This book is intended primarily for the use of students reading for an honours degree in mathematics, and our aim in writing it has been to give a rigorous and systematic account of projective geometry, which will enable the reader without undue difficulty to grasp the fundamental ideas of the subject and to learn to apply them with facility.
Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods. We have accordingly taken for granted acquaintance with the elements of linear algebra and the calculus of matrices, and, except in one instance, we have not gone into the proofs of purely algebraic theorems. The exception is a theorem which is fundamental in our system but is possibly not met with in quite the same form outside geometry, and this theorem we have proved in the Appendix.
In spite, however, of treating geometry algebraically, we have tried never to lose sight of the synthetic approach perfected by such geometers as von Staudt, Steiner, and Reye. If one consideration has been more prominent in our minds than any other it is that of giving precedence to the geometrical content of the system and the geometrical way of thinking about it. Nothing, in our opinion, could be more undesirable than that this traditionally elegant subject should be allowed to take on the appearance of being merely a dressing-room in which algebra is decked out in geometrical phraseology. We have, therefore, tried to show that although the basis of the formal structure is algebraic, the structure itself is thoroughgoing geometry, inasmuch as its concepts, its methods, and its results are all essentially dependent on geometrical ideas. ....
J G Semple, G T Kneebone
THE CONCEPT OF GEOMETRY
Our main purpose in this book is to construct and develop a systematic theory of projective geometry, and in order to make the system both rigorous and easily comprehensible we have chosen to build it on a purely algebraic foundation. In adopting such a course, however, we may run the risk of appearing to reduce our subject to an ingenious manipulation of symbols in accordance with certain arbitrarily prescribed rules. Although the axiomatic form is the proper one in which to present a mathematical theory, we must not lose sight of the fact that an abstract system can only be fully appreciated when seen in relation to a more concrete background; and this is the reason why we have prefaced the formal development of projective geometry with two introductory chapters of a more informal character. The present chapter is devoted to a rather general consideration of the nature of mathematics and, more specifically, of geometry, while Chapter II contains an outline of the intuitive treatment of projective geometry from which the axiomatic theory has gradually been disentangled by progressive abstraction.
The growth of geometrical knowledge in the past has been marked by a gradual shifting away from empirical observation towards rational deduction; and we shall begin by looking for a moment at this process.
Geometry is commonly regarded as having had its origins in ancient Egypt and Babylonia, where much empirical knowledge was acquired through the experience of surveyors, architects, and builders; but it was in the Greek world that this knowledge took on the characteristic form with which we are now familiar. The Greek geometers were not only interested in the facts as such, but were intensely interested in exploring the logical connexions between them. In other words, they wished to raise the status of mathematics from that of a mere catalogue to that of a deductive science - and the Elements of Euclid is an embodiment of this ideal. In the Elements we have the systematic derivation of a large body of geometrical theorems by strict deduction from a small number of axioms. The system, as is now known, is not altogether perfect, and modern mathematicians have shown how it needs to be amended; but the modifications required are comparatively slight, and there is perhaps no easier way for a student to learn to appreciate mature mathematical reasoning than by studying the first book of Euclid and observing the way in which it is constructed.
Now for the Greeks, we must remember, geometry meant study of the space of ordinary experience, and the truth of the axioms of geometry was guaranteed by appeal to self-evidence. This view persisted for a very long time, and was still accepted without question at the end of the eighteenth century - when Kant, for example, made it an integral part of his philosophy. But about that time mathematicians were already beginning to see their subject in a new light, as a branch of study not directly dependent on experience, and this change of outlook was encouraged by the discovery, early in the nineteenth century, of the non-euclidean geometries, systems consistent within themselves but incompatible with Euclid's system. Since then it has become a commonplace that the mathematician is free to study the consequences of any axioms that interest him, whether or not they have any application in experience, provided only that they are not mutually contradictory.
We see, then, that in the period which elapsed between the first beginnings of mathematics and the conscious adoption of the modem axiomatic method, two major revolutions took place in mathematical thinking. First, the mere collecting of useful or interesting facts gave place to the rational deduction of theorems; and then, much later, mathematicians began to detach themselves from experience and to concentrate on the study of formal axiomatic systems. Neither of the revolutions came about suddenly, and the second is in a sense still in progress. Mathematics, as conceived today, is fundamentally the study of structure. Thus, although arithmetic is ostensibly about numbers and geometry about points and lines, the real objects of study in these branches of mathematics are the relations which exist between numbers and between geometrical entities. As mathematics develops, so it becomes more abstract, until at last it is seen to be concerned with networks of formal relations only, and not with any particular sets of entities between which the relations hold. The process of abstraction whereby the formal structure is by degrees detached from the concrete systems in which it is exhibited is of so great importance to the understanding of the nature of mathematics as to justify closer examination of the manner in which it takes place.
