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

Dom Turner
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    Could you please explain a little more what you mean? – kimchi lover Jan 17 '20 at 01:36
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    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. – Dom Turner Jan 17 '20 at 02:24
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    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. – AChem Jan 17 '20 at 04:15
  • The science history around the 17th-18th century is reliable enough but before that take everything with a grain of salt. – AChem Jan 17 '20 at 04:17
  • 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. – Dom Turner Jan 17 '20 at 05:58
  • If you mean the "concept" as presented in modern textbooks to introduce modern trigonometric functions it is fairly modern. But the original trigonometric functions, chords, were introduced using a circle at least since Hipparchus c. 150 BC. – Conifold Jan 17 '20 at 06:44
  • The OP should edit their question so that one can understand what is being asked without reading the comments. – kimchi lover Jan 17 '20 at 12:02
  • @Coniford Hipparchus didn't use a unit circle, though, for his table of cords but rather one with a large diameter (wikipedia says 3438 units). Presumably so that he could give his results in tables of integers rather than fractions. IIRC, Ptolemy did something similar, though I think just to make his math easier rather than avoid fractions. It seems an interesting question when people switched to using the more natural seeming unit circle for trig tables. – simplicio Jan 17 '20 at 19:43
  • If your unit circle requires a coordinate system, then we must wait until after Descartes and Fermat - really until at least Euler. Amongst the ancients, Ptolemy's use of a circle of radius 60 for his table of chords is the most famous. I seem to recall reading that there is a growing opinion that Hipparchus may never have produced a table of chords. – nwr Jan 17 '20 at 20:45
  • This 1884 paper by Cayley is the earliest Google Books result, but it reads as if the term is already well known. – Spencer Jan 17 '20 at 23:31

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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.

Conifold
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  • Huh? How did Euler know $2 \pi$ is irrational? – vonbrand Feb 06 '20 at 22:32
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    @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. – Conifold Feb 06 '20 at 23:34