Measurements of the Earth meridian before 1791

Measuring the size of Earth had been a challenging problem for ages, since it was first realized that Earth is spherical, i.e. at least by the sixth century B.C. [24]. The most famous ancient estimate is that due to Eratosthenes (276-195B.C.), who reported a value of 250000 stadia, i.e. about $\approx 40 000$ kilometers, if we take 159m per stadium.²⁹

Table: Some milestones in measuring the Earth meridian. $l_m$ stands for the length of the meter calculated as the 40000000th part of the meridian (for some important cases, $l_m$ is also given in lignes -- see Table 1 for conversion). In the results expressed in the form ` $\mbox{xxxxx}\times 360^o$' xxxxx stands for the length of one degree meridian arc ($s/\alpha$ in the text). Ancient estimates have to be taken with large uncertainty (see e.g. Ref.[24]).

Author(s)	Year	Value	Unit	km value	$l_m$ (m)
					{lignes}
Eratosthenes	(III B.C.)	250000	stadium$^a$	$\approx 40000$	$\approx 1.0$
Caliph Al-Mamun	820	56 $^2$/3	Arab mile$^b$	39986	0.9997
Fernel	1525	$56746\times 360^o$	toise	39816	0.9954
Snellius	1617	$55100\times 360^o$	toise	38661	0.9665
Norwood	1635	$57300\times 360^o$	toise	40204	1.0051
Picard	1670	$57060\times 360^o$	toise	40036	1.0009
J. Cassini	1718	$57097\times 360^o$	toise	40062	1.0016
Lacaille and	1740	$57027\times 360^o$	toise	40013	1.00033
Cassini de Thury					{443.44}
[Lapland expedition	1736	57438$\times$360$^o$	toise	40302	1.0075]$^c$
[Peru expedition	1745	56748$\times$360$^o$	toise	39817	0.9954]$^c$
Delambre and	1799	20522960	toise	40000	1
Méchain					{443.296}
		[57019$\times$360$^o$	toise	40008	1.00019 ]$^d$
					{443.38}
present value		40009152	m	40009.152	1.000229
					{443.3975}
$a)$ Stadium estimated to be 159 m.
$b)$ Arab mile estimated to be 1960 m [24].
$c)$ Values obtained at extreme latitudes, very sensitive to Earth ellipticity.
$d)$ The entries of this line assume a spherical model for Earth, as for older estimates.
The value of 57019 toises per degree is obtained dividing the $551 584.74$ toises of the
meridian arc Dunkerque-Barcelona by their difference in latitude, $9^o 40^\prime 25.40^{\prime\prime}$[35] .

The principles of measurement of the Earth parameters, exposed at an introductory level, as well as milestones of these achievements, can be found in Ref. [24]. The basic idea is rather simple: if one is able to measure, or estimate somehow, the length of an arc of meridian ($s$) and its angular opening ($\alpha$), the length of the meridian can be determined as $360^o\times s/\alpha$, if a circular shape for the meridian is assumed (i.e. for a spherical Earth). The angle $\alpha$ can be determined from astronomical observations. The measurement of $s$ is bound to the technology of the epoch and varies from counting the number of steps in the early days to modern triangulation techniques [24]. Indeed, the rather accurate measurements between the 16th and 18th centuries provided the results in terms of $s/\alpha$, expressed e.g. in toises/degree (see Table 3).

As we can see in Table 3, not exhaustive of all the efforts to pin down the Earth dimensions, there was quite a convergence on the length of the unitary meridian arc in Europe, as well as a general consistency with older measurements. For example, taking the Lacaille and Cassini de Thury result, based on the about 950 km arc of the Paris meridian³⁰ across all France, it is possible to calculate a value of 443.44 lignes for the new unit of measurements, assuming a spherical Earth ( $57027\times 90^o / 10 000 000 \times 864=443.44$). Even a conservative estimate would give a value of 443.4 lignes, with an uncertainty on the last digit -- a difference of three lignes (about 6 mm) with respect to the length of the second pendulum. That was definitely known to Borda and colleagues.

