This article looks at the problems surrounding systems of measurement which grew up over many centuries, and looks at the introduction of the metric system. Let us first comment on what, in broad terms, is the meaning of measurement. It is associating numbers with physical quantities and so the earliest forms of measurement constitute the first steps towards mathematics. Once the step of associating numbers with physical objects has been made, it becomes possible to compare the objects by comparing the associated numbers. This leads to the development of methods of working with numbers.

The earliest weights seem to have been based on the objects being weighed, for example seeds and beans. Ancient measurement of length was based on the human body, for example the length of a foot, the length of a stride, the span of a hand, and the breadth of a thumb. There were unbelievably many different measurement systems developed in early times, most of them only being used in a small locality. One which gained a certain universal nature was that of the Egyptian cubit developed around 3000 BC. Based on the human body, it was taken to be the length of an arm from the elbow to the extended fingertips. Since different people have different lengths of arm, the Egyptians developed a standard royal cubit which was preserved in the form of a black granite rod against which everyone could standardise their own measuring rods.

To measure smaller lengths required subdivisions of the royal cubit. Although we might think there is an inescapable logic in dividing it in a systematic manner, this ignores the way that measuring grew up with people measuring shorter lengths using other parts of the human body. The digit was the smallest basic unit, being the breadth of a finger. There were 28 digits in a cubit, 4 digits in a palm, 5 digits in a hand, 3 palms (so 12 digits) in a small span, 14 digits (or a half cubit) in a large span, 24 digits in a small cubit, and several other similar measurements. Now one might want measures smaller than a digit, and for this the Egyptians used measures composed of unit fractions.

It is not surprising that the earliest mathematics which comes down to us is concerned with problems about weights and measures for this indeed must have been one of the earliest reasons to develop the subject. Egyptian papyri, for example, contain methods for solving equations which arise from problems about weights and measures.

A later civilisation whose weights and measures had a wide influence was that of the Babylonians around 1700 BC. Their basic unit of length was, like the Egyptians, the cubit. The Babylonian cubit (530 mm), however, was very slightly longer than the Egyptian cubit (524 mm). The Babylonian cubit was divided into 30 kus which is interesting since the kus must have been about a finger's breadth but the fraction ^{1}/_{30} is one which is also closely connected to the Babylonian base 60 number system. A Babylonian foot was ^{2}/_{3} of a Babylonian cubit.

Now we commented in the previous paragraph about a subdivision of a Babylonian unit which was closely related to their number system. This presents a problem as we look at developing systems of measures. Many early number systems tended to be based on ten for the obvious reason that we have ten fingers on which to count. Most such systems were not positional systems, so the reason to use multiples of ten in measurement subdivision was less strong. Also ten is an unfortunate number into which to divide a unit of measurement since it only divides naturally into ^{1}/_{2} , ^{1}/_{5} , ^{1}/_{10} . Basing subdivisions on 12, mean that ^{1}/_{2} , ^{1}/_{3} , ^{1}/_{4} , ^{1}/_{6} , ^{1}/_{12} are natural subdivisions, giving much more range for trading quantities. However, since most measuring systems seem to have grown up as a combination of different "natural" measures, no decision about a number to subdivide by would arise. One exception, and the earliest known decimal system of weights and measures, is the Harappan system.

Harappan civilisation flourished in the Punjab between 2500 BC and 1700 BC. The Harappans appear to have adopted a uniform system of weights and measures. An analysis of the weights discovered in excavations suggests that they had two different series, both decimal in nature, with each decimal number multiplied and divided by two. The main series has ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch". Of course ten units is then 13.2 inches (33.5 centimetres) which is quite believable as the measure of a "foot", although this suggests the Harappans had rather large feet! Another scale was discovered when a bronze rod was found to have marks in lengths of 0.367 inches. It is certainly surprising the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches (93 centimetres) which is about the length of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in their construction.

European systems of measurement were originally based on Roman measures, which in turn were based on those of Greece. The Greeks used as their basic measure of length the breadth of a finger (about 19.3 mm), with 16 fingers in a foot, and 24 fingers in a Greek cubit. These units of length, as were the Greek units of weight and volume, were derived from the Egyptian and Babylonian units. Trade, of course, was the main reason why units of measurement were spread more widely than their local areas. In around 400 BC Athens was a centre of trade from a wide area. The Agora was the commercial centre of the city and we know from the plays of Aristophanes the type of noisy dealing which went on there. Most disputes would arise over the weights and measures of the goods being traded, and there a standard set of measures kept in order that such disputes might be settled fairly. The size of a container to measure nuts, dates, beans, and other such items, had been laid down by law and if a container were found which did not conform to the standard, its contents were confiscated and the container destroyed.

