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© 2000-2019   Gérard P. Michon, Ph.D.

Measurements & Units
How Far is a League?

Man is the measure of all things.
Protagoras of Abdera (c.490-c.420 BC)

Articles previously on this page:


Related articles on this site:

Metric System  &  SI   (Système International)

Current definitions of the SI base units  (NIST).
The International System of Units (SI), (BIPM, Paris, France).
General Tables of Units of Measurement (Office of Weights and Measures, NIST).
Guide for the Use of SI Units  (NIST Physical Measurement Laboratory).
Metrics the Right Way  by  George H. Sudikatus  (1996-07-03).
ISO  |  BBC  |  The International System of Units, by  Robert A. Nelson.
Tout sur les unités de mesure, by  Thierry Thomasset  (in French).
A more fundamental International System of Units  by  David Newell  (2014).

Conversion of Traditional or Specialized Units

A Dictionary of Units of Measurement  by  Russ J. Rowlett.
Dictionary of Units by the late Frank Tapson (CMIT, University of Exeter).
Conversion Factors (Process Associates of America).  |  www.sizes.com
Typography  |  Some Ancient Measures  |  Old French Measurement Units
Glossary of Ancient Weights and Measures  |  Anglo-Saxon Units
Viking Units   |   History of Measures by Livio C. Stecchini (1913-1979)
Wikipedia  |  KnowledgeDoor  |  Convert-Me  |  Online Conversion  |  Freebie

Metrication History & Controversies

In praise of the pound? (lb)  |  Metrication  |  Chronology  |  Enforcement
UKmetrication  |  weights-and-measures.com

Wikipedia :   Systems of measurement   |   Obsolete Spanish and_Portuguese units of measurement
Units of measurement   |   Unusual units of measurement   |   Humorous units of measurement
International System of Units (SI)   |   Units conversion by factor label

Are Imperial Measurements outdated? (2:50)  by  Matt Parker  (2013-11-26).
Bizarre Units used by Scientists (18:50)  by  Becky Smethurst  (2018-10-30).
Measuring to a Millionth of an Inch (12:53)  by  Destin Sandlin  (2018-12-21).


Measurements and Units

(Calvin of Farina, IL. 2000-11-05)   Current and Deprecated Prefixes
What are  all  of the metric prefixes?  (Only 20 are official ones.)

Official SI metric prefixes (largest to smallest)
and deprecated metric prefixes (obsolete or bogus)
SIValueRemarks Obsolete Bogus
  1033  una, vendeka (V)
1030  dea, weka (W)
1027  nea, xenna (X)
yotta-Y 1024 Adopted in 1991. otta
zetta-Z 1021 Adopted in 1991. hepa
exa-E1018  Adopted in 1975.
peta-P1015  Adopted in 1975.
tera-T1012  Adopted in 1960. megamega (MM)
giga-G109  Adopted in 1960.kilomega (kM)
mega-M1000 000  CGS system since 1874.  Legal in France since 1919.
 100 000  hectokilo (hk)
10 000 myria (ma, my) 1795
kilo-k1000  Since 1793.
hecto-h100  Since 1793.
deca-da10  Since 1793. Also deka.dk
  1  Unprefixed.
deci-d1/10  Since 1793.
centi-c1/100  Since 1793.
milli-m1/1000  Since 1793.
  1/10 000 decimilli, dimi (dm)myrio (mo)
1/100 000 centimilli (cm)
micro-m,u10-6  Within CGS system since 1874 (BAAS).
nano-n10-9  Adopted in 1960.millimicro (mm)
pico-p10-12  Adopted in 1960.micromicro (mm)
femto-f10-15  Adopted in 1964.
atto-a10-18  Adopted in 1964.
zepto-z 10-21 Adopted in 1991. ento
yocto-y10-24  Adopted in 1991. fito
 10-27   syto, xenno (x)
10-30  tredo, weko (w)
10-33  revo, vendeko (v)

The use of metric prefixes dates back to the inception of the French metric system, in 1793. It was originally decided that the submultiples of all basic units would be prefixed with a Latin root, corresponding to the decimal divisor (deci for 10, centi for 100, milli for 1000), whereas the decimal multiples would be prefixed with a Greek root, corresponding to the decimal multiplier (deca for 10, hecto for 100, kilo for 1000). In 1795, the Greek root myria for 10000 was added to the latter list (it's now officially obsolete, see below).

There was soon an obvious need to extend the system beyond its original limited range. The prefix micro (from the Greek mikros, small) was introduced to denote one millionth of the basic unit. The prefix mega (from the Greek megas, great) appeared around 1870 to denote a million times the basic unit.

It used to be acceptable to combine two prefixes (see above "obsolete" column). In 1960 however, it was decided to name only powers of 1000, not intermediary powers of 10, except for the original 1793 prefixes (the popular myria prefix was thus deprecated in the process). Four additional prefixes were introduced at that time: pico (Spanish pico beak, small quantity), nano (Greek nanos, little old man, dwarf), giga (Greek gigas, giant), tera (Greek teras, monster).

It was then decided that the names of future prefixes should serve as reminders of the relevant power of 10. This started in 1964, with the introduction of femto and atto (Danish or Norwegian: femten for 15, atten for 18). The former prefix was particularly convenient, because it made the widespread abbreviation fm (for "fermi") correspond correctly to the the officially endorsed femtometer. After that, however, it became clear (!?) that since only powers of 1000 were to be named, the prefixes should reflect the ranks of the powers of 1000 involved. This is why, in 1975, the prefix exa (Greek hex, 6) was chosen for 1018=10006, whereas peta (Greek pente, 5) was picked to represent 1015=10005. The four latest prefixes, which were made official in 1991, are also supposed to remind an international audience of the relevant powers of 1000: yocto (1000-8), zepto (1000-7), zetta (10007), and yotta (10008); the trend being that the ending "a" is used for large powers, while "o" is used for small ones. The 5 exceptions to this modern rule are all the 1793 prefixes, except deca (for these 6 "low" prefixes, the long forgotten Greek/Latin distinction applies, as mentioned above).

The last column of the above table lists as bogus 10 extreme prefixes (revo, tredo, syto, fito, ento, hepa, otta, nea, dea, una). The larger of these follow the etymological pattern described above, and 4 of them "compete" with the latest official SI prefixes.  These bogus prefixes have apparently not been used by anyone and are not endorsed by anybody,  but they show up in tables which have been floating around in Cyberspace...  This is probably the result of a minor hoax perpetrated sometime around 1996.  [2003-06-22 update:]  Other dubious prefixes also appeared (vendeka, xenna, xenno, vendeko) which I discuss elsewhere.  Please, tell me whatever you know about the issue...

Note (2002-05-01) : Usenet Archives show Alejandro López-Ortiz posting 3 times, between 1998 and 2000, a bogus list of prefixes ["7.5" dated 1998-02-20] whose previous version ["7.1", dated 1995-12-31, last posted 1996-10-09] didn't include any bogus information...

In a 2002-01-14 post, Robi Buecheler plagiarized the above text...

On 2004-12-14, Robi Buecheler apologized:   [edited summary]
I should have given you credit [and/or posted a] link.  Sorry.

Bogus prefixes are no longer spreading out of control.  However, at least one (careless) science-fiction writer has been fooled:  In his 2003 novel entitled  Schild's Ladder,  Greg Egan uses the two bogus prefixes  xenno  and  vendeka  as if they were legitimate.  (Thanks to Tom Alcorn for pointing that out, 2007-12-05.)

(J. B. of New Lenox, IL. 2001-02-09)
How many kilobytes [kB or "K"] in 2 "megs" [megabytes, MB]?

For units of information that are multiples of the bit (and only these), the multiplicative prefixes kilo- mega- giga- tera- etc. do not have their usual meaning as powers of 1000.  They're powers of 1024 (2 to the power of 10).

Thus, a kilobyte (kB) is 1024 bytes and a megabyte (MB) is 1024 times that (namely 1048576 bytes).  Therefore, 2 "megs" is 2048 kilobytes.

A gigabyte (GB) is 1024 MB  (1073741824 bytes) and a terabyte (TB) is 1024 GB  (1099511627776 bytes).  A petabyte (PB) would be 1024 times as large, namely 1125899906842624 bytes  (9007199254740992 bits or 86.2 nJ/K).

The situation may be quite confusing for several reasons.  In particular, a few commercial designations have wrongly ignored the above binary-based convention (powers of 1024) and used the standard decimal one (powers of 1000) in some cases.  Even worse, the two have been mixed to create a special type of digital macaronic terms like the "megabyte of storage" which turns out to be worth 1024000 bytes, but is only used commercially for some removable storage media.  This came about (sadly) when the capacity of the so-called 3½" IBM microfloppies doubled from 720 kB to 1440 kB and the larger capacity was widely advertised as "1.44 MB" (instead of "1.40625 MB" or "1.4 MB").

In December 1998, the International Electrotechnical Commission (IEC)  attempted  to clear things up by introducing a kilobinary system, in which we would no longer use kilobyte to designate 1024 bytes, but kibibyte (KiB).

The IEC proposal is slowly gaining some ground.  However, it should only be a way to disambiguate the customary exception which has been universally used for multiples of the bit (b) and the 8-bit byte (B), as far as addressable computer memory is concerned.  Ideally, acceptance of the IEC proposal should only replace "kB" or "K" by "KiB" to mean 1024 bytes.  It should never be construed as the permission to use "kB"  concurrently  to mean 1000 bytes.  (Current usage does not allow unrestricted use of metric prefixes anyway:  It's not permissible to use "kiloinch" for 25.4 m, is it?)  Otherwise, ambiguity and confusion would be increased, not decreased.  Arguably, manufacturers of digital storage who use the abbreviation "GB" for 1000 000 000 bytes would still be shortchanging their customers by  7.4%,  even if the unambiguous IEC binary prefixes gain wider acceptance.

Warning:   1 kb/s  =  1 Kib/s  =  1.024 kbps

Be aware that the binary exception only applies to multiples of the bit, not to  derived units  like the "bps" (bit per second), so that 56 kbps is exactly 56000 bps. This may not look so bad until you realize that a transfer speed of "1 kilobit per second" is actually equal to 1.024 kbps.  The latter should only be pronounced "kilo-bee-pee-ess" to avoid confusion with the former!

Likewise, the related  baud rates  have always used decimal prefixes:  A kilobaud  (kBd)  is  1000 Bd.  A megabaud  (MBd)  is  1000000 Bd.

That's the current mess we've built for ourselves.  Careless standardization efforts could make the situation even worse before it gets better.

"Brontobyte"  [ hoax alert ]

This unit is just a joke  (2004)  nothing more!  Unfortunately, the word caught the fancy of many unsuspecting webmasters and is now often listed among "serious" units of information  (even more dubious is the  geobyte  of 1024 brontobytes).

1 brontobyte  =  (1024) 9  bytes  =  9903520314283042199192993792 bits

In terms of entropy, this huge amount of information is only  1.7668 J/K.

( L. K. of Owen, WI. 2000-10-10)
What has a density of 1 ?

Proper units (g/cc, lb/ft 3, etc.) are used to express an absolute density.

A relative density is the ratio of an absolute density to the absolute density of "water".  For the utmost precision, it's important to specify what kind of "water" is meant.

For example, SMOW ("Standard Mean Ocean Water") at its densest  (around 3.98°C)  has an absolute density of about 0.999975 g/cc.

