A Treatise on the Theory of Screws

 434 THE THEORY OF SCREWS. [397- All objects whose co-ordinates satisfy one linear homogeneous equation we shall speak of as an extent. All objects whose co-ordinates satisfy two linear homogeneous equations we shall speak of as a range. It must be noticed that the content, with its objects, ranges, and extents, have no necessary connexion with space. It is only for the sake of studying the content with facility that we correlate its several objects with the points of space. 398. The Intervene. In ordinary space the most important function of the co-ordinates of a pair of points is that which expresses their distance apart. We desire to create that function of a pair of objects which shall be homologous with the distance function of a pair of points in ordinary space. The nature of this function is to be determined solely by the attributes which we desire it to possess. We shall take the most fundamental pro- perties of distance in ordinary space. We shall then re-enunciate these properties in generalized language, and show how they suffice to determine a particular function of a pair of objects. This we shall call the Intervene between the Two Objects. Let P, Q, R be three collinear points in ordinary space, Q lying between the other two; then we have, of course, as a primary notion of distance, PQ + QR = PR. In general, the distance between two points is not zero, unless the points are coincident. An exception arises when the straight line joining the points passes through either of the two circular points at infinity. In this case, however, the distance between every pair of points on the straight line is zero. These statements involve the second property of distance. In ordinary geometry we find on every straight line one point which is at an infinite distance from every other point on the line. We call this the point at infinity. Sound geometry teaches us that this single point is properly to be regarded as a pair of points brought into coincidence by the assumptions made in Euclid’s doctrine of parallelism. The existence of a pair of infinite points on a straight line is the third property which, by suitable generalization, will determine an important feature in the range. Ihe fourth property of ordinary space is that which asserts that a point at infinity on a straight line is also at infinity on every other straight line passing through it. This obvious property is equivalent to a significant law of intervene which is vital in the theory. If we might venture to enunciate it in an epigrammatic fashion, we would say that there is no short cut to infinity. i .'4