Discovering methods to cleverly conceal


One might form the impression that relations between contemporary mathematicians and scientists is one glorious brotherhood, where individual egos are subsumed into a single scientific zeitgeist. Not a bit of it.

It is important for scientists to establish priority of discovery, and this frequently leads them into a dilemma when they are developing a theory that they know will be regarded as a breakthrough. If they publish what they have of the fledgling theory in order to be credited with the discovery, they run the risk of disclosing a missing piece of the jigsaw to their rivals.

To resolve this dilemma, scientists and mathematicians have sometimes resorted to publishing the gist of their theory at the earliest possible stage in the form of an anagram or other cryptic conundrum. The anagram is usually dated and handed to a reliable third party for verification. When the fully developed theory is published by the discoverer, the key is revealed, thus crediting its author with at least a partial share of the spoils.

Newton and Leibniz both claimed to have invented the calculus at more or less the same time. In a letter to Leibniz regarding the calculus in 1677, Newton 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."

This conundrum is in fact an anagram. Its solution is widely believed to be the statement of the fundamental theorem of calculus, which like all the scholarly writing at the time was in Latin.

It reads: "Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa." When translated this becomes: "Given an equation involving any number of fluent quantities, to find the fluxions, and vice versa." The Latin sentence is an exact anagram, save for a missing `t', assumed to be an error on Newton's part.

Ironically, if Newton had been up front about the calculus in the letter, he could have made an inarguable case for himself as the first developer of the method of calculus, as the letter was passed through an intermediary. In a letter to Johann Kepler (and others) in 1610, Galileo also used an anagram to establish his priority of discovery regarding what he thought were the moons of Saturn. His encoded description of his discovery was presented as: smaismrmilmepoetaleumibunenugttauiras.

After painstaking work Kepler came up with the following translation: "Salve umbisteneum geminatum Martia proles." This when translated becomes "Hail, twin companionship, children of Mars." This is a classic case of someone seeing what they want to see, even to the point of overlooking a single character difference to Galileo's string. Kepler himself had predicted that Mars had two moons by means of a mathematical series, and took the Galilean anagram as confirmation.

In fact, the anagram had nothing to do with Mars, and its correct reconstruction turned out to be: "Altissimum planetam tergeminum observavi." When translated it becomes: "I have observed the highest of the planets (Saturn) three-formed." Galileo assumed he was seeing two moons on either side of Saturn. In fact he was seeing the now famous rings around the planet.

Galileo can be excused for concealing certain discoveries regarding planetary motion, given that the Catholic church was opposed to the Copernican model of the solar system. In a later letter to Kepler he posted the following anagram. "Haec immatura a me iam frustra leguntur oy." This sentence roughly translates to: "This was already tried in vain by me too early."

The sentence has no particular context, and Kepler wrote back to Galileo pleading not to be left too long in doubt, assuring him that he was "dealing with real Germans". I presume what Kepler meant was that he and the other recipients of the letter were Protestants and not subject to the anti-intellectualism of the Roman Church at that time.

Galileo relented and on New Year's Day in 1611 he forwarded the unscrambled version of the anagram to Kepler. "Cynthiae figuras aemulatur mater amorum."

This refers to the planet Venus, the goddess of love. Cynthia is an alias of Diana, the goddess of the moon. So the reconstructed anagram translates to: "Venus emulates the phases of the moon." Galileo, using the newly invented telescope, had observed the compelling evidence that Venus was a satellite of the sun, and hence that the Ptolemaic view of an earth-centred solar system was mistaken.