Appendix B

**Units and Systems of Measurement**

**Their Origin, Development, and Present Status**

**1.  Introduction**

The National Institute of Standards and Technology (NIST) (formerly the
National Bureau of Standards) was established by Act of Congress in 1901
to serve as a national scientific laboratory in the physical sciences,
and to provide fundamental measurement standards for science and
industry. In carrying out these related functions the Institute conducts
research and development in many fields of physics, mathematics,
chemistry, and engineering.  At the time of its founding, the Institute
had custody of two primary standards--the meter bar for length and the
kilogram cylinder for mass.  With the phenomenal growth of science and
technology over the past century, the Institute has become a major
research institution concerned not only with everyday weights and
measures, but also with hundreds of other scientific and engineering
standards that are necessary to the industrial progress of the Nation. 
Nevertheless, the country still looks to NIST for information on the
units of measurement, particularly their definitions and equivalents.

The subject of measurement systems and units can be treated from several
different standpoints.  Scientists and engineers are interested in the
methods by which precision measurements are made. State weights and
measures officials are concerned with laws and regulations that assure
equity in the marketplace, protect public health and safety, and with
methods for verifying commercial weighing and measuring devices. But a
vastly larger group of people is interested in some general knowledge of
the origin and development of measurement systems, of the present status
of units and standards, and of miscellaneous facts that will be useful
in everyday life. This material has been prepared to supply that
information on measurement systems and units that experience has shown
to be the common subject of inquiry.

**2.  Units and Systems of Measurement**

The expression \"weights and measures\" is often used to refer to
measurements of length, mass, and capacity or volume, thus excluding
such quantities as electrical and time measurements and thermometry.
This section on units and measurement systems presents some fundamental
information to clarify the concepts of this subject and to eliminate
erroneous and misleading use of terms.

It is essential that the distinction between the terms \"units\" and
\"standards\" be established and kept in mind.

A [unit]{.underline} is a special quantity in terms of which other
quantities are expressed. In general, a unit is fixed by definition and
is independent of such physical conditions as temperature. Examples: the
meter, the liter, the gram, the yard, the pound, the gallon.

A [standard]{.underline} is a physical realization or representation of
a unit. In general, it is not entirely independent of physical
conditions, and it is a representation of the unit only under specified
conditions. For example, a meter standard has a length of one meter when
at some definite temperature and supported in a certain manner. If
supported in a different manner, it might have to be at a different
temperature to have a length of one meter.

**2.1. Origin and Early History of Units and Standards.**

> **2.1.1.  General Survey of Early History of Measurement Systems.** ‑
> Weights and measures were among the earliest tools invented by man.
> Primitive societies needed rudimentary measures for many tasks:
> constructing dwellings of an appropriate size and shape, fashioning
> clothing, or bartering food or raw materials.

Man understandably turned first to parts of the body and the natural
surroundings for measuring instruments. Early Babylonian and Egyptian
records and the Bible indicate that length was first measured with the
forearm, hand, or finger and that time was measured by the periods of
the sun, moon, and other heavenly bodies.  When it was necessary to
compare the capacities of containers such as gourds or clay or metal
vessels, they were filled with plant seeds which were then counted to
measure the volumes. When means for weighing were invented, seeds and
stones served as standards. For instance, the \"carat,\" still used as a
unit for gems, was derived from the carob seed.

