CHA P. VĮ. Of Indeterminate Problems. T was formerly observed, (Chap. III.) IT It was unknown quan that if there are more tities in a question, than equations by which their relations are expreffed, it is indetermined; or it may admit of an infinite number of answers. Other circumftances, however, may limit the number in a certain manner; and these are various, according to the nature of the problem. The contrivances by which fuch problems are refolved are fo very different in different cafes, that they cannot be comprehended in general rules. EXAMPLE I To divide a given square number into two parts, each of which shall be a Square number. There are two quantities fought in this queftion, and there is only one equation expreffing their relation; but it is required alfo, that they may be rational, which circumftance cannot be expreffed by an equation; another condition therefore must be affumed in fuch a manner as to obtain a folution in rational numbers. Let the given fquare be a2, let one of the fquares fought be x2, the other is a2x2. Let rx-a alfo be a fide of the last square, therefore By tranfp. r2x2—2rxa+a2 —a2—x2 r2x2+x2=2rxa x=2ra 2ra Let r, therefore, be affumed at pleasure, and and 2ra a, which muft always be rational, will be the fides of the two squares required. Thus, if a2=100. Then, if r3, the fides of the two fquares are 6 and 8, for 36+64 100. = Alfo, let a2=64. Then, if r=2, the fides The reafon of the affumption of rx-a as a fide of the fquare a2-x2, is that being fquared and put equal to this laft, the equation manifeftly will be fimple, and the root of fuch an equation is always rational. EXAMPLE II To find two fquare numbers whofe difference is given. Let Let x2 and y' be the fquare numbers, and a If x and y are required only to be rational, then take v at pleasure, and x= whence x and y are known. But, if x and y are required to be whole numbers, Take for z and v any two factors that produce a, and are both even or both odd numbers, and this is poffible only where a is either an odd number greater than 1, or 2 a number divifible by 4. Then and are the numbers fought. 2 V For the product of two odd numbers is odd, and that of two even numbers is divifible by 4. Also, if x and v are both odd, odd, or both even, and must be in tegers. Ex. 1. If a=27 take v=1, then x= 27; and the squares are 196 and 169: or 2 may be 9 and v3, and then the fquares are 36 and 9. 2. If a=12, take v=2, and ≈=6; and the squares are 16 and 4. To find a fum of money in pounds and fillings, whofe half is just its reverse. Note. The reverse of a fum of money, as 81. 12 s. is 12 l. 8 s. Let x be the pounds and y the fhillings. |