# Why count in tens rather than twelves?

Martin Vlietstra looks at our counting system and explains why we count in tens and not in twelves. Our counting system and the metric system are both decimal based (i.e. they use base 10). The entire metric system is based on tens and powers or ten, which fits in neatly with our counting system and makes calculations easy.

One of the strengths of the metric system is the way in which sub-divisions of the base units are handled. Apologists for the Imperial System point out that 10 is only divisible by 2 and 5 whereas 12 is divisible by 2, 3, 4 and 6. While this might be true, it is worthwhile looking deeper into the way in which the metric system was developed.

Each of the different ancient civilisations had their own way of representing numbers, though of course there were overlaps.  Since Roman numerals influenced the post-Renaissance culture more than any other system, we will only look at that system. It had seven symbols to represent whole numbers: I, V, X, L, C, D and M. The largest number that could be represented using them is MMMDCCCLXXXVIII (3888). Various complex systems were developed to cope with larger numbers but there was no one standard way of doing so.  Although the Romans used a decimal system for counting whole numbers, they used a duodecimal system for representing fractions: they used the letter “S” to represent a half (or rather 6/12) and a system of between one and five dots, arranged much like those found on a die which represented the fractions 1/12 up to 5/12. If a number had a fractional part, then these symbols were added to the end of the number, for example 25½ was written XXVS while 2⅙ was written “II:” (literally one plus one plus two twelfths). There were further symbols to enable representation of fractions as small as 1/288, but again these were rather cumbersome and not always standard.

The biggest drawback for representing large numbers was the use of numeral notation (such as Roman numerals) rather than positional notation such as we use today. In the First Century AD, Hindu mathematicians first started using the positional notation whereby they had nine symbols representing the number 1 to 9. This value was then multiplied by 1, 10, 100 etc, depending on its position in the number string.  They had to add a tenth character as a place-holder to denote an unused space. Today we call that tenth character “zero”. This system was developed further by Arabic mathematicians and by the Ninth Century AD, it had reached a form which is very similar to the system used in Arabic countries today.

This system of numbers was introduced into Europe during the first decade of the thirteenth century by Leonardo Bonacci, better known as Fibonacci, the son of a merchant from Pisa who travelled to North Africa where he learned the concepts of the Hindu-Arabic notation.

How then did the ancients represent small units of measure?  In general, they developed smaller and smaller units of measure which is why the imperial system of weights is based on the grain. Until the advent of metrication, chemists (apothecaries) used a system whereby there were 20 grains in one scruple, 3 scruples in one dram and 8 drams in one troy ounce (about 31 g). Precious metal dealers however used a pennyweight of 24 grains, 20 of which made one troy ounce.  They seldom had to use units of measure that were less than a grain (about 65 mg).  For general purposes, there were 16 drams in an avoirdupois ounce (about 28.5 g). When it came to small lengths, the line (1/12 of an inch) was often used and sometimes the point (1/6 line) was also used, but these were never included in the official list of units that comprised the imperial system.

The observant reader will notice the use of specific units for small quantities placed a limit on measing very small quantities. It also place a limit on highly accurate measurements – for example, the bore of a car’s cylinder must be machined to a very high precision. The Americans got around this problem by using decimals of an inch while the British used multiples of successive binary fractions of an inch (1/2, 1/4, 1/8 etc down to 1/64 or in some cases 1/128).  If the British required finer measurements than that, they switched to using decimals of an inch.

In 1585, Simon Steven, a native of Bruges who had fled to the Netherlands to escape religious persecution, published his book De Thiende (“the tenth”) in which he advocated representing fractions in tenths, hundredths and so on. Steven forecast that in time, both money and units of measure would be based on decimal values rather than the complex systems then in place. His original notation was a little clumsy, but it was refined by, amongst others, John Napier who, in 1614, published his book about logarithms, a system that could not exist without the use of decimal numbers.

In 1668, Wilkins proposed a decimal-based unified system of units using a one-second pendulum as the basis for a unit of length and the mass of one cubic unit of water as the basis for the unit of mass. In his system, he recycled existing names of units and left the question of how to represent smaller numbers open. Two years later, in 1670, Gabriel Mouton proposed a system for defining units of length based on the minute of arc of the earth’s circumference. Unlike Wilkins, he developed a totally new set of names which contained the roots “centi and deci” thereby laying the foundations for what was to become the metric system.

