238 



and correspondence, and he left many undemonstrated theorems, whose 

 proofs battled succeeding mathematicians for 50. 100, and even 200 years. 



The Quadrature of Curves, published in 1704, and the Principia, are 

 the proper sources for Newton's matured ideas on the calculus, and not 

 his earlier manuscript, published by Wallis. The earlier paper adopts 

 the infinitesimal method of neglecting small quantities which is now 

 associated with Leibnitz's calculus, not, however, with the latter's dis- 

 regard of logic, but in connection with the idea of a limit which is the 

 modern foundation of that method. 



Newton states in the Quadrature of Curves that "in mathematics the 

 minutest errors are not to be neglected." Also, 



"I consider mathematical quantities in this place, not as consisting 

 of very small parts, Init as described by continuous motion. Lines are 

 described and therel>y generated, not by the apposition of parts, but 

 by the continued motion of points; superficies by the motion of liues; 

 solids by the motion of superficies; angles by the rotation of sides; por- 

 tions of time by continual liux; and so on in other quantities. These 

 geneses really take place in the nature of things and are daily seen in 

 the motion of bodies." 



He then goes on to define fluxions, or as we would now call them, 

 differentials: 



"Fluxions are as near as we please, as the increments of fluents, gen- 

 erated in times which are the same and as small as possible, and to 

 speak accurately, they are in the prime ratio of nascent increments; 

 yet they can be expressed by any lines whatever which are proportional 

 to them." 



Newton immediately illustrates this definition by the abscissa and 

 ordinate of a curve, whose differentials are shown to be any correspond- 

 ing increments of abscissa and ordinate along the tangent line. This, 

 and numerous similar illustrations in the Principia, show that Newton 

 meant by the ultimate ratio of vanishing quantities, tJic limit of the 

 ratio of ail!/ finite proportioiiaJs to the viiiisliiiKj quantities. See, for ex- 

 ample, Princ. Bk. 1, Lemma 1, Art. 12. "Ultimate Ratio of Vanishing 

 Quantities." Also, Lemmas 7, 8, 9. Newton did not consider the modern 

 (piestion as to whether or not this ratio was definite, and the answer to 

 that question is not pertinent to his definition. In other words, differen- 

 tials can exist when such ratio is indeterminate. Translated into its 

 exact modern equivalent, his definition is: 



