At several points we have had occasion to note the fact that logic in smooth worlds differs in certain subtle respects from the classical logic with which we are familiar. These differences have not been obtrusive, and in developing the calculus and its applications in a smooth world it has not been necessary to pay particular attention to them. Nevertheless, it is a matter of logical interest to examine these differences a little more closely, and also to formulate an explicit description of the logical system which underpins reasoning in smooth worlds.

As explained in the Introduction, any smooth world may be taken to be a certain type of category called a topos, which may be thought of as a model for mathematical concepts and operations in much the same way as the universe of set theory serves as such a model. In particular, will contain an object, which we shall denote by Ω, playing the role of the set of truth values. In set theory, Ω is the set 2 consisting of two distinct individuals true and false; we assume that, in, Ω contains at least two such distinct individuals. Now the key property of Ω in any topos is that (just as in set theory) maps from any given object X to Ω correspond exactly to parts of X, the map with constant value true corresponding to X itself, and the map with constant value false corresponding to the empty part of X.

In this chapter we turn to some of the traditional applications of the calculus, namely, the determination of areas, volumes, arc lengths, and centres of curvature. The arguments here take the form of direct computations, based on the analysis of figures: they can be rigorized quite easily by introducing the definite integral function over a closed interval, which will be deferred until Chapter 6.

In these applications, as well as in the physical applications to be presented in subsequent chapters, the role of infinitesimals will be seen to be twofold. First, as straight microsegments of curves, they play a ‘geometric’ role, enabling each infinitesimal figure to be taken as rectilinear, and as a result, ensuring that its area or volume, as the case may be, is a definite calculable quantity. And second, as nilsquare quantities, they play an ‘algebraic’ role in reducing the results of these calculations to a simple form, from which the desired result can be obtained by the Principle of Microcancellation.

Areas and volumes

We begin by determining the area of a circle. The method here – which was employed by Kepler (see Baron 1969) – is to consider the circle as being composed of a plurality of small isoceles triangles, each with its base on the circumference and apex at the centre (Fig. 3.1). Thus let C(x) be the area of the sector OPQ of the given circle, where Q has abscissa x.