BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 299 



And what are these fluxions? The velocities of evanescent in- 

 crements. And what are these same evanescent increments? They 

 are neither finite quantities, nor quantities infinitely small, nor yet 

 nothing. May we not call them ghosts of departed quantities ? 



In regard to the controversy between the friends of Newton 

 and those of Leibnitz as to priority in the invention of the calculus, 

 Newton himself says in a celebrated scholium : 



The correspondence which took place about ten years ago, be- 

 tween that very skilful geometer G. G. Leibnitz and myself, when I 

 had announced to him that I possessed a method of determining 

 maxima and minima, of drawing tangents, and of performing similar 

 operations, which was equally applicable to surds and to rational 

 quantities, and concealed the same in transposed letters, involving 

 this sentence, (Data Aequatione quotcunque Fluentes quantitates in- 

 volvente, Fluxiones invenire, et vice versa), this illustrious man re- 

 plied that he also had fallen on a method of the same kind, and 

 he communicated his method, which scarcely differed from my own, 

 except in the forms of words and notation (and in the idea of the 

 generation of quantities). 



Of the controversy as a whole Newton's biographer Brewster 

 remarks : 



The greatest mathematicians of the age took the field, and states- 

 men and princes contributed an auxiliary force to the settlement of 

 questions upon which, after the lapse of nearly 200 years, a verdict 

 has not yet been pronounced. 



Although the honour of having invented the calculus of fluxions, 

 or the differential calculus, has been conferred upon Newton and 

 Leibnitz, yet, as in every other great invention, they were but the 

 individuals who combined the scattered lights of their predecessors, 

 and gave a method, a notation, and a name to the doctrine of quanti- 

 ties infinitely small. 



In studying this controversy, after the lapse of nearly a century 

 and a half, when personal feelings have been extinguished, and national 

 jealousies allayed, it is not difficult, we think, to form a correct esti- 

 mate of the claims of the two rival analysts, and of the spirit and 

 temper with which they were maintained. The following are the 

 results at which we have arrived : 