However, the universally accepted conversion factor between "absolute" and "relative" density is 0.999972 g/cc !  This is one number which has acquired the unofficial status of a  defined  exact conversion factor, which has  ultimately  little to do with actual water or SMOW.

In other words, the short answer to this question is: "Water."
A more precise (somewhat cynical) long answer is:
"Anything with an absolute density of exactly 0.999972 g/cc."

Videos:   7-Layer Density Tower  |  9 Layers  |  Density Rainbow  |  Soda Density  |  Astronomical Densities

(Michael of United Kingdom. 2001-02-12)
What's the difference between normal [1N] and molar [1M] solutions in acid chemistry? Particularly for  sulfuric acid.

Each liter of a molar solution (1M or 1000mM) contains a mole of a given compound (a mole of H2SO4 is about 98.08 grams of it). A normal acid (1N), on the other hand, contains the solute(s) that could produce a mole of H+ ions.

In the case of sulfuric acid, you'd have 2 H+ ions per molecule, so that a normal (1N) solution of sulfuric acid is actually a 1/2 molar solution (0.5M or 500mM).

A mole of "objects" [atoms, molecules, ions, electrons] is defined to be as many of these as there are atoms in 12 grams of carbon-12.
      The "number of things per mole of stuff" is a universal constant known as Avogadro's Constant : 6.02214129(27)´1023 per mole. [Here, the parenthesized 27 indicates an uncertainty whose standard deviation is 27 times the weight of the last decimal position shown.]
Free protons  (H+ ions)  in  water  are mostly a convenient fiction, since such ions would quickly combine with nearby molecules of water to form  hydronium ions.  The dissociation of water molecules into ions is thus best described by the following reversible chemical reaction:

2 H2O    «    H3O+  +  OH-

(J. M. of College Station, TX. 2001-02-11)   Definition of the Calorie
How much energy is required to raise the temperature of one kilogram of water [by] one degree Celsius?

If the calorie was still defined as the energy required to raise a gram of water by 1°C, the answer would be "1000 calories" (or 1 kcal).

However, that definition of the calorie was dependent on the starting temperature and wasn't good enough for metrological purposes.

Several more precise definitions have been given  (see table below)  including the  fifteen degree calorie  which is still defined as the energy that raises a gram of water from 14.5°C to 15.5°C.  This type of calorie must be  measured  to be equal to 4.1855 J  at a fairly modest level of precision  (an uncertainty of about 0.0003 J).  It's only good for casual use...

Since 1935, the current (thermochemical) calorie has been defined as exactly equivalent to  4.184 J.  No other conversion factor should be used in Science.  The recommendation is to use joules primarily.

The energy which raises a kg of water by 1°C  (under  1 atm = 101325 Pa)  is a function of temperature which features a minimum of about  4178 J  around  34.5°C.  It's about the same at the ice point  (4218 J at 0°C)  and the steam point  (4216 J at 100°C).  All values between  4178 J  and  4216 J  are correct for  two  temperatures  (one below  34.5°C, one above that).

Various "calories" competing with the thermochemical calorie of  4.184 J
NameTemperatureSymbolValue  (J)Date
Bunsen calorie0°Ccal0 4.218 
 100°C  4.216 
"therm" (deprecated)3.48°C - 4.48°Ccal4 4.20451888-1896
Mean calorie0°C - 100°Ccalm 4.19002 
Steam tables calorie(13°C or 67°F)calIST 4.1868   exactly1956
British calorie   (180 J  =  43 cal)  4.1860465...1926
American calorie15°C = 59°F  4.1858 
15° calorie14.5°C - 15.5°Ccal15 4.1855 (3) 
calorie(16°C or 61°C)cal 4.184     exactly1935
Rossini calorie62°F  4.18331933
20° calorie19.5°C - 20.5°Ccal20 4.18190 
[ minimum ]34.5°C  4.178 

The dubious "IST calorie" (or "steam tables calorie") was created merely for compatibility with the official definition of the Btu,  discussed next.

British Thermal Unit (Btu)  Therm  and  Quad :

The  Btu  itself is never used in Science and it seems to be utterly unknown outside of the US and UK.  So is the obscure unit dubbed  therm,  defined as  100000 Btu.  (Warning:  The name "therm" was used to denote the  4° calorie  between 1888 and 1899,  as mentioned in the above table.)

The 5th  International Conference on the Properties of Steam  (London, July 1956)  gave the (final) modern definition of the  Btu  (British thermal unit)  based on their nominal heat capacity for water:  2326 J/kg/°F  (which is equivalent to  4186.8 J/kg/°C).  This authoritative definition of the Btu had the unfortunate side-effect of creating a bogus  "IT calorie" of 4.1868 J  (IT or IST stands for "International [Steam] Tables")  which has infected many computers and most handheld calculators.  It is off by 669 ppm and shouldn't be used in connection with the precise thermochemical data which has been lovingly compiled by our elders since 1935  (using an exact equivalent of 4.184 J  to the calorie).
The Btu  had an historical definition similar to that of the calorie:  Before 1820,  it had been introduced by Michael Faraday (1791-1867)  as the quantity of heat which would raise a pound of water from  63°F  to  64°F  (presumably under normal atmospheric pressure)... In 1876, it was re-defined as the energy required to raise the temperature of one pound (lb) of water by 1°F, around the point of maximum density  [about 3.98°C].
All told, it's best to use the IST definitions (1956) for the Btu (1 Btu is 1055.05585262 J, namely the ratio of the pound to the kilogram multiplied by 2326 J) and to use the standard 1935 thermochemical definition for the calorie (1 cal is 4.184 J).  Unfortunately, you may also encounter a "thermochemical Btu" (» 1054.35 J) and an "IST calorie" (4.1868 J = 2326 * 0.0018 J).

Other definitions of the Btu are still floating around/  and the ensuing confusion extend to the  therm  of 100000 Btu  and the  quad  of  1015 Btu.

Definitions of the  Btu  competing with the standard one  (1055.05585262 J)
Value  (J)QualifierRemarks
  1055.206 778.28 foot-pounds
  1055.05585262ITIST definition (1956)
  1054.80459°FUS  (1 cal = 4.1858 J)
  1054.68 Canada
  1054.350264488888... Thermochemical (unused)

Water Structure and Science  by  Martin Chaplin  (Emeritus Prof., London South Bank University).
British thermal unit explained
Videos :   Joule and calorie conversions   |   Specific Heat

cdw239 (2001-08-23)
What is the equation for converting horsepowers to watts?

The horsepower and the watt are both units of power; there's just a conversion factor between them.  The way power is delivered  (voltage, etc.)  is completely irrelevant.

A horsepower (hp) is about 745.7 watts (W), but many metric countries use another  closely related unit  [best abbreviated "ch"]  of nearly 735.5 W.

The horsepower unit (hp) was originally defined by James Watt (1736-1819) as exactly equal to 550 ft-lbf per second  (lbf = "pound-force") . Since January 1, 1959, the foot and the pound have been defined in metric terms (1 ft = 0.3048 m and 1 lb = 0.45359237 kg, both exactly).

Furthermore, since the third CGPM of 1901, the standard (or conventional) acceleration of gravity has been defined as  exactly  equal to  9.80665 m/s2.  That's the proper "conversion factor" to use to transform  (not "convert", please)  a unit of mass like the  pound  (lb)  into the unit of force best called  pound-force  (lbf).  The resulting  exact  conversion factor has 12 digits:

1 lbf   =   (0.45359237 kg) (9.80665 N/kg)   =   4.4482216152605 N

Multiply this by the length corresponding to 550 ft (exactly 167.64 m) and you have the equivalence of a horsepower in watts  (as a watt "W" is simply a meter-newton per second).  This gives an  exact  modern conversion factor which requires no fewer than  17  digits:

1 hp   =   745.69987158227022 W

Needless to say that everybody usually rounds this up in the most obvious way  (which is appropriate except in computerized conversion tables):

1 hp   »   745.7 W

In countries where the metric system has been around for a while, the horsepower (ch) is a 1.37% smaller unit, called Pferdestärke (PS) in German, paardekracht (pk) in Dutch, hästkraft (hk) in Swedish, caballo de vapor (CV) in Spanish, cavalo-vapor in Portuguese and cavalli vapore in Italian.  The French call it cheval-vapeur (ch) or cheval (plural is chevaux).

This "metric" horsepower (ch) is defined as 75 kgf-m/s, which engineers used to abbreviate as 75 kgm/s, using the obsolete symbol kgm for a "technical" unit of energy called kilogrammetre or kilogram-meter and worth 9.80665 J  (that same unit of energy was also called  kilopond-meter  and abbreviated  kpm ).  A metric horsepower (ch) is thus  exactly :

1 ch   =   (75 kg) (9.80665 N/kg) (1 m/s)   =   735.49875 W

French readers should not confuse this cheval-vapeur (ch) unit with the French cheval fiscal (CV) which is a nonlinear rating of a motor vehicle for tax purposes (registration cost is about $30 per CV, as of this writing).  The CV rating, or fiscal power [sic], is  (P/40)1.6+ U/45,  where P is the maximum DIN power (in kW) and U is the amount of CO2 emitted per unit of distance (in g/km).
From 1909 to 1947, British car taxation was based on another  unphysical  rating called  RAC horsepower  (introduced by the  Royal Automobile Club )  numerically equal to  0.4  times the square of the  bore  (the diameter of each cylinder in inches)  multiplied by the number of cylinders.  Several generations of British engines had artificially  long strokes  which produced a larger displacement (in cubic inches) for the same tax rating.  The assumption behind the RAC formula was that short stroke engines would work at higher rpm:  The maximum product of stroke and rpm was taken to be a constant  (6000 in/min)  equal to  half  the largest acceptable mean piston speed, (estimated to be  1000 ft/min).  Paying no attention to British tax law, American automakers designed  short stroke engines.

Adding to the confusion, a so-called electric horsepower is defined as exactly equal to  746 W  (it's clearly a rounded-up version of the "hp").

Finally, there's an unrelated unit of power called the boiler horsepower, defined in 1884 as the power it takes to boil 34.5 lb of water per hour (under 1 atm, when water is already at 100°C = 212°F).  So defined, the boiler horsepower is approximately 9809.91 W, or about 13.155 hp.  However, this is so close to 1000 kgf-m/s (which is 9806.65 W) that I suspect such a "metric" definition of the boiler horsepower may have been given...  (The quotes around "metric" are a reminder that "technical" units of force, named after units of mass, are not official SI units.)  I'd be grateful if anyone could tell me if this is so...

Why is 9.80665 m/s2 [1 G] the standard acceleration of gravity?

To an actual measurement of 9.80991 m/s2 in Paris, a theoretical correction factor of 1.0003322 was applied which gives a sea-level equivalent at 45° of latitude.  The result (9.80665223...) was rounded to five decimals to obtain the value officially enacted by the third CGPM, in 1901.

The above includes a centrifugal component due to the rotation of the Earth, whereas the gravitational field at altitude zero has a slightly larger value, used when tracking satellites outside the atmosphere in nonrotating coordinates  (9.82025048(2) m/s)  which is the ratio of the  Earth's gravitational constant  (3.986004415(8) 1014 m3/s2 ) to the square of the conventional Earth radius  (R = 6371000 m).  The  centripetal acceleration  of a satellite orbiting the Earth at a distance  R+z  from its center is thus:

(9.82025 m/s/  (1+z/R) 2

(2018-02-27)   Periodic change in phase  vs.  random  activity.
One is expressed in hertz  (Hz)  or  rad/s,  the other in  becquerels  (Bq).

