The discovery of stellar aberration was empirical evidence that the Earth orbits the Sun; finding empirical evidence that the Earth rotates daily on its axis proved, perhaps surprisingly, difficult. The first indirect evidence for diurnal rotation in interesting in two ways. Firstly, it is based, not on a single theory but on a chain of interdependent theories. Secondly, it is an interdisciplinary proof involving physics, astronomy, geophysics and geodesy.

That the Earth was a sphere had been accepted in educated European circles since at least the fifth century BCE. The acceptance of this knowledge automatically led to attempts to estimate or in fact measure the size of that sphere. Aristotle claimed that mathematicians had measured the circumference of the Earth to be 400,000 stadia (between 62,800 and 74,000km) which is far to large. Archimedes set an upper limit of 3,000,000 stadia (483,000km), making Aristotle look almost reasonable. One of the earliest serious attempts to measure the circumference of the Earth was that of Eratosthenes, which now has legendary status. It is reported that he calculated a figure of 250,000 stadia. What is not known is which stadium he was using so the error in his value lays somewhere between about 2% and 17%. Eratosthenes was by no means the only thinker in antiquity to give a calculated figure for the Earth’s circumference. Posidonius produced a value, which varies considerably in size in the literature in which it is quoted. Ptolemaeus gives two completely different values 252,000 stadia in his *Mathēmatikē Syntaxis* and later 180,000 stadia in his *Geōgraphikḕ Hyphḗgēsis*. In the Middle Ages, the Indian mathematician, Aryabhata, calculated a value for the Earth’s diameter of 12,500km. Islamic scholars also produced varying figures, most famously al-Khwarizmi and al-Biruni. Up till the Early Modern Period nobody could actually say, which of the various values, that were floating around in the available literature, was the correct one, Columbus famously chose the wrong value.

The basic method of determining the circumference of the Earth is to determine the length of a stretch of a meridian, a line of longitude through both poles, and then determine how many degrees of latitude this represents. From this data it is then possible to determine the circumference. This process took a major turn in accuracy with the invention, by Gemma Frisius (1508–1555), of triangulation in the sixteenth century. This meant that it was now possible to exactly measure the length of a stretch of a meridian and by taking the latitudes of the ends of the stretch to determine the length of one degree of latitude.

The first mathematicus to try and determine the circumference of the Earth using triangulation was the Dutchman Willebrord Snel (1580–1626), who carried out a triangulation of the Netherlands in the early part of the seventeenth century. He published the results of survey in his *Eratosthenes Batavus, De Terræ ambitus vera quantitate *in 1617.

The first part of the title translates as the Dutch Eratosthenes. Taking the distance between Alkmaar and Breda, which almost lie on the same meridian, he calculated one degree of latitude to be 107.37km giving a circumference of 38,653km, an error of about 3.5%.

Later in the seventeenth century the French astronomer Jean-Félix Picard (1620–1682) now triangulated a meridian arc through Paris, between 1669 and 1670, calculating a value for one degree of latitude of 110.46km producing values for the Earth’s polar radius and circumference with more than 99% accuracy.

In 1672 Jean-Dominique Cassini (1625–1712) made an attempt to measure the parallax of Mars in order to determine the astronomical unit, the distance between the Earth and the Sun.

He sent his assistant Jean Richer (1630–1696) to Cayenne in French Guiana, so that he and Cassini could make simultaneous observations of Mars during its perihelic opposition. We shall return to this in a later episode, but it is another experiment or better said discovery of Richer’s, whilst in Cayenne, that is of interest here. Richer was equipped with all the latest equipment including a state-of-the-art pendulum clock with a seconds pendulum, that is a pendulum whose period is exactly two seconds, or at least it was a seconds pendulum when calibrated in Paris. Richer discovered that in Cayenne that he needed to shorten the pendulum by 2.8mm. As gravity is the driving force of a pendulum clock this meant that the Earth’s gravity was different in Cayenne to in Paris or that Cayenne was further from the Earth’s centre than Paris. The Earth was not, after all, a sphere.