> Our present knowledge of early weights and measures comes from many
> sources. Archaeologists have recovered some rather early standards and
> preserved them in museums. The comparison of the dimensions of
> buildings with the descriptions of contemporary writers is another
> source of information. An interesting example of this is the
> comparison of the dimensions of the Greek Parthenon with the
> description given by Plutarch from which a fairly accurate idea of the
> size of the Attic foot is obtained. In some cases, we have only
> plausible theories and we must sometimes select the interpretation to
> be given to the evidence.
>
> For example, does the fact that the length of the double‑cubit of
> early Babylonia was equal (within two parts per thousand) to the
> length of the seconds pendulum at Babylon suggest a scientific
> knowledge of the pendulum at a very early date, or do we merely have a
> curious coincidence? By studying the evidence given by all available
> sources, and by correlating the relevant facts, we obtain some idea of
> the origin and development of the units. We find that they have
> changed more or less gradually with the passing of time in a complex
> manner because of a great variety of modifying influences. We find the
> units modified and grouped into measurement systems: The Babylonian
> system, the Egyptian system, the Phileterian system of the Ptolemaic
> age, the Olympic system of Greece, the Roman system, and the British
> system, to mention only a few.
>
> **2.1.2. Origin and Development of Some Common Customary Units.** ‑
> The origin and development of units of measurement has been
> investigated in considerable detail and a number of books have been
> written on the subject. It is only possible to give here, somewhat
> sketchily, the story about a few units.
>
> Units of length:  The [cubit]{.underline} was the first recorded unit
> used by ancient peoples to measure length. There were several cubits
> of different magnitudes that were used. The common cubit was the
> length of the forearm from the elbow to the tip of the middle finger.
> It was divided into the span of the hand (one‑half cubit), the palm or
> width of the hand (one sixth), and the digit or width of a finger (one
> twenty-fourth). The Royal or Sacred Cubit, which was 7 palms or
> 28 digits long, was used in constructing buildings and monuments and
> in surveying. The [inch]{.underline}, [foot]{.underline}, and
> [yard]{.underline} evolved from these units through a complicated
> transformation not yet fully understood. Some believe they evolved
> from cubic measures; others believe they were simple proportions or
> multiples of the cubit. In any case, the Greeks and Romans inherited
> the foot from the Egyptians. The Roman foot was divided into both 12
> unciae (inches) and 16 digits. The Romans also introduced the mile of
> 1000 paces or double steps, the pace being equal to five Roman feet.
> The Roman mile of 5000 feet was introduced into England during the
> occupation. Queen Elizabeth, who reigned from 1558 to 1603, changed,
> by statute, the mile to 5280 feet or 8 furlongs, a furlong being 40
> rods of 5‑1/2 yards each.
>
> The introduction of the [yard]{.underline} as a unit of length came
> later, but its origin is not definitely known. Some believe the origin
> was the double cubit, others believe that it originated from cubic
> measure. Whatever its origin, the early yard was divided by the binary
> method into 2, 4, 8, and 16 parts called the half‑yard, span, finger,
> and nail. The association of the yard with the \"gird\" or
> circumference of a person\'s waist or with the distance from the tip
> of the nose to the end of the thumb of Henry I are probably
> standardizing actions, since several yards were in use in Great
> Britain.
>
> The [point]{.underline}, which is a unit for measuring print type, is
> recent. It originated with Pierre Simon Fournier in 1737. It was
> modified and developed by the Didot brothers, Francois Ambroise and
> Pierre Francois, in 1755. The point was first used in the United
> States in 1878 by a Chicago type foundry (Marder, Luse, and Company).
> Since 1886, a point has been exactly 0.3514598 millimeters, or about
> 1/72 inch.
>
> Units of mass:  The [grain]{.underline} was the earliest unit of mass
> and is the smallest unit in the apothecary, avoirdupois, Tower, and
> Troy systems. The early unit was a grain of wheat or barleycorn used
> to weigh the precious metals silver and gold. Larger units preserved
> in stone standards were developed that were used as both units of mass
> and of monetary currency. The [pound]{.underline} was derived from the
> mina used by ancient civilizations. A smaller unit was the shekel, and
> a larger unit was the talent. The magnitude of these units varied from
> place to place. The Babylonians and Sumerians had a system in which
> there were 60 shekels in a mina and 60 minas in a talent. The Roman
> talent consisted of 100 libra (pound) which were smaller in magnitude
> than the mina. The Troy pound used in England and the United States
> for monetary purposes, like the Roman pound, was divided into
> 12 ounces, but the Roman uncia (ounce) was smaller. The
> [carat]{.underline} is a unit for measuring gemstones that had its
> origin in the carob seed, which later was standardized at 1/144 ounce
> and then 0.2 gram.
>
> Goods of commerce were originally traded by number or volume. When
> weighing of goods began, units of mass based on a volume of grain or
> water were developed. For example, the talent in some places was
> approximately equal to the mass of one cubic foot of water. Was this a
> coincidence or by design? The diverse magnitudes of units having the
> same name, which still appear today in our dry and liquid measures,
> could have arisen from the various commodities traded. The larger
> avoirdupois [pound]{.underline} for goods of commerce might have been
> based on volume of water which has a higher bulk density than grain.
> For example, the Egyptian hon was a volume unit about 11 percent
> larger than a cubic palm and corresponded to one mina of water. It was
> almost identical in volume to the present U.S. pint.
>
> The [stone]{.underline}, [quarter]{.underline},
> [hundredweight]{.underline}, and [ton]{.underline} were larger units
> of mass used in Great Britain. Today only the stone continues in
> customary use for measuring personal body weight. The present stone is
> 14 pounds, but an earlier unit appears to have been 16 pounds. The
> other units were multiples of 2, 8, and 160 times the stone, or 28,
> 112, and 2240 pounds, respectively. The hundredweight was
> approximately equal to two talents. In the United States the ton of
> 2240 pounds is called the "long ton." The "short ton" is equal to 2000
> pounds.
>
> Units of time and angle: We can trace the division of the circle into
> 360 degrees and the day into hours, minutes, and seconds to the
> Babylonians who had a sexagesimal system of numbers. The 360 degrees
> may have been related to a year of 360 days.