Any periodic quantity is only a function of its  phase.  Phases add up like  circular angles  do;  returning to the same position after a whole number of turns.  They are naturally measured in angular units,  the most obvious of which is variously called  turncycle  or  revolution.

For a metronome or a pendulum  (only)  the  beat  has been historically defined as  half  of the period in time  (the motion is composed of two symmetrical halves).  The beat  of a  simple pendulum  of length  L = 1 m  is nearly equal to one second.  More precisely,  its period  T  in a  standard gravitational field  g = 9.80665 N/kg  is equal to:

T   =   2 p   Ö   L     =   2.006409292589... s

The meter was first officially defined as one ten-millionth of a  quarter-meridian.  Before that decision was made,  the idea was tossed around that it would be more convenient to define it as the length of a simple pendulum which would  "beat one second".  This,  however,  would have made the definition dependent on exactly where on Earth the measurement was made.  The inceptors of the metric system would have none of that.  It didn't fit their idea of "Universality".

In all other cases,  a  beat  is equal to a complete time period.  Thus,  beat  and  period  are almost always synonymous.  (This general definition does applies to the  sound  made by a symmetrical pendulum or a symmetrical metronome,  albeit  not  to their visual appearances.)

Musicians express a  tempo  in  beats per minute  (bpm)  in reference to the  sound  of a metronome.  The same unit outside of music is  called a  revolution per minute  (rpm).  Notably in the  automotive industry.

Physicist most often use the ratio of two lengths to quantify an angle  (the ratio of a subtended circular arc to the radius of the circle involved).  Thus,  an angle looks superfically like a dimensionless number because it seems to be just the ratio of two quantity having the same dimensionality.  However,  this ain't quite so.  What makes all the difference is that  one  of the two aforementioned lines is  curved.

There are two ways to curve  (to the left or to the right).  Angles can be negative or positive,  ultimately depending on a  conventional  orientation of the Euclidean plane  (the universal convention today is that counterclockwise is positive and clockwise is negative).

An angle  (planar angle or solid angle)  measures a quantity with  axial  symmetry,  whereas so-called  pure numbers  only measure quantities with  radial symmetry  (which is the technical way to state that theirs signs do not depend on the orientation of the frame of reference).

Unfortunately,  the peculiarity of axial quantities is no longer a popular thing to teach.  Before 1995,  angular units were usefully singled out as  supplementary  SI units.  In a dubious move,  the  CGPM  has effectively dropped all SI units of angle  (planar or solid).  Formerly,  the  radian  (rad)  was the SI unit of planar angle and the  steradian  (sr)   was the SI unit of solid angle.  The reader is hereby advised to ignore the misguided new status of angular units as dimensionless derived units.  Failing to do so would obscure the procedures which makes certain physical formulas numerically reliable.  In particular,  the formulas of  photometry  rely heavily on the distinction between total luminous power and luminous power per unit of solid angle  (compare the  definition of the lumen  to the definition of the  candela).  Likewise,  any numerical application involving  Planck's constant  requires a clear specification of which "natural" angular unit is understood for phases;  (either cycles or radians).

Numericana :   Proper units of measurement for Plank's constant.

Frequency  or  Pulsatance :

The rate at which phase changes with time is called  frequency,  especially when the angular unit of phase is the  cycle  (turn  or  revolution,  same thing).  Otherwise the more precise terms of  pulsatance  or  angular frequency  are preferrable.

All those termes denote strictly the same physical thing.  This is clear only if the  hertz  (Hz, the official SI unit of frequency)  is properly defined as  one cycle per second  and not as  "the reciprocal of the second"  (the absence of any reference to angular phase in the latter definition would make it better suited for the  becquerel,  as discussed next).

Activity of a random phenomenon.  Radioactivity.

stochastic process  which occurs at  irregular  time intervals  (as opposed to a regular  period)  is characterized by a quantity called  activity,  which is the expected  (average)  number events occuring per unit of time.

Such phenomena have  no phase,  which is to say that the time it will take until the next event  doesn't  depend on the time elapsed since the last one.  Radioactive decay  is the prime example of this.

As a result the unit of activity  (the SI unit is the  becquerel,  symbol Bq)  is just the reciprocal of the unit of time  (1 Bq  is one event per second,  on average).  It's fundamentally different from the aforementioned units of  frequency  which ultimately measure a rate of  change of phase.  In spite of a superficial similarity,  one random event per second  and  one cycle per second  are two very different things!



(Bob J.of Clarksville, TN. 2000-09-28)
What is the term for 1/1000 of an attosecond?  (This would be 10-21 s.)

That's one  zeptosecond  (zs).
One thousandth of  that  is a  yoctosecond :   1 ys  = 10-24 s
Both terms were officially adopted by the CGPM in 1991.

Fred Berman, Ph.D., P.E. (2002-11-29; e-mail)   What's a  jiffy ?
Is a  jiffy  really the time for light to travel one centimeter in a vacuum?

A formal definition of the  jiffy  as a  light-centimeter  (roughly equal to 33.3564 picoseconds) was first proposed, in physical chemistry, by Gilbert Newton Lewis (1875-1946), the American chemist who isolated heavy water and defined a  Lewis acid  as an  acceptor  of electron pairs  (1916).

In 1926, the same Gilbert N. Lewis also coined the term  photon  which is now used to denote the  quantum of electromagnetic radiation  introduced by Albert Einstein in 1905, under the German name of  Lichtquant.  (Ironically, Lewis originally intended the word "photon" to mean something else entirely.)

Informally, a  jiffy  can be any short period of time, though.  The word was commonly used before 1785 Jiffy  meant "lightning" in  thieves' cant  (possibly as early as 1530)  but its early etymology is otherwise unknown.  The  jiffy  has been given  several  definitions in various contexts:

In the quaint context of computer engineering, a  jiffy  may denote the period of the system's main clock (e.g., 10 ns for a 100 MHz clock)  but it can also be the interval between two regular  timer interrupts,  which is usually something between  1 ms  and  20 ms  (most commonly  4 ms).

In electrical engineering, a  jiffy  used to be the period of the electrical power grid, namely:  20 ms in Europe (50 Hz) or about 16.6667 ms in the US (60 Hz).  Nowadays, this flavor of  jiffy  has all but disappeared;  a modern  jiffy  is usually equal to  10 ms  (the resolution of an ordinary stopwatch).

On 2008-12-30, Dr. Robin Whitty wrote:   [edited summary]
I love it!  Where but  Numericana  could you find something so minute given such Johnsonian treatment?    ;-)

A much smaller obsolete unit  [about 9.3996392(13) 10-24 s]  is related to the above  jiffy  of physical chemists:  The  tempon  is defined as the time required for light to travel a distance of one classical electron radius.

The smallest recognized unit of time is called  chronon,  or Planck time:

( hG / 2pc5 ) ½   »   5.3912 ´ 10-44 s

(G is Newton's constant, h is Planck's constant, c is Einsteins' constant.)  Expressed in chronons, the  age of the Universe  is about  8 ´ 10 60.

Electric Power & Transmission & Distribution Forum:  Jiffy? (2011-04-18)

(2014-11-28)   shake  is  10 ns.
That's one informal nuclear time unit coined for the  Manhattan Project.

The saying  two shakes of a lamb's tail  denotes any short time interval...

The typical time required for each step in a nuclear chain reaction is roughly  one shake.  A nuclear explosion takes less than 100 shakes  (1 us).

Shake (time unit)

(B. D. of Australia. 2000-05-01)
How long is one second?
(J. F. of Memphis, TN. 2000-10-20)
Who determined the length of a second?

The "SI second" (formerly called "atomic second") is now defined as equal to 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of Cesium-133. (Until recently, surprisingly, nobody seemed to care about general relativistic effects, which are becoming relevant:  Are we talking about cesium atoms in free fall or not?)

In 1967, this replaced officially the "Ephemeris Second", which was based on the orbital motion of the Earth around the Sun.  An earlier definition was based on the mean solar day instead, and was thus tied to the Earth's rotation around its own polar axis, although fluctuations in this rotation make it a poor basis for the definition of a precise unit of time (as was first shown by Simon Newcomb).

The Full Story:

Originally, the second was defined as 1/86400 of the mean solar day.  In other words, there are 24 hours of 3600 seconds in a day.  It is necessary to specify "mean" solar day because the length of the day varies throughout the year, as the angular speed of the Earth varies in its elliptical motion around the Sun.  (It is this angular speed which determines how soon the Sun will be seen again at the same longitude in the sky, after roughly one revolution of the Earth on its axis.)

This mean solar second came under international scrutiny by the CGPM in 1954, and the BIPM proposed (in 1956) a new official definition of the second:  The definition of the so-called ephemeris second is based entirely on the orbital period of the Earth, which is steadier than its spin.  It is specified, as explained below, by equating to 31 556 925.9747 ephemeris seconds the instantaneous value at epoch 1900.0 of the tropical year.  This definition was ratified by the CGPM in 1960, but it originated in the 19th century:

  Simon Newcomb

The American astronomer Simon Newcomb (1835-1909) discovered that there are significant irregularities in the rotation of the Earth on its own axis (this was apparent to him when he analyzed the ephemerides of the Moon published by Hansen in 1857).  Newcomb came up with a famous equation giving L, the so-called "mean geometrical longitude of the Sun", as a function of the time T expressed in the number of centuries [of exactly 31 557 600 000 seconds each] elapsed since  "January 0.5 1900"  [this is either 24:00 GMT on 1899-12-31 or 0:00 GMT on 1900-01-01].  That "longitude" is measured against the vernal point [which means it integrates the wobbling of the Earth's spin which influences the length of the tropical year and causes the precession of equinoxes].  The qualifier "geometrical" is a reminder that the equation gives the immediate position of the Sun, not its apparent location, as perceived from solar light emitted about 499 seconds before.  Finally, the qualifier "mean" is a reminder of the averaging made necessary by the variable angular speed of the Earth, in its elliptical orbit around the Sun.

L     =     279° 41' 48.04"   +   129602768.13" T   +   1.089" T 2

A tropical year is the time it takes for L to increase by a full turn (360° or 1296000"), we may thus state that the instantaneous tropical year at time T is a full turn divided by dL/dT.  To obtain the duration Y of this year expressed in seconds (rather than Julian centuries), we simply multiply by 3155760000.  This boils down to:

Y(T)   =   227214720000000000 / ( 7200153785 + 121 T )
=   31556925.9747415242... - (0.5303203455...) T + O( T 2 )

It turns out that Newcomb's equation can be used backwards to define the unit of time with far greater precision than anything based on the rotation of the Earth.  By specifying the value of Y(0) in some unit of time, that unit is very precisely defined in terms of the orbital motion of the Earth around the Sun, rather than on the less precise rotation of the Earth about its own axis.  This is precisely how the so-called ephemeris second was defined, by making exact de jure Newcomb's value of Y(0) rounded [down] at the fourth position after the decimal point:

Y(0)   º   31556925.9747 ephemeris seconds

This definition makes the ephemeris second very slightly longer than whatever we may call the "second" used by Newcomb himself to establish his equation. 

The above rounding corresponds to a relative precision of 1.31585 10-12 (roughly 50 microseconds per year).  This is lower than the combined precision of the observations used by Newcomb, which were made between 1750 and 1892.  The solar second and the ephemeris second were identical around 1820 or 1826. Since then, the mean solar day has been slightly longer than 86 400 ephemeris seconds, as the rotation of the Earth is slowing down under the braking effect of the tides.  It may be amusing to record that, according to Newcomb's original equation, the instantaneous tropical year was exactly 31556925.9747 "seconds" about 247097 seconds after T=0: January 3, 1900, at 20:38:17 GMT.

