Why ten and not twelve?

This is a question that often arises during discussions on the merits of the metric system. Martin Vlietstra, one of our regular readers, provides some thoughts on the matter, coincidentally on 10 October.

Critics of the metric system of cite the difficulty of dividing ten by three or four, something which would be easy in a duo-decimal system.  This note examines why the original developers of the metric system used ten and not twelve.

Mankind has developed various counting systems over the ages.  The Babylonians inherited a sexagesimal (base 60) system of counting from the Sumerian civilisation.  They did not however have 60 different symbols, but 14 symbols: symbols for 1 to 9 and symbols for 10, 20, 30, 40 and 50, implicitly defining 60 as 6×10 rather than 5×12.  Although the Hebrews, Greeks and Romans used a decimal system of counting, the sexagesimal system still used when telling the time or measuring angles while the IT world often uses hexadecimal (base-16) as a shorthand for grouping set of binary numbers.

A few civilisations have used the duodecimal (base-12) system of counting, but their influence in western scientific and philosophical thinking has been minimal.

The duodecimal system has however been widely used in measurements. The French royal system of measurement of length user the pied or foot (324.8 mm) which had twelve pouces or inches (27.07 mm). The pouce had 12 lignes or lines (2.25 mm) and the ligne had 12 points (0.188 mm).

The duodecimal system was not extended to units greater than one pied, though the toise or fathom was six pieds.  The French did not use a duodecimal system weight or volume, though many European countries had a “light pound” of 12 ounces (as opposed to a “heavy pound”).  In Britain, the “light pound” was the troy pound made up of 12 ounces each of 31.1 g while the “heavy pound”, more correctly known as the avoirdupois pound, has 16 ounces, each of 28.35 g. As is rightly pointed out by supporters of the duodecimal system, 12 can easily be divided by 3 and 4 – a boon to those who do not have calculators.

Simon Stevin, who in 1585 published the basis of what we today know as decimal fractions, forecast that one day units of measure would one day be divided decimally. In 1620 Edmund Gunter developed Gunter’s chain which was 22 yards in length and which had 100 links, each of 7.82 inches (782 inches = 66 feet = 22 yards).  This chain enabled him to measure the lengths of fields in decimals of a chain and so facilitate the calculation of the area of the field (one square chain had an area of 0.1 acres).  In practice, when using Gunter’s chain, the area of the field was measured in acres and decimals of an acre and then, to keep the lawyers happy, the decimal part was converted into rood, perches and square yards.

In 1668 John Wilkins published a proposal that was remarkably similar to the metric system as we know it.  He proposed that the base unit of length should be a “seconds pendulum” (which turned out to be about 993 mm in length) and that multiples and sub-multiples of this base unit should be integer powers of 10.  He made similar proposals for area, volume and weight, defining one unit of weight as being equal to the weight of one cubic volume of rainwater (this compares with a litre of water at 4°C having a mass of one kilogram). Unlike the modern metric system, he reused names of units that were already in use.

One of the most forward-looking European rules of the pre-revolutionary era was Peter the Great (1672 – 1728), Czar of Russia.  He dragged Russia from being a medieval state to being an equal of other European states.  In 1698 he reformed the Russian system of weights and measures and took the English foot as the new Russian fut.  In 1704 he reformed the Russian currency, introducing a “new” rouble which was divided into 100 kopecks, probably the first decimal currency in the world.

One of the eighteenth century’s biggest projects, as least insofar and the amount of computation that was involved, was the survey of France undertaken between 1756 and 1789 by César-François Cassini and his son Jean-Domenica Cassini.  The Cassinis enlisted the help of the British establishment to extend the map to the Greenwich Observatory thus providing a correlation between British and French maps.  The theodolites used were able to measure angles down to about one second of arc, an accuracy that would have required seven-figure trigonometric tables.

When the French Revolution broke out (1789), the royal system of measure was not mandatory (except possibly for assessing the king’s taxes). Merchants could (and did) define their own systems of units which resulted in a wholesale exploitation of the peasants.  In 1790 the French Assemblée set up a new committee under the auspices of the Académie de Sciences to investigate weights and measures. The members were five of the most able scientists of the day: Jean-Charles de Borda, Joseph-Louis Lagrange, Pierre-Simon Laplace, Gaspard Monge and the Marquis de Condorcet.

The Committee quickly agreed that the system of weights and measures needed to be overhauled rather than be “fine-tuned” and that there should be a universal multiplier for units of the same quantity.  There was a debate as to whether the universal multiplier should be 10 or 12. Those connected with trade and commerce favoured a duodecimal system of weights and measures but those who were connected with astronomy and surveying, especially Monge, an associate of the Cassinis, favoured a decimal system.  They argued that nice as the duodecimal system was for simple calculations, it impractical in the fields of heavy computation and would remain so unless the decimal system of counting was replaced by a duodecimal system of counting. Eventually the decimal argument won the day as the introduction of a duodecimal system of counting was totally impracticable.

In 1792 the United States introduced a decimal-based currency as have most, if not all the countries in the world, thereby eschewing the “divisibility” argument of a decimal-based system.

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7 Responses to Why ten and not twelve?

