Zero (16 page)

Rearranging the terms yields

Since
y
=
x
²
+
x
+ 1, we can subtract
y
from the left side of the equation and
x
²
+
x
+ 1 from the right side of the equation and leave the system unchanged. That leaves us with

Now comes the dirty trick. Newton declared that since
o
was really, really small, (
o
)
²
was even smaller: it vanished. In essence, it was zero, and could be ignored. That gives us

which means that
o
/
o
= 2
x
+ 1, which is the slope of the tangent line at any point
x
on the curve (Figure 26). The infinitesimal time period
o
drops right out of the equation,
o
/
o
becomes
/
, and
o
need never be thought of again.

The method gave the right answer, but Newton's vanishing act was very troubling. If, as Newton insisted, (
o
)
²
and (
o
)
³
and higher powers of
o
were equal to zero, then
o
itself must be equal to zero.
*
On the other hand, if
o
was zero, then dividing by
o
as we do toward the end is the same thing as dividing by zero—as is the very last step of getting rid of the
o
in the top and bottom of the
o
/
o
expression. Division by zero is forbidden by the logic of mathematics.