|Photo: Catherine Meyers|
|Photo: Courtesy of 20th Century Fox|
Math plays a starring role in the movie "Hidden Figures," which is nominated for three Oscars, including Best Picture, at this weekend's Academy Awards.
Adapted from a book of the same name by Margot Lee Shetterly, the movie chronicles the grit and ultimate triumphs of three African-American women working as "human computers" for NASA in the segregated south during the space race. The film's standout math whiz is Katherine Goble Johnson. During a pivotal scene, Johnson and a team of white, male engineers are staring at a blackboard, trying to solve equations for the trajectory of astronaut John Glenn's space capsule. They're stumped until Johnson hits upon a solution: "Euler's Method," she says. "That's ancient," says one of the engineers incredulously. "Yes. But it works," she counters. "It works numerically."
The scene -- and indeed the timing of key details in the film -- is clearly dramatized for Hollywood, but it raises the questions: What exactly is Euler's Method? And did Katherine Johnson actually use it to help send astronauts into orbit in the 1960s?
An "ancient" approximation technique
First off, Euler's Method is indeed pretty old, if not exactly ancient. It was developed by Leonhard Euler (pronounced oy-ler), a prolific Swiss mathematician who lived 1707-1783. "He was one of the greatest in history," said Po-Shen Loh, a mathematician at Carnegie Mellon University in Pittsburgh. What has come to be known as Euler's Method is just a tiny fraction of his legendary contributions.
The method tackles what many people may not realize is a common challenge in math -- often the equations just can't be solved exactly. When that happens, mathematicians must figure out ways to approximate the answers for specific situations.
Euler's method is one such technique applied to what is called a differential equation. These equations often show up, among many other places, in physics problems that describe the path of a moving object subject to changing forces. For example, when a capsule is flying through space, gravity is constantly tugging at it. How hard gravity pulls is related to distance, so as the spacecraft gets nearer to or farther away from Earth, the forces on it also change.
One way to visualize the meaning of the equations could be as ocean currents, Loh said. As you travel through the water, the currents change direction and speed. If you're planning to navigate from a remote island on a raft, you'd want to determine exactly how you'd float through the water, which is a pathway made up of infinitely many points -- analogous to an exact solution to a differential equation.
Sometimes, however, it's impossible to get this comprehensive, ideal solution. Instead, as you drift along, you could measure the current at regular time intervals. By knowing your starting point and assuming the current stays roughly constant between readings, you could plot an approximate trajectory. This process of calculating solutions at discrete points and connecting them is essentially Euler's Method, Loh said.
The method works best when the points are close together and when the solution changes slowly and smoothly, because errors can accumulate at each step of the process.
Approximating a pathway made up of infinitely many points, by linking together a finite number of calculations, is an example of something called a numerical approach in mathematics. That's what Johnson's character in the movie means when she says Euler's Method works "numerically."
Euler's Method is one of the simplest of many numerical methods that now exist for solving differential equations.
Euler method - Wikipedia, the free encyclopedia
Euler's method | First order differential equations | Khan Academy
Source: Inside Science News Service and Khan Academy Channel (YouTube)