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I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other Outsiders; An Introduction to Algebra, Topology, and Functional Analysis by Roman, Paul (1974).

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Another common example, is that of representing convex sets which all over the web:

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Note that the use of circles in logical discussions is several centuries years old (1690, as found by SE reader (Mauro Allegranza) in Google Books. This is in Latin Book Image.

This is from Nicolas Bourbaki, Elements of Mathematics, General Topology Chapters 1-4, Translated Image

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  • $\begingroup$ I am afraid no, Venn diagram history of using circles is rather well documented. I am searching about irregular shapes like the ones shown in the query. $\endgroup$
    – ACR
    Commented Apr 19 at 0:11
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    $\begingroup$ Branko Grunbaum has written several papers related to Venn diagrams that might be of interest. faculty.washington.edu/moishe/branko/… $\endgroup$
    – Joseph Malkevitch
    Commented Apr 20 at 0:48
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    $\begingroup$ This question seems more appropriate for HSM. $\endgroup$
    – LSpice
    Commented Apr 21 at 4:39

2 Answers 2

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This may not be the sort of answer you intended, but (following up on Joe's comment) Venn's original 1880 paper includes figures using multiple shapes for sets. Specifically, page 7 shows the four ellipse configuration with a fifth set, an elliptical annulus, superimposed to show 32 nonempty subsets.

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Also, page 8 shows the familiar three circle configuration with a rounded crescent for a fourth set arranged to show 16 nonempty subsets. Then a very irregular fifth set giving 32 nonempty subsets.

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Citation for Venn's article: J. Venn, On the diagrammatic and mechanical representation of propositions and reasonings, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series Five, Volume 10, Issue 59 (1880), 1-18, https://doi.org/10.1080/14786448008626877.

See also Ruskey & Weston's Venn Diagrams in the Electronic Journal of Combinatorics Dynamic Surveys for several other figures where sets are represented with noncircular shapes.

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  • $\begingroup$ Thank. Those diagrams/graphics appear to be quite recent. I believe the set diagrams shown in the original query originate from 1930s -1950s. I am still searching the Internet Archive and old books on set theory. Early books don't have any. Bourbaki's Topology has similar representation (will update the figures soon). $\endgroup$
    – ACR
    Commented Apr 24 at 3:55
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    $\begingroup$ Added DOI etc. for the 1880 Venn article with the graphics I mentioned. $\endgroup$
    – Brian Hopkins
    Commented Apr 24 at 9:01
  • $\begingroup$ Thanks. I added an image from Bourbaki's book as well. $\endgroup$
    – ACR
    Commented Apr 24 at 18:44
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The article contains discussion of a concave piecewise linear comb diagram by Venn (supplementing the example given by Brian Hopkins in another answer).

  • Moktefi, Amirouche, and Sun-Joo Shin, "A history of logic diagrams." Handbook of the History of Logic. Vol. 11. North-Holland, 2012. 611-682.

Contains discussions of various historical logic diagrams, including Leibnitz's line-diagrams which are not necessarily circular or convex:

enter image description here (Image from: https://iep.utm.edu/leib-log/)

Lewis Carroll discusses his proposed graphical diagram for logic in his book "Symbolic Logic" (MacMillan 1897) and discusses both Euler's and Venn's proposal, noting the Venn has problems with scaling.

Carroll's diagrams are rectilinear but can be concave, see p177ff.

enter image description hereLeibnitz Linear Logic Diagram

(Images from: https://www.gutenberg.org/files/28696/28696-h/28696-h.htm)

Carroll also uses concave subspaces from intersection to indicate certain situations such as if "some element exists" that meets a set condition.

enter image description here

(Image from: https://www.gutenberg.org/files/28696/28696-h/28696-h.htm)

Probably not germain but fun to look at:

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(Image from: https://web.mit.edu/redingtn/www/netadv/SP20150413.html)

"The Front of a Collar" is an example of an illustration of a 3-term syllogism, apparently not completely circular. It is noteworthy that many illustrations are much stranger than this, see more illustration at this MIT blog.

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    $\begingroup$ The Swinburne is charming, thanks for including it. $\endgroup$
    – Brian Hopkins
    Commented Apr 25 at 16:05

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