Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/693

Rh are called the coordinates of the point. In a similar though more extended sense we may use the word &quot; coordinates &quot; to express the group of eight magnitudes which we have found to be adequate to the complete specification of the field. By the measurement of the field in the most complete sense of the term we mean the measurement of its eight coordinates. Suppose that an object is completely specified by n coordi nates, then every different group of n coordinates will specify a different object. The entire group of such objects will form what is called a continuously extended manifold- ness. The singly extended manifoldness may be most conveniently illustrated by the conception of time, the various epochs of which are the elements in the manifold- ness. Space is a triply extended manifoldness whereof the elements are points. All conceivable spheres form a quadrnply extended manifoldness. All conceivable triangles in space form a manifoldness of nine dimensions. The number of coordinates required to specify the position of an element in a manifoldness is thus equal to the order of the manifoldness itself. It is important to observe that the elements of the manifoldness may be themselves objects of no little complexity. Thus, for instance, the conies forming a confocal group constitute the elements of a singly ex tended manifoldness. The essential feature of a singly extended manifoldness is that a continuous progress of an element can take place only in two directions, either forwards or backwards. But a singly extended manifoldness may be regarded as itself an element in a manifoldness of a higher order. Thus the points on a circle form a singly extended manifold- ness, while the circle itself is one element of the manifold- ness which consists of a series of concentric circles. The system of concentric circles may in like manner be regarded as an element in the manifoldness which embraces all systems of concentric circles whose centres lie along a given line. We are thus led to conceive of a multiply extended manifoldness as made up by the successive com position of singly extended manifoldnesses. It follows from the conception of a manifoldness that in the case of a singly extended manifoldness the position of every element must be capable of being completely specified by a single quantity. It becomes natural to associate with each element of the manifoldness a special numerical magni tude. These .magnitudes may vary from -co to + oo ; to each magnitude will correspond one element of the manifold- ness, and conversely each, element of the manifoldness is completely specified whenever the appropriate number has been assigned. It is quite possible to have this association of numerical magnitude with the actual position of an element independent of any ordinary metrical relations of the system; it will, however, most usually be found that the numerical magnitudes chosen are such as admit of direct interpretations for the particular manifoldness under con sideration. Thus, for instance, in the case of the system of concentric circles it will be natural to associate with each circle its radius, and the position of each circle in the manifoldness will thus be completely defined by the radius. So also in the case of that singly extended manifoldness which consists of colours, it will be natural to employ as the number which specifies each particular colour the wave-length to which that particular colour corre sponds. If the elements of such a manifoldness can receive a simultaneous displacement, then it is plain that to each element in the original position will correspond an element in the second position. Let x and y bo the numerical magnitudes correlated to these two elements. Then, since the relation must be of the one-to-one type, it is necessary that the magnitudes x and y must be connected by an equation of the type axy + Ix + cy + d = 0. It follows from this that there are a pair of elements which are common to both systems, for if x=y we have the equation The original equation may be written in the form axy + (b- ta)x + (c + w)y + d*= , and whatever value o&amp;gt; may have this equation will lead to the same quadratic for the two common elements. We thus have a singly infinite number of displacements which are compatible with the condition that the two fundamental elements shall remain unaltered, and it is displacements of this kind which express the movements of a rigid system. The position of a point is to be defined by three coordinates. In our ordinary conception of coordinates the position of the point is defined by certain measure ments, and thus it would seem that the very mention of coordinates had already presupposed the idea of distance. This, however, need not be the case. We can assume a point in space to be completely defined by three purely numerical quantities. It will be supposed that to each group of three coordinates corresponds one point, and that conversely to one point will correspond three coordinates and no ambiguity is to be present. This latter considera tion will exclude from our present view such cases as those where the position of a point is defined by a line and two angles, because angles are subject to a well-known ambiguity amounting to any even multiple of TT. In this case it would not be true that to one point corresponds one set of coordinates, although the converse may be correct. It is necessary to understand clearly the nature of the suppositions which are made with regard to space by this assumption. Let x, y, z and x, y, z be the coordinates of two points a and a. Now x, y, z can change continu ously by any law into x, y , z. Each intermediate stage will give the coordinates of a point. It must thus be possible to pass continuously in an infinite number of ways from the point a to the point a. We thus assume that space is continuous when we have assumed that its points are represented by coordinates. It must be observed that we predicate nothing as to space which is not involved in the fact that to each point corresponds one group of three coordinates. To some extent the considerations now before us will apply to any other continuous manifoldness which requires three coordinates for the complete specification of its elements. Take, for instance, a musical note. It can be specified accurately by its pitch, intensity, and timbre. These three quantities may be regarded as the three coordi nates which will discriminate one sound from the rest. The manifoldness comprising all musical notes is, however, very different from the manifoldness which embraces all the points of space. Each of these manifoldnesses is no doubt continuous, and each of them is of three dimensions, but the conception of distance can have no place in the musical manifoldness. This is due to the absence from the musical manifoldness of anything parallel with the conception of rigidity in the space manifoldness. These remarks will show that the conception of &quot; distance &quot; is something of a special type even in a three-dimensioned continuous manifoldness. There are also other three-dimensioned and continuous manifoldnesses from which the conception of distance is also absent. Take, for instance, the manifoldness which embraces all the circles that can lie in a given plane. The points of such a manifoldness are the circles. It is three- dimensioned, for two coordinates will be required for the centre of each circle and one for its radius. It is obviously a continuous manifoldness, for one circle may by infinitely graduated modifications pass into any other. Yet from this manifoldness also the conception of distance is absent. There is no intelligible relation of one circle to another 