126 LIMITS AND FLUXIONS 



on the ground that the continued curve *'is not to 

 be described, but by an endless number of para- 

 bola's " (of which the curve is the envelope) ; thus, 

 Philalethes gave *'as an equation expressing the 

 nature of a single curve, one which in reality 

 includes an infinite series." " Philalethes supposed 

 a last form of the inscribed figures, that was equal 

 to the curve." Robins observed "that equality 

 implies the things, which bave that property, to be 

 distinct from each other. For to say a thing is 

 equal to itself is certainly no proper expression." 

 But ''there is no such last form distinct from the 

 curve," as Philalethes admits ; hence Philalethes 

 **gives up the point. " 



136. In the Principia^ Newton does not deliver 

 the doctrine of fluxions, but the doctrine of prime 

 and ultimate ratios. ''The understanding of this 

 book is attended with difificulty. " The expression 

 ultima sumina is defective : **Can any sum of a set 

 of quantities, whose number is supposed infinite, in 

 strict propriety of speech be called their last sum ? " 

 Later, Robins says : *' Let Philalethes reconcile 

 the actual arrivai of these quantities to the ratio 

 supposed, and at the same instant vanishing away. 

 Is not this saying, that the two quantities become 

 nothing, and bear proportion at the same instant of 

 time ? " (p. (14)). Philalethes " has thought himself 

 unjustly accused by Mr. Robins of supposing a 

 nascent increment to be some intermediate state of 

 that increment between its finite magnitude, and 

 its being absolutely nothing. To bave proved this 



