WE PROCEED to solve the geodesic equations in the Schwarzschild solution and use the solution to describe the classical tests of general relativity. These are the precession of the perihelion of planetary orbits and the bending of light by the sun, effects that arise from the small differences between orbits in Newtonian gravitation and orbits, i.e. geodesies, in general relativity.

Whatever conception of the freedom of the will one may form in terms of metaphysics, the will's manifestations in the world of phenomena, i.e. human actions, are determined in accordance with natural laws, as is every other natural event. History is concerned with giving an account of these phenomena, no matter how deeply concealed their causes may be, and it allows us to hope that, if it examines the free exercise of the human will on a large scale, it will be able to discover a regular progression among freely willed actions. In the same way, we may hope that what strikes us in the actions of individuals as confused and fortuitous may be recognised, in the history of the entire species, as a steadily advancing but slow development of man's original capacities. Thus marriages, births, and deaths do not seem to be subject to any rule by which their numbers could be calculated in advance, since the free human will has such a great influence upon them; and yet the annual statistics for them in large countries prove that they are just as subject to constant natural laws as are the changes in the weather, which in themselves are so inconsistent that their individual occurrence cannot be determined in advance, but which nevertheless do not fail as a whole to sustain the growth of plants, the flow of rivers, and other natural functions in a uniform and uninterrupted course. Individual men and even entire nations little imagine that, while they are pursuing their own ends, each in his own way and often in opposition to others, they are unwittingly guided in their advance along a course intended by nature. They are unconsciously promoting an end which, even if they knew what it was, would scarcely arouse their interest.

Since men neither pursue their aims purely by instinct, as the animals do, nor act in accordance with any integral, prearranged plan like rational cosmopolitans, it would appear that no law-governed history of mankind is possible (as it would be, for example, with bees or beavers). We can scarcely help feeling a certain distaste on observing their activities as enacted in the great world-drama, for we find that, despite the apparent wisdom of individual actions here and there, everything as a whole is made up of folly and childish vanity, and often of childish malice and destructiveness.

Immanuel Kant was born on 22 April 1724 in Königsberg (now Kaliningrad) in East Prussia which, except for occasional journeys into the immediate vicinity, he hardly ever left during the whole of his long life of almost eighty years. Königsberg in the eighteenth century was a lively city which, owing to its flourishing trade, was by no means isolated from the world at large. Kant, who was anything but a recluse, enjoyed social life and intelligent conversation. He was friendly with many Königsberg merchants, among whom there were also Englishmen, two of whom, Green and Motherby, were particularly close friends. Although he was meticulous and regular in his habits, punctual to a fault, he was also a man of urbanity and wit.

Kant's parents were not rich. His father was a harness-maker who lived in Königsberg. His family was steeped in Pietism, the Protestant religious movement which stressed emotional religiosity and the development of the inner life. The pietistic atmosphere of his parents’ household was a formative influence in his childhood, and he was particularly impressed by his mother's simple piety. After the early death of his parents (his mother died in 1738, his father in 1746), Kant's relations with his family were not very close.

Kant's outstanding intellectual gifts were recognised at school. It was made possible for him to enter the University of Königsberg, where he was a brilliant student. In 1755 he was granted the right to lecture as Magister legem or Privatdozenty i.e. as an unsalaried lecturer who depended on his lecture fees for his income. Since his lectures were popular and since he gave a large number of them—twenty a week at least—he was able to eke out a meagre living. He lectured on many subjects—logic, metaphysics, ethics, theory of law, geography, anthropology etc. He began to make his name as a scholar and scientist by his writings. In his General History of Nature and Theory of the Heavens (1755), he Put forward a highly original account of the origin of the universe similar to the one later elaborated by the French scientist Laplace.

Cartesian tensors: an invitation to indices

LOCAL DIFFERENTIAL GEOMETRY consists in the first instance of an amplification and refinement of tensorial methods. In particular, the use of an index notation is the key to a great conceptual and geometrical simplification. We begin therefore with a transcription of elementary vector algebra in three dimensions. The ideas will be familiar but the notation new. It will be seen how the index notation gives one insight into the character of relations that otherwise might seem obscure, and at the same time provides a powerful computational tool.

