Galileo's Anagrams and the Moons of Mars

The origin of the telescope is unknown.  A Dutch spectacle maker
named Zacharias Janssen seems to have constructed one in 1604, but 
this was based on one he had acquired from an unknown source in 
Italy, dating back at least to the 1590's.  We're unable to trace 
the origin of the idea any further.  Janssen exhibited and sold his 
telescopes widely.  In 1608 another Dutch spectacle maker, named 
Hans Lippershey, claimed to have independently discovered the fact 
that distant objects appeared greatly magnified if viewed through two 
lenses separated by a suitable distance.  He mounted two lenses in 
a rigid tube at the right distance and sold the device to the Prince 
of Holland for military purposes.  Still another Dutchman, James 
Metius, also applied for a "patent" on the telescope about this 
same time, but the application was refused because the idea was 
already regarded as common knowledge.

Early in 1610 Galileo, then a teacher at Padua, wrote

   About 10 month's ago a report reached my ears that a certain
   Fleming had constructed a spyglass by means of which visible
   objects, though very distant from the eye of the observer,
   were distinctly seen as if nearby...[this] caused me to apply
   myself wholeheartedly to investigate means by which I might
   arrive at the invention of a similar instrument.

Galileo apparently had no detailed description of the device, but
he reasoned that it must involve two lenses (since the device was in
the shape of an elongated tube), and those could only be convex,
concave, or flat.  He knew a flat lens wouldn't have any effect, so
he decided that a combination of a convex and concave lens must be
the answer (although Kepler subsequently showed how a telescope 
could be made with two convex lenses).  This was the basis for the 
telescopes that Galileo began to build, first with only 3x or 4x 
magnifying power, but by November of 1609 he had an instrument that
magnified 15 times, and by March of 1610 he had built another 
that magnified 30 times.  

These were by far the most powerful telescopes ever built up to that 
time, and Galileo immediately (even as he continued to build new and 
stronger models) began making a series of spectacular discoveries in 
astronomy.  As Swerdlow describes it, "In about two months, December 
and January [1610], he made more discoveries that changed the world 
than anyone has before or since."  Among these was his discovery of 
the four major moons of Jupiter, about which he wrote in The Sideral
Messenger in April 1610.  News of this soon reached Kepler in Germany,
who was an instant and enthusiastic supporter of Galileo's findings 
(unlike many others, especially since several people who had tried 
to verify Galileo's findings had been unable to see the new moons).

In July of 1610 Galileo was still making discoveries faster than he
could publish descriptions of them.  On the 25th he discovered that
Saturn was apparently situated between two smaller companions that
always moved together.  Wanting to establish his priority of discovery,
but not yet ready to reveal what he had found, he sent to Kepler (and
others) the following jumble of letters, which he informed them was 
a coded description of his latest discovery:

          smaismrmilmepoetaleumibunenugttauiras

It was not uncommon in those days for scientists to communicate (or
rather, to avoid communicating) their discoveries by means of coded
expressions.  Kepler, of course, was a born riddle solver, and made
strenuous efforts to decipher Galileo's string of characters.  It's 
fair to say that Kepler had an unusual aptitude for seeing patterns, 
to the extent that he sometimes saw patterns that weren't even there
(as witness his perception that the planets were arranged in accord 
with the dimensions of circumscribed Platonic solids).  A good example 
is that when he learned of Jupiter's four moons, andcompared this with
the Earth's one Moon, he concluded that Mars must have two moons (by 
the geometrical progression 1,2,4...).  This suggestions seems to 
have been taken up by other people, including Voltaire and the English
author Jonathan Swift, whose Gulliver's Travels (written in 1726) 
contains the remarkable description of the astronomical prowess of 
the fictional Laputan astronomers:

  They have made a Catalogue of ten Thousand fixed Stars, whereas
  the largest of ours do not contain above one third Part of that
  Number.  They have likewise discovered two lesser Stars, or 
  Satellites, which revolve about Mars; whereof the innermost is 
  distant from the Center of the primary planet exactly three of 
  his diameters, and the outermost five; and the former revolves 
  in the space of ten hours, and the latter in twenty-one and an 
  half, so that the squares of their periodical times are very near
  in the same proportion with the cubes of their distances from 
  the center of Mars; which evidently shews them to be governed 
  by the same Law of Gravitation that influences the other
  heavenly bodies.