Jean-Dominique Cassini and later his son Jacques (1677–1756) extended Picard’s Paris meridian northwards to Dunkirk and southwards to the Spanish border.

They split the meridian into two and compared lengths for one degree of latitude thus obtained, combining the results with Richer’s pendulum discovery, they proposed and defended the theory that the Earth was not a sphere but a prolate spheroid or an ellipsoid created by rotating an ellipse along its major axis; put in simple terms the Earth was lemon shaped. Jacques Cassini published these results and this theory in his *De la grandeur et de la figure de la terre* in 1723.

Both Newton and Huygens interpreted Richer’s pendulum discovery differently. Newton arguing from an assumption of diurnal rotation and his theory of gravity theorised that the Earth was in fact flattened to the poles and a bulge at the equator. That is the Earth is an oblate spheroid or ellipsoid created by rotating an ellipse along its minor axis, put in simple terms the Earth was shaped like an orange. Huygens also arguing from an assumed diurnal rotation but Descartes’ vortex theory, rather than Newton’s theory of gravity, arrived at the same conclusion. What is important here is that the theory depended on the existence of diurnal rotation.

Given the already strident philosophical debate between the largely French supporters of Descartes and the largely English supporters of Newton, this new dispute between the Cassini, Cartesian, model of the Earth and the Newton-Huygens, Newtonian model, Huygens actually a Cartesian was here viewed as a Newtonian, rumbled on into the early decades of the eighteenth century. Finally, in the 1730s, the Académie des sciences in Paris decided to solve the issue empirically. They equipped and sent out two scientific expeditions to Lapland and to Peru, now part of Ecuador, to measure one degree of latitude.

The expedition to Meänmaa or Torne Valley in Lapland

under the leadership of Pierre Louis Maupertuis (1698–1755)

took place successfully in 1736-37, despite atrocious conditions, and their results combined with the results of the Paris meridian showed that the Newton-Huygens model was indeed correct.

Maupertuis published his account of the expedition *La Figure de la Terre, déterminée par les Observations de Messieurs Maupertuis, Clairaut, Camus, Le Monier & de M, L’Abbé Outhier accompagnés de M. Celsius*. (Paris, 1738).

Jacques Cassini launched a last-ditch attempt to defend his father’s honour and wrote a scathing criticism of the expeditions work in his *Méthode de déterminer si la terre est sphérique ou non* (*Method to determine if Earth is a sphere or not*) in 1738. However, the Swedish scientist Anders Celsius (1704–1744), who had also taken part in the expedition completely demolished Cassini’s paper and the Newtonians, of whom Maupertuis although a Frenchman was one, carried the day. Celsius’ *De observationibus pro figura telluris determinanda* (*Observations on Determining the Shape of the Earth*) from 1738 made his reputation.

The second expedition to Peru under the leadership of Pierre Bouguer (1698–1758)

and Charles Marie de La Condamine (1701–1774)

actually left Paris a year earlier that the Lapland expedition in 1735. This team had even more difficulties than their northern colleagues and only returned to Paris in 1744. Their results, however confirmed those of the Lapland expedition and the Newton-Huygens oblate spheroid. Bouguer published his account of the expedition in his *La figure de la terre* (1749),

La Condamine his *Journal du voyage fait par ordre du roi, a l’équateur, 1751*.

Through these two expeditions the Earth had acquired a new shape, it was no longer a sphere but an oblate spheroid, an important advance in the history of geodesy. However, possible more important, because the prediction of the Newton-Huygens model was based on the assumption of diurnal rotation, these results produced the first indirect empirical evidence that the Earth rotates around its own axis. This result combined with the return of Comet Halley in 1759 also led to the final general acceptance of Newtonian theory over Cartesian theory.

Voltaire said of Pierre Louis Maupertuis, the leader of the Lapland expedition, that he had “flattened both the earth and the Cassinis.”

Wonderful quote, can you give a source for that?

Since I had already seen the quote floating around, I was intrigued by this and looked for the source. I could not find the exact formulation, but Voltaire does greet Maupertuis in a letter with “my dear flattener of worlds and Cassinis”.

