The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in section 2.1 of <A HREF="http://arxiv.org/abs/1204.4716">A case of mathematical eponymy: the Vandermonde determinant</A> (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.

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

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Here is the original in French, from <A HREF="http://gallica.bnf.fr/ark:/12148/bpt6k35711/f690.image.r=vandermonde">Mémoire sur l'Élimination, M. Vandermonde, Histoire de L'Academie Royale des Sciences, 1772, part two, pages 516-532.</A>

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