The Earth based units of length and the birth of the metric system

In the middle of 1790 there was quite an international convergence towards a unit of length based on the pendulum. Nevertheless, at the beginning of spring of the following year a different unit of length was chosen by the French Academy of Sciences, thus signing the end of the seconds pendulum as length standard and causing a setback of the international cooperation on units of measurements.

Table: Chronology of units of length based on nature: from the seconds pendulum to (a fraction of) the light second.

1644	Mersenne writes about basing the unit of length upon a pendulum length.
1660	The length of the seconds pendulum is proposed for unit of length
	by the Royal Society, upon a suggestion of Huygens and Rømer.
1668	Picard proposes the universal foot, equal to 1/3 of the length
	of the seconds pendulum.
1670	Mouton proposes a unit of length equal to one minute of Earth's arc
	along a meridian (equal to present nautical mile).
1675	The catholic (i.e. universal) meter, equal to the length of a seconds
	pendulum, is proposed by Burattini, together with a a complete
	system of inter-related units based on the second.
1720	J. Cassini proposes the radius of Earth as unit of length (whose
	practical unit would be its ten millionth part).
1774	La Condamine proposes a unit of length equal to the length
	of the seconds pendulum at the equator (439.15 lignes).
1790, April	Talleyrand proposes to the National Assembly a unit of length equal
	to the seconds pendulum at $45^\circ$ latitude (440.4 lignes).
1790, July	Jefferson proposes to the U.S.A. House of Representative a unit of
	length equal to a second rod at $45^\circ$ latitude.
1790, August	Talleyrand's proposal is recommended to king Louis XVI by the
	Committee on Agriculture and Commerce. As a result, the Academy of
	Science is entrusted of the reform of measures and weights.
1790, October 27	A first commission recommends a decimal scale for all measures,
	weights and coins.
1791, March 19	A second commission, set up on February 16, chooses the quarter of
	meridian as natural unit of length, and its $1/10 000 000$ as practical unit.
1791, March 26	The National Assembly accepts the recommendations of the commissions.
1791, April	The Academy of Sciences initially confides the new measurement of the
	meridian to Méchain, Legendre and J.-D. Cassini (the latter two resigned).
1792, June	Led by Delambre and Méchain, the meridian expedition finally begins
1793, May 29	A third commission provides a provisional length of the meter as 443.44
	lignes (based on the 1740 measurements by Lacaille and Cassini de Thury).
1793, August 1	The provisional meter is adopted by decree.
1795, April 7	The Decimal Metric System (also including including units of weight,
	surface and volume -- but not the unit of time) is established by law.
1798,	The meridian expedition ends.
1799, March	An international commission fixes the length of the meter in 443.296 lignes.
1799, June 22	The meter standard, a platinum bar, is officially presented in Paris.
1812	A hybrid system of measurements is decreed by Napoleon, in which
	old names can be used for non-decimal multiples and sub-multiples
	of the meter (e.g. 1 toise for `2 meter', etc.).
[1832	The second is adopted in the Metric System, breaking the decimal scheme.]
1837	The 1812 decree is repealed, banning old names in France.
1840	Non metric system units become penal offence in France.
1889	The meter is redefined as the length of the 1799 platinum bar, with
	no longer reference to the Earth meridian.
1960	The meter is redefined as $1 650 763.73$ wavelengths of the
	radiation corresponding to the $2p_{10}$-$5d_{5}$ transition of Krypton 86.
1983	The meter is redefined as the distance traveled in vacuum by
	electromagnetic waves in $1/299 792 458$ of a second.

According to revolution style, the pace was very rapid (see central frame of Table 2). In August 1790 the French National Assembly entrusted the reform to the Academy of Sciences. The Academy nominated a preliminary commission,¹⁶which adopted a decimal scale¹⁷ for all measures, weights and coins. The commission presented its report on 27 October 1790. A second commission¹⁸was charged with choosing the unit of length. The commission was set up on 16 February 1791 and reported to the Academy of Sciences on 19 March 1791. On 26 March 1791 the National Assembly accepted the Academy's proposals of the decimal system and of a quarter of the meridian as the basis for the new system and the adoption of the consequent immediate unit.

