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I was reading an old paper (specifically, the first appearance of the Pearcey function, here) and I was struck by the beauty of the plots it contains, particularly for a paper from 1945-46:

Pearcey goes into significant depth of analysis about the integral $$ I(X,Y) = \int_{-\infty}^\infty \exp\mathopen{}\left(i\left(Yt+Xt^2+t^4\right)\right)\mathclose \: \mathrm dt , $$ and he also includes the plots above. He does not go into any detail regarding the numerical computation of the function, but the acknowledgement at the end,

The author is greatly indebted to the staff of the Cambridge University differential analyser for collaboration in the computation involved in this work.

is very clear on what device (the Cambridge differential analyzer) was used for this. I've had a read through this nice blog post describing the technology, and I feel quite OK in understanding how the technology available in 1945 would be enough to allow for quite accurate calculations of this integral, using mechanical differential-equation integrators to calculate the incomplete $t$ integral and therefore its eventual asymptotic value.

However, the actual production of the plots in the paper is still murky to me. How does one go from the output of a mechanical differential analyzer to a contour plot of the corresponding function? Would the expectation be that the (human) computer tabulate the function for a dense grid along $X$ and $Y$, from there extract relevant points at which the desired contours were reached, and then draw the diagram by hand from those? Or would it have been possible to use the output of each individual $t$ integration to feed back into the machine and use that to make it graph the contours somehow?

Emilio Pisanty
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    As far as I know, scientific iconography at that time was using photographic processes (including corrections) a lot. Drawing such a diagram by hand looks doubtful. – Jean Marie Becker Mar 29 '22 at 20:49
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    Don’t underestimate a good technical artist. I started grad school just as they were phased out at the university, IBM Research, and Bell Labs. They made beautiful graphics. Still would have needed lots of data. Not sure how that differential analyzer worked though. – Jon Custer Mar 29 '22 at 20:52
  • I seem to recall a previous question here (or one of our companion sites) how the complicated contour plots of the gamma function of complex argument, the Bessel functions, etc in the 1933 edition of Jahnke and Emde were drawn. I cannot find that Q&A right now, but as I recall, the answer was that these plots were drawn manually by experienced drafters, likely with the help of tools like French curves. – njuffa Mar 29 '22 at 23:28
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    See https://hsm.stackexchange.com/questions/3634/how-did-scientists-plot-complicated-graphs-in-the-19th-century/3635#3635 – Alexandre Eremenko Mar 30 '22 at 01:05
  • @AlexandreEremenko Thanks, that could be the question I remembered (if so, not quite correctly its seems). Based on Emde's forword to the 1933 edition, the graphs added in that edition seem to have been pretty much the work of one man, Rudolf Rühle (September 29, 1907 - April 21, 2001). He was a student of physics at the time and seems to have worked for Emde between 1930 and 1933. – njuffa Mar 30 '22 at 02:16
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    To me, figure 2 looks clearly hand drawn. Figure 1 is a bit cleaner, but both seem to be definitely hand lettered, suggesting the whole of both drawings are by hand. – Todd Wilcox Mar 30 '22 at 10:35
  • @njuffa I thought about including the J&E gamma-function plots into this question. But it is my understanding that that book predates the widespread adoption of differential analyzers (which IIUC dates from ca. 1930), so the methods in question might be different (or, again, they might not). – Emilio Pisanty Mar 30 '22 at 13:43
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    @ToddWilcox - hand lettering would be added in a second iteration. Copy (trace) in pencil, erase bits to add lettering, then finish with ink for the final version. Graphic artists had to work hard before computers. – Jon Custer Mar 30 '22 at 20:21
  • @JonCuster Yeah I’ve done work like this with pencil on vellum myself, although I did not erase. I used F or 2H graphite to do guide lines and lightly draw it and then went over it with 2B, leaving gaps where I knew I would letter. If guide lines showed up on repro, then very gentle use of drafting powder usually took care of it. I avoided deliberately erasing and mainly used it for mistakes. – Todd Wilcox Mar 30 '22 at 22:05
  • I really don’t miss those days. So much easier to redo a plot at the push of a button… – Jon Custer Mar 31 '22 at 00:01

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