thank you Hans, thank you Todd for your replies. This was the problem and the reason why John Harrison had to wait for 60 years in order to catch the prize according to the Act of Queen Ann 1714. There was the mathematical method based on the observation of the moons of Jupiter for the calculation of the longitude (Nevil Maskerville, the royal astronomer, chair of the Board of Longitude). The disadvantage of this method was that when the sky is clouded the observaton was not possible and the calculation had to be done by hand and took some hours (not practical on sea!, the ship was already at another location). Maskerville and his colleagues had the concept that the moons of the Jupiter were the celestial clock, provided by the Creator. They could not imagine that a simple clockmaker could solve in a mechanical way the problem of the longitude. My question is 1. what was the method the navigator used to ´calculate´ the longitude by means of the moons of Jupiter, before the common use of the clocks of Harrison, I guess at the end of the 18th century? 2. are there any records of this calculation method?

I am not an expert on the history, but I believe that this method was never used by navigators because reliable predictions of the moons' positions came in the 20th century. Before the models coming from simulations, there were functions published in the form of series, but these require a computer to be used. With some effort a calculator, and by hand, quite a bit of effort. In any case, it would not have been used in practice. After sea-worthy accurate clocks were invented, those were used and tables could not have been produced before the modern era. Trying to use the average periods of their orbits is just not reliable. (Maybe some navigator tried it and never made it back to tell the tale?)

For fun though, one can go through the how the calculation would have worked, but only with the computer to mimic the table of moon positions using Wolfram Alpha.

Make your observation of the Jupiter moons. Then find the right time so that Wolfram Alpha gives the same image as the one you see with queries like "Galilean moons configuration 11:15pm" Just change the time until you get close. Of course, the point here is to figure out the time using the moons' configuration, not using the clock on your computer which is not available to our hypothetical navigator.

With the time of day, that corresponds to the angle the Earth has rotated through since the sun was overhead, then use the right ascension of Jupiter (which, hopefully, would be roughly constant for your journey) measure the angle Jupiter makes with the sun (Jupiter is close to the ecliptic plane) and use the sun's right ascension (known from the time of year) to calculate the longitude. This step is just simple addition and subtraction.

I am not an expert, but I believe that the basic idea was that if you knew the exact time of day then with the position of the sun and stars, you can then infer longitude. The problem of latitude is much simpler, as you can use the highest altitude of the sun in the sky.

For example, take the sun's transit (highest elevation) in Champaign

http://www.wolframalpha.com/input/?i=sun+transit+Champaign

and sun's transit in Peoria

http://www.wolframalpha.com/input/?i=sun+transit+Peoria

which today is 12:02pm versus 12:08pm They are in the same timezone, the difference comes from the difference in longitude.

All you would need was a good watch to know which city you were in. The problem for the early navigators was that they did not have a good watch.

See the article in Wikipedia on the longitude prize

The basic idea was that if you could see the Galilean moons (which is the only connection to Galileo that I know of) and you had a table for predicting their positions, you could infer what time it was. An elegant idea but too elegant. Turned out that they did not obey the simplest model for orbital dynamics, and seemed unpredictable.

While the prize's intention was to have an astronomical solution, I believe the winner was the one who made a good clock.

Later the reason for their unpredictability was given the name orbital resonance. I think rather it is a good example of unpredictability in nature (see Wolfram's A New Kind of Science ) and the only decent predictions come from large computer calculations that simulate their motions.

Of course nowadays you can get the arrangement of Galilean moons on Wolfram Alpha

http://www.wolframalpha.com/input/?i=Galilean+moons+configuration