Ferguson Tidal Clock

Originally posted on the RetroMat website, but fits better here.

Source: The Antique Clock Company

In the August 2022 edition of Clocks Magazine, there is an article by Brian Loomes about the astronomical clocks with tidal indications of Phillip Lloyd based on a mechanism devised by astronomer James Ferguson.

One of the clocks shown in the article - the Hotwells one - was for sale as at August 2022 (and still seems to be so as at May 2026). The dial is a thing of great beauty, although perhaps not intuitive to understand.

There is a very good explanation of the dial and mechanism used by Ferguson, albeit in dated language, in the Encyclopaedia Londinensis published in 1811, which I've copied into this document for ease of reading, with some alterations:

• the low-resolution black and white diagrams accompanying the text have been replaced with high resolution images of the original colour plates by J. Pass, 1809, from the Wellcome Collection
• annotations have been made to one of the figures and some footnotes added
• updated spelling of words where the archaic long 's' is used.

With regard to the Hotwells clock by Lloyd: the indicator used by Ferguson to show high water is not present, perhaps having been replaced by the pair of 'spikes' on opposite sides of the central disk representing the Earth. If this is so, then because these are offset to the 'tidal bulge' around the Earth, this depiction corrects the misrepresentation of high water in the Ferguson clock.

However, such a device would seem to be less accurate in use, being some distance from the dial on the sun plate.

Also, the Hotwells clock has two additional, centrally-mounted hands. Presumably the straight hand is for showing seconds, with the other for minutes, the hours being indicated by the short hand with the sun symbol mounted on the sun plate.

Plimpton 322 clay tablet

Originally posted on the RetroMat website as part of a page on programming the Sinclair Spectrum to generate Pythagorean Triples.

Source: wikipedia

Plimpton 322 is a Babylonian clay tablet containing a list of triangles composed of Pythagorean Triples, and their associated properties, in the form of a table with 4 columns and 15 rows ordered in descending sequence of (short side/long side) squared. 

These triples can be generated by using a method, such as the one we know as Euclid's Formula, and the following analysis refers to using that formula. Presumably the Babylonians used a similar method.

There are 15 triples implied on the tablet (it lists the short sides and hypotenuses, but not the longer sides). Of these, 13 are primitive. Eight have c > 1000 and the largest has a = 12709 and c = 18541. These are large triangles! (To generate the largest, m=125 and n=54). 

(I wonder what method they used to determine that these are co-prime and if they knew about what we now call Euclid's Algorithm? (Not to be confused with the above similarly-named formula!))

The tablet is broken and it is possible that there are missing columns at the left. Angles aren't used, but can be determined. If we define the angle as being next to the long side and hypotenuse, then angles* range from ~45 degrees at the top of the table to ~32 degrees at the bottom. The difference between the angles associated with successive rows is just under one degree on average, but actually varies from less than half a degree to nearly two and there seems to be no pattern to this.

Why were those particular triples chosen? I found that by increasing the maximum value for m to the 125 needed for the largest triangle in the tablet, the formula generates 441 triples in this range, providing many more suitable candidates for a more evenly-spaced table.

Why this range of angles? Perhaps it is one of a set - four such tablets could cover angles from 0 to 45 degrees (albeit with 60 rather than 45 entries!). For 45 to 90 degrees, the triples could be reused, switching the short and longer sides.

Was the tablet intended to be used as a trigonometric table, as we understand that term, with angles being present originally in a column now missing? (Presumably, angles would have had to be obtained by direct measurement of suitably scaled-down triangles). If so, it predates those of the Ancient Greeks, who are usually regarded as having developed the first ones, by 1500+ years! However, such a table would have been of limited practical use:

• the angles, if present, would not have been 'tidy' integers or rational numbers 
• there is an uneven difference in angle between successive entries
• the number of entries (15) also counts against (would have expected 14 for the range given).

Perhaps instead they were searching for triples that could produce (near) whole integer angles and this was a work in progress, a prototype of trig. tables to come?

Regardless, what I find truly incredible, perhaps even more than the information it contains, is that in spite of its great antiquity, it is in a form that we recognise today - an ordered numbered table consisting of rows and columns!

* angles of the triangles associated with the triples: 44.76, 44.25, 43.79, 43.27, 42.07, 41.54, 40.32, 39.77, 38.72, 37.44, 36.87, 34.98, 33.86, 33.26, 31.89.

Earth's Keplerian system & towards universal gravitation

Updated 31 May 2026

Having investigated the Solar, Jovian and Saturnian Keplerian Systems, let's take a closer look at home and start with trying to determine the value of its Kepler 'constant' (should it exist) by looking at the Moon's orbit about the Earth.

