THE HISTORY OF MATHEMATICS 413 



Morley which showed that the velocity of light was apparently inde- 

 pendent of the velocity of the medium in which it travelled, and obser- 

 vations on the motions of certain particles with very high velocities, 

 started a reconstruction of ideas. It was possible to explain the re- 

 sults on the assumption that the length of a body depended on its 

 velocity. It was then that Einstein sought to generalize Newton's equa- 

 tions of motion by making them entirely relative, not only for uniform 

 velocity, but also for accelerated motion. By adding the assumption 

 that the laws of nature should refer to all such systems of reference 

 and by making the velocity of light a fundamental constant of nature, 

 he was finally able to generalize the whole subject. Matter appears 

 simply as a form of energy. Gravitation can be exhibited as due to a 

 warping of space without the introduction of force, but if this is so, 

 the Newtonian law requires a minute correction. The motion of the 

 perihelion of Mercury and the bending of a light ray as it passes near 

 the sun have given remarkable confirmation of this theory. His work 

 crosscuts several subjects which previously had an interest only for 

 the pure mathematician, in particular the theory of extensible vectors 

 (tensors) and the theory of invariants. His differential equations for the 

 gravitational field should supply mathematicians with problems of 

 great difficulty and interest for some time to come. Those for the elec- 

 tric field are unchanged. A further interesting product of this work 

 on relativity lies in the question of what we can or cannot observe and 

 in what may be deduced from observation without the assumption of 

 hypotheses. This is, of course, fundamental in all experiments, but it 

 has received little attention as an exact science. Given that we can only 

 observe certain properties of a function, what limitations is it possible 

 to make in the construction of the function? 



Finally the quantum theory of Planck, according to which energy 

 is not infinitely divisible but is always received or emitted in exact 

 multiples of a fundamental unit, is bringing forward the necessity for 

 a calculus allied to that of finite differences as against the differential 

 and integral calculus which depends in general on continuity. It is 

 even suggested that not only energy, but also space and time have 

 ultimate parts which cannot be divided. At present the mechanics of 

 this theory is in a very nebulous state but as a statement of the results 

 of observation it has had very considerable success. The construction 

 of the atom is now generally exhibited as a kind of minute solar system, 

 but there is as yet no indication how such a system can only permit of 

 the limited number of motions required by the quantum hypothesis. 

 It may perhaps be due to fundamental instabilities for we know little 

 of the ultimate stability of most of the motions in the problems of 

 even three particles. In any case, the field of work has approached one 

 of the oldest of mechanical problems and the reaction of celestial me- 

 chanics on that of the atom should prove stimulating to both. 



