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Final Answers
© 2000-2005 Gérard P. Michon, Ph.D.

Measurements and Units

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)

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 Physics Lab).
Metrics the Right Way, at the Pacific Northwest National Laboratory.
ISO  |  BBC  |  The International System of Units, by Robert A. Nelson.

Traditional or Specialized Units

A Dictionary of Units of Measurement by Russ Rowlett (UNC at Chapel Hill).
A Dictionary of Measures by Frank Tapson (University of Exeter, UK).
Conversion Factors (Process Associates of America).  |  www.sizes.com
Some Ancient Measures  |  Old French Measurement Units
Anglo-Saxon Units  |  History of Measures by the late Livio C. Stecchini.
Convert-Me Converter   |   Online Conversion Converter

Metrication History & Controversies

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

Measurements and Units

(Calvin of Farina, IL. 2000-11-05)
What are all of the metric prefixes?

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)
1/100 000 centimilli (cm)
micro-m,u1/1000 000  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 are also shown (vendeka, xenna, xenno, vendeko) which we discuss elsewhere.  Please, tell us 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:
[...]  I should have given you credit [and/or posted a] link.  Sorry.

(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 are powers of 1024 (2 to the power of 10) instead.

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

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 up the situation by introducing a kilobinary system, in which we would no longer use kilobyte to designate 1024 bytes, but kibibyte (KiB). This proposal has failed so far;  we've never seen this in actual use!

Building on the accepted convention that "b" is for bit and "B" is for byte (8 bits), some individuals keep proposing the use a lowercase "k" for 1000 and an uppercase "K" for 1024.  For now, this must be rejected for several reasons:  The idea applies only to written text, cannot be generalized to other commonly used prefixes, and is in direct conflict with the current usage described above...  [Note also that "K" by itself denotes a kilobyte, kB, 1024 bytes or 8192 bits.] Within a specific technical document, ambiguities could conceivably be lifted by stating  first what is meant (e.g., "k=1000, K=1024") before using compound prefixes like:  kk (for 1000000), kK (for 1024000) or KK (for 1048576).

Warning:   1 kb/s  =  1.024 kbps

Be very aware that the binary exception only applies to multiples of the bit and 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, so that the latter should only be pronounced "kilo-bee-pee-ess" to avoid confusion with the former!

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

( 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, whereas a relative density is defined as the ratio of an absolute density to the absolute density of "water".  When top precision is called for, it would be important to specify what kind of "water" is meant.  For example, SMOW ("Standard Mean Ocean Water") at its densest point (around 3.98°C) has an absolute density of about 0.999975 g/cc.  However, the accepted conversion factor between "absolute" and "relative" density is 0.999972 g/cc !  This is one number which has acquired (unofficially at least) the 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."

Note:   The original batch of SMOW came from seawater collected by Harmon Craig on the equator at 180 degrees of longitude.  After distillation, it was enriched with heavy water to make the isotopic composition match what would be expected of undistilled seawater (distillation changes the isotopic composition, because lighter molecules are more volatile).  In 1961, Craig tied SMOW to the NBS-1 sample of meteoric water originally collected from the Potomac River by the National Bureau of Standards (now NIST).  For example, the ratio of Oxygen-18 to Oxygen-16 in SMOW was 0.8% higher than the corresponding ratio in NBS-1. This "actual" SMOW is all but exhausted, but water closely matching its isotopic composition has been made commercially available, since 1968, by the Vienna-based IAEA (International Atomic Energy Agency) under the name of VSMOW or "Vienna SMOW".

(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 : 6.022 141 99(47)´1023 per mole. [Here, the parenthesized 47 indicates an uncertainty whose standard deviation is 47 times the weight of the last decimal position shown.]

(J. M. of College Station, TX. 2001-02-11)
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 to this question would be "1000 calories" (or 1 kcal).

However, the historical definition of the calorie was dependent on the starting temperature and it's been deprecated.  Since 1935, the current (thermochemical) calorie has been defined as exactly 4.184 J. 