The ephemeris second was the official definition of the second from 1960 to 1967.  During that period, the credit for determining the "length of a second" would clearly have gone to Simon Newcomb...

Since 1967, the official definition of the second has been in "absolute" atomic terms rather than astronomical ones.  It was decided to define the second in terms of a number of standard transitions of the Cesium atom.

In 1958, it had been determined that there were 9192631770 such transitions (give or take 20) in an ephemeris second.  This was the result of a three-year collaboration between William Markowitz at USNO (U.S. Naval Observatory, in Washington, DC) and Louis Essen (1908-1997) at NPL (National Physical Laboratory, in Teddington, England).  USNO contributed accurate astronomical time measurements, using a dual-rate Moon camera (invented by Markowitz in 1951) which was compensating simultaneously for sidereal and lunar motions.  Occultations of stars by the Moon provided the best estimate of Ephemeris Time.  On the other hand, NPL provided the World's first caesium clock standard which had been perfected by Louis Essen and Jack Parry since 1953.  (The two clocks were compared using synchronizing radio transmissions from the WWV station operated by the National Bureau of Standards, now called NIST.)

This value of 9192631770 Cesium transitions per second was ultimately accepted as the de jure value. Therefore, the guys who really determined "the length of a second" are the authors of that particular measurement.  It was a team effort, by Markowitz, Hall, Essen and Parry [See "Frequency of Cesium in Terms of Ephemeris Time" by W. Markowitz, R. Glenn Hall, L. Essen, and J.V.L. Parry in Physical Review Letters, Volume 1, pp. 105-106 (1958)].  That's our final answer, as long as the "Cesium standard" remains the basis for the official definition of the second.

The international body which is responsible for making such definitions official is the CGPM.  However, the CGPM should not be credited for the work on which its decisions are based.  Instead, we ought to remember the accomplishments of great scholars like Simon Newcomb, Louis Essen, or William Markowitz...

(C. V. of Indianapolis, IN. 2000-10-23)
How many seconds in a day?

The short answer is 86400 (24 hours of 3600 seconds).

At a higher level of accuracy, it may be useful to point out that there are 3 kinds of standard days, but we may still say that there are exactly 86400 "solar" seconds in a "mean solar day" and 86400 "ephemeris" seconds in an ephemeris day.  The "day" used in modern science is also defined as exactly equal to 86400 SI seconds (officially defined in terms of the cesium atomic standard). 

When the "day" of one system is expressed in terms of the "second" of another, the numbers are slightly off.  For example, the mean solar day "at epoch 2000.0" is about 86400.002 SI seconds.

Now, the so-called "sidereal day" is another matter entirely because it is significantly different from the above 3 "standard" days and has never been used as a standard unit of time. A sidereal day is about 86164.09 SI seconds.

It is interesting to notice a weird point of etymology about "sidereal" (which is often misspelled "sideral", as would be correct in French and/or a few other languages). "Sidereal" should mean that a "sidereal day" refers to the rotation of the Earth with respect to the fixed stars (as is the case with other "sidereal" motions, by the way). This is the definition most dictionaries will give you.  However, that's not quite so. Historically, astronomers have most often used the term "sidereal day" to refer to the rotation with respect to the slowly moving "vernal point" (which rotates a full turn in about 25772 years, the period of "precession of the equinoxes").

When motion with respect to the fixed stars is meant, the unambiguous term "Galilean day" should be used. In other words, the Earth rotates on its axis once per Galilean day (i.e., once in each period of 86164.1 s).  The Galilean day is longer than the sidereal day by 0.0084 s. The Galilean day increases by about 0.00164 s per century because of the braking effects of tides.  Both the sidereal day and the mean solar day also increase at almost exactly the same rate (so the differences between these three remain roughly constant).  The drift rates are almost exactly the same because all the other relevant astronomical motions are far more stable than the spin of the Earth on its axis.  The SI "atomic" day, on the other hand, is absolutely stable in principle (assuming only that the laws of physics themselves do not change over time).

(2000-11-03)   The scientific year:   1 a  =  365.25 D  =  31557600 s
(P. H. of Concord, CA. 2000-11-03 and I. I. of Canada. 2001-02-05)
How many seconds in a year?
(John of Springville, AL. 2000-10-08) What is a "scientific year" ?
jwill123 (2002-05-05) A light-year is the distance that light travels in one year.  How many seconds in [such] a "year"?
" The recognised symbol for a year is the letter  a  [annum]  rather than yr, which is often used in papers in English.  [...]  Although there are several different kinds of year  (as there are several different kinds of day), it is best to regard a year as a julian year of 365.25 days  (31.5576 Ms)  unless otherwise specified. "
The IAU Style Manual  (1989)  5.15,  page 24.

The only recognized "year" unit in scientific practice is a year of exactly 365.25 days, based on a day of exactly 86400 seconds (these are standard SI seconds, formerly known as "atomic seconds").  Therefore:

The number of seconds in a year is exactly 31557600.

This is the number you should use, for instance, to compute precisely the number of meters in a light-year (which is exactly 9460730472580800).

Some scientists like to memorize the duration of a year in seconds as approximately equal to "p times ten to the seventh".

This scientific year is longer than the average calendar year, the Gregorian year of 365.2425 mean solar days, and it's extremely close to the Julian year of 365.25 mean solar days.  As the mean solar day slowly drifts in duration, so do both the Gregorian year and the Julian year.  The related tropical year is more stable than either of these calendar years, because it is based on the orbital motion of the Earth, which is steadier than its spin.  The wobbling period of the Earth's axis (responsible for the precession of equinoxes) affects this tropical year but not the sidereal year which is measured with respect to the "fixed stars" (more precisely, the background of galactical nebulae).  However, even this sidereal year is not absolutely stable, since the orbit of the Earth does decay...

By contrast, the scientific year of 31557600 seconds is rock stable [more stable than any rock will ever be, actually]; it's a true unit of time.  It will never change, unless the laws of physics themselves change.  Finally, it is properly based on a local [atomic] definition, as any unit of time should be: According to Special and General Relativity, there is not such thing as an absolute time which would "flow" the same for all observers, irrespective of their motions and/or surrounding gravitational fields.



nara (2000-04-11)   How long is a meter?
I know it is not the same system, but how many inches are in a meter?

There are (very) sligthly more than 39.37 inches in a meter (a more precise number is 39.37007874).

Since January 1, 1959, the International inch has been defined to be exactly equal to 25.4 mm (0.0254 meter). Now, the inch and the meter are thus almost part of the same system (well, kinda)...

Since 1866, the US Coast and Geodetic Survey has been using another metric definition of the inch, equating a meter to 39.37 inches.  This "US Survey" inch (of about 25.4000508 mm) was confirmed for general use by the Mendenhall ordinance of April 5, 1893, but it's been restricted to US surveying since 1959.

There's a noteworthy  numerical coincidence  concerning the ratio of these two different "types" of inches, since (254/10000)/(100/3937) turns out to be exactly 999998/1000000, so that it can be stated that the modern International inch is exactly 2 ppm less than the 1893 "US Survey" inch, whose value in mm has the following expansion:  25.400050800101600203200406400812801625603...

You may notice a pattern in the above decimal expansion which allows you to write dozens of decimals very quickly. It comes from the fact that 1000000/999998 is the sum of a geometric progression of ratio 0.000002 and is thus equal to 1.000002000004000008000016...

The 1824 Imperial inch was based on the actual British standard yard, which kept shrinking  (the 1760 brass artifact was lost in an 1834 fire; new ones were made of Baily's Metal, after 1841).  This obsolete inch was "calibrated" to be:

  • 25.399978 mm   in 1895.
  • 25.399956 mm   in 1922.
  • 25.399950 mm   in 1932.
  • 25.399931 mm   in 1947.

The 1895 and 1922 calibrations are still quoted today in an historical context, whereas the others are all but forgotten.  The preliminary 1819 equivalence of 39.3694" to the meter describes a larger inch (of about 25.400438 mm) which may best match the yard made by Bird in 1760 (after an old Tower standard).

(2007-05-22)   The Typographical Point
The typographer's point is  exactly   0.013837"  =  0.3514598 mm.

Typographers use several specific traditional units of length.

The ATA point and its spinoffs

The typographer's point was defined in term of the inch in 1886.  Thus, when the inch evolved and was redefined in terms of the meter in 1959, so was the typographer's point.

The  point  is exactly 0.013837 inches.  That's precisely  0.01 ppm  below what is implied by the conversion factor of  72.27 points  to the inch, which has been advocated by  Donald Knuth  in connection with his "TeX" computerized typesetting system.  The difference between the genuine typographical point and the "TeX point" is so minute that the two are interchangeable, even in the most exacting typographical work.  (There are 72.2700007227... points to the inch.)

Not so with the  coarse  equivalence of 72 points to the inch, which was part of the original specification of PostScript, the page description language championed by the Adobe Corporation, which was instrumental in launching the "DeskTop Publishing" (DTP) industry in the mid 1980's.  This rough equivalence gave birth to a new set of "DTP" units for computerized typography:  There are, for example, exactly 6 DTP picas to the inch, and 72 DTP points to the inch.

Other points.  155520 Didot points to the arpent...   ;-)

Although the above now dominates computerized publishing worldwide, some typographic systems are based on other unrelated "point" units.

Most notably, the  Didot point  was introduced by  François Ambroise Didot  (1730-1804)  with the system of font measurements that we still use today  (regardless of a slight difference in scale).  Didot's father  (also named François, 1689-1757)  was the founder of the printing and publishing enterprise which survives to this day (in Paris, France) under the name of  Firmin-Didot & Cie.

The original Didot point was defined as the 72nd part of the French Royal inch  (pouce).  The foot corresponding to 12 pouces was the  pied de roi.

Two (very close) metric equivalences can be given for the pied de roi.  The earlier one goes back to the very inception of the metric system itself, since the French scientists who conceived the new system were actually using the  toise  of 6  pieds de roi  in their preliminary work.  The metrological equivalence they gave now stands as a metric definition of the old unit:  0.513074 toise to the meter.  (That would make a toise approximately equal to  1.9490366 m.)

However, the Canadians can be considered to be the rightful heirs to the ancient French system, as they still use the  arpent  of  30 toises.  The modern Canadian definition is thus just as relevant as the current definition of the International inch  (of exactly 25.4 mm)  regardless of previous definitions...  The Canadian arpent is now defined to be  191.835 ft  or  50.471308 m.  This makes the  toise  exactly equal to 1.9490436 m.  The  pied de roi  is  0.3248406 m  and the  pouce  is exactly  27.07005 mm,  The proleptic value of the Didot point is 72 times smaller than that, namely:

1  Didot point   =   0.3759729166666... mm
1  Cicéro  (12 Didot points)   =   4.511675 mm   ( exactly )

The French  Imprimerie nationale  (IN)  now uses a metric point of exactly  0.4 mm.  The obsolete  Truchet point  was exactly half of a  Didot point.

Other traditional units of length pertaining to newsprint include the line  (the agate line of 1/14 in, called a "ruby" in the UK)  and the "SAU column width" of 36/16 in  (i.e., 2-1/16 inches of print and a 1/8" gutter space between columns).

