Non-Standard Analysis

For the first hundred and fifty years of its existence, differential and integral calculus were known as the analysis of the infinitely small. Euler’s influential textbook, for example, is entitled “Introductio in Analysin Infinitorum” (Lausanne, 1748). The infinitely small magnitudes which we encountered as “atoms of straight lines” in Cavalieri’s integration continue to play an important role. The precursors of Newton and Leibniz found a new use for vanishingly small quantities in problems of determining tangents and in finding maxima and minima. Then they were introduced systematically by Leibniz in the form of differentials. Throughout he intended that differentials should be understood as legitimate elements of the range and domain of functions, but neither he nor his successors could provide them with a solid mathematical foundation. Long into the Enlightenment, analysis (through its stormy development) lived with this somewhat makeshift and precarious notion among its basic concepts. In his witty and well-informed polemics Bishop Berkeley could accuse the intellectually arrogant scientists themselves of harboring dubious assumptions. Very much to the point, he referred to differentials as “ghosts of departed quantities”.