Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN9780883853481, pp. 112–113
clarku.edu
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Proposition 8 in Book XIII of Euclid's Elements proves by similar triangles the same result: namely that length a (the side of the pentagon) divides length b (joining alternate vertices of the pentagon) in "mean and extreme ratio".
And in analogous fashion Proposition 9 in Book XIII of Euclid's Elements proves by similar triangles that length c (the side of the decagon) divides the radius in "mean and extreme ratio".
cut-the-knot.org
An interesting article on the construction of a regular pentagon and determination of side length can be found at the following reference [1]
To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the Harvard copy of De Revolutionibus to that for the derivation of sin(A-B) found in the above cut-the-knot web page
In De Revolutionibus Orbium Coelestium, Copernicus does not refer to Pythagoras's theorem by name but uses the term 'Porism' – a word which in this particular context would appear to denote an observation on – or obvious consequence of – another existing theorem. The 'Porism' can be viewed on pages 36 and 37 of DROC (Harvard electronic copy)
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To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the Harvard copy of De Revolutionibus to that for the derivation of sin(A-B) found in the above cut-the-knot web page
