Coherence (units of measurement)
A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.
A coherent derived unit is a derived unit|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 the proportionality factor being one.
If a system of quantities has equations that relate quantities and the associated system of units has corresponding base units, with only one unit for each base quantity, then it is coherent if and only if every derived unit of the system is coherent.
The concept of coherence was developed in the mid-nineteenth century by, amongst others, Kelvin and James Clerk Maxwell and promoted by the British Science Association. The concept was initially applied to the centimetre–gram–second in 1873 and the foot–pound–second systems of units in 1875. The International System of Units was designed in 1960 to incorporate the principle of coherence.
Examples
In the SI, the derived unit is a coherent derived unit for speed or velocity but is not a coherent derived unit. Speed or velocity is defined by the change in distance divided by a change in time. The derived unit uses the base units of the SI system. The derived unit requires numerical factors to relate to the SI base units: and.In the cgs system, is not a coherent derived unit. The numerical factor of is needed to express in the cgs system.
History
Before the metric system
The earliest units of measure devised by humanity bore no relationship to each other. As both humanity's understanding of philosophical concepts and the organisation of society developed, so units of measurement were standardized—first particular units of measure had the same value across a community, then different units of the same quantity were given a fixed relationship. Apart from Ancient China where the units of capacity and of mass were linked to red millet seed, there is little evidence of the linking of different quantities until the Enlightenment.Relating quantities of the same kind
The history of the measurement of length dates back to the early civilization of the Middle East. Archaeologists 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 civilizations, 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 with prime factors aside from 2, 3 and 5 were sometimes used—the Egyptian royal cubit being 28 fingers or 7 hands. In 2150 BC, the Akkadian emperor Naram-Sin rationalized the Babylonian system of measure, adjusting the ratios of many units of measure to multiples of which the only prime factors were 2, 3 and 5; for example there were 6 she in a shu-si and 30 shu-si in a kush.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 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 other than 1, the pascal is a coherent unit of pressure, but the bar is not.Note that coherence of a given unit depends on the definition of the base units. Should the standard unit of length change such that it is shorter by a factor of, then the bar would be a coherent derived unit. However, a coherent unit remains coherent if the base units are redefined in terms of other units with the numerical factor always being unity.
Metric system
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 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.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 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.