ipS On the RESOLUTION of 



Suppose ^ = 8, and m = 2, then x = — —^ — =: 7, and 

 y zz - — — — — 4, and the numbers are 48 and 15 ; but 



48+15+1 = 64 = (8)^ 



PROBLEM V. 



'to find twofquares which, diminijhed by unit, pall be in a given 

 ratio. 



By hypothefis, a:b::x'^ — 1 : y'^ — i; whence the equation, 

 ay'^ — a =z bx^- — b, and by refolution, {_ay-^d){y — i) :=. 

 {bx-]-b^{x — i) J wherefore by afTumption, ay-\-a = mx — m, and 

 my — m =: bx-\-b. Tranfpofing the firft, ay — mx — ;// — a, and 



dividing y = . Tranfpofing the fecond, my zz 



hx-\-b-\-m, and dividing, y = — - — , wherefore, ^ =: 



— , and reducing m'^x — ot* — ma zz abx-\-ab-\-ma, that 



is, m'^x — abx zz m'-^ ab-\-2ma, and therefore, x zz 

 -- „^Jab ' ^^^y - '~ln ' confequently J/ = «._,a • 



- , 1 9+6+12 



Suppose a zz 1, b zz ^, and wz = 3 ; then x = — _^ zz. 9, 



and y zz -g — =11; but 2 : 3 : : 80 : 1 20. 



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

 vious that the ratio of x- — i to j/^ — i, will be nearly equal to 

 that of x'^ to y'^ ; and conf^quently the ratio of ^a to ^/b will 

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



be 



