Systematic units of measurement

One can, with a modicum of searching, find many places on the internet where anglophones argue with one another about the relative merits of their customary units and the Système International (SI) units. All sorts of fascinating rationales are advanced on both sides (such as the relative precision of the relationship between various volumes of water and their masses, under certain conditions of temperature and pressure) but there is one simple fact thanks to which I vastly prefer SI: it places fewer demands on my memory.

I should note that one must distinguish between SI and the metric system, since the latter may equally refer to the cgs system, in which the centimetre and gramme are used in place of the metre and the kilogramme. Quite why no-one uses the metre, gramme, second system I am at a loss to discern, but adding such a system now wouldn't really help anyone much: it would merely add to the confusion. The cgs system yields derived units that are power of ten multiples of SI units; but I seem to remember that the electrical units used with it are defined sufficiently differently as to yield rather more complications in converting between the two. But at least it still only has one unit for each kind of quantity, for the most part.

In SI, there is one unit of length, one unit of time (although we continue using the customary units of time in practice) and one unit of mass. Likewise there's one unit for each of the small number of other independent kinds of quantity. For each kind of quantity dependent on these, there is a natural way to derive a single base unit for that kind of quantity from these base units. For each kind of quantity, one may use some standard quantifiers (milli, kilo, micro, mega and so on) or one can use powers of ten directly. There are indeed some minor quirks (the unit of mass uses one of the quantifiers in its name, for example) but they are few and far between. It's not perfect, but it is at least largely systematic.

A few dozen of the things I was expected to memorise as a child

The customary units of the anglophone world, on the other hand, require me to remember more than the entire SI base system merely to describe lengths. I still remember being taught these units when I was a child: the teacher expected me to memorise a vast vocabulary of names for units of lengths and stated the value of each in terms of some earlier one (not necessarily the last, nor all in terms of one start-point), expecting me to remember this bewildering mass of ratios. Let me just list part of that litany:

• a line is six points
• a pica is two lines, hence twelve points
• a barleycorn is two picas
• an inch is three barleycorns, hence twelve lines
• a hand is four inches; a span is nine inches
• a cubit is two spans, an ell is five spans
• a foot is three hands, hence twelve inches
• a yard is three feet, which is two cubits; a rope is twenty foot
• a fathom is two yards
• a chain is eleven fathoms
• a rod is a quarter chain; a link is one hundredth of a chain
• a furlong is ten chains, so a thousand links
• a mile is eight furlongs

That's seventeen distinct units for lengths in the range from a third of a millimetre to less than two km. Every number from two to eleven, except for seven, shows up among the eleven factors used, ignoring the uses of twelve in hence clauses. We also had to remember that an acre was a chain by a furlong and, just to confuse us all a bit more, our teacher threw in the hectare (a metric unit, but the teacher neglected to point this out), which is a little under two and a half acres but I can't remember what value our teacher gave us for it. It's probably a good thing she didn't tell us that some engineers take the chain to be 100 foot (instead of the 66 feet you'll get if you work your way through the above).

Six of those units aren't relevant to the sequence from point up to mile; and we can shed another four by simplifying the sequence to: twelve points make a pica, six pica make an inch, twelve inches make a foot, six feet make a fathom, eleven fathoms make a chain, ten chains make a furlong and eight furlongs make a mile, but that's still eight unit names and five distinct multipliers to remember; and none of that remembered information does me any good in learning the kindred mess of complexity involved for weights (seven units, five distinct ratios) or for volumes (I forget which units we were taught). Given that I've never been much good at memorising things, this was all too much for me. These days I have a python module, study.value.archaea in my study package, to remember all this arcane complexity for me.

Having to remember so many things increases the over-head of learning to use such a system, quite apart from the complications that arise from different nations having different versions of some of these units – the US differs significantly from the UK, hence much of the commonwealth, on volumes and on some weights; and many of these units also existed in other nations, with usually (but not always) the same ratios among values and similar values. It gets even more confusing when some of the names show up for different kinds of unit: the timber foot is a unit of volume, one cubic foot; the water ton is a unit of volume (and, at the right temperature and pressure, this much water does, indeed, have a mass of one UK ton; only be sure not to confuse the water ton with the tun, which is a slightly smaller volume) and the bushel is a unit of volume except when it's a unit of mass – the wheat bushel and similar for other grains – doubtless approximately equal to the mass of a bushel of each grain involved. And then of course there's the use of pound both as unit of weight (i.e. force) and as unit of mass.

