The metre is now defined as the distance travelled by light in a vacuum during 1/299,792,458 of a second. When the French Academy of Sciences proposed the metre in 1791, however, they took the definition to be 1/10,000,000 of the distance between the Equator and the North Pole (measured via Paris). But what was that distance? Ken Alder, in The Measure of All Things, explains how it was calculated (I’ve been reading the Abacus paperback edition).
Between 1792 and 1798, astronomers Jean-Baptiste Delambre and Pierre Méchain undertook to measure the distance from two sea-level locations: Dunkerque and Barcelona. Astronomical observations were used to calculate the latitude of the endpoints, which were connected by a network of triangles with precisely measured angles. An additional 84 days of back-breaking effort with special platinum rulers (pp. 228, 241) gave precise measurements of the lengths of two of the sides (one would have been sufficient), from which all the other distances could be derived. The required distance between the Equator and the North Pole could then be calculated by extending the arc to an ellipse, factoring out the heights of the mountains, of which there were more than a few:
That is, Delambre and Méchain intended to calculate one quarter of the Earth’s circumference, by measuring the marked sector in this diagram, and extrapolating (mountains exaggerated 400 × in the picture):
It was a huge task, involving surveying on a massive scale, as well as performing complex trigonometric calculations. They had disease to contend with (Méchain later died of malaria), and the French Revolution was going on around them (they were imprisoned several times for looking suspicious). There was war with Prussia in the north and war with Spain in the south (the latter made operating in Spanish Barcelona a little difficult). And, in spite of the new high-precision surveying instruments they used (“Borda repeating circles”), there were two fundamental flaws with the enterprise.
The first problem was that Méchain and Delambre had no real understanding of scientific error. All measurements have both systematic error and random error. Random error can be eliminated by averaging many measurements, but systematic error is more insidious. Imagine an experiment to measure the speed of sound with a stopwatch, by timing the visible “flash” and the audible “bang” of a distant explosion. The systematic error comes from the reaction time of about 0.2 seconds taken to press the stopwatch button. Since this applies for both the “flash” and the “bang,” it mostly cancels out, except that people react about 0.04 seconds faster to auditory stimuli – which means the measured times will vary randomly around a value which is 0.04 seconds too small. Méchain and Delambre did not understand the systematic errors in their “Borda repeating circles,” nor how to compensate for them (p. 316). The strain of worrying about it not only gave Méchain a nervous breakdown (described very well by Ken Alder), but led him to give in to the temptation to fudge his data (p. 307).
The second problem with the whole scheme was the shape of the Earth. They had begun by assuming the Earth was an oblate spheroid, flattened at the poles, with an eccentricity of 1/300. That was contradicted by additional latitude measurements which they took, and this caused some consternation. Today, surveyors define the geoid to be an oblate spheroid with about that eccentricity (1/298.257, actually), but with several adjustments that move “sea level” up or down by up to 100 metres in different part of the Earth. Unfortunately for Méchain and Delambre, France is in a part of the globe that “bulges out” slightly, and this made them over-estimate the eccentricity (p. 262):
Differences between the EGM 96 geoid and the reference oblate spheroid
However, Méchain and Delambre did eventually finish, although Méchain’s mental and moral struggles meant that Delambre did most of the work.
In the end, Méchain and Delambre were out by only 0.02% (the circumference of the Earth through the poles is actually 40,007.86 km, ignoring the “bulges”). A platinum bar with length approximately equal to their calculated metre was placed in the French National Archives in 1799. It remained the standard of length until 1889, when it was replaced by a new X-shaped platinum-iridium bar. However, this new standard (like all the redefinitions since) was intended to be as close to the original bar as possible.
This fascinating and very readable story of the Metric System’s birth is one for both science buffs and history buffs. I couldn’t put it down, and it deserves at least four stars.