# History of the study of rational points on the circle

What is the first known instance of a mathematician parameterizing rational points on the unit circle by the slopes of rational lines going through a rational point on the circle?

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Weil writes in his book on the history of number theory that Fermat scoffed at British mathematicians for giving all solutions of Pell's equation in rational numbers, when he wanted all integer solutions. He essentially wrote that even the lowest type of mathematician could find all rational solutions to Pell's equation, so the rational parametrization of conics went back at least to Fermat. –  KCd Oct 10 '12 at 2:28
Thanks KCd. I wonder if the Greeks knew it... –  Jonah Sinick Oct 10 '12 at 2:51
I believe this has to be Diophantus, who uses this method to find rational solutions to second degree equations in $x$ and $y$. Thomas Heath summarizes (p. 68) some of these results by saying that Diophantus solves the equation $$Ax^2+c^2=y^2$$ by putting $y=mx\pm c$, obtaining $$x=\pm{2mc \over A-m^2}$$ Of course, Diophantus would only consider the positive solution.