718 Fundamental Principle of the Higher Calculus 
“Tn 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 ; 
which 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 
true 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 two quantities tend continually to equality, and approach to 
one another nearer than by any assignable difference, their Ratio is 
ULTIMATELY a Ratio of assouute equality. Let L, L/ denote the 
Limits, whatever they are, towards which the quantities L+1, L/+-V 
continually converge, and suppose their difference, in any state of the 
convergence, tobe D. Then 
or L-—-L/+1—r —~pD=06 
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 
L 
L—L/=0, or [Tv =!. accurately. Q. e. d. 
