# Tony Finch – blog

Nautical miles and metres were both originally defined in the same way: each of them was the distance on the surface of the earth subtended by a particular angle. For nautical miles, the angle is a minute of arc; for the metre, it is 1/40,000,000 of a turn.

I was idly wondering about the history of this way of defining a unit of length, and how it led to these two particular units, and what (if any) influence they had on each other. The In Our Time episode on Pierre-Simon Laplace briefly discussed his involvement in the definition of the metric system in revolutionary France, which nudged me to actually do some reading.

## nautical miles

Wikipedia says Robert Hues wrote in 1594 that the distance along a great circle was 60 miles per degree, that is, one nautical mile per arcminute.

At that time, a mile was not a firmly fixed distance: the English Statute Mile was defined in 1593, and different countries had their own definitions. So I think Hues’s statement should be understood as an approximation.

The size of the earth was known, but not to very high precision. In 1637, Richard Norwood measured a nautical mile to be 2,040 yards (6,120 feet); this was an over-estimate: compare the mid-1800s Admiralty nautical mile of 6,080 feet.

The nautical mile did not settle on a single international standard length until 1970.

## geodesy

In his Principia (1687) Isaac Newton argued that (if measured precisely) the Earth would turn out to be an oblate spheroid. So then the question was not just the size of the earth but also its shape. Now we know, because the earth is flattened, a minute of arc has different lengths at different latitudes: 6,106 feet at the poles, and 6,044 feet at the equator. But it took a long time to obtain those measurements.

To measure the earth’s size and shape, there was a steady development of improved techniques and instruments. In 1615 Willebrord Snell pioneered the technique of triangulation, using a quadrant accurate to 0.1 degree. In 1631, Pierre Vernier described his clever scale that allowed him to construct a quadrant accurate to a minute of arc.

Later on, in 1784, Jean-Charles de Borda and his assistant Étienne Lenoir improved the reflecting circle, creating the repeating circle that was used by Delambre and Méchain to survey the Paris meridian.

So, by the end of the 1700s, we were able to measure with some precision the relationship between degrees of latitude and distance along a meridian, and how that varies from the equator to the poles.

One of the major scientific problems of the 1700s was how to measure longitude accurately and conveniently. The eventual answer was the marine chronometer invented by John Harrison, but before clocks became good enough there were a couple of astronomical methods.

In France, the preferred system was to use the orbits of Jupiter’s moons as a clock. The Paris Observatory was set up to make observations for an almanac containing predictions of when Jupiter’s moons would be eclipsed. (During these observations, Ole Rømer made the first measurement of the speed of light.)

In England, the preferred system was to use the motion of the moon, observing which stars were occluded when, to establish the current time. The Greenwich Observatory was set up to make the star atlas that was necessary for this to work.

(For more, see Longitude by Dava Sobel.)

Giovanni Domenico Cassini was the first director of the Paris Observatory; his son Jacques Cassini and great-grandson Jean-Dominique Cassini made measurements of the Paris meridian that were the starting point for Delambre and Méchain.

## the state of the art

Most of the answer to my idle ponderings seems to be that an educated philosopher-savant of the Académie des sciences in the late 1700s would be familiar with the links between astronomy, geodesy, and navigation, as one of the most theoretically interesting and practically important fields of knowledge.

Better measurements of distance on land were interesting for geodesy and cartography, but I infer that it wasn’t particularly important to know the precise length of the nautical mile as such. Part of the evidence for this is that it took a long time to standardise its length. I think what mattered for navigation at sea was being able to establish a ship’s location on a chart, and exact distances were a lesser concern.

## the metric system

Ken Alder’s book The Measure of All Things describes (in chapter 3) the discussions around the design of the metric system in the Académie des sciences.

One possible basis for a unit of length was a one-second pendulum, proposed by Charles Maurice de Talleyrand-Périgord and Nicolas de Condorcet, but this was not satisfactory for a number of reasons: it was known that the force of gravity varied depending on your latitude (so it would be hard to define a convincingly universal standard), and they would have a different standards if they defined it using the Babylonian base-60 second or a new decimal unit of time.

The rationale for the final design was explained by Borda, as chair of the French commission on weights and measures.

Borda was a fan of decimalisation, so his repeating circle measured gradians (1/100 of a quarter-turn) and centigrades (1/10,000 of a quarter turn) instead of degrees and minutes. So a kilometre was defined as a centigrade of arc along a meridian from the north pole to the equator.

To establish the precise length of a metre, the commission argued that some suitably objective and universalist criteria should be followed: at least 10 degrees of arc of the chosen meridian should be measured, and it should span the 45th parallel. This would allow the surveyors to get a good measurement of the oblateness of the earth, and thus the length of the entire meridian. It just so happens that the Paris meridian from Dunkirk (51° north) to Barcelona (41° north) satisfies these completely objective and universalist criteria.

So that was the line that was surveyed by Jean Baptiste Joseph Delambre and Pierre Méchain, using Borda’s repeating circle, and described at length by Ken Alder.

## angular time

It’s a bit awkward that a whole turn of the earth is 24 hours, but as an angle it is 360°, so there’s an awkward factor of 4 or 15 when converting between longitude and time of day.

The French revolutionary system of decimal time had a similar problem: it was based on 1000 minutes per day, although there were 400 gradians in a circle. So (if it had not been thoroughly unpopular) decimal time would have had awkward factors of 4 or 25 when converting between decimal longitude and time of day. Oops.

## design vs reality

Snark aside, after the design process in the Académie the story of the metre has a certain amount of tragic irony.

Méchain’s survey of the southern section of the meridian did not go well; he made a number of mistakes, which he tried to cover up by cooking his books. Delambre perpetuated this cover-up, to preserve Méchain’s reputation and the reputation of the project as a whole.

So the original measurement was not, in the end, objective and repeatable. But its universalism was also based on a mistaken assumption: the earth is even more wonky than they thought it was, so one meridian is not equivalent to every other.

Alder suggests that Delambre satisfied himself by reasoning that it is more important for a standard to be widely accepted than to be completely perfect. And that seems to have been correct for the metric system: it has improved in precision over the decades, and maintained backwards compatibility, even if the original prototype units were slightly erratic.

And at last, nearly 230 years after the Académie’s commission on weights and measures, the SI base units have fully universal definitions - universal in the modern sense of the universe, not just the earth-bound enlightenment universalism. The new SI units are not based on any particular unique artefact, not even the earth.

## postscript

I’m amused that Condorcet and Borda are also known for their work on voting: the Condorcet method and the Borda count.

## credits

Thanks to Rachel for locating my copy of The Measure of All Things!