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HOW DO WE KNOW?

THE LENGTH OF

ONEBY ANDREW ROBINSON Defining the length of a metre meant a revolution in accurate scientific measurement. From metal bars to spectrums of light to today’s laser-based system, discover how pioneers rewrote the rules

HE EARLIEST UNITS for to the , a length of 1,760 yards line. So there were 12 lines in an measuring length were based (1,609m). The milliare equalled mille inch, 12 inches (and 36 barleycorns) on the human body – most passuum, a ‘thousand paces’ of a length in a foot, and three feet in a yard. obviously the arm and the taken by a Roman legionary on a long In , before the 1789 foot – and on seeds and march. This makes a ‘pace’ about Revolution paved the way for the grains. An Ancient Greek 1.6m – clearly an impossible average introduction of the , a relief sculpture dated to about length for walking. The Roman ‘pace’ staggering 250,000 measures were in 450BC, kept at the Ashmolean must therefore refer to the full cycle of use under the guise of some 800 names, Museum in Oxford, shows left-right-left or right-left-right, giving including the aune for length. Since T a male figure with two arms a reasonable single pace of about 0.8m. the time of Charlemagne, eight French stretched out and a foot-shaped In Medieval England, the barleycorn kings – including Louis XIV, the ‘Sun depression above one of the arms. defined the inch, the foot and the yard. King’ – had attempted to lay down the The sculpture may have been set up During the reign of Edward II, in the law regarding standard weights and in a public place as a set of standard 14th Century, the inch was defined as measures. All failed, because uniformity measurements. If its damaged arm ‘three grains of barley, dry and round, did not please the feudal aristocracy, is completed by symmetry, then the placed end to end, lengthwise’. For who habitually manipulated the full arm span, known as the fathom, smaller lengths, the barleycorn was diversity of customary units to the measures 2.08 (m); the split into four equal parts, to create the disadvantage of their peasantry. forearm, known as the cubit, 0.52m; Inconsistency particularly and the foot 0.297m. bedevilled the cubit, a unit In the Roman Empire, the approximately equal to the milliare gave its name distance from a man’s

This Ancient Greek relief from 450BC shows an early attempt at standardising measurement PHOTO: NPL, ASHMOLEAN MUSEUM OXFORD PHOTO:

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The Length Bar Interferometer at the National Physical Laboratory uses laser light to define a given length up to a metre

> IN A NUTSHELL

From early forms of defining length with seeds and parts of the body, to defining a length based on accurate measurements of the size and shape of the Earth, it’s taken hundreds of years for a standard metre to be used throughout the world.

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elbow joint to the farthest on its walls – that is, the summer modern conversion of 250,000 stadia fingertip of his extended hand, solstice – , further north would give 39,690km (24,662 ) or about half a metre. In ancient Egypt, in Alexandria, measured the angle – very close to the current value of the short cubit equalled six palms, created between the obelisk and its Earth’s equatorial circumference, while the royal cubit (used in the shadow. Since Syene and Alexandria 40,075km (24,901 miles). construction of the Pyramids) equalled lay almost on the same of seven palms. The eight different cubits , this angle measured the known in the early civilisations vary in difference in between Syene TRUSTED TRIANGULATION length from Roman 0.444m to and Alexandria. As the Earth’s geography became Palestinian 0.641m. If the Earth were a perfect , clearer through the great sailing Clearly a measurement of length the angle at the Earth’s centre voyages of the 16th Century, so did the defined by a universally agreed subtended by the well and the obelisk accuracy of surveying techniques on standard was needed, especially – by definition the difference in their land. Triangulation was first reported for international trade. The Greek – had to equal the angle of in a book published in Antwerp mathematician and geographer the shadow, from simple geometry. in 1533. In the 17th Century came Eratosthenes, who became director of (This assumed, reasonably, that the telescopes with cross-hairs for sighting the library at Alexandria in 235BC, was Sun was so far away that all its rays the stations being triangulated. the first to attempt such a definition by were parallel at the Earth.) The angle These were introduced in the 1670s determining the Earth’s circumference was 7.2°. Given the distance by camel in France, the earliest country to using and geometry. from Alexandria to Syene, known to be attempt an exact survey of itself, under Eratosthenes employed a well in about 5,000 stadia, the circumference the direction of Jean-Dominique far-off Syene (modern Aswan) and a of the Earth (subtending 360°) could Cassini, the first of four generations vertical obelisk in the grounds of the be calculated. Multiply 5,000 by the of Cassini surveyors of France. The Alexandria library. On the day when ratio 360:7.2 and you get 250,000 survey shifted the western coastline the overhead Sun shone directly stadia. The length of the stadium of existing maps about one-and- into the well, casting no shadow is unfortunately disputed, but one a-half degrees of longitude east in

