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Official poster for the film Hidden Figures

I was 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.

I'm trying to understand more about what it was like, before electronic computers, for these “human computers” to do “rocket science”. What would an everyday problem and calculation look like? What roles would be involved, and how would the workers accomplish them?

In the article I found an example of the sort of calculations that these amazing human computers carried out. It is signed at the bottom "nmg - 2-3-53". 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:

But I'm still curious to know what the day-to-day work of the human computers looked like. I'm guessing some of them would fill out tables like the one shown in the example, and in my gist, mostly by looking up logarithms and doing addition and subtraction with them. They would then use these new tables to do the calculations for specific problems.

But were there other clever tricks they used? Are there any more detailed descriptions of the everyday life of a human computer?


$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 ...

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If you are interested in descriptions of “everyday life” of human computers, here is an excerpt from Stan Ulam’s autobiography, Adventures of a Mathematician (University of California Press, 1991) concerning the years 1949–1952, when he was a part of the team working in Los Alamos on thermonuclear explosion. The account of the mathematical problems involved is rather general, but it mentions the female human computers – and shows some chauvinistic attitude on part of the team members.

My calculations with Everett concerned the first phase of the explosion, the problem of the initial ignition. (...) The numerical work was done on desk computers with the assistance of a number of programmers from the laboratory's computing group, managed by a good-humored New Yorker, Max Goldstein. Much to Max's annoyance, Fermi wanted to encourage the girls to use slide rules; the machine precision was not really warranted because of our simplifications. But Max insisted on the usual routine with desk calculators. Reading from slide rules and using logarithms as Fermi did was much less accurate, but with his marvelous sense he had the ability to judge the right amount of accuracy which would be meaningful. The girls, who were merely making calculations without knowing the physics or general mathematics behind them could not do that, of course, so in a way Max was quite right in insisting on the standard routines.

I particularly remember one of the programmers who was really beautiful and well endowed. She would come to my office with the results of the daily computation. Large sheets of paper were filled with numbers. She would unfold them in front of her low-cut Spanish blouse and ask, ‘How do they look?’ and I would exclaim, ‘They look marvelous!’ to the entertainment of Fermi and others in the office at that time.

Ulam's wife, Francoise, also worked as a “computer” for some time.

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    $\begingroup$ Thanks! Do you have any sense for how many significant digits they could get with a slide rule? I doubt it would work for the table I reproduced above. $\endgroup$ – nealmcb Jan 25 '17 at 1:47
  • $\begingroup$ No, Ulam does not get into this level of detail. Writing for general readership, and on something that might still have been classified material, he is also vague about the number of operations involved. $\endgroup$ – Margaret Friedland Jan 25 '17 at 15:35
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    $\begingroup$ A slide rule had three digits at best for a single calculation. But if you reuse that value 10 different ways in some calculation then the error could be much worse. Imagine two partial calculations yielding 10.4 and 8.37. Now what you really want is the difference between the two numbers which is 2.03. You really end up with more like 2.0 +/- 0.1 that 2.03 +/- 0.1. $\endgroup$ – MaxW Jan 26 '17 at 7:38
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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 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.

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....

Equipment

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....

Type of Work

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.

I wish the film had had just a bit more on those mechanical calculators. Here is a video of one of the models they used in action, a 1956 Friden STW 10 taking a square root.

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  • $\begingroup$ Note that I haven't yet found details on what Friden or Marchant models were available in 1942, and what they could do, with how many significant figures. But adding and multiplying seem to be included. $\endgroup$ – nealmcb Mar 12 '17 at 15:08
  • $\begingroup$ Aha - see the demo of a 1956 Friden STW 10 which I added in an update. $\endgroup$ – nealmcb Mar 29 '17 at 14:35

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