Metre-based systems of measurement

For a topical guide to this subject, see Outline of the metric system

For a more accessible and less technical introduction to this topic, see Introduction to the metric system

A metric system is a system of measurement that succeeded the decimalised system based on the metre introduced in France in the 1790s. The historical development of these systems culminated in the definition of the International System of Units (SI), under the oversight of an international standards body.

The historical evolution of metric systems has resulted in the recognition of several principles. Each of the fundamental dimensions of nature is expressed by a single base unit of measure. The definition of base units has increasingly been realised from natural principles, rather than by copies of physical artefacts. For quantities derived from the fundamental base units of the system, units derived from the base units are used–e.g., the square metre is the derived unit for area, a quantity derived from length. These derived units are coherent, which means that they involve only products of powers of the base units, without empirical factors. For any given quantity whose unit has a special name and symbol, an extended set of smaller and larger units is defined that are related in a systematic system of factors of powers of ten. The unit of time should be the second; the unit of length should be either the metre or a decimal multiple of it; and the unit of mass should be the gram or a decimal multiple of it.

Metric systems have evolved since the 1790s, as science and technology have evolved, in providing a single universal measuring system. Before and in addition to the SI, some other examples of metric systems are the following: the MKS system of units and the MKSA systems, which are the direct forerunners of the SI; the centimetre–gram–second (CGS) system and its subtypes, the CGS electrostatic (cgs-esu) system, the CGS electromagnetic (cgs-emu) system, and their still-popular blend, the Gaussian system; the metre–tonne–second (MTS) system; and the gravitational metric systems, which can be based on either the metre or the centimetre, and either the gram(-force) or the kilogram(-force).

Background [ edit ]

The French revolution (1789–99) provided an opportunity for the French to reform their unwieldy and archaic system of many local weights and measures. Charles Maurice de Talleyrand championed a new system based on natural units, proposing to the French National Assembly in 1790 that such a system be developed. Talleyrand had ambitions that a new natural and standardised system would be embraced worldwide, and was keen to involve other countries in its development. Great Britain ignored invitations to co-operate, so the French Academy of Sciences decided in 1791 to go it alone and they set up a commission for the purpose. The commission decided that the standard of length should be based on the size of the Earth. They defined that length to be the 'metre' and its length as one ten-millionth of the length of a quadrant on the Earth's surface from the equator to the north pole. In 1799, after the length of that quadrant had been surveyed, the new system was launched in France.[1]:145–149

The units of the metric system, originally taken from observable features of nature, are now defined by seven physical constants being given exact numerical values in terms of the units. In the modern form of the International System of Units (SI), the seven base units are: metre for length, kilogram for mass, second for time, ampere for electric current, kelvin for temperature, candela for luminous intensity and mole for amount of substance. These, together with their derived units, can measure any physical quantity. Derived units may have their own unit name, such as the watt (J/s) and lux (cd/m2), or may just be expressed as combinations of base units, such as velocity (m/s) and acceleration (m/s2).[2]

The metric system was designed to have properties that make it easy to use and widely applicable, including units based on the natural world, decimal ratios, prefixes for multiples and sub-multiples, and a structure of base and derived units. It is also a coherent system, which means that its units do not introduce conversion factors not already present in equations relating quantities. It has a property called rationalisation that eliminates certain constants of proportionality in equations of physics.

The metric system is extensible, and new derived units are defined as needed in fields such as radiology and chemistry. For example, the katal, a derived unit for catalytic activity equivalent to a one mole per second (1 mol/s), was added in 1999.

Principles [ edit ]

Although the metric system has changed and developed since its inception, its basic concepts have hardly changed. Designed for transnational use, it consisted of a basic set of units of measurement, now known as base units. Derived units were built up from the base units using logical rather than empirical relationships while multiples and submultiples of both base and derived units were decimal-based and identified by a standard set of prefixes.

Realisation [ edit ]

[3] The metre was originally defined to be one ten millionth of the distance between the North Pole and the Equator through Paris.

