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quantities as made up of particles, or should use little curve lines for right ones; I would not
be understood to mean indivisibles, but evanescent divisible quantities; not the sums and
ratio s of determinate parts, but always the limits of sums and ratio s: and that the force of
such demonstrations always depends on the method lay d down in the foregoing lemma s.
Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities;
because the proportion, before the quantities have vanished, is not the ultimate, and when
they are vanished, is none. But by the same argument it may be alledged, that a body arriving
at a certain place, and there stopping, has no ultimate velocity; because the velocity, before
the body comes to the place, is not its ultimate velocity; when it has arrived, 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; that is, that velocity with which the body arrives at its last place, and with
which the motion ceases. An in like manner, by the ultimate ratio of evanescent quantities is
to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with
which they vanish. In like manner the first ratio of nascent quantities is that with which they
begin to be. And the first or last sum is that with which they begin and cease to be (or to
be augmented or diminished.) There is a limit which the velocity at the end of the motion
may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all
quantities and proportions that begin and cease to be. And since such limits are certain and
definite, to determine the same is a problem strictly geometrical. But whatever is geometrical
we may be allowed to use in determining and demonstrating any other thing that is likewise
geometrical.
It may also be objected, that if the ultimate ratio s of evanescent quantities are given,
their ultimate magnitudes will be also given: and so all quantities will consist of indivisibles,
which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th
book of his Elements. But this objection is founded on a false supposition. For those ultimate
ratio s with which quantities vanish, are not truly the ratio s of ultimate quantities, but limits
towards which the ratio s of quantities, decreasing without limit, do always converge; and to
which they approach nearer than by any given difference, but never go beyond, nor in effect
attain to, till the quantities are diminished in infinitum. This thing will appear more evident
in quantities infinitely great. If two quantities, whose difference is given, be augmented in
infinitum, the ultimate ratio of these quantities will be given, to wit, the ratio of equality;
but it does not from thence follow, that the ultimate or greatest quantities themselves, whose
ratio that is, will be given. Therefore if in what follows, for the sake of being more easily
understood, I should happen to mention quantities as least, or evanescent, or ultimate; you
are not to suppose that the quantities of any determinate magnitude are meant, but such as
are conceiv d to be always diminished without end.
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