The BBC documentary series *Precision: The Measure of all Things* is about measurement and its history. The cinematography is excellent, with that BBC polish which is not often found in other technical documentaries. The presenter (host) is Professor Marcus du Sautoy who is a mathematician. He is the Simonyi Professor for the Public Understanding of Science at the University of Oxford.

What does Professor du Sautoy see as the purpose of this series? He states that: “In this series I want to explore why we measure. What drives us to try and reduce the chaos and complexity of the world to just a handful of elementary units?” Du Sautoy states that from childhood he has been obsessed with measuring things, and wondered who decided that a kilo is a kilo and a second is a second?

The first episode is called *Time and Distance*, which examines the second and the meter. It is pointed out that time and distance are interrelated in people’s minds, so much so that people use the phrase “length of time.” This statement is quite ironic when one sees it within the context of the information presented in the episode.

The origins of time measurement as a correlation between the stars above and the seasons is discussed. The abstraction of the motions of the sky into terrestrial clocks from sundials to atomic clocks is detailed. The origins of distance measurement is interleaved with the time discussion. The cubit rod (ruler), used to aid the construction of the Egyptian pyramids, points to the earliest notion of the importance of length standardization. He then comments on the contemporary situation:

Despite the obvious logic of having one international system, it hasn’t been completely embraced. Take me, for example. I’m going to the airport in this cab which measures its speed in kilometres per hour and miles per hour. When I’m up in the air, they’ll be measuring my altitude in feet. My clothes are measured in inches and my shoes are measured in …. well, frankly I’ve never quite understood what the unit for shoe size is!

Well, I would have hoped that the host of a documentary on measurement would not confess ignorance about the unit for shoe size. It is the barleycorn in the US. I have written about the confusion known as shoe size previously. There is a metric alternative which works rather elegantly, it is called mondopoint.

The farrago of pre-metric measurements in England is touched upon. The Professor then discusses France and argues that matters were even worse there. Professor du Sautoy states that the French decided to stop basing measurements on the human body, and instead based them on the Earth. This is indeed true, but it omits the fact that a British scientist, John Wilkins (1614-1672), invented the system part of the metric system, which was also not based on the human body, but rather upon a scientific phenomenon. Wilkins argued that the length of a seconds pendulum should be used as a universal standard of length. He then took one-tenth of that length, and created a cube from it to use as a volume standard, which in modern terms would be called a liter. This volume was then filled with water, and that amount was to be the standard for mass, which would later be called the kilogram. This proposed system was published by the Royal Society in a 1668 book and was known in Britain and Europe.

Du Sautoy does not mention the fact that using a seconds pendulum, as Wilkins suggested, had been “debated” along with using the Earth for a distance standard by the French. The seconds pendulum appeared to be the default choice for the new system. The addition of decimal divisions of these units, with linguistic prefixes, produced the original version of the metric system. At the last moment it was decided by the French committee to use a meridian of the Earth as a standard. This came as quite a surprise to many at the time.

The mathematics professor argues that by using the Earth as a length standard no other countries could see it as belonging to one country. This seems doubtful. The meridian chosen went through Paris, and Thomas Jefferson saw the abandonment of a seconds pendulum (he argued for a rigid “seconds rod” to also be considered) as an act of French nationalism and no longer endorsed the metric project. It appears that du Sautoy seems to

be creating a kumbaya consensus among the delegates where from my reading of history, one did not exist.

Pierre Mechain and Jean Baptiste Delambre are identified as the men who would measure the Earth and determine the length of the meter. The controversy over the “hidden error” in this measurement is mentioned and seems at odds with our current understanding:

In fact, due to errors that Mechain made early on in his survey it’s fractionally wrong. The errors that Mechain made were pretty much irrelevant, because for the first time, the world had a unit of length [the meter] that was based on something they believed was permanent and unchanging – the Earth.

