- BRANCHES AND ROOTS OF THE CALCULUS 485
The foreign mathematicians are supposed by some, even of our
own, to proceed in a manner less accurate, perhaps, and geomet-
rical, yet more intelligible... Now to conceive a quantity infinitely
small, that is, infinitely less than any sensible or imaginable quan-
tity or than any the least finite magnitude is, I confess, above my
capacity. But to conceive a part of such infinitely small quantity
that shall be still infinitely less than it, and consequently though
multiplied infinitely shall never equal the minutest finite quantity,
is, I suspect, an infinite difficulty to any man whatsoever.Berkeley analyzed a curve whose area up to ÷ was x^3 (he wrote xxx). If æ - ÷
was the increment of the abscissa and z^3 — x^3 the increment of area, the quotient
would be z^2 + zx + x^2. He said that, if æ = ÷, of course this last expression is 3x^2 ,
and that must be the ordinate of the curve in question. That is, its equation must
be y = 3x^2. But, he pointed out,
[Hjerein is a direct fallacy: for, in the first place, it is supposed
that the abscisses æ and ÷ are unequal, without which supposi-
tion no one step could have been made [that is, the division by
æ — ÷ would have been undefined]; which is a manifest incon-
sistency, and amounts to the same thing that hath been before
considered... The great author of the method of fluxions felt this
difficulty, and therefore he gave in to those nice abstractions and
geometrical metaphysics without which he saw nothing could be
done on the received principles... It must, indeed, be acknowledged
that he used fluxions, like the scaffold of a building, as things to
be laid aside or got rid of as soon as finite lines were found pro-
portional to them... And what are these fluxions? The velocities
of evanescent increments? And what are these same evanescent
increments? They are neither finite quantities, nor quantities in-
finitely small, nor yet nothing. May we not call them the ghosts
of departed quantities?The debate on the Continent. Calculus disturbed the metaphysical assumptions of
philosophers and mathematicians on the Continent as well as in Britain. L'Hospital's
textbook had made two explicit assumptions: first, that if a quantity is increased
or diminished by a quantity that is infinitesimal in comparison with itself, it may
be regarded as remaining unchanged. Second, that a curve may be regarded as an
infinite succession of straight lines. L'Hospital's justification for these claims was
not commensurate with the strength of the assumptions. He merely said:
[T]hey seem so obvious to me that I do not believe they could
leave any doubt in the mind of attentive readers. And I could even
have proved them easily after the manner of the Ancients, if I had
not resolved to treat only briefly things that are already known,
concentrating on those that are new. [Quoted by Mancosu, 1989,
p. 228]The idea that ÷ + dx = x, implicit in l'Hospital's first assumption, leads alge-
braically to the equation dx — 0 if equations are to retain their previous meaning.
Rolle raised this objection and was answered by the claim that dx represents the