All that diversity of base units complicates matters even further when we come to deal with derived units. I have had occasion to work with software in which volume flow-rates could emerge from calculation in either cubic feet per second or US gallons per minute (and, to the credit of the US gallon, unlike the UK one, there is at least a defined ratio of whole numbers between these); and forces could come out in pound (mass) feet per second per second or in pounds weight (requiring a factor of 32.174 and a bit to convert between them). The code sporadically involved stray random-looking multipliers that resulted from conversions between units; and it often wasn't obvious what conversion was being done (because it wasn't always obvious what units any given quantity was measured in).

So the true virtue of SI is its simplicity. All arguments about how convenient it is (or isn't) that one system's units of volume and mass give handy values to the density of water (or any other physical constant) are essentially irrelevant: no matter how handy such a datum is, it's a physical property of a material and it changes with circumstances (e.g. temperature and pressure), so any practical work with the given material requires you to have the correct value for the conditions you're in, at which point it isn't really very important that the value is close to one, except when one is doing rough calculations – at which point, the systems tend to do roughly as well as one another.

Improving SI

It should be noted that SI could be made better, although the upheaval in doing so would probably be more effort than it's worth. Getting anglophones to abandon an archaic system in favour of one that's simpler by a factor of several dozen has taken us two centuries and we're not done yet: Britain was inching towards metrication when I was a child and Ronald Reagan derailed the USA's limited progress in my youth. Getting the whole world to change over to a marginally simpler system is probably too much to expect. Still, it amuses me to consider what improvements we could consider.

There would be a certain virtue to switching from base ten to base two, if only for the sake of our computations. There are already quantifiers ready to serve if we do: Kibi, Mibi and Gibi have been introduced for the powers of 1024 = two^ten, although I'd be inclined to suggest some other factor than (roughly) a thousand would be more suitable; 256 = two^eight fits better with how computers actually manage data; but sixteen = two^twotwo or 65536 = 256² = two^sixteen would fit more elegantly with base two. Still, our units are meant for our use, so it's perhaps better to stick with base ten, as it's embedded in most of our languages and the computers can handle the arithmetic better than we can anyway.

The metre is now defined in terms of the second: light travels exactly 299792458 metres through a vacuum in a second. This distance is called a light second just as a light year is the distance light travels in a year. This effectively makes the metre a derived unit and could be construed as an argument for replacing the metre with the light second as unit of length. Of course, that's rather a large distance for most terrestrial purposes; but the light nanosecond is quite a practical length – in fact, it's roughly a foot; slightly more than the archaic Swedish foot and slightly less than most of its cousins, including the anglophone foot. It would thus be natural to introduce the light nanosecond as the SI foot. (The point = pica / 12 = inch / 6 / 12 = foot / 12 / 6 / 12 is 1.17674 light picoseconds (almost exactly), while the caliber = inch / 100 is 0.847 (and a bit) light picoseconds; they're off 1 light ps by roughly equal amounts and not really close enough to make compelling candidates to serve as a nick-name for the light picosecond; the situations for the light microsecond and light millisecond are even worse. The light kilosecond is actually fairly well approximated by the diameter of Earth's orbit, 2 AU, but we tend to use minutes and hours rather than kiloseconds.) The original specification of the metre tried hard (and failed) to be one forty-millionth of the Earth's circumference (which isn't really a very good kind of definition for a unit, hence the recent switch to a definite fraction of the light second), so there's no particularly strong reason to be attached to it; and an SI foot has a certain comic charm to it ;^>

Then again, the second isn't a natural unit in the first place. We arrived at it by dividing the day and night into a dozen watches each then subdivided those by sixty to get minute subdivisions of them and by sixty again to get a second minute subdivision. What we actually started with is the day, because that's the rhythm of our waking, eating, working, sleeping; even if we leave our home planet, with its daily cycle of light and dark, we'll still have the legacy of our past wanting us to live our lives to roughly this rhythm. The cycle time might shift to a bit longer or shorter, but we'll be living with something like the day as a practical unit in our lives for the foreseeable future. So let's use the day as unit of time and take a look at what our SI quantifiers do to it. We get deciday (2 hr 24 min) intervals at which to take breaks during our working day (of three or four decidays). The scale of those breaks is the centiday (nearly quarter hour); a few for lunch, a third of one for the breaks to stretch our legs and get some fresh air. We get a milliday of 86.4 seconds, which is of order a minute; the microday is then a fairly good instant of time, about a twelfth of a second, the sort of amount of time that could fairly be called an eyeblink. The hectoday is a bit longer than a season, the kiloday a bit less than three years; the megaday is almost 2738 years, roughly how long the Eurasian cultures have had philosophy; our species is about a gigaday old, cyanobacteria started polluting our planet with oxygen about a teraday ago and the universe is impressively close to five teradays old.

