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I am wondering how did Maxwell in the 19th century draw such figures as the one shown? What tools or procedures did he need?

enter image description here

Is it all compass and ruler drawing?

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I do not know how exactly was this picture made, but there are at least two methods. The first one is to compute this potential (which is not too difficult), plot sufficiently many points and connect them by smooth curves. This was still quite common in pre-computer era, in 1970s when a special drawing tool was used, in the shape of a curved ruler of variable curvature. (I just found that it is called a "French curve" in English:-) You attach an appropriate part of this "ruler" to several points to draw a smooth curve.

Here is how it looks:

https://en.wikipedia.org/wiki/French_curve

The second method which could be used was experimental. The picture of equipotential lines can be obtained with a metal dust on a properly charged plate. Then the plate can be photographed and carefully copied from the photo. Or even drawn without photographing on a transparent paper layed over the metal dust. I've seen such method used on more complicated potentials (which would be difficult to compute numerically). An example of such photo is here:

https://en.wikipedia.org/wiki/Field_line

Both methods were still used in the 1970s.

Remark. It is sometimes difficult to tell by eye whether a picture in an old book really represents the exact data, or is just made by an artist, that is whether the shapes and distances are true. I recomputed some pictures from old books using a computer, and sometimes they look very different, which suggests that they are just hand drawings, showing essential features but not real representations of an object. The picture you show seems to be real, because it represents quite a simple potential which is not difficult to compute.

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    $\begingroup$ Other possible tools: Lesbian rule, splines, pantograph $\endgroup$
    – Michael E2
    Commented Apr 8, 2016 at 12:40
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    $\begingroup$ Pantograph will not help much here, but thanks for the Lesbian rule. English names of these things are funny indeed:-) $\endgroup$
    – Alexandre Eremenko
    Commented Apr 8, 2016 at 12:45
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    $\begingroup$ To me the French curve seems the most likely and a spline the least likely tools. (I'm not at all sure about the use of splines in printing.) A pantograph could be used to scale a drawing to the size needed by the printer. $\endgroup$
    – Michael E2
    Commented Apr 8, 2016 at 12:57
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Adding to Alexandre's answer. My father, educated in the 1930s, used a set of "French curves" like this

French curves

Secondly ... perhaps the picture in the post was not drawn by Maxwell himself, but by a professional technical artist.

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    $\begingroup$ Indeed - as difficult as it might seem to the younger folks, into the 1980's most laboratories (IBM, Bell Labs) and univerisities had professional technical artists on staff to make all of the figures and illustrations for articles and books. I was a grad student at the tail end of that era, and I can testify that they were really, really good. But, computer graphics got 'good enough', and the funding for the artists faded away. Sad really - I see lots of badly thought out plots these days. $\endgroup$
    – Jon Custer
    Commented Apr 8, 2016 at 15:52
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    $\begingroup$ I used "French curves" of exactly the same shape in 1970s, probably they were standard. For illustrations in my own papers of that period I asked help of a professional draughtsman. This was a very common profession, and all research and engineering facilities had plenty of them. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 8, 2016 at 19:08
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    $\begingroup$ When I searched for that image, I found that French curves are still for sale. $\endgroup$
    – Gerald Edgar
    Commented Apr 8, 2016 at 19:11
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    $\begingroup$ With the French curve images visible in this answer, I feel anyone reading this who had not previously known about French curves will understand now the funny story Feynman told about them (which I tell to my first-semester calculus students): read the excerpt in the question at math.stackexchange.com/questions/884317/… $\endgroup$
    – KCd
    Commented Apr 9, 2016 at 0:44

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