>> So that's the assignment.
5,000 steps by hand...
or write a computer program
that will do it in a minute or two.
It's interesting historically
that in the years before computers, of course,
they didn't have that choice.
It was literally 5,000 steps by hand.
And the introduction of Euler's Method,
largely by Katherine Johnson and her group at NASA,
changed what scientists were able to do
by making these numerical computations possible.
The clip from the movie "Hidden Figures"
illustrates that scene very dramatically,
with Taraji P. Henson playing Katherine Johnson.
>> It's when the capsule moves
from an elliptical orbit to a parabolic orbit.
There's no mathematical formula for that.
Yes, we can calculate launch and landing
but without this conversion, the capsule stays in orbit.
We can't bring it back home.
>> Maybe we've been thinking about this all wrong.
>> How's that?
>> Maybe it's not new math at all.
>> It could be old math.
Something that looks at the problem numerically
and not theoretically.
Math is always dependable.
>> For you, it is.
(gentle music)
>> Euler's Method. >> Euler's Method?
>> Yes. >> But that's ancient.
>> But it works.
It works numerically.
♪ ♪
>> That's it.
>> So that's Hollywood.
Let's actually take a second
and look at the real Katherine Johnson,
who is gonna describe the type of calculation she made
for the Apollo and the Mercury astronaut missions.
♪ ♪
>> (narrator) Commander Alan B. Shepard
was to become the first man sent into suborbital flight.
The Mercury capsule is right on course.
>> (Katherine Johnson) Our office computed
for every mission that went out at that time.
The height, the speed, and so on.
It became a geometry problem.
>> (announcer) At ignition sequence start.
>> (Katherine) I felt most proud of the success
of the Apollo mission.
They were going to the moon!
I computed the path that would get you there.
You determined where you were on Earth when you started out,
and where the moon would be at a given time.
We told them how fast they would be going,
and the moon will be there by the time you got there.
>> (man) Beautiful, just beautiful.
>> So what Katherine Johnson was doing
was solving a differential equation
by use of Euler's Method.
And in this case, the differential equation
is here is the Earth, and here is the moon,
and what you want to do is you want to compute a trajectory.
Here's your little takeoff point from the Earth.
And you wanna compute a trajectory
that goes like that
and lands you exactly on the moon.
Then, when you come home,
you have to have another complicated trajectory
calculated.
And the Euler's Method here is just lots and lots and lots
of tiny little finite steps
approximating this complicated orbit.
And she had been doing that by hand.
And then, there's a scene, which is accurate,
in the movie where they first wheel in these giant machines,
room-size machines, that they call "computing machines."
And this was a computing machine.
Previously, the word "computer" referred to a person.
It was an occupation.
"What do you do?"
"Oh, I'm a computer for NASA... I work at Langley."
And the interesting thing-- there's a very good book
which I recommend called, "When Computers Were Human"
by David Alan Grier.
And he tells the whole story
of the army of people called "computers."
And, by the way, 98% of them were women
who actually carried out hand calculations
of artillery trajectories for the Army,
ballistic trajectories for the Navy,
pursuit problems for the Navy.
Rooms full of women with hand calculators,
calculating complicated trajectories.
And in this case, of course,
the differential equation that she's solving
has to do with the positions and velocities,
and there's an X-position of the satellite,
of the rocket, and a Y-position and a Z-position.
And there's an X-prime, Y-prime, Z-prime.
And Newton's Laws give you the forces of gravity that apply.
And you just have to apply Euler's Method,
ehh, 50,000 times.
And because you want a short time step,
because the longer time step saves you time,
but you could very well, with a bigger time step,
end up doing that,
which you really don't wanna do.
So you need a short time step to stay accurate
but that means you have a lot of time steps.
And so, the problem was, these are massive computations
We have rooms full of women carrying them out,
but it takes a long time.
And then-- and this is historically accurate--
there's a scene in the movie "Hidden Figures"
where they wheel in this big computer,
and the Octavia Spencer character
looks at the computer, and they tell her what it's for,
and she goes back to her room full of women,
and she says, "Ladies, we are either computer programmers
"or we are unemployed."
And she goes-- you remember, she steals a book on computing
from the library because they wouldn't lend it to her
'cause she's African American.
She got that book on computing languages,
and they learned computing.
And these things, calculations,
started to be done with these giant new IBM computers.
But as a tiny little footnote,
there's a scene in the movie,
which is, I checked, it's historically accurate,
where John Glenn, the great John Glenn,
is going up into an orbital flight,
and they tell him that they computed his trajectories
with these new computers,
and he was accustomed to working with Katherine Johnson
and trusted her,
and John Glenn says to the head of the program at Houston,
he said, "Did the girl check the calculations?
"I'm not flying unless she checked the calculations."
And they asked her to do it.
And she did some steps by hand to test the method.
And she verified that it was doing it correctly.
And Glenn said, "Okay, I will fly."