Peirce's systems of existential graphs (Alpha: classical propositional calculus; Beta: first-order purely relational logic; Gamma: modal calculi and second-order logic) are presented both from an historical perspective and succinctly for the modern reader. Peirce's alternative continuum, with its main non-Cantorian properties (genericity, reflexivity, modality), is also presented both historically and synthetically. The blend of Peirce's existential graphs and his non-Cantorian continuum gives rise to a thoroughly original logical approach to the "labyrinth of the continuum". We explain why such an approach was set aside in the main developments of logic in the $\,\hbox{\tiny XX}^{\hbox{\tiny th}}\,$ century, and we hint to a possible renewal of interest for Peirce's continuity logic from the viewpoint of contemporary developments in category theory and geometric logic.