# How were contour plots of complex functions produced in the days of mechanical differential analyzers?

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?

• 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. Mar 29 at 20:49
• 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. Mar 29 at 20:52
• Mar 30 at 1:05
• 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. Mar 30 at 10:35
• @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. Mar 30 at 20:21