Precision: The Measure of All Things

By The Metric Maven

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/s2 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

20 thoughts on “Precision: The Measure of All Things

  1. > 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,

    In my experience, there are some people who are very good at estimating the mass of the produce they work with. When I go to a butcher and ask for some amount of meat, I’ve seen some of the more experienced ones get surprisingly close. My guess is that people base their esimates of mass more on the size of familiar objects, which they can’t do so well with unfamiliar objects.

  2. Your analysis of the modern electronic scale as a force balance which must be calibrated in situ with a known mass is correct.

    You appear to not notice that this is precisely what’s wrong with the seconds pendulum as a standard. It’s period depends on local gravity, being about 2*pi*sqrt(L/g) for small amplitudes. Thus it would vary between the top and bottom of your tall building, and it is hard to measure local gravity without standards for both length and time (it being an acceleration). A standard that varies with latitude and elevation above sea level (there are also microgravity variations from local density anomalies in the earth) is not much of a standard. It would have immediately led to a master artifact too. The period also varies with the angular amplitude of the swing.

    In addition, it varies from the modern meter by about 0.55%, whereas 10^-4 meredians differs by about 0.02% (using the modern WGS ellipsoid, the distance is slightly under 10 002 km). No idea what precision repeated physical measurements would yield, but the bias in the mean is smaller.

    Incidentally, check your “weight” change on the tall building, 10.9 mg x 9.81 m/s² is 107 mN on my calculator, not 107 µN..

    • The acceptance of a “base unit” that has a prefix, makes it easy to forget to convert a value in grams to Kilograms. This needs to be done prior to computing a value for force in SI. Force has units of Kilogram-meter per second squared. My computed value in micronewtons is correct.

      I conferred with Sven to make certain I had not missed a detail somewhere in the force computation, and he noted that I had the wrong magnitude on the energy calculation. It should be 981 gigajoules, but was 981 megajoules. It has been corrected. It appears I remembered the Kilogram conversion in my original energy computation, but put on the wrong prefix in the blog.

      • Curse you, kilogram. Metric Maven, you are correct. Sorry.

        • John:

          Don’t feel bad about this — last summer I had originally made the same error when expressing the mass of the Higgs particle in yoctograms instead of the silly electron volts we found in newspaper articles and elsewhere but realized it shortly thereafter and corrected it. (If I recall it correctly, its (rest) mass came to about 225 yg.)

          • Correction on this posting of mine: “last summer” should be “two summers ago”.

            Also, here’s the calculation with a good deal of precision:

            Since many newspapers gave the mass of the Higgs Boson as 125.3 GeV, we first change such to joules, giving 20.075441 nJ.

            Thus, we have

            mass = 20.075 441 nJ / (299 792 458 m/s)^2

            After changing joules into N*m = kg*m^2/s^2 and cancelling, all that remain is the kg, and so

            mass = 20.075 441 x 10^-9 kg / (299 792 458)^2 = 2.2337 x 10^-25 kg = 223.37 x 10^-24 g = 223.37 yg

            Since the 125.3 GeV was apparently a lower bound, the 225 yg for the mass cited in my original posting was probably an upper bound. (Comments welcome from those more familiar with this…)

            • David,

              I went to see the documentary Particle Fever, which I liked, but I must confess the values they presented in electron-volts did not really provide me with any feeling for the mass of the Higgs Boson. When I saw the value you just presented, 225 yg it immediately struck me that, unless I’ve missed my mark, this is a very large mass for a subatomic particle. If I recall correctly a proton has a rest mass of 1.67 yg. When I saw your value, I did a double take, and checked the computation you provided—looks right to me.

              The energy gained by an electron through a one-volt potential difference I understand (my background is electromagnetism), but because I don’t use it in my everyday work it has no real meaning. It really surprised me how readily I could understand the mass of the Higgs Boson when it was translated into yoctograms. Very interesting—Thanks.


              • MM:

                What you write bothers me too as the mass of the Boson would be about 135 times that of a proton!

                Thus, my guess is that that 125 GeV was Not the rest mass/energy, and so the common Einstein formula that I used would have to be adjusted for such.

                Nevertheless, I do recall a physicist comparing it to an element whose mass was supposedly about the same. Since 225/1.67 equals about 135, then Barium would probably be that element.

                Anyway, intuitively, something seems amiss with this, and so it’s probably that adjustment that needs to be made…

                Anyone else want to chime in here?…


  3. Two papers that cover the most important terms in gravity variation:

    Note the graph in the 2nd showing variation with latitude. Gravity varies most rapidly in the temperate latitudes between the tropics and polar regions, precisely where most of the metrology powerhouse nations are located. There would have been war over the seconds pendulum.

