The dynamical interaction of material bodies does not consist in their gravitational attraction alone. Electric and magnetic forces between them have been known for a very long time and have by Faraday and Maxwell been reduced to the notion of the electromagnetic field. In ordinary circumstances electromagnetic forces are, whenever they are observed at all, very much stronger than the gravitational pull, which is exceedingly weak unless at least one of the interacting pieces of matter is very large, of the size of a celestial body. Of late, one has been induced to admit that between the elementary particles (nucleons) which go to build up the nucleus of an atom there is a force (called the nuclear force) which is perceptible only at very small distances, but outweighs there even the strong electric repulsion between some of those particles. The field of this force is usually referred to as the meson field, for reasons on which we will not enter at the moment.

Ever since Finstein discovered his theory of the gravitational field in 1915, there have been unceasing attempts to generalize it so as to account in the same natural way for the electromagnetic field as well. Since the latter is in empty space described by an antisymmetric tensor of the second rank, the idea suggests itself at once that one should take the fundamental tensor gik to be non-symmetric, hoping that its skew part ½(gik − gki) should have something to do with electromagnetism. But this plan meets with a certain difficulty. We had established Einstein's field equations in two steps.

In Einstein's theory of gravitation matter and its dynamical interaction are based on the notion of an intrinsic geometric structure of the space-time continuum. The ideal aspiration, the ultimate aim, of the theory is not more and not less than this: A four-dimensional continuum endowed with a certain intrinsic geometric structure, a structure that is subject to certain inherent purely geometrical laws, is to be an adequate model or picture of the ‘real world around us in space and time’ with all that it contains and including its total behaviour, the display of all events going on in it.

Indeed the conception Einstein put forward in 1915 embraced from the outset (and not only by the numerous subsequent attempts to generalize it) every kind of dynamical interaction, not just gravitation only. That the latter is usually in the foreground of our mind—that we usually call the theory of 1915 a theory of gravitation—is due to two facts. First, its early great successes, the new phenomena it predicted correctly, were deemed to refer essentially to gravitation, though that is, strictly speaking, true only for the precession of the perihelion of Mercury. The deflexion of light rays that pass near the sun is not a purely gravitational phenomenon, it is due to the fact that an electromagnetic field possesses energy and momentum, hence also mass. And also the displacement of spectral lines on the sun and on very dense stars (‘white dwarfs’) is obviously an interplay between electromagnetic phenomena and gravitation.

It is far beyond the scope of these lectures to report on the development of the ideas, first of Restricted, then of General, Relativity and to show how they are logically built on the outcome of a number of crucial experiments, as the aberration of the light of fixed stars, the Michelson-Morley experiment, certain facts regarding the light from visual binary stars, the Eötvös-experiments which ascertained to a marvellously high degree of accuracy the universal character of the gravitational acceleration—that is to say that in a given field it is the same for any test-body of whatever material.

Yet before going into details about the metrical (or Riemannian) continuum, I wish to point out the main trend of thought that suggests choosing such a one as a model of space-time in order to account for gravitation in a purely geometrical way. In this I shall not follow the historical evolution of thought as it actually took place, but rather what it might have been, had the idea of affine connexion already been familiar to the physicist at that time. Actually the general idea of it emerged gradually (in the work of H. Weyl, A. S. Eddington and Einstein) from the special sample of an affinity that springs from a metrical (Riemannian) connexion—emerged only after the latter had gained the widest publicity by the great success of Einstein's 1915 theory. Today, however, it seems simpler and more natural to put the affine connexion, now we are familiar with it, in the foreground, and to arrive at a metric by a very simple specialization thereof.

The subject-matter of the previous chapter is called tensor algebra. It is characterized by the fact that only relations between invariants, vectors or tensors referring to the same point of the continuum are contemplated. From the point of view taken here, algebraic relations between vectors and tensors referring to different points are meaningless.

