It is scarcely necessary to apologise for writing a book on n-dimensional geometry. One should regret rather the comparative neglect which the subject has suffered at the hands of British mathematicians. [In the twenty-seven volumes of the new series of the Proceedings of the London Mathematical Society there are barely a dozen papers dealing with higher space. On the other hand, it is interesting to notice that there are about an equal number in the three volumes of the journal; this seems to indicate a revival of interest.] Yet one may almost say that this country was its home of origin, for, with the exception of a few previous sporadic references, the first paper dealing explicitly with geometry of n dimensions was one by Cayley in 1843, and the importance of the subject was recognised from the first by three of our most famous pure mathematicians - Cayley, Clifford, and Sylvester. On the Continent the classical works of Grassmann and Schläfli attracted at first no attention. Schläfli's remarkable memoir, in fact, failed to secure publication, and in spite of Cayley's gallant attempt at rescue by translating and publishing part of it in the "Quarterly journal" it remained unknown until it was found and published several years after the author's death, and fifty years after it was written. By that time Schlegel and others in Germany had made the subject well known, but mostly in its metrical aspect. The wonderful projective geometry of hyperspace has been almost entirely the product of the gifted Italian school of geometers; though this branch also was inaugurated by a British mathematician, W K Clifford, in 1878.
The present introduction deals with the metrical and to a slighter extent with the projective aspect. A third aspect, which has attracted much attention recently, from its application to relativity, is the differential aspect. This is altogether excluded from the present book.
In writing this book I have not attempted to produce a complete systematic treatise, but have rather selected certain representative topics which not only illustrate the extensions of theorems of three-dimensional geometry, but reveal results which are unexpected and where analogy would be a faithless guide.
The first four chapters explain the fundamental ideas of incidence, parallelism, perpendicularity, and angles between linear spaces; and in Chapter I there is an excursus into enumerative geometry which may be omitted on a first reading. Chapters V and VI are analytical, the former projective, the latter largely metrical. In the former are given some of the simplest ideas relating to algebraic varieties, and a more detailed account of quadrics, especially with reference to their linear spaces. In the latter there are given, in addition to the ordinary Cartesian formulae, some account and applications of the Plücker-Grassmann co-ordinates of a linear space, and applications to line-geometry. The remaining chapters deal with polytopes, and contain, especially in Chapter IX, some of the elementary ideas in analysis situs. Chapter VIII treats of the content of hyperspatial figures, and the final chapter establishes the regular polytopes.
A number of references have been given at the ends of the chapters. Some of these are the original works in which the various theories were first expounded, others are a selection of more recent works in which a fuller account may be found. Reference may be made to the author's Bibliography of Non-Euclidean Geometry, including the theory of parallels, the foundations of geometry, and space of n dimensions (London: Harrison, for the University of St Andrews. 1911), which contains, in addition to a chronological catalogue, a detailed subject index and an index of authors.
I am indebted in particular to Schoute's Mehrdimensionale Geometrie (Leipzig, 2 Vols., 1902 and 1905), Bertini's "Introduzione alla geometria proiettiva degli iperspazi" (Pisa, 1907), and the various articles of the "Encyklopädie der mathematischen Wissenschaften."
For assistance in correcting the proofs I have to thank Mr F F Miles, M.A., Lecturer in Mathematics at this College.
D M Y Sommerville.
VICTORIA UNIVERSITY COLLEGE,
1. Origins of Geometry.
Geometry for the individual begins intuitionally and develops by a co-ordination of the senses of sight and touch. Its history followed a similar course. The crude ideas of shape, bulk, superficial extent, and length became analysed, refined, and made abstract, and led to the conception of geometrical figures. The development started with the solid; surface and line, without solidity, were later abstractions. Witness the inability of most animals and some primitive races of men to recognise a picture. Having no depth, except such as is imitated by the skilfulness of the drawing or shading, it conveys to the undeveloped intelligence only an impression of flat regions of contrasted colouring. When the power of abstraction had proceeded to the extent of conceiving surfaces apart from solids, plane geometry arose. The idea of dimensionality was then formed, when a region of two dimensions was recognised within the three-dimensional universe. This stage had been reached when Greek geometry started. It was many centuries, however, before the human mind began to conceive of an upward extension to the idea of dimensionality, and even now this conception is confined to the comparatively very small class of mathematicians and philosophers.