The base units used in a measurement system must be realisable. Each of the definitions of the base units in the SI is accompanied by a defined mise en pratique [practical realisation] that describes in detail at least one way in which the base unit can be measured.[4] Where possible, definitions of the base units were developed so that any laboratory equipped with proper instruments would be able to realise a standard without reliance on an artefact held by another country. In practice, such realisation is done under the auspices of a mutual acceptance arrangement.[5]

In the SI, the standard metre is defined as exactly 1/299,792,458 of the distance that light travels in a second. The realisation of the metre depends in turn on precise realisation of the second. There are both astronomical observation methods and laboratory measurement methods that are used to realise units of the standard metre. Because the speed of light is now exactly defined in terms of the metre, more precise measurement of the speed of light does not result in a more accurate figure for its velocity in standard units, but rather a more accurate definition of the metre. The accuracy of the measured speed of light is considered to be within 1 m/s, and the realisation of the metre is within about 3 parts in 1,000,000,000, or a proportion of 0.3x10−8:1.

The kilogram was originally defined as the mass of a man-made artefact of platinum-iridium held in a laboratory in France, until the new definition was introduced in May 2019. Replicas made in 1879 at the time of the artefact's fabrication and distributed to signatories of the Metre Convention serve as de facto standards of mass in those countries. Additional replicas have been fabricated since as additional countries have joined the convention. The replicas were subject to periodic validation by comparison to the original, called the IPK. It became apparent that either the IPK or the replicas or both were deteriorating, and are no longer comparable: they had diverged by 50 μg since fabrication, so figuratively, the accuracy of the kilogram was no better than 5 parts in a hundred million or a proportion of 5x10−8:1. The accepted redefinition of SI base units replaced the IPK with an exact definition of the Planck constant, which defines the kilogram in terms of the second and metre.

Base and derived unit structure [ edit ]

The metric system base units were originally adopted because they represented fundamental orthogonal dimensions of measurement corresponding to how we perceive nature: a spatial dimension, a time dimension, one for inertia, and later, a more subtle one for the dimension of an "invisible substance" known as electricity or more generally, electromagnetism. One and only one unit in each of these dimensions was defined, unlike older systems where multiple perceptual quantities with the same dimension were prevalent, like inches, feet and yards or ounces, pounds and tons. Units for other quantities like area and volume, which are also spatial dimensional quantities, were derived from the fundamental ones by logical relationships, so that a unit of square area for example, was the unit of length squared.

Many derived units were already in use before and during the time the metric system evolved, because they represented convenient abstractions of whatever base units were defined for the system, especially in the sciences. So analogous units were scaled in terms of the units of the newly established metric system, and their names adopted into the system. Many of these were associated with electromagnetism. Other perceptual units, like volume, which were not defined in terms of base units, were incorporated into the system with definitions in the metric base units, so that the system remained simple. It grew in number of units, but the system retained a uniform structure.

Decimal ratios [ edit ]

Some customary systems of weights and measures had duodecimal ratios, which meant quantities were conveniently divisible by 2, 3, 4, and 6. But it was difficult to do arithmetic with things like ​1⁄ 4 pound or ​1⁄ 3 foot. There was no system of notation for successive fractions: for example, ​1⁄ 3 of ​1⁄ 3 of a foot was not an inch or any other unit. But the system of counting in decimal ratios did have notation, and the system had the algebraic property of multiplicative closure: a fraction of a fraction, or a multiple of a fraction was a quantity in the system, like ​1⁄ 10 of ​1⁄ 10 which is ​1⁄ 100 . So a decimal radix became the ratio between unit sizes of the metric system.

Prefixes for multiples and submultiples [ edit ]

In the metric system, multiples and submultiples of units follow a decimal pattern.[Note 1]

Metric prefixes in everyday use Text Symbol Factor Power tera T 1 000 000 000 000 1012 giga G 1 000 000 000 109 mega M 1 000 000 106 kilo k 1 000 103 hecto h 100 102 deca da 10 101 (none) (none) 1 100 deci d 0.1 10−1 centi c 0.01 10−2 milli m 0.001 10−3 micro μ 0.000 001 10−6 nano n 0.000 000 001 10−9 pico p 0.000 000 000 001 10−12

