Men of Mathematics (20 page)

If all modern refinements are similarly ignored in the manner of the seventeenth century it is easy to see how the calculus finally got itself invented. The underlying notions are those of
variable, function,
and
limit.
The last took long to clarify.

A letter, say
s,
which can take on several different values during the course of a mathematical investigation is called a
variable;
for example
s
is a variable if it denotes the height of a falling body above the earth.

The word
function
(or its Latin equivalent) seems to have been introduced into mathematics by Leibniz in 1694; the concept now dominates much of mathematics and is indispensable in science. Since Leibniz' time the concept has been made precise. If y and
x
are two variables so related that whenever a numerical value is assigned to
x
there is determined a numerical value of y, then y is called a (one-valued, or
uniform)
function of
x,
and this is symbolized by writings
y
=
f
(
x
).

Instead of attempting to give a modern definition of a
limit
we shall content ourselves with one of the simplest examples of the sort which led the followers of Newton and Leibniz (the former especially) to the use of limits in discussing rates of change. To the early developers of the calculus the notions of variables and limits were intuitive; to us they are extremely subtle concepts hedged about with thickets of semimetaphysical mysteries concerning the nature of numbers, both rational and irrational.

Let
y
be a function of
x,
say
y = f(x
). The rate of change of y with respect to x, or, as it is called, the
derivative of y with respect to x,
is defined as follows. To
x
is given any increment, say
Δx
(read, “increment of
x”),
so that
x
becomes
x
+ Δ
x
, and
f
(x), or
y,
becomes
f
(
x
+ Δ
x
). The corresponding increment, Δ
y,
of
y
is its
new
value
minus
its initial value; namely, Δ
y
=
f
(
x
+ Δ
x
)
—f(x
). As a crude approximation to the rate of change of
y
with respect to
x
we may take, by our intuitive notion of a rate as an “average,” the result of dividing the increment of
y
by the increment of
x,
that is,

But this obviously is too crude, as both of
x
and
y
are varying and
we cannot say that this average represents the rate for
any particular
value of x. Accordingly, we decrease the increment Δx
indefinitely,
till, “in the limit” Δx approaches zero, and follow the “average”
all through the process:
Δy
similarly decreases indefinitely and ultimately approaches zero; but
does not, thereby, present us with the meaningless symbol
but with a definite
limiting value,
which is the required rate of change of
y
with respect to x.

To see how it works out, let
f(x)
be the particular function x
²
, so that y = x
²
. Following the above outline we get first

Nothing is yet said about limits. Simplifying the algebra we find

Having simplified the algebra as far as possible, we
now
let Δx approach zero and see that the limiting value of
is 2
x
. Quite generally, in the same way, if
y = x
ⁿ
,
the limiting value of
is
nx
ⁿ
–
^l
,
as may be proved with the aid of the binomial theorem.

Such an argument would not satisfy a student today, but something not much better was good enough for the inventors of the calculus and it will have to do for us here. If
y
=
f
(
x
), the
limiting value
of
(provided such a value exists) is called the
derivative of y with respect to
x, and is denoted by
This symbolism is due (essentially) to Leibniz and is the one in common use today; Newton used another (
) which is less convenient.