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Avoiding the Metric System

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. 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

Although a change to counting in base-12 seems highly unlikely, 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. Could base-12 associate itself with the decimal system in some similar way, so that the two systems could coexist?

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; and then there's an approximate correspondence at 12^13, too high to be useful.

Perhaps some other base might lend itself to providing the benefits of the duodecimal system, while having a connection to the decimal system that would allow a coexistence similar to that provided by 1,024 with the binary system.

 96 = 2 * 2 * 2 * 2 * 2 * 3
108 = 2 * 2 * 3 * 3 * 3

so 12 could be multiplied either by 8 or by 9 to approximate 100. Both 96 and 108 are too large to serve as bases, though; and neither is a power of a smaller base.

6 * 12 * 12 * 12 = 10,368 but a mixed-radix system is hardly an option.


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