International politics in the seventeenth and eighteenth centuries is usually seen as the high water mark of the “traditional” view of the European states-system. It is in this period, as seen in the previous chapter, that the notion of the “balance of power,” so central to realist thinking, first emerges, self-consciously at least, and it is the writings of scholars such as Vattel (whose The Law of Nations was first published in 1758) that establish or cement the meanings of some of the central terms of modern international thought (such as sovereignty).

However, it is also in this period that some of the major challenges to conventional understanding of the states-system also emerged. Most specifically, in the thinking of some of the writers associated with what is usually called the “European Enlightenment,” some of the basic assumptions underlying the emerging states-system were challenged, a challenge which has lasted until our own day and is still continuing.

What is the “Enlightenment”?

At its simplest, the “Enlightenment” is a phrase that represents the collective, overlapping (and not always congruent) views of a group of scholars, writers, activists, and campaigners in eighteenth-century Europe. Especially prominent in France, it had representatives in almost every major European country and, as the eighteenth century went on, it became increasingly important both intellectually and politically. Figures especially associated with it include David Hume and Immanuel Kant, Montesquieu and Jean-Jacques Rousseau, Edward Gibbon and Adam Smith, Voltaire and Denis Diderot.

This is a comparatively late stage to introduce the subject of differentials but there are good reasons for the delay. Readers will probably already have met the notation dx and dy in applied subjects and have been told that these represent ‘very small changes in x and y ’. There are even those who tell their students that dx and dy stand for ‘infinitesimally small changes in x and y ’. Such statements do not help very much in explaining why manipulations with differentials give correct answers. Indeed, in some contexts, such statements can be a positive hindrance. In this chapter we have tried to provide a more accurate account of the nature of differentials without attempting anything in the way of a systematic theoretical discussion. In studying this account, readers may find it necessary to put aside some of the preconceptions they perhaps have about differentials. Those who find this hard to do may take comfort in the fact that everything which is done in this book using differentials may also be done by means of techniques described in other chapters. For instance, three alternative methods of solving a problem are given in Example 1 before the method of differentials is used. However, this is not a sound reason for neglecting differentials. Their use in both applied and theoretical work is too widespread for this to be sensible.

Difference equations and the shift operator E are introduced in this chapter. The chapter is mainly concerned with higher order linear differential and difference equations with constant coefficients. The analogies between the two types of equation are exploited by showing that essentially the same techniques work for both differential and difference equations. For this reason, the symbol x is used to denote the variable in both types of equation, though in the case of difference equations, it only takes discrete values, which are nonnegative integers. The techniques are easy, provided some mastery of complex numbers has been garnered from the previous chapter. A brief theoretical justification for the techniques described appears in §14.3 and §14.4, which will be of particular interest to those with a knowledge of linear algebra. But readers impatient with theory will find it adequate to confine their attention in these sections to the manipulation of operators as described in the examples of §14.1 and §14.2 and to the assertions of §14.3 and §14.4 without delving into the reasons why these are correct.

It should be noted that the technique for finding particular solutions of nonhomogeneous equations described in §14.7 is only one of several. We have chosen to present this technique because it involves no essentially new ideas. But it is often quicker to find particular solutions using operators in a more adventurous way than described here.

In §14.8 there is a short discussion on the stability of the solutions of equations.

This book takes for granted that readers have some previous knowledge of the calculus of real functions of one real variable. It would be helpful to also have some knowledge of linear algebra. However, for those whose knowledge may be rusty from long disuse or raw with recent acquisition, sections on the necessary material from these subjects have been included where appropriate. Although these revision sections (marked with the symbol ◊) are as self-contained as possible, they are not suitable for those who have no acquaintance with the topics covered. The material in the revision sections is surveyed rather than explained. It is suggested that readers who feel fairly confident of their mastery of this surveyed material scan through the revision sections quickly to check that the notation and techniques are all familiar before going on. Probably, however, there will be few readers who do not find something here and there in the revision sections which merits their close attention.

The current chapter is concerned with the fundamental techniques from linear algebra which we shall be using. This will be particularly useful for those who may be studying linear algebra concurrently with the present text.

Algebraists are sometimes neglectful of the geometric implications of their results. Since we shall be making much use of geometrical arguments, particular attention should therefore be paid to §1.3 onwards, in which the geometric relevance of various vector notions is explained.

Ancient accountants laid pebbles in columns on a sand tray to help them do their sums. It is thought that the impression left in the sand when a pebble is moved to another location is the origin of our symbol for zero.

The word calculus has the same source, since it means a pebble in Latin. Nowadays it means any systematic way of working out something mathematical. We still speak of a calculator when referring to the modern electronic equivalent of an ancient sandtray and pebbles. However, since Isaac Newton invented the differential and integral calculus, the word is seldom applied to anything else. Although there are pebbles on its cover, this book is therefore about differentiating and integrating.

Students who don't already know what derivatives and integrals are would be wise to start with another book. Our aim is to go beyond the first steps to discuss how calculus works when it is necessary to cope with several variables all at once.

We appreciate that some readers will be rusty on the basics, and others will be doubtful that they ever really understood what they can remember. We therefore go over the material on the calculus of one variable in a manner that we hope will offer some new insights even to those rare souls who feel confident of their mathematical prowess. However, we strongly recommend against using this material as a substitute for a first course in calculus.

This chapter is chiefly about integrating real valued functions of one real variable. Because an integral is a generalisation of the idea of a sum, it is also convenient to include some discussion of summation in this chapter. It is assumed that readers have some previous knowledge of these subjects. In particular, §10.2– §10.9 inclusive consist of an accelerated account of elementary integration theory which a newcomer to the topic would find difficult to assimilate adequately unless they had an unusual aptitude for mathematics. However, experience shows that students are often very rusty on the techniques of integration and it is strongly advised that Exercises 10.10 be used as a check on how well the reader recalls the relevant material before moving on to more advanced work. It is also suggested that all readers study §10.4 with some care. The notation for indefinite integrals can be very confusing if imperfectly understood.

The material given in §10.11– §10.12 about infinite series and integrals over infinite ranges will probably be new to most readers of this book. It is not essential for most of what follows and is best omitted if found at all difficult to understand. The same applies to §10.14 on power series. This is not to say that these are unimportant subjects: only that it may be advisable to leave their study until a later date. There remains §10.13 on differentiating integrals. Although this will be new, the technique can be very useful in some contexts.