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I asked my supervisor about how plots used to be made before computer graphics were a thing, and during this discussion, he showed me a plot from a 1933 book "Tables of Functions with Formulae and Curves" by Jahnke and Emde. Here is the plot: enter image description here

My question is, how were these plots made? In the preface they mention this is calculated by taking the modulus of the function at every coordinate, but it still seems unclear how they got the perspective. Did they manually calculate the perspective, or did they carve it out in wood and use that as reference?

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  • $\begingroup$ Is that not more about speed and efficiency than anything else? Is it not simply the case that Jahnke and Emdem and their plots laid the foundation for all that followed? $\endgroup$
    – Robbie Goodwin
    Commented 11 hours ago

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Drawing and graphing was an important skill that was cultivated in many disciplines leading up to Jahnke and Emde. "Linear drawing", "technical drawing" or "descriptive geometry" had been cultivated and taught in at the middle to highschool level in Germany in the late 19th century.

An example textbook is:

which teaches how to draw 3D objects (e.g. platonic solids) as well as various projection techniques. Standard tools are ruler and compass drawn in ink though strings and other helping tricks can also be involved (for example the use of a string between the legs of a compass to draw an ellipse, see p. 36). I'm including two examples:

Saddle handdrawn by Holzmüller in 1886

Construction of a 3D setup by Holzmüller 1886

Holzmüller's and other texts are discussed in a book by Felix Klein, which discusses pedagogy of mathematics at the middle to highschool level:

(Klein and his students were skilled illustrators as is documented by the amazing drawings that were developed by him and his students as part of his seminar).

The foundational techniques go back to Monge. In his Géométrie descriptive (1799), p.133 one can already find rendering of function elevation over a plane, that in much more elaborate form is then found in the figures of Jahnke and Emde.

Monge's function in the plane of 1798

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    $\begingroup$ (+1) Couldn't find the earlier Q&A I thought I remembered. I think isometric projection as the basic technique for transforming 3D into 2D in technical drafting deserves a mention, as do the templates called French curves as an important tool for curved lines. In fact, Emde himself contributed to the state of the art: Fritz Emde, "Kurvenlineale." Zeitschrift für Instrumentenkunde, Vol. 58, 1938, pp. 409-411. B. Hague, "New method of constructing French curves." Engineering, Vol. 149, No. 3868, Mar. 1, 1940, pp. 215-217. $\endgroup$
    – njuffa
    Commented Dec 12 at 1:03
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    $\begingroup$ E. Jahnke & F. Emde first published "Funktionentafeln mit Formeln und Kurven" with Teubner in Leipzig in 1909. A 2nd edition (revised and enlarged) followed in 1933, a 3rd (revised and enlarged) in 1938. During WW2 the Office of Alien Property Custodian expropriated the copyright leading to publication by Dover in 1943. Between 1930 and 1933 the physicist Rudolf Rühle (Stuttgart: Sep. 29, 1907 - Apr. 21, 2001) prepared the well-known drawings for the book while he worked for Fritz Emde at the University of Stuttgart. He worked for Robert Bosch GmbH from 1934 to 1971, rising to head of R&D. $\endgroup$
    – njuffa
    Commented Dec 12 at 1:15
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    $\begingroup$ @AlexBenitez Other useful tools are the flexible ruler, French curves, and radius aids. Note that the image can be created in pencil, inked over, and then the construction lines can be erased. $\endgroup$
    – Andrew Morton
    Commented 2 days ago
  • $\begingroup$ I still have my drafting spline somewhere. But yes, the pertinent part of this is that this was a skill that was taught and professionally valued. $\endgroup$
    – A rural reader
    Commented yesterday
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That's a straightforward 45-degree orthographic projection. I never got far enough into technical drawing to learn the fancy techniques for projecting geometric shapes (and I'm not sure those would apply once you get away from the simple polynomials and conic sections), so here's my brute-force method for drawing those:

  1. Preparation work: calculate coordinate points, pick my scale, place a blank sheet of paper on the drafting table, etc.
  2. Using a high-hardness pencil, 45-degree triangle, and parallel rule, sketch out the coordinate axes.
  3. Working from front to back, for each point, use a scale and 45-degree triangle to measure out where the point will appear: first measuring horizontal (x-coordinate), then on the slant (y), then vertical (z). Draw a faint dot at this location.
  4. Once I've got a set of three adjacent points, at least one of which is non-hidden, use a French curve to draw a smooth curve connecting them. A flat spline drawing between the middle two of a set of four points would produce a smoother curve, but these drawings aren't large enough to justify the extra effort.
    Working front-to-back makes for cleaner handling of hidden surfaces in this step: back-to-front would require less thinking, but would also require erasing hidden surfaces.
  5. Patch up any missing lines from steps 2 and 3, and erase any that shouldn't have been drawn because they were hidden.
  6. Finishing work: ink the lines, add hatching and labels, and so on.
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