A common set of decimal-based prefixes that have the effect of multiplication or division by an integer power of ten can be applied to units that are themselves too large or too small for practical use. The concept of using consistent classical (Latin or Greek) names for the prefixes was first proposed in a report by the French Revolutionary Commission on Weights and Measures in May 1793.[3]:89–96 The prefix kilo, for example, is used to multiply the unit by 1000, and the prefix milli is to indicate a one-thousandth part of the unit. Thus the kilogram and kilometre are a thousand grams and metres respectively, and a milligram and millimetre are one thousandth of a gram and metre respectively. These relations can be written symbolically as:[6]

1 mg = 0.001 g

1 km = 1000 m

In the early days, multipliers that were positive powers of ten were given Greek-derived prefixes such as kilo- and mega-, and those that were negative powers of ten were given Latin-derived prefixes such as centi- and milli-. However, 1935 extensions to the prefix system did not follow this convention: the prefixes nano- and micro-, for example have Greek roots.[1]:222–223 During the 19th century the prefix myria-, derived from the Greek word μύριοι (mýrioi), was used as a multiplier for 10000.[7]

When applying prefixes to derived units of area and volume that are expressed in terms of units of length squared or cubed, the square and cube operators are applied to the unit of length including the prefix, as illustrated below.[6]

1 mm2 (square millimetre) = (1 mm)2 = (0.001 m)2 = 0.000 001 m2 1 km2 (square kilometre) = (1 km)2 = (1000 m)2 = 1 000 000 m2 1 mm3 (cubic millimetre) = (1 mm)3 = (0.001 m)3 = 0.000 000 001 m3 1 km3 (cubic kilometre) = (1 km)3 = (1000 m)3 = 1 000 000 000 m3

Prefixes are not usually used to indicate multiples of a second greater than 1; the non-SI units of minute, hour and day are used instead. On the other hand, prefixes are used for multiples of the non-SI unit of volume, the litre (l, L) such as millilitres (ml).[6]

Coherence [ edit ]

James Clerk Maxwell played a major role in developing the concept of a coherent CGS system and in extending the metric system to include electrical units.

Each variant of the metric system has a degree of coherence—the derived units are directly related to the base units without the need for intermediate conversion factors.[8] For example, in a coherent system the units of force, energy and power are chosen so that the equations

force = mass × acceleration energy = force × distance energy = power × time

hold without the introduction of unit conversion factors. Once a set of coherent units have been defined, other relationships in physics that use those units will automatically be true. Therefore, Einstein's mass–energy equation, E = mc2, does not require extraneous constants when expressed in coherent units.[9]

The CGS system had two units of energy, the erg that was related to mechanics and the calorie that was related to thermal energy; so only one of them (the erg) could bear a coherent relationship to the base units. Coherence was a design aim of SI, which resulted in only one unit of energy being defined – the joule.[10]

Rationalisation [ edit ]

Maxwell's equations of electromagnetism contained a factor relating to steradians, representative of the fact that electric charges and magnetic fields may be considered to emanate from a point and propagate equally in all directions, i.e. spherically. This factor appeared awkwardly in many equations of physics dealing with the dimensionality of electromagnetism and sometimes other things.

Common metric systems [ edit ]

A number of different metric system have been developed, all using the Mètre des Archives and Kilogramme des Archives (or their descendants) as their base units, but differing in the definitions of the various derived units.

Gaussian second and the first mechanical system of units [ edit ]

In 1832, Gauss used the astronomical second as a base unit in defining the gravitation of the earth, and together with the gram and millimetre, became the first system of mechanical units.

Centimetre–gram–second systems [ edit ]

The centimetre–gram–second system of units (CGS) was the first coherent metric system, having been developed in the 1860s and promoted by Maxwell and Thomson. In 1874, this system was formally promoted by the British Association for the Advancement of Science (BAAS).[11] The system's characteristics are that density is expressed in g/cm3, force expressed in dynes and mechanical energy in ergs. Thermal energy was defined in calories, one calorie being the energy required to raise the temperature of one gram of water from 15.5 °C to 16.5 °C. The meeting also recognised two sets of units for electrical and magnetic properties – the electrostatic set of units and the electromagnetic set of units.[12]

The EMU, ESU and Gaussian systems of electrical units [ edit ]

Several systems of electrical units were defined following discovery of Ohm's law in 1824.