During the seventeenth and eighteenth centuries there were a number of further proposals for a decimal based system of units: In 1699, John Locke (1632-1704) proposed a decimal system of units of measure when drafting a constitution for the colony of Carolina.  James Watt (1736-1819) who, in 1783, was having problems communicating with German scientists also proposed a system and in 1788 Antoine Lavoisier (1743 – 1794) commissioned a set of decimal weights for his chemical experiments. In 1790 Thomas Jefferson (1743-1826) presented a document to the United States Congress in which he proposed decimal-based systems of currency and units of measure.

It is estimated that when the French Revolution broke out in 1789, there were eight hundred or so different units of measure in France: the king’s units applied to royal taxes, the various ducal units of measure applied to ducal dues while many merchants had their own units of measure, many of which went by the same name. It was not unknown for a merchant to have two units of measure with the same name – one for use when buying and the other for use when selling a product.  In 1790, a panel of five leading French scientists Jean-Charles de Borda, Joseph-Louis Lagrange, Pierre-Simon Laplace, Gaspard Monge and Nicolas de Condorcet formed the Commission for Weights and Measures.

This was the Age of Enlightenment and nothing was off limits in their deliberations. One of the first activities of the Commission was to decide on a base for the new system of units. Parts of the existing French system of units, like the English system of units, used a base of 12, but, as in England, this was by no means universal.  By 1790 a considerable amount of experience in using decimal numbers had been achieved – a century earlier both Wilkins and Mouton had demonstrated the power of decimal numbers. One of the first decisions that the committee came to was that the system of units should match the system of counting. From a mathematical point of view this suggested that a duodecimal [base 12] system of counting and of measuring was the ideal. The duodecimal system was not perfect – if one was performing precision work there was nothing to be gained in using a duodecimal system.

Furthermore, from a practical point of view, the committee came to the conclusion that changing from a decimal to a duodecimal system of counting would be totally unacceptable to the population at large, so in order to produce a useful system of units, one should use a decimal based system. Thus it was that in 1792 they produced the first draft of what we now call the metric system. Possibly Charles Dickens had the last word when he wrote “Let us measure how we count – in tens”.

https://en.wikipedia.org/wiki/Roman_numerals

https://en.wikipedia.org/wiki/Positional_notation

https://en.wikipedia.org/wiki/Numeral_system

https://en.wikipedia.org/wiki/Fibonacci

https://en.wikipedia.org/wiki/Plan_for_Establishing_Uniformity_in_the_Coinage,_Weights,_and_Measures_of_the_United_States

John Wilkins (1668) – An Essay Towards a Real Character, and a Philosophical Language – https://www.google.co.uk/books/edition/An_Essay_Towards_a_Real_Character_and_a/BCCtZjBtiEYC (pages 190 – 193)

https://mathshistory.st-andrews.ac.uk/Biographies/Mouton/

Gabriel Mouton (1670) – Observationes diametrorum solis et lunae … https://www.google.co.uk/books/edition/Observationes_diametrorum_solis_et_lunae/zcArxp6qVRgC

Peter R. Anstey (2015) – Locke on Measurement – https://www.rotman.uwo.ca/wp-content/uploads/anstey_lockeonmeasurement.pdf

Andrew Carnegie (1905) – James Watt – https://www.gutenberg.org/files/26131/26131-h/26131-h.htm

Ken Adler (2002) – The Measure of All Things – ISBN 0-349-11507-9

Dicken (1863) – All the Year Round; A Weekly Journal, Vol IX, pg 223 https://books.google.co.uk/books?id=RdUNAAAAQAAJ&pg=PA233

## 2 thoughts on “Why count in tens rather than twelves?”

1. metricviewer says:

Very interesting.

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2. Daniel says:

“Apologists for the Imperial System point out that 10 is only divisible by 2 and 5 whereas 12 is divisible by 2, 3, 4 and 6. ”

There is no rule from the BIPM or any other organisation that states only the divisors of 2 and 5 must be used. The BIPM leaves what divisors are used up to the industry using SI units. In the construction industry, the 100 mm module is the standard with increments of 300 mm common for the reason of a large amount of divisors. In other industries sizes are based on the Renard series.

In FFU, the only place 12 appears is in the inch to foot relationship. Every other relationship uses a myriad of factors, with no consistency.

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