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The entry on Alexandre-Théophile Vandermonde at the MacTutor History of Mathematics archive ends with the description of the contents of Vandermonde's fourth and last mathematical paper, concluding with the sentence

Finally he gave a remarkably clever notation for determinants which has not survived.

Sadly, it does not tell us what this remarkably clever notation was! I've searched a bit for the actual paper, Mémoire sur l'élimination, with no success.

Maybe someone knows what this notation was?

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The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in section 2.1 of A case of mathematical eponymy: the Vandermonde determinant (2010). It seems Lebesgue didn't like it because it could have induced a mix-up between indices and exponents, and that may be a reason it did not survive. Leibniz used a similar notation.

The following quotation from his 1772 paper shows how Vandermonde used this positional notation for coefficients to construct determinants:

Here is the original in French, from Mémoire sur l'Élimination, M. Vandermonde, Histoire de L'Academie Royale des Sciences, 1772, part two, pages 516-532.

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    $\begingroup$ This is rather a notation for the entries of a matrix. So maybe MacTutor is stating this not quite correctly? $\endgroup$
    – Ivan Izmestiev
    Commented Sep 28, 2016 at 8:11
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    $\begingroup$ I' d guess that the remarkable notation is that of the second screen shot (which is surely of a translation! I am not being able to get Gallica to give me the pages) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 28, 2016 at 8:56
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    $\begingroup$ @MarianoSuárez-Álvarez --- added link to the French original, with screenshot. (I had some difficulty finding it, because in the 2010 article it is cited as from the 1771 volume, whereas actually it is from the 1772 volume, with a 1771 submission date.) $\endgroup$
    – Carlo Beenakker
    Commented Sep 28, 2016 at 11:28
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    $\begingroup$ Non-mathematical comment: I find it awesome to look at that typography! If one considers that it dates back to $1772$ and that nowadays, with LaTeX and all the rests, there are editors complaining if one creates too bizarre, or too unusual, a symbol—well, this makes me think that the beautiful art of typography is getting lost. I find it sad. $\endgroup$
    – Filippo Alberto Edoardo
    Commented Sep 28, 2016 at 12:39
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    $\begingroup$ @FilippoAlbertoEdoardo --- Laplace = Marquis de la Place = Le Comte Laplace, both spellings with and without a space between "La" and "Place" were used concurrently, compare this reference from 1829 and this other reference from 1814. $\endgroup$
    – Carlo Beenakker
    Commented Sep 29, 2016 at 10:15

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