    However, I am not sure there was any understanding of these variations at the time of the proposal, or, if so, how detailed that understanding was, circa 1800. However, the “seconds-meter” would have varied around ±1.5 mm from equator to pole.

  4. I have not seen the series, but from your account it is somewhat surprising that an Oxford Professor didn’t notice that the difference in the masses between ground level and the top of a building were wrong and then offer a truthful explanation. I don’t know if the scale he used was user calibratable but if it wasn’t, he should have known enough to use a true balance to show that the mass of an object does not change with location. A real professor would know that mass and weight are not the same.

    Of course, if this so-called professor is in fact a metric opponent, his actions are predicable.

    From 2013-06-12:

    Oxford Professor’s view of metric.

    Marcus du Sautoy is the Charles Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. He presents a new three-part series about the history of measurement on BBC Four, “Precision: The Measure of All Things”

    Here’s what he said about the metric system, speaking on the BBC Radio 4 programme “Midweek”:

    “… I think actually the metric system is mathematically flawed … (the originators of metric) made us all go decimal basically because we have ten fingers. But actually the system we were using before, which had units divided into twelve or sixteen, are much better units actually because they are more easily divided.”
    He went on to say that an hour of sixty minutes is better than one of a hundred, and to praise the Babylonian use of the number sixty.

    What is truly flawed is Marcus du Sautoy and the institution that awarded him a degree. This for sure doesn’t throw a positive light on Oxford’s credentials.

    • If a professor isn’t pro-metric then they aren’t a real professor? Really?

      • Well, perhaps not in the sciences (or STEM, to be trendy). Perhaps not the best candidate for Professor of Public Understanding of Science. English lit., wouldn’t matter much.

        In his field, I feel his remarks raise some concerns, although I’m not quite as upset as Ametrica.

  5. Tracing the metre to John Wilkins doesn’t go back far enough. Pre-Norman Britain used a length called the wand (as in the magic wand we hear about from folklore) that is very close to the modern metre. This system was also decimal based as is the modern metre. The yard replaced this “metre” and only in our time has the metre replace the yard. Who is to say if John Wilkins was aware or not of pre-Norman British measures, but the fact remains the metre predates the yard and is the true British unit of length.

    The old English unit of 1007 millimetres was called a ‘wand’, and although the ‘yard’ was created to replace the wand, the wand was still used for some centuries because of its convenience as part of an old English decimal system that included:

    * 1 digit (base of long finger) about 20 millimetres
    * 10 digits = 1 small span (span of thumb and forefinger) 200 millimetres
    * 10 small spans = 1 fathom (1 arm-stretch from finger tip to finger tip) about 2 metres
    * 10 fathoms = 1 chain about 20 metres
    * 10 chains = 1 furlong about 200 metres
    * 10 furlongs = 1 thus-hund of about 2000 metres

    • While I was disappointed with the series, there is nothing in it to suggest Professor du Sautoy is secretly metric-subversive. If anything, the problem was almost the reverse: he’s certainly informed enough to know there are no other standards today, but he focuses on iconography, rather than what is truly revolutionary. The photography of his visit to the archive where the original meter bar is kept suggested a pilgrimage to a secular shrine. When he was finally shown the bar, I was almost surprised that he didn’t genuflect and kiss the reliquary.

      That he apparently has a thing for so-called highly composite numbers is interesting, but not surprising. It reinforces the Maven’s point that he’s a mathematician, not a scientist or engineer—since only a mathematician would suppose that HCNs are helpful in real-world measurement. For all his assertions of being fascinated by measurement, he gives the impression of having done very little measuring. The series might have been better if he had spent some time in the kitchen first.

      You’ve brought up this “wand” business before. Frankly, it’s as puzzling and unimpressive as your inability to cope with post-1824 usages of the word imperial.

      John Wilkins is of interest because he proposed a system of measurement in the seventeenth century that, far from looking to historical precedents—such as your supposed wand—deliberately rejected them (apart from the second, which could be reproduced by any astronomer, anywhere on the globe). His system was independent of any sort of artifact, whether the king’s foot or the Paris meridian (du Sautoy misses this entirely). If you could demonstrate that an old unit called the wand was approximately a modern meter, all you would have is a not-very-remarkable coincidence. Wilkins’ system was based entirely on what today would be called reproducible laboratory models. It’s a vision we in the twenty-first century have yet to realize, since we’re still ostensibly dependent on an artifact, in a rather tacky-looking cabinet, on the outskirts of Paris.

      It isn’t that there couldn’t have been such a unit, but few such tribal or regional units are known today with any more accuracy than that of a cubit. I’ve tried searching the internet for combinations of terms like wand and measurement, and all I’ve got to show for it are Harry Potter quotations.

