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I'm doing research on the history of the Lagrange inversion theorem. The earliest predecessor I've found is the one referenced by De Morgan; viz. Jo. H. Lambert's construction in Observationes Variae in Mathesin Puram, Acta Helvetica, Vol. 3, 1758, pp. 128-168.

If anyone knows of an earlier construction I'd greatly appreciate hearing about it.

Thanks in advance.

Cheers, Scott

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If you count any inversion of a power series as a predecessor of Lagrange inversion, then I believe the earliest examples are Newton's inversion of the log series to obtain the exponential series, and inversion of the inverse sine series to obtain the sine series. The exponential series is in his De methodis (1671), p.61, and the sine series is in his De analysi (1669), p. 233, 237.

Also, de Moivre, Philosophical Transactions 20 (1698), pp. 190-193, gave a more general formula in a paper entitled "A method of extracting the root of an infinite equation."

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  • $\begingroup$ Newton produced at least the first few terms of the general compostional inversion series for a formal power series that we now relate to the combinatorics of associahedra. See ref in oeis.org/A133437. $\endgroup$
    – Tom Copeland
    Commented Nov 12, 2018 at 8:26

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