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

MORE ABOUT LIMITS 287
empire,-what had been seemingly unconnected, reciprocally alien provinces of thought.
The meaning and justice of what I have just now said
may be made evident by means of simple examples.
Three or four little ones will suffice, and we can both
shorten the task and enliven it by speaking of our variables in the customary dynamic fashion.
In the first example, I am going to ask you to imagine
that we have arrived at a stage of mathematical evolution
where we are familiar with the ordinary fractions, or
ratios, including such as 3,,..., which for convenience
we will write, 2,...; and that we know nothing of socalled irrational numbers. Let S be the sequence of the
ratios arranged in the natural order of increasing magnitude. Let K, represent any ratio less than 2 (i.e., r)
and let F2 present any one greater than 2. You immediately see that, under either D3 or D4, 2 is a limit of kÇ
and also of J, JF approaching it from below and F
from above. Observe that neither of the V's can reach
the limit; one of them is always less, the other always
greater, than 2; they can, however, so close in upon 2
as to make the difference between them less than any
preassigned positive ratio, however small,-we can make
the V's as near together as we please if only we do not
please to make them meet-between them stands their
common limit, 2, fringed on both sides with a row of
ratios which the F's in their race towards 2 can never run
through. Now consider very carefully two other variables, V and V', the former representing any one of the
ratios whose square is less than 2, and the latter any one
whose square is greater than 2. Note that the new V's,
like the old ones, can come indefinitely near together;
observe that as they approach each other, one of them