Of course, it wasn't until 1877, using a telescope hundreds of times
more powerful than Galileo's, that Aseph Hall discovered the two
small moons of Mars.  The inner one he named Phobos (Fear) and the 
outer Deimos (Terror), after the horses that drew the chariot of Mars 
in mythology.  The distances of these from the center of Mars are 
2.743 and 6.891 times the radius of Mars, respectively, and the 
periods of revolution about Mars are 7 hr, 39 min, and 30 hr, 17 
minutes.  The orbital periods are roughly similar to those guessed
by Swift, but the radii were too big by about a factor of two.  As
a result, the mass of Mars that one would infer from Swift's data
is about (3.86)10^24 kg, which is roughly 64% of the EARTH's mass,
whereas the actual mass of Mars is only (6.43)10^23 kg, which is
almost exactly 1/6 the mass "predicted" by Swift's Laputans.

Incidentally, Swift was obviously familiar with Kepler's third law,
so he knew the orbital periods T and radii R needed to give the same
value of R^3/T^2.  To give the results in round numbers he needed
to solve the diophantine equation (at least approximately)

                 R1^3 T2^2  =  R2^3 T1^2

Since 21.5 is 43/2, his solution reduces to the fact that (5^3)(20^2) 
= 50000 is nearly equal to (3^3)(43^2) = 49923.

Returning to Kepler and his struggle with the mysterious coded message,
it so happens that after a great deal of effort he actually succeeded 
in making sense out of Galileo's jumbled string of character:

          Salve umbistineum geminatum Martia proles.

which means "Be greeted, double knob, children of Mars."  In other 
words, Kepler deduced that Galileo had in fact discovered two moons 
of Mars, exactly as he (Kepler) had inferred from the geometric
series!  Unfortunately Kepler's deciphering result differs by one 
character from those in Galileo's string, but surely he had uncovered
the substance of Galileo's discovery - or so Kepler imagined.  It's 
truely fascinating that Kepler somehow managed to interpret this 
message to signify precisely the discovery (or premonition of a 
discovery) that he himself had made!

Of course, Galileo's message actually had nothng to do with Mars. When
he finally revealed its meaning (reportedly because he learned that
His Majesty the Emperor wished to learn it), the message turned out
to be
           Altissimum planetam tergeminum observavi.

which means "I have observed the highest of the planets [Saturn]
three-formed".  We have no record of Kepler's reaction when he learned
the true interpretation.  As to Saturn's peculiar shape, Galileo
was chagrined to see the feature disappear over the next few years,
although it subsequently reappeared.  Christain Huygens (and his
brother) built some very powerful telescopes, and used one of them
to finally give an accurate description of the rings that surround
Saturn.  Since these rings are not aligned precisely with the ecliptic
they periodically become almost invisible from Earth, because we are
viewing them "edge on".

The discovery of Saturn's peculiar shape wasn't the only ocassion 
when Galileo resorted to coded messages.  Perhaps even more significant
was the discovery which he expressed to Kepler and others in the form
of an anagram

        Haec immatura a me jam frustra leguntur oy

which, translated, means roughly "This was already tried by me in 
vain too early".  Max Caspar's biography reports that Kepler wrote
to Galileo after receiving this second coded message

  I adjure you not to leave us long in doubt of the meaning.
  For you see you are dealing with real Germans.  Think in
  what distress you place me by your silence.

Galileo relented and on 1 January 1611 forwarded the unscrambled 
version of the anagram

        Cynthiae figuras aemulatur mater amorum.

which means "Venus imitates the phases of the Moon".  Galileo had
discerned the phases of Venus as it circles the Sun, proving that,
like the Moon, its light is purely reflective from the Sun's light.
More importantly, the correlation between its phases and it positions
showed rather conclusively that Venus (and by implication Mercury as 
well) is indeed circling the Sun, in accord with the new Copernican
model.  This became, along with Galileo's other discoveries, one of
the strongest pieces of evidence in favor of Sun-centered models for
the solar system.

At that time the University of Padau was a stronghold for 
Aristotelianism, and several of the philosophy professors issued
a sharp rebutal to Galileo's claimed discoveries, refuting the
possibility of the existence of any "new" planets or stars on
what they regarded as logical grounds, although their arguments
don't look very logical to modern readers.  Galileo challenged them
to come and look in his telescope for themselves, to see with their
own eyes the things he was describing, but they declined, saying
that they would not look in his telescope, because it showed merely
appearances, which were illusions, not to be trusted when they 
conflicted with logic and reason.