  1. John Frewen-Lord says:

    That is a great piece of history from Martin, but that is where any argument on the relative merits of 12 vs 10 should remain. Twelve is only applicable if the imperial 12-inches-in-a-foot becomes the basis for measurement. Otherwise it could be any number.

    As for divisibility, well if we take 12 inches as a benefit by being divisible by 2, 3, 4 and 6, then its metric equivalent - 300 mm - is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 100, and 150. No contest.

    Finally, as I've said before, all these arguments are irrelevant when 95% of the world's population knows and uses only SI.

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  2. Stan says:

    Even if it were practicable for the world to move away from the base ten system of numbers for general arithmetic, base twelve would probably not be the preferred choice.

    As far as anyone knows base ten was adopted because of the number of digits on human hands, so nothing magical about it mathematically.

    However, consider what it would be like if we adopted base twelve. We'd have twelve different symbols to contend with when doing arithmetic and that many more combinations to remember. When we do arithmetic, even on paper, we have to remember what say 8 and 5 add up to or 9 and 7 and so on. In a base twelve system we'd have to carry far more of that sort of thing in our heads.

    If we used a base smaller than ten then things would be more cumbersome because numbers would need a larger number of digits to represent them (try doing arithmetic in binary).

    On the whole base ten is about right for human arithmetic and there isn't a strong enough reason to change it.

    Computers use binary because it is easier to construct devices or create storage media elements that have two distinct states - a hole or not a hole in punched tape, electric charge or no electric charge in a static cell, magnetised region or no magnetism on the surface of tape or disc, reflection or no reflection on the surface on an optical disc, and so on. Future technology may change this but system adopted is not directly for human consumption.

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  3. Han Maenen says:

    Just consider the practical aspects: the more than atrononomical costs to change the number base from 10 to 12 and having to start literally from scratch in mathematics and counting; then we would have to design an entirely new system of units, a duodecimal SI. Imperial has much more binary divisions, there are only two duodecimal ones : 12" = 1' and 1 lb troy = 12 oz troy. That was used in the old weights used in medicine. The troy pound is no longer used as far as I know. I also think 10 is in practice divisible by 4 as the outcome is 2.5.

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  4. jackthesmilingblack says:

    Geometry and navigation: 360 degrees in a circle.
    Another can of worms.
    Will this have to wait until "we" colonise another planet?

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  5. Martin Vlietstra says:

    During the French Revolution, the circle was divided decimally, a quarter-circle being called a "grade". An angle of 0.01 grades was called a "centigrade" which is why, in 1948, the "degreee centigrade" was renamed the "degree Celsius". The French persisted with the decimal circle until well after the second world war, but with a 24-hour clock which was used elsewhere, plus time zones of 15 degrees, the decimal circle became less and less popular. Mathematicians also decried it - they preferred a circle of 2*pi radians.

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  6. Stan says:

    There is one other point to bear in mind with regard to SI.

    The SI does not depend on the number ten in its basic principles of measurement. The number ten only features in its system of multiples and submultiples. It could easily be adapted if we did happen to count base twelve instead of ten. A prefix such as kilo would represent twelve to the power 3 instead if ten to the power 3, milli would be twelve to the power minus 3 instead of ten to the power minus 3 and so on. But the convenience would be unaffected because the (duo)decimal point would behave exactly the same when it comes to scaling up or down.

    Critics of the metric system often fail to understand this. When it was invented it was based on the idea of rationalising all measurements to a single base unit for all distinctly different types of physical measurement independently of the application. So for example when you measure the capacity of a container you measure volume based on cubic units of length no matter what it is used for. You don't need special units like pints instead of cubic inches for example, especially with such an awkward relationship. Of course there is the litre in metric but it is only a convenient name for a cubic decimetre (IMHO not really necessary).

    It is important to understand why the SI is organised into base and derived units. It ensures that all base units can be defined, where possible, to a unique combination of physical constant of nature, and linking them to derived units without the need for separate experimental determination. The way we count the units has been designed only to make the system convenient to use, a feature sadly lacking in both respects for traditional units like imperial.

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  7. Mark Williams says:

    @John Frewen-Lord:

    To be fair, the 12 vs. 10 argument is applicable in a far wider context than the narrow one focused on measurement. For imperial to truly capitalise on this, it would still have to switch everything to twelves; 12 yards to the chain, 12 chains to the furlong, 12 furlongs to the mile, 12 ounces to the pound, 12 pounds to the stone, etc. This is probably why you rarely hear imperial fans advocating it. There is no such difficulty in having a dozenal metric system---the prefixes would just be powers of 12 and the base units are already coherent. The committee which initially developed it did consider this, but estimated that changing the way people counted and do arithmetic had no better than a one in a do-mo chance of success ;-).

    @Stan:

    John Wilkins made the case for base 8 (and, more generally, bipartitioning) in 1668---but noted that `because general custom hath already agreed upon the decimal way, therefore I shall not insist upon the change of it'.

    @Martin Vlietstra:

    A grad (female gender in the French language?) is 1/400 of a revolution. If you are navigating along a great circle on Earth, a centigrad of arc is helpfully close to 1 km on the surface.

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