"The 'root cause' of the loss
of the spacecraft was the failed
translation of English units into
metric units in a segment of ground-based,
navigation-related mission software."
— Arthur Stephenson,

NASA chairman of the Mars Climate Orbiter Mission Failure
Investigation Board, 1999

Units are of vital importance in every civilisation, as they are an
essential means of communication. Trade and science make extensive
use of them and they also play a significant role in everyday
life. But they are in no way an attainment of modern times. Whether
a caveman wanted to pose with the size of the fish he had caught or
a physicist wants to determine the relaxation time of an electron
gas – units are needed everywhere on all levels of technological
and cultural development.

However in order to have fun with them, a system of units must fulfil
some basic requirements:

• All people have to use the same units.
• The set of units must be minimal.
• The units must be defined as exactly as possible.
• The units should be connected with each other in a way that
  formulae become as simple as possible.

It turned out to be a rocky road to such a system.

A Short History of Units

For the very first units, sizes of body parts were used:
Ulna (cubit), foot, thumb (inch). However people don't share the
same body sizes and it was
unbearable to have measured data depending on the measurer.

To overcome this, cultures began to keep standards of their units
(standard bars, standard weights) in temples or – later –
in government buildings. The oldest known unit is the Babylonian foot from
2100 BC with a length of 264.5 mm. The political leaders
often took the definition very
personally: The distance between the tip of the nose of
Henry I. and his thumb was the British yard.

Under the rule of Charlemagne in the 9^th century Europe
had yet a uniform unit scheme (introduced 789 AD).
Unfortunately the disunity of Europe during the Middle Ages led to
ubiquity of completely unstandardised units, differing even from
town to town. It is reported that in Baden (now SW Germany)
in 1810 there were 112 different yards, 92 units of square
measure, 65 units of volume, 183 units for cereals and 80
different pounds! This may have been fun if you wanted to cheat the
farmer of your neighbour village, however it made trade very
difficult and was useless for science.

But the solution had been already found – in Paris.

The SI

In 1791 Delambre and Méchain measured the distance between
Dunkerque (France) and Barcelona (Spain) which was used by a
pan-European group of scientists to estimate the length of one Earth
quadrant (1/4 of the circumference of the earth).
Its 10⁷th part was called one metre.
Furthermore 1/1000 cubic metre was one litre and one litre of
water had a weight of one kilogram. These new units were
introduced in France in 1795 and in Germany in 1868. Seven years
later, in 1875, 17 countries joined the 'metric union', called CGPM.

In 1960 the CGPM decided to build a unit system, consisting of six
(later seven) base units and their derivatives and called it
Système International d'Unités, the SI.
Since then, only the definitions of the base units have been more
and more improved, besides that the SI is state-of-the-art. It
dominates the 'metric' countries and does this even more in
science. Its home is near Paris at the
Bureau International
des Poids et Mesures (BIPM).

So much for history.

Base Units of the SI

The SI developed from the metric system but today both terms can be used
synonymously. It is worth mentioning that all metric countries have a
different set of legal units, including some non-metric. Mostly the SI
represents its core though.

The science that deals with measuring quantities and definition of
units is called metrology. Many countries run metrological
institutes¹. Partly they do real science
there (eg development of extremely accurate clocks) partly they
work as a standardising authority in their respective country.

In regular international meetings these institutes decide about
(re-)definitions of SI units. Since 1983 the seven base units
are given as follows.

As you can see, 'base unit' doesn't imply that it is independent of any
other (base) unit. For example, the definition of the mole obviously
contains the kilogram, and the ampère depends on three other base
units: second, metre and kilogram. To some extent the set of base units is
arbitrary, and in this context it is discussed to abandon the candela.

Derived Units of the SI

Another great convenience of the SI is that all units are either base
units or simple products of them (according to their dimension)
– no coefficients, no zero point shifts. This is called a
coherent system. Therefore it's perfectly superfluous to
give any conversion factors. There aren't any.

