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

VARIABLES AND LIMITS 243
finite class composed of the terms, 2, 3, 4, 5, 6, 7, 8, 9:
we can indicate the range by saying, let x represent any
one of the verifiers of the function-n is a cardinal whose
value is between I and Io. What is n's range? What
would the range of x be if we simply said " x is a cardinal
between I and Io "?
(3) If we say, let x be any verifier of the function
-n is a cardinal between 5 and 7-, the range ofx is-, not
5-but the class whose sole member is 5; in this case the
varible is a constant.
(4) Consider the infinite (endless) series: I+2+3+
4+5+..; the sum, Sn, ofthe first n terms is ~n(n +),
so that S~=~-n(n+I). Here the language implies that
n is being regarded as a variable whose range is the class
of all positive integers; perhaps some one doubts the
implication; very well, let us explicitly agree to let n be
such a variable; you see at once that we then have
another variable on our hands, namely, Sn-an ordinary
(not a propositional) function of n, for to each value, as
we say, of n (i.e., to each number in n's range) there
corresponds a definite sum, a definite value of S.; plainly
these sums are: I, 3, 6, o1, I5,...; the class of these is
the range of the variable S,. Adopting the usual figurative speech, we may say that, as the variable n runs along
the row I, 2, 3,...., the dependent variable, or function
S,, glides along the sequence I, 3, 6,... But we must
not let such talk make us forget what the ranges are:
these are classes and not rows of their members or terms;
the terms of either range appear in many different rows,
but the range is one thing and each of the rows is another.
If we so desire we can make explicit the propositional
function involved in determining the range of S,; we can
say, for example, that the symbol S. represents any one