NEWTON 33 



used the infinitely small increment as a means of 

 determining it ; while, with Leibnitz, the relation 

 of the infinitely small increments is itself the object 

 of determination. " 



(2) As early as 1665, Newton speaks of describing 

 an ''infinitely little line" in " one moment," and 

 then uses the expression "in the next " moment.^ 

 Here ** moment" cannot mean a point of time, 

 destitute of duration ; it means an infinitely small 

 duration, an infinitesimal of time. Doubtless this 

 use of "moment" with reference to time suggested 

 the more extended and general use of the term 

 " momentum " or " momenta " as found in the 

 Principia ^ and later publications. 



(3) The use of dots, " prickt letters," to indicate 

 velocities or fluxions goes back to 1665,^ but they 

 are not used by Newton in print until 1693 ^" 

 Wallis's Algebra ; they are used extensively in 

 Newton's Quadrature of Curves of 1704.* 



(4) Newton first used the word " fluxion " in 

 print in 1687 in the Principiai 



(5) The first refinement of the doctrine of fluxions 

 is found in Newton's Principia, where he speaks of 

 "prime and ultimate ratios"^ and of "limits. "^ 



(6) The high-water mark of Newton's efforts to 

 place the doctrine of fluxions upon a thoroughly 

 logicai basis is found in his Quadrature of Curves, 

 1704. It indicates the almost complete exclusion 



^ See our § 48. - See our §§ 16, 18, 21, 24. 



^ See our § 48. * See our §§ 22, 25, 44, 45. 



•"' See our §§ 16, 18. « See our §§ io, 13. 



' See our §§ 4, 6, 8, io, 13. 