The disadvantage of a coherent system is that some units are tiny (eg Pascal) whilst
others are huge (eg tesla). Therefore there is a couple of units in use that are very
closely related to SI units – namely by powers of ten. These are:

Metric Prefixes

A much more general solution for titchy or gigantic units are prefixes.
Put them directly before the unit and change their value by powers of ten. For example,
1 nF (one nano farad) is equal to 10⁻⁹ F =
0.000 000 001 F.

Although there are no further rules of using them there is some sort of
etiquette:

• Yotta, zetta, zepto and yocto are not regarded as part of
  the SI.
• Use deci only with metre.
• Use centi only with litre and metre.
• Use hecto only with litre and pascal.
• Don't use deka.

In information theory the same prefixes are applied to its fundamental
unit, the byte (B). However mostly 1 MB represents
1024·1024 B = 1,048,576 byte rather than
1,000,000 B. Therefore some metric purists called for clear distinction.
The result was published in 1998: Now 1024 B are one kibibyte
(1 KiB), 1024² B are one mebibyte (1 MiB) and
1024³ B are one gibibyte (1 GiB). This well-meant
albeit slightly academic approach, while approved by many standardisation
authorities, hasn't yet made it into everyday practice.

Minutes and Hours

Minutes and hours as measures of time are not part of the core of the SI
but they are very closely intertwined with it: 60 seconds are one
minute (min) and 60 minutes are one hour (h).

Most people
think that 24 hours are one day which is only an approximation though.
The day is adjusted to the course of the sun which is a little bit
unregular.
The same is true for week and even more for
month and year. So the realm
of actual units ends at the hour.

Non-SI Units in Use

Maybe life would be too boring if all people used SI units. Sometimes
non-SI units are used because they make a certain specific calculation
easier, however in most cases people are simply not bold enough to
break with tradition or reluctant to 'learn' the SI unit. A car with
eg 150 hp has much power but a car with 112 kW? Well, as
you probably guessed, exactly the same.

Even scientists frequently prefer non-SI units. Tradition plays a role,
too, but sometimes they have a real excuse:
Natural units. They are useless for
practice but have the effect
that natural constants become equal to one, which keeps formulae simple. For
example, Einstein's famous equation E=mc² is reduced
to a very elegant E=m.

In the following tables numbers in bold face are exact per
definitionem.

United Kingdom

Being part of the European Union the UK was urged to
abandon all imperial units. Only the pint as a measure
of beer is still legal, but you can hire the other
units for a decent fine of £ 5000.

However practically they are still widely used. A big
supermarket
chain tested metric units for six months, then switched
back to pounds and ounces because the customers had felt
'puzzled and bemused'.

USA

The USA are bottom of the metric league. The metric system is legal national
standard since 1866, though nobody cares. Curiously enough metric units are
used in most of their science fiction series. Hopefully they are no complete
fiction. The National Institute for Standards and Technology
(NIST) tries hard to enforce the SI
in the USA but hitherto with little success.

Partly the following table refers to obsolete UK units, see below.

Germany

Since the old days in Baden (see above) the situation has greatly improved.
Like in most continental countries, non-SI units have become very rare here.
Responsible is the PTB in Braunschweig calling itself the
'Guardian of Units'.

('Counterpart' doesn't imply that it has the same value.)

Peculiar Units in Use

Temperature

Temperature is basically a rather simple concept. There is a point of absolute
zero temperature (0 K, ie no heat at all) and from there you can measure
temperature just like length or mass. Unfortunately this leads to very odd values
for everyday temperatures (around 300 K). Therefore people use other
scales with other zero points (eg °C, °F). This makes it necessary to
give formulae rather than mere factors to convert between temperature units.

Let's assume you wanted to know how much Fahrenheit 20 °C is. The table
says (5th column, 4th row)
'°F = 1.8 · C + 32'. This yields
68 °F. Voilà.

