...m/s.¹

A meter is the distance covered by light in vacuum in $1/299 792 458$ of a second.

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... wave.²

A second is equal to the duration of $9 192 631 770$ periods of the radiation corresponding to the transition between two hyperfine levels (F=4, M=0 and F=3, M=0 of the fundamental status $^2S_{1/2}$) of the atom Cesium 133.

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... quantity'' ³

All English quotes not referring to English bibliography are translation by the authors.

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... '' ⁴

The earliest measurement standard we have evidence of is the Egyptian cubit, the length of the forearm from elbow to fingers, realized around 2500 B.C. in a piece of marble of about 50 centimeters [5].

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... system,⁵

With this respect, changes of currencies are easily ruled by central banks, especially in modern times in which banknotes and coins have no intrinsic value.

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... fiasco.⁶

In history there are plenty of examples of this kind, like the difficulty of France to establish the decimal metric system or the failed Carolingian and Renaissance reforms [9]. Just to make a recent and practical example, Italy has adopted the International System (SI) since the mid seventies, and other units were banned by law. Nevertheless, after thirty years, though the SI unit of power is the Watt, car power is quoted in HP (yes, official documents do have kW, but drivers, sellers and media only speak of HP), centralized home heating power in kCal/h, (but small electric heaters are given in Watt) and air conditioning cooling power in Btu/h. What is bad is that, contrary to centimeters and inches, where people know that the units measure the same thing but citizens of different countries have a mental representation in either unit, the average Italian does not even know that all the above units measure the same thing and practically none has a mental representation of a Btu/h, arrived to us in mass with air conditioners in the last few years. Therefore, people don't even suspect that it is possible to convert Btu/h to Watt to have a better perception of what 7000 Btu/h might mean (a cooling power of about 2kW).

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... others.⁷

For example, this was one of the possibilities envisaged in France before the metric system: to extend to all France the system used in Paris. This was still the proposal of Joseph-Jérôme Lalande in April 1789. And he strenuously defended it later against the meter. Though his figure is often presented as conservative, for his opposition to the meter, we have to admit that Lalande was quite right in proposing to base the unit of length on a physical standard, like the Paris toise, rather than on the size of Earth. This is exactly what happened one century later, when in 1889, having metrologists realized that a practical unit based on Earth was not accurately reproducible as required for precision measurements and, most important, the definition itself was basically flawed, as we shall see at the end of this paper. The definition of the unit of length was then solely based upon the platinum standard, with no reference to Earth any more. Perhaps what Lalande underestimated was the psychological driving force of standards taken from nature that, with all the problems he correctly spotted (see Ref. [10]), was crucial to reach, soon or later, some national and international agreement.

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... destroyed.⁸

It is a matter of fact that ancient standards are lost forever and the interpretation of data taken with those units can only be guessed somehow. A relative recent episode of an important set of standards lost by accident is the fire in the British Houses of Parliament in 1834 (see e.g. [11]).

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...naturally ⁹

The adjective `natural' has been quite misused in the context of choosing the fundamental unit of length, calling natural what seems absolutely arbitrary to others, as we shall see in the sequel.

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... scale¹⁰

It is not by chance that the smallest historical unit of time with proper name is approximately of the order of magnitude of the human heart pace.

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... solution.¹¹

This choice is not surprising. Try to build yourself a 25 cm and a 100 cm pendula with a piece of string and a little weight, and you do not need to be an great experimenter to realize that, if you want to use one of them to define a unit length, you would prefer to work with the longer one.

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... Jefferson¹²

Jefferson states to have read the Talleyrand's report to the National Academy of France when his report was practically ready. Yet the ``proposition made by the Bishop of Autun'' -- this way Talleyrand was known -- convinced him to change the reference latitude of the pendulum from 38$^o$, ``medium latitude of the United States'', to $45^o$ [22].

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... preferred.¹³

A homogeneous rod of length $l$ oscillating from one end behaves as a simple pendulum of length $2/3 l$. Therefore Jefferson's second bar was 3/2 the seconds pendulum, i.e. about 150 cm.

