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 Sidereal 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:




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, and compared 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)1024 kg, which is roughly 64% of the Earth's mass, whereas the actual mass of Mars is only (6.43)1023 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 R3/T2. To give the results in round numbers he needed to solve the Diophantine equation (at least approximately)



Since 21.5 is 43/2, his solution reduces to the fact that (53)(202) = 50000 is nearly equal to (33)(432) = 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 truly 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 nothing 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. Christiaan 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 occasion 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 rebuttal 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.


Return to MathPages Main Menu