The standard Cartesian coordinates of 3-dimensional space with respect to a fixed origin will be denoted xi (i = 1,2,3) and we shall write A = Ai to indicate that the components of a vector A with respect to this coordinate system are Ai . The magnitude of A is given by A · A = AiAi . Here we use the Einstein summation convention, whereby in a given term of an expression if an index appears twice an automatic summation is performed: no index may appear more than twice in a given term, and any ‘free’ (i.e. non-repeated) index is understood to run over the whole range.

Enlightenment is man's emergence from his self-incurred immaturity. Immaturity is the inability to use one's own understanding without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. The motto of enlightenment is therefore : Sapere aude! Have courage to use your own understanding!

Laziness and cowardice are the reasons why such a large proportion of men, even when nature has long emancipated them from alien guidance (naturaliter maiorennes, nevertheless gladly remain immature for life. For the same reasons, it is all too easy for others to set themselves up as their guardians. It is so convenient to be immature! If I have a book to have understanding in place of me, a spiritual adviser to have a conscience for me, a doctor to judge my diet for me, and so on, I need not make any efforts at all. I need not think, so long as I can pay ; others will soon enough take the tiresome job over for me. The guardians who have kindly taken upon themselves the work of supervision will soon see to it that by far the largest part of mankind (including the entire fair sex) should consider the step forward to maturity not only as difficult but also as highly dangerous. Having first infatuated their domesticated animals, and carefully prevented the docile creatures from daring to take a single step without the leading-strings to which they are tied, they next show them the danger which threatens them if they try to walk unaided. Now this danger is not in fact so very great, for they would certainly learn to walk eventually after a few falls. But an example of this kind is intimidating, and usually frightens them off from further attempts.

Thus it is difficult for each separate individual to work his way out of the immaturity which has become almost second nature to him. He has even grown fond of it and is really incapable for the time being of using his own understanding, because he was never allowed to make the attempt.

NO SINGLE theoretical development in the last three decades has had more influence on gravitational theory than the discovery of the Kerr solution in 1963. The Kerr metric is a solution of the vacuum field equations. It is a generalization of the Schwarzschild solution, and represents the gravitational field of a special configuration of rotating mass, much as the external Schwarzschild solution represents the gravitational field of a spherical distribution of matter.

However, unlike the Schwarzschild case, no simple non-singular fluid ‘interior’ solution is known to match onto the Kerr solution. There is, nevertheless, no reason a priori why such a solution shouldn't exist.

Fortunately such speculations are in some respects beside the point, since the real interest in the Kerr solution for many purposes is its characterization of the final state of a black hole, after the hole has had the opportunity to ‘settle down’ and shed away (via gravitational radiation and other processes) eccentricities arising from the structure of the original body that formed the black hole.

To put the matter another way, suppose someone succeeded in exhibiting a good fluid interior for the Kerr metric. Well, that would be in principle very interesting; but there is no reason to believe that naturally occurring bodies (e.g. stars, galaxies, etc.) would tend to fall in line with that particular configuration.

However exalted we may wish our concepts to be, and however abstract we may make them in relation to the realm of the senses, they will continue to be associated with figurative notions. The proper function of these is to make such concepts, which are not in other respects derived from experience, suitable for use in the experiential world. For how else could we endow our concepts with sense and significance if we did not attach them to some intuition (which must ultimately always be an example derived from some possible experience)? If we then subtract the figurative associations from this concrete act of the understanding— first those of fortuitous sense-perception, and then the pure sensuous intuition itself—we are left with the pure concept of the understanding, but with its scope now enlarged so as to constitute a complete rule of thought. This is the way in which even universal logic came into being; and in the application of our understanding and reason to experience, there may still lie hidden certain heuristic methods of thought which, if we could carefully extract them from experience, might well enrich philosophy with useful maxims, even in abstract thought.

