Abstract
This chapter provides an example of how an alternative proof may be used to provide a rational reconstruction of a historical practice. It concerns the following well-known
Theorem: \(\sqrt{n}\) is rational if and only if it is integral, that is, if and only if n is a perfect square.
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Notes
- 1.
As noted already in Zeuthen (1896), pp. 156–157, Euclid’s proof of that proposition is faulty. Proposition VII,20 does, however, follow by reductio from the division algorithm and proposition VII,17 (that \(b/c = ab/ac\) for any natural numbers a, b, c).
References
Heath, T.: The Thirteen Books of Euclid’s Elements (3 vols.). Dover, New York (1956)
Itard, J.: Les livres arithmétique d’Euclide. Hermann, Paris (1961)
Knorr, W.: The Evolution of the Euclidean Elements. A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry. Reidel, Dordrecht (1975)
Zeuthen, H: Geschichte der Mathematik im Altertum und Mittelalter. Andr. Fred. Höst & Sön, Copenhagen (1896)
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Dawson, J.W. (2015). Quadratic Surds. In: Why Prove it Again?. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17368-9_4
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DOI: https://doi.org/10.1007/978-3-319-17368-9_4
Publisher Name: Birkhäuser, Cham
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