i86 LIMITS AND FLUXIONS 



gives an account of his Treatise of Fluxions. On 

 p. 330 of these Transactions it is pointed out that 

 "the Theory of Motion is rendered applicable to 

 this Doctrine with the greatest Evidence, without 

 supposing Quantities infinitely little or having 

 recourse to prime or ultimate Ratios." Again 

 (P- 336): *'There is, however, no Necessity for 

 considering Magnitude as made up of an infinite 

 Number of small Parts ; it is sufficient, that no 

 Quantity can be supposed to be so small, but it 

 may be conceived to be diminished further ; and it 

 is obvious, that we are not to estimate the Number 

 of Parts that may be conceived in a given Magni- 

 tude, by those which in particular determinate 

 Circumstances may be actually perceived in it by 

 Sense ; since a greater Number of Parts become 

 visible in it by varying the Circumstances in which 

 it is perceived." Of importance is the lollowing 

 (p. 336): " We shall therefore observe only, that 

 after giving some plain and obvious Instances, 

 wherein a Quantity is always increasing, and yet 

 never amounts to a certain finite Magnitude (as, 

 while the Tangent increases the Are increases but 

 never amounts to a Ouadrant)." That a variable 

 need not reach its limit is also emphasised in other 

 passages, as for instance (pp. 337, 338): '* In like 

 manner a curvilineal Area . . . may increase, while 

 the base is produced, and approach continually to a 

 certain finite Space, but never amount to it. . . . 

 A Spirai may in like manner approach to a Point 

 continually, and yet in any Number of Revolutions 



