Coherence (units of measurement)

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

A coherent derived unit is defined as a derived unit that, for a given system of quantities and for a chosen set of base units, is a product of powers of base units with no other proportionality factor than one.[1] The concept of coherence was developed in the mid-nineteenth century by, amongst others, Kelvin and James Clerk Maxwell and promoted by the British Association for the Advancement of Science. The concept was initially applied to the centimetre–gram–second (CGS) and the foot–pound–second systems (FPS) of units in 1873 and 1875 respectively. The International System of Units (1960) was designed around the system of coherence.

Basic concepts

In SI, which is a coherent system, the unit of power is the "watt" which is defined as "one joule per second".[2] In the US customary system of measurement, which is non-coherent, the unit of power is the "horsepower" which is defined as "550 foot-pounds per second" (the pound in this context being the pound-force), similarly the gallon is not equal to a cubic yard (nor is it the cube of any length unit).

Before the metric system

The earliest units of measure devised by man bore no relationship to each other. As both man's understanding of philosophical concepts and the organisation of society developed, so units of measurement were standardised - first particular units of measure had the same value across a community then different units of the same quantity (for example feet and inches) were given a fixed relationship. Apart from Ancient China where the units of capacity and of mass are linked to red millet seed, there is little evidence of the linking of different quantities until the Age of Reason.[3]

Relating quantities of the same kind

The history of the measurement of length dates back to the early civilisations of the Middle East (8000 – 10000 BC). Archeologists have been able to reconstruct the units of measure in use in Mesopotamia, India, the Jewish culture and many others. Archaeological and other evidence shows that in many civilisations, the ratios between different units for the same quantity of measure were adjusted so that they were integer numbers. In many early cultures such as Ancient Egypt, multiples of 2,3 and 5 were not always used—the Egyptian royal cubit being 28 fingers of 7 hands.[4] In 2150 BC, the Akkadian emperor Naram-Sin rationalised the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of 2, 3 or 5, for example there were 6 she (barleycorns) in a shu-si (finger) and 30 shu-si in a kush (cubit).[5]

Measuring rod on exhibition in the Archeological Museum of Istanbul (Turkey) dating to the (3rd millennium BC) excavated at Nippur, Mesopotamia. The rod shows the various units of measure in use.

Relating quantities of different kinds

Non-commensurable quantities have different physical dimensions which means that adding or subtracting them is not meaningful. For instance, adding the mass of an object to its volume has no physical meaning. However, new quantities (and, as such, units) can be derived via multiplication and exponentiation of other units. As an example, the SI unit for force is the newton, which is defined as kg m s−2. Since a coherent derived unit is one which is defined by means of multiplication and exponentiation of other units but not multiplied by any scaling factor, the pascal is a coherent unit of pressure (defined as kg m−1 s−2), but the bar (defined as 100000 kg m−1 s−2) is not.

Note that coherence of a given unit depends on the definition of the base units. Should the meter's definition change such that it is 100000 longer, then the bar would be a coherent derived unit. However, a coherent unit remains coherent (and a non-coherent unit remains non-coherent) if the base units are redefined in terms of other units with the constant of proportionality always being unity.

Metric system

Rational system and use of water

The concept of coherence was only introduced into the metric system in the third quarter of the nineteenth century; in its original form the metric system was non-coherent - in particular the litre was 0.001 m3 and the are (from which we get the hectare) was 100 m2. A precursor to the concept of coherence was however present in that the units of mass and length were related to each other through the physical properties of water, the gram having been designed as being the mass of one cubic centimetre of water at its freezing point.[6]

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, equivalent to the g·cm2/s2) could bear a coherent relationship to the base units. By contrast, coherence was a design aim of the SI, resulting in only one unit of energy being defined -- the joule.[7]

Dimension-related coherence

Work of James Clerk Maxwell and others

Each variant of the metric system has a degree of coherence – the various derived units being directly related to the base units without the need of intermediate conversion factors.[1] 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
power = energy / time

hold without the introduction of constant factors. Once a set of coherent units have been defined, other relationships in physics that use those units will automatically be true - Einstein's mass-energy equation, E = mc2, does not require extraneous constants when expressed in coherent units.[8]

Catalogue of coherent relations

This list catalogues coherent relationships in various systems of units.

SI

The following is a list of coherent SI units:

Frequency (hertz) = reciprocal of time (s−1)
force (newtons) = mass (kilograms) × acceleration (m/s2)
pressure (pascals) = force (newtons) ÷ area (m2)
energy (joules) = force (newtons) × distance (metres)
power (watts) = energy (joules) ÷ time (seconds)
potential difference (volts) = power (watts) ÷ electric current (amps)
electric charge (coulombs) = electric current (amps) × time (seconds)
equivalent radiation dose (sieverts) = energy (joules) ÷ mass (kilograms)
absorbed radiation dose (grays) = energy (joules) ÷ mass (kilograms)
Radioactive activity (becquerels) = reciprocal of time (s−1)
Capacitance (farads) = electric charge (coulombs) ÷ potential difference (volts)
Electrical resistance (ohms) = potential difference (volts) ÷ electric current (amperes)
Electrical conductance (siemens) = electric current (amperes) ÷ potential difference (volts)
Magnetic flux (weber) = the magnetic flux which, linking a circuit of one turn, would produce in it an electromotive force of one volt if it were reduced to zero at a uniform rate in one second
Magnetic flux density (tesla) = magnetic flux (webers) ÷ area (square metres).

CGS

The following is a list of coherent centimetre–gram–second (CGS) system of units:

acceleration (gals) = distance (centimetres) ÷ time2 (s2)
force (dynes) = mass (grams) × acceleration (m/s2)
energy (ergs) = force (dynes) × distance (centimetres)
pressure (barye) = force (dynes) ÷ area (cm2)
dynamic viscosity (poise) = mass (grams) ÷ (distance (centimetres) × time (seconds))
kinematic viscosity (stokes) = area (cm2)÷ time (seconds)

FPS

The following is a list of coherent foot-pound-second (FPS) system of units:

force (poundal) = mass (pounds) × acceleration (ft/s2)

See also

References

  1. 1 2 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 2012-04-12
  2. ↑ SI Brochure, Table 4, pg 118
  3. ↑ McGreevy, Thomas (1995). Cunningham, Peter, ed. The Basis of Measurement: Volume 1—Historical Aspects. Chippenham: Picton Publishing. Chapter 1: Some Ancient Units. ISBN 0 948251 82 4.
  4. ↑ Clagett, Marshall (1999). Ancient Egyptian science, a Source Book. Volume Three: Ancient Egyptian Mathematics. Philadelphia: American Philosophical Society. p. 7. ISBN 978-0-87169-232-0. Retrieved 2013-05-02.
  5. ↑ Melville, Duncan J. (2001). "Old Babylonian Weights and Measures". St. Lawrence University. Retrieved 2013-05-02.
  6. ↑ "La loi du 18 Germinal an 3 la mesure [républicaine] de superficie pour les terrains, égale à un carré de dix mètres de côté" [The law of 18 Germanial year 3 "The republican measures of land area equal to a square with sides of ten metres"] (in French). Le CIV (Centre d'Instruction de Vilgénis) - Forum des Anciens. Retrieved 2010-03-02.
  7. ↑ SI brochure - §1.2 Two classes of SI Units - p92
  8. ↑ Michael Good. "Some Derivations of E = mc2" (PDF). Retrieved 2011-03-18.
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