One of the simplest illustrations of the process is provided by the evolution of the concept of number. Our first rudimentary idea of number is arrived at by simple abstraction from the processes of counting and measuring ordinary objects, and this idea is adequate at the level of school arithmetic. At a more advanced stage, numbers are seen to require redefinition in purely logical terms, and several alternative definitions have, in fact, been given. In whatever way numbers are defined, however, they obey the same formal 'laws of algebra' - the associative law of addition (a + b) + c = a + (b + c), the distributive law a(b + c) = ab + ac, etc. - and many of the standard theorems of arithmetic and algebra can be deduced directly from these laws, without any need to specify further the nature of the numbers that are represented by the symbols a, b, etc. But this is not all. When studying elementary algebra one soon becomes aware of the close analogy that exists between the algebra of polynomials and the arithmetic of whole numbers; and it is now easy to account for this analogy by pointing out that polynomials, as well as numbers, satisfy the 'laws of algebra'. This is tantamount to saying that the system of numbers and the system of polynomials have a common structure; and when once this fact is recognized it is a natural step to undertake the study of an abstract system whose nature is unspecified beyond the fact that it has this particular structure. Such a system is known in algebra as a ring. If, on the other hand, we apply a similar process of abstraction to the system of rational numbers or the system of rational functions, we arrive at the abstract system known as a field.
There is no need for recognition of structural similarity to come to an end, even at this stage. Thus we might observe, for instance, that addition of rational numbers and multiplication of non-zero rational numbers obey similar laws; and we could then verify that the additive structure of a field and its multiplicative structure (when the element zero is excluded) are formally alike. Carrying the process of abstraction one stage farther, we could now introduce the abstract system known as a group.
Mathematics, then, is concerned with abstract systems of various kinds, each defined by a suitable set of axioms, which serves to characterize its structure. But although, from the point of view of pure mathematics, each structure is regarded as self-contained, the mathematical scheme usually has one or more concrete realizations; that is to say, the structure is usually to be found (possibly only to a certain degree of approximation) in a more concrete system. Abstract euclidean geometry of three dimensions, for instance, has as one of its realizations the structure of ordinary space. Indeed this is what led to its discovery, as well as what makes it so much more interesting than other systems which are logically of equal status with it. We do not, of course, always have to go all the way back to everyday experience for a realization of a mathematical formalism, since one is usually provided, as in the arithmetical example already considered, by a more concrete part of mathematics itself. One of the most important instances is the widespread occurrence of the group structure, which is found not only in additive and multiplicative groups of numbers, but also in groups of transformations and groups of matrices. Since this type of structure pervades much of mathematics, we may say that it is especially significant.
In this book we shall study the structure of projective geometry which, as is well known, is closely associated with certain simple algebraic structures, and with linear algebra particularly. Since the relevant algebra is part of every mathematician's essential equipment, we shall take it for granted that the reader is already familiar with it.
What we have said so far about the nature of mathematics holds quite generally, but when we limit the discussion to geometry we are able to be rather more specific. The structures studied in this branch of mathematics occur in experience as spatial structures, and from this alone we can infer something of their general character. If, in fact, we turn back once again to Greek geometry, we may recall that the geometrical knowledge with which the Greeks began was derived ultimately from measurements made upon rigid bodies, and was therefore essentially a knowledge of shapes. Now the shape of a body can be conceived as determined by those relations between its parts which remain unaltered when the body is moved about in space. Whenever one body can be made in this way to take the place of another, the two bodies have the same shape; and they are then equivalent as regards their geometrical properties, or, in the language of elementary geometry, 'equal in all respects'. It will be remembered that in order to prove that certain sets of conditions are sufficient to ensure the congruence of two triangles Euclid showed that, if the conditions are satisfied, one triangle may be placed so as to bring it into coincidence with the other.
The idea of studying those properties of bodies which remain unaltered when the bodies are displaced in any way is most suggestive to a modern mathematician. In the language now in use, we would say that the geometrical (or, more accurately, the euclidean) properties of a body are invariant with respect to the operation of displacement in space; and invariance with respect to a certain kind of operation at once suggests the existence of an underlying group of operations. In the present instance the appropriate group is not far to seek. The totality of all displacements in space is a group of transformations; two bodies are congruent if and only if one can be made to take the place of the other by an operation of the group; and the shape of a body is determined by those of its spatial characteristics which are invariant with respect to the whole group. This, then, is the nature of euclidean geometry - it is the invariant-theory of the group of displacements.
