**“Two Bits, Four Bits,
Six Bits, a**

**Dollar…all for **

**Pascal, Pythagoras,**

**and**** **

Jerry Woodfill

Former NASA Apollo
Warning System Engineer

**Reflections on the Space Race and how mathematics helped **

**Contact Jerry Woodfill to schedule this free 30 minute talk
for your school or professional organization at 281-474-2974**

*** * * * * * ***

**For more about Space Mathematics try:
**

This recollection was prompted by an
invitation to keynote the opening of a math and science center at *How America Got to the Moon
First*. The well-rehearsed program promised to
be altogether perfunctory. However, a brief closing comment from the
University academic coordinator of the event changed all that: “Could you include something about how math
benefited you, your career, and NASA?”
Taking no thought of how I might do that, I half-heartedly agreed.

Forgotten until a few days before the
engagement, that promise had a way of launching feelings about my character,
ethics, and trustworthiness. My thought
had been to conveniently brush aside the request in the heat of my Space Race
presentation. But I hadn’t been invited to inaugurate a space education center. These educators and attendees would come to
hear about math learning and its familiar friend science education.

My wife’s off-hand comment heightened my
guilt, “I can’t remember one time that you’ve mentioned using math on your job
at NASA.” I’d only worked at the Johnson
Space Center 37 years. Her follow-up
remark simply buried any hope of waving off the professor’s challenge, “For
that matter, I don’t know that you’ve ever talked about any engineering
either.” (I hold two degrees from

The first page of this treatise begins in
the year 1948. The scene opens at the
blackboard of Mrs. Geisen’s first grade class at

Though five decades
past, that moment remains as humiliating as it was that morning so long ago…a
paralysis of fear consumed me. Sweating hands were the least of my
afflictions. No need to grip the chalk
anyway. The thought of defeat clouded my
thinking. Confused by anxiety, I lost on the first problem of the contest. What happened next will never be forgotten
by my classmates. Slowly placing the wet
chalk stick on the blackboard’s metal ledge, I burst into tears. Head bowed, weeping aloud, I returned to my
seat wholly dejected, embarrassed, and forlorn.

Sensing I needed a measure of
encouragement, Mrs. Geisen invited me to have lunch at her desk, even giving me
her desert in hopes of restoring my confidence and self worth. I can still remember that small hexagonal
white paper cup of Mrs.
Geisen’s cherry desert.
This was my humble introduction to the world of mathematics.

Flash forward to the fall of 1955, two years
prior to the October 4, 1957 launch of the Soviet Sputnik. Those 30 classmates
and I had separated. Of all those who
graduated from Lincoln School, I was the lone Highland student among 500 freshmen
attending the 2000 student Hammond High School, a large metropolitan school in
northwest Indiana. Hammond High was on my father’s route to work. His daily drive to the office made attending
the city school possible while hometown friends took a school bus to the much
smaller

But the recollection which comes to mind
that fall occurred on the final day of my freshman Algebra class. I opened my report card. Though I’d mostly recovered from that first
grade humiliation by making an A on three successive periods, my teacher Katie
Williams had written something in the margin.
It said, “Jerry
is a careful thinker. He will make
something of himself.” I was
vindicated! I had made a comeback from
the depths of mathematical defeat. “I
had a future in math after all, despite my dismal beginnings!”

The building of the first high school in

Perhaps, it was Sputnik, the advent of Communist
Cuba, or the Cold War which led to the establishment of

So distressing was my showing at that
first math contest that had my first grade classmates been present, Mrs.
Geisen’s desert would have been needed once more. But the same resolve

By March of 1960, both the state math
contest and basketball tournament were imminent: 700 schools competing,
thousands of students
vying for recognition and victory, academically and
athletically. And the formula
worked! No, we won neither contest as
far as becoming number one in the state, but our basketball team finished the
season at 19-1 ranked in the top twenty among those 700 plus schools…and took
the ultimate winning team to task in the final game of the sectional, the
fourth game of the state tournament, scoring more points against them than any
other team in the tournament.

