Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

INFINITY 309
Let us recall the current definition of an infinite class:
An infinite class is a class having a sub-class, or part,
equivalent to the whole-equivalent, that is, in the sense
that a one-to-one correspondence can be set up between
the elements of the part and those of the whole. This
definition of infinity was not given by the poet for, as
we have seen, he gave no formal definition of it at all.
We may ask, however, whether Lucretius was aware of
the fact that an infinite multitude, as conceived by him,
contained parts, or sub-multitudes, equivalent, as we now
say, to the whole. The answer is, yes: not only was he
aware of it but he repeatedy employed this characteristic property of infinite multitudes correctly and effectively.
This rather astonishing fact is sufficiently interesting to
justify citation of one or two passages supporting my assertation of it. If we bear in mind that one of the fundamental Lucretian infinites was the succession of time units
(days, say, or generations or other finite stretches) beginning at any given instant and together composing what
is called the future, the following famous passage makes
it perfectly clear that, according to its author, the removal of any finite multitude of elements from an infinite multitude of them leaves a remainder-a partexactly equal (or equivalent, as we say) to the whole:
Nor by prolonging life do we take one tittle from
the time past in death nor can we fret anything away,
whereby we may haply be a less long time in the condition of the dead. Therefore, you may complete as
many generations as you please; none the less, however, will that everlasting death await you; and for no
less long a time will he be no more in being, who beginning with today has ended his life, than the man who
has died many months and years ago.l
1Munro's translation, 4th ed., p. 83.