62 LIMITS AND FLUXIONS 



8i. Berkeley proceeds to show that correct results 

 are derived from false principles by a cornpensation 

 of errors^ a view advanced again later by others, 

 particularly by the French critic L. N. M. Carnet. 

 Taking y'^—px, Berkeley says that the subtangent 

 is not ydx j dy \i dy is the true increment of y 

 corresponding to dx ; the accurate subtangent, 

 obtained by similar triangles, \sydx / {dy-{-.z), where 

 z — dydy / (2j/). That is, if dy is the true increment, 

 then in ydx I dy there is an "error of defect. " But 

 in ydx / dy, as used in the differential calculus, the 

 dy is not its true value, viz. dy=pdx / (2y) — 

 dydy I (2y) (obtained by writing x-\-dx for x and 

 y + dy for j, in the equation r^=/;i'), but its erroneous 

 value, pdx I {2y). There is bere an " error of 

 excess." " Therefore the two errors being equal 

 and contrary destroy each other (§ 21); . . . by 

 virtue of a twofold mistake you arrive, though not 

 at science, yet at truth." Berkeley gives other 

 illustrations of cases where " one error is redressed 

 by another. " 



82. "A point may be the limit of a line : a line 

 may be the limit of a surface : a moment may 

 terminate time. But how can we conceive a velocity 

 by help of such limits ? It necessarily implies both 

 time and space, and cannot be conceived without 

 them. And if the velocities of nascent and evan- 

 escent quantities, i.e. abstracted from time and 

 space, may not be comprehended, how can we 

 comprehend and demonstrate their proportions ; or 

 consider their rafiones prima^ and ultimceì For, to 



