Today, let's do calculus. The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
Math is an odd business.
Anyone who's literate knows arithmetic. Anyone with
a high school education has been exposed to some
algebra. Almost all of you have done the arithmetic
of algebra in which numbers are replaced with
But far fewer people have taken the next step and
studied calculus. So the word calculus
sounds pretty arcane to most of us. In fact, it's
not any harder than algebra, but it has quite a
different look and feel. Algebra still looks a
little like arithmetic, while the calculus goes off
into a different place entirely.
To see the difference, think about the kind of
questions that each answers: Algebra gives us a
foolproof way to answer questions like this one:
"We have 240 dollars to give away. We want to give
Mary twice as much as Bill and Jake three times as
much. How much money do we give Bill?"
Calculus, on the other hand, gives us a
foolproof way to answer a question like this one:
"I know how my drag racer's acceleration changes
with its speed. If I go from a standing start to
eighty miles an hour in ten seconds, how far have I
Algebra is a language in which we solve problems by
equating numerical elements in symbolic form. We
assign a symbol to Bill's payment, equate all the
payments to 240 dollars, and do the arithmetic. We
get the number that'll make the symbol satisfy the
equation, and we find that Bill gets forty dollars.
Calculus, on the other hand, deals with instants in
time, or points in space. It deals with sequences
of fleeting moments or places. Go back to that
dragster: At one instant it's going thirty miles an
hour. But what does that mean? It doesn't spend an
hour on a thirty-mile track. The time it spends at
that speed is actually zero. It merely passes
through thirty miles an hour.
Instantaneous speed is a pure calculus idea, yet we
all understand it. That's because the language of
the calculus has percolated into everyday life. So,
how far did the dragster go in its acceleration
test? It traveled the sum of distances that each of
those instantaneous speeds took us during the ten
The calculus is a simple language that lets us talk
about things that change from moment to moment or
point to point. It serves us when we deal with
smooth movements or smooth shapes. It is the
language by which we can calculate the capacity of
an oddly shaped gas tank, or the trajectory of a
space probe aimed at Mars.
It's also a language that codifies intuitive ideas
that are deeply felt and understood. We all see our
lives as fleeting moments and as the sum of
fragments almost too small to notice. Indeed, Joan
Didion captures the calculus perfectly in her dark
and passionate novel, Run River. She asks,
Was there ever in anyone's life span a point
free in time, devoid of memory, a night when choice
was any more than the sum of all the choices gone
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
The number of calculus textbooks is vast. Simply go
to your library and browse them. In the Library of
Congress catalog system, they are to be found under
the call number QA303.
I am grateful to UH colleagues Lewis Wheeler and
Ralph Metcalfe for their counsel on this episode.
If we let Bill's payment be x, then: x + 2x + 3x =
$240 or 6x = $240. Thus x= $40. We cannot solve the
drag racer problem until someone first tells us
just how the acceleration varies with speed.
The calculus becomes downright dramatic in these
graphs of elliptic integrals
The Engines of Our Ingenuity is
Copyright © 1988-2003 by John H.