Here is a link to the letter (in french, August 10, 1741). The relevant part is in the first sentence. https://fr.m.wikisource.org/wiki/Correspondance_de_Voltaire/1741/Lettre_1465

A couple of quick-fix suggestions:

– You say of Eratosthenes measurement that “his value lays somewhere between about 2% and 17% of the correct value” but I think you mean that the ERROR in his value is between 2% and 17% of the correct value. I know you are trying to say that he almost got it right, but a literal reading of your words would imply that has value was far too small.

– Your image for La Condamine’s

Journalis actually just a repeat of the image for Bouguer’sLa figure.Setting aside the issue of which stadium Eratosthenes used, I have always found it fascinating that he made two errors that almost perfectly cancelled. Alexandria and Syene (modern Aswan) are not quite on the same meridian, but also Syene was not directly on the Tropic of Cancer as Eratosthenes thought based on the sunlight reaching the bottom of the well story. Neither of those errors would have been particularly large so he would have gotten a decent result anyway (assuming he had an accurate distance from Syene to Alexandria) but the fact that they almost perfectly cancel is neat.

I’d love to know more about how the Cassini’s came up with their prolate spheroid theory. As you say, Richer found that he had to shorten his seconds pendulum in Cayenne which is much closer to the equator than Paris. With a modern understanding of how pendulum’s work (I taught a lab on this yesterday) we know that means the (effective) gravitational field strength is slightly weaker in Cayenne than in Paris. I would think that would suggest that Cayenne is farther from the center than Paris is, which would mean an oblate Earth. I know the Cassinis were not using Newtonian gravity, but I’d love to know how they thought this implied a prolate Earth. Or did they mostly just use their geodetic measurements and not pay much attention to the pendulum results?

Thanks for this great summary of a fascinating episode in the history of science.

There is a Italian scholar (Lucio Russo) that some times ago suggested a new hypothesis on how the measure of Eratosthenes can be so accurate (spoiler: it is not that the errors miraculously cancels each other) .

The paper is https://www.researchgate.net/publication/236629491_Ptolemy's_longitudes_and_Eratosthenes'_measurement_of_the_Earth's_circumference.

Russo’s work is at best dubious, at worst complete crap

Why do you say so? I’m quite a fan of the Russo’s works (I just like history of science/mathematics, I’m not absolutely a storician of science, professional nor amateur), so it surprise me to read such an opinion. Can you elaborate more? I’m really interested in it (it would be wonderful if you write a post on this theme but I don’t think you will do so).

P. S. Sorry for my bad English but it’s not my first language.

Thx for the heads up on the illustrations! That I’m capable of overseeing that when editing is remarkable!

The big problem with the whole Eratosthenes story is that we only have secondary sources written literally centuries after the event.

You are of course right on the Cassinis. If the Earth were indeed prolate, then Paris would be further from the centre than Cayenne, so Richer’s result actually contradicts their model.

In context, Archimedes’ upper bound isn’t so off-the-wall, as an upper bound. It’s from

The Sand Reckoner:Having set this goal, Archimedes lays out his premises. To make his argument utterly compelling, he begins:

Actually, Richer didn’t, strictly speaking, discover the need to shorten the pendulum, since finding out the length of the seconds pendulum at Cayenne was one of the tasks he was given. This had been suggested by Auzout in 1667 and done by Picard on an expedition to Uraniborg in 1671. What Richer did was to find the length of a simple pendulum (mass on a string) that would match his seconds-beating clock, this simple pendulum was simply allowed to decay over time. (My sources for this are a 1943 Isis article by Olmstead, and Richer’s own memoir–the pendulum results are at the end). Trying to find the length of a simple pendulum from the length of a clock pendulum is, well, not simple, because a clock pendulum has a complicated mass distribution, a spring at the top, and an escapement driving it.

As always, thanks for the lucid account. I hadn’t ever thought of the pendulum results as proving rotation (I would have thought of Foucault instead) but you are quite right.