This study was conducted to apply the Atume’s series in the absolute prediction of melting and freezing points of a wide range of saturated liquid and solid hydrocarbons. The calculated results that were obtained, shows that a range of 92% to 99% accuracy was theoretically achieved when compared to experimental results. Basically, precise interpolations were deployed by equating the energies released by frozen liquid molecules to the energies absorbed by their corresponding boiling molecules; which represents their lower and upper energy fixed points respectively. These two fixed points were also found to vary infinitesimally and inversely to each other. The energies absorbed or released, molar masses, and trigonometric properties formed the basis for this method. In branched chain hydrocarbons, the melting points of their corresponding linear molecules were also used as reference points to determine their melting and freezing points; indicating the mathematical relationships between their fixed points and trigonometric properties.