This Gridiiron Demonstaration gives an insight into the construction of Harrison's clocks

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.

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?