42 ILLINOIS ACADEMY OF SCIENCE 



ample, it is assumed that if we divide a distance-interval of a 

 moving point by the corresponding time-interval, giving an 

 average velocity, that in all actual motions such a ratio must, 

 as the intervals are decreased, have a limiting value called the 

 instantaneous velocity. So long as all phenomena of motion 

 were supposed to be continuous this seemed to be a necessary 

 assumption. But we know now that such a thing as instan- 

 taneous velocity may not exist at all. For Weierstrass in- 

 vented a continuous function which does not have a derivative 

 anywhere, and since his time a great many others have been 

 invented. There is no reason at all why change of position 

 may not take place continuously and yet with no definable ve- 

 locity anywhere, although there could be an average velocity 

 over any interval which for a smaller interval might run up as 

 near oo as we like. If we remember the fact that a point which 

 changes its position in time may assume two given values of x, 

 and during the moment between two given instants of time, 

 may assume every value between x x and x 2 , and do this discon- 

 tinuously, and the fact that it may do the same thing continu- 

 ously and yet with no definable velocity at each intermediate 

 instant of time, we certainly offer the scientists a chance to ac- 

 cept even radical modern atomistic conceptions of matter, elec- 

 tricity and energy, and yet not pass outside the range of the de- 

 finable function. If he were to do this, however, he would be 

 dependent entirely upon the mathematician for the development 

 of such laws and the study of their consequences. That the 

 phenomena of nature do not actually take place in this way we 

 have no right to assert. While we have deduced many laws on 

 the hypothesis of differentiability as a basis, we must recognize 

 that it is not a necessary hypothesis. We may assert confi- 

 dently that, for example, there may be motions in which there 

 is a change of place, a definable velocity, and yet no definable 

 acceleration, even though there may be a field of force in which 

 the particle moves. What becomes of Newton's second law in 

 such case? Again the ordinary laws of dynamics are based 

 upon the assumption that there are practically no shocks or 

 collisions, but if we suppose that there is a relatively very large 

 number of collisions or an infinite number, then the solutions 

 of the actual dynamic laws would have to be functions with a 

 relatively dense set of discontinuities, or even infinitely dis- 

 continuous functions. That such are possible every mathema- 

 tician knows. 



