Editing Doubling the cube (section)

==History==
The problem owes its name to a story concerning the citizens of [[Delos]], who consulted the oracle at [[Delphi]] in order to learn how to defeat a plague sent by [[Apollo]].<ref name="Zhmud">{{Cite book |last=Zhmudʹ |first=Leonid I︠A︡kovlevich |url=https://books.google.com/books?id=oX28qf7LKdoC&pg=PA84 |title=The Origin of the History of Science in Classical Antiquity |date=2006 |publisher=Walter de Gruyter |isbn=978-3-11-017966-8 |pages=84, quoting Plutarch and Theon of Smyrna |language=en}}</ref>{{r|Kern1934|p=9}} According to [[Plutarch]],<ref>{{Cite web |title=Plutarch, De E apud Delphos, section 6 386.4 |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2008.01.0243:section=6 |access-date=2024-09-17 |website=www.perseus.tufts.edu}}</ref> however, the citizens of [[Delos]] consulted the [[oracle]] at [[Delphi]] to find a solution for their internal political problems at the time, which had intensified relationships among the citizens. The oracle responded that they must double the size of the altar to Apollo, which was a regular cube. The answer seemed strange to the Delians, and they consulted [[Plato]], who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of [[Delos]] to occupy themselves with the study of geometry and mathematics in order to calm down their passions.<ref>[[Plutarch]], De genio Socratis 579.B</ref>

According to [[Plutarch]], Plato gave the problem to [[Eudoxus of Cnidus|Eudoxus]] and [[Archytas]] and [[Menaechmus]], who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using [[pure geometry]].<ref>(Plut., ''Quaestiones convivales'' [https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 VIII.ii], 718ef)</ref> This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic ''[[Sisyphus (dialogue)|Sisyphus]]'' (388e) as still unsolved.<ref>Carl Werner Müller, ''Die Kurzdialoge der Appendix Platonica'', Munich: Wilhelm Fink, 1975, pp. 105–106</ref> However another version of the story (attributed to [[Eratosthenes]] by [[Eutocius of Ascalon]]) says that all three found solutions but they were too abstract to be of practical value.<ref>{{citation|title=The Ancient Tradition of Geometric Problems|title-link= The Ancient Tradition of Geometric Problems |series=Dover Books on Mathematics|first=Wilbur Richard|last=Knorr|author-link=Wilbur Knorr|publisher=Courier Dover Publications|year=1986|isbn=9780486675329|at=[https://books.google.com/books?id=sgQS5BaTWl4C&pg=PA4 p. 4]}}.</ref>

A significant development in finding a solution to the problem was the discovery by [[Hippocrates of Chios]] that it is equivalent to finding two [[geometric mean]] proportionals between a line segment and another with twice the length.<ref name="Heath">T.L. Heath ''[[A History of Greek Mathematics]]'', Vol. 1</ref> In modern notation, this means that given segments of lengths {{math|''a''}} and {{math|2''a''}}, the duplication of the cube is equivalent to finding segments of lengths {{math|''r''}} and {{math|''s''}} so that
:<math>\frac{a}{r} = \frac{r}{s} = \frac{s}{2a} .</math>

In turn, this means that
:<math>r=a\cdot\sqrt[3]{2}.</math>
But [[Pierre Wantzel]] proved in 1837 that the [[cube root]] of 2 is not [[constructible number|constructible]]; that is, it cannot be constructed with [[straightedge and compass]].<ref>{{Cite journal |last=Lützen |first=Jesper |date=24 January 2010 |title=The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.2009.00160.x |journal=Centaurus |language=en |volume=52 |issue=1 |pages=4–37 |doi=10.1111/j.1600-0498.2009.00160.x}}</ref>