Remember, however, that we based the notion of tensors on that of vectors, and the latter on the notion of the gradient, and there is hardly any simple and natural alternative to this procedure. Now in forming the gradient we actually had to compare the values of an invariant at different points, and at the same time we made the first step at introducing analysis into our continuum. In this and the following chapters we shall have to extend it. Analysis will involve derivatives and integrals. We shall have to study both from the point of view of general invariance. However, this does not mean to look out only for invariants, but also for entities with tensorial character, because, as we have seen, an equation between them (or in other words a system of equations saying that a tensor vanishes) is conserved on transformation. We begin with space-time-integrals. That leads to a certain extension of the notion of tensors, viz. to tensor densities.

We had emphasized that there is no point in adding (or, more generally, in forming linear aggregates of) tensors or vectors referring to different points. This would have no simple meaning.

The triumph of mathematics during the Enlightenment can be judged by the strangely conflicting testimony of those mathematicians who helped to create it. In a famous letter of September 21, 1781, Joseph-Louis Lagrange (1736–1813) wrote to his mentor Jean d'Alembert that he feared mathematics had reached its limit. He compared mathematics to a mine whose precious minerals had been pursued deeper and deeper into the earth to the limit of human accessibility. “Unless new seams of ore are discovered, it will be necessary to abandon it sooner or later.” Bernard Fontenelle had sounded the same warning as early as 1699, and Diderot used the exhaustion of mathematics as the best argument for turning to the more descriptive sciences of natural history, anatomy, chemistry, and experimental physics. He argued that like the pyramids of Egypt, the creations of mathematicians would stand for centuries but that like the pyramids, they could have little added to them and little practical use could be made of them. D'Alembert and the Marquis de Condorcet, on the other hand, urged mathematicians to keep the faith and trust to the future, even though the future for mathematics was uncertain.

The Meaning of Analysis

One wonders why the mathematicians of the eighteenth century, who had witnessed the spectacular success of their discipline during their lifetimes and who had seen mathematics become the prime exemplar of reasoned thought and the model against which the other sciences were to be judged, were so uncertain about its future.

In Herbert Butterfield's famous book The origins of modern science (1300–1800) (New York, 1956), there is a chapter entitled “The postponed scientific revolution in chemistry.” The title refers to the fact that the new chemistry associated with the oxygen theory of Antoine Lavoisier did not emerge until the 1770s and 1780s, a century after Newton's Principia had put the capstone on the Scientific Revolution of the seventeenth century. It is also significant that Lavoisier's contemporaries were conscious of and frequently mentioned a “revolution” that was occurring in chemistry, and that Lavoisier himself stated in 1773, in a private memorandum, that he believed the experiments he was undertaking would “bring about a revolution in physics and chemistry.” Scientists at the time and historians since have concurred in identifying chemistry as a subject that enjoyed its “revolution” during the Enlightenment.

In fact, the Chemical Revolution was more the creation of a new science than a change in an existing one. Before 1750, chemistry could not be regarded as an independent discipline. It had long antecedents, but they were ancillary to other fields. Alchemy was a source for many of the recipes and much of the apparatus of chemistry, but this information was concealed in intentionally ambiguous and allegorical language. Alchemy sought to complicate nature, not to rationalize it, and the alchemists' search for the philosopher's stone that would allow them to change base metals into gold was as much a spiritual quest as it was a scientific one.

In 1774, Turgot — the same Turgot who wrote the article entitled “Expansibilité” for the Encyclopédie — became controller general of France under the new king, Louis XVI. For the first time the most important ministerial position in the kingdom was held by a friend of the philosophes. France had suffered a series of financial crises that had grown in severity throughout the century. The cause was not a general decline in prosperity but a tax system that made it impossible for the king to tax the real sources of wealth in the kingdom. The financial crisis had at its root a social crisis. The clergy, the nobility, and the parlements (traditional judicial bodies that claimed the right to approve taxes) jealously guarded their prerogatives and sought to extend their powers with little thought for the state as a whole. By 1774, France was approaching disaster. The failure of fiscal and social reform at that juncture (and Turgot did fail; he stayed in office for only twenty months) meant that special interests would prevail and that future ministers would be chosen not for their reforming skills but for their ability to borrow money. Fifteen years after Turgot began his ministry, France collapsed in a decade of revolution that eventually brought the needed reforms, but only at the expense of war within and without, and protracted political chaos.