Together with Agustín de Betancourt, he developed the classification of mechanisms begun by Jean Nicolas Pierre Hachette, and counts as one of the founders of industrial kinematics (machine construction).
He was born in Campeche in the south-east of Mexico, then Viceroyalty of New Spain, and studied at Instituto Campechano, where he was later also a teacher.
In 1789 he made a tour through Europe with Josef de Mendoza y Ríos, visiting France, England, Germany, Sweden, Poland and Russia.
Soon after the establishment in 1794 of École Polytechnique in Paris, Monge had proposed a class on the analysis of mechanisms and described the subject to fellow teachers.
Lanz revised his lecture notes and published them together with Betancourt as Essai sur la composition des machines (1808; 2nd ed.