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... Miller¹⁴

Indeed, there were contacts between Talleyrand and Miller to collaborate towards a common solution. But due to technical and political events, the most relevant among them being certainly the French choice of the meridian, the projects based on the pendula were put aside in France, Great Britain and United States between 1790 and 1791, as we shall see later.

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... States.¹⁵

Jefferson had already accepted the French proposal of the 45th parallel, because ``middle term between the equator and both poles, and a term which consequently might unite the nations of both hemispheres, appeared to me well chosen, and so just that I did not hesitate a moment to prefer it to that of $38^0$'' [22].

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... commission,¹⁶

The Commission was made up of Jean Charles Borda, Condorcet, Joseph Louis Lagrange, Pierre Simon de Laplace and Mathieu Tillet.

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... scale¹⁷

Historians generally agree that Mouton attempted the first metric system in 1670 when he proposed that all distances should be measured by means of a decimal system of units [25].

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... commission¹⁸

This commission was made up of Borda, Lagrange, Laplace, Gaspar Monge and Condorcet. They worked in close contact with Antoine Laurent Lavoisier[26].

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... determined.¹⁹

Any person can easily determine the length of a seconds pendulum within the percent level. Surveying the Earth is a problem more difficult by orders of magnitude.

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... 1852 m.²⁰

The unit of speed consistent with it is the knot, corresponding to 1 nautical mile per hour. It is particularly suited in navigation (and hence the name). For example, a ship that sails at 30 nodes along a meridian travels one degree in latitude in two hours.

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... commission.²¹

The Mouton's minute of meridian is just one of the possible subdivision of the meridian, namely the 324000-th part of the quarter of the meridian: we understand that the subdivision of the right angle in 90 degrees and each degree in 60 minutes and 60 seconds was judged `unnatural' by the académiciens because not decimal. We might guess that the radius of Earth was not considered for the `obvious difficulty' to make an immediate measurement from the center of Earth to its surface. However, it should be similarly obvious that it was also impossible to measure all other quantities (diameter, meridian, equator) in an immediate way. We shall come back to this point in section 7.

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... measure²²

The difficulty in measuring arcs of the equator is not only related to perform measurements in central Africa or South America. It would have required precise measurements of differences in longitude along the equator, and measurements of longitude are intrinsically much more difficult than measurements of latitude, because the former rely on absolute synchronizations of clocks in different places, and that was not an easy task at that time. (For a novelized account of those difficulties, see Dava Sobel's Longitude [28]. A classical novel in which practical ways to measure latitude and longitude are well described is Jules Vernes's Mysterious Island.)

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... democratic'.²³

Several of the claims and the slogans of the académiciens show a certain degree of naïveness (frankly a bit too much for such extraordinary clever people they were: the suspicion that they had hidden purposes in mind is almost unavoidable). We shall come back to this point in section 7.

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... lengths''.²⁴

Actually, such a length is not `precisely' the average over the lengths at all parallels, but only a very good approximation. In fact, the net gravitation acceleration $g$ does not vary linearly with the latitude, but follows the following law [24]:

$$g/(\mbox{m/s}^2) = 9.7803185(1+0.005258895 \sin^2\phi- 0.000023462 \sin^4\phi) ,$$

where $\phi$ is the latitude (the theoretical formula is valid at the ellipsoid surface and, as is customary in geodesy, $g$ is the sum of the effects of gravitation and centrifugal forces).

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... day.²⁵

Ten billion is the square of one hundred thousand (the length of the pendulum is proportional to the square of its period). Seen with modern eyes, it looks a bit bizarre that this `hypothetical pendulum', sized almost twenty times the distance Earth-Moon, would have been natural, while the second wouldn't. (By the way, for those who like to understand all digits: 73cm comes from rounding to the pouce a length that, directly rounded to the centimeter, would be 74cm.)

