# Metric system

Four metric measuring devices: a tape measure in centimetres, a thermometer in degrees Celsius, a kilogram mass and a multimeter that measures potential in volts, current in amperes and resistance in ohms

A metric system is any one of the systems of measurement that succeeded the decimalised metre-based system introduced into France in the 1790s. The most recent such system is the internationally recognised International System of Units (SI).

Metric systems have developed to comply with the following basic concepts. There should be a single base unit of measure for quantities of each of the fundamental dimensions of nature (e.g. the metre is the base unit for length). To measure quantities derived from the fundamental dimensions, units derived from the base units should be used (e.g. the square metre is the derived unit for area, a quantity derived from length). For the sake of universality, the base unit standards should be realisable anywhere from natural phenomena, rather than having to be copied from a single centralised standard physical artefact. A set of standard prefixes should be used to define units which are decimal multiples and sub-multiples of the base and derived units. No conversion factors should be required to convert between units, thus ensuring coherence.

Metric systems have evolved since the 1790s as science and technology has evolved, to provide a single universal measuring system which can be used for all measurement applications.

## Background

Pavillon de Breteuil, Saint-Cloud, France, the home of the metric system since 1875

The French revolution (1789-99) provided an opportunity for the French to reform their unwieldy and archaic system of 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

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

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

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

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

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 ​14 pound or ​13 foot. There was no system of notation for successive fractions: for example, ​13 of ​13 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 ​110 of ​110 which is ​1100. So a decimal radix became the ratio between unit sizes of the metric system.

### Prefixes for multiples and submultiples

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 1000000000000 1012
giga G 1000000000 109
mega M 1000000 106
kilo k 1000 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 0.000001 10−6
nano n 0.000000001 10−9
pico p 0.000000000001 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 = 0.000001 m2 1 km2 (square kilometre) = (1 km)2 = (1000 m)2 = 1000000 m2 1 mm3 (cubic millimetre) = (1 mm)3 = (0.001 m)3 = 0.000000001 m3 1 km3 (cubic kilometre) = (1 km)3 = (1000 m)3 = 1000000000 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

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

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

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.

Variants of the metric system
Quantity SI/MKS CGS MTS
distance, displacement,
length, height, etc.
(d, x, l, h, etc.)
metre (m) centimetre (cm) metre
mass (m) kilogram (kg) gram (g) tonne (t)
time (t) second (s) second second
speed, velocity (v, v) m/s cm/s m/s
acceleration (a) m/s2 gal (Gal) m/s2
force (F) newton (N) dyne (dyn) sthene (sn)
pressure (P or p) pascal (Pa) barye (Ba) pièze (pz)
energy (E, Q, W) joule (J) erg (erg) kilojoule (kJ)
power (P) watt (W) erg/s kilowatt (kW)
viscosity (μ) Pa⋅s poise (P) pz⋅s

### Gaussian second and the first mechanical system of units

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.

### The EMU, ESU and Gaussian systems of electrical units

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

### Centimetre–gram–second systems

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]

### International System of Electrical and Magnetic Units

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]

### MKS and MKSA systems

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.[14]

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

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.[15][16]

### Gravitational systems

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.[17] 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).[18]

### International System of Units

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 ​1299,792,458 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.[19]

## Notes

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

## References

1. ^ a b McGreevy, Thomas (1997). Cunningham, Peter (ed.). The Basis of Measurement: Volume 2—Metrication and Current Practice. Chippenham: Picton Publishing. ISBN 978-0-948251-84-9.
2. ^ "The International System of Units (SI), 9th Edition" (PDF). Bureau International des Poids et Mesures. 2019.
3. ^ a b Alder, Ken (2002). The Measure of all Things—The Seven-Year-Odyssey that Transformed the World. London: Abacus. ISBN 978-0-349-11507-8.
4. ^ "What is a mise en pratique?". BIPM. 2011. Retrieved 11 March 2011.
5. ^ "OIML Mutual Acceptance Arrangement (MAA)". International Organisation of Legal Metrology. Archived from the original on 21 May 2013. Retrieved 23 April 2013.
6. ^ a b c International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 121, 122, ISBN 92-822-2213-6, archived (PDF) from the original on 14 August 2017
7. ^ Brewster, D (1830). The Edinburgh Encyclopædia. p. 494.
8. ^ Working Group 2 of the Joint Committee for Guides in Metrology (JCGM/WG 2). (2008), International vocabulary of metrology – Basic and general concepts and associated terms (VIM) (PDF) (3rd ed.), International Bureau of Weights and Measures (BIPM) on behalf of the Joint Committee for Guides in Metrology, 1.12, retrieved 12 April 2012
9. ^ Good, Michael. "Some Derivations of E = mc2" (PDF). Archived from the original (PDF) on 7 November 2011. Retrieved 18 March 2011.
10. ^ a b International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 111–120, ISBN 92-822-2213-6, archived (PDF) from the original on 14 August 2017
11. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), p. 109, ISBN 92-822-2213-6, archived (PDF) from the original on 14 August 2017
12. ^ Thomson, William; Joule, James Prescott; Maxwell, James Clerk; Jenkin, Flemming (1873). "First Report – Cambridge 3 October 1862". In Jenkin, Flemming (ed.). Reports on the Committee on Standards of Electrical Resistance – Appointed by the British Association for the Advancement of Science. London. pp. 1–3. Retrieved 12 May 2011.
13. ^ "Historical context of the SI—Unit of electric current (ampere)". The NIST Reference on Constants, Units and Uncertainty. Retrieved 10 April 2011.
14. ^ "In the beginning... Giovanni Giorgi". International Electrotechnical Commission. 2011. Retrieved 5 April 2011.
15. ^ "System of Measurement Units". IEEE Global History Network. Institute of Electrical and Electronics Engineers (IEEE). Retrieved 21 March 2011.
16. ^ "Notions de physique – Systèmes d'unités" [Symbols used in physics – units of measure] (in French). Hydrelect.info. Retrieved 21 March 2011.
17. ^ Michon, Gérard P (9 September 2000). "Final Answers". Numericana.com. Retrieved 11 October 2012.
18. ^ "Resolution of the 3rd meeting of the CGPM (1901)". General Conference on Weights and Measures. Retrieved 11 October 2012.
19. ^ "The World Factbook, Appendix G: Weights and Measures". Central Intelligence Agency. 2010. Retrieved 26 February 2020.