"SAU" stands for "Standard Advertising Unit" and is also called "column-inch".  The SAU is actually a unit of surface area equal to 36/16 square inches, namely the surface of one inch of a standard column, as described above.  That unit is used to charge advertisers for commercial space in printed media.

cheftell (Wilmington. 2001-02-11)
How far in miles is 20000 leagues? One league equals how many miles?
creus (2001-07-06)
How far is a league?

A (land) league used to be defined as an hour's walk.  It's now defined as exactly  3  statute  miles  (4828.032 m).

However, a nautical league is 3 nautical miles (5556 m, or about 3.452 miles), and that's the league Jules Vernes refers to in the title of his book "20000 Leagues under the Sea". So, if you are a fan of Jules Vernes and Captain Nemo, 20000 nautical leagues is 60000 nautical miles. That's about 69047 statute miles, 111120 km or almost 3 times around the Globe [at the Equator].

Originally, the  nautical mile  was defined as an  average minute of latitude.

(D.W. of Orangevale, CA. 2000-10-07)
What is the circumference of the Earth at the equator?
(D.N. of Grass Valley, CA. 2000-10-09)
What is the radius of the Earth?

The irregularities of the Earth are charted with respect to a perfect ellipsoid whose dimensions were precisely defined (not measured) once and for all in 1980, by the IUGG (International Union of Geodesy and Geophysics). The equatorial radius of that ellipsoid is exactly 6378137 meters, which makes the circumference at the equator equal to 40075016.685578486...m down to the nearest (ludicrous) nanometer. That's about 24901.46 statute miles (these are "land miles" of 1609.344 m; the circumference may also be expressed as 21638.7779 "nautical miles", the modern nautical mile being exactly 1852 m).

The conventional "radius of the Earth" is a unit defined to be 6371000 m. This is almost the radius of a sphere having the same volume as the reference ellipsoid (6371000.79 m) or the radius of a sphere with the same area as the ellipsoid (6371007.181 m).

(Michael of Nashville, TN. 2000-10-03)   [2012 update]
Are there any units longer than a lightyear, or shorter than an ångström?

A list of extreme units of length that have actually been used, largest first:

Big ones :

  •   gigaparsec (Gpc).  Over 3 thousand million light-years.
  •   hubble.  A thousand million light-years, by definition,
  •   megaparsec (Mpc).
  •   kiloparsec (kpc).
  •   parsec (pc)   =  3.261563378 light-years.
  •   light-year  (exactly 9460730472580800 m).

The parsec (pc) is actually a very specific  (irrational)  multiple of the astronomical unit (au) since it's defined as the radius of a circle for which an arc of one second has a length of one astronomical unit (au).  In other words, a parsec is exactly 648000/p au (about 206265 au).

The astronomical unit (au) was once defined as the radius of the circular trajectory of a tiny mass orbiting an hypothetically isolated Sun with a period equal to a  Gaussian year.  Since August 2012, the astronomical unit is simply defined to be  149597870700 m  (exactly).  The mean distance between the Earth and the Sun was about  1.00000003641 au  at  J2000.0.
A light-year is about 63241.07708 au...  In 1911, the Swedish astronomer  Carl Ludwig Charlier (1862-1934) proposed a unit of stellar distances worth one million au  (about 15.8125  light-years)  which is nearly twice the distance to Sirius.  This never caught on.

The radius of the observable Universe itself is about 4 Gpc, so there is no need for units larger than the gigaparsec.

Small ones :

  •   ångström  (Å).  10-10 m   =   0.1 nm.
  •   picometer (pm).  1/100 of an ångström.
      Formerly known as a  micromicronbicron,  or  stigma.
  •   femtometer  or  fermi  (fm).  1/100000 ångström.
  •   microångström.  1/1000000 ångström.

Only the picometer (pm) and femtometer (fm) are official SI units.  The ångström and its submultiples aren't.

The prevalent US spelling has been used in the above, but the British spelling seems to be gaining ground for the "metre" and the various standard multiples of the "metre": "picometre", "femtometre", etc.  This happens to be closer to the original French spelling: "mètre", "kilomètre", "centimètre", "millimètre", etc.

Well below all of these is a truly minuscule "unit", the  Planck length,  which is about 1.616 10-35 m and describes a scale at which space itself is thought to lack any kind of  smoothness.  At the Planck scale, the very concept of length measurement becomes meaningless.


Surface Area

 One acre is 
 10 square chains. robster (2001-04-15)   10 sq. chains to the acre.
How many square inches in one acre?

An acre [Greek agros, field] is precisely 1/10 of a square furlong.  A furlong being 660 feet, a square furlong is 6602 = 435600 square feet and an acre is 43560 square feet.  There are 122 = 144 square inches in a square foot, so an acre is 43560 times 144 square inches, or exactly 6272640 square inches.

1 acre   =   0.1 fur2   =   43560 ft2   =   4046.8564224 m2

A (Gunter) chain is 66 ft (1/10 of a furlong).  The chain is itself divided into 100  Gunter links  (each of those is  0.001 fur,  7.92 in  or  20.1168 cm).  An acre is thus the area of a rectangle whose length is one furlong and whose width is one chain.

Historically, the relation is reversed:  The furlong ["furrow-long"] was a basic unit so strongly favored by the Tudors that they redefined the mile so that it would be exactly 8 furlongs.  This statute mile of 8 furlongs or 5280 ft thus displaced the previous London mile of 5000 ft, which had a definition similar to that of the Roman mile of 1000 strides (double-paces) of 5 Roman feet each.  The acre was thus defined to be 1/10 of a square furlong well before Edmund Gunter introduced the chain (in 1620) as the "width" of an acre.  Gunter's invention of the chain [divided into 100 links of exactly 7.92 inches] actually made it much easier to work out land areas expressed in acres.

A Gunter chain is also 4 poles.  Nowadays, a pole is an odd unit of exactly 16½ feet.  However, it is a much older unit which was defined as exactly 15 Saxon feet (also called "Drusus feet").  A furlong was exactly 600 Saxon feet or 40 poles (a Saxon foot was thus exactly 11/10 of a modern foot), which made a lot more sense in the old days.
      So, the original furlong was the Saxon equivalent of the ancient Greek stadion, which was similarly divided into 600 feet (the length of a Greek foot varied from one city-state to the next).  On the other hand, the related Roman stadium was 1/8 of a Roman mile, which may explain why the Tudors wanted 8 furlongs to their [statute] mile...

A lot of the bizarre conversion factors which are now floating around were once perfectly sensible.  The way to (numerical) hell is paved with good intentions...

Sulung  =  4 yokes (of land)   |   Hide   |   Carucate   |   Virgate   |   Oxgang (Bovate)


Volume, Capacity

( Lacy of Fort Walton Beach, FL. 2000-12-03)
Why is the abbreviation for liter "L" instead of [a lowercase] "l" ?

This is the only metric symbol which you may choose to capitalize or not.

You also have a choice between the US spelling and the British one: "Litre" is becoming acceptable in US English the same way "metre" is gaining ground as a favored (not "favoured") spelling for the SI unit of length. Not so with the original French spellings of mass units like "gramme" and "kilogramme", which remain confined to British English.

For all other metric units, the symbol is capitalized if, and only if, it has been named in honor of a person, whereas the unit name is never capitalized: V for volt, Hz for hertz, A for ampere, E for erlang..., but m for meter and g for gram because these two were not named after anybody!

Up until a few years ago, the recommendation was indeed a lowercase "l" for liter, according to the common rule, but cursive script became commonly used to make a clear distinction between a lowercase "L" and the numeral "1". When typing, a cursive "L" may not be an option and a capital "L" became acceptable.

The lowercase symbol was the only symbol adopted by the CIPM in 1879 and this was confirmed in 1948, by the 9th CGPM.  In 1979 however, Resolution 6 of the 16th CGPM recognized that both "L" and"l" should be accepted until actual practice could be monitored by the CIPM, so the 18th CGPM could rule further... In 1990, on the recommendation of the CIPM, the 18th CGPM declined to do so. 

(Joan of Norwell, MA. 2000-11-05)
What are the formulas for changing ounces or teaspoons into drops?
(T.S. of Clarksboro, NJ. 2001-01-25)
How many drops are in a milliliter?
Note : Due to the two "li" syllables, the incorrect spelling "mililiter" is more common than the wrong spelling "milimeter".  The standard SI prefix is "milli", so it's "milliliter", not "mililiter"... This note should make search engines deliver this page to anybody with a "mililiter" query who may be surprised to have so few pages to choose from!  We apologize for quoting the wrong spelling "mililiter" 4 times.   Just a joke!

In either the US (Winchester) or the UK (Imperial) system of liquid measures, a drop is another name for a minim and there are 480 of these in a fluid ounce.  Thus, if you have a volume in ounces, multiply by 480 to have the number of drops in it.

However, since the US and UK ounces are slightly different, a  drop  is about  0.0616 cc  in the US and only  0.0592 cc  in the UK.

The  metric drop  is exactly 0.05 cc.  Nowadays, this is the conventional value worldwide:  20 metric drops to a cubic centimeter  (or milliliter).

Similar distinctions hold for teaspoons :  A teaspoon is 1/6 of a fl oz (about 4.929 cc in the US and 4.7355 cc in the UK).  So, there are exactly 80 drops in a teaspoon (in either the Imperial or the Winchester system).

The metric teaspoon is slightly larger (5 cc) and the metric drop slightly smaller (0.05 cc) than the nonmetric counterparts, so there are exactly 100 metric drops in a metric teaspoon.

In a cubic centimeter or milliliter (cc, ml, or mL), there are exactly 20 metric drops and about 16 Winchester drops or 17 Imperial drops (more precise values being 16.23 and 16.89 respectively).

Note that all of the above are conventional values, which are only loosely related to the results you would actually get by using a thin dropper. So, don't be disappointed at the lack of "accuracy" if you do.

dbsafe (2001-06-21)
How do I convert milliliters into ounces?

Roughly speaking, divide a number of milliliter by about 30 to express that volume in fluid ounces (fl oz).  For example, 300 mL is about 10 fl oz.

Actually, the fluid ounce has different values in the Winchester (US) system and in the Imperial (UK) system.  The US ounce is about 4% larger than the British ounce  (the ratio is  1.04084273078623608419542947895884...);  about  29.6 mL  (29.6 cc)  to the  US fl oz  and  28.4 mL  to the  UK fl oz.

More precisely:

There are  exactly  29.5735295625 milliliters in a US ounce.  In the US, the Winchester system is used and the basic unit of capacity for fluids is the US gallon, defined to be exactly 231 cubic inches. Since 1959, the inch has been defined to be exactly 2.54 cm, and the number of milliters in a cubic inch is thus 2.543=16.387064. Now, there are 128 US ounces in a US gallon, so the number of milliliters in a US ounce is exactly (i.e., legally) 231/128 multiplied by 16.387064, which is the number advertised above.

There are  exactly  28.4130625 milliliters in a UK ounce (Imperial fl oz).
The British Imperial gallon was first introduced in 1824 as the volume occupied by 10 pounds of water at 62°F.  Unlike the US gallon, it is divided into 160 fluid ounces.  (Since there are also 160 ounces of mass in 10 avoirdupois pounds, this equated a fluid ounce with the volume of one  avoirdupois  ounce of water at  62°F.)  The Imperial gallon was later redefined in metric terms as 4.54609 L, making the number of milliliters in a fluid ounce exactly equal to 4546.09/160 = 28.4130625, as advertised.