How about the distance units ? The light day is about twice the diameter of the Kuiper belt, so we'll definitely be needing to apply quantifiers to it. We get a light picoday of just under 26 metres (85 foot; just short of a shackle and a bit over a chain and a quarter); and that gives us a light femtoday of 25.9 mm, which is a respectably good approximation to the inch (25.4 mm). So we could use the day as unit of time and have a light femtoday as common unit of length, in which to measure the diagonals of visual display screens. Sprinters can, in fractions of a milliday, race over a few (kinches or) light picodays while longer distance runners practice for hundred light picoday (2.59 km or just over 1.609 miles, almost the same ratio as the mile to km) races and the marathon runners adjust to one or two (meginch or) light nanoday races, that take a few decidays. The Earth's diameter is about half a (giginch or) light micro-day, the Moon is 14.8 micro-days away, the Sun is 5.78 (terinches or) light milli-days away and the distances to other nearby stars are small numbers (1.56 and up) of light kilodays; the parsec is 1.19129 light kilodays. The Milky Way is about 38 light megadays across and we're nine and a half light megadays from its centre. The Small Magelanic cloud orbits the Milky Way at a distance of about 77 light megadays, Andromeda is about 0.85 light gigadays away and the Virgo cluster about 17.5 light gigadays away.

Just as the metre can be defined in terms of the second via the speed of light, c, the kilogramme could be defined in terms of the second and metre via Planck's constant, h, if we knew it with suitably great precision (which may well happen before 2020). The resulting unit would, however, be spectacularly tiny: even neutrino masses are (probably) huge when measured in it. Worse yet, I can't even find (despite their diversity) an archaic unit of mass that's close to a neat power of ten times it. But, in any case, Planck's constant appears to describe real physics, where the speed of light is merely an artefact of our relationship with space and time, a conversion factor between two units conventionally used for displacements in different space-time directions; not fundamentally different from a slope of 110 fathoms per furlong, the conversion factor between units of archetypically horizontal and vertical lengths. I'd be more willing to take Newton's constant as a unit conversion factor: this time the unit is rather big but not wildly so. Using seconds and metres with G, we get the mass of a few mega tonnes; intermediate between astronomical and everyday masses. Using day and light femto-day (a.k.a. inch), we get a mass of 35 mg, which is a bit over half a grain (the mass of a barleycorn; seven thousand of these make a pound) and on the order of some medical doses (of active ingredient in a pill); with day and light pico-day, we're up to 35 tonnes. So there's potential for a half-way decent system of units based on the day, using the speed of light and Newton's constant as calibration for length and mass; but we don't know Newton's constant anything like accurately enough.

I've explored these ideas further on someone else's blog.

See also

If you need to convert between diverse units, module study.value.archaea in my study package can do quite a lot (and pulling in study.space.home can get you a few others, like the Astronomical Unit and some other properties of the Earth that get used as units sometimes). You can also use the web:

• If what you type in Google's search box looks enough like a calculation, which may involve units, the search results will start with an answer to the calculation; so typing 27 mile / km will give you an answer of 43.45 and a bit, telling you that 27 miles equal almost 43 and a half kilometres. If you've got scripting enabled, you'll even get the results in a drop-down, as you type, without having to submit the form.
• The International unit conversion calculator covers conversion between various units. It's easy enough to use and covers plenty of kinds of quantity – especially on the cooking page. I'm particularly tickled by this last's inclusion of both megaliter (the volume of a cube ten metres on each side – that might just qualify as a catering unit, for a large enough music festival or military campaign) and microliter (you'll be hard-pressed to see – let alone notice any taste effect of – this much of any ingredient your guests wouldn't be freaked out to know you were using – I seem to remember there are some neurotoxins that are lethal in doses this small, but please don't treat that as a suggestion).

Finally, here's what Lord Kelvin said (sadly over-optimistically) to some USAish audience:

You, in this country, are subjected to the British insularity in weights and measures; you use the foot, inch and yard. I am obliged to use that system, but must apologize to you for doing so, because it is so inconvenient, and I hope Americans will do everything in their power to introduce the French metrical system. … I look upon our English system as a wickedly, brain-destroying system of bondage under which we suffer. The reason why we continue to use it, is the imaginary difficulty of making a change, and nothing else; but I do not think in America that any such difficulty should stand in the way of adopting so splendidly useful a reform.

Thank you, Sir William Thomson, 1st Barron Kelvin of Largs – wickedly, brain-destroying system of bondage – indeed !

Written by Eddy.