240 



independent increments. In his method, the principle quantities were the 

 differential coefficients, and if z were a function of x, y, he wrote 



, dz , , dz 



dz = ^r- dx -\- -=- dy 

 d.r ' dy ^ 



where dz/d.c was a whole symbol for the coefficient of dx in dz, and not the 



quotient of dz by dx; and similarly for dz/dy. 



This idea was not received with favor, partly because it made the cal- 

 culus depend upon expansions in series, whereas, one important feature of 

 the calculus was the determination of such expansions. 



At present, we have a derivative calculus, with a differential notation, 

 in which differentials have significance only in quotient forms; in fact the 

 derivative is Lagrange's differential coefficient, and the two terms are used 

 interchangeably. The student is taught that the quotient form is an in- 

 separable symbol, but the notation, and the calculus itself, eventually 

 require their separation. The explanations which have been devised for 

 such separation of inseparable symbols are sometimes remarkable. The 

 method of rates is simply to define the derivative dy dx as tlie rate at which 

 y is changing, and dy, dx, as any quantities whose ratio is dyulx. This is 

 not the same as Newton's metliod, who makes dy the amount which y 

 changes in its state of change when x changes by dx, and thence dy/dx is 

 the change of y per unit change of x. It does matter whether we make dif- 

 ferentials the prime quantities, and thence deduce the significance of their 

 ratios, or whetlier we make the ratios the prime quantities, and thence 

 deduce differentials. For, two variables can have differentials, with no 

 ratio tliat is definite, i. e., independent of the values of the differentials 

 themselves. 



In a calculus in which the derivative is the prime quantity, the differ- 

 ential notation creates numerous (irtificial difficulties which would be elim- 

 inated by a proper derivative notation; but this would limit the scope of 

 the calculus and alter many of its time-honored developments. Nor is it 

 necessary to make a change of notation, because tlie present notation is 

 made completely significant by Newton's definition. 



"When we consider the weight that attaches to the name of Newton, it 

 would seem that his views on the calculus were worthy of being considered, 

 even today. When we add tliat he is the original inventor, and that his 

 fundamental idea of the differential is the very one tliat is needed to give 

 the differential calculus an intelligent and rigorous matliematical basis, it 

 is certainlv time that he came into liis own. 



