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I am interested in looking up P. G. Tait's flawed proof of the four-colour theorem, published in 1880. The citation that I have seen is:

However, this is just an abstract of the full paper; several details are omitted, and it is indicated that these will be given in full detail somewhere else, presumably the full paper.

I am not familiar with how this particular journal works; perhaps it is not unusual for only abstracts to be published, and for the full paper to appear elsewhere? If so, where can I find the full paper of an abstract published in this journal?


What follows is a summary of my search for details regarding this. Perhaps those who are knowledgeable on this matter can confirm or deny how close my guesses are.

In this case, I found the following paper by Tait (again, actually an abstract) that was published later in the same volume of this journal:

In this paper, Tait writes,

But a remark made incidentally in the abstract of my former paper has led me to a totally different mode of attacking the question, which puts its nature in clearer light. I have therefore withdrawn my former paper, as in great part superseded by the present one.

So, perhaps the first paper never appeared in full anywhere, since Tait says he is withdrawing it? Then, the question becomes whether the paper "Remarks on the previous communication" was published in full anywhere else. I cannot find any other paper by Tait with the same title, but admittedly this is a very unenlightening title. If the full paper appeared elsewhere, possibly it would have a different title attached to it: with this guess, a search among other papers of Tait in the same year turned up the following:

This paper references the paper in Proc. Roy. Soc. Edinburgh 10, no. 106, and concerns the proof (albeit flawed) of the remark that was noted in Proc. Roy. Soc. Edinburgh 10, no. 107, but it does not mention the latter paper. However, from the content it does appear that this might be the one that contains the full details of the latter; another reason for this suspicion is that both of these were read on July 19, 1880.


To conclude, these are the questions that I am specifically looking to be answered:

  1. For an abstract published in Proc. Roy. Soc. Edinburgh, is the complete paper then published in Trans. Roy. Soc. Edinburgh?

  2. Is Tait's paper in Trans. Roy. Soc. Edinburgh 29 the full paper whose abstract was published in Proc. Roy. Soc. Edinburgh 10, no. 107?

  3. What is the correct citation for Tait's 1880 flawed proof of the four-colour theorem? Is it the Remarks on … paper, or the Note on … paper, or something else altogether?


[1]: The "previous communication" mentioned in the title is the following paper by Frederick Guthrie: F. Guthrie, Note on the colouring of maps, Proc. Roy. Soc. Edinburgh 10 (1879–80), no. 107, 727–728, doi:10.1017/S0370164600044631. This paper in turn references the first paper by Tait mentioned above, and provides some further background on the problem.

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  • $\begingroup$ You do know a good research librarian, right? If not, this is a good time to start using one. They love things like this. $\endgroup$
    – Jon Custer
    Commented Mar 3, 2021 at 18:18
  • $\begingroup$ @JonCuster Actually, I've never tried asking anyone stuff like this before! Perhaps once this pandemic eases I can visit my institute's library and see if anyone is interested and can help me :) $\endgroup$
    – user13915
    Commented Mar 3, 2021 at 19:05
  • $\begingroup$ They are amazingly good. You think you know how to use Web of Science, right? Well, not nearly as well as they do... Go sit with one (post-pandemic) and let them build you a search that gets you 5 to 15 completely relevant papers rather then 100 mostly irrelevant ones to sort through. Plus finding the old or odd papers like above. $\endgroup$
    – Jon Custer
    Commented Mar 3, 2021 at 19:11
  • $\begingroup$ My understanding is that the "previous communication" refers to his own previous communication, that is the one titled "On the colouring of maps". It is probably only incidental that the literal previous communication also concerns the four color theorem. Indeed, Tait does not reference Guthrie's comments anywhere in his "Remarks …" paper/abstract, but only speaks about his own work that was presented earlier. $\endgroup$ Commented Jun 17, 2022 at 23:47

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I am not sure about Q1., but Q2. is answered in Biggs, Norman L.; Lloyd, E. Keith; Wilson, Robin J. Graph theory 1736–1936. Oxford University Press (1976). Reprinted, with corrections (1977). Zbl 0335.05101, Zbl 0904.05001.

From Chapter 6: The four-colour problem—early history, on page 103 (you might be able to view the relevant pages on Google Books):

Kempe's improved 'proof' aroused the interest of P. G. Tait…On 15 March 1880 he read a paper on the map-colouring problem to the Royal Society of Edinburgh and an abstract of this paper was published in the Proceedings of the society.

This is the one titled "On the colouring of maps". Further, on the same page:

It was the published abstract of Tait's paper which prompted Frederick Guthrie to publish his account of the origin of the problem in the same Proceedings [1]. Immediately following this account is the abstract of another paper by Tait, delivered on 19 July 1880 [6C].

The paper [1] by Guthrie is the one titled "Note on the colouring of maps", which is mentioned in a footnote in the question. The paper [6C] by Tait is the one titled "Remarks on the previous communication". Following this, on page 104:

The full version of Tait's second paper was eventually published in the Transactions of the Royal Society of Edinburgh

This is the one titled "Note on a theorem in geometry of position".

So, Tait's full 'proof' of the four-colour theorem is the one that appeared in the Transactions of the Royal Society of Edinburgh.

Since Tait himself mentions in his "Remarks…" paper that he has withdrawn his earlier paper, I expect that the brief outline in the first abstract is all that we have of his initial attempt at proving the four-colour theorem.


To explicitly answer Q3., I would cite the full paper published in the Transactions of the Royal Society of Edinburgh when needing to refer to Tait's 'proof' of the four-colour theorem.


I'll also pull down my comment from below the question, since I think it's relevant:

My understanding is that the "previous communication" refers to his own previous communication, that is the one titled "On the colouring of maps". It is probably only incidental that the literal previous communication also concerns the four color theorem. Indeed, Tait does not reference Guthrie's comments anywhere in his "Remarks …" paper/abstract, but only speaks about his own work that was presented earlier. – The Amplitwist Jun 17 at 23:47

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