The “hidden error” became well enough known among anti-metric persons, that John Quincy Adams could not help but chortle about it in his Report on Measures. It is hard to characterize what occurred as an error. There was an implicit assumption in that era that the equipment used would produce exact measurements. There was no common understanding that all measurements have an error associated with them, nor had statistics been developed to quantify these measurement uncertainties. When two repeated measurements deviated, it was assumed that an error had been made.

It seems quite surprising how the presenter simply dismisses the importance of the error in a “measurement standard” as irrelevant. In the 19th Century, James Clerk Maxwell would make sport of the idea of using the Earth as a measurement standard. It took Mechain and Delambre seven years to measure the distance from Dunkirk to Barcelona.

What is not mentioned by du Sautoy is that a complex seconds pendulum was constructed near Paris, apparently as a backup option. From June 15th to August 4th of 1792 twenty sets of measurements were made and its length was computed to be 994.5 mm (440.5593 lines). The chosen alternative of taking seven years to measure a section of the Earth seems quixotic. Then for du Sautoy to immediately dismiss the “errors” introduced into this measurement as irrelevant, after such a large undertaking, is hard to understand.

The original meter was in actual fact, an artifact. The surprising similarity between the length of a seconds pendulum, and that of one ten millionth of the distance from the north pole to the equator, seems never to be mentioned in many histories. When the episode is entitled *Time and Distance*, it would seem that the seconds pendulum would be a natural place to start. One uses the best second available with a pendulum length to define the meter. Indeed this is almost a “length of time” as he mentioned earlier. One could then go through the history of the meter, which would lead up to the modern definition of the meter as the distance that light travels in a vacuum in 1/299 792 458 of a second. One can link the original definition by Wilkins which had time and length intertwined, and the modern definition which also involves both.

The second episode, *Mass and Moles*, opens with a viewing of *Le Grande K*, The Kilogram, which is kept near Paris.

To explain the importance of the kilogram, Du Sautoy goes to a British market and talks about weight when all the produce is in grams and kilograms. He offers an engaging demonstration of how people can perceive weights incorrectly. People are offered different sized objects and are asked to determine which weighs the most. These people, who are in a market with produce sold by mass all around them, fail the test over and over, and of course, that is why we need to use a scale, and most importantly, one which has been calibrated.

The almost certainly fabricated story of Newton seeing an inspirational apple drop from a tree^{[1]} is related to introduce the difference between weight and mass. The host calls the difference a subtle but vital one, and indeed, mass and weight is constantly confused. In a quite interesting demonstration du Sautoy measures a cylinder of metal with an incredibly sensitive scale. The mass of the metal is 368.7025 grams according to the read out. The scale and its metal test mass is then moved to the top of a nearby tall building. The metal now weighs 368.6916 grams for a 10.9 milligram difference.

The Professor does not see the irony that he just pointed out the subtle difference between mass and weight, and has a scale that indicated that the *mass* (in grams) of the object changed with location. Of course the scale is actually *weighing* the mass, which is the force exerted on the mass. The scale assumes a gravitational force value and calculates the mass based on this assumption. The *gravitational force has changed because of its location with respect to the center of the earth, and not the mass of the object*. If there had been a 10.9 milligram change in mass, then we must account for about 981 gigajoules of missing energy, and the presenter would probably have been injured in some manner by the experiment. A stick of dynamite releases about 2 megajoules of energy.

The unit for the measurement of force (weight) in the metric system is newtons. This is the value which is being indirectly measured and not the mass. Du Sautoy correctly points out that number of grams contained in the object did not change, but that gravity did. Certainly the amount of *gravitational force did*. Well, I’m not sure this is as clear an explanation of the difference between mass and weight as the presenter thinks it is. The measurement device reads in grams in both cases, which is a mass. The way to mass an object is to use the oldest technology available, a mass balance. The force on each side of a balance is the same, which cancels out, which allows for a true mass comparison to take place. To use the force scale correctly du Sautoy would have needed to calibrate the scale with a known mass at street level, and then again at the top of the building. The measured mass would then remain constant as it must. Assuming 9.8 m/s^{2} is the gravitational acceleration at street level, the difference in “metric weight” is about 107 μN (micronewtons). For comparison, the “metric weight” of a human being with a mass of 70 kg is about 686 N.

