(a) Foundations of Geometry. The usual history of the develop- ment of the ideas of a student of geometry to-day is somewhat as follows. After a more or less prolonged (and highly desirable) course of experimental geometry, very largely (and undesirably) limited to a plane, in which a line is a mark made on paper, and a straight line is a mark which agrees with a physical object (a'ruler), the student passes through a course in which he is shown that there is a logical connex- ion between the geometrical conceptions his experience may have led him to form. At first, and for a long time, often permanently, lines and circles are regarded as objects of perception and, for instance, there is no hesitation in accepting the idea of two lines being in the same direction, and it appears intuitive that two points must have a certain distance, a result of familiarity with the rigid bodies which the student has had put before him. This teaching, after a certain knowledge has been obtained of the detailed relations of circles and lines, often painfully acquired and difficult to remember, is continued, on the same plan, for the so-called geometrical properties of conic sections, though these are apt to appear at first as much less concrete than circles. After this, as soon as some facility with algebraical computation is acquired, the student learns that a straight line has an equation, and that, e.g. the co.nmon points of two circles depend on the solution of a quadratic equation, while the common points of two conies depend on a quartic equation. If his instruction is pursued far enough, he learns, with the expenditure of much time and energy, a vast number of algebraical devices, and is now, if apt in using them, capable of proving algebraically al.nost any question that his usual examinations are likely to require of him. For his further efficiency to this end he is probably taught towards the end of his career something about har.nonic relations, about homography , and about projections. In particular, for exa nple, he may be taught that the equations which give the foci of a conic are obtainable by applying the analytical conditions for a circle to the equation of the pair of tangents to the conic fro.n any point. If he is fortunate it may be pointed out to hi.n, near the end of his laborious drilling in detail, that a circle behaves as if it were a conic with two definite (albeit imaginary) points; and if he must in any case know the properties of conies he may, for econony of memory, seize hold of this remark, and come also to a geometrical description of the property of foci just referred to and pursuing this course, if cir- cumstances allow, he may finally reach a framework of hypothetical constructions including the so-called circular points and the circle at infinity, from which, looking back, as fro n a hill-top, he sees the whole country of geometrical fact, with which he has so laboriously become acquainted, shrink into a landscape dominated by very few main routes. He may now be at the stage of the third year university student. With continued consideration he may be led finally, even if only with the purpose of summarizing his geometrical outlook in the fewest possible ideas, to regard as working hypotheses such as the following: (a) there is no funda nental di.Terence between points at infinity and those not at infinity; (6) there is no difference in reality between real and imaginary points; (c) there is no gain but great loss in refusing to consider space of more than three dimen- sions; (d) distance, as a fundamental conception, is unnecessary- And with these will come a recognition that the so-called non-Eu- clidian geometries are, logically, prior to the Euclidian geometry.
Leaving aside now the tempting pedagogic question of whether he has been justly treated in being so long denied the synthesis which, if he could have appreciated it, would so much have lightened his task of becoming familiar with the details, we re:tiark that he finally works with a conceptual scheme, which includes the perceptual experiences by which it has been suggested but discards many ideas which at earlier stages his perceptions see.ned to suggest as necessary. For instance, the points of a line are not now in (linear) order, and lines have lost their straightness, the lines of threefold space being for many purposes better regarded as points of a quaclric in five dimensions. Questions then arise such as: Isgeonetry unique in thus replacing the first crude ideas of physical experience by a concep- tual scheme of entities, whose properties are determined logically, not from a set of definitions which tell us what these entities are, but from a set of fundamental propositions or statements of rela- tions between them? And connected there .vith, are the ideal entities of such a conceptual scheme less real than those, for example, which the physicist employs, say the aether, or electrons, to explain his con- ceptions? May the statement that distance is not necessary as a fundamental conception be fairly replaced by the statement that distance in the abstract is an illusion? It would seem that the difference is one only of the degree of abstractness of the conceptual scheme employed. We may in geometry itself have different levels of abstractness; for example, we may in the first instance regard the points of a line as conforming to our idea of an abstract order of such a kind that the so-called Dedekind's axiom is applicable, although finally, when we allow the so-called imaginary points, we discard this notion of order and use the word line in a still more abstract sense. It would seem that every science as it advances in comprehensiveness must similarly evolve for itself a conceptual scheme of ideal entities; and that even in strict logic, no proposition can be asserted to be true or false except in reference to entities whose fundamental relations are made explicit.
