# Numerical Integration of Differential Equations. Part I.: Katherine Johnson and the Euler’s Method.

This article was inspired by the beautiful 2016 movie Hidden Figures (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

In the movie, the mathematician Creola Katherine Johnson (or Globe) (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method [UPDATE, May 2022: I just come across another excellent youtube video by the Tibees about the mathematical work of K. Johnson at NASA, and at the end of the video she reveal that the book shown in the movie is the classic H Margenau & G. M Murphy, The Mathematics of Physics and Chemistry. Published by Van Nostrand, 1956]. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Johnson is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

# The Calculation of the Lattice Energy: The Born-Haber Cycle

My blog in italian on this topics is very popular and for this reason I decided to add an English translation (when I have some free time, I will also translate the text in the Figure and Table). So be tune and more will come!

The stability of a crystal lattice at constant T and P conditions is linked to the Gibbs free energy of lattice formation by the relations

$M^+ (g) + X^- (g) \rightarrow MX (s) \hfill (1)$

$\displaystyle \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \hfill (2)$

If ${\Delta G^{\circ}}$ is more negative for the formation of the ${I}$ structure than for the ${II}$ structure, the ${II \rightarrow I}$ transition will be spontaneous and the solid will have that structure.