FRANCISCO DE VITORIA (1486–1546), a Spanish Dominican, was one of the founders of the revival of scholastic philosophy during the sixteenth and seventeenth centuries. Vitoria's ideas are preserved in notes on his lectures at the University of Salamanca, where he held the most prestigious chair of theology in late medieval Spain. Vitoria argued that the rights of sovereigns are derived from the universal laws governing the human community, and that the conduct of sovereigns must be judged by those laws. In his lecture “On the American Indians” (1539), Vitoria considers claims made on behalf of the Spanish conquest from the standpoint of this universal law.

From “On the American Indians”

Relection of the Very Reverend Father Friar Francisco de Vitoria, Master of Theology and Most Worthy Prime Professsor at the University of Salamanca, Delivered in the Said University, a. d. 1539

The text to be re-read is ‘Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost’ (Matt. 28: 19). This raises the following problem: whether it is lawful to baptize the children of unbelievers against the wishes of their parents? The problem is discussed by the doctors on Lombard's Sentences IV. 4. 9, and by Aquinas in ST [Summa Theologiae] II–II. 10. 12 and III. 68. 10.

SAMUEL PUFENDORF (1632–94), German philosopher and jurist, from 1662 professor of “the law of nature and of nations” at Heidelberg. Rejecting Hobbes' theory of natural law in favor of Grotius', Pufendorf built upon the latter a comprehensive system of private, public, and international law. He also wrote extensively on the public law of the German states, in the course of which he was forced to grapple with the distinction between a state and a system of states, making him a pioneer theorist of international relations as well as of international law. On the Duties of Man and Citizen (1673), a distillation of his system of natural law which he wrote for his students, is his most accessible work.

From On the Duties of Man and Citizen

On men's natural state

1. We must next inquire into the duties which fall to man to perform as a result of the different states in which we find him existing in social life. By ‘state’ (status) in general, we mean a condition in which men are understood to be set for the purpose of performing a certain class of actions. Each state also has its own distinctive laws (jura).

2. Men's state is either natural or adventitious. Natural state may be considered, in the light of reason alone, in three ways: in relation to God the Creator; or in the relation of each individual man to himself; or in relation to other men.

ADAM SMITH was born at Kirkcaldy in Fife in 1723, the son of a government official. At the age of three he was stolen by gypsies but before long was returned to his parents and educated in Glasgow and from the age of seventeen at Oxford as an exhibitioner at Balliol College. He stayed in Oxford for six years, immersing himself in philosophy and literature. In 1748 he was appointed lecturer in literature at Edinburgh and then, in 1751, professor of literature at Glasgow. From 1752, he held the chair of Moral Philosophy at Glasgow. During the next ten years he worked on and eventually published his first major book, the Theory of Moral Sentiments, and lectured on a range of topics in moral philosophy (under which he included political economy). In 1763, he resigned his chair and became private tutor to the duke of Buccleuch and in that capacity spent some years travelling in Europe with the duke, during which time he got to know the major French school of economic thinkers (or physiocrats) well. He returned to Kirkcaldy in 1767, where he lived quietly writing the second of his major books, moving to London in 1773, and publishing An Inquiry into the Nature and Causes of the Wealth of Nations in 1776. He moved back to Scotland a couple of years later and accepted a post of Commissioner of Customs in Scotland, where he lived until his death in 1790.

DAVID HUME was born at Edinburgh in Scotland in 1711. His family hoped he would follow a legal career, though Hume from quite a young age was obsessed with literature and philosophy. However, the family was not wealthy and so Hume could not initially follow this path. After a brief stint working in Bristol, Hume went to France determined to live as frugally as possible and to become a writer. In France he composed the first of his major works, his Treatise on Human Nature, which was published in three volumes (1738–40) but which met with little critical success (at the time). After returning from France in 1737 Hume lived with his family in Scotland and in 1741–2 he published Essays, Moral and Political, which was much more successful. After this, Hume rewrote much of the Treatise (which was later published as An Enquiry Concerning Human Understanding and An Enquiry Concerning the Principles of Morals in 1748 and 1751) and wrote a series of Political Discourses (1752), which became his most successful book to date. That same year he became Librarian to the Faculty of Advocates in Edinburgh and set up house with his sister in the city. Helped by his access to a magnificent library, he turned to writing a mammoth History of England (1756, 1759, 1761). In 1763 Hume traveled to France as secretary to the British Ambassador and while there he became a major figure among the French philosophes (becoming especially friendly with Diderot), returning to England in 1766, while trying (and failing) to help the increasingly paranoid Jean-Jacques Rousseau.

