Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s Theorem from analytic Euclidean geometry into the Poincar´e ball model of analytic hyperbolic geometry, which is based on M¨obius addition. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of the hyperbolic trigonometry, called gyrotrigonometry, to which the Poincar´e ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

This research article provides an unconditional proof of an inequality proposed by \textit{Srinivasa Ramanujan} involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for every real $x\geq \exp(1486)$, using specific order estimates of the \textit{Mertens Function}, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the \textit{Second Chebyshev Function}, $\psi(x)$, in addition to applying the meromorphic properties of the \textit{Riemann Zeta Function}, $\zeta(s)$ with an intention of deriving an improved approximation for $\pi(x)$.