A remarkable twentieth century development in mathematics has been the construction of smooth inifintesimal analysis (SIA) and its extension, synthetic differential geometry (SDG), realizing a non-punctiform conception of continua in contrast with the dominant classical (set-theoretic) conception. Here one is concerned with “smooth worlds” in which all functions (on or between spaces) are continuous and have continuous derivatives of all orders. In this setting, the once discredited notion of “infinitesimal quantity” is admitted and placed on a rigorous footing, reviving intuitive and effective methods in analysis prior to the nineteenth century development of the limit method. The infinitesimals introduced, however – unlike the invertible ones of Robinsonian non-standard analysis – are nilsquare and nilpotent. While they themselves are not provably identical to zero, their squares or higher powers are set equal to 0.