"To solve [Book I] Prop. XI algebraically, or to find the point H in AB such that the rectangle contained by the whole line AB and the part HB shall be equal to the square of the other part AH. Let AB contain a linear units, and AH one of the unknown parts contain x units, then the other part HB contains a - x units. And \therefore a(a - x) = x^2, by the problem, or x^2 + ax = a^2, a quadratic equation.Hence [by the ] x = \frac{\pm\, a \sqrt{5} - a}{2}.The former of these values of x determines the point H. So [substituting a = AB] that x = \frac{\sqrt{5}-1}{2} \cdot AB = AH, one part, and a - x = a - AH = \frac{3 - \sqrt{5}}{2} \cdot AB = HB, the other part. It may be observed that the parts AH and HB cannot be numerically expressed by any rational number. Approximation to their true values in terms of AB, may be made to any required degree of accuracy, by extending the extraction of the square root of 5 to any number of decimals."
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Robert Potts, Notes to Book I (1845) Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes, p. 72.
https://en.wikiquote.org/wiki/Euclid%E2%80%99s_Elements
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Euclid’s Elements
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