THE KEY ’s theory of predicted the true shape of the Earth, and therefore the DISCOVERY varying length of a of latitude. It would prove a crucial step in defining the metre

Newton found that NEWTON USED THE noted that a clock the Earth was in fact trigonometrical survey of the carried to the should an oblate meridian in his calculation beat slightly slow, since of the force of gravity. His gravity was weaker there theory led him to predict, in his – which had indeed Principia Mathematica of 1687, been observed when that the Earth could not be a a French savant took perfect sphere. Centrifugal such a clock to the force, caused by axial spinning, Caribbean in 1672. was balanced by gravitational Third, he pointed force. But since the equator out that moved faster than the poles, astronomers the equator must bulge very had observed slightly, while the poles must be Jupiter to be slightly flattened, resulting in an flattened at oblate spheroid like a flattened the poles. tomato. Gravitational attraction Finally, he at the equator must be slightly showed how less than at the poles, since the gravitational gravity weakens with distance pull of the Sun from the centre of the Earth. and the Moon on To prove this, Newton first a bulging equator re-analysed the French survey could account for the data to show that a degree of swivel in the Earth’s latitude appeared to lengthen axis that had been known slightly as one moved north — to be the cause of the an increase to be expected precession of the equinoxes from a bulging equator and since the time of the flattened poles. Second, he ancient astronomers. PHOTO: ALAMY X2, THINKSTOCK X2, AKG IMAGES, NASA IMAGES, AKG X2, THINKSTOCK ALAMY X2, PHOTO:

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relation to the , and the southern coastline about half a degree of latitude to the north. Brest moved CAST OF From an Ancient Greek to a French emperor, it’s 177km (110 miles), Marseilles 64km CHARACTERS taken an eclectic mix to standardise measurement (40 miles). When in 1682 Louis XIV paid a visit to the and saw the new map, he exclaimed Eratosthenes to Cassini: “Your journey has cost me (c. 276BC - a major portion of my realm!” c. 195/194BC) was a The French survey data also adjusted Greek mathematician, the shape of the Earth: from the perfect astronomer and sphere imagined by Eratosthenes to a geographer, as well prolate spheroid: that is, a sphere as a poet and music slightly flattened at the equator and theorist, who became slightly bulging at the poles. But Isaac director of the library at Newton, working across the English Alexandria in 235BC. He Channel, disagreed with Cassini (see is principally known for his accurate calculation ‘The Key Discovery’). Newton Isaac Newton of the circumference of calculated an oblate Earth, bulging at (1642-1727), the son the Earth and his map the equator and flattened at the poles. of an English yeoman of the world based on Not until the -40s, after Newton’s farmer who needs parallels and meridians. death, was the question resolved in no introduction as a Newton’s favour. Two gruelling mathematician and expeditions led by French savants physicist, also studied measured a degree of latitude in alchemy and religion. Lapland, near the pole, and in Peru, at Although he never left the equator. The results were conclusive the shores of England, Pierre François André enough, said the French writer he took a keen interest Méchain (1744-1804), Voltaire – a devotee of Newton – “to in calculating the shape a French astronomer flatten both the poles and the Cassinis”. of the Earth from and hydrographer, he both survey data and surveyed the southern gravitational theory. THE FRENCH CONNECTION part of the meridian The very first proposal to relate arc from to all length measurements to a during the decimal system based on the Earth’s 1790s in collaboration dimensions came from a French with Jean Baptiste churchman, the Abbé Gabriel Mouton, Joseph Delambre. writing in 1670. He suggested that the Director of the Paris primary length unit should equal 1 Observatory from minute of arc of a great circle (ie the 1799, Méchain died in circumference) of the Earth, a length Jean Baptiste while checking not far short of 2,000m. But instead, a Joseph Delambre his survey, which was standard length based on the length of (1749-1822), a completed by François a pendulum was considered. French astronomer, Arago in 1806-09. By the late 18th Century, scientists he surveyed the were aware that the rate of swing of northern part of the a pendulum depended not at all on from Napoléon Bonaparte the weight of the bob, only on the Dunkirk to Barcelona (1769-1821) strongly length of the pendulum. Its period in collaboration with encouraged the could therefore be used to define Méchain. After the development of science length. In fact, the so-called seconds death of Méchain, in France. This began pendulum, swinging once every Delambre became with his invasion second, has a length of 0.994m at director of the Paris of Egypt in 1798, sea level and latitude 45°, halfway Observatory and accompanied by a party between the equator and the poles, published three of scientific savants, under conditions of standard gravity. volumes defining the and continued after However, the pendulum basis of the metric he proclaimed himself measurement of the metre was system in 1806-10. emperor in 1804. rejected, partly because the period Although he was a depended on gravity, which was leading supporter of known to vary with altitude and the new metric system, latitude, and partly because the he personally refused units of time were themselves to use it.

