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Is there any evidence showing that a unit circle approach was used by early mathematicians?

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    $\begingroup$ Could you please explain a little more what you mean? $\endgroup$ Jan 17, 2020 at 1:36
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    $\begingroup$ At what period of time was the concept of the 'unit circle' invented or created. Was it back in the Roman era? During Isac Newton's time? Do we know a specific date in which it was created or do we not know at all? That's what I'm trying to say. $\endgroup$
    – Dom Turner
    Jan 17, 2020 at 2:24
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    $\begingroup$ The problem with very old history is that it is hard to find out the truth. Indians claim they invented these, and then Arabs took it forward, as if they were mere translators of Greek and Indian science. Finally, Europeans learned from the translations of Arabic works. There is so much historical bias, depending on who wrote the history and in which part of the world. The lack of original works, and extensive nationalistic nonsense out there will never make it possible to find out the truth. $\endgroup$
    – AChem
    Jan 17, 2020 at 4:15
  • $\begingroup$ The science history around the 17th-18th century is reliable enough but before that take everything with a grain of salt. $\endgroup$
    – AChem
    Jan 17, 2020 at 4:17
  • $\begingroup$ oh ok, so I guess there just isn't enough evidence to precisely or at least reasonably predict who and when created the concept of the unit circle. $\endgroup$
    – Dom Turner
    Jan 17, 2020 at 5:58

1 Answer 1

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The modern convention, like much of modern notation, goes back to Euler's Introductio in Analysin Infinitorum (1748), chapter VIII, there is an English translation by Blanton. The unit circle was not invented to extend trigonometric functions, power series already accomplished that even for complex values, it was rather an illustrative device. At the beginning of the chapter Euler writes:

"After having considered logarithms and exponentials, we must now turn to circular arcs with their sines and cosines. This is not only because these are further genera of transcendental quantities, but also since they arise from logarithms and exponentials when complex values are used. This will become clearer in the development to follow. We let the radius, or total sine, of a circle be equal to 1, then it is clear enough that the circumference of the circle cannot be expressed exactly as a rational number."

He then denotes half of the circumference by the familiar $\pi$, and gives the standard suite of trig identities.

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  • $\begingroup$ Huh? How did Euler know $2 \pi$ is irrational? $\endgroup$
    – vonbrand
    Feb 6, 2020 at 22:32
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    $\begingroup$ @vonbrand Because nobody succeeded at writing it as a ratio of integers since antiquity, and those who tried included Euclid, Archimedes, Huygens and Euler himself. They were not hung on formal proofs back then, it was "clear enough". Lambert proved it, more or less rigorously, soon after, in 1760s. $\endgroup$
    – Conifold
    Feb 6, 2020 at 23:34

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