There are twelve inches in a foot, and sixteen ounces in a pound. Three feet make a yard, four quarts make a gallon.
While these various scale factors are what has led to the complicated structure of the traditional English system of measurement, they also show that a need was felt to be able to easily divide quantities up, without resorting to fractions, into halves and thirds and quarters.
Dividing something into quarters takes two places of decimals, not just one, and dividing something into thirds is not possible in decimals. Thus, the metric system of measurement, relying exclusively on factors of 10 to relate unis of different size, doesn't provide units that can be divided up as well as it seems to be desired to divide up units of measure.
How could one have a system of measurement that offers the divisibility found in traditional systems of units, and yet one which conforms in the kind of simple and direct manner to arithmetic that the metric system conforms in?
There is one simple answer to this, one way in which these two conflicting requirements can be reconciled. We can change the way we do arithmetic, using the duodecimal system instead of the decimal system.
Of course, that is such a far-reaching change that it is extremely unlikely ever to happen. Yet, it's true that the prefixes kilo-, mega-, giga-, and so on now carry a double meaning, since two to the tenth power is 1,024, which is not far from a thousand. Unfortunately, one has to wait until twelve to the thirteenth power to get 106,993,205,379,072 - which exceeds ten to the fourteenth power by seven per cent - so there is no trivial way for the duodecimal system to serve as an approximation to the decimal system in an analogous manner.
Still...
1 12 2 144 3 1 728 4 20 736 5 248 832 6 2 985 984 7 35 831 808 8 429 981 696 9 5 159 780 352 10 61 917 364 224 11 743 008 370 688 12 8 916 100 448 256 13 106 993 205 379 072 14 1 273 817 464 548 864 15 15 407 021 574 586 368
12 to the fourth power is approximately 20,000; 12 to the sixth power is approximately three million;
Still, one can briefly consider how this might allow the traditional system of English measures to be transformed so as to become as rational as the metric system.
As there are twelve inches to the foot, both these units could fit directly into a duodecimal system of weights and measures - using the prefixes proposed by Tom Pendelbury, who proposed a more sophisticated duodecimal meaurement system (and there was also the Do-Metric system, which is not the same as this one either), a brief table of correspondences can be exhibited:
quedrafoot 3 miles 7 furlongs 4 chains 4 yards trinafoot 2 furlongs 6 chains 4 yards dunafoot 2 chains 4 yards zenafoot 4 yards; 2 fathoms foot foot inch inch zeniinch line; 1/4 barleycorn; 1/2 pica duniinch 1/2 point
Thus, a quedrafoot is just a bit less than four miles (there are eight furlongs to a mile, ten chains to a furlong, and 66 feet to a chain). The printer's units shown, the pica and the point, are predicated on the Selectric Composer point of exactly 1/72 inch.
There are sixteen ounces to a pound. However, the avoirdupois pound is often considered to be a unit of force (with the slug as its associated unit of mass) although it can also be considered a unit of mass (with the poundal as its associated unit of force). Also, an avoirdupois pound is equal to 7000 grains.
On the other hand, the troy pound, used mainly to specify the price of gold, is clearly a unit of mass, just like the kilogram, there are twelve troy ounces to the troy pound, and a troy ounce is 480 grains. So this is the English customary unit of weight that would be well suited to a duodecimal system:
troy pound troy pound troy ounce troy ounce zeniounce 40 grains duniounce 3 1/3 grains
The metric system did not attempt to modify how we tell time; the hour, the minute, and the second remain the units of time, with the second being the basis of other metric units.
However, since there are 24, twice twelve, hours in a day, it would seem that the opportunity of making the clock fully duodecimal does exist. Thus, the units of time might be:
zenahour 1/2 day hour hour zenihour 5 minutes dunihour 25 seconds trinihour 2 1/12 seconds
Note that while there are twelve trinihours in a dunihour, the trinihour is a bit over two seconds and the dunihour is a bit under half a minute, so trinihours would still take the place of seconds and dunihours the place of minutes.
Just as the meter, the kilogram, and the second allow the other units of SI to be derived, or the centimeter, the gram, and the second allow the units of the alternative cgs system to be derived, one can derive a complete system of units from the foot, the troy ounce, and the hour; replacing the troy ounce by the troy pound, and the hour by the trinihour, allows the magnitudes of the units to remain in somewhat the same ballpark as those in the SI or MKS system.
12 trinihours = 25 seconds; 1 trinihour = 2.08333... seconds 1 troy pound = 5760 grains = 373.2417216 grams = 0.3732417216 kilograms 1 foot = 30.48 centimeters = 0.3048 meters troy_pound foot / trinihour^2 = 0.4937676942 Newtons = 4937.676942 dynes troy_pound / foot trinihour^2 = 5.314871227 Pascals troy_pound foot^2 / trinihour^2 = 0.1505003932 Joules = 150500.3932 ergs troy_pound foot^2 / trinihour^3 = 0.3135424858 Watts
While there are 5,280 feet to a mile, and therefore 1,760 yards to a mile, and, thus, 1,728 yards to a mile instead, with 1,728 being twelve to the third power, is not very much different, going over to base-12 numeration is rather too extreme a measure to take to make the British Imperial system of measures comparable to the metric system in rationality.
