The History of Mathematics: A Brief Course

484 16. THE CALCULUS

Perhaps it may be objected, that there is no ultimate proportion of

evanescent quantities; because the proportion, before the quanti-

ties have vanished, is not the ultimate; and when they are vanished,

is [not defined]. But by the same argument it may be alleged that

a body arriving at a certain place, and there stopping, has no ul-

timate velocity, because the velocity before the body comes to the

place, is not its ultimate velocity; when it has arrived, there is

none. But the answer is easy; for by the ultimate velocity is meant

that with which the body is moved, neither before it arrives at its

last place and the motion ceases, nor after, but at the very instant

it arrives.

Was this explanation adequate? Do human beings in fact have any conception
of what is meant by an instant of time? Do we have a clear idea of the velocity
of a body at the very instant when it stops moving? Or do some people only
imagine that we do? We are here very close to the arrow paradox of Zeno. At
any given instant, the arrow does not move; therefore it is at rest. How can there
be a motion (a traversal of a positive distance) as a result of an accumulation of
states of rest, in each of which no distance is traveled? Newton's "by the same
argument" practically invited the further objection that his attempted explanation
merely stated the same fallacy in a new way.
That objection was raised in 1734 by the philosopher George Berkeley^11 (1685-
1753, Anglican Bishop of Cloyne, Ireland), for whom the city of Berkeley^12 in
California is named. Berkeley first took on Newton's fluxions:

It is said that the minutest errors are not to be neglected in math-

ematics: that the fluxions are celerities [speeds], not proportional

to the finite increments, though ever so small; but only to the mo-

ments or nascent increments, whereof the proportion alone, and

not the magnitude, is considered. And of the aforesaid fluxions

there be other fluxions, which fluxions of fluxions are called sec-

ond fluxions. And the fluxions of the second fluxions are called

third fluxions: and so on, fourth, fifth, sixth, &c. ad infinitum.

Now, as our sense is strained and puzzled with the perception of

objects extremely minute, even so the imagination, which faculty

derives from sense, is very much strained and puzzled to frame clear

ideas of the least particles of time... and much more so to compre-

hend. .. those increments of the flowing quantities... in their very

first origin, or beginning to exist, before they become finite par-

ticles. .. The incipient celerity of an incipient celerity, the nascent

augment of a nascent augment, i.e., of a thing which hath no mag-

nitude: take it in what light you please, the clear conception of it

will, if I mistake not, be found impossible.

He then proceeded to attack the views of Leibniz:

(^11) Pronounced "Barkley."
(^12) Pronounced "Birkley".