The Fundamental Anagram of Calculus
Here's an interesting quote from the correspondence of Isaac Newton:
6accdae13eff7i3l9n4o4qrr4s8t12ux
This is from the 2nd letter that Newton wrote to Leibniz (via
Oldenburg) in 1677. He was responding to some questions from
Leibniz about his method of infinite series and came close to
revealing his "fluxional method" (i.e., calculus), but then
decided to conceal it in the form of an anagram. After describing
his methods of tangents and handling maxima and minima, he wrote
"The foundations of these operations is evident
enough, in fact; but because I cannot proceed with
the explanation of it now, I have preferred to
conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux.
On this foundation I have also tried to simplify
the theories which concern the squaring of curves,
and I have arrived at certain general Theorems".
The anagram expresses, in Newton's terminology, the fundamental
theorem of the calculus: "Data aequatione quotcunque fluentes
quantitates involvente, fluxiones invenire; et vice versa", which
means "Given an equation involving any number of fluent quantities
to find the fluxions, and vice versa."
Arranging the characters in his Latin sentence in alphabetical
order (and assuming he counted the dipthong "ae" as a separate
character, and u's and v's are counted as the same character), the
number of occurrances of each character are as follows
--6--- -----13------ ---7--- -3- ----9---- --4- --4-
aaaaaa cc d ae eeeeeeeeeeeee ff iiiiiii lll nnnnnnnnn oooo qqqq
--4- ----9---- -----12-----
rr ssss ttttttttt uuuuuuuvvvvv x
This agrees with Newton's anagram
6a cc d ae 13e ff 7i 3l 9n 4o 4q rr 4s (8t) 12u x
9t?
except that I count nine t's instead of eight. Possibly Newton's
original Latin spelling used one fewer t's, although I can't see
which one of them could plausibly be omitted. It could also be that
the anagram has been incorrectly copied, but it agrees with the
version in both Westfall's and Christianson's biographies, as well
as the transcription of Newton's letter contained in Calinger's
Classics of Mathematics. Another possibility is that Newton simply
mis-counted. This isn't as implausible as it might seem at first,
since there is a well-known psychological phenomenon of overlooking
the second letter in short connective words (like the f in "of") when
quickly counting the number of occurences of a certain letter in a
string of text. It's very easy, when counting the number of t's in
Newton's latin phrase, to neglect the "t" in the word "et".
Ironically, neither Leibniz nor Newton had published anything on
calculus at the time this letter was exchanged, although both are
believed to have been in possession of the calculus, so if Newton
had just come right out with a complete and explicit statement of
his calculus he would have placed Leibniz in a VERY difficult
position, and would have established his own priority beyond doubt
(since the letter passed through Oldenburg). Instead, Newton's
very protectiveness and secrecy caused him to lose whatever
unambiguous claim to priority he might have had (and led to an
acrimonious priority dispute that embittered both his and Leibniz's
later lives).
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