# Revising the Metric System

Relationship between the new SI units (image produced using the igraph package of R)

On May 20, a major redefinition of SI (metric) units comes into force. In particular, the second, metre, ampere, mole, kilogram, kelvin, and candela will be defined as follows:

## The second (unit of time)

As it is now, the second will be defined using ultra-precise caesium clocks. Specific microwave radiation from caesium atoms is defined to have a frequency of exactly 9.192 631 770 GHz. That is, counting 9,192,631,770 waves will take exactly one second.

## The metre (unit of length)

As it is now, the metre will be defined using the speed of light, which is defined to be exactly 299,792,458 metres per second. That is, the metre is the distance travelled by light in one 299,792,458th of a second (where the second is defined as above).

## The ampere (unit of electric current)

The definition of the ampere (amp) has been greatly simplified, taking account of the connection between electricity and electrons. The ampere is a coulomb of electric charge flowing past a given point per second, and the charge on a single electron is now defined to be 1.602 176 634 × 10−19 coulombs. Thus an ampere is about 6,241,509,074 billion electrons flowing past a given point in a second.

As a consequence of this new definition, two important natural constants which used to have defined values (the permeability of free space and the permittivity of free space) now have experimentally determined ones. This will require rewriting pretty much every physics and electrical engineering textbook.

## The mole (unit of amount of substance)

The mole represents Avogadro’s number of atoms, molecules, or other particles. Previously, Avogadro’s number was defined to be the number of carbon atoms in 12 grams of pure carbon-12. It is now defined to be exactly 6.022 140 76 × 1023.

## The kilogram (unit of mass)

Until 2019, the kilogram was defined by the mass of a specific metal cylinder held in Paris. This has been felt to be unsatisfactory for many years. The current definition uses the fact that the energy of a light photon (in joules) is its frequency times Planck’s constant h, which is defined to be exactly 6.626 070 15 × 10−34.

In practice, a Kibble balance will be used to measure weights by balancing them against an electrically produced force. Units derived from the kilogram include:

• The newton (unit of force): the force needed to accelerate 1 kilogram at a rate of 1 metre per second squared
• The pascal (unit of pressure): 1 newton of force per square metre
• The joule (unit of energy): the energy used in applying a force of 1 newton over a distance of 1 metre
• The watt (unit of power): 1 joule of energy per second
• The volt (unit of electric potential): the amount of electric potential across a resistance producing 1 watt of heat per ampere of current
• The ohm (unit of electrical resistance): the resistance which produces 1 ampere of current when 1 volt of electric potential is applied

## The kelvin (unit of temperature)

Temperature in degrees Celsius was originally measured on a scale with 0 °C being the freezing point of water and 100 °C the boiling point (at standard pressure). The lowest possible temperature turned out to be absolute zero, −273.15 °C. In 1954, the two fixed points on the scale were changed to −273.15 °C (0 kelvins) and the triple point of water, 0.01 °C (273.16 kelvins).

This definition proved unhelpful for calibrating thermometers intended for very high temperatures, and the current definition uses the fact that the average translational kinetic energy (in joules) of a moving atom of a monoatomic ideal gas is (3/2k T, where T is the temperature of the gas in kelvins, and the Boltzmann constant k is defined to be exactly 1.380 649 × 10−23.

## The candela (unit of luminous intensity in a given direction)

The definition of the candela remains what it has been, except that it is influenced by the change in definition of the kilogram (and hence the watt). A light source that emits monochromatic yellowish-green light at a frequency of 540 THz (roughly 555 nm wavelength) is taken to emit 683 lumens per watt, and a light source that uniformly radiates 1 candela in all directions has a total luminous flux of 4π lumens (the constant 683 reflects the human ability to perceive light). The lux is a lumen per square metre.

## The dream

When the metric system was first introduced, the metre was defined in terms of the world (1/10,000,000 of the distance between the Equator and the North Pole, measured via Paris). Today, the metric system carries that philosophy to its ultimate conclusion, with all units except the candela defined in terms of the universe. Five of the units are defined in terms of fundamental physical constants: the speed of light (first measured by Rømer in 1676), the charge on the electron (first measured directly by Robert A. Millikan in 1909), the Avogadro constant (measured several ways by Jean Perrin around 1910), and the Planck and Boltzmann constants (first defined by Max Planck around 1900).

The redefined metric system is a little difficult to grasp without understanding modern physics, but fortunately most of us will just keep on using exactly the same measurement instruments as we have done for years.

# 2019 in science so far

This year in science so far (click to zoom). Clockwise from top left:

# Dunkerque to Barcelona, in metres: a review of The Measure of All Things

The Measure of All Things by Ken Alder

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:

Click for interactive map by Ken Alder

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.

A “Borda repeating circle”

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.

The two ends of the survey (photos by “Romainberth” and Maria Rosa Ferre)

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.

The Measure of All Things by Ken Alder: 4 stars