408 THE SCIENTIFIC MONTHLY 



in which the human brain can think. But without going into this ques- 

 tion it is possible to indicate the general method followed at the pres- 

 ent time. At the outset a set of statements, usually called axioms or 

 postulates (there is a difference of opinion as to the exact meaning to 

 be attached to these words), is made. These statements may be re- 

 dundant but must not be contradictory : a complete system is one which 

 contains all the statements necessary for the object in view but which 

 has nothing unnecessary or redundant so that no statement or combi- 

 nation of statements can result in another of the set. To connect and 

 deduce, a system of reasoning is also required. From this it will be 

 seen that mathematical science has no necessary relation to natural 

 phenomena, but that it can be regarded as solely a product of the brain 

 and that its results are simply consequences which may be deduced 

 from ideas without external assistance. The last half century has seen 

 great progress made in clarifying our ideas and in the introduction of 

 rigorous methods of argument. It has now extended to the phenomena 

 of nature, particularly in the direction of a reconsideration of our 

 ideas of time and space and in an examination of our powers of ob- 

 servation, as will be illustrated below. 



The idea of invariants has permeated every branch of pure and 

 applied mathematics. In its elementary forms it is not difficult to un- 

 derstand. Natural processes are subject to change but we can nearly 

 always find certain features of them which seem to remain the same 

 in the conditions under which we observe them, as for instance, mass 

 and energy. In geometry, the ratio of the circumference of a circle to 

 its diameter is constant, the ratio of the sections of any system of 

 straight lines cut by three parallel lines is the same, certain properties 

 of a system of curves remain unchanged under given conditions of de- 

 formation, and so on. In analysis, one of the commonest modes of in- 

 vestigation is to find out what expressions remain unchanged when a 

 specified change is made in the symbols. It has been said that all 

 physical investigations are fundamentally a search for the invariants of 

 nature. The various terms which occur in modern algebra, such as 

 discriminant, Jacobian, covariant, are special forms of the same funda- 

 mental idea. 



The idea of permutations and combinations which few of us fail 

 to meet with in our every day experience, was chiefly developed in 

 earlier times from the point of view of the number of arrangements 

 which could be made of a set of objects under certain specified con- 

 ditions. The modern theory of groups is the natural successor of this 

 subject, but as has so often happened, the point of view and its develop- 

 ment have changed. If we have a set of symbols and replace one by 

 another according to a specified law, we can consider what changes 

 will leave unchanged certain combinations of those symbols, as well as 



