How Long Is a Metre?

Abstract

To start at the end (!) we can definitively pronounce a metre to be as long as the distance travelled by light in a vacuum in 1/299,792,458 of a second. This is the current definition of the metre as established in 1983. As a standard, it has the particular virtue of not being identified in terms of an artefact (a manufactured exemplar), but rather in terms of universal constants of nature. In principle, this reference can be reproduced at any time and in any place; you just need to make room in your garage for one or more stabilised lasers, an interferometer and an optional atomic clock (for extra accuracy). But why 1/299,792,458 of a second particularly rather than any other time interval? This new question requires us to start at the beginning …

In 1740, Cassini de Thury had measured the Paris meridian from Dunkirk to Perpignan using the prevailing French unit of length, the toise. (The toise was divided into 6 feet (pieds du roi), each foot into 12 inches (pouces), each inch into 12 lines (lignes) and each line into 12 points.) There were two references: the toise du Pérou (named for its use in geodetic measurements in Peru) and the toise du Nord, which were regarded as identical. In 1747, the toise du Pérou was identified by the Académie des Sciences in Paris as the prototype for the unit of length; it became known as the toise de l’Académie. Much of the drive for this sprang from geodesy, the science which includes the measurement of the earth.

It has long been a vision that measurement should be referenced to some universal and invariant standard. In 1791, the Académie proposed 1/10,000,000 of the length of the Paris meridian from the North Pole to the equator, and they commissioned a new measurement. The measurement was performed by triangulation, the measurement of angles to fixed landmarks coupled with trigonometry, in a process that allows the lengths of the sides of a succession of triangles to be calculated with reference to an originating baseline length.

To facilitate this measurement, a provisional metre was identified as 1/10,000,000 of the meridian as measured in 1740 (3 feet and 11.44 lines of the toise du Pérou), and a brass bar was made corresponding to this length. Four bars of platinum were then made, each 12 pieds long and measured in terms of the provisional metre. (Seems perverse to have made these bars with a length based on the toise, but there we are. Another example of going metric ‘inch by inch’?) These bars were then used to measure the baselines for the triangulation.

The measurement was undertaken by Jean Baptiste Joseph Delambre and Pierre Méchain, during 1792–1799, and required heroic commitment on their part to overcome obstacles of terrain, arrest, war and political turmoil.

The actual measurement was made from the belfry in Dunkirk to the fortress at Mount Juic in Barcelona (within sight of which this article was drafted on a hotel terrace with 0.0005 m³ of ale to hand). This measurement was then extrapolated to identify the total length of the quarter meridian. This calculation included provision for the flattening of the earth, which turned out to be in error, although, in the final analysis, this may be said to be of no practical concern.

Once the new measurement of the meridian was available, the metre was identified as 443.296 lines of the toise du Pérou and a new artefact was prepared: the ‘metre of the archives’. This is a platinum bar end standard, the very first metre ‘ruler’ (traditionally, rulers are line standards rather than end standards), dedicated to ‘all times and all men’. It might be unkindly said that the Académie might simply have engraved two lines on any old bar of platinum–iridium alloy they had to hand, separated by a distance of something like a yard but a bit longer, but the value of this development was largely in the idea rather than the realization: the notion that the standard should be independent of national pride or interests and linked to something considered universal and constant.

It was later acknowledged that the abstract meridional definition of the metre was impracticable and that the prototype held in the archives ‘… can be considered invariable and as belonging to all nations …’, the question was rather how to propagate this standard to all the nations. In 1869, the metre commission was established to develop and internationally promote the metric system with new standards for the metre and the kilogram. It proposed to make a new international line standard based on the metre of the archives, which, being an end standard, was more susceptible to damage to the ends of the bar. It also instigated the creation of the Bureau Internationale des Poids et Mesures as the body to act as guardians for the standards.