It's at a distance (measured with reasonable accuracy by Ptolemy nearly 2000 years ago) of about 60 Earth radii and takes about 27 days to complete one orbit.

From this we can say the value of the Kepler 'constant' for the Earth is ~ 60³/27² ~ 296.

In the case of the other systems, there are many orbiting bodies and we can calculate an average, but in this case, if we go back many decades at least, there is only one such body orbiting the Earth.

However, let's take inspiration from Newton, who realized that an apple falling to the ground also experiences an acceleration towards the centre of the Earth. This acceleration is commonly referred to as g and can be measured accurately by means of a pendulum, length l, the period of which is 2π√(l/g).

We can generalize the formula from the previous post and say that the acceleration at distance d, regardless of whether the object is orbiting, is:

4π²k/d².

At the Earth's surface, we state in this case that d = 1 (the same as if all the Earth's mass were concentrated at its centre) and g = 4π²k, where k is the Kepler 'constant' for the Earth. If we know g, we can therefore determine that:

k = g/(4π²).

In metric units g ~ 9.8 m/s² and we need to convert this to the units used when calculating the value from observation of the Moon, i.e. Earth radii/day²:

• the radius of the Earth (also measured, albeit in different units, with reasonable accuracy by the Ancient Greeks) is ~ 6.4 x 10⁶ m and we divide our value of g by this number
• there are 86400 seconds in a day, squared is ~ 7.5 x 10⁹ and we multiply the above result by this number.

In these new units g ~ 11719 and dividing by 4π² gives ~ 297, which is very close indeed to the value we derived earlier.

We can conclude therefore that the Earth, as with the Sun, Jupiter and Saturn also has its own Keplerian system, where the central body exerts a gravitational force that follows an inverse-square law.

From earlier considerations, that force appears to be related to the mass of the central object (Sun, Jupiter etc), possibly simply proportional to it, in which case the Kepler 'constant' for the system is also proportional to the mass of the central object.

We can surmise therefore that all massive bodies, including those bodies without satellites of their own, exert such a gravitational force on surrounding matter. (Surface features on the Moon, for example, and the near spherical shape of all planets, including those without moons, provide strong evidence for this).

This was to be confirmed experimentally by Henry Cavendish in the laboratory at the end of the 18th century - an incredible achievement for the time given the weakness of the gravitational force. 

From Kepler's third law to the inverse-square law

When a small object orbits a massive one, such as a moon going about its planet, say, it experiences an acceleration towards it. In perfect circular motion (an approximation), that acceleration is: r⍵²,where r is the radius of the orbit and ⍵ is its angular speed, which is 2π/t, where t is its orbital period. We know this from Newton.

We know also from Kepler's third law that

r³/t² = k,

where k is the Kepler 'constant' for that system. 

Hence the acceleration 

r⍵² = r4π²k/r³ = 4π²k/r²

This is the famous inverse-square law for the gravitational force (also from Newton of course).

The Saturnian Keplerian system

Decades after the discovery of Jupiter's moons, Saturn was observed to have its own satellites: the first being Titan, the largest, in 1655.

Let's try to compute the Kepler value for the three largest, using the same method as for Jupiter, noting however, that accuracy and precision would likely have been low in the early days, so the figures below (especially its density) may not have been achievable until much later.

There is no orbital resonance with these, so the table shows the period in days.

(Rhea's orbit is actually about 4.5 Saturnian diameters. Were we able to estimate to this precision, then we should expect the Kepler value to be ~4.5).

We see that the values in the last column suggest a Kepler 'constant' for Saturn of ~ 4 and hence that Saturn, as well as the Sun and Jupiter, also has its own gravitational system obeying the same laws.

Converting this to Solar system units:

• as Saturn's angular diameter at opposition is ~ 1/3 minute of arc and it's 8.4 AUs away, it's diameter is ~ 0.0008 AUs
• we cube this:  ~ 5 x 10^(-10) and multiply our Kepler constant by it
• 1 day is 1/365 years, so we square this ~  7.5 x 10^(-6) and divide the above result by it

Hence, the average Kepler 'constant' for Saturn is ~ 0.0003.

We deduce that its gravitational pull is ~ 0.0003 times that of the Sun and ~ 0.3 times that of Jupiter. 

We suspect its mass is also in the same proportions. That being the case, as we know its diameter is ~ 0.8 of Jupiter's, its volume is therefore around half of Jupiter's and its density ~ 0.6 times Jupiter's, indicating a different composition.

