470 



SCIENCE 



[N. S. Vol. XLV. No. 1168 



would require not less than a year's course 

 of lectures. We can not hope to do more 

 than indicate some of the central provi- 

 sions. 



Let us begin with a consideration of the 

 early stages in the development of particu- 

 lar sciences. Bach separate experimental 

 science passes through a period of infancy 

 in which it is not able to stand the strong 

 meat of mathematics; and mathematical 

 ideas initially find an application in it by 

 slow processes. The first essential is to 

 gather data — descriptive results, measured 

 quantities, or what not; and only after 

 much labor does law become apparent and 

 the mathematical tool acquire its character- 

 istic power. 



But even in these preliminary stages 

 mathematics makes an essential contribu- 

 tion in a preliminary way. It furnishes the 

 only language in which exact information 

 can be expressed, recorded and conveyed; 

 and in this respect is a necessary element of 

 that collaboration which is essential to such 

 rapid progress as has been usual in recent 

 years. But it does more than this. It en- 

 ables us to record observations in such a 

 way that we are able readily to grasp the re- 

 lations of the various measured elements in- 

 volved. I refer here particularly to the use 

 of graphs which present data in so compact 

 a manner and in a way so well adapted to 

 our intuitive realization of their signifi- 

 cance. 



It would now probably be impossible to 

 lay the foundations of any new experimen- 

 tal science without the collection of much 

 numerical data, that is to say, without the 

 use of statistics. But how are these to be 

 interpreted? Clearly it must be by the 

 methods of statistical mathematics. 



Let us suppose now that we have made a 

 record of our measurements of phenomena, 

 their juxtapositions, their magnitudes, their 

 order in time; let us assume (as we always 

 do and must) that they are connected by 



law. How shall we ascertain what that law 

 is ? By what criterion shall we judge of the 

 accuracy of our hypothetical explanation? 

 Certainly not on any absolute grounds ; we 

 can only select the explanation which seems 

 to us most probable. And for this our best 

 and surest guide is and must be the mathe- 

 matical theory of probability. 



A science in the stage now being exam- 

 ined would properly be called non-mathe- 

 matical, notwithstanding the preliminary 

 use which it makes of mathematical science. 

 Among those divisions of systematic 

 thought which are at present in this stage 

 of development one would probably in- 

 clude political science, economics, biology, 

 psychology and geology. 



At the nest stage of development one 

 would think of the individual science as 

 having come to the period of vigorous youth 

 but not yet mature. As preeminently the 

 example of such a science I would select 

 chemistry. By no means is it reduced es- 

 sentially to mathematical form ; and yet its 

 laws are so stated as to be subject to the 

 sharpest experimental verification. It em- 

 ploys the mathematical tools which are 

 common in the earlier stages of science and 

 also some additional ones. It may even em- 

 ploy the first derivative as a measure of 

 rate of the reactions which it considers ; but 

 I believe that it seldom or never makes use 

 of the second derivative. 



It may be taken as a mark of the more 

 advanced development of physics that it 

 finds constant use for derivatives of the 

 second order and sometimes for those of 

 higher orders. Some one has expressed 

 this increasing order of complexity and cer- 

 tain accompanying dependencies by saying 

 that behind the artisan is the chemist, be- 

 hind the chemist is the physicist, and be- 

 hind the physicist is the mathematician — a 

 pleasing climax, at least if you are a mathe- 

 matician. 



"When we look upon physics, for instance, 



