Professor Joël Merker studied Mathematics and Philosophy at the Ecole Normale Supérieure in Paris where he received his Ph. D. in Mathematics (1996), followed by his habilitation in Mathematics (2006) and Ph. D. in Philosophy (2012). He was a CNRS researcher (1997-2010) and is currently Professor of Mathematics at Paris-Sud-Orsay University.

• Part I Modern Presentation.- 1 Three Principles of Thought Governing the Theory of Lie.- 2 Local Transformation Equations and Essential Parameters.- 3 Fundamental Differential Equations for Finite Continuous Transformation Groups.- 4 One-Term Groups and Ordinary Differential.-  Part II English Translation.-  5 Complete Systems of Partial Differential Equations.- 7 Determination of All Systems of Equations Which Admit Given Infinitesimal Transformations.- 8 Complete Systems Which Admit All Transformations of a One-term Group.- 9 Characteristic Relationships Between the Infinitesimal Transformations of a Group.- 10 Systems of Partial Differential Equations the General Solution of Which Depends Only Upon a Finite Number of Arbitrary Constants.- 11 The Defining Equations for the Infinitesimal Transformations of a Group.- 12 Determination of All Subgroups of an r-term Group.- 13 Transitivity, Invariants.- 14 Determination of All Systems of Equations Which Admit a Given r-term Group.- 15 Invariant Families of Infinitesimal Transformations.- 16 The Adjoint Group.- 17 Composition and Isomorphism.- 18 Finite Groups, the Transformations of Which Form Discrete Continuous Families.- 19 Theory of the Similarity [AEHNLICHKEIT] of r-term Groups.- 20 Groups, the Transformations of Which Are Interchangeable With All Transformations of a Given Group.- 21 The Group of Parameters.- 22 The Determination of All r-term Groups.- 23 Invariant Families of Manifolds.-  24 Systatic and Asystatic Transformation Groups.- 25 Differential Invariants.- 26 The General Projective Group.- 27 Linear Homogeneous Groups.- 28 Approach [ANSATZ] towards the Determination of All Finite Continuous Groups of the n-times Extended Space.- 29 Characteristic Properties of the Groups Which are Equivalent to Certain Projective Groups.- Glossary of significantly used words.-  Index./p>