In 1956, a competing definition gave rise to a slightly different "calorie" unit:  The "IT calorie" is 4.1868 J  (IT or IST stands for "International [Steam] Table").  This conversion factor is consistent with the definition of the Btu (British thermal unit) adopted at the 5th International Conference on the Properties of Steam (London, July 1956), which equates 1 Btu/lb and 2326 J/kg (incidentally, a therm is 100000 Btu).  The Btu had an historical definition similar to that of the calorie: In 1876, it was defined as the energy required to raise the temperature of one pound (lb) of water by 1°F, from the point of maximum density [around 3.98°C].  All told, it's best to reserve the 1956 IST definition to the Btu (1 Btu is 1055.05585262 J, namely the ratio of the pound to the kilogram multiplied by 2326 J) and 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).
The fifteen degree calorie (also known as gram-calorie or "g-cal") is still defined as the energy which raises a gram of water from 14.5°C to 15.5°C.  It has been measured to be equal to 4.1855 J (with an uncertainty of 0.0005 J).

The energy which raises a kg of water by 1°C is thus "about 4185.5 joules".

cdw239 (2001-08-23)
What is the equation for converting horsepower 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 irrelevant. 

A horsepower (hp) is about 745.7 watts (W), but many metric countries use another closely related unit  [best abbreviated "ch"]  which is 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", see below). 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, which is thus the "conversion factor" to use to transform units of mass (like the pound, lb) into their common namesakes as units of force (pound-force, lbf):  1 lbf is (0.45359237)(9.80665), or 4.4482216152605 N exactly.  Multiply this by the length corresponding to 550 ft (exactly 167.64 m) and you have the equivalence of a horsepower in watts (since a watt "W" is simply a meter-newton per second), namely 1 hp = 745.69987158227022 W exactly.  There's (almost) no need to say that everybody usually rounds this up in the most obvious way:  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, cavalli vapore in Italian...

The French call it cheval-vapeur (ch) or simply 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.  A metric horsepower (ch) is thus (75)(9.80665), or exactly  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).

Adding to the confusion, a so-called electric horsepower has also been 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 value includes a centrifugal component due to the rotation of the Earth, whereas the value of the gravitational field at altitude zero is equal to 9.82025048(2) m/s2.  Such a larger value is relevant when tracking satellites outside the atmosphere in nonrotating coordinates.  It's simply the ratio of the Earth's gravitational constant (3.986004415(8) 1014 m3/s2 ) to the square of the conventional radius of the Earth (R = 6371000 m).


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

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

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

Yes, kinda.  Informally, a jiffy can be any short period of time (the etymology is unknown).  The term's been given several formal definitions in various contexts...

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 (also remembered for defining a Lewis acid as an acceptor of electrons).

In the quaint context of computer engineering, however, a jiffy may denote the period of the system's main clock (e.g., 10 ns for a 100 MHz clock).

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 is considered equal to 10 ms instead.

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

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

( hG / 2pc5 ) ½   »   5.39 ´ 10-44 s

(G is Newton's constant, h is Planck's constant, c is Einsteins' constant.)

(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 the 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:

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] ellapsed since "January 0.5 1900" [which means either 24:00 GMT on 1899-12-31 or 0:00 GMT on 1900-01-01].  This "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, however, 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 transistions 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).

(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 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, by the way).

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).

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 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].

(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)
Are there any units longer than a lightyear, or shorter than an ångström?

Here's 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), megaparsec (Mpc), kiloparsec (kpc), parsec (pc = 3.261563378 light-years), light-year. Only the parsec (pc) and its multiples are official SI units (the light-year is not): By definition, the parsec is 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 206264.8 AU).

The astronomical unit (AU or UA) itself is close to the average radius of Earth's orbit around the Sun. More precisely, it's equal to what would be the radius of the circular orbit of a small mass going around an hypothetically isolated Sun at the same rate as what is observed for the Earth (the same sidereal year). The 1992 ephemerides list 1 AU as equal to precisely 149597870660 m. An astronomical unit is also very close to the average distance from the center of the Earth to the center of gravity of the Earth-Sun system. Because the mass of the Earth is not totally negligible, the average distance from the Earth to the Sun is slightly larger than one astronomical unit, it's about 1.0000010573 AU, or 149598028825 m, using the above value of the AU.
A light-year (exactly 9460730472580800 m) is about 63241.077 AU...
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, picometer (1/100 of an ångström, formerly known as a micromicron, bicron, or stigma), femtometer or fermi (1/100000 ångström) and microångström (1/1000000 ångström). Only the picometer (pm) and femtometer (fm) are official SI units (the ångström is not).