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under Revolutionary review. The unit of length finally selected TIMELINE Becoming ever more accurate, there have been many by the French Academy of Sciences in methods of defining length over the centuries 1791 was one ten-millionth of a quarter of a great circle, in other words one ten-millionth of the distance from the equator to the (assuming the Earth to be spherical). Since Eratosthenes, from Sun the circumference of the Earth was measurements of the Sun’s 40,075km (24,901 miles), a quarter shadow in Alexandria and 3RD was just under 10,019km. Dividing this Alexandria Syene, calculates Earth’s figure by 10 million gave just over 1m. Syene circumference as 250,000 CENTURY BC stadia, approximately 39,690km In 1792, two French scientists, Jean —very close to the current Baptiste Joseph Delambre and Pierre value of Earth’s equatorial François André Méchain, set out to circumference, 40,075km. measure the length of the Paris meridian from Dunkirk to Barcelona by triangulation; the latitudes of Dunkirk and Barcelona were figured Isaac Newton, in his Principia by astronomical observations. This Mathematica, proposes challenging task, conducted at a time that Earth is not perfectly of political turmoil in France and war spherical. It is an oblate spheroid, flattened at the between France and Spain, took more 1687 poles, bulging at the equator. than seven years to complete, ending An equatorial degree of in the death of Méchain. latitude is slightly shorter than Its accuracy was formidable. Even a polar degree. so errors crept in because of the oblate Earth, instrument error and human error in making endless finicky Two astronomers, Jean Baptiste Joseph Delambre observations. But the great scientific and Pierre François André Méchain, measure the expedition did give legitimacy to meridian arc from Dunkirk to Barcelona, which 1792-99 the metre and prestige to the metric serves to fix the length of the metre. system, which was officially imposed on France by Napoléon Bonaparte’s government in 1801. For all its initial The metric system is unpopularity, was imposed in France by inevitable in the long run. As Napoléon Napoléon Bonaparte. All farsightedly congratulated Delambre measures, except hours, in 1806: “Conquests will come and go, minutes and seconds, but this work will endure.” are decimalised. 1801 However, conflict with pre-metric systems causes its partial GOING GLOBAL withdrawal in 1812. For a century and a half, until the introduction of the Système International (SI) in 1960, the survey- based metre was embodied in the In Paris, representatives length of a metal bar kept in a vault at of 17 nations and empires the International Bureau of Weights sign the Convention Of The and Measures in Sèvres near Paris, Metre, ‘desiring international copies of which had been distributed uniformity and precision in 1875 standards of weight and to national standards institutions in measure’. Britain signs in 1884. other countries. In 1889, a new prototype bar was made from a dense platinum-iridium alloy. It had an X-like cross-section intended to minimise sag and distortion when the bar was properly supported. On the polished facets at both ends, there were fine horizontal The Système International d’Unités, generally known graticules, lines made for visual as the SI system, is established. It consists of seven settings by micrometer, and thicker 1960 basic units, including the metre, kilogram and second. The metre is redefined in terms of a spectrum line of vertical lines to monitor the expansion krypton, and later in terms of the speed of light. of the metal in the temperature range PHOTO: ALAMY X2, MARY EVANS PICTURE LIBRARY, WIKIPEDIA X2 LIBRARY, PICTURE EVANS MARY ALAMY X2, PHOTO:

