Early historical examples of when apparently very different definitions or propositions turned out to describe the same objects or facts.

One of the characteristic traits of mathematics is the possibility of representing in vastly different ways what is — in some sense — one and the same object or structure, or of expressing what is — in some sense — one and the same fact. Some examples are quite trivial, like the infinite possibilities of representing the same natural number, others are quite profound, like the equivalence of `Poncelet projectivities’ and `von Staudt projectivities’ established in the fundamental theorem of projective geometry. I’m looking for some nice examples of this from the earlier history of mathematics, preferably ancient Greek mathematics and early modern mathematics (up to the late 18th century).

Many thanks in advance.

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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