The Romans adapted the Greek system. They had as a basis the foot which was divided into 12 inches (or ounces for the words are in fact the same). The Romans did not use the cubit but, perhaps because most of the longer measurements were derived from marching, they had five feet equal to one pace (which was a double step, that is the distance between two consecutive positions of where the right foot lands as one walks). Then 1,000 paces measured a Roman mile which is reasonably close to the British mile as used today. This Roman system was adopted, with local variations, throughout Europe as the Roman Empire spread. However, if one looks at a country like England, it was invaded at different times by many peoples bringing their own measures. The Angles, Saxons, and Jutes brought measures such as the perch, rod and furlong. The fathom has a Danish origin, and was the distance from fingertip to fingertip of outstretched arms while the ell was originally a German measure of woollen cloth.

In England and France measures developed in rather different ways. We have seen above how the problem of standardisation of measures always presented problems, and in early 13^{th} century England a royal ordinance *Assize of Weights and Measures* gave a long list of definitions of measurement to be used. On one hand it was an extremely successfully attempt at standardisation for its definitions lasted for nearly 600 years. The Act of Union between England and Scotland decreed that these standards would hold across the whole of Great Britain. Locally, however, these standards were not always adhered to and districts still retained their own measures. Of course, although an attempt had been made to standardise measures, no attempt had been made to rationalise them and Great Britain retained a bewildering array of measures which were defined by the ordinance as rather strange subdivisions of each other. Scientists had long seen the benefits of rationalising measures and those such as Wren had proposed a new system based on the yard defined as the length of a pendulum beating at the rate of one second in the Tower of London.

In France, on the other hand, there was no standardisation and as late as 1788 Arthur Young wrote in "*Travels during the years *1787, 1788, 1789" published in 1793:-

In fact it has been estimated that France had about 800 different names for measures at this time, and taking into account their different values in different towns, around 250,000 differently sized units. To a certain extent this reflected the powers which resided in the hands of local nobles who had resisted all attempts by the French King over centuries to standardise measures. Diderot and d'Alembert in theirIn France the infinite perplexity of the measures exceeds all comprehension. They differ not only in every province, but in every district and almost every town.

This proposal was not designed to bring in a French system of measurement but to design an international system of measurement, so agreement was sought from other countries. An immediate problem was that the pendulum length depended on the latitude at which the experiment was performed so a latitude had to be chosen. The French proposed 45° which conveniently fell in France, the British proposed London, and the United States proposed the 38th parallel which was conveniently close to Thomas Jefferson's estate. Diplomatic wording allowed an international agreement to be reached, but in March 1791 Borda, as chairman of the Commission of Weights and Measures, proposed using instead of the length of a pendulum, the length of ^{1}/_{10,000,000} of the distance from the pole to the equator of the Earth. They might have got international agreement on this had they not declared that this distance would be determined by an accurate survey of the distance between Dunkerque and Barcelona. The Royal Society in London declared this was based on a measurement of France, the Americans were not prepared to accept the word of the French mathematicians for its length and even in France it was claimed that the whole project was really proposed in order to gain information on the shape of the Earth. Indeed, probably Laplace and others were more interested in finding the shape of the Earth rather than the length of the metre.

Delambre and Méchain measured the meridian from Dunkerque and Barcelona between 1792 and 1798. However between these dates the French Revolution progressed to the stage where the Académie des Sciences was abolished in August 1793 but before that Borda, Lagrange and Laplace had computed a provisional value for the metre based on the survey carried out by Cassini de Thury in 1740. The metric system was passed into law by the National Assembly and a metre bar together with a kilogram weight were dispatched to the United States in the expectation that they would adopt the new measures. Congress hesitated because the standards were provisional. Britain became hostile to the metre as did Germany which wanted a standard based on the pendulum.

An International Commission began work in September 1798 to replace the provisional values with precise ones computed from the data collected by Delambre and Méchain. By June of the following year the Commission had produced a platinum bar which became the official definition of the metre, and in September 1799 the metre was required by law to be used in the Paris region. However, as one might expect, introducing the new measure was easier said than done. Part of the problem was that Greek and Latin prefixes like kilo- and centi- had been proposed to help make the new system internationally acceptable but were strongly disliked in France. It was also a law which was essentially impossible to enforce and, again as one might expect, many traders took the opportunity to cheat their customers. Teaching the metric system became compulsory in schools and the hope was that at least the next generation would accept it even if the current generation would not.