G.W.F. HEGEL (1770–1831), along with Kant the most important of German philosophers. Hegel's idealism – his belief that ideas, in particular the notion of Geist (spirit), shaped reality – was highly influential throughout the nineteenth century partly in its own terms, but also in stimulating critics such as Marx and Nietzsche. Through most of the twentieth century Hegel's influence waned; he was regarded (quite wrongly) as a supporter of German militarism and forerunner of fascism, and best known as the philosopher whom Marx “stood on his head” by demonstrating the material basis of thought. However, with the ending of communism, Hegel's idea of civil society as a set of institutions to be distinguished from both the family and the state and which form the basis for human freedom became very influential, and he has now regained his place as one of the greatest of political philosophers. Elements of the Philosophy of Right was published in 1821; it is a set of lecture notes setting out the way in which the individual is formed by the family, civil society, and the state. In the extract printed below, the relations of states are described.

From Elements of the Philosophy of Right

International Law (Das äußere Staatsrecht)

International law (das äußere Staatsrecht) applies to the relations between independent states. What it contains in and for itself therefore assumes the form of an obligation, because its actuality depends on distinct and sovereign wills.

JOHN STUART MILL (1806–73) was a leading philosopher, political economist, and social critic of mid-Victorian Britain. His essay “A Few Words on Non-Intervention” (1859) was largely designed to contest the view that non-intervention was, for Great Britain, a self-serving policy which either betokened indifference to suffering, or was downright hypocritical in so far as the British empire was very obviously the product of active intervention in the affairs of the rest of the world. In the following extended extract, Mill sets out the reasons why non-intervention is generally a principled policy, but also why it is not necessarily appropriate when dealing with “primitive” peoples.

From “A Few Words on Non-Intervention”

HUGO GROTIUS (1583–1645), Dutch humanist, theologian, and jurist. Grotius achieved fame first as a poet and later for his efforts to reconcile Protestant and Catholic Christianity. He is known to students of international relations for his defense of the principle of freedom of the seas and, even more, as the author of The Law of War and Peace (1625), which he wrote while living as an exile in France. Despite its baroque style and almost complete neglect of contemporary international practice, Grotius' famous work continues to be read as a statement of the view that the jurisdiction of morality extends even to war (the law of war, for Grotius, is natural, not positive, law). Although the habit of regarding Grotius as the founder of international law both ignores the contributions of his predecessors and reads back into his work ideas that belong to a later period, his writings contain, in embryo, a powerful theory of international justice.

From The Law of War and Peace

Prolegomena

1. The municipal law of Rome and of other states has been treated by many, who have undertaken to elucidate it by means of commentaries or to reduce it to a convenient digest. That body of law, however, which is concerned with the mutual relations among states or rulers of states, whether derived from nature, or established by divine ordinances, or having its origin in custom and tacit agreement, few have touched upon. Up to the present time no one has treated it in a comprehensive and systematic manner; yet the welfare of mankind demands that this task be accomplished.

ST. THOMAS AQUINAS (c. 1225–74), the greatest theologian of the medieval, and perhaps of any, era. His efforts to reconcile Christian and Aristotelian understandings of humanity's place in the world yielded a powerful synthesis that continues to inspire moral and political theorizing. Aquinas rejects the view, powerfully articulated by Augustine in antiquity and reiterated in modern times by Luther, that human nature is inherently corrupt. Like Aristotle, he regards human beings as capable of self-control and self-government. One of the tasks of theology, then, is to articulate principles to guide human conduct. Because these moral principles apply in every sphere of life, their authority extends even to war.

From Summa Theologiae

II–II, Question 40, Of War

First Article: Is it Always Sinful to Wage War?

We proceed thus to the First Article:

Objection 1. It would seem that it is always sinful to wage war, because punishment is not inflicted except for sin. Now, those who wage war are threatened by Our Lord with punishment, according to Mt. 26:52: “All that take up the sword shall perish by the sword.” Therefore, all wars are unlawful.

Obj. 2. Further, whatever is contrary to a divine precept is a sin. But war is contrary to a divine precept, for it is written: “But I say to you not to resist evil,” and “Do not defend yourselves, my dearly beloved, but yield to [God's] wrath.” Therefore, war is always sinful.

Obj. 3. Further, nothing except sin is contrary to an act of virtue. But war is contrary to peace. Therefore, war is always a sin.

SAMUEL RACHEL (1628–91), German jurist, diplomat, and professor of moral philosophy. In contrast to Pufendorf, who, like Grotius, based international law (or “the law of nations,” as it was then called) on natural law, Rachel argued that the law of nations consists of positive laws jointly enacted by sovereigns and expressed in treaties between them. This international law must be distinguished from natural as well as civil law. In rejecting natural law as a source of international law and emphasizing the importance of treaties, Rachel adopts a positivist view of international law as resting on state practice, anticipating mainstream international legal doctrine by more than a century.

From “On the Law of Nations”

I. Not only has Nature provided its own Law for men, whereby, as if by a world-wide chain, they are bound to one another in virtue of being men, but mankind has itself also laid down various positive laws for its own guidance, not merely those by which in every State the government binds its subjects to itself or by which these bind themselves to one another, but also those which the human race, divided up as it is into independent peoples and different States, employs as a common bond of obligation; and peoples of different forms of government and of different size lie under the control of these rules, which depend for their efficacy upon mutual good faith.

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.