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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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up vote 5 down vote accepted

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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Thanks much, John –  Scott Guthery Feb 6 '11 at 18:20
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