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Arnold, in his paper The underestimated Poincaré, in Russian Math. Surveys 61 (2006), no. 1, 1–18 wrote the following:

``...Puiseux series, the theory which Newton, hundreds of years before Puiseaux, considered as his main contribution to mathematics (and which he encoded as a second, longer anagram, describing a method of asymptotic study and solution of all equations, algebraic, functional, differential, integral etc.)...''

Arnold says this is several other places as well.

As I understand, the "first anagram" is this


You can type this on Google to find out what this means. Or look in Arnold's other popular books and papers.

Question: what is the "second anagram" Arnold refers to?

P.S. This was my own translation from Arnold's original. The original is available free on the Internet, but the translation is not accessible to me at this moment. I hope my translation is adequate.

P.P.S. I know the work of Newton where he described Puiseux series, probably it was unpublished. But there is no anagram there.

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You wrote Poincare where you meant to write Newton. – Ben McKay Aug 24 '13 at 20:39
"Arnold says this is several other places as well" -- where? – Boris Novikov Aug 24 '13 at 20:50
@Ben --- corrected – Carlo Beenakker Aug 24 '13 at 21:41
@Boris --- – Carlo Beenakker Aug 24 '13 at 21:42
There is no "a" in "Puiseux". – Rob Arthan Aug 24 '13 at 23:02
up vote 25 down vote accepted

Newton's anagram on his method to solve differential equations is contained in his letter to Leibniz dated October 24, 1676, as described here

At the end of his letter Newton alludes to the solution of the "inverse problem of tangents," a subject on which Leibniz had asked for information. He gives formulae for reversing any series, but says that besides these formulae he has two methods for solving such questions, which for the present he will not describe except by an anagram which, being read, is as follows, "Una methodus consistit in extractione fluentis quantitatis ex aequatione simul involvente fluxionem ejus: altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua caetera commode derivari possunt, et in collatione terminorum homologorum aequationis resultantis, as eruendos terminos assumptae seriei."

You can read the anagram in the published letter:

5accdæ10effh11i4l3m9n6oqqr8s11t9y3x: 11ab3cdd10eæg10ill4m7n6o3p3q6r5s11t8vx, 3acæ4egh5i4l4m5n8oq4r3s6t4v, aaddæcecceiijmmnnooprrrsssssttuu.

as well as the translation of the latin text:

"One method consists in extracting a fluent quantity from an equation at the same time involving its fluxion; but another by assuming a series for any unknown quantity whatever, from which the rest could conveniently be derived, and in collecting homologous terms of the resulting equation in order to elicit the terms of the assumed series".

A curiosity: the anagram is flawed, there are two i's too few and one s too many. Errare humanum est.

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Thanks. This is exactly what I was looking for. – Alexandre Eremenko Aug 24 '13 at 23:28

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