A nice metrological opportunity was missed by British lawmakers in 1985 when they enacted a final metric equivalence for the Imperial gallon.  From 1976 to 1985, the official British equivalence had been  4.546092 L  to the gallon.  For a while, the Canadians had already been using a rounded-down equivalent of  4.54609 L,  on which the British decided to align themselves.  If both countries had chosen the nearest multiple of 16  (namely, 4.54608 L)  the Imperial ounce would be exactly  28.413 mL  for all eternity.  We wouldn't be stuck with four extra decimals and a ludicrous conversion factor that nobody will ever care to memorize.  Too bad.

Since 1963, the  Imperial gallon  has ben defined purely in metric terms.  The 1963 definition merely translate a metrological renovation of the 1824 naive definition  (volume of 10 pounds of water at  62°F )  according to which an Imperial gallon ought to be equal to the   space occupied by 10 lb of distilled water of density 0.998859 g/mL, weighed in air of density 0.001217 g/mL against weights of density 8.136 g/mL.

  • 4.545964591 L   from 1963 to 1976.
  • 4.546092 L    from 1976 to 1985.
  • 4.54609 L     since 1985.
  • 4.53608 L     future reform (?) to enforce:   1 fl oz   =   28.413 mL

( A. B. of Saint George, UT. 2000-05-02)   5 Different Gallons...
How many milliliters [ml or mL] in a gallon?

The US gallon is the  Winchester gallon, now defined as exactly equal to 231 cubic inches  (this odd value comes from rounding up the volume of a cylindrical measure 7 inches in diameter and 6 inches in height, which dates back to the days of the Magna Carta).  Since 1959, the inch is exactly 25.4 mm. Therefore, there are  exactly  3785.411784 ml  in a US gallon.

If the British Gallon is meant, the answer is 4546.09 ml, also an exact value according to the 1985 British "Weights and Measures Act" (in 1963, the British Parliament had decided to redefine all British units in metric terms).  There are about 277.42 cubic inches in this modern  Imperial gallon.

Originally (in 1819), the Imperial gallon was meant to be the volume occupied by 10 pounds of water at 62°F.  It was thus intermediate in value between the two British units it replaced in 1824, namely the  corn gallon  of 272¼ cubic inches (4461.378174 ml) and the  ale gallon  of 282 cubic inches (4621.152048 ml).  The old British  wine gallon  of 231 cu in survives as the US gallon (see above).

Finally, a US dry gallon is defined as 1/8 of a US bushel (or Winchester bushel, see below) and is thus exactly equal to 268.8025 cu in (4404.88377086 ml).  This unit was once known in England as the Winchester corn gallon.

(Gérard Michon. 2000-11-2)     Bushels and Gallons
A US bushel (bu) is defined to be exactly 2150.42 cubic inches.
How many bushels in a cylindrical container 74 inches in diameter and 50 inches deep?  Explain the "curious" numerical result...

With ludicrous precision:  100.000007969708869510499316219846+ bu.
There are very nearly 100 bushels in such a container! Here's why:

The US system of capacity is based on the Winchester system whose two basic units are the gallon for liquids and the bushel for dry goods.

The ancient Celtic city of Winchester was once an important Roman community, and it became the capital of England in the 9th century, when the kings of Wessex ruled the country.  It seems that the Winchester bushel was originally equivalent to  4 Roman modii  (or 4/3 of a Roman cubic foot).

On the other hand, there does not seem to be any direct link between Roman measures and the Winchester gallon for liquids.
      In the Roman system, the congius was the basis for liquid measures; There were 8 congii to the amphora (defined as precisely one Roman cubic foot) and the culleus of 20 amphorae was the largest liquid unit.  For dry goods, the basic unit was the sextarius (so named because it was 1/6 of a congius).  The modius ("peck" » 8.8 L) of 16 sextarii was the largest dry measure unit.  In other words, there were 3 modii to the amphora, but the modius was not used at all for liquids.  Unlike larger units, the submultiples of the sextarius were used for both liquids and dry goods:  hemina (1/2 of a sextarius), quartarius (1/4), cyathus (1/12), cochlear ("spoonful"; 48 cochlearia to the sextarius).  Note that the Roman talent was the mass of an amphora of water and was divided into 80 librae (Roman pounds).

Henry VII [Tudor] reigned from 1485 to 1509.  In 1495, the Winchester bushel was legally defined as the capacity of actual standard bushels bearing his seal and kept at the Exchequer.  In 1696, these were measured to be about 2145.6 cubic inches, under the supervision of members of the British House of Commons who were discussing some excise duty on malt.  It was then suggested that the bushel itself be defined as a simple circular measure roughly equivalent to this.

This was enacted in 1701 (during the reign of William III of Orange)  when the Winchester bushel was legally redefined, under the name of corn bushel, as the capacity of "any round measure with a plain and even bottom, being 18½ inches wide throughout and 8 inches deep" (there would have been exactly 100 of these in the above container).  Sometime before 1795,  this volume was rounded down from  2150.420171... to exactly 2150.42 cubic inches,  which is how the so-called malt bushel was normally defined.  (I couldn't determine the exact point at which the older cylindrical definition of this bushel faded from view.  Please, tell me whatever you may know.  Thanks.)

The same thing happened to the US gallon, which is a descendant of the old Winchester wine gallon, a cylindrical measure from the days of the Magna Carta: 7" in diameter and 6" deep, or about 230.90706 cubic inches.  This capacity was statutorily rounded to 231 cubic inches in 1707, by  Anne Stuart  (it was thus once known as the  Queen Anne wine gallon).

Both Winchester units are thus tied to the inch and have, in effect, been redefined every time the inch was.  The current units of capacity are based on the 1959 international inch, which is now forever defined in metric terms (1" = 25.4 mm).

The US adopted the Winchester system for capacities in 1836, using the above equivalences. The British had adopted the competing Imperial system in 1824, on the totally different basis of an Imperial gallon then introduced as the volume occupied by 10 lb of water at 62°F  (later redefined in metric terms, as exactly equal to  4.54609 L)  and an  Imperial bushel  equal to exactly  8  of these gallons  (36.36872 L).

Nowadays, agricultural goods are no longer sold by volume.  Instead, weight equivalents of the bushel are used for various commodities: 60 lb to the US bushel for wheat and potatoes, 56 lb for rye, 53 lb for tomatoes, 48 lb for barley, 32 lb for oats, 20 lb for spinach, etc.

jlj3394 (2001-01-15)   Kegs and Barrels
How many 12 oz beers are in a keg?

The US government defines (for tax purposes and such) a barrel of beer as exactly equal to 31 US gallons (these are Winchester gallons of exactly 231 cubic inches, not the Imperial gallons used in the UK).

The US brewing industry calls a  [full]  keg a quantity of beer equal to half of such a barrel, namely 15.5 gallons (half a keg is called a "pony-keg" and equals 7.75 US gallons). A US gallon being divided into 128 oz, the above implies that a keg equals 1984 oz, or 165 and 1/3 times a "12 oz beer".

The 12 oz size (can or bottle) is most commonly sold in "packs" of 6 or 12 ("6 pack" or "12 pack"), but retail packs of 18, 20, 24 or 30 are also widely available. Traditionally, a case of beer consists of 24 cans or 24 bottles. There are thus almost 7 cases of beer (which would be 168 cans) to the keg.

The above (modern) US "barrel of beer" has nothing to do with the international barrel (of oil), which is used to measure crude oil and is defined to be exactly equal to 42 US gallons, or 9702 cubic inches (158.987294928 liters). This unit is best abbreviated "bo" (barrel of oil) to distinguish it from the many other types of "barrels" which are all abbreviated "bbl". It is acceptable to use metric prefixes with the symbol "bo", but not with "bbl", which is far too ambiguous...
Besides the international barrel (42 US gallons) and the above US barrel of beer (31 US gallons), there's also a US barrel of wine (most commonly 31.5 US gallon) and a "barrel bulk" of 5 cubic feet. The US "dry barrel" is 7056 cubic inches; it was so defined in 1912 as the US "apple barrel" (it's thus almost exactly equal to 105 "dry quarts", or 105/32 US bushels). The "barrel of cranberries" is 5826 cubic inches. All this covers only modern US usage... The Imperial system formerly used in the UK included a larger barrel ("dry barrel" or "barrel of beer") of 36 Imperial gallons (163.65924 L).
Worse, the "barrel" is also used as a measure of mass, which comes in several flavors as well: The "barrel of cement" (4 bags) is 376 lb (376 avoirdupois pounds, or about 170.55 kg). The "barrel" used in the US for pork, beef or fish is 200 lb (90.718474 kg), whereas a "barrel of flour" is only 196 lb (88.90410452 kg)...


Mass, "Weight"

(C. B. of Philadelphia, PA. 2000-10-25)
Is there a [unit of] measurement smaller than a milligram?

Here's a list of the smaller official units of mass in "concrete" terms:

  • gram (g):  A paper clip.
  • milligram (mg):  Cubic millimeter of water. Mass of a typical ant.
  • microgram or gamma:  Dust mite (dermatophagoides pteronyssinus).
  • nanogram (ng)
  • picogram (pg):  A typical bacterium (Escherichia coli).
  • femtogram (fg)
  • attogram (ag):  A typical virus, or 20 prions.
  • zeptogram (zg, 10-21g):  3 gold atoms, or 33 water molecules.
  • yoctogram (yg, 10-24g):  60% of a hydrogen atom.
  • unnamed (10-27g):  110% of an electron.
  • unnamed (10-30g):  561 eV/c2
  • unnamed (10-33g):  40% of a neutrino.

The zeptogram and yoctogram have been officially recognized by the CGPM since 1991.  An atom of hydrogen is about 1.66 yg.

An electron is about 0.00091 yg.  This is roughly equal to the next unit down the list (namely, yg/1000 or 10-27g), which doesn't yet have an official name.

(2012-11-07)   Solar mass :   1.98855(24)  1030 kg     [ CODATA 2010 ]
The unit of mass in the  astronomical system of units.

The gravitational influence of a celestial body is measured by its  standard gravitational parameter, equal to the product of its mass into  Newton's universal constant of gravity.  Both factors may only be known at a modest level of precision in SI units, but the product can be determined with  astronomical  precision  (literally).  For example:

Heliocentric gravitational constant   1.32712440042(8)  1020  m3/s2
Geocentric gravitational constant 3.986004415(8)  1014  m3/s2

Thus, the ratio of the mass of the Sun to that of the Earth  (atmosphere included)  is known with excellent precision, namely:  332946.0438(8).

Although the Sun loses  millions of tons per second, it will take more than  2000 years  for this to affect the least significant digit of that last ratio.  This is good enough to use the changing mass of the Sun as a very practical unit which allows the mass of large celestial bodies in the solar system to be expressed with much more precision than SI units  (kilograms)  would allow, using the values of their  relative  gravitational constants, as defined above.

Sun  11
Jupiter  9.5479194 (74)   10-4 1047.3486(8)
  Earth + Moon    3.040432685(9)  10-6 328900.5558(11)
Earth  3.003489661(7)  10-6   332946.0438 (8)  
Moon  3.69430242(46)  10-8 27068710 (34)

The Earth is  81.30059(1)   times as massive as the Moon.

Astrodynamic Constants  (NASA / JPL)
Basic Astronomical Data for the Sun  (BADS)  by  Eric Mamajek.

("Biker" of Jerome, ID. 2000-10-09)
What is a slug, in the [engineering] weight measurement system?

