642 SCIENCE PROGRESS 



discrete number, while the science of quantity is founded on the totally- 

 different hypothesis of continuity, is an example of the question we are now 

 considering. This involves an important point, for physical science is con- 

 cerned with quantity, and this condition necessitates the use of infinitesimals 

 in the geometry of the world. 



Then, again, if all the points of space are to play the same rdle, that space 

 must be homogeneous ; if all the directions from any one point are to play 

 the same role, that space must be isotropic. If the space is to be homogeneous 

 it must be without limits, for if it were finite in extent it could not be homo- 

 geneous, since its boundary could not play the same part as the centre in the 

 scheme of things. By this we do not mean that space is necessarily infinite ; 

 this apparent paradox may be overcome by regarding the sphere as having a 

 surface without boundary, yet finite in extent. 



In the past it has been customary to look upon force as the great fact 

 that lies at the bottom of all things ; but this is far from the final act. Later, 

 the conception of force disappears, and whatever happens is regarded as the 

 fact to be observed. It will be evident from this examination of the point- 

 events of nature that an infinitesimal geometry is required to explore the 

 properties of space ; this attitude now brings us to a new and all-important 

 view-point of phenomena. The hypothesis of continuity involves such an 

 interdependence of the facts of the universe as forbids us to speak of one 

 fact, or a group of facts, as the cause of another fact, or group of facts. From 

 this hypothesis of space-time a knowledge of the whole history of a single 

 particle is shown to be involved in a complete knowledge of its state at any 

 moment. While dealing with this point, the reader's attention might be 

 drawn to a statement in the admirable book of Viscount Haldane, The 

 Reign of Relativity, p. 114 (3rd edition), where the author states that the 

 " laity have been taught that infinitesimals are now banished out of mathe- 

 matics, except as symbols for limiting relations of order in quantity." The 

 raison d'etre of the infinitesimal geometry hes deeper than would at first appear 

 from this quotation, in so far as the element of length, ds, implies that a process 

 involving the average value underlies the consequent mathematical treatment 

 of the event under consideration. Mathematically stated, Taylor's Theorem 

 gives the entire past and future history of the movement of a point-event, 

 provided that there is no infinite change in the derivatives involved. 



As it is easy to show that no velocity can exceed that of light, so long as the 

 facts of nature are recognised in their true sequence, it may be reasonably 

 asked, What would happen in the case of two particles passing each other with 

 a relative velocity greater than that of light ? The answer to this query, in the 

 light of the new theory of time and space, would be that the presence of the two 

 particles would alter the neighbouring space in such a way as to make the 

 relative velocity in four-dimensional space less than that of Ught. This explana- 

 tion is very useful in interpreting the variation of the mass of high-velocity 

 particles, and it also suggests that gravitational energy has mass. 



Thus it is seen that Einstein's Theory makes for a great unification in 

 mechanics, for it brings together inertia and gravitation — further, it does more, 

 it identifies these two forms of forces as belonging to one common force, and 

 not, as heretofore, calling them opposing forces. Henceforth, a body travell- 

 ing through a frictionless medium has a non-uniform motion, since its path 

 in space is parabolic in form ; to describe a uniform motion a suitable mesh 

 system of the space-time continuum would have to be used. 



The tendency of a moving body is to foUow a natural track, or geodesic, 

 and in the presence of matter a force will be called into play which acts on the 

 body in question ; this force is composed of gravitation and inertia elements. 

 In other words, a gravitational field necessitates a curved, or grooved, con- 

 tinuum, and this consequent groove in space can be shown to be of the first 

 order of curvature. From the earlier remarks, it is now clear that this dis- 