In November 1800 an attempt was made to make the system more acceptable by dropping the Greek and Latin prefixes and reinstating the older names for measures but with new metric values. In September of the following year it became illegal to use any other system of weights and measures anywhere in France but it was largely ignored. It did not last long for, on 12 February 1812, Napoleon returned the country to its former units. The metre standard was still used in the sense that a fathom was declared to be 2 metres, there were 6 feet in a fathom and 12 inches in a foot.

Now, despite this retrograde move, Napoleon had a major effect on the spread of the metric system. French conquests of the Low Countries had seen the metric system introduced there and, on the defeat of Napoleon and the restoring of monarchy in those countries, they retained the system. The decimal metric system was required to be used by law in the Low Countries in 1820. In 1830 Belgium became independent of Holland and made the metric system, together with its former Greek and Latin prefixes, the only legal measurement system. Perhaps the fact that the French had scrapped the system they invented, helped its acceptance in other European countries. In 1840 the French government reintroduced the metric system but it took many years before use of the old measures died out.

In the 1860s Britain, the United States and the German states all made moves towards adopting the metric system. It became legal in Britain in 1864 but a law which was passed by the House of Commons to require its use throughout the British Empire never made it through its final stages on to the statute books. Similarly in the United States it became legal in 1866, although its use was not made compulsory. The German states passed legislation in 1868 which meant that on the unification of these states to form Germany, use of the metric system was made compulsory.

It is interesting that many leading British scientists were opposed to the introduction of the metric system in Britain in 1864, which is one reason that it only became legal but not compulsory. George Airy and John Herschel argued strongly against it, as did William Rankine who composed the poem *The Three-Foot Rule* :-

In 1870 an International Conference was convened by the French in Paris. Invitations had been sent to scientists from countries around the world with the aim of improving international scientific cooperation by having the metric system as the world-wide standard. War broke out between France and Prussia just before the delegates were due to arrive, however, and the German delegation did not attend. Wishing that any decision be a truly international one, the conference was postponed and met again in 1872. The outcome was the setting up of the International Bureau of Weights and Measures, to be situated in Paris, and the Convention of the Metre of 1875 which was signed by seventeen nations. Further countries signed up over the following years.Some talk of millimetres, and some of kilograms,

And some of decilitres, to measure beer and drams;

But I'm a British Workman, too old to go to school,

So by pounds I'll eat, and by quarts I'll drink, and I'll work by my three foot rule.

A party of astronomers went measuring the Earth,

And forty million metres they took to be its girth;

Five hundred million inches, though, go through from Pole to Pole;

So lets stick to inches, feet and yards, and the good old three foot rule.

In 1889 the International Bureau of Weights and Measures replaced the original metre bar in Paris by a new one and at the same time had copies of the bar sent to every country which had signed up to the Convention of the Metre. The definition now became the distance between two lines marked on a standard bar made from 90 percent platinum and 10 percent iridium. This remained the standard until 1960 when the International Bureau of Weights and Measures adopted a more accurate standard for international science when it defined the metre in terms of the wavelength of light emitted by the krypton-86 atom, namely 1,650,763.73 wavelengths of the orange-red line in the spectrum of the atom in a vacuum. The metre was redefined again in 1983, this time as the distance which light travels in a vacuum in ^{1}/_{299,792,458} seconds. This remains the current definition. Note that in all these redefinitions, the length of the metre was always taken as close as possible to the value fixed in 1799 by data from the Delambre-Méchain survey.

Notice that the current definition defines the metre in terms of the second. Now Borda had argued against using the length of a pendulum which beats at the rate of one second to define the metre in 1791 on the reasonable grounds that the second was not a fixed unit but could change with time. Indeed the second, then defined as 1/86,400 of the mean solar day, does change but a fixed definition was introduced in 1956 by the International Bureau of Weights and Measures, as ^{1}/_{31,556,925.9747} of the length of the tropical year 1900. Although this fixed the value, it was seen as an unsatisfactory definition since the length of the year 1900 could never be measured after 1900. It was changed in 1964 to 9,192,631,770 cycles of radiation associated with a particular change of state of the caesium-133 atom. By 1983 when the metre was defined in terms of the second, Borda's objection was no longer valid as the definition of the second by then did not have the astronomical definition which was indeed variable.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Measurement.html]