In "Stetigkeit und irrationale Zahlen" section 4, Dedekind introduces the Theorem of Theaetetus (Eu. X.9): if a whole number D is not the square of a whole number, it is not the square of any rational number. He gives an indirect proof summarized below. What is the source of this proof?
Robert Eldon Taylor philologos at mindspring dot com
First, if D is a whole number, but not the square of a whole number, then it lies between two squares, i.e. there is a whole number, n, such that nn < D < (n + 1)^2.
Assume there is a rational number whose square is = D, then there are two positive whole numbers, t, u which satisfy the equation tt - Duu = 0, and one may assume that u is the smallest positive number which possesses the property that its square may be transformed into the square of a whole number t by multiplication with D. Now, since apparently, nu < t < (n + 1) u
thus the number u' = t - nu is a positive whole number, and indeed smaller than u. Further if one sets t' = Du - nt
t' is likewise a positive whole number and there results
t't' - du'u' = (nn - D)(tt - Duu) = 0, which stands in contradiction to the assumption about u.