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the elements to receive its dynamical explanation by means of an analysis of the 

 detailed possibilities of relatively stable types of motion, the general schematic 

 character of which will have been indicated by the physicist? It is undoubtedly 

 true that the cleft between Pure Mathematics and Physical Science is at the present 

 time wider than formerly. That is, however, a result of the natural development, 

 on their own lines, of both subjects. In the classical period of the eighteenth 

 century, the time of Lagrange and Laplace, the nature of the physical investiga- 

 tions, consisting largely of the detailed working out of problems of gravitational 

 Astronomy in accordance with Newton's law, was such that the passage was 

 easy from the concrete problems to the corresponding abstract mathematical 

 ones. Later on, mathematical physicists were much occupied with problems 

 which lent themselves readily to treatment by means of continuous analysis. In 

 our own time the effect of recent developments of Physics has been to present 

 problems of molecular and sub-molecular Mechanics to which continuous analysis 

 is not at least directly applicable, and can only be made applicable by a process 

 of averaging the effects of great swarms of discrete entities. The speculative 

 and incomplete character of our conceptions of the structure of the objects of 

 investigation has made the applications of Dynamics to their detailed elucidation 

 tentative and partial. The generalised dynamical scheme developed by 

 Lagrange and Hamilton, with its power of dealing with systems, the detailed 

 structure of which is partially unknown, has however proved a powerful weapon 

 of attack, and affords a striking instance of the deep-rooted significance of 

 mathematical form. The wonderful and perhaps unprecedentedly rapid dis- 

 coveries in Physics which have been made in the last two decades have given 

 rise to many questions which are as yet hardly sufficiently definite in form to be 

 ripe for mathematical treatment; a necessary condition of which treatment con- 

 sists in a certain kind of precision in the data of the problems to be solved. 



The difficulty of obtaining an adequate notion of the general scope and aims 

 of Mathematics, or even of special branches of it, is perhaps greater than in the 

 case of any other science. Many persons, even such as have made a serious and 

 prolonged study of the subject, feel the difficulty of seeing the wood for trees. 

 The severe demands made upon students by the labour of acquiring a difficult 

 technique largely accounts for this ; but teachers might do much to facilitate the 

 attainment of a wider outlook by directing the attention of their students to the 

 more general and less technical aspects of the various parts of the subject, and 

 especially by the introduction into the courses of instruction of more of the 

 historical element than has hitherto been usual. 



All attempts to characterise the domain of Mathematics by means of a formal 

 definition which shall not only be complete, but which shall also rigidly mark 

 off that domain from the adjacent provinces of Formal Logic on the one side and 

 of Physical Science on the other side, are almost certain to meet with but doubt- 

 ful success ; such success as they may attain will probably be only transient, in 

 view of the power which the science has always shown of constantly extending 

 its borders in unforeseen directions. Such definitions, many of which have been 

 advanced, are apt to err by excess or defect, and often contain distinct traces 

 of the personal predilections of those who formulate them. There was a time 

 when it would have been a tolerably sufficient description of Pure Mathematics 

 to say that its subject-matter consisted of magnitude and geometrical form. 

 Such a description of it would be wholly inadequate at the present day. Some 

 of the most important branches of modern Mathematics, such as the theory of 

 groups, and Universal Algebra, are concerned, in their abstract forms, neither 

 with magnitude nor with number, nor with geometrical form. That great 

 modern development, Projective Geometry, has been so formulated as to be 

 independent of all metric considerations. Indeed the tendency of mathema- 

 ticians under the influence of the movement known as the Arithmetisation of 

 Analysis, a movement which has become a dominant one in the last few decades, 

 is to banish altogether the notion of measurable quantity as a conception neces- 

 sary to Pure Mathematics ; Number, in the extended meaning it has attained, 

 taking its place. Measurement is regarded as one of the applications, but as no 

 part of the basis, of mathematical analysis. Perhaps the least inadequate 

 description of the general scope of modern Pure Mathematics — I will not call it 

 a definition — would be to say that it deals with form, in a very general sense of 



