How is it known that Phidias used the golden ratio in the Parthenon?
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The first use of the term golden section (Der goldenen schnitt) was by the German mathematician Martin Ohm in his book "Pure Elementary Mathematics" (1835). Since Ohm, various authors have theorized about the presence of the ratio between the extreme and mean as defined by Book 2, Proposition 11 of Euclid, notably Jay Hambridge in his book "Dynamic Symmetry: The Greek Vase" (1920). The mysticism of the section dates back to the Renaissance and is best exemplified by the book Divina Proportione (The Divine Proportion)(1509) by Luca Pacioli and Leonardo Da Vinci. In modern times, the theory was developed by Theodore Cook in his 1914 book, The Curves of Life: being an account of spiral formations and their application to growth in nature, to science and to art; with special reference to the manuscripts of Leonardo da Vinci., which was 400 pages of showing how the "golden ratio" was supposedly everywhere. Cook is the first one to make the connection to Phidias, even calling the golden section, the "ratio of Phidias." The actual relation to Phidias, however, appears to have been popularized, not by Cook, but by Le Corbusier and is mentioned in several of his books, for example, Towards a New Architecture (1923) in which he writes:
Phidias felt in this way: the entablature of the Parthenon is a witness.
This was a new idea. Traditionally, the ancients considered the ratios of the whole numbers to be the ideal proportions, as described in Plato's Timaeus and detailed by Vitruvius. Le Corbusier, however, was probably channeling Hambridge and Cook who had written several years before him. Note that many sources mention the mathematician Mark Barr as originally making the association to Phidias, but this is purely apocryphal because Barr never published any such thing, and the only one who ever claimed that it was Barr's idea was Cook in his book Curves of Life, who may have just made this completely unsupported remark about Barr to give credence to his own ideas.
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