However, apart from experimental errors in the determination of the 57027 toise/degree, the value of 443.44 was still affected by uncertainties due to the shape of Earth. At that time the scientists were rather confident on an elliptical shape of the meridians, resulting from the Earth flattening at the poles. In fact, centrifugal acceleration due to rotation is responsible for the bulge of the Earth at the equator. The resulting Earth shape is such that the total force (gravitational plus centrifugal) acting on a body at the Earth surface is always orthogonal to the `average' surface of the Earth. If that were not the case, there would be tangential forces that tended to push floating masses towards the equator, as eloquently stated by Newton: ``...if our Earth were not a little higher around the equator than at the poles, the seas would subside at the poles and, by ascending in the region of the equator, would flood everything there.'' (Cited in Ref. [33].) Newton had estimated an Earth ellipticity of 1/229[34].

Several measurements had been done during the 18th century to determine the value of Earth flattening. In particular, there had been an enormous effort of the French Academy of Sciences, that supported measurements in France as well as expeditions at extreme latitudes, up to the arctic circle and down to the equator.³¹

The latter measurements were essential in order to gain sensitivity on the flattening effect. In fact, the unitary arc length $s/\alpha$ gives the local curvature along the meridian around the region of the measurements. As a consequence, $\rho=s/\alpha \times 360^o/2\pi$ is equal to the radius of the circle that approximates locally the meridian ellipse.

Figure: An exaggerated representation of the ellipsoidal Earth shape, showing local circles at the equator and at the pole (the ellipse is characterized by the semi-axes $a$ and $b$). The ellipse of the figure has a flattening of about 1/2, i.e. an eccentricity of 0.87. (A flattening of 1/298, corresponding to the Earth one, i.e. a minor axis being 0.3% smaller than the major axis, is imperceptible to the human eye.) Note how the equatorial local circle underestimates the ellipse circumference, while the polar one overestimates it.

As it can be easily understood from figure 1, the curvature decreases with the latitude: the radius of the `local circle' is minimum at the equator and maximum at the pole. The measurements of arcs of meridian at several latitudes (two distant latitudes are in principle sufficient) can yield the ellipse parameters. Comparing the result of Lacaille-Cassini from Table 3 with the results of the Lapland and the Peru expeditions from the same table, we see that $s/\alpha$ is indeed increasing with the latitude (note these measurements were quite accurate -- for a very interesting account of the Peru expedition see Ref. [37]). The combination of these and other measurements gave values of the Earth flattening in the range 1/280-1/310 [37], with a best estimate around 1/300, very close to the present value of 1/298 (see Table 4).

Table: Earth data [38]. The geometrical data refer to the WGS84 ellipsoid [37]. Note that the generic `radius of Earth' $R$ refers usually to the equatorial radius, but sometimes also to the `average' equivolume radius. In literature the name `ellipticity' is often associated to what is called `geometric flattening' in this table, and even to the difference of equatorial and polar radii divided by their average, i.e. $(a-b)/((a+b)/2)$. Anyway, the three different `ellipticities' give with good approximation the same number, about $1/298$, because of the little deviation of our planet from a perfect sphere. Note also that sometimes the flattening is even confused with the ellipse eccentricity, that differs quite a lot from flattening and ellipticity.

equatorial radius, $a$	6378137m
Polar radius, $b$	6356752m
Equivolume sphere radius	6371000m
Geometric flattening, $f=(a-b)/a$	1/298.26
Ellipticity, $(a^2-b^2)/(a^2+b^2)$	1/297.75
Eccentricity, $e=\sqrt{1-b^2/a^2}$	0.08182 = 1/12.2
(for $f\ll 1$, $e$ is about $\sqrt{2 f}$)
Mass, $M$	$5.97369\times 10^{24}$kg
Mean density	$5.5148\times 10^3$kgm$^{-3}$
Normal gravity at equator	$9.7803267$ms$^{-2}$
Normal gravity at poles	$9.832186$ms$^{-2}$
$G M$ (where $G$ is the gravitational constant)	$3.986005\times 10^{14}$m$^3$s$^{-2}$

However, given such a tiny value of the flattening (imagine a soccer ball squeezed by 0.7 mm), its effect on the circumference of the ellipse is very small, of the order of a few parts in $10 000$.

To summarize this subsection, we can safely state that the length of the meridian, and hence of any of its subdivisions, was known with a relatively high accuracy decades before the report that recommended the unit of length equal to $1/10 000 000$ part of the quarter of meridian was produced. In particular, for what this paper is concerned, it was well known that the new standard was equal to the length of the one seconds pendulum within about half percent.