The confusion continues when it is pointed out by the professor that the original kilogram was to be equal “to the weight of one cubic decimeter of water.” Well, it’s not the weight, it’s the mass definition. He also credits Lavoisier as coming up with this kilogram definition. John Wilkins predated this “redefinition” cited by du Sautoy. Lavoisier is also “credited” with using this to define the liter—worse and worse. The weight/mass statement confusion is ubiquitous, even among contemporary engineers and scientists. Du Sautoy interviews a person who has a device that weights single biological cells. Indeed, he describes how small the *weights* are that he can measure with this device in picograms and femtograms.

The third and final episode is *Heat, Light and Electricity*. It has a number of interesting images of the attempts to measure these less directly experienced phenomenon. The overall series has a considerable number of scientific demonstrations that are quite interesting and bring the issues at hand to life.

Du Sautoy has also hosted another series called *The Code*. It is about the mathematics which describes our world. In my view, du Sautoy consistently displays a belief that numerical expression is incidental to explanation. This may reflect his background as a mathematician who has not done much design and fabrication during his career.

In *The Code,* an example of this measurement system neglect occurs when the professor decides to measure a neolithic circle. The tape he uses determines the diameter of the ancient relic to be approximately 27 meters and 90 centimeters. He then measures the circumference at 91 meters and 70 centimeters. He wrote down meters and centimeters, in the same way an Englishman (or American) might write down feet and inches. One could write down 27.90 meters and 91.70 meters. He could have written down 27 900 mm and 91 700 mm. I don’t see anything objectionable in either of these expressions, meters or millimeters. He then divides 917 by 279 to get the ratio of the circumference to the diameter. Clearly he does this to show us the ratio is near the value of π, at about 3.3. So, he is using integer values of decimeters for his computation, which does make the act of dividing by hand easier. One cannot say that what he has done is “wrong” but it certainly makes one suspect that his view is that numerical expression for computation and clarity is not a priority in his world.

It does not really surprise me that a mathematician would jumble around numbers like this, they tend to see computation and numerical expression as incidental to a point that they might be making. This is why I have argued strenuously that mathematicians should not be tasked with teaching the metric system. They have very little acquaintance with fabrication and measurement.

Unfortunately, mathematicians seem be unaware of this deficiency. They appear to think that because they understand prime numbers, irrational numbers, integer numbers and so on, they automatically have an understanding of the optimum use of numbers used to describe and compute the world. This may be precipitated by the idea that this type of mathematical expression is “just a detail.” It could also be due to the fact that mathematicians don’t do a lot of actual applied work.

Du Sautoy then takes a much more modern circular object, a dinner plate, and measures it. The diameter is called out as 26.4 centimeters, and the circumference is about 82.9 centimeters. Why the decimal point suddenly? He seemed fine with integer decimeters. Why not 82 centimeters 9 millimeters?—which is consistent with the meter and centimeter unpacking he used previously—or use an integer?—like 829 millimeters?

Without further hand computation he announces the answer for the circumference to the diameter as 3.14 for the plate. He performs the same measurement on a roll of cloth tape and gets 3.14 as the ratio. His point is that all circles have π as the ratio of the circumference to the diameter. The computation and how to go about it is incidental, just something to be accomplished to make a general mathematical point. This is a habit of mind, and one which was not broken for *Precision*.

I have very high expectations for BBC programs, because I’ve seen so many high quality science documentaries. The series *Light Fantastic* for example, is an excellent explanation of the history of light and electromagnetic radiation. What was offered by Professor du Sautoy is not up to the high standards of the science documentaries offered by the BBC. I can only hope that someday a more worthy series on measurements is produced.

[1] *Isaac Newton The Last Sorcerer*, Michael White, 1998 pg 87

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