The guiding ideas of the French scientists are well expressed in the introduction to the document presented to the Academy:

The idea to refer all measures to a unit of length taken from nature has appeared to the mathematicians since they learned the existence of such a unit as well as the possibility to establish it: they realized it was the only way to exclude any arbitrariness from the system of measures and to be sure to preserve it unchanged for ever, without any event, except a revolution in the world order, could cast some doubts in it; they felt that such a system did not belong to a single nation and no country could flatter itself by seeing it adopted by all the others.
Actually, if a unit of measure which has already been in use in a country were adopted, it would be difficult to explain to the others the reasons for this preference that were able to balance that spirit of repugnance, if not philosophical at least very natural, that peoples always feel towards an imitation looking like the admission of a sort of inferiority. As a consequence, there would be as many measures as nations. (Ref. [2], pp. 1-2)

Three were the candidates considered by the commission:

• the seconds pendulum;
• a quarter of the meridian;
• a quarter of the equator.

The latter two units are based on the dimension of Earth. Indeed, Earth related units had had also quite a long history, though they were not as popular as the seconds pendulum, probably because their intrinsic difficulty to be determined.¹⁹Mouton had suggested in 1670 [25] the unit that we still use in navigation and call now nautical mile: the length of one minute of the Earth's arc along a meridian, equal to 1852m.²⁰ In 1720 the astronomer Jean Cassini had proposed the radius of Earth [27], a `natural' unit for a spherical object (he had also indicated the one ten-millionth part of the radius as the best practical unit). However, neither of these old, French proposals are mentioned in the report of the commission.²¹

The quarter of the equator was rejected, mainly because considered hard to measure²² and somehow `not democratic'.²³

So, we believe we are bound to decide to assume this kind of unit of measure and also to prefer the quarter of the meridian to the quarter of the equator. The operations that are necessary to establish the latter could be carried out only in countries that are too far from ours and, as a consequence, we should have to undertake expenditures as well as to overcome difficulties that would be superior to the advantages that seem to be promised. The inspections, in case somebody would like to carry them out, would be more difficult to be accomplished by any nation, at least until the progress of the civilization reaches the peoples living by the equator, a time that still seems to be unfortunately far away. The regularity of this circle is not more assured than the similarity or regularity of the meridians. The size of the celestial arc, that corresponds to the space that would be measured, is less susceptible to be determined with precision; finally it is possible to state that all peoples belong to one of Earth's meridians, while only a group of peoples live along the equator. (Ref [2], pp. 4-5 )

The pendulum was rejected after a long discussion (two pages over a total of eleven of the document -- the quarter of equator is instead ruled out in less than half a page). As a matter of fact, the commission acknowledges that

the length of the pendulum has appeared in general to deserve preference; it has the advantage of being the easiest to be determined, and as a consequence to be verified. (Ref [2], p. 2)

Then the report specifies that the pendulum should be a simple pendulum at the reference latitude of $45^o$, because ``the law followed by the lengths of simple pendula oscillating with the same time between the equator and the poles is such that the length of the pendulum at the forty-fifth parallel is precisely the mean values of all these lengths''.²⁴ (Ref. [2], p. 3)

There was still the problem of the reference time of the pendulum, since the second was considered an ``arbitrary subdivision of this natural unit [the day]'' [2]. But a possible way out was envisaged:

In reality, we could avoid the last inconvenience taking as unit the hypothetical pendulum that made just one oscillation in one day; its length, divided by ten billion parts, would give a practical unit of measurement of about twenty seven pouces [27pouces$\approx$73cm]; and this unit would correspond to the pendulum that makes one hundred thousand oscillations in one day.²⁵ (Ref. [2], p. 4)

Essentially, the report of the commission does not provide any specific weakness of the pendulum²⁶ and, finally, the choice of the quarter of the meridian is justified only in terms of `naturalness', as it was perceived by Borda and colleagues:

[...] one would still have to include an heterogeneous element, time, or what is here the same thing, the intensity of the gravitational force at the Earth's surface. Now, if it is possible to have a unit of length that does not depend on any other quantity, it seems natural to prefer it.²⁷
[...]
Actually, it is much more natural to refer the distance between two places to a quarter of one of the terrestrial circles than to refer it to the length of the pendulum.
[...]
The quarter of the Earth meridian would become then the real unit of length; and the ten million-th part of this length would be its practical unit. (Ref [2], pp. 4-5)

It must be stressed that, however, there was still a strong resistance from those scientists who preferred the pendulum [10,29].

The new unit was officially called `meter'²⁸ only two years later (see section 5.2), in the occasion of a report of a new commission, in which also for the first time an estimate of its length was made public by Borda, Lagrange and Monge [32,14].