Moon	Orbital radius (r) in Saturnian diameters	r^3	Period (t) days	t^2	r^3/t^2
I. Rhea	4	64	4.5	20	3.2
II. Titan	10	1000	16	256	4
III. Iapetus	30	27000	79	6241	4.3

Jupiter vs. the Sun

To compare the Jovian Kepler constant with the Solar one, we must convert it to the same units (AUs cubed over years squared):

• we determine the diameter of Jupiter is ~ 0.001 AUs by estimating* its angular diameter is about 3/4 of one minute of arc at opposition when we know the planet is 4.1 AUs away (its orbital radius minus that of the Earth's)
• we cube this: ~ 10⁻⁹ and multiply our Jovian Kepler constant by it
• we determine the period of moon I (Io) is ~1.8/365.25 or 0.0049 years
• we square this: ~ 2.4 x 10⁻⁵ and divide the above result by it

(* If there were two stars of known angular separation visible in the eyepiece of the telescope at the same time as Jupiter, their presence could provide a useful yardstick against which to make a reasonably accurate estimate).  

Hence, the average Kepler constant for the Jovian system is ~  0.001, i.e. 1/1000th of the value (1) for the Solar System. This means that for two different objects orbiting at the same distance from the Sun and from Jupiter, in the former case the object's speed is 10√10 times that of the latter. (Similarly, if they have the same orbital period, then the object orbiting the Sun does so at 10 times the distance of that orbiting Jupiter). 

We conclude that the Sun's gravitational pull is much stronger than Jupiter's and that the ratio of their Kepler constants is a measure of this.

Given its angular diameter at opposition is as stated earlier, then we conclude that at the distance of the Sun (1 AU), it would be 4.1 times larger, i.e. ~ 0.05°, compared with the Sun's  ~ 0.5°. That is, Jupiter's diameter is about 10% that of the Sun and its volume is about one-thousandth, i.e. about the same as the ratio of its Kepler constant to the Sun's.

Here then is a first inkling about what may be behind the strength of a body's gravitational pull. Volume seems an unlikely candidate as we shouldn't expect a diffuse cloud of gas to have the same effect as a solid object of the same size. Perhaps the gravitational pull is simply proportional to its mass then? If so, then Jupiter's density is about the same as the Sun's. And yet, what different bodies they seem to be!

The Jovian Keplerian system

Updated 11 May 26

The discovery of Jupiter's moons through the newly-invented telescope back in the early 1600s must have caused great excitement at the time and a wish to investigate further to see if the Jovian system follows Kepler's laws.

However, imagine trying to track these moons individually over many days and weeks in an attempt to measure the orbital radius and orbital period for each!

The orbital radius could be estimated crudely (being imprecise and prone to human error) using the diameter of the planet as the base unit, e.g. the innermost moon being found to orbit at around 3 Jovian diameters from the centre of the planet.

I've tried this myself on just a few occasions with binoculars and it was difficult! Doing this to an acceptable level of accuracy and precision in the early 17th century would have demanded the best optics of the time, a solid mount, good skies over many weeks and an experienced and patient observer. And then no doubt repeating the exercise again and again. 

The orbital period could be measured with more precision by timing the intervals between successive identical locations in a moon's orbit (e.g. maximum elongation), although we should bear in mind that this was before the invention of the pendulum clock.

(Would we have to allow for the movement of the Earth introducing errors due to parallax?)

See the table at the bottom of the page for an investigation. (The table had to be imported and in this case, if not placed there, messes up the font of the text following it!). The moons are numbered according to their proximity to Jupiter.

It turns out that the orbital periods of moons I, II and III are in the ratio 1:2:4, so I'm using the orbital period of moon I as the unit of time here.

The average estimate of the Jovian Kepler 'constant' (last column) is 26, but with a large range of values, no doubt due to the lack of precision in measuring orbital radii in whole Jovian diameters, which are then cubed. (In practice, there would be even greater variation with this method, because, unlike the case here, estimates would likely be inaccurate, as well as imprecise). However, these values are of a similar order of magnitude and a more precise and more accurate method should yield results that are more consistent*. Therefore, we tentatively assert that these moons do indeed obey Kepler's third law.

(* If instead we were to estimate accurately using the radius of Jupiter as our measuring unit, then we should find that the Jovian Kepler constant has an average of 202 with a proportionately smaller range of 182 to 216. However, I expect this advantage would have been easily removed by increased human error, in estimating, say, that Callisto orbits at 26 units from Jupiter! I imagine that the later invention of the reticule would have been revolutionary for making accurate and precise measurements). 

Moon	Orbital radius (r) in Jovian diameters	r^3	Period (t) as a multiple of that of moon I	t^2	r^3/t^2
I. Io	3	27	1	1	27
II. Europa	5	125	2	4	31
III.Ganymede	7	343	4	16	21
IV. Callisto	13	2197	9.4	89	25