our purpose, and if, according to what has been taught, we substitute 1+y for z, we shall arrive at another biquadratic formula equally deceptive. In fact, this is one of those embarrassing formulas which often mock our ingenuity; nothing, therefore, remains for us to do, but to prove the impossibility of the problem proposed.

We succeed better in attempting to assign x such that x2+2, and x2+11 may both be squares, and we find x=1, we may easily find as many other values as we please.

35. There is also a particular and useful method of determining such that a*x+bx+c, and d2x2+ ex+f may both be squares, which we must not omit. We should first, when necessary, reduce by multiplication or division, the square coefficients of x2 tọ an equality, which is sufficiently easy; the operation afterwards will be plain from an example. To find x such that x2-x+7, and x2-7x+1 may both be squares. Suppose x2-x+7=A2, and x2 −7x+1= B2; whence, by subtraction 6x+6=A2-B2, or (2x +2)×3=(A+B)×(A−B): now assume 2x+2=A +B, 3=A-B, whence x+-A, and x-1=B. Each of these will give the same value of x; if we use A, we have x2-x+7=x2+5x+25, from which we obtain x=, a value which will make both the formulas squares.

36. Let us find x such that ax+b, cx+d, and ex +f may all be squares.

yz-b

Put ax+b=y2, whence x= and this value of x

a

,

being substituted for it in the remaining formulas

cx+d, and ex+f, they become

aey2+a2f-abe

acy-a2d-abc

and

: the question therefore is reduced to

the finding of x such that the two quadratic formulas acy2+a2d-abc, and aey2+a2f-abe may both be

squares.

By applying this method to the formulas 1-x, 2 -x, 8-x, we find x=

37. Such are some of the most general methods of resolving formulas involving only one unknown quantity; but by far the most difficult and curious part of my subject yet remains, I mean the solution of Diophantine problems, in which two or more numbers are required. On this inquiry I cannot enter at present, for my paper has already swelled beyond its intended limits, I must, therefore, defer the farther prosecution of this research to another opportunity.

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