July 9, 1920] 



SCIENCE 



35 



from teacliing fellow to associate in medical 

 entomology. 



Dr. Louis J. Gillespie, professor of physical 

 chemistry at Syracuse University, who was 

 formerly with the Department of Agriculture, 

 Washington, D. C, has resigned to go to the 

 Massachusetts Institute of Technology as as- 

 sistant professor of physico-chemical research. 



Dr. Arthur F. Buddington, Ph.D. (Prince- 

 ton, '16), and Dr. Benjamin F. Howell, Ph.D. 

 (Princeton, '20), have been appointed assist- 

 ant professors of geology at Princeton Univer- 

 sity. 



DISCUSSION AND CORRESPONDENCE 



MODERN INTERPRETATION OF DIFFER- 

 ENTIALS 



In an advance copy of a note to Science, 

 which Professor Huntington has kindly sent 

 to me, he says that " some indication as to the 

 manner in which N is to vary " is necessary 

 to define dy = lim NAy. This is not true. 

 Of course, there must be some relation be- 

 tween N and Ay, in order that, for example, 

 lim NAy = 5, but the number of such rela- 

 tions is infinite, and it is only necessary to 

 know that they exist. For example, if Ay=- 

 (5/N) ■+ (.8/N^), then NAy = 5 -f (8/iV), and 

 for lim N = oa, lim Ay = 0, lim NAy = 5. It 

 was stated in my note which Professor Hunt- 

 ington is criticizing^ that N varies from zero 

 to infinity. We are not concerned with the 

 method of approach, but only with the possible 

 value of the limit. The preceding illustration 

 shows that if y be an independent variable, 

 such limit dy exists, and in any value we 

 please to name. It is difFerent if y be depend- 

 ent, and my note in Science of May 7, con- 

 tained a demonstration that df(x) exists when 

 the graph of /(x) has a tangent, and deter- 

 mines its construction, corresponding to any 

 value of dx, including in particular, dx = Ax, 

 which is, of course, not always true. 



The problem of diilerentiation is larger 

 than that of a single value, since it determines 

 an infinite number of corresponding values. 

 We have the analogy of the infinite number of 

 corresponding values of the derivative variable 



1 Science, February 13. 



and its argument x. We justify this variable 

 as a limit on the ground that it is a true limit 

 for each numerical value of x. The example 

 having been set, its extension to differentials 

 can not he denied. 



The infinite number of corresponding differ- 

 entials (dx, dy, dz) pertain to the one set of 

 corresponding variables (x, y, z), just as the 

 increments (Ax, Ay, Az) pertain to it, and are 

 corresponding increments of the instantaneous 

 state of the variables, also, increments in the 

 first ratio (Newton's "prime" ratio), etc. 

 This is not a vague idea but one which, in 

 numerical cases, determines numerical values. 

 The source of this terminology is the physical 

 idea that equimultiples of very small simulta- 

 neous increments are approximately incre- 

 ments of the instantaneous state. The differ- 

 ential analysis of Newton, which carries this 

 idea to its logical conclusion, is therefore the 

 mathematical foundation for such physical 

 idea. 



It is easy to make statements appear vague 

 by separating them from the facts on which 

 they are based, and such facts appear in the 

 article from which Professor Huntington 

 quotes, with a figure showing the finite equi- 

 multiples which are becoming exact differ- 

 entials — differentials which his " modern " 

 method can not represent, since they pertain 

 to a system of two independent variables, and 

 of which the derivative calculus can give no 

 adequate idea, although they are of great 

 practical importance. 



Such so-called modern method is crude in 

 its limitation dx = Ax, narrow in its applica- 

 tion only to plane curves in rectangular 

 coordinates. A natural extension to space is 

 impossible, but Newtonian differentials are 

 coordinates of tangent planes, from their 

 points of contact as origin. By Newton's 

 method, all kinds of continuously variable 

 quantity, in plane or space, lines, areas, 

 volumes, forces, may have corresponding 

 differentials represented in finite quantities 

 of the same kind, and by the limits of finite 

 and visible values. 



ARTHtm S. Hathaway 



EosB Polytechnic Institute 



