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## 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 kilometers, if we take 159m per stadium.29

 Author(s) Year Value Unit km value (m) {lignes} Eratosthenes (III B.C.) 250000 stadium Caliph Al-Mamun 820 56 /3 Arab mile 39986 0.9997 Fernel 1525 toise 39816 0.9954 Snellius 1617 toise 38661 0.9665 Norwood 1635 toise 40204 1.0051 Picard 1670 toise 40036 1.0009 J. Cassini 1718 toise 40062 1.0016 Lacaille and 1740 toise 40013 1.00033 Cassini de Thury {443.44} [Lapland expedition 1736 57438360 toise 40302 1.0075] [Peru expedition 1745 56748360 toise 39817 0.9954] Delambre and 1799 20522960 toise 40000 1 Méchain {443.296} [57019360 toise 40008 1.00019 ] {443.38} present value 40009152 m 40009.152 1.000229 {443.3975} Stadium estimated to be 159 m. Arab mile estimated to be 1960 m [24]. Values obtained at extreme latitudes, very sensitive to Earth ellipticity. 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 toises of the meridian arc Dunkerque-Barcelona by their difference in latitude, [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 () and its angular opening (), the length of the meridian can be determined as , if a circular shape for the meridian is assumed (i.e. for a spherical Earth). The angle can be determined from astronomical observations. The measurement of 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 , 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 ( ). 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 gives the local curvature along the meridian around the region of the measurements. As a consequence, is equal to the radius of the circle that approximates locally the meridian ellipse.

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 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).

 equatorial radius, 6378137m Polar radius, 6356752m Equivolume sphere radius 6371000m Geometric flattening, 1/298.26 Ellipticity, 1/297.75 Eccentricity, 0.08182 = 1/12.2 (for , is about ) Mass, kg Mean density kgm Normal gravity at equator ms Normal gravity at poles ms (where is the gravitational constant) ms

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 .

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 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.

Next: The 1793 provisional meter Up: Establishing the length of Previous: Establishing the length of
Giulio D'Agostini 2005-01-25