I'm very interested to learn about historical phenomenology of proportion and ratios, i.e. I want to know how and why people in the past introduced these concepts. As far as I understand symmetric objects or patterns in geometric figures found in nature suggest the idea of proportion and ratios to human mind. But can someone explain how exactly they came to be studied with examples and details?
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$\begingroup$ The word "proportion" has at least two meanings. The technical meaning used in Mathematics, and a different meaning in art etc. If you are asking about mathematical meaning, the answer is readily available: read Euclid. If it is about the other meaning, this is perhaps out of scope of this list. $\endgroup$– Alexandre EremenkoCommented Feb 13, 2015 at 21:15
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$\begingroup$ Non-trivial Greek mathematics seems to have been entirely founded on the concept of proportion (Euclid, starting in book... 4? Maybe 5?). The Egyptians had fractions (Rhind papyrus) but seemed to think of them more as units chopped into smaller pieces, rather than as as a kind of relation between magnitudes. $\endgroup$– Jack MCommented Feb 13, 2015 at 21:54
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2$\begingroup$ The origin of the concept in mathematics and art is the same, relating different magnitudes, and it certainly long predates Euclid. And the notion of proportion in painting, sculpture, and architecture is pretty much the same as in elementary geometry, which is explicitly used there. $\endgroup$– ConifoldCommented Feb 15, 2015 at 3:46
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It seems that originally ratio and proportion emerged not so much "from nature", as from human activities, first practical and later more theoretical. We often find things in nature only after we already know what to look for. I will give several early examples.
In construction of the pyramids ancient Egyptian builders needed to maintain constant slope. Special measure called seked was introduced for this purpose, which measured it as rise over run. The use of seked implies practical understanding of the fact that sides of similar triangles are in the same ratio.
According to the tradition, the same fact was used more consciously by Thales of Miletus to measure the height of a pyramid, and determine the distance to ships at sea, described here. Although the attribution may be anecdotal, it shows that Greeks were aware of such methods already in 5th century BC.
According to Xenocrates (4th century BC), Pythagoras noticed that different sounds appear in consonance to us when the lengths of strings producing them are $2:1$, and in dissonance when they are $9:8$, etc. Attribution to Pythagoras in this case is very likely fabricated, but Pythagoreans did develop a system of musical tuning based on such observations. This one is the closest to coming from nature, but still involves instruments.
The famous golden ratio, that some find everywhere nowadays, does not seem to have emerged from direct observations of nature, but rather from Pythagorean obsession with "mystical figures". Namely, the diagonals in the regular pentagon (which form the pentagram) cut each other in the golden ratio. The idea that the ratio represents "ideal proportions" of the human body was introduced into sculpture by Phidias only after the fact.
Around 350 BC Eudoxus of Cnidus wrote a book, On Speeds, where he was trying to reproduce motions of the planets using combined rotations of several spheres. He noticed that the path of the planet depends not on the rotation speeds of the spheres, but only on their ratios. Riddell speculates that this work led Eudoxus to his famous theory of proportion, presented in Book V of Euclid's Elements, which became a cornerstone of Greek mathematics.