The ubiquitous use of infinitely small quantities in mathematics dates back at least to the seventeenth century. Despite continuing qualms as to their legitimacy and their supposed elimination as a result of the thoroughgoing reform movement of the nineteenth century, "infinitesimals" have continued to be used, especially in applied mathematics. The logician Adolf Fraenkel gave what was no doubt the widely accepted view when he stated, "The infinitely small is only to be understood as a manner of speaking based on the limit concept, hence a potential infinite; it is a matter of variable … [positive] numbers or quantities that can ultimately decrease below any arbitrarily small positive value. A fixed [positive] number different from zero that can serve as a lower bound to all finite positive values is not possible" (1928, p. 114, my translation, emphasis in original). In 1960 Fraenkel's one-time student Abraham Robinson showed how to obtain just such...