Euclidean geometry, however, is not the whole of geometry. Early in the nineteenth century it was realized that other systematic collections of geometrical properties are possible besides that of Euclid, and in 1822 Poncelet published his Traité des propriétés projectives des figures, the first systematic treatise on projective geometry. In constructing this system Poncelet was fully conscious that his classification of geometrical theorems was based upon a new kind of fundamental operation, namely conical projection. A projective property of a figure is, in fact, simply a property that is invariant with respect to projection, and this enables us easily to identify the associated group of transformations. Confining ourselves, for simplicity, to two-dimensional geometry, we may consider the totality of all those transformations of the plane into itself which can be resolved into finite chains of projections from one plane on to another; and it is clear that this totality of transformations is a group and that it has plane projective geometry as its invariant-theory. Since the euclidean group, consisting of all displacements of the plane, may be shown to be a proper subgroup of the projective group, it follows at once that every projectively invariant property is also a euclidean invariant, whereas not every euclidean property is projective.
If we were now to take any arbitrarily chosen group of transformations of the plane into itself (containing the group of displacements as a subgroup) we could use this group in order to define an associated system of geometry; and all such systems are, mathematically speaking, of equal status. This was the general principle laid down by Klein in his famous Erlangen Programme of 1872. Some of the geometries that can be obtained in this way, such as euclidean geometry, affine geometry, and projective geometry, are very well known; others, such as inversive geometry (which arises from the group of all transformations that can be resolved into finite sequences of inversions with respect to circles) are known but not usually studied in much detail; and yet others are presumably ignored altogether.
We shall confine our attention to the three geometries first mentioned - the geometries of the projective hierarchy - and since this restriction is somewhat arbitrary from a purely mathematical point of view, we should perhaps give some indication of why we choose to impose it. In the first place, euclidean geometry is of particular interest on account of its close connexion with the space of common experience, and this alone is sufficient to single it out for special attention. It so happens, however, that euclidean geometry is complicated; and we can appreciate it better when we relate it to projective geometry, where the structure is very much simpler. Projective geometry is more symmetrical than euclidean, by virtue both of the existence of a principle of duality and also of the fact that it may be handled by means of homogeneous coordinates. When homogeneous coordinates are used for this purpose, the algebra has the merit of being either already linear or else readily made so. Thus the system of projective geometry is easy to work out and equally easy to comprehend when it has been worked out. Furthermore, projective transformations have the property of transforming conics into conics; and this means that the conic takes its place as naturally in projective geometry as does the circle in euclidean geometry. Finally, the essentials of euclidean geometry may be treated projectively by the simple artifice of introducing the line at infinity and the circular points. We thus have two geometries, projective geometry and euclidean geometry, which fit naturally together and which between them include most of the classical geometrical theorems. It is convenient to take in conjunction with them affine geometry, an intermediate geometry that is more general than euclidean but less so than projective; and the projective hierarchy is then complete.
What has been said so far concerns the subject-matter of our book, and it still remains for us to say something of the kind of approach that we shall use. It is customary to distinguish between two modes of reasoning in geometry, commonly referred to as synthetic and analytical. In a synthetic treatment we argue directly about geometrical entities (points, lines, etc.) and geometrical relations between them, whereas in an analytical treatment we first represent the geometrical entities by coordinates or equations, in order to be able to use the technique of algebraic manipulation. Since the discussion of projective geometry which follows in Part II is to be analytical, we shall conclude this chapter by touching upon the use of coordinates; but it should be realized, nevertheless ' that we are under no logical compulsion to introduce a coordinate system at all. In the Elements, as in all Greek treatises, euclidean geometry is treated synthetically, and synthetic treatments of projective geometry are to be found in a number of modern books on the subject. (The first work of this kind was von Staudt's Geometrie der Lage (Nuremberg, 1847). A standard text-book, written in a similar spirit, is Veblen and Young's Projective Geometry (Boston, 1910).)
Coordinates were first introduced into geometry by Descartes, in the seventeenth century, and the fruitfulness of the innovation soon became apparent. The older method of labelling figures was by letters of the alphabet, as in 'the triangle ABC', but such labels were in fact no more than arbitrarily assigned names. Descartes's new technique of coordinates, on the other hand, made use of a system of labels which itself possesses a mathematical structure capable of reflecting the structure of the system labelled. This method of labelling has since become indispensable in mathematics, and the domain in which it can be applied now extends far beyond that originally envisaged by Descartes. In geometry itself, not only points but also lines and other entities can be represented by sets of coordinates; and in dynamics - to take an instance of another kind - the configuration of a system is ordinarily specified by n coordinates q1, q, ..., qn .
We have now seen how mathematics may be looked upon as a study of formal structure, and how geometry may be fitted into the general scheme. What has been said so far has been of a rather general character, and we must now turn more specifically to the details of the geometries of the projective hierarchy. This will be the topic of the second chapter of Part I, in which our purpose will be to recall enough of the elementary treatment of projective geometry to enable the reader to appreciate the process of abstraction which leads to the formal system of Part II.