How was longitude determined in the 1700s?

I'm going through the journals of Alexander Mackenzie (ca 1790) and I came across this passage:

I gather that he's determining his latitude and longitude but I'm not clear on what units he's using, or the procedure for that matter. For example, is 29. 23. 48 an angle in degrees, minutes, seconds? In context 3. 5. 53 seems to be a time, but again I'm not clear on the units.

A secondary question is about his use of the moons of Jupiter to help determine the longitude. Was this a common technique? And how did (does) it work? It's amazing to me that using a late 18th century portable telescope he could not only view the moons but distinguish between the first and the third ones.

Here is a link to the full text (volume 2, page 284).

A reprint of Mackenzie's diary prints the table as

Assuming this is correct (not exactly sure what m[h] and h[m] mean), I think it's more clear. Here's how I understand the procedure:

1. He measures the altitude of sun (i.e. degrees above horizon) at 3h 5m 53 in the afternoon by his watch
2. By a table, he knows that this altitude should occur 1h 22m 38s later (so his watch is slower than it should be)
3. He'd done a similar measurement in the morning and computed 1h 21m 44s slow, so he takes the mean of these measurements to get 1h 22m 11s slow
4. Difference of nine hours slow – I suppose this is to put his watch time back to GMT (he's near the west coast of North America), not sure why he's adding 8 to the seconds column
5. He observes an emersion of Jupiter's third satellite (some time later, at night) and finds in a table the exact GMT time that it should occur at. From this (and his local time) he computes his longitude to be ~8d more than some previous reference, or ~128d west of Greenwich.

Is that close to right? Step 4 is a bit unclear to me, especially regarding what he's adding into the second column.

1. On the units. He mixes time and angle units. In the first sentence the altitudes are in angular degrees, minutes and seconds. The main calculation in hours, minutes and seconds of time. Then he converted the final result into angular units. For example, $$1^h22^m38^s$$ and the following columns are time units, while $$8^\circ 32'21$$ are angle units.

2. On the method of determining longitude by Jupiter satellites. Determination of longitude is equivalent to determination of time difference between the local time and the time at some "zero meridian". For the local time, many observations are available, in this example he uses Sun observations in the first line.

For the time at the zero meridian, three methods were available in 1790. The text that you posted illustrates the oldest method based on timing of appearance and disappearance of Jupiter satellites (the method was proposed by Galileo who discovered these satellites, and immediately understood this important application). These times were tabulated for the reference meridian. Since these events do not depend on the place of the observer on Earth, their observation permits to determine the time at the zero meridian. This was the first practical astronomical method of determining longitude.

In the middle of 18 century two new, more convenient methods were introduced (chronometer and Lunar distances). Unlike Jupiter satellites, these new methods could be used at sea. But on land, Jupiter satellite could give better results.

Remark. The method based on Jupiter satellites does not give good results at sea. (The main reason is that there is no way to stabilize a telescope on a moving ship.) So one cannot say that this was a "common method" at sea. But it was common among astronomers/geographers, and remained in use until the middle of 19 century.

Could you please give a precise reference on the text that you cite? I suspect the observation was made on land, since in 18th century there was no instrument for measuring angles at sea which could give a reading to angular seconds.

Here is my interpretation of his specific calculation: He (or his assistant) observed Sun's altitude $$29^\circ23'48''$$ at $$3^h5^m52^s$$ by his watch. This is actually the average of 5 observations in a very quick succession.

From the tables (for given latitude, which he already knew knew) he obtained that this altitude must be at $$1^h22^m38^s$$ for the beginning of his observations and $$1^h21^m44^s$$ for the end (Sun's altitude changes quickly, his 5 observations apparently took about 1 minute, which is not unusual, if you have an assistant who records the time. Or 5 observations can be made almost simultaneously by several people). All this part is called a "time sight".

Then he computes the arithmetic average of these two times (adds them and divides by 2), and obtains $$1^h22'11''$$, and corrects additionally for the error of his watch (8 sec) accumulated in 9 hours from the time when he checked it by the above Sun observation till the time of Jupiter satellites observation. This is not surprising since he observed Sun near noon, and Jupiter's satellites in the evening. (And his watch was apparently just an ordinary watch rather then chronometer).

He does not tell the detail of Jupiter satellite observations, but apparently he managed to time the moment of emergence of the 3-d and 1-st satellites, and comparing this time with the tables obtained two longitude differences, of which he takes the average.

• A glance at en.wikipedia.org/wiki/Alexander_Mackenzie_(explorer) makes it clear these observations were done on land. I think 128W marks as far west as he went, to the coast of British Columbia. Commented Jan 26, 2022 at 2:23
• @kimchi lover: thanks. After reading this article I noticed that the date of observation was 1790 (not 1700 as I first thought). I edited my answer accordingly. Commented Jan 26, 2022 at 4:45
• Thanks for the answer! I've added a link to the full text. So $1^h 22^m 38^s$ would be the time in GMT? What does 3. 5. 53 correspond to? Commented Jan 26, 2022 at 6:17
• @Likas Bystrisky: 3.5.53 is the time of his observation of solar altitudes by his watch. His watch is only used to measure the time difference between the observation of the Sun and observation of Jupiter (evidently they cannot be done simultaneously). Commented Jan 26, 2022 at 14:03
• Ok, I'm starting to understand. So I should read that as 3 hours, 5 minutes, and 53 seconds after the sun reached its peak? What do the 29. 23. 48 and the other times refer to? Commented Jan 26, 2022 at 18:49

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).

• I do not understand why did you repeat my answer in an expanded form. Commented Jan 27, 2022 at 3:00
• @Alexandre Eremenko : well with all respect, I don't see in your post an answer to the question 'how did it work' (i.e. the Jupiter-moon eclipse method), so the details and working reference-doc that I offered don't seem to repeat what you said. For the rest, I apologise for overlooking anything. Commented Jan 27, 2022 at 16:10
• @terry-s Thank you very much, the references are quite useful. Commented Feb 1, 2022 at 20:29