The only text in which Descartes explicitly mentions the expression mathesis universalis is in a passage from the Rules for the Direction of the Mind, where it is described as a “general science that explains everything that it is possible to inquire into concerning order and measure, without restriction to any particular subject-matter” (AT X 378, CSM I 19). The philosophical importance of this concept, the interpretation of which is rather problematic, has been appreciated only since the end of the nineteenth century, notably by Paul Natorp, then by Cassirer, Husserl, and Heidegger. By contrast, the term was understood in a mathematical sense in Descartes’ lifetime, since Principia Matheseos Universalis is the title that Frans van Schooten – the professor of mathematics and intimate of Descartes – gave to one of the introductory essays to his Latin translation of Descartes’ Geometry. This work explains in an elementary manner the principles of the new algebraic calculus, using the literal notation and exponential sign of numerical powers, applying arithmetic operations to all sorts of quantities, and it is the origin of numerous mathematical works throughout the seventeenth and eighteenth centuries.

Whether Descartes’ expression characterizes only the first works where he introduces a new algebra or whether it is a later innovation is still an open question. Also debated is whether mathesis universalis is restricted or whether it can be extended more generally and identified with Cartesian method. But it is certain that the expression mathesis universalis appears neither in Descartes’ published works nor in his correspondence.

In keeping with received usage, Descartes employs in certain texts the term mathesis in the sense used in mathematical disciplines at that time, and he distinguishes those disciplines from physics. Mathesis and mathematics do not differ in meaning, and like the authors of the Renaissance, Descartes uses the two terms interchangeably. The sciences that composed the mathesis (or quadrivium) were traditionally divided into pure mathematics, pure mathesis (arithmetic and geometry), mixed or applied mathematics, and mixed mathematics (music, astronomy, mechanics, optics, etc.).