The extract you reproduced involves the observation of eclipses of one or more of Jupiter's satellites (the 'Galilean' or 'Medicean' satellites).
This was in origin an invention and improvement of Galileo and others during the 17th century. It was facilitated by successive tables of the motions and eclipses of Jupiter's satellites, initially by J-D Cassini in Paris in the 1660s, and improved through the 1690s and afterwards. It still remained in use through the 1700s (an example of eclipse tables from the 1690s in English and how to use them is here).
During the 18th century, the improved tables of Wargentin were often used for Jupiter's satellites. A lecture about the satellite motions was given by W de Sitter in 1931 here.
The principle of the longitude method based on the eclipses was that local times of satellite eclipse were to be telescopically observed at times also astronomically determined. These eclipse times were to be compared with times of predictions of the same eclipses at a standard meridian given by the special tables, usually times for a well-known observatory meridian. The result was a time-difference for one and the same eclipse as seen locally, and as predicted for the standard meridian/observatory. From this result a longitude-difference for the same two places follows, in the proportion of 15 degrees for each 1 hour of time-difference. There is more description of the use of eclipses of Jupiter's satellites here, and the de Sitter lecture already cited has information showing that the timing accuracy of satellite eclipse observations was intrinsically limited to no better than about +/- 10 seconds, a limitation that ensured the eventual obsolescence of the method.
But the 1700s also make up a period of revolutionary changes in many methods of longitude determination, changes both of accuracy or of principle. A number of main stages are noticeable.
In the early 1700s, traditional methods from the 17th century dominated. Then the mid-century, to about the 1760s, was a period of specially notable experiments: in measurements on land, by the initial development of accurate clocks (chronometers), and by the improvement of tables of lunar motion against the stars. In the later 1700s the new and improved methods largely took over from the old.
A general view of traditional techniques for determining longitude to the mid-1700s can be seen for example from entries in the first (Edinburgh) edition of the Encyclopedia Britannica, produced 1768-1771 at a point when traditional methods especially at sea were just becoming effectively superseded by the new methods.
(1) The early 1700s:
On land, longitude could be determined astronomically with telescopes and tables of progressively improving accuracy -- some details of this already given above.
At sea, techniques most used were still limited and traditional, often based on dead-reckoning with log lines when out of sight of land, and often with poor results and accidents -- that motivated public rewards for adequate methods of finding longitude at sea.
(2) Mid-century, to the 1760s, saw much experimentation and increasing hopes of accurate longitude determination. On land, astronomical
techniques and tables for measuring differences of meridian were improving. For use at sea, a winnowing of numerous proposals took place. Some promising methods turned out infeasible. For example, the ship's motions made it impossible in practice to control a telescope so as to observe Jupiter's satellites, in spite of attempts at stabilisation, such as suspended observing chairs. Pendulum clocks, suggested for this use by Huygens in the 17th century, turned out unsuitable for stable timekeeping at sea, because the ship's motion interfered with the motion of the pendulums. Eventually, only two methods still looked to have reasonable prospects:
(a) Observation of the moon's positions against the stars, astronomically observed at local times also astronomically determined, at places reached on voyage. These lunar positions and times were to be compared with tabulated predictions of the same moon positions for the local times of a standard observatory meridian. Hence, by difference of times, the difference of longitude was obtained.
Many astronomers produced tables of the moon, but the first to be successful and accurate enough were those of Tobias Mayer (1753), and their incrementally improved versions of 1762 onwards through the end of the century, expecially in the tables of Mason (also known for his part in surveying the 'Mason-Dixon line') and finally by Buerg (1806). The necessary calculations were unwieldy, but they were reduced to feasible size in stages. One of the pioneering advocates was Nicolas-Louis Lacaille, whose role is described for example in this paper of Guy Boistel. The method was adapted for practical use at sea by Nevil Maskelyne in his book 'The British Mariner's Guide' (1763), and it was still further simplified or streamlined by the methods of the 'Nautical Almanac and Astronomical Ephemeris' (annually 1767 onwards) and its associated 'Tables requisite...'.
(b) The other longitude method, which eventually superseded the lunar method, required adequately stable and regular clocks or watches (with balanced timing components, not pendulums) to keep the time of a standard observatory or port of departure. Again this was to be compared with local time determined on voyage by astronomical observation. Here too, the difference between local time, astronomically observed, and the time (for the same physical instant) at the standard observatory or port, kept by chronometer, gave the difference of meridians. The development of suitable clocks and chronometers is described for example in 'The Quest for Longitude' (ed. W J H Andrewes, 1996).
In the later 18th-century, from 1760s onwards, the lunar method became widely used, partly it seems, because the books and tables needed for it were printed in large quantities and thus were available and affordable. The chronometer method eventually caught up, when clockmakers had competed to produce accurate stable chronometers of simpler design (and larger quantity) than John Harrison's earliest and very complex pioneering single examples. It took some further advances in horological technique before robust accurate instruments were made and supplied in large enough quantity to satisfy demand, and at affordable prices (see Andrewes (1996) cited above).