The slug is a unit of mass.  The word was coined in a 1902 textbook by the British physicist A.M. Worthington to designate the British engineer's unit of mass, which appeared in engineering calculations late in the 19th century.

The slug is defined as the mass which would accelerate at a rate of 1 ft/s2 under a force of one pound-force (lbf).  Since 1 lbf is the force exerted on a mass of one pound by a standard gravitational field (of exactly 9.80665 meters per square second), a slug is thus exactly equal to 196133/6096 pounds (about 32.1740485564 lb or 14.593902937206 kg).

It's worth making a few technical points about this:

  • The slug is the unit of mass in a coherent system called either "British engineering system" or "English gravitational system".  On the other hand, the  Imperial  (formerly "English") unit of mass is the pound (lb), which is now defined in metric terms (0.45359237 kg exactly). 
  • The "metric equivalent" of the slug is the  hyl  of exactly  9.80665 kg  which is the unit of mass of the so-called "metric-technical system".  The  hyl  is also called "metric slug" or designated by the German acronym  TME  (Technische Mass Einheit ).  A mass of one hyl gets accelerated at a rate of one meter per square second by a force of one kilogram-force (namely, 9.80665 N). 
  • The SI unit of mass is the kilogram, not the gram or the hyl. 
  • Both the pound and the slug are units of mass.  The latter weighs about 32 times as much as the former, even on the surface of the moon.  On the moon, however the weight of a pound-mass (lb or lbm) is only about one sixth of a pound-force (lbf).

(2007-05-13)   Extant customary units of mass, in the electronic age.
What are the units of mass available on modern electronic balances?

The customary units listed below are mostly kept alive by gold traders.

A common feature of electronic analytical and/or precision balances is the ability to use various customary units of mass.  Copying each other over the years (often misspelling "baht" and/or "mesghal") manufacturers have picked from the following limited catalog of units, which caters to all international traders.

In East Asia, the  catty  is to the  tael  (TL)  what the pound  (lb)  is to the  ounce  (oz).  There are  16 taels  to the  catty...  The  Taiwanese  tael  (37½ g)  thus corresponds to a  catty  of  600 g,  whereas the "tael of Singapore"  (defined as 1/12 lb or 4/3 oz)  corresponds to a  catty  of 4/3 lb  (about 604.79 g).

1000 grams (g) to the kilogram (kg).
7000 grains (gn) to the avoidupois pound (lb).
Note that the abbreviation  "gr"  is best shunned
(as it could stand for either grams or grains).
NameSymbolValue in grams (g)Definition Usage
(kan)kw 3750 g      (since 1891) 1000 momJapan
poundlb 453.59237 g  (1959 treaty) 7000 gnavoirdupois
(troy pound)lbt 373.2417216 g   (12 ozt) 5760 gnillegal 1879
(newton-mass)"N" 101.97162129779282... 1 N/Gdubious (*)
tael  (S)
tael  (M)
tael  (C)
tael  (T)
cay, luong
tael  (H)
TL   37.79936416666666...
  37.429 g
1/12 lb
37½ g
10 chi
tael troy
troy ounceozt   31.1034768 480 gnprecious metals
ounceoz   28.349523125 437½ gnavoirdupois
baht, tical    15.244 g   (or 15.2 g) (235¼ gn)Thailand
tolatol   11.6638038 180 gnIndia
mesghalMs     4.6083       (24 nukhuds) 1 g / 0.217Iran
(mithgál)      4.25 moderngold dinar
dram (**)dr.   3     3.8879346  (3 scruples) 60 gnapothecaries'
mom     3.75 0.001 kwJapan / pearls
South Korea
       2.975 0.7 mithgálsilver dirham
drachm (**)dm  (dr)     1.7718451953125 1/16 ozUS ammo
pennyweightdwt     1.55517384 24 gntroy (1/20 ozt)
gramg  (gr)     1 0.001 kgSI
caratct     0.2 g       (CGPM, 1907) 200 mgprecious stones
graingn, gr     0.06479891 1 lb/7000troy & avdp
milligrammg     0.001 0.001 gSI

(*)  A "newton-mass" unit was (improperly) introduced by some instrument makers as the mass  (about 102 g)  on which a standard gravitational field of 9.80665 m/s2  would exert a force of exactly  1 N  (1 newton).  This is yet another offspring of the ongoing confusion between mass and weight (the latter being the force exerted by gravity on a given mass).
(**)   The apothecaries' dram  (dr)  of 60 grains is still commonly available on modern electronic precision balances.  It's about twice as big as the avdp  drachm  (dm)  of 1/16 oz  which is still used in the US for loading ammunition  (best done in grains).  The difference in spelling can't be relied upon,  as both spellings have been used in either system and the symbol dr can denote either unit  (I recommend dm for the avdp drachm).  The clear distinction between the two units made in some popular conversion tables  has been historically butchered in practice.  Unfortunately.

Ohaus   |   Shimadzu   |   A&D   |   Mettler Toledo   |   Sartorius (Acculab)

(2018-12-06)   Final British systems of weights:  Avoirdupois  and  Troy.
The  troy ounce  (ozt)  still dominates the  precious metals trade.

The word  troy  is so strongly associated with the gold trade that it's now employed  (as a  suffix)  for the unrelated Asian units used to trade gold in Hong-Kong and elsewhere  (manufacturers of electronic scales use the term  tael of Hong-Kong  for the  tael troy):

100  candareen troy   =   10  mace troy   =   1 tael troy   =   37.429 g

The proper British  troy system  itself is fully compatible with the deprecated  apothecaries' weight  system of  pharmacists,  which has been illegal for trade in the UK since 1985.  This is to say that units with the same name  (pound, ounce, grain)  have identical values in both system.  However,  there are units which are used in only one of those two systems:

Units of the Troy System and Apothecaries' Weights
NameSymbolTroyApothecaries'GrainsGrams (g)
Troy ton29400 ozt  =  2450 lbt  =  2016 lb 14112000914442.21792
Error poundNo16 ozt  7680497.6556288
Poundlbt12 ozt12 ounces5760373.2417216
Ounceozt20 dwt8 drams48031.1034768
Dramdr 3 scruples603.8879346
Pennyweightdwt OK 241.55517384
Scruple OK201.2959782
Graingn  (or gr)OK OK10.06479891

Apothecaries' weight was used for dispensing medicine in the UK until the Medical Act of 1858 adopted the  avoirdupois  system,  which could then be used for everything but precious metals,  pearls and gemstones.

Units of the Avoirdupois System of Weights  (avdp)
(Long) tont20 cwt156800001016.0469088 kg
(Short) tont (US)20 cwt (US)2000 lb907.18474 kg
Hundredweightcwt4 qr78400050.80234544 kg
Short hundredweightcwt (US)100 lb70000045.359237 kg
Quarterqr2 st19600012.70058636 kg
Stonest14 lb980006.35029318 kg
Poundlb16 oz7000453.59237 g
Ounceoz16 dm437.528.349523125 g
Drachmdm  (dr)OK27.343751.7718451953125 g
Graingn  (or gr) OK164.79891 mg

In the US,  the  stone  (14 lb)  and its multiples never caught on and larger units are  not  multiple of that.  Thus,  Americans use  short  versions of the  hundredweight  (100 lb instead of 112 lb)  and the  ton  (2000 lb instead of 2240 lb).  Whenever there is a risk of confusion,  the proper qualifiers  short  and  long  should be used for those units.  Be aware that  many other  ton  units are in use!

The  grain  unit of  64.79891 mg  is shared by all of the above systems.

Pound   |   Avoirdupois system (etymology)   |   Troy weight   |   Apothecaries' system
Mendenhall Order (US, 1893)   |   International yard and pound (1959)

(2007-06-03)   Royal French weights:  18827.15 grains  to the  kg.
The ancient  livre de Charlemagne  and the  poids de marc  system.

From the early definitions of the kilogram survives only an exact equivalence between the old French units of mass poids de marc and the metric system; there are exactly 18827.15 French grains  to the kilogram.

Once it had been realized that a definition of the kilogram as the mass of a cubic decimeter of water was not satisfying  (by the metrological standards of the late 18th century)  the kilogram was evaluated using the best system of weights then available.  In pre-revolutionary France, that was based on a famous artifact known as the pile de Charlemagne, which is still preserved in the  Musée National des Techniques  in Paris, France.

In spite of its name, the "Pile de Charlemagne" was built in the 14th century—half a millenium after the days of Emperor Charlemagne.  It consists of 13 copper weights in the form of truncated cones (larger base on top).  Except for the smallest one, all of those are hollow, so that the next smaller weight may fit in it snugly.  The largest weight has a handle and a lid and serves as a box for the whole thing, which stands at a height of about 9cm, with a top diameter of about 15.5cm, and a lower diameter around 14cm (for a volume of about 1.54L, and a mass of about 12.2376 kg).  The nominal mass of this revered standard was exactly 50 marcs (of 4608 French grains each).  The royal French livre  (better known as  livre poids de marc )  was once defined as 1/25 of the "pile de Charlemagne" mass.

When the kilogram (then called the grave) was first defined (on August 1, 1793), it was equated to  18841 grains  of the above  poids de marc  system, from a single measurement by Lavoisier and Haüy Jean-Valentin Fabbroni 
 1752-1822  Early in 1799, an accurate equivalence of  18827.15 grains  to the kilogram was established...

That last measurement was due to the French academician  Louis Lefèvre-Gineau (1751-1829)  and the Italian engineer  Jean-Valentin Fabbroni (1752-1822)  [elected to France's  Corps Législatif  in 1809].  Defining the kilogram as the mass of a cubic decimeter of water at 4°C  (close to the densest point at 3.984°C)  they weighed a hollow brass cylinder of known dimensions, first in the air, then in water at 4°C.

The new determination was enacted on May 30, 1799, and it became the  final  legal equivalence between the kilogram and the "old" French units.

Because of that, the metric equivalent of the French  grain  no longer depends on the actual mass of the "pile de Charlemagne".  However, we may remark that, at 9216 grains  to the  livre , the  Pile de Charlemagne  has a nominal mass of  230400 / 18827.15  or about 12.237646165...kg  whereas its  actual  mass has been measured to be 12.2376429 kg.  Interestingly, the 0.27 ppm difference translates into  0.005 grains/kg,  which goes to show that the above 18th century determination  (which settled, once and for all, the conversion factor between grains and kilograms)  was indeed  fully accurate.

Since the newer equivalence was quite different from Lavoisier's original one  (13.85 grains is about 3/4 of a gram)  the standard weights that had been sent to all departmental  chef-lieus  had to be recalled.  New ones were made.

The same thing did not happen for prototypes of the meter because the difference with revised standards of length was considered acceptable.

The old French system of 18 onces to the livre  had been introduced in the wake of Charlemagne's monetary reform.  The  once  was understood to be  exactly  the same as the Roman  uncia  but there were  18  of those to Charlemagne's  livre  (French pound,  poids de marc )  as opposed to  12 unciae  to the  libra  (Roman pound).  So, a French pound was  exactly  1½  Roman pounds.

The above thus provides a  paper trail  to what may be construed as a "legal" value of the ancient Roman pound in metric terms, namely:

1 Roman pound (libra)   =   12 onces  (of 512 grains)   =   0.326337231... kg

(J. W. of Tustin, CA. 2001-02-07)   Biblical Units
How many pounds was a talent?   How many ounces was a shekel?

