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
6accdae13eff7i3l9n4o4qrr4s8t12ux
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P.P.S. I know the work of Newton where he described Puiseux series, probably it was unpublished.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.
Best Answer
Newton's anagram on his method to solve differential equations is contained in his letter to Leibniz dated October 24, 1676, as described here
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.