2. Extension of the Dimensional Idea.
There are two main ways in which we may arrive at an idea of higher dimensions: one geometrical, by extending in the upward direction the series of geometrical elements, point, line, surface, solid; the other by invoking algebra and giving extended geometrical interpretations to algebraic relationships. In whatever way we may proceed we are led to the invention of new elements which have to be defined strictly and logically if exact deductions are to be made. A great deal is suggested by analogy, but while analogy is often a useful guide and stimulus, it provides no proofs, and may often lead one astray if not supplemented by logical reasoning. If we follow the geometrical method the only safe course is that which was systematically laid down for the first time by Euclid, that is to lay down a basis of axioms or assumptions. When we leave the field of sensuous perception and can no longer depend upon intuition as a guide, our axioms will no longer be "self-evident truths," but simply statements, assumed without proof, as a basis for future deductions.
3. Definitions and Axioms.
In geometry there are objects which have to be defined, and relationships between these objects which have to be deduced either from the definitions or from other simpler relationships. In defining an object we must make reference to some simpler object, hence there must be some objects which have to be left undefined, the indefinables. Similarly, in deducing relations from simpler ones we must arrive back at certain statements which cannot be deduced from anything simpler; these are the axioms or unproved propositions. The whole science of geometry can, thus be made to rest upon a set of definitions and axioms. The actual choice of fundamental definitions and axioms is to a certain extent arbitrary, but there are certain principles which have to be considered in making a choice of axioms. These are:-
(i) Self-consistency. The set of axioms must be logically self-consistent. No axiom must be in conflict with deductions from any of the other axioms.
(2) Non-redundance or Independence. This condition is not a necessary one, but in a logical scheme it is desirable. Pedagogically the condition is frequently ignored.
(3) Categoricalness. This means not only that the set of axioms should he complete and sufficient for the development of the science, but that it should be possible to construct only a unique set of entities for which the axioms are valid. It is doubtful whether any set of axioms can be strictly categorical. If any set of entities is constructed so as to satisfy the axioms, it is nearly always, if not always, possible to change the ideas and construct another set of entities also satisfying the axioms. Thus with the ordinary ideas of point and straight line in plane geometry the axioms can still be applied when instead of a point we substitute a pair of numbers (x, y), and instead of straight line an equation of the first degree in x and y; corresponding to the incidence of a point with a straight line we have the fact that the values of x and y satisfy the equation. It is desirable, in fact, that the set of axioms should not be categorical, for thereby they are given a wider field of validity, and propositions proved for the one set of entities can be transferred at once to another set, perhaps in a different branch of mathematics.
4. The Axioms of Incidence.
As indefinables we shall choose first the point, straight line, and plane. With regard to these we shall proceed to make certain statements, the axioms. If these should appear to be very obvious, and as if they might be taken for granted, it will be a good corrective for the reader to replace the words point and straight line, which he must remember are not yet defined, by the names of other objects to which the axioms may be made to apply, such as "committee member" and "committee." Following Hilbert we divide the axioms into groups.
AXIOMS OF INCIDENCE OR CONNECTION
I.1. Any two distinct points uniquely determine a straight line.
We imagine a collection of individuals who have a craze for organisation and form themselves into committees. The committees are so arranged that every person is on a committee along with each of the others, but no two individuals are to be found together on more than one committee.
I.2. If A, B are distinct points there is at least one point not on the straight line AB.
This is an "existence-postulate."
I.3. Any three non-collinear points determine a plane.
I.4. If two distinct points A, B both belong to a plane a, every point of the straight line AB belongs to a.
From I.1 it follows that two distinct straight lines have either one or no point in common. From I.4 it follows that a straight line and a plane have either no point or one point in common, or else the straight line lies entirely in the plane; from I.3 and 4 two distinct planes have either no point, one point, or a whole straight line in common.
I. 5. If A, B, C are non-collinear points there is at least one point not on the plane, ABC.
I.2 and 5 are existence-postulates; 2 implies two-dimensional geometry, and 5 three-dimensional.
The next of Hilbert's axioms is that if two planes have one point A in common they have a second point B in common, and therefore by I.4 they have the whole straight line AB in common. If this is assumed it limits space to three dimensions.
5. Projective Geometry.
There is a difficulty in determining all the elements of space by means of the existence postulates and other axioms, for while Axiom I.1 postulates that any two points determine a line, there is no axiom which secures that any two lines in a plane will determine a point. In fact, in Euclidean geometry this is not true since parallel lines have no point in common. For the present therefore we shall confine ourselves to a simpler and more symmetrical type of geometry, projective geometry, for which we add the following axiom:
I.1'. Any two distinct straight lines in a plane uniquely determine a point.
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