Given the pervasiveness and versatility of computers in our world of 2025 it is easy to ignore that 80 years ago they were initially conceived as what we would call today a calculator, specifically created to solve tedious mathematical problems. Of course, users of TikTok, Call of Duty, YouTube, Visual Studio Code, or even readers of this website through some web browser like Firefox are usually (and blissfully) unaware of the gazillion computations per second that it takes to produce everything you see on screen.
The quest of using computers to solve more and more complicated mathematical problems with ever simplified usability and accessibility is as old as the computer themselves. Among major step stones worth mentioning along the way we can find MATH-MATIC (created by a team under the supervision of Grace Hopper around 1955), FORTRAN (all in uppercase, please, created by John Backus of IBM just a few years after), FORMAC (created by none other than the one and only Jean Sammet in the early 1960s), MATLAB® (it is a trademark, after all, written in FORTRAN in the early 1970s), Maple and Mathematica (both very commercially successful products since the 1980s), GNU Octave (mostly compatible with MATLAB, and whose development started in the 1990s), the R programming language (a rising star, also from the 1990s), NumPy and Jupyter Notebooks (the kings of the first quarter of the 21st century, the era of machine learning and LLMs)… and of course, spreadsheets.
You might have watched an often-shared short clip on social media, taken from the Academy Award-nominated 2016 movie “Hidden Figures”, where Katherine Johnson, played by the extraordinary Taraji Henson, tries to calculate the re-entry point of John Glenn’s first orbital flight in 1962; needless to say, a critical piece of information. Given the shenanigans of space travel and orbital dynamics, an algebraic solution for those complex differential equations is out of question (or at least, out of grasp for modern mathematicians).
To solve this conundrum, and to the surprise of Paul Stafford (a surprisingly dramatic role played by Jim Parsons of “Big Bang Theory” fame), Ms. Johnson proposes the use of Euler’s Method:
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book “Institutionum calculi integralis” (published 1768–1770).
The scene ends with Mr. Stafford famously claiming “that’s old!” as if the Pythagorean theorem was not useful at the venerable age of 3500 years old… Mr. Stafford’s reaction is canonical and very appropriate; it is exactly the same reaction that most software engineers have upon learning the fact that COBOL is running most credit card transactions in the planet, or when frontend engineers discover that static or server-rendered HTML websites do not need 10 megabytes of JavaScript on the browser.
(As a person with dual Argentine nationality, at this point I must include a reference to “El Eternauta”, a popular Netflix show released a few months ago, where Favalli, upon rescuing his friend Juan Salvo from death, happily claims “Lo viejo funciona, Juan!”, or, in English, “Old things work, Juan!”)
My first exposure to spreadsheets as tools of mathematical resolution was while studying physics in the early ’90s. Although my high school in Geneva had a lab for “analyse numérique”, furnished with brand-new Apple Macintosh II machines running some early version of Microsoft Excel, I was not able to join that class as it was, understandably enough, fully booked. So I had to wait until my debut at the University of Geneva to be exposed to the resolution of differential equations without resorting to algebraic analysis, but rather, with pure numbers.
This month’s Library entry is, precisely, the book which provided me with the required skills to find numerical solutions to various problems during my studies: an exploration into the realms of Euler’s Method (among other strategies) with a spreadsheet, in this case, Lotus 1-2-3 for DOS. The book in question is “Spreadsheet Physics” by Charles Misner and Patrick Cooney, published by Addison-Wesley in 1991.
Dr. Charles Misner, who passed away in 2023 at the age of 91, was a professor of physics at the University of Maryland from 1963 to 2000, and a renowned expert in General Relativity. He was the recipient of the Albert Einstein medal in 2015, and was the co-author of “Gravitation”, one of the best-selling books in the subject of General Relativity, featuring none other than Kip Thorne on the cover (spoiler alert: if you enjoyed Nolan’s movie “Interstellar”, well, Mr. Thorne was behind the CGI simulation of the black hole).
Misner and Cooney’s book was hardly the only one about the application of spreadsheet technology in the resolution of physics problems; more or less simultaneously, we had “Dynamic Models in Physics” (1989) by Potter, Peck, and Barkley; “Quattro Pro for Scientific and Engineering Spreadsheets” by Robert G. Parks (published by Springer-Verlag in 1992), and “The Excel Spreadsheet for Engineers and Scientists” by Irvin H. Kral (published by Prentice-Hall in 1992).
In some ways, Misner and Cooney’s book seemed outdated when it was released in 1991; the rise of Lotus 1-2-3 was stalling, and Microsoft Excel was the rising star on the horizon at the time. But the book provides simple yet effective approaches to the resolution of numerical problems with a spreadsheet: use a simple table, with data on the left column and formulae on the right, pointing to a constant increase step (a “delta”) in some fixed-reference cell, and then repeat the formula field downwards ad nauseam. Let the PC do its work (you can press F5 to refresh the spreadsheet in old versions of Lotus 1-2-3), and then display a nice graph with the results. Change the delta, recalculate, re-plot the graph; wash, rinse, repeat.
We take such interactivity for granted nowadays, but merely half a century ago such possibility was unimaginable. Spreadsheets still serve very well the purpose of modeling non-linear systems, without the need of developing any kind of ad hoc software package nor writing code in any programming language. Bar the Lotus 1-2-3 specific sections, Dr. Misner and Cooney’s book stands the test of time, providing a straightforward mental model, useful to solve pretty much any mathematical problem that can be modeled as a differential equation.
Of course, we are in 2025, and if you are using Excel you can use thePDSOLVE() function to… you guessed it, solve partial differential equations right away.
Cover photo by the author.