36 LIMTTS AND FLUXIONS 



form : " But, by the same argument, it might as 

 well be maintained, that there is no ultimate velocity 

 of a body arriving at a certain place, when its motion 

 is ended : because the velocity, before the body 

 arrives at the place, is not its ultimate velocity ; 

 when it has arrived, is none. But the answer is 

 easy : for by the ultimate velocity is meant that 

 . . . at the very instant when it arrives." If 

 ''instant," as used here, is not an infinitesimal, the 

 passage would seem to be difficult or impossible of 

 interpretation. 



(7) A return to the open use of the infìnitely 

 small quantities is seen in writings of Newton after 

 the year 1704. It might be argued that such a 

 return was necessary in the second edition of the 

 Principia, 17 13, unless the work were largely re- 

 written. Newton's Anafysis per cequationes numero 

 terniinorum infinitas was first printed in 171 1, and 

 might bave been rewritten so as to exclude infini- 

 tesimals as fully as was done in the Quadrature of 

 Curves of 1704. But the infìnitely little is per- 

 mitted to remain.^ There is no disavowal of such 

 quantities either in the Commercium Epistolicum, 

 with the editors of which Newton was in touch, 

 or in Newton's own account of this publication, 

 contributed to the Pìiilosophical Transactions.^ 



(8) The theory of limits is involved in the first 

 lemma of the Principia,^ and in the explanation of 

 prime and ultimate ratios as given in that work. 



1 See our § 66. - See our § 47. 



3 See our §§ 4, 6, 8, 9, io, 12, 13, 15. 



