ABSTRACT
This chapter reviews the basic theory of length, area, and volume, and see how infinitesimals insinuate themselves into the study of curved figures. In finding tangents, areas, and the length of curves such as the circle, it seems that the shortest route to the truth passes through the impossible. The geometric equivalent of the assumption is to suppose that the infinitesimal arc of the curve between P and Q is the same as the infinitesimal line segment PQ. Since an infinitesimal is required to be smaller than any nonzero number, but not zero, infinitesimals are not numbers. They can be functions of time however, and this seems in the right spirit. One of the fundamentals of mathematics is the relationship between length and area. The great challenge of ancient mathematics was to square the circle, that is, to construct a square of the same area as the disk bounded by the unit circle.
