8. To find a number x such, that x+1 and x-1 shall be both square numbers. (t) Ans. x= 9. To find a number x such, that x+128 and x+192 shall be both squares. Ans. x 97 10. To find a number x such, that x2+x and x2 -X shall be both squares. Ans. 11. To find two numbers such, that if each of them be added to their product, the sums shall be both squares. Ans. and 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, 840, and 1560 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. and 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 18. It is required to resolve 4225, which is the square of 65, into two other integral squares. Ans. 2704 and 1521 (t) The answers to many of the questions here given, cannot be found in whole numbers. 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,, 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 22. To find three square numbers such, that their sum when added to each of their three sides, shall be all square numbers. Ans., 351, and 12831= roots required 629 6292 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. 40, 18, and 1 969 26. To find three square numbers such, that the sum of their squares shall also be a square number. 27. To find three square numbers such, that the difference of every two of them shall be a square number. Ans. 485609, 34225, and 23409 28. To divide any given cube number (8) into three other cube numbers. Ans. 1, 4 and 13,5 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, 2413, 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 OF THE SUMMATION AND INTERPOLATION OF 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 |