Such questions as these arise when it is assumed that it ought to be possible to ascertain by observation whether the world is finite, or
still more whether space (in the abstract) is Euclidian or non-Eu- clidian. If the attitude which has been suggested is sound, the most that can be done is to inquire what would be the modifications in our statements of perceptual regularity which would follow if we adopted a particular fecheme of conceptions in regard to the extent of the world, or the character of space.
Of such conceptions, those which have reference to a method of measurement are of fundamental importance. And if measurement is possible at all, it must presumably be based upon a scheme for assigning identification numbers to the points of bodies which are to be measured. This is not the same as assigning numbers to points of space, nor even if this could be done would the method of measure- ment be determined uniquely thereby. A way of assigning identi- fication numbers to the points of a figure must be conditioned (a) by the fundamental theorems of incidence of the elements of the figure (as that a line is determined by two points, or that two planes meet in a line); (6) by the nature of the numbers to be used (whether they allow commutative multiplication for example); (c) by the freedom of the assignment, that is the number of points of the figure for which the corresponding numbers may be assigned arbitrarily, the numbers belonging to any other point being then determinate; (d) which is in fact included under (a), by the character of the " infinity " of the figure (as whether the space of the figure is open or closed); and even then (e) it appears to be necessary to assume one or more definite limiting theorems of incidence. In the way which has been studied most in detail, as being that which is most naturally suggested by the Euclidian scheme in which geometrical thought has developed, the numbers being taken to be those of ordinary arithmetic, it is possible to assume arbitrarily the numbers for three points of a line, so long as this is considered by itself, the numbers of four points of a plane regarded as isolated, and the numbers of five points of a three-dimen- sional space; this space is regarded as closed, the numbers belonging to a point are regarded as ratios of numbers, and infinite values of numbers are thereby excluded from consideration and the assump- tion is made that four lines of which no two intersect have two com- mon transversals. It is shown that the introduction of this assump- tion is equivalent (other things being equal) to assuming that the numbers used are commutative in multiplication. The number space of Descartes, in which each point is represented by three ordinary numbers, one or more of which may be infinite, may be regarded as a particular case of the so-called projective space thus described.
In his famous Habilitationsschrift (1854), when 28 years old, B. Riemann considered a Cartesian space in which each point is specified not by three but by n numbers or coordinates, and proposed to measure the distance between two neighbouring points by means of a quadratic function of the small differences of their corresponding coordinates. He remarked then that such a space has not necessarily any rigid bodies capable of movement without change of linear dimensions. For this to be possible it is necessary and sufficient that certain functions of the coefficients in the quadratic form and their differerttial coefficients should be constant. The number of these functions isVijM 2 (n 2 i); for instance for n = 2, this number is I, and for n = 3, it is 6. When these conditions are satisfied the space is said to be of constant curvature. But it is to be remarked that a Cartesian space of n dimensions, such as that considered by Riemann, is in reversible, point-to-point correspondence with a quadratic manifold (also of n dimensions), in a projective space of w + l dimen- sions. In such a space of n + i dimensions, as was first remarked by Cayley, we can set up a measurement of distance between any two points by taking, quite arbitrarily, a quadric manifold of ref- erence. It is then the case that Riemann's definition of distance, when his space is of constant curvature, and allow srigid bodies capa- ble of movement, is so obtainable, after Cayley's manner.
These details appear to bring out very clearly that even when the difficult step has been made of passing from the descriptive proper- tics of a geometrical figure to the assignment of coordinates, it is a further step of much artificiality to introduce a measure of the dis- tance between any two points.
In recent years, under the stimulus of A. Einstein, H. Minkowski, H. Weyl and others, Riemann's dearest dream of a uniform formula-' tion of all phenomena of physics, has, it would seem, been brought appreciably nearer to realization, in what is known as a Theory of Relativity. An event, occurring in a definite place at a definite time, is regarded as depending on three coordinates for its position, ami one for its time, and these four together are spoken of as its coordi- nates in a Cartesian space of four dimensions. As formulated by Einstein, there is an interval between two neighbouring events, given by a quadratic in the differences of their corresponding coordinates; this quadratic will then have ten coefficients. It can be shown that there exist functions of these coefficients and of their derivatives in regard to the point coordinates, which are unchanged in value if calculated for the quadratic form into which the given one is trans- formed by any transformation of the coordinates; for instance, the 20 functions which, as has been stated, arise in the consideration of what is called the curvature, are such functions. It is clear that the vanishing of such an invariantive function expresses a fact which is not altered by any simplification that may be possible in the form of the quadratic expression; for instance if the 20 functions above re- ferred to all vanish the quadratic expression has a form the same as in Cartesian Euclidian geometry; and if they are all equal to the same
