The [entry](http://www-history.mcs.st-andrews.ac.uk/Biographies/Vandermonde.html) 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?



Vandermonde's remarkably clever notation for determinants
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](http://arxiv.org/abs/1204.4716) (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.


![](https://ilorentz.org/beenakker/MO/Vandermonde.png)


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


![](https://ilorentz.org/beenakker/MO/Vandermonde_2.png)


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.](http://gallica.bnf.fr/ark:/12148/bpt6k35711/f690.image.r=vandermonde)


![](https://ilorentz.org/beenakker/MO/Vandermonde_3.png)


