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About the choice of the quarter of the meridian

It is clear that, once the académiciens were bound to the decimal system, decided by the first commission, and the unit of length had to be related to Earth dimensions, the unit of length of practical use had to be a small decimal sub-multiple of an Earth dimension. But why the quarter of the meridian, instead of the meridian itself? The Rapport [2] does not give any justification of the choice, as if all other possibilities were out of question. And this is a bit strange. The meridian as unit of length had no tradition at all, and there had been no discussion about which submultiple to use. Evidence against the naturalness of the quarter of meridian seems to us provided by the fact that the vulgarization of the definition of the meter, as it is often taught at school and as it is memorized by most people, is the forty millionth part of `something', where this `something' is often remembered as the `equator' or the `maximum circle'.

It could be that we have nowadays a different sensitivity to the subject (we have made a little poll among friends and colleagues, and our impression has been unanimously confirmed), but we find it hard to be rationally convinced by the arguments of the following kind:

Once it has been chosen as base, will either the whole meridian or a sensible part of it be taken as a unit? The wholeness? Out of question! The half, that stretches from one pole to the other, may not be easily conceived by our mind because of the part which is located ``below'', in the other hemisphere. This is not the case of the quarter of the meridian that, on the contrary, can be easily imagined: it stretches from ``one pole to the equator''. In the future it will be said: France opened the divider and pointed it on one pole and the equator, a sentence that will be greatly successful. There is another reason, that is really scientific and supports the meridian: its quarter is the arc intersected by the right angle. That's right: however, why should it be considered as a further advantage? Simply because the right angle is considered as the natural angle, the angle of the vertical and the gravity. It is the unit-angle, the degree is nothing but its subdivision. (Ref. [35], p. 55 of the Italian translation)
What would be the alternatives? As an exercise, we show in table 5 some possible `natural' choices of units of length based on the dimensions of Earth, together with a reasonable decimal sub-multiple as practical unit.

Table: Some possible choices of units of length based on the dimensions of Earth, assumed to be spherical, together with a reasonable decimal sub-multiple as practical unit and the half period of the simple pendulum of such practical unit. (Analogous quantities can be defined assuming an ellipsoid).
unit decimal practical unit $T/2$
  sub-multiple (cm) (s)
radius 1/10000000 64 0.803
diameter 1/10000000 128 1.135
meridian 1/100000000 40 0.635
$1/2$ meridian (pole-pole) 1/10000000 200 1.419
$1/4$ meridian (pole-equator) 1/10000000 100 1.004
45th parallel 1/100000000 28 0.534
one radiant along the meridian 1/10000000 64 0.803
(same as radius)      
1 degree of Earth's arc 1/100000 111 1.057
1 minute of Earth's arc$^{(*)}$ 1/1000 185 1.367
1 second of Earth's arc 1/100 31 0.558
$^{(*)}$Equal to 1 nautical mile, that is 1852m.

Sub-multiples of the length of the meridian, e.g. one part over 10000000 or one part over 100000000, had led to a `meter' of 400 or 40 of `our' centimeters. The former is certainly too large, but the latter is quite appropriate for daily use, and, indeed, it falls in a range of length that is better perceived by people (one of the criticisms to the meter is its unnaturality, at least compared for example to the foot or even to the first standardized unit, the cubit). Even the pole-to-pole arc would have yield a better practical unit, very close to the toise.

We see from table 5 that the 10000000 part of the quarter of meridian is the closest to the length of the seconds pendulum. So, when the French scientists proposed the new unit of length, we think it is possible, among the many `defensible natural units', they chose the closest to the seconds pendulum. The reason could be a compromise with the strenuous defenders of the seconds pendulum. Or it could have happened that, since they had in mind some `cooperation' between the new unit and ``a pendulum having a determined length''[2], choosing a unit close to the well studied seconds pendulum would have simplified the intercalibrations.

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Next: About the naturalness of Up: Conclusions and discussion Previous: About the choice of
Giulio D'Agostini 2005-01-25