Euler’s method & Hidden Figures – is it really ‘old maths’?

At some point in a Mathematician’s life, they will hear Mathematics being referred to as a ‘Universal Language’. In your world, $5+5=10$, and if you go over to your neighbour’s house, $5+5$ will still be equal to $10$. Flying halfway across the world? If it’s not too obscure, ask a local and they will (hopefully) tell you that $5+5$ does indeed equal $10$.

So, it is clear that Mathematics does not depend on distance, but what about time? 

The ‘Ancient’ Euler’s Method… 

Depending on when you were born, Euler’s Method may be considered ‘old’. When Leonhard Euler published his 3-volume book ‘Institutionum Calculi Integralis‘ in 1768, Euler’s method (included somewhere within the 479 pages of the riveting read) was considered to be ‘brand new Maths’ at the time. The writing entails some of the very first discoveries on first-order Differential Equations, and a new perspective on a first-order step method.  

When I first studied the method, it seemed like a little bit of guess work. Take an initial point on a curve, calculate the tangent to the curve at said point, then take a small step to the next point. Similar to the phrase ‘eat, sleep, work, repeat’: the method in question is repetitive, and dependent on the previous calculation. So, unfortunately, there are no error carried forward marks here.  

Take this analogy: imagine you are asked to walk on a tightrope, blindfolded. How big is your first step going to be so that you don’t fall? The trivial answer would be to take as small of a step as possible, so that mentally, you still have a general idea of where the tightrope is, but the logic is guesswork. The worst method would be to take a giant leap, miss dramatically, and fall off. So, the ‘step length’ must be small, but not too small so that it requires thousands of little baby steps to reach the final position (a happy middle). 

This is Euler’s Method!

 (Sort of, if only it were that easy in Rocket Science and the 60’s Space Race)

In Mathematical terms, take a known initial condition, i.e., $x=t_0$, and substitute into a first-order Differential Equation to calculate the tangent at $t_0$. The new coordinate is the product of the step length, $h$, and the function of the new point: 

$$ y_{k+1} = y_k + hu(x_k,y_k) \ \text{ for } \ k=0,1,2,… $$

This graph has no domain/range scale and shows only the 1st quadrant.  Shown in the graph is a sample solution function, labeled y(t).  It is looks to be a parabola with vertex in the 1st quadrant and opening downward.  The left most point on the solution curve is labeled at $\left(t_{0},y_{0}\right)$.  Also on the solution curve is another point (between the leftmost point and the vertex) labeled $\left(t_{1},y(t_{1})\right)$.  Included on the graph is the tangent line to the curve at $\left(t_{0},y_{0}\right)$.  It rises over the solution curve and over the point $\left(t_{1},y(t_{1})\right)$ on the solution curve is a point on the tangent line labeled $\left(t_{1},y_{1}\right)$.
Euler’s Method

Now, the biggest question on everyone’s mind would be, how do we apply this to a real-life situation? 

Hidden Figures – A Point In Time 

While my mum was driving me home for Christmas back in 2019, she couldn’t get over this movie she had seen a couple of nights before, and insisted I watched it as soon as we got back. “It’s got Maths in it” was her reasoning for why she was so adamant I watch this film with her. So, as keeping to my promise, we sat down after dinner that night and started the movie. To my surprise I became absorbed into the TV, mind-blown and on the edge of my seat throughout.  

Hidden Figures is based on a true story about 3 female African-American Mathematicians working at NASA: Katherine Johnson, Dorothy Vaughan, and Engineer Mary Jackson. These incredible women were behind the scenes, working on a variety of assignments such as the Heat Shield, the introduction of electronic computing, and the Space Task group, which calculated trajectories and launch windows for the Mercury Project missions. The calculations these women were doing (by hand) were crucial in order to succeed in putting man in Space, and the world didn’t even know they existed. 

Mary Jackson was one of the "human computers" portrayed in the film "Hidden Figures."
Mary Jackson working as the first black female engineer at NASA

Being a woman in STEM has its restrictions at times, but nothing compares to the sexism and racism they faced, along with the general pressure of trying to achieve something that had never been done before. 

They thought creatively, found a new angle, and found answers to questions they didn’t even know how to ask. As history repeated itself, a new perspective is found once again. Euler’s Method was brought in to calculate the trajectory coordinates for Alan Shepard, the first American in space. And it was Katherine Johnson who Shepard, and later John Glenn, would turn to for confirmed coordinates in which she calculated.  

As the movie came to its end, my mum, as any aspirational mother would, asked if I had understood any of the Maths that they had shown. Unfortunately, I did not at the time. However, it wasn’t until I found myself sitting in an online lecture (during a pandemic as per) for Inviscid Fluid Flow, where I recognised a certain type of Runge-Kutta Numerical Method: 

Timeline

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Lecture Notes from Inviscid Fluid Flow

From the mid 1700’s, to the ongoing prejudice in the Southern States in the 1960’s, to my University lecture in 2021. The Euler Method has lasted the test of time, proving that Maths is indeed never truly finished.  

Related websites
  1. More on Euler’s Method: https://tutorial.math.lamar.edu/classes/de/eulersmethod.aspx
  2. Hidden Figures on Wikipedia: https://en.wikipedia.org/wiki/Hidden_Figures
  3. Professor Alan Garfinkel has two helpful videos on YouTube:

Leanne is a 2nd year Mathematics Student at UEA. Her learning consists of Inviscid Fluid Flow, Linear Algebra and other complex subjects, though it was the Differential Equations module that won her over! Integration and Differentiation are her cup of tea, and the methods are ‘just fascinating’ (when it works!)

On the side of her degree, Leanne plays for the Norwich Devils Women’s American Football team. It’s intense but has an incredible fast-paced environment.

She believes the best way to grow is to speak to new people, ask questions, and to become what you respect. She loves writing about people, and telling their stories.