We present the essential theoretical basis and prove concrete practical formulas to compute the image of a point on the terrestrial sphere under Peirce quincuncial projection. We also develop a numerical method to implement such formulas in a digital computer and illustrate this method with examples. Then, we briefly discuss the criticism of Pierponton the correctness of Peirce’s formula for the projection. Finally, we draw some conclusions regarding the generalization of Peirce’s original idea by means of Schwarz-Christoffel transformations. Keywords: Peirce quincuncial projection, elliptic functions, geographic maps, numerical conformal mapping, tessellations. To cite this article: L. Solanilla, A. Oostra, J.P. Yanez, Peirce quincuncial projection, Rev. Integr. Temas Mat. 34 (2016), No. 1, 23–38.