198 On the RESOLUTION of 



Suppose a — 8, and m zz 2, then # zz — — — zz 7, and 



4.8 — 4—8 

 y tz — +I zz 4, and the numbers are 48 and 15 ; but 



48+15+1 = 64 = (8)*. 



PROBLEM V. 



To find two fquares which, diminijhed by unit, jhall he in a given 

 ratio. 



By hypothefis, a:b::x 7 — 1 :y 2 — 1; whence the equation, 



ay % — a zz bx 1 — b, and by refolution, (ay-\-a)(y — 1) — 

 [bx-\-b)(x — 1) ; wherefore by aflumption, ay-\-a zz mx — m, and 

 my — m zz bx-\-b. Tranfpofing the firft, ay zz mx — m — a, and 



dividing y zz . Tranfpofing the fecond, my zz 



...... bx+b+m . _ mx—m—a 



bx-\-b-\-m, and dividing, y zz — - — , wherefore, zz 



, and reducing m 2 x — m % — ma zz abx-{-ab-\-ma t that 



is, m 2 x — abx zz m 2 -\- ab-\-ima, and therefore, x zz 



m x +ab+2ma . bx+b+m m*+ab+2mb 



ffla'-k '> huty = ~ 1 confequently y = w ,^ . 



Suppose # zz 2, 3 zz 3, and m zz 3 ; then * zz — 6 zz 9, 



9 +<+i8 



and j/ zz 6 — zz 1 1 ; but 2 : 3 : : 80 : 120. 



Cor. 1. When the numbers x and y are very great, it is ob- 

 vious that the ratio of x* — 1 toy — 1, will be nearly equal to 

 that of x z to y % ; and confequently the ratio of ^/a to s/b will 

 be ftill more nearly equal to that of x to y. If a and b, befidesj 



be 



