(This is adapted from my question/answer at Day-to-day tasks of human computers, ala Hidden Figures movie - History of Science and Mathematics Stack Exchange)
I was also fascinated by the film Hidden Figures, and a related article from New Scientist magazine "Gifted and black: The brilliant woman who got the US into space".
Compressible Flow
In the article I found an example of the sort of calculations that these amazing human computers carried out. It is from https://crgis.ndc.nasa.gov/crgis/images/b/b7/CompressibleFlow.pdf, which is signed at the bottom "nmg - 2-3-53". Hmmm - I wonder who that was, and how this fit in to the work at the time.
First, I put together a modern re-working of it, with a Jupyter notebook and Python code. A bit of it is below, and you can see the whole thing, and the IPython notebook behind it, at my gist:
A bit of googling turned up some relevant background material on the math itself, which includes:
$M = { V \over a} $
${H - p \over q}
= {F_c}
= {1 + \eta}
= { {2 \over {\gamma M^2}} { \left[{( {1 + {{\gamma - 1} \over 2}} M^2)}^{\gamma \over {\gamma - 1} } - 1\right]} }
= {1.42857 \over M^2} \left[(1 + 0.2 M^2) ^ {3.5} - 1\right] $
...
M H−p/q T/T0 A1/A rho/rho0 Psonic p/H q/H
0.001 1.0000 1.0000 0.0017 1.0000 -673870.092 1.0000 0.00000
0.002 1.0000 1.0000 0.0035 1.0000 -168467.127 1.0000 0.00000
0.003 1.0000 1.0000 0.0052 1.0000 -74873.985 1.0000 0.00001
...
Orbits
The first memo, on orbit determination, that Katherine Johnson did, with a co-worker, is The Determination of Azimuth Angle at Burnout for Placing a Satellite over a Selected Earth Position 1960. T.H. Skopinski, Katherine G. Johnson, NASA TN D-233
This example from the (most excellent) Hidden Figures book provides some details of the actual computations that she did, checking the IBM 7090 trajectory calculations for the first US orbital mission. E.g. she was working to 8 significant digits:
[Katherine Johnson] worked thru every minute of what was programmed to be [John Glenn's] three-orbit mission, coming up with numbers for eleven different output variables, each computed to eight significant digits. It took a day and a half.
Update: See more on Glenn's orbit here: Where to look for historical or reconstructed orbit data for early NASA missions - Mercury-Atlas 6 for example
Equipment
The book also notes that the human computers used mechanical calculators. Some great detail on their working environment comes from a 1942 memo reproduced in Hidden Figures and Human Computers | National Air and Space Museum, including these brief excerpts:
... the work of a computer required skill and judgment. Computers gathered data by reading pressure values from manometers placed in the wind tunnel. Depending on the application, the data were smoothed, plotted, and interpolated. Data reduction and analysis were carried out with the help of calculators, slide rules, planimeters, drafting tools, and other instruments. The women in these roles knew how to organize computational work and how to do so quickly without making mistakes. This knowledge was unique to them....
...
The automatic computing machines and Comptometers cost over $500.00 each....
The automatic calculator is usually the Friden or Marchant.... The computers were also furnished with 20 inch (log-log duplex) slide rules....
... for the type of computer who has some mathematical background the type of work required of her could probably be better done on the calculating machine....
Each computing section has one light-table for tracing purposes.... A set of five-place logarithmic tables and trigonometric tables is also provided, together with a few scale, triangle and French curves....
I wish the film had had just a bit more on those mechanical calculators. But I found a video of one of the models they used in action, a 1956 Friden STW 10 - Youtube taking a square root.
Type of Work
One more excerpt:
There is a large amount of simple calculation required in the work here at NACA. Most [wind] tunnels have means for taking photographs of banks of nearly a hundred manometers at a time. The computers read off the liquid levels and complete the analysis. On the other hand, some of the calculations are sent to the computers in the form of complicated formulas which necessitate a knowledge of trigonometry and sometimes of mathematics involving the calculus. In general, however, the group head would reduce this more complicated work down to tabular form requiring rather routine operations before it would be given to the machine operator. Most of the work coming from the engineers is accompanied by a memo of calculating instructions or word-of-mouth explanations. The computers in any one section soon learn what the usual type of calculation required of them would be.
Many of the calculations which involved ordinary arithmetic could have been handled on the mechanical calculators. But I'm guessing that when it involved trigonometry or logarithms, they would have been using slide rules or looking results up in tables. Tables which had, most likely, been computed years earlier by other human computers as part of the Mathematical Tables Project.
And, as in the example I started with, they were creating their own tables for things like compressible flow, that others could in turn use in the future.