  6. This all strikes me as a bit funny – such academic wondering so far from the real world. The idea of the metric second was not a bad one – and was precise enough for many things – and removed the power of the government to inflate prices with fudging – a really good idea – not perfect – reminds me of bitcoin.

    The mathematician seems to think we should switch to base 12?

    This is a learning moment – if it is worth forcing people to change to metric – is it also worth forcing them to change to base-12? Freedom of choice is what I advocate and I also advocate voluntary metric. Force is for dictators.
    I really like freedom. Let the people focus on forcing the government to use metric – not the other-way-round.

    So what about switching to base-12? It has definite advantages – One now has to acknowledge that due to our history of having ten fingers it isn’t likely to ever happen. I just think we should have a little sympathy for those that favor tradition, as we also do – as we are not going to switch to base-12. I do believe we are switching to metric – slowly, but freely.

    And – the title, Precision, had me thinking that this column was going to be about the constant confusion of the the terms accuracy, resolution, and precision.
    OT – When I was in a ‘metric country’ in Central America a few months ago – I had a hard time finding ANY metric screws after going to several stores.. Las ferreterías (hardware stores) had NO metric selection in machine screws – my local stores here in unsophisticated Kansas have a wide metric selection. In Central America, one needs to order the parts in from the United-States (that terribly non metric country?) I think the ‘metric country’ narrative needs some sunshine.

    • So Karl,

      If you like voluntary so much:
      *Why do I have to tell the DMV my height in feet and inches
      *Why do I have to buy beer in fl oz, pints, quarts, etc.
      *Why do I have to buy meat in pounds
      *Why do I have to buy produce and deli in pounds.
      *Why can I only buy gas in gallons
      *Why are 99+% of all road signs Customary.

      These units are proscribed as mandatory for these things under US law (Federal and/or state) and metric is not acceptable. (Unlike standard packaged goods where dual is compulsory). Why am I not allowed to be as metric as I want under the Metric Act of 1866. Why do they FORCE me to be Customary.

      The real facts are that Customary is usually compulsory, and since metric is voluntary, those who don’t like it can deny it to those who do. “Metric is voluntary” is a crock of repurposed bovine waste.

      Since by law metric is the preferred system of weights & measures in trade and commerce (1988 act of Congress), shouldn’t EITHER party in any transaction be able to insist on metric and ONLY if BOTH parties to the transaction agree to Customary, can the transaction be denominated in Customary? That would be voluntary in my mind.

      • Totally agree that the government should not be forcing us to us customary units. And where the government meets the public I’m in favor of metric – I don’t want the government forcing the use of any unit on private transactions.

        If I want to sell fish to the public in cubic cubits – I should have that freedom. People should be free to do stupid things. Freedom to not have our behavior forced by bureaucrats is a really cool thing – we should never give it up.

        • Not sure I agree. There are two issues. Is it an individual private transaction or you selling to the general public. Do I want to buy in cubic cubits? Do we agree on how big a cubit is?

          If you and I agree on the cubits, I agree it is then none of the government’s business. If you want cubits and I want metric, or my idea of a cubic is bigger than your’s (I’m pretty tall), then I think the government has a trading standards role.

          The government can’t support a large set of unit choices and should be able to set the system of weights and measures and set the accuracy requirements for trade, to minimize the dishonesty in the marketplace. If both buyer and seller want to go rogue, the government can ignore the transaction.

          By encouraging dual, I’m not sure if our government is doing a half-assed job or a double-assed job, but I am pretty sure it is not doing a good job.

          • Again, I have no problem with government defining standards (as long as they aren’t fudging them to increase tax revenues). An I wish our government only set bids in metric.

            But we should be free to do business with whoever we want – if you don’t want to buy fish in cubits you can go to a different store.

            In a free enterprise system, both parties only engage in a transaction if they feel they are better off afterwards.

            I would walk away if someone was selling in cubits. I would also prefer doing business at a store that supports metric.

            If someone is selling in small cubits – one can sue – or call the cops. A store that gets in the news for cheating customers isn’t going to last long. We really don’t want to support a army of regulators to force the use of metric.

            You should read the history of why the meter pendulum came to be. Governments are corrupt – physics isn’t.

  7. A couple of comments:

    Re: “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.”

    Besides the common confusion of weight and mass, this should also indicate something else to us: The Periodic Table should now give atomic masses in yoctograms!

    Re: Du Sautoy’s view of the metric system.

    As The Maven and Ametrica indicated, his view is biased. Why? Simple explanation: As with many other mathematicians, he adores non-decimal fractions and mixed numbers! (BTW, just about the opposite is true among statisticians beyond elementary probability, as density functions (for example, the normal distribution) are continuous, and so probabilities are naturally decimal numbers…)

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