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A standard metre from the late 18th Century in Paris. People would bring their metre rulers and place them within this standard to see if they were accurate

0-20°C. The standard length was Union, which metricated in 1924 after NEED TO KNOW always measured at 0°. the Russian Revolution. These terms will help you The disadvantages are obvious, Britain officially committed itself understand the story of the metre and during the first half of the 20th in 1965, then dragged its feet and Century scientists made progressive abolished the Metrication Board in attempts to find techniques to redefine 1979. Since 1974, the metric system the metre in terms of the wavelength has been taught in British schools and MERIDIANS of light – an invariable standard that metric packaging has been gradually 1 Circles of constant longitude, passing could be measured in any laboratory introduced alongside imperial. But through a given place and through the with the right equipment. In 1960, there is no plan to convert road terrestrial poles. The French meridian the metre was redefined in terms signs. As for the United States, there passing through Paris was used to define of a spectrum line of krypton. Then is little political will to metricate. the metre, while the British meridian in 1983, the current definition was Even in science, the old measures are passing through Greenwich still defines adopted, based on the speed of laser sometimes used alongside the metric the world’s time zones. light. The metre is now the length of system – as became embarrassingly the path travelled by light in vacuum obvious in 1999 when a NASA probe during a time of 1/299,792,458 second. sent to Mars was lost because one of its OBLATE/PROLATE design teams had used the traditional 2 SPHEROID units while the other had used metric. An oblate spheroid is a sphere that A METRIC WORLD Since 1791, when the metre was is flattened at the poles, bulging at After France, the next countries to measured as one ten-millionth of the the equator, like a tomato. A prolate metricate were its neighbours, which quarter meridian of the Earth to within spheroid, by contrast, is elongated at had come under direct French rule. 0.06mm, scientists have made drastic the poles, flattened at the equator, like Spain went metric in the 1850s, strides in accuracy. When you’re told a lemon. Earth is an oblate spheroid, as followed by Germany and Italy as part the length of something in metres, first demonstrated by Isaac Newton. of their political unifications. In 1875, you can be confident that we now 17 nations and their empires signed the know the length of a metre to within Convention Of The Metre. By 1900, 0.00000002mm, or one-hundredth of TRIANGULATION well over half of European nations had the width of DNA’s double helix. „ 3 Triangular surveying proceeds by gone metric. The first Asian nation to building up a chain of interlinked triangles. convert was Mongolia in 1918. Before The first triangle is constructed from a this, Japan and China had both shown Andrew Robinson is the author of The Story precisely measured horizontal baseline. interest, but neither nation moved Of Measurement (Thames & Hudson) Its two base angles are measured through until the 1920s. In Japan, there was sighting instruments aimed at a prominent popular opposition and conversion Find out more landmark — the triangle’s third vertex. was shelved until the 1950s; in China, These angles are then used to calculate metrication did not revive until Watch Marcus du Sautoy the lengths of the triangle’s other two 1959, 10 years after the Communist explain the quest to measure sides by simple trigonometry, after Revolution. Political upheaval the metre in Time And Distance allowing for any altitude di!erence promoted metrication, as it had in http://youtu.be/x1DT8fIqqZAc between the vertex and the baseline. France, Italy, Germany and the Soviet

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