This proof can be generalized to show that any square root of any natural number that is not the square of a natural number is irrational. See quadratic irrational or infinite descent for a proof that the square root of any non-square natural number is irrational. One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true. Because there is a contradiction, the assumption (1) that is a rational number must be false. This means that is not a rational number; i.e., is irrational. This proof was hinted at by Aristotle, in his "Analytica Priora", §I.23. It appeared first as a full proof in Euclid's "Elements", as proposition 117 of Book X. However, since the early 19th century historians have agreed that this proof is an interpolation and not attributable to Euclid. As with the Proof by Infinite Descent, we obtain formula_4. But now the factor 2 appears an odd number of times on the right, but an even number of times on the left, a contradiction. A simple proof is attributed by John Horton Conway to Stanley Tennenbaum when the latter was a student in the early 1950s and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of "American Scientist". Given two squares with integer sides respectively "a" and "b", one of which has twice the area of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1. Another geometric reductio ad absurdum argument showing that is irrational appeared in 2000 in the American Mathematical Monthly. It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the algebraic proof of the previous section viewed geometrically in yet another way.