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. 

 The 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. 