Could the British Imperial system of measures be modified slightly, to remove its worst parts, so that the additional complexities of using it instead of the metric system would not be particularly severe?
I think that is possible, although it's quite unlikely that even a rationalized Imperial measure system could replace the metric system at this late date.
In the case of the mile, the solution is obvious. Take the literal meaning of the Roman unit, one thousand paces, and thus define a new, rationalized mile to be 6,000 feet, or 2,000 yards.
Twelve inches to the foot, three feet to the yard, and two thousand yards to the mile; that is a small number of multiplications, hardly worth complaining about.
In the case of units of mass, sixteen ounces to the pound, and two thousand pounds to the ton, are also not that much. However, 7,000 grains to the pound (avoirdupois), but 480 grains to the pound (troy) with four troy ounces to the troy pound is definitely annoying; and, worse yet, the avoirdupois pound and the troy ounce are both so fundamental that neither one seems reasonable to change.
One thousand is divisible by eight. So one avoirdupois ounce is 437 1/2 grains. A unit of 2 1/2 grains is common to both the troy ounce (192, or 3 * 64, of them) and the avoirdupois ounce (175, or 25 * 7, of them) but they are relatively prime.
The division of an ounce into 8 drams, and each dram into 3 scruples, is not only applied to fluid ounces in liquid measure, but also to troy ounces in the apothecaries' scale of weight.
If a troy ounce is 24 scruples, then each scruple is 20 grains in weight.
Take 25 scruples; this is 500 grains, or 12 1/2 grains times 40. Whereas 437 1/2 grains is 12 1/2 grains times 35.
And so I could suggest that the troy ounce and the avoirdupois ounce be brought into harmony through the following device:
In addition to the troy ounce (480 grains), the dram (60 grains), and the scruple (20 grains), also define a unit of 24 grains, 20 of which make a troy ounce.
Wait a moment; dividing a troy pound into 12 parts to make troy ounces, and then into 20 parts... haven't I seen something like this before? Why, yes, but in the opposite order; a pound sterling is 20 shillings, each one of which is 12 pence. And, indeed, 24 grains is the unit of mass known as the pennyweight.
But now let us call the existing pennyweight the short pennyweight, and also define a long pennyweight of 25 grains. And an avoirdupois ounce is 17 1/2 long pennyweights, and the relationship between the troy ounce and the avoirdupois ounce is now clear and obvious!
Finally, there is liquid measure.
The weight of a fluid ounce of water, in the British Imperial system, as opposed to the U.S. customary system, is very close to one avoirdupois ounce. Unfortunately, while a pint is 16 fluid ounces in the U.S. customary system, it is 20 fluid ounces in the British Imperial system. The U.S. fluid ounce may be larger than the British fluid ounce, but it's not large enough for a U.S. fluid ounce of water to equal one troy ounce in weight.
If one calls the British Imperial pint of 2 1/2 cups a long pint, and also has a short pint of 16 fluid ounces, there finally would be a pint that's a pound (the world around!) creating a link between liquid measure and weight comparable to that in the metric system.
However, it is the U.S. gallon, at 231 cubic inches, that has some relation to the cubic inch; the British Imperial system of liquid measure has no such relation.
A British Imperial fluid ounce of water is 2/9 of 1% heavier than an ounce, so there is a bit of wiggle room. Could that fluid ounce be changed so as to be closer to an ounce in weight, and provide a link to linear measure as well?
A British Imperial gallon is 4.54609 litres, and a cubic inch is 16.387064 millilitres. So a British Imperial gallon is some 277.41943279... cubic inches. Change that to 277.375 cubic inches, and one has a whole multiple of the volume of a cube one-half inch on a side, at least. (Unfortunately, 2,219, while not a prime number, is 7 times 317, and so that does not lead to convenient dimensions for a rectangular prism with a volume of one gallon.) That would be 4.545361877 litres, making the modified fluid ounce equal to 28.40851173125 millilitres.
Even if the British Imperial gallon were simply defined as 277 cubic inches, an Imperial gallon of water would still weigh slightly more than 10 pounds; so under that system, an Imperial fluid ounce would be 28.37010455 ml; compare that to an avoirdupois ounce being 28.349523125 grams, and then take into account the fact that a kilogram of water occupies 1.000028 cubic decimetres (the old value of the litre, redefined in 1964 to be exactly a cubic decimetre), a British Imperial fluid ounce would weigh about 29.34955... grams.
There: a system of measurement, based on the British Imperial system of measures, so simple and rational that there is hardly any reason to prefer the metric system to it for the slight additional simplifications that it, admittedly, would still offer!
Of course, one could be a bit more radical. Abolish the avoirdupois system of weights, and define the gallon as 243 cubic inches, to weigh about the same as 128 fluid ounces of water, corresponding to the U.S. customary system of liquid measure, but somewhat larger in size. Here, a fluid ounce of water would weigh slightly less than one troy ounce, but a pint would be a pound of 16 of those ounces rather than a troy pound of 12 troy ounces.