EXAMPLE IV. The fum of two quantities is a, and the fum of their fquares b. Qu. the quantities? Suppofe x+ya... and xay, ..... · x2 + y2 = b ... ... ... ... ... ..... x2 = b — y′′, whence a-2ay+y2=b—y2; tranfp. and di S23 vide b y2 ay= 2 add —, y2 — ay+ extr. √, y — a a2 = A company dining together in an inn, find their bill amounts to 175 billings; two of them were not allowed to pay, and the rest found that their fbares amounted to 10 s. a man more than if all bad paid. Qu. How many were in company? Suppofe their number x; then if all had paid, each man's share would have been 175, seeing 2 is the number of thofe that pay. It is therefore, by the question, and 175x175x+350 = 10x2-20%; that is, 10x- 20x=350, and x2- 2x=355 It is obvious that the pofitive value 7 gives the folution of the question; the negative va5 being, in the prefent cafe, useless. lue -- There are three numbers in continual geometrical proportion; the fum of the first and fecond is 10, and the difference of the fecond and 3d is 24. Qu. the numbers? Let the first be x, and the second will be 10-x, and the third, 34-x; therefore, x: 10-x: 10-x: 34-x, and 34x tranfp. and divid. { 54x=100+2x2, add 27×27...x2—27x+729 — 729 — 4 50529 4 4 So the three continued proportionals are 28: 32, or 2515 9. $91. Any equation of this form yam +aym = b, where the greatest index of the unknown quantity y is double to the index of y in the other term, may be reduced to a quadratic z2+az=b, by putting yz, and confequently y2m 22. y2m = x2. And this quadratic refolved as above, gives The product of two quantities is a, and the fum of their fquares b. Qu. the quantities? Put now y2=z... and confequently yz", and it is To find a number from the cube of which if you fubtract 19, and multiply the remainder by that cube, the product shall be 216. Call the number required x; and then, by the question, x319X3216, Put x3z....xz2, and it will be 19 and √ ̄ ̄..x-12= +35; whence z = 19±35 = 27, or —— 8. 3 2 But =√; wherefore =+ 3, or—2. x= Z; To find the value of x, fuppofing that x3- 3 But x3 = x2, and x = √22 = √64 = 4. IF fo as to leave no remainder, as 2a measures 10a, being found in it five times, it is faid to be an aliquot part of it, and the greater is faid to be a multiple of the leffer. The leffer quantity in this cafe is the greatest common meafure of the two quantities; for as it measures the greater, fo it also measures itself, and no quantity can measure it that is greater than itself. When a third quantity measures any two propofed quantities, as 24 measures 6a and 10a, it |