PRINTED BOOKS, ETC, BEFORE 1734 43 



of Indeterminate or variable Quantities." He 

 cautions the reader : " But we must take great 

 heed, not to consider the Fluxions, or Increments, 

 or Decrements as finite Quantities " (p. 4). He 

 rejects xèy and xèy ''as being incomparably less " 

 than xzy. 



The same year in which Hayes wrote this first 

 English hook on fluxions which could make any 

 claim to attention, saw the appearance of Newton's 

 Quadratuì-a Curvarum. The contrast in the defini- 

 tion of " fluxion " was sharp. Hayes called it " an 

 infinitely small increment " ; Newton called it a 

 "velocity, " a finite quantity. 



62. William Jones, in his Synopsis P almariot-uni 

 Matheseos, London, 1706, devotes a few pages to 

 fluxions and fluents, using the Newtonian notation. 

 On p. 225 he gives, in substance, Newton's lemma, 

 in these words : "Quantities, as also their Ratio's, 

 that continually tend to an Equality, and therefore 

 that approach nearer the one to the other, than 

 any Difìference that can possibly be assign'd, do 

 at last become equal." Then he says : "Hence ali 

 Curved Lines may be considered as composed of 

 an Infinite Number of Infinitely little right Lines." 

 He uses "infinitely small" quantities, but defines 

 a fluxion as "the Celerity of the Motion," fluxions 

 being "in the first Ratio of their Nascent Aug- 

 ments." Jones represents bere the Newton of the 

 Principia, and of the Quadrature of Curves as given 

 in 1793. 



63. The earliest hook exhibiting a careful study 