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... pendulum²⁶

It should be noted that a unit of length based on the pendulum has intrinsic problems, like the dependence of its period on temperature, latitude and above sea level, plus other more technical issues, considered also in Jefferson's document [22]. But, in the part of the report in which the seconds pendulum is discussed and rejected as unit of length, the French commission does not seem concerned at all with this kind of physical questions. Only later (p. 9), when they propose the pendulum as ancillary reference of length, they specify that the pendulum should beat ``at the sea level, on vacuum and at the temperature of melting ice.'' [2]

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... it.²⁷

Our note: to be precise, that is not right ``the same thing.'' The pendulum relates space and time via the net gravitational acceleration $g$. Therefore the included heterogeneous elements are two: time and acceleration. A similar situation happens now, where the meter is related to the second via the speed of light.

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... `meter'²⁸

The name `meter' comes from Greek metron, meaning `measure'. The first who proposed the name meter in the context of the French Academy work is acknowledged to be the mathematician Leblond in 1790 [30,31]; still, some historians (see e.g. Ref. [29]) maintain the idea has to be originally attributed to Borda. However, the name meter for a unit of length (that practically coincides with the French Academy meter) was proposed more than one century earlier by Burattini (see Appendix B).

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... stadium.²⁹

Due to uncertainties in the conversion factor stadium-meter, approximations and errors in evaluating distances and differences in latitude, the usually quoted value of 40000 kilometers obtained by Eratosthenes has to be considered fortuitous, being the uncertainty on that number of the order of 10% [24]. Anyway, not bad for that time (in frontier physics a completely new measurement that provides a result with 10% uncertainty is considered a good achievement).

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... meridian³⁰

The Paris meridian, now $2^0 20^\prime 14^{\prime\prime}$East, had been the oldest zero longitude, until in 1884 it was replaced by the Greenwich meridian, even though France and Ireland adopted the new zero only in 1911.

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... equator.³¹

The Lapland expedition measured an arc of $57^{\prime}$ crossing the north polar circle in northen Finland, at an average latitude of $66^o 19^{\prime}$N. The Peru expedition measured an arc of $3^0 7^{\prime}$ at an average latitude of $1^0 31^{\prime}$S (see [36] for a nice web site dedicated to the expeditions).

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... Perpignan,³²

More precisely, the lowest latitude was about Collioure, a small town close to the Spanish border.

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... flattening.³³

The Dunkerque-Barcelona arc is sufficiently large to allow an estimate of the meridian curvatures in several sub-arcs and to make an independent estimate of Earth flattening. That came out to be about one half of that based on many data sets from equator to Lapland. The flattening based on the latter information was finally preferred, As Alder puts it ``...1/150 offered the best description of the arc as it passed through France, but they knew that the older data offered a more plausible picture of the overall curve of the Earth. They could choose consistency or plausibility. And after some heated discussion, they chose plausibility and the old data.'' Actually, the discrepancy between the values of flattening in different sub-arcs was a first indication that Earth has a more complicate shape than just a rotational ellipsoid, giving rise to the concept of Geoid (see e.g. [24]).

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... meridian.³⁴

We would like to point out that the two results would come to a better agreement if they were treated in the same way. In fact, correction for flattening was not applied to the Lacaille-Cassini result. In Table 3 we have done the exercise of calculating the length of the meridian from $s/\alpha$ of Delambre-Méchain, equal to 57019 toises/degree. We obtain a resulting meter of 443.379 lines. Since the Lacaille-Cassini arc was roughly similar to the Delambre-Méchain one, we can use the ratio 443.296/443.379 as an approximate correction factor to take into account Earth flattening in the Lacaille-Cassini data. After the correction the meridian length becomes 40006 km and the corrected provisional meter would be 443.36 lignes (and $l_m=1.00014$m), i.e. a difference of only 0.064 lignes (1.4 mm) with respect to the 1799 final meter. Anyway, though the two results get closer, the one based on the Lacaille-Cassini measurements gets also slightly closer to the present value of the meridian length.