Rankine and reaumur are obsolete units. Rankine was intended to be for
Fahrenheit what kelvin is for Celsius (ie a thermodynamic temperature
scale). However Celsius made the race which meant the end of the rankine.

Historical remark: Probably you have noticed all the degree
signs in the temperature units. They have nothing to do with angles
of course, but with a problem of the early days
of temperature measurement: What does
zero temperature
mean really?

• Fahrenheit (1714) said 'temperature of a mixture of water, ice and
  ammoniac'.
• Reaumur (1730) said 'temperature of a mixture of water and ice'.
• Celsius (1742) said 'temperature of boiling water'. Later people
  found this potty and changed it to Reaumur's definition.

Because all of this was fully arbitrary these units were disfigured with
degree signs. The real zero point remained a mystery until
W. Thompson
(= Lord Kelvin) determined
it in 1848. Hence the kelvin has no '°' and is allowed to
wear metric prefixes like mK or µK.

Pureness of Precious Metals

Not only that outside the SI there are many units for the same quantity,
sometimes it's even vice versa: The carat is a unit of mass
(see above) but it can also be used to indicate
how much gold, silver etc is contained in a given alloy:
(25/6 · carat)
gives the percentage of pureness. Thus a say 22-carat gold chain contains gold by
92 %. Pure gold has 24 carat, so higher values would be a
metallurgical miracle (= a lie).

Logarithmic Units

The Weber-Fechner law (simplified) says that our perception of external
stimulation is proportional to the logarithm of that stimulation. It is only
a very rough approximation but it explains the need for logarithmic units,
especially if our senses are involved.

Mathematical note: Unfortunately the logarithm can't stand units.
In order to get rid of them one has to divide the quantity by another
quantity of the same unit. This second quantity is part of the
definition and usually very small to avoid negative values.

decibel (dB): Actually the decibel is a far more general thing but
mostly it is used to indicate the loudness of sound: Let p be the
sound pressure, then 10·log(p/20µPa) is the loudness
in decibel. 40 dB is normal small talk,
140 dB is a jet engine and
180 dB is lethal. However
our ears are not equally sensitive for all frequencies; the maximum is at
approx. 1000 Hz. The phon scale takes this into account (at
1000 Hz, phon and dB are identical).

magnitudines (^m): Astronomers specify the brightness of
stars in magnitudines (or magnitudes) which bases on an ancient Greek
system. There the brightest stars had 1^m and the faintest
6^m. Today's exact definition is
m=−2.5·log(s/s₀). Here s
indicates the illuminance of the star seen from earth and s₀
is adjusted in a way that the Pole Star has exactly 2.12^m. Later they
found out that its brightness is slightly variable. But astronomers are very
patient people.

Richter scale: It's a measure for the intensity of earthquakes.
Let A be the maximal amplitude of the ground
oscillation 100 km away from the epicentre of an earthquake, then
log(A/1µm) is its value on the Richter scale. Earthquakes beyond 6
are regarded as really big ones, for them A can be several metres.

pH scale: Let n be the molar concentration of
H⁺-ions in a solution, then −log(n/(1mol/l)) is its
pH value. In water, values from 0 (strong acid) up to 14 (strong
alkali) are possible. Pure water has a pH of 7 (neutral),
yet drinking water is a little bit lower.

bit: The bit can be seen as a logarithmic measure of
information. If a certain data container can represent N different
possible contents (eg numbers) it has
(3.322·log N) bit. For example, a variable that can
have values from 0 up to 255 is a 3.322·log 256 =
8 bit variable. 8 bit form one byte. There are even larger
units like word, double word and paragraph, however their sizes are not
clearly standardised. See also the discussion of binary metric prefixes
above. (By the way 3.322 is an approximation for 1/log 2.)

