Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow

he theories as developed by European mathematicians prior to 1870 differed from the modern ones in that none of them used the modern theory of limits. Fermat develops what is sometimes called a “precalculus ” theory, where the optimal value is determined by some special condition such as equality of roots of some equation. The same can be said for his contemporaries like Descartes, Huygens, and Roberval. Leibniz’s calculus advanced beyond them in working on the derivative function of the variable x. He had the indefinite integral whereas his prede-cessors only had concepts more or less equivalent to it. Euler, following Leibniz, also worked with such functions, but distinguished the variable (or variables) with constant differentials dx, a status that corresponds to the modern assignment that x is the independent variable, the other variables of the problem being dependent upon it (or them) functionally. Tiziana Bascelli is an independent researcher in history and philosophy of science. Her email address is tiziana