RELATIONS OF APPLIED MATHEMATICS 593 



error, and in such cases the explanations of the errors follow soon, 

 and they are not of great influence on the structure of the science as 

 a whole. 



It is, indeed, strongly emphasized that every established and 

 logically recognized truth must remain incontrovertible. Although 

 this cannot be doubted, experience teaches that the structure of our 

 theories is by no means composed entirely of such incontrovertibly 

 established truths. They are composed rather of many arbitrary 

 pictures of the connections between phenomena, of so-called hypo- 

 theses. 



Without some departure, however slight, from direct observation, 

 a theory or even an intelligibly connected practical description for 

 predicting the facts of nature cannot exist. This is equally true of 

 the old theories whose foundations have become questionable, and 

 of the most modern ones, which are resigning themselves to a great 

 illusion if they regard themselves as free from hypotheses. 



The hypotheses may perhaps be indefinite, or may be in the shape 

 of mathematical formulae, or the thought may be equivalent to the 

 latter, but expressed in words. In the latter cases the agreement 

 with given data may be checked step by step; a complete revolu- 

 tion of that previously constructed is indeed not absolutely impos- 

 sible, as, for example, if the law of the conservation of energy should 

 turn out to be incorrect. But such a revolution will be exceedingly 

 rare and highly improbable. 



Such an indefinite, slightly specialized theory might serve as a 

 guiding thread for experiments whose purpose is a detailed develop- 

 ment of knowledge previously acquired and which is proceeding in 

 barren channels ; beyond this its usefulness does not reach. 



In contradistinction to these are the hypotheses which give the 

 imagination room for play and by boldly going beyond the material 

 at hand afford continual inspiration for new experiments, and are 

 thus pathfinders for the most unexpected discoveries. Such a theory 

 will indeed be subject to change, a very complicated mass of inform- 

 ation will be brought together and will then be replaced by a new 

 and more comprehensive theory in which the old one will be the pic- 

 ture of a limited type of phenomena. Examples of this are the theory 

 of emission in regard to the description of the phenomena of catoptrics 

 and dioptrics, the hypothesis of an elastic ether in the representation 

 of the phenomena of interference and refraction of light, and the 

 notion of the electric fluid in the description of the phenomena of 

 electrostatics. 



Moreover the theories which proudly designate themselves as free 

 from hypotheses are not exempt from great revolutions; thus, no one 

 will doubt that the so-called theory of energy will have completely 

 to alter its form if it desires to remain effective. 



