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

MORE ABOUT LIMITS 289
ought to have one for the sake of our convenience, and, as
none exists, we will create one; we will call the creature
" the square root of z," denote it by the symbol, V/2,
regard it as a number, and describe it as irrational to
distinguish it from the old sort of numbers,-the ratios,to be henceforth described as rational. (C) Yet others of
us, not so numerous but harder-headed and more critical,
will say: it is evident that to escape decently from our
predicament we must somehow enlarge our conception of
number; not, however, by asserting that V and F' have
a mysterious sort of common limit, for they evidently
have no common limit; nor by pretending to " create "
one for them, which we can not do; but by discovering
that certain existing things, not hitherto regarded as
numbers, ought to be so regarded-a discovery that,
briefly sketched, runs as follows: we reflect that the
ranges of 1F and F (we could equally well use J2 and F')
are classes of ratios ordered by S; we observe that neither
of the ranges contains a maximum term, a largest ratio,
though one of them (or its variable Fi) has an upper
limit, 2, and the other has no upper limit; giving the
name segment to such ratio ranges, that is, to such of them
as have no maximum, we see that a segment may or may
not have an upper limit; we readily see that segments
have certain properties (summability, and so on) very
like the properties of what we have been calling numbers;
we accordingly and naturally agree to call the segments
themselves numbers; they are a new kind of numbers-not ratios, but certain classes thereof; we call the new
numbers rational if the segments have upper limits and
irrational if they have not; thus the segment represented
by JF is a rational number while that represented by F
is irrational; we denote the former by z because the