whole series will be r÷(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 ri÷(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, they can never be the occasion of any assignable error, in any operation or demonstration where it is used in that sense; because, if you say that it exceeds that value, it is demonstrable 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 rl÷(r−−1) may be looked upon as expressing the adequate and just value of the series, continued to infinity. = But we are further satisfied of the reasonableness of this doctrine by finding, in fact, that a finite quantity actually converts into an infinite series, as appears in the case of circulating decimals. Thus,, turned into a decimal, is .6666, &c. 10 + 100 + 1000 + 10000, &C. continued ad infinitum. But this is plainly a geometric series, beginning from, in the continued ratio of 10 to 1, and the sum of all its terms, continued to infinity, will evidently be equal to, or the number from whence it was originally derived. And the same may be shown of many other series, and of all circulating decimals in general. |