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 square chains...
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. 

A (Gunter) chain is 66 ft (1/10 of a furlong).  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...


Volume, Capacity

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

In fact, 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 by the 9th CGPM in 1948. However, the 16th CGPM (in 1979) recognized that both "L" and"l" should be accepted, at least until actual practice could be monitored and a further ruling could be issued by the 18th CGPM. In 1990, the 18th CGPM declined to issue such a ruling, so that both symbols are now offically recognized (although "L" seems more popular in print).

(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 here.  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.

That's your first answer: 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 0.0592 cc in the UK.

The so-called metric drop is exactly 0.05 cc (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 to a teaspoon (in either the Imperial or the Winchester system).  If you have a volume expressed in teaspoons, multiply by 80 to have the number of drops.

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 obtain the equivalent capacity expressed in fluid ounces (fl oz); 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 American ounce is about 4% larger than its British counterpart  (the ratio is 1.04084273078623608419542947895884...);  about 29.6 mL (29.6 cc) to the US fl oz and 28.4 mL (28.4 cc) to the UK fl oz.

More precisely:

There are exactly 29.5735295625 milliliters in a US ounce (Winchester fl oz).
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 given 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 pounds avoirdupois, this equated a fluid ounce with the volume of one ounce avoirdupois of water at 62°F.)  The Imperial gallon was later officially 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. B. of Saint George, UT. 2000-05-02)   5 Different Gallons...
How many milliliters [ml or mL] in a gallon?

The US gallon, the so-called Winchester gallon, is 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. This means that 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 so-called Imperial gallon.

Originally (in 1819), the Imperial gallon was meant to be the volume occupied by 10 pounds of water at 62°F.  It's 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 has been argued that the Winchester bushel was originally defined as equal 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 following 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).  This volume was later rounded from  2150.420171... down to exactly 2150.42 cubic inches, which is how the so-called malt bushel has been normally defined since at least 1795.  (We couldn't determine the exact point at which the older cylindrical definition of this bushel faded from view.  Please, tell us 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, under the reign of 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 8 of these gallons.

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)
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-sized) 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.

Note that 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.

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 does not yet have an official name.

("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 some engineering calculations late in the 19th century.

The slug is defined as the mass which would accelerate at a rate of 1 ft/s per second 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 basic unit of mass in a coherent system called either "British engineering system" or "English gravitational system".  On the other hand, the basic 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 (also called "metric slug" or TME), which is the unit of mass of the so-called "metric-technical system".  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).

(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"), which would imply a value of about 0.2975 m for the Roman foot.  For some obscure reason, a  foot whose length is derived backwards from a given value for 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 (bishekel hachodesh) 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.

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 used 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 only slighly more than an ounce.  It's defined to make 1 milligram per assay ton equivalent to one troy ounce (ozt) per ton (this ton is usually the short ton of 2000 lb, but it may also be a long ton of 2240 lb so that there are, in fact, two different assay tons; the usual short assay ton and the rarer long assay ton).  Well, the troy ounce is 480 grains, the short ton is 14000000 grains (since the avoirdupois pound is defined to be 7000 grains) and the long ton is 15680000 grains.  The ratio of a troy ounce to a ton is therefore either 3/87500 or 3/98000, which makes the (short) assay ton exactly equal to 175/6 g (29.16666... grams), whereas the rarer long assay ton is 98/3 g (32.66666... grams).

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 (2000 lbt, 746.4834432 kg, or about 1645.714 lb).

In all of this, 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) has been officially abandoned since January 6, 1879 (unlike the troy ounce "ozt", which remains in use today for precious metals): The troy ton was probably obsolete well before that date, and may never have enjoyed any significant use anyway...

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 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:  All the physical quantities we measure fall into two broad categories: The so-called extensive ones, which are added when several separate physical things are considered as a whole (volume, mass, etc.) and the intensive ones which are not (temperature, pressure, ratios of extensive quantities, etc.). Choosing some "stuff" of reference --water under normal conditions, say-- establishes a coefficient of proportionality (a so-called "conversion factor") linking any pair of extensive quantities, thus creating new "practical" units ad nauseam. Some such units are called tons and are thus defined as a given extensive property 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.)

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 when the ice melts in the same amount of time.  This would be equal to 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).

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