202. Much labor has been formerly employed by some mathematicians to find two biquadrates, whose sum or difference might be a square, but in vain; and at length it has been demonstrated, that neither the formula x4 + y4 nor the formula x4 − y4, can become a square, except in these evident cases; first, when x = 0, or y = 0, and, secondly, when y = x. This circumstance is the more remarkable, because it has been seen, that we can find an infinite number of answers, when the question involves only simple squares.

203. We shall give the demonstration to which we have just alluded; and, in order to proceed regularly, we shall previously observe, that the two numbers x and y may be considered as prime to each other: for, if these numbers had a common divisor, so that we could make x = dp, and y = dq, our formulæ would become d4p4 + d4q4, and d4p4 − d4q4: which formulæ, if they were squares, would remain squares after being divided by d4; therefore, the formulæ p4 + q4 and p4 − q4, also, in which p and q have no longer any common divisor, would be squares; consequently, it will be sufficient to prove, that our formulæ cannot become squares in the case of x and y being prime to each other, and our demonstration will, consequently, extend to all the cases, in which x and y have common divisors.