ABSTRACT
This chapter provides a critical discussion of the belief that continuity implies the infinitesimal. The distance between consecutive points presupposes that there are consecutive points—a view which there is every reason to deny. And so with most instances—they afford no precise definition of what is meant by the infinitesimal. As regards magnitudes, the axiom of Archimedes is the only way of defining, not only the infinitesimal, but the infinite also. The points on a limited portion of a line obtainable by the quadrilateral construction form a collection which is infinitesimal with respect to the said portion; this portion is ordinally infinitesimal with respect to any bounded area; any bounded area is ordinally infinitesimal with respect to any bounded volume; and any bounded volume is ordinally infinitesimal with respect to all space.
