Houston, Tranquillity Base here. The Eagle has landed.

Neil Armstrong, 20 July 1969

This article is also dedicated to the memory of Margaret Hamilton (1936–2026), whose pioneering work in software engineering contributed to the success of the Apollo missions.

This is the second article in a series introducing numerical methods for solving differential equations, inspired by one of humanity’s greatest technological achievements: the journey to the Moon.

In July 2019, I wrote the Italian version of this article to celebrate the 50th anniversary of the Apollo 11 mission. On 20 July 1969, Neil Armstrong and Edwin “Buzz” Aldrin became the first humans to walk on the lunar surface, while Michael Collins remained in lunar orbit aboard the Command Module Columbia.

More than half a century later, the Moon continues to inspire scientists, engineers, and explorers. NASA’s Artemis programme, together with initiatives from other space agencies and private companies, represents a renewed effort to explore our nearest celestial neighbour. In 2026, the Artemis II mission marked another milestone with a crewed journey around the Moon, recalling the historic flight of Apollo 8 in 1968. These developments renewed my interest in the mathematical and computational techniques that made the Apollo missions possible. How could the computers of the 1960s, with their extremely limited memory and processing power, calculate the trajectories necessary to navigate between the Earth and the Moon?

To explore this question, I returned to historical NASA documentation and the scientific literature on numerical integration. In this article, I summarise some of these findings and introduce the Runge–Kutta method, one of the most widely used techniques for solving differential equations numerically. As usual, I will also include a simple computer program to illustrate how these mathematical ideas can be put into practice.

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