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This MacTutor biography gives approximate dates and discusses the surviving evidence for Diocles' work on the cissoid of Diocles and the parabola. The footer identifies John Joseph O'Connor and Edmund Frederick Robertson as authors and April 1999 as its revision date.

Online text: mathshistory.st-andrews.ac.uk/Biographies/Diocles/

Diocles was a Greek mathematician associated with the cissoid of Diocles and the geometry of the parabola. The dates of his life are approximate.^[ref]

Biography: mathshistory.st-andrews.ac.uk/Biographies/Diocles/

Substituting $x=r\cos \theta$, $y=r\sin \theta$ into the circle equation gives $r^2-2r\cos \theta =0$. Thus its nonzero radial intersection is $r_Q=2\cos \theta$, with the origin also on the circle. The tangent to a circle at the rightmost point is $x=2$, or $r_R\cos \theta =2$.

Take rays with $-\pi /2<\theta <\pi /2$. The difference of the two radial distances gives the cissoid of Diocles:Multiplying by $\cos \theta$ gives $x=2{\sin }^2\theta =2y^2/(x^2+y^2)$, hence the Cartesian equation $y^2(2-x)=x^3$. A convenient parametric curve, with $t=\tan \theta$, isThe real curve has $0\le x<2$, a cusp at the origin, symmetry about the $x$ axis, and the vertical asymptote $x=2$. Near the origin, $x\sim 2t^2$ and $y\sim 2t^3$, so its cusp has horizontal tangent direction. As $|t|\to \infty$, $x\to 2^{-}$ while $|y|\to \infty$. The circle and curve meet away from the origin at $(1,\pm 1)$.