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... number.³⁵

The task specified in the 4th point was later committed to Borda and Charles Augustin de Coulomb.

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... pendulum³⁶

This name is not appropriate for the pendulum mentioned in the 1791 document, as the name 'meter' had still to be made official. Nevertheless, let us call it so hereafter.

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... measurement:³⁷

That could be the reason why the world best selling encyclopedia erroneously reports that the meter ``was originally defined as one ten-millionth of the distance from the equator to the North Pole on a line running through Paris.''[44] This could be just a minor flaw due to superficiality, but it could also be a heritage of the anglo-saxon reaction to what was perceived as a French imposition.

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... problems.³⁸

As a side remark, we would like to point out that even the very interesting Alder's book The Measure of All Things [10], that has been very useful to us in this research, is a proof the dreams of the académiciens are still far from coming true. In fact most lengths are given in feet and inches, used by to American and British readers, but hard for the others, especially when, in translation, unavoidable mistakes happen, as the 25 feet of page 188, that becomes 672 meters at page 289 of the Italian edition. Nothing compared to the Mars Climate Orbiter disaster, but this is symptomatic of the troubles that disuniformity of units of measures still causes, made worse by globalization.

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... Earth.³⁹

This remains true also if an elliptic, rather than spherical model, is considered, though two parameters have to be taken into account, instead then just one. One might argue that the meridian has the advantage that it is bound only to the assumed rotational symmetry of Earth, and not to its particular shape (sphere, ellipse, or even something more complicate). But, apart from the fact the second commission report [2] speaks explicitly of measuring an arc of meridian, this possibility would imply to envisage a campaign of triangulation from the pole to the equator, that would have been infeasible at that time. (Remember that before 1909 the north pole was just an hypothetical place never reached by human beings.) As a matter of fact, the preferred parameters of modern geodesy to characterize the Earth ellipsoid (also called `spheroid') are the equatorial radius and the flattening. In particular, the latter is the best determined Earth parameter, given with 12 significant digits (the value of table 4 has been rounded): $1/f=298.257 223 563$ (WGS84)[24,46].

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... time,⁴⁰

With this respect, a suggestion put forward in 1889 by Max Planck has been particularly influential. He proposed that systems of units should be based on values assigned conventionally to certain fundamental physical constants. The first (partial) realization of Planck's idea took place in 1983 when the constancy of the speed of light in different inertial frames, adopted by Albert Einstein as the grounding principle of special relativity, was finally used to relate the unit of length to the unit of time.

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... mission.⁴¹

``The new measures are being adopted for the commerce independent of the new measure of the earth; so there is little need for you to push yourself too hard to bring your result now'', wrote Lalande to Delambre (cited in Ref. [10]).

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... unique.⁴²

Even not taking into account asperities of the ground, hills and mountains, the concept of spheroid (or solenoid), is just a first approximation of the Earth shape (the 'zero-th order' approximation is the sphere). The equipotential surface of Earth has a complicate shape called Geoid, of which the spheroid is a kind of best fitting curve (see e.g. Ref.[46] for an introduction, that also contains a visual representation of the Geoid).

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... length.⁴³

That is due neither to ``le difficoltà ad applicare tale definizione, causate dalla forma sferica della terra'' (the difficulties to apply such a definition, caused by the spherical shape of the Earth) [47], nor because ``later it was discovered that the Earth is not a perfect sphere''[44].