But most impressive was the Trojan math
team. I, as a senior, finished 14^{th} in the state
in my division and my teammate, a junior, in the top 10 of his. So based on our
collective performance,
we just might have won the state math contest for

Having the best game of my career in the state
tournament led to a basketball scholarship at

One might ask, “How can you make an F minus?” The answer is by never scoring, i.e., in Math
310, as on the hardwood, I never scored!
I’m zero for twelve. There were three one
hour tests with three problems each and a final with three problems. That’s 12 altogether, and I never answered
one correctly. The fact is, I GAVE UP AFTER THE
FIRST TEST. Where was Mrs. Geisen when I
needed her?

Despite this, a merciful electrical
engineering professor allowed me to take a summer make-up course elsewhere at a
less difficult university math-wise. Nevertheless, I did graduate from Rice in
1965.

Lest you think too ill of me as a math and
science speaker, I
did make an A on the make-up course at the anonymous college, a school well
thought of in engineering circles.
Though you may be thinking of ^{th} out of 13 in my Rice electrical engineering class was
a divine sign of my work as the Apollo 11 and Apollo 13 warning system
engineer.

These reflections have departed from their
original intent which was to refute my wife’s doubt about mathematics in my
career at NASA. Additionally, I have not
shown math’s contribution
to the Space Race. In the interest of
restoring a measure of respectability to my narrative, I want to show proof
that some kind of cloud hung over my Rice basketball and math endeavors.

There is this evidence: The national
Scholastic Aptitude Test (SAT) math achievement exam had 800 as the
highest possible score. Shortly before
entering Rice, I
made 770 which registered me well above the top 1% of those taking the
test. Years after graduation, my oldest
son brought a math problem home from high school with the comment, “The math
instructor said no parent in his years of experience at

As for basketball: Entering the NASA industrial league comprised
of numerous recent college players, some of whom were all-conference
performers, I averaged over 20 points per game as one of the three best players
in the league. It was as though a curse had existed during my Rice years, a curse
which had lifted on graduation. But I believe the academic
and athletic foundations established during high school carried me in those
early years at NASA, years when I needed help from the past in designing the
alarm systems for the manned spacecraft which would carry men to the Moon.

How was math used? I remember the warning system used Boolean Algebra and voting logic to determine which of the lander’s
sixteen thrusters failed. Then there
were those probability equations which determined what parts were criticality
ONE, i.e., apt to threaten the lives of the crew. How can I ever forget the mathematics of
program control where flow charts integrated the intricate schedules of
thousands of components and systems through a math framework to determine which
were “pacing” items? My warning system
often appeared as that pacing item in the early years.

Though those ancient electrical
engineering text books were half vacuum tube - half transistor technology,
there was one digital chip in the warning system. It was a binary counter. The
mathematics of set theory and binary counting were used to denote hardware
failures. In reality, mathematics was so
very much a part of going to the Moon that most failed to consider its value in
our quest to win the Moon Race.

The science of going to the Moon, the
engineering of the *Columbia* and *Eagle*, the ultimate winning of
the space race could not have succeeded without the bedrock of mathematics:
From the simplest of arithmetic problems through intricate geometric principles
to the farthest reaches of the Calculus with its unfathomable concepts of
infinity, all contributed to putting the first men on the Moon.

No astronaut who ever rode a rocket into
space could not help but appreciate the power of mathematics to make it happen.
The very discovery of the “Rocket Equation” and the measure of specific impulse
tells the story, “Will
It Make Orbit?” Without mathematics, the answer is
assured, “No it won’t.” So that every
Russian cosmonaut, NASA astronaut, and future spacefarer will, like my

**“Two Bits, Four Bits,
Six Bits, a Dollar…**

**all**** for Pascal, Pythagoras,
and Euclid stand up and holler!”**

…and thank you Mrs.
Geisen. Wherever you are.

* * * * * * * *

Jerry Woodfill

Jerry Woodfill graduated from *Eagle* and
Jim Lovell’s Apollo 13 craft when his warning system sounded alarms.

For his role in the rescue of Apollo 13,
Jerry shared the Presidential Medal of Freedom as part of the Apollo 13 Mission
Operations Team at the