International System of Electrical and Magnetic Units [ edit ]

The CGS units of electricity were cumbersome to work with. This was remedied at the 1893 International Electrical Congress held in Chicago by defining the "international" ampere and ohm using definitions based on the metre, kilogram and second.[13]

Other early electromagnetic systems of units [ edit ]

During the same period in which the CGS system was being extended to include electromagnetism, other systems were developed, distinguished by their choice of coherent base unit, including the Practical System of Electric Units, or QES (quad–eleventhgram–second) system, was being used.[14]:268[15]:17 Here, the base units are the quad, equal to 107 m (approximately a quadrant of the earth's circumference), the eleventhgram, equal to 10−11 g, and the second. These were chosen so that the corresponding electrical units of potential difference, current and resistance had a convenient magnitude.

MKS and MKSA systems [ edit ]

In 1901, Giovanni Giorgi showed that by adding an electrical unit as a fourth base unit, the various anomalies in electromagnetic systems could be resolved. The metre–kilogram–second–coulomb (MKSC) and metre–kilogram–second–ampere (MKSA) systems are examples of such systems.[16]

The International System of Units (Système international d'unités or SI) is the current international standard metric system and is also the system most widely used around the world. It is an extension of Giorgi's MKSA system – its base units are the metre, kilogram, second, ampere, kelvin, candela and mole.[10] The MKS (metre–kilogram–second) system came into existence in 1889, when artefacts for the metre and kilogram were fabricated according to the Metre Convention. Early in the 20th century, an unspecified electrical unit was added, and the system was called MKSX. When it became apparent that the unit would be the ampere, the system was referred to as the MKSA system, and was the direct predecessor of the SI.

Metre–tonne–second systems [ edit ]

The metre–tonne–second system of units (MTS) was based on the metre, tonne and second – the unit of force was the sthène and the unit of pressure was the pièze. It was invented in France for industrial use and from 1933 to 1955 was used both in France and in the Soviet Union.[17][18]

Gravitational systems [ edit ]

Gravitational metric systems use the kilogram-force (kilopond) as a base unit of force, with mass measured in a unit known as the hyl, Technische Masseneinheit (TME), mug or metric slug.[19] Although the CGPM passed a resolution in 1901 defining the standard value of acceleration due to gravity to be 980.665 cm/s2, gravitational units are not part of the International System of Units (SI).[20]

International System of Units [ edit ]

The International System of Units is the modern metric system. It is based on the metre–kilogram–second–ampere (MKSA) system of units from early in the 20th century. It also includes numerous coherent derived units for common quantities like power (watt) and irradience (lumen). Electrical units were taken from the International system then in use. Other units like those for energy (joule) were modelled on those from the older CGS system, but scaled to be coherent with MKSA units. Two additional base units, degree Kelvin equivalent to degree Celsius for thermodynamic temperature, and candela, roughly equivalent to the international candle unit of illumination, were introduced. Later, another base unit, the mole, a unit of mass equivalent to Avogadro's number of specified molecules, was added along with several other derived units.

The system was promulgated by the General Conference on Weights and Measures (French: Conférence générale des poids et mesures – CGPM) in 1960. At that time, the metre was redefined in terms of the wavelength of a spectral line of the krypton-86[Note 2] atom, and the standard metre artefact from 1889 was retired.

Today, the International system of units consists of 7 base units and innumerable coherent derived units including 22 with special names. The last new derived unit, the katal for catalytic activity, was added in 1999. Some of the base units are now realised in terms of invariant constants of physics. As a consequence, the speed of light has now become an exactly defined constant, and defines the metre as ​1⁄ 299,792,458 of the distance light travels in a second. Until 2019, the kilogram was defined by a man-made artefact of deteriorating platinum-iridium. The range of decimal prefixes has been extended to those for 1024, yotta, and 10−24, yocto, which are unfamiliar because nothing in our everyday lives is that big or that small.

The International System of Units has been adopted as the official system of weights and measures by all nations in the world except for Myanmar, Liberia, and the United States, while the United States is the only industrialised country where the metric system is not the predominant system of units.[21]

See also [ edit ]

Notes [ edit ]

^ Non-SI units for time and plane angle measurement, inherited from existing systems, are an exception to the decimal-multiplier rule ^ A stable isotope of an inert gas that occurs in undetectable or trace amounts naturally

References [ edit ]