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... meridian.⁴⁴

One might also think of a simplification due to the fact the length of the pendulum is unitary. To readers with little physics background we would like to make clear that what matters for the pendulum period is the length, and not the unit in which the length is expressed. The value of a generic quantity $Q$ is given by $Q=\{Q\}\cdot [Q]$, where $[Q]$ is the unit of measurement and $\{Q\}$ the numerical value, e.g. $l=1.38$m. If we change unit of measurement from system $A$ to system $B$ , $\{Q\}$ and $[Q]$ change, preserving $Q$ invariant:

$$Q=\{Q\}_A\cdot [Q]_A = \{Q\}_B\cdot [Q]_B .$$

The two numerical values are then related by

$$\{Q\}_B = \{Q\}_A \cdot \frac{[Q]_A}{[Q]_B} .$$

If we call M a different unit of length, such that $1 M = \alpha$m, we get $\mbox{m/M} = 1/\alpha$. Therefore, a length $l = n$m will be expressed as $l= N \mbox{M} = n/\alpha \mbox{M}$ in the new unit (but the length is the same: you will not grow up, if your height is expressed in centimeters or millimeters rather than in meters). As far as Eq. (1) is concerned, the numerical value of $g$ will be transformed in the same way as $l$, namely $g=(9.8/\alpha) M/s^2$. As a consequence, the conversion factor $\alpha$ simplifies in Eq. (1) and the period will remain the same, as it must be.
On the other hand, if we consider a different length, that has unitary numerical value in the new unit M, we get a different period of the pendulum, namely $T^\prime = 2 \pi \sqrt{1 \mbox{M}/(9.8/\alpha) \mbox{M}/\mbox{s}^2} = \sqrt{\alpha} T$. (For example, if the new unit of length is twice the meter, the half period of a simple pendulum of unitary length will be 1.419s).

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... radius⁴⁵

The generic `radius of Earth' $R$ refers usually to the equatorial radius and it implies that Earth is considered sufficiently spherical for the purpose of the calculations.

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... rotation.⁴⁶

At the equator, the negative centrifugal acceleration gives a contribution to $g$ equal to $\Delta g_c = - v^2/R = -4 \pi^2 R/T^2_{rot}=-0.034 \mbox{m}/\mbox{s}^2$ ( $T_{rot}=86400$s stands for the rotation period). Note that, however, in geodesy `gravitational acceleration' $g$ indicates the overall free fall acceleration experienced by a body and takes into account the genuine gravitational force and the centrifugal one.

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... century,⁴⁷

Tito Livio Burattini (born 1617 in Agordo, Belluno, Italy and died 1681 in Krakow, Poland) was an Italian Egyptologist, inventor, architect, scientist, instrument-maker, and traveler. He was an extremely versatile person (he even designed ``flying machines''!), with interests in mathematics, physics, astronomy, geodesy and economics. He spent a few years in Egypt, where he prepared a triangulation map of this country (he was also an excellent cartographer), made measurements of many pyramids and obelisks, copied monuments and tried to classify them. After some stay in Germany, he finally settled in Krakow, where he served as the King's architect. There he performed optical experiments and contributed to the discovery of irregularities on the surface of Venus, in collaboration with the astronomers Stanislaw Pudlowsky, a former student of Galileo, and Girolamo Pinocci. He became also a highly regarded maker of microscope and telescope lenses, sending some of them as gifts to Cardinal Leopold de' Medici. In 1645, he published Bilancia Sincera, where he proposed an improvement to the hydrostatic balance described by Galileo in his Bilancetta.

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... WORLD.'' ⁴⁸

``Trattato nel qual si mostra come in tutti li Luoghi del Mondo si può trovare una MISURA, & un PESO UNIVERSALE senza che habbiano relazione con niun'altra MISURA, e niun altro PESO, & ad ogni modo in tutti li luoghi saranno li medesimi, e saranno inalterabili, e perpetui sin tanto che durerà il MONDO.'' (The original is for the pleasure of Italian readers.)

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... it.'' ⁴⁹

``Dunque li Pendoli saranno la base dell'opera mia, e da quelli cavarò prima il mio Metro Cattolico, cioè misura universale, che così mi pare di nominarla in lingua Greca, e poi da questa cavarò un Peso Cattolico.''

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... Clocks. ⁵⁰

``Il Metro Cattolico non è altro che la lunghezza di un Pendolo, le di cui vibrazioni siano 3600 in un hora [...]

ch'io intendo d'un Pendolo libero, e non di quelli che sono attaccati agli Horologi.''

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