A talent was the mass of a cubic foot of water.  The exact value of the talent thus depended on what foot was in use in a specific part of the world at a certain period in history.  If there was such a thing as a modern  Imperial talent  (based on water at 62°F)  it would be about 62.288 lb (or 28.25 kg).

The Roman talent was also defined as 80 Roman pounds ("librae", plural of "libra").  The above value of the libra, from the days of Charlemagne, makes the Roman talent equal to about  26.107 kg.

Incidentally, this would imply a value of about 0.2969 m for the Roman foot  (water at 62°F has a density of 10 lb per Imperial gallon).  For some obscure reason, a  foot whose length is derived backwards from a given value of the talent is called a geometric foot.

The ancient Sumerian talent is estimated at about  28.8 kg  (about 63.5 lb) from the mass of surviving standard weights (basalt statuettes in the form of sleeping ducks).  Outside of Rome, the talent was normally divided into 60 minas; a  mina  (or maneh)  was thus roughly equal to a modern avoirdupois pound.

The shekel was always some submultiple of this mina:  The Babylonian shekel was 1/60 mina, the Phoenician shekel was 1/25 mina, the Egyptian shekel was 1/100 mina, whereas the "modern" Palestinian or Syrian shekel is 1/50 of a mina.

Solomon's mina of gold  (1 Kings 10:17) was divided into 100 units (unnamed in the Hebrew text of 2 Chr. 9:16) not necessarily related to the Biblical shekel of the sanctuary  or  holy shekel  (cf.  bishekel hakodesh)  whose value ought to be determined by the last words of  Ezekiel 45:12.  Unfortunately, Bible scholars have been advocating at least two contradictory renditions of that verse, namely:

  • 50 shekels to a mina (Septuagint, according to Walther Zimmerli):
    "[...] 5 shekels are to be 5, and 10 shekels are to be 10, and 50 shekels are to amount to a mina with you." 
  • 60 shekels to a mina (King James and other English versions, also supported by Rabbi Nosson Scherman, in the  Stone Edition Tanach): "[...] 20 shekels, 25 shekels, and 15 shekels shall be your mina." 

The latter may have exhorted traders to check their minas against smaller standard weights...  If you know for sure, please tell me.

Beqa' Weight (half a shekel? = 5.99 g)  7th-6th Century BC

ginapa (2001-06-11)
How many pounds in 1 ton?

There are many different kinds of tons.  In the US, you're most likely to encounter the short ton (2000 lb, or about 907.185 kg)  unless you're primarily concerned with ships, for which the displacement ton and the gross ton are in fact units of mass both equivalent to the British long ton of 160 stones (2240 lb, or about 1016 kg).  The long ton is retained in this international context because it's almost exactly equal to the mass of a cubic meter of seawater.  This is a prime example of crossbreeding between the metric and Imperial systems.

Another example of interbreeding between the metric system and the Imperial system (and the troy system) involves a much smaller "ton", the assay ton, which is slighly more than an ounce.  It's defined to make 1 milligram per assay ton equivalent to one troy ounce (ozt) per ton.  There are 2 or 3 kinds of assay tons, depending on which reference "ton" is used.  The most common one seems to be the short assay ton of 29.1666... g, which corresponds to the ton of  2000 lb.
A troy ounce (ozt) per ton is
a milligram (mg) per assay ton
Ton, in lbAssay ton, in g
  short ton    2000 lb    175 / 6   29.16666... g  
long ton  2240 lb 98 / 3 32.66666... g
troy ton  2016 lb 147 / 5  29.4 g

Other types of tons include the very important metric ton (better spelled tonne, which corresponds to 1000 kg or about 2204.62 lb) and the totally unimportant and unused troy ton of 144 stones (2450 lbt = 2016 lb = 914.44221792 kg).

The pound is understood to be the common avoirdupois pound ("lb" or "avdp lb") of exactly 0.45359237 kg (a 1959 international statute now defines the pound in metric terms).  For the record, the troy pound (lbt) was officially abandoned on January 6, 1879  (175 lbt = 144 lb).  However, the troy ounce (ozt) is still widely used for precious metals.

As if this were not bad enough, a few units of volume are also called tons:  This includes, most notably, the international register ton of 100 cubic feet (2831.6846592 L).  Of lesser importance is the British water ton of (exactly) 224 Imperial gallons, which originally corresponded to the volume occupied by a long ton (2240 lb) of distilled water at 62°F, when the Imperial gallon was still defined in like terms as a "10 pound gallon".  (Under the modern definition of the Imperial gallon, in metric terms, the British water ton is exactly 1018.32416 L.) On the other hand, the unit variously called shipping ton, freight ton or marine ton is 40 cubic feet (1132.67386368 L), which happens to be equal to the so-called ton of timber (of 480 board feet).  There's also a fluid ton of 32 cubic feet (906.139090944 L), a corn ton of 32 bushels (which means exactly 1127.65024534016 L in the US and 1163.79904 L in the UK), and a British tun, spelled with a "U", of two pipes or 252 Imperial gallons (1145.61468 L).


 Explosion On 2001-10-26, Darren Finck wrote:
I just wanted to drop you a line to tell you that I enjoyed your treatise on the "ton"(s). [above]
For completeness, it would be interesting if you were to also mention and/or describe the origination/relation of the "refrigeration ton" and/or the "explosion ton" units.
Regards, Darren Finck

Thanks for the kind words, Darren.

First a general remark:  The adjective "extensive" qualifies (loosely speaking) physical quantities for which the measure of the whole is the sum of the measures of the parts.  Volume and mass are examples of  extensive  quantities  (pressure and temperature are not).  Choosing some "stuff" of reference, like water under normal conditions, establishes a "conversion factor" (coefficient of proportionality) between any pair of extensive quantities and/or the units which measure them.  New "practical" units may thus be created ad nauseam,  including many flavors of tons which correspond to various extensive properties of a ton of "stuff".

This is how some of the "tons" mentioned above as units of mass gave rise to units of volume (a volume of one ton being the volume occupied under standard conditions by a mass of one ton of water).  This is also how a unit of mass may become a unit of force (the corresponding weight in a standard gravitational field, equal to 9.80665 m/s). In particular, the ton of thrust is a unit of force equal to the standard weight of a metric ton/tonne, namely 9806.65 N.  [The newton (N) is the SI unit of force.  Applying for 1 second a force of 1 N to a mass of 1 kg, initially at rest, will make it move at a speed of 1 m/s.]

First Nuclear Test ('Trinity' Site) 
 July 16, 1945 at 5:29:45 am
 0.16 s after explosion (18.6 kilotons).   The ton unit pertaining to nuclear explosions is a unit of energy equal to 1000 000 000 thermochemical calories (of exactly 4.184 J) and is thus exactly equal to 4184 000 000 joules.  (The kiloton and megaton are a thousand and a million times as large.)
Video :   How G.I. Taylor (1886-1975) obtained the classified tonnage of the Trinity bomb from dimensional analysis.

Detonating 1000 kg of TNT  (227.134 g/mol)  yields only 64% of such a ton:

C7H5N3O6     ®     6 CO  +  5/2 H2  +  3/2 N2  +  C  +  608.8 kJ

The carbon (C) produced appears as black smoke.  Some residues may subsequently burn in air to give more energy  (393.51 kJ per mole of carbon, 241.826 kJ per mole of hydrogen gas, 282.98 kJ per mole of CO).  The total heat of combustion of TNT is thus about 3305 kJ/mol, which translates into 3½ of the above tons of energy for 1000 kg of TNT (227.13 g/mol)...  What's wrong?  Well, to optimize the energy of the initial blast, an oxidizer (ammonium nitrate = AN) must be added to TNT to form a balanced high explosive, called amatol.  The optimal proportion for a given total weight is 78.7% AN and 21.3% TNT, matching the stoichiometry of the following reaction.  (A slight excess of AN seems better for dynamic reasons, so the usual mix is 80/20.)

2 C7H5N3O6 + 21 NH4NO3   ®   47 H2O + 14 CO2 + 24 N2 + 9088.6 kJ

This yields 1.0174 tons of energy when 1000 kg of the mix are detonated, which justifies quantitatively the term "ton of TNT " commonly used for the above ton of energy, although "ton of amatol" would have been more proper...

Other types of "tons" are used to measure energy in a more peaceful context: Burning a ton of crude oil releases about 10 times as much energy as exploding a ton of TNT/amatol.  On the other hand, the best grade of coal (anthracite) is supposed to be about 30% less efficient than oil.

Burning pure carbon completely into carbon dioxide would release about 393.51 kJ/mol, which is more than 7800 cal/g (a mole of carbon is 12 g).  However, actual coal can be much less efficient; see below

This gave rise to two other "ton" units for measuring energy, the ton oil equivalent (toe) and the ton coal equivalent (tce): 1 tce = 0.7 toe.  Both refer to metric tons (1000 kg) but, unlike the ton of TNT, they are usually defined as round multiples of the IT calorie (International Steam Table calorie of exactly 4.1868 J instead of 4.184 J):

1 tce = 29 307 600 000 J       1 toe = 41 868 000 000 J

Natural gas is an important source of energy as well, so that the toe has also been given the following equivalences in term of gas quantities, using the different units of measurement preferred in various regions of the Globe (these values are, unfortunately, slightly incompatible with each other and with the above):

  • USA : 42900 cubic feet (about 1214.8 cubic meters).
  • Europe : 1270 cubic meters.
  • Japan : 0.855 metric tons of LNG ("Liquefied Natural Gas").
Standard Calorific Values :   Coal = 7000 cal/g.  Oil = 10000 cal/g.
Actual Calorific Values (CV):       [ NB:  1 cal/g = 1.8 Btu/lb ]
Brown coal = 2250 cal/g.  Firewood = 4300 cal/g (= 7740 Btu/lb).
Bituminous coal = 6000 cal/g.  Crude oil = 10800 cal/g.

Now, the ton of refrigeration or ton of cooling is a unit of power (which can't be compared with any of the above units of energy).  It was first defined as the power released by a ton (2000 lb) of water when it freezes in one day (86400 seconds) or, conversely, the power absorbed by a ton of ice which melts in a day.  This would be about 3502.6 W (watts), but the ton of cooling is now conventionally defined as exactly 12000 Btu/h (about 3516.852842 W), based on the rounded value of 144 Btu/lb for the latent heat of fusion of water.  In the United Sates, air conditioning units are now rated using the Btu of cooling, which is a unit of power simply equal to a Btu per hour (about 0.293 W, more precisely 0.2930710701722222...W).  The labeling of A/C units is in terms of thousands of Btu [per hour] (typically: 024, 030, 036, 042, 048, or 060), but betrays its origin in terms of tons of cooling (2, 2½, 3, 3½, 4, or 5 tons of refrigeration).

The electrical energy fed to the motor of an A/C unit may allow the transfer of a greater energy "uphill", from cold to hot.  The ratio of these two energies is called the coefficient of performance (COP), which is normally much more than 100%.  This would be clear if refrigeration and electrical powers were both expressed in the same units (W/W), but this fact is obscured in the US, where the so-called EER (Energy Efficiency Ratio) is used instead:  An EER of 10 means 10 Btu/h/W, (a COP of about 2.93 W/W, or 293%).  An EER of 15 is 439.6%.

Last, and probably least, we're told that the "ton" is also an informal British unit of speed equal to 100 mph (160.9344 km/h or 44.704 m/s).  [Colloquially, in the UK, a  ton  can be 100 times as large as any commonly understood unit.]

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