To this category belongs that principle to which the late Moses Mendelssohn expressly declared his allegiance—but only, so far as I know, in his last writings (see his Morgenstunden ﹛Morning Hours), pp. 164f. and his letter An die Freunde Lessings (To Lessing's Friends), pp. 33 and 67): namely the maxim that it is necessary to orientate oneself in the speculative use of reason (which Mendelssohn, on other occasions, credited with considerable powers in the cognition of supra-sensory objects, and even with the power of conclusive proof) by means of a certain guideline which he sometimes described as common sense (in his Morgenstunden), sometimes as healthy reason, and sometimes as plain understanding (in An die Freunde Lessings). Who would have thought that this admission would not only have such disastrous effects on his favourable opinion of the power of speculative reasoning in theological matters (which was in fact inevitable), but also that even ordinary healthy reason, given the ambiguous position to which he relegated the use of this faculty in contrast to speculation, would risk becoming the basic principle of zealotry and of the complete subversion of reason?

IT IS INEVITABLE that with the passage of time Einstein's general relativity theory, his theory of gravitation, will be taught more frequently at an undergraduate level. It is a difficult theory—but just as some athletic records fifty years ago might have been deemed nearly impossible to achieve, and today will be surpassed regularly by well-trained university sportsmen, likewise Einstein's theory, now over seventy-five years since creation, is after a lengthy gestation making its way into the world of undergraduate mathematics and physics courses, and finding a more or less permanent place in the syllabus of such courses. The theory can now be considered both an accessible and a worthy, serious object of study by mathematics and physics students alike who may be rather above average in their aptitude for these subjects, but who are not necessarily proposing, say, to embark on an academic career in the mathematical sciences. This is an excellent state of affairs, and can be regarded, perhaps, as yet another aspect of the overall success of the theory.

A fresh look at anti-symmetric tensors

WE have introduced local differential geometry in a notation that makes great use of indices. This is the classical route and it does have a great deal of merit. There is a parallel development in an index free notation that is more generally used by pure mathematicians. The different approaches have their separate advantages and drawbacks: a calculation with indices may be cumbersome and sprawling; conversely an index-free notation may labour what is easily written with indices.

Kant's standing as a political thinker has been substantially enhanced in the English-speaking world since this volume went to the printers just over two decades ago. More and more scholars are willing to rank him among the leading figures in the history of political thought. John Rawls's important and much discussed treatise A Theory of Justice is indebted to him, and that has certainly made an impact. (Even legal historians and jurists have taken note of his writings, but to discuss their findings would go beyond the scope of this edition.) The secondary literature on his political thought has grown appreciably, and not only in Germany where research on Kant flourishes as always. Yet much of this writing, perhaps inevitably, covers well-tilled ground; there has been no revolution in the interpretation of Kant's political thought. Nevertheless, in view of this growing interest, it is perhaps justifiable not only to raise some new issues but also to elaborate some of the features which were mentioned only briefly, or merely alluded to, in my original introduction. Some of the following remarks are of a general nature, and others refer to specific issues. For ease of reference, they are grouped under the following headings: ‘the nature of rational discourse in polities’; “the nature of mature political judgement”; ‘property as the basis of the legal order’; ‘morality and polities’; ‘the republican constitution: representation and the separation of powers’; ‘Kant and the French Revolution’; ‘Kant's rejection of the right of rebellion’; ‘the rejection of the right of rebellion and twentieth-century totalitarianism’; ‘the limits of obedience to goverment’; ‘the Prussian context’; and ‘Kant's argument against world government’.

THE NATURE OF RATIONAL DISCOURSE IN POLITICS

To tackle the more general issues first: Hans Saner, in his challenging, and, on the whole, well-received study Kants Weg vom Krieg zum Frieden. Widerstreit und Einheit. Wege zu Kants politischem Denken (which has been translated into English with the somewhat misleading title Kant's Political Thought. Its Origins and Development), has argued that Kant's metaphors indicate a profound interest in politics from the very beginnings of his academic career; he points out that, from 1755 onwards, Kant continuously uses images of war and peace in his writings. Saner overstates a good case; for metaphors have to be interpreted with much care.