There was much debate about whether the new metre should be based on a theoretical definition, the metre of the archives or some other standard. Debate also included such varied and abstruse matters such as

• The bar cross section (to improve its stability and rigidity),

• The finish (upon which the lines would be engraved),

• The comparison techniques (to allow propagation from the new standard to derivative measures),

• The coefficient of thermal expansion,

• The supports and their positioning (to reduce the influence of flexure),

• The alloy to be used (platinum with 10% iridium being selected),

• The line marking,

• The temperature measurement,

• The temperature stability.

The full story of the development of the prototypes has filled books with some fascinating detail (my main reference being From Artefacts to Atoms, by Terry Quinn ¹ ) and is beyond the scope of this article. For my purposes, suffice it to say that an order was eventually placed with Johnson Matthey of London for 30 off metre bars which would be distributed as national prototypes. Lines marking the metre were engraved on a specular (mirror) finish which was eventually demonstrated to be the better approach. Of these 30 (designated, not unreasonably, Nos 1–30), No. 6 was found to be closest in length to the metre of the archives and was chosen as the new international prototype, now designated M and identified as having a length of exactly 1 m at 0 °C. This then superseded the ‘metre of the archives’ as the international standard. The others were distributed as national prototypes (the UK holding No. 16).

Eighty years later, in 1959, the British and American yards were redefined to equal 0.9144 m exactly so that the international inch became 25.4 mm exactly.

In a prophetic statement of 1870, James Clerk Maxwell said,

Yet after all, the dimensions of our earth and its time of rotation, though, relatively to our present means of comparison, very permanent, are not so by physical necessity … If, then we wish to obtain standards of length, time and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wavelength, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.

It was to be many years of course before science and technology had developed sufficiently to allow such a standard, but in 1960, in fulfilment of Maxwell’s 1870 vision, the metre was redefined as ‘the length equal to 1650763.73 times the wavelength in vacuum of the radiation corresponding to the transition between the 2p₁₀ and 5d₅ levels of the atom of krypton 86’. The detail of the physics of this definition need not trouble us; the key point is that the metre was now defined in relation to the wavelength of a particular light. The ratio of the wavelength (in a vacuum) of the krypton emission line to that of the cadmium emission line was determined and then related to the metre through the wavelength of the cadmium line in air (with a correction for the refractive index of air) that had been determined by Fabry and Pérot in 1906.

In 1983, the metre was again redefined as the ‘length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second’. This exploits the very low uncertainty of the atomic clock and is not dependent on the specification of a light source. The speed of light was nominated by consensus to be EXACTLY 299,792,458 m/s so that there would be no appreciable discontinuity in the unit of length. (The choice was based on previous measurements of the speed of light using the 1960 standard for the metre.) Instead of defining the length of a metre and measuring the speed of light as so many metres per second, the speed of light was declared as an exact figure and the metre defined as the corresponding distance travelled by light in a specified time. (This sounds suspiciously like picking yourself up by your bootlaces, but there is a logic to it!)

We can now bring our story full circle, so to speak, and estimate that it takes 10,000,000 m/299,792,458 m/s ≈ 0.0333 s for light to travel from the North Pole to the equator.

Had the physics been known in 1791, the Académie might well have arbitrarily nominated 300,000,000 m/s as the exact speed of light and declared the metre to be the distance light travels in vacuum in 1/300,000,000 of a second (giving us a slightly shorter metre). It is remarkable that they were so very close, or rather that the earth just happens to be of a size that takes light very nearly exactly 1/300,000,000 of a second to travel the distance of one 10 millionth a quarter meridian.

Each time the standard has changed, the uncertainty has reduced, but the new length has remained within the uncertainty of the previous standard. In this way, there is no loss of continuity in the measurement system; all existing measurements referenced to previous standards remain valid. For this reason, although the ‘metre of the archives’ has long since been superseded, it, and the heroic efforts of Delambre and Méchain, may be said to be enshrined ‘for all times and all men’ in the not quite perfectly round figure for the speed of light.