Galileo dedicates Jupiter’s stars to Cosimo de’ Medici
« […] Behold, therefore, four Stars reserved for your illustrious name, and not of the common sort and multitude of the less notable fixed stars, but of the illustrious order of wandering stars, which, indeed, make their journeys and orbits with a marvelous speed around the star of Jupiter, the most noble of them all, with mutually different motion, like children of the same family, while meanwhile all together, in mutual harmony, complete their great revolutions every twelve years about the centre of the world , that is, about the Sun itself.  

Indeed, it appears that the Maker of the Stars himself, by clear arguments, admonished me to call these new planets by the illustrious name of Your Highness before all others. For as these stars, like the offspring worthy of Jupiter, never depart from his side except for the smallest distance, so who does not know the clemency, the gentleness of spirit, the agreeableness of manners, the splendour of the royal blood, the majesty in actions, and the breadth of authority and rule over others, all of which qualities find a domicile and exaltation for themselves in Your Highness? Who, I say, does not know that all these emanate from the most benign star of Jupiter, after God the source of all good? It was Jupiter, I say, who at Your Highness’s birth, having already passed through the murky vapours of the horizon, and occupying the mid-heaven and illuminating the eastern angle from his royal house, looked down upon Your most fortunate birth from that sublime throne and poured out all his splendour and grandeur into the most pure air, so that with its first breath Your tender little body and Your soul, already decorated by God with noble ornaments, could drink in this universal power and authority. But why do I use probable arguments when I can deduce and demonstrate it from all but necessary reason?  

It pleased Almighty God that I was deemed not unworthy by Your Serene parents to undertake the task of instructing Your Highness in the mathematical disciplines, which task I fulfilled during the past four years, at that time of the year when it is the custom to rest from more severe studies.  

Therefore, since I was evidently influenced by divine inspiration, to serve Your Highness and to receive from so close the rays of your incredible clemency and kindness, is it any wonder that my soul was so inflamed that day and night it reflected on almost nothing else than how I, most desirous of Your glory (since I am not only by desire but also by origin and nature under Your dominion), might show how very grateful I am toward You.  

And hence, since under Your auspices, Most Serene Cosimo, I discovered these Stars unknown to all previous Astronomers, I decided by the highest right to adorn them with the very august name of Your Family.  

For since I first discovered them, who will deny me the right if I also assign them a name and call them the MEDICEAN STARS, hoping that perhaps as much honour will be added to these Stars by this appellation as was brought to other stars by the other Heroes? For, to be silent about Your Most Serene Highness’s ancestors to whose eternal glory the monuments of all histories testify, Your virtue alone, Great Hero, can, by Your name, impart immortality to these stars. 

Indeed, who can doubt that You will not only meet but also surpass by a great margin the highest expectation raised by the most happy beginning of your reign, so that when You have surpassed Your peers You will still contend with Yourself, which self and greatness You are daily surpassing. […] »
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The detailed description of the observations carried out by Galileo
« […] IAccordingly, on the seventh day of January of the present year, 1610, at the first hour of the night, when I inspected the celestial constellations through a spyglass, Jupiter presented himself. And since I had prepared for myself a superlative instrument, I saw (which earlier had not happened because of the weakness of the other instruments) that three little stars were positioned near him - small but yet very bright. Although I believed them to be among the number of fixed stars, they nevertheless intrigued me because they appeared to be arranged exactly along a straight line and parallel to the ecliptic, and to be brighter than others of equal size. And their disposition among themselves and with respect to Jupiter was as follows: 
 
That is, two stars were near him on the East and one on the West; the more eastern one and the western one appeared a bit larger than the remaining one. I was not in the least concerned with their distances from Jupiter, for, as we said above, at first I believed them to be fixed stars. 
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Galileo’s infinite amazement
But when, on the eighth, I returned to the same observation, guided by I know not what fate, I found a very different arrangement. For all three little stars were to the west of Jupiter and closer to each other than the previous night, and separated by equal intervals, as shown in the adjoining sketch:
 
Even though at this point I had by no means turned my thought to the mutual motion of these stars, yet I was aroused by the question of how Jupiter could be to the east of all the said fixed stars when the day before he had been to the west of two of them. I was afraid, therefore, that perhaps, contrary to the astronomical computations, his motion was direct and that, by his proper motion, he had bypassed those two stars. For this reason, I waited eagerly for the next night, But I was disappointed in my hope, for the sky was everywhere covered with clouds. Then, on the tenth, the stars appeared in this position with regard to Jupiter:
 
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Galileo’s first doubts: were they fixed stars?
Only two stars were near him, both to the east. The third, as I thought, was hidden behind Jupiter. As before, they were in the same straight line with Jupiter and exactly aligned along the Zodiac. When I saw this, and since I knew that such changes could in no way be assigned to Jupiter, and since I knew, moreover, that the observed stars were always the same ones (for no others, either preceding or following Jupiter, were present along the Zodiac for a great distance), now, moving from doubt to astonishment, I found that the observed change was not in Jupiter but in the said stars. And, therefore, I decided that henceforth they should be observed more accurately and diligently. And so, on the eleventh, I saw the following arrangement:
 
First measurements of the orbital periods
There were only two stars on the east, of which the middle one was three times as far from Jupiter than from the more eastern one, and the more eastern one was about twice as large as the other, although the previous night they had appeared about equal. I therefore arrived at the conclusion, entirely beyond doubt, that in the heavens there are three stars wandering around Jupiter, like Venus and Mercury around the Sun. This was at length seen clear as day in many subsequent observations, and also that there not only three, but four wandering stars making their revolutions about Jupiter: the following is an account of the changes in their positions, accurately determined from then on. I also measured the distances between them with the glass, by the procedure explained above. I have added the times of the observations, especially when more than one were made on the same night, for the revolutions of these planets are so swift that the hourly differences can often be perceived as well. […] »
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Galileo summarises the discoveries made during his observations
« […] These are the observations of the four Medicean Planets recently, and for the first time, discovered by me. From them, although it is not yet possible to calculate their periods, something worthy of notice may at least be said. And first, since they sometimes follow and at other times precede Jupiter by similar intervals, and are removed from him toward the east as well as the west by only very narrow limits, and accompany him equally in retrograde and direct motion, no one can doubt that they complete their revolutions about him while, in the meantime, all together they complete a 12 – year period about the centre of the world. Moreover they whirl around in unequal circles, which is clearly deduced from the fact that at the greatest separations from Jupiter two planets could never be seen united while, on the other hand, near Jupiter two, three, and occasionally all four planets are found crowded together at the same time. It is further seen that the revolutions of the planets describing smaller circles around Jupiter are faster. For the stars closer to Jupiter are often seen to the east when the previous day they appeared to the west, and vice versa, while from a careful examination of its previously accurately noted returns, the planet traversing the largest orb appears to have a semi-monthly period. We have moreover an excellent and splendid argument for taking away the scruples of those who, while tolerating with equanimity the revolution of the planets around the Sun in the Copernican system, are so disturbed by the attendance of one Moon around the Earth while the two together complete the annual orb around the Sun that they conclude that this constitution of the universe must be overthrown as impossible. For here we have only one planet revolving around another while both run through a great circle around the Sun: but our vision offers us four stars wandering around Jupiter like the Moon around the Earth while all other with Jupiter traverse a great circle around the Sun in the space of 12 years. […] »
Selected passages from: Galilei, Galileo, «Sidereus Nuncius» 

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Bibliography: Galileo Galilei - Sidereus Nuncius - translated and commented by Albert Van Helden - University of Chicago Press