78 Fundamental Principle of the Higher Calculus 



" In the Scholium at the end of this Section he is more explicit. 

 He says, The ultimate Ratios, in which quantities vanish, are not in 

 reality the Ratios of Ultimate quantities; but the Limits to which 

 the Ratios of quantities continually decreasing always approach ; 

 ivhich they never can pass beyond or arrive at, unless the quantities 

 are continually and indefinitely diminished. After all, however, 

 neither our author nor any of his commentators, though much has 

 been advanced upon the subject, has obviated this objection. Bishop 

 Berkeley's ingenious criticisms in the Analyst remain to this day un- 

 answered. He therein facetiously denominates the results, obtained 

 from the supposition that the quantities, before considered finite and 

 real, have vanished, the " Ghosts of Departed Quantities;" and it 

 must be admitted there is reason as well as wit in the appellation. 

 The fact is, Newton himself, if we may judge from his own words 

 in the above cited Scholium, where he says, "If two quantities, 

 whose DIFFERENCE IS GIVEN are augmented continually, their Ulti- 

 mate Ratio will be a Ratio of Equality," had no knowledge of the 

 t7-ue nature of his Method of Prime and Ultimate Ratios. If there 

 be meaning in words, he plainly supposes in this passage, a mere 

 approximation to be the same with an Ultimate Ratio. He loses 

 sight of the condition expressed in Lemma I. namely, that the quan- 

 tities tend to equality nearer than by any assignable difference, by 

 supposing the difference of the quantities continually augmented to 

 be given, or always the same. In this sense the whole earth, com- 

 pared with the whole earth minus a grain of sand, would constitute 

 an Ultimate Ratio of equality ; whereas so long as any, the minutest 

 difference exists between two quantities, they cannot be said to be 

 more than nearly equal. But it is now to be shown, that 



"if tivo quantities tend continually to equality, and approach to 

 one another nearer than by any assignable difference, their Ratio is 

 ULTIMATELY « Ratio of ABSOLUTE equality. Let L, U denote the 

 Limits, whatever they are, towards which the quantities L + 1, L'+l' 

 continually converge, and suppose their difference, in any state of the 

 convergence, to be D. Then 



L + 1 _ 17 - 1' == D, 

 or L - L' + 1 - 1' - D -= 0, 

 and since L, L' are fixed and definite, and 1, ]', D always variable, 

 the former are independent of the latter, and we have by the funda- 

 mental principle 



li — L' = 0, or ^ =1, accurately. Q. e. d. 