Angular Units

The only mathematically legitimate unit of plane angle is the
radian (rad).
It's the length
of the corresponding arc in the unit circle. For all non-mathematicians:

2π rad is the full angle, π/2 rad is a right angle.
(You can omit the 'rad'.)

For solid angle the steradian (sr) is used. It's the
area of the corresponding sphere segment of the unit sphere. The full
sphere has 4π sr.

Actually there aren't any other sensible possibilities to measure angles.
Well, mankind is little sensible yet highly imaginative. One
degree (°) is (π/180) rad (so 360° are
the full angle). Every degree is
divided into 60 arc minutes (') end every arc minute
into 60 arc seconds ('').

One gon is
(π/200) rad. Gon is also called grade or
new degree (^g). Every new degree is
divided into 100 new minutes (^c)
(you may also call it centigon) and every new minute
into 100 new seconds (^cc).
Silly enough the gon is a legal unit, however almost nobody uses
it.

There are angular units that are even more bizarre. Just have a look at
the following table.
All these units – legal or not –
must be regarded as obsolete units, excluding degree and its derivatives.

Obsolete Units

The following passages contain units not used anymore. However you may come
across them sometimes.

It is completely impossible to list all units, there being almost as many
as human beings. If you encounter an unknown unit and look for its value
a rather good starting point is one of the national genealogy institutes,
especially if it is probably a historical unit. Otherwise the metrological
authorities are worth a try (see referenced sites).

United Kingdom

Germany

The following units are all but one Prussian. (Prussia was the
north-eastern part of Germany.) The exception is the Deutsche Meile, a general
(historical) German unit.

Russia

Today Russia uses consequently the metric system. Even in their airplanes which
is pretty dangerous since western towers normally use imperial units. Note that
the Russian dujm and fut are exactly equal to the inch and foot. The
names are mere transcriptions from Cyrillic. Other sources may differ in
spelling.

Status and Future of the SI

Today, 94.5 % of the world's population uses the metric
system.³
On the whole only three countries haven't included it into their industrial
standards: The USA, Burma and Liberia.

Most other systems have been connected to the SI. Since 1959, the inch is
defined via the metre, and the avoirdupois, troy and apothecaries' systems
via the kilogram.

The Convention of the Metre of 1875 is still the basis of all international
agreement on units of measurement. There are now 48
member countries, including all major industrialised
nations.⁴ However
practical enforcement varies heavily from country to country.

The US National Institute of
Standards and Technology (NIST) has launched a
'Metric Program' under the banner 'Toward a Metric America'.
Hopefully they will succeed.
However this would cost hundreds of billions of dollars.

Fortunately the German
Physikalisch-Technische Bundesanstalt
(PTB) needn't do this. They focus on more accurate definitions
of units. For example the kilogram is defined in a rather anachronistic
way by a prototype that is likely to change over the centuries.
Their project 'Avogadro' tries an overhaul.

Please don't take your encyclopaedias and throw further units on me.
I dream of them! I know that there are tons of them, and
believe me, I know the sources. I could add hundreds (no lie) of German and
hundreds (no lie too) of French units but I think the article is
long enough already.

Of course, if you think that I really omitted something you're welcome.

There can be severe display problems with Netscape, ie some characters are
substituted with '?'. I can't do something about that, nobody can. My way
to include special symbols like minus or Greek letters is according to
the guidelines for sub-editors (have a look at
Mark Moxon's comment on
that) and according to HTML 4. We have to wait
for better days when Netscape supports these sequences correctly. So
don't convert all minus signs to dashs; this would produce terrible line
breaks in IE.

I violated the guidelines in two respects:

1. 
   I put a space between figure
   and measurement unit.
   Readability would have lost drastically if I hadn't
   done this. However I used a   entity for this, so bad line
   breaks are impossible.
   
2. 
   I inserted commas in numbers greater than 999999 rather than 999. The
   article had so many mantissas that I wanted to avoid confusion.