**2.2. The Metric System.**

> **2.2.1.  Definition, Origin, and Development.** ‑ Metric systems of
> units have evolved since the adoption of the first well defined system
> in France in 1791. During this evolution the use of these systems
> spread throughout the world, first to the non-English speaking
> countries, and more recently to the English speaking countries. The
> first metric system was based on the centimeter, gram, and second
> (cgs) and these units were particularly convenient in science and
> technology. Later metric systems were based on the meter, kilogram,
> and second (mks) to improve the value of the units for practical
> applications. The present metric system is the International System of
> Units (SI). It is also based on the meter, kilogram and second as well
> as additional base units for temperature, electric current, luminous
> intensity, and amount of substance. the International System of Units
> is referred to as the modern metric system.

The adoption of the metric system in France was slow, but its
desirability as an international system was recognized by geodesists and
others. On May 20, 1875, an international treaty known as the
International Metric Convention or the Treaty of the Meter was signed by
17 countries including the United States. This treaty established the
following organizations to conduct international activities relating to
a uniform system for measurements:

1\) The General Conference on Weights and Measures (French initials:
CGPM), an intergovernmental conference of official delegates of member
nations and the supreme authority for all actions;

> 2\) The International Committee of Weights and Measures (French
> initials: CIPM), consisting of selected scientists and metrologists,
> which prepares and executes the decisions of the CGPM and is
> responsible for the supervision of the International Bureau of Weights
> and Measures;
>
> 3\) The International Bureau of Weights and Measures (French initials:
> BIPM), a permanent laboratory and world center of scientific
> metrology, the activities of which include the establishment of the
> basic standards and scales of the principal physical quantities and
> maintenance of the international prototype standards.
>
> The National Institute of Standards and Technology provides official
> United States representation in these organizations. The CGPM, the
> CIPM, and the BIPM have been major factors in the continuing
> refinement of the metric system on a scientific basis and in the
> evolution of the International System of Units.
>
> Multiples and submultiples of metric units are related by powers of
> ten. This relationship is compatible with the decimal system of
> numbers and it contributes greatly to the convenience of metric units.
>
> **2.2.2. International System of Units.** - At the end of World War
> II, a number of different systems of measurement still existed
> throughout the world. Some of these systems were variations of the
> metric system, and others were based on the customary inch-pound
> system of the English-speaking countries. It was recognized that
> additional steps were needed to promote a worldwide measurement
> system. As a result the 9th GCPM, in 1948, asked the ICPM to conduct
> an international study of the measurement needs of the scientific,
> technical, and educational communities. Based on the findings of this
> study, the 10th General Conference in 1954 decided that an
> international system should be derived from six base units to provide
> for the measurement of temperature and optical radiation in addition
> to mechanical and electromagnetic quantities. The six base units
> recommended were the meter, kilogram, second, ampere, Kelvin degree
> (later renamed the kelvin), and the candela.
>
> In 1960, the 11th General Conference of Weights and Measures named the
> system based on the six base quantities the International System of
> Units, abbreviated SI from the French name: Le Système International
> d\'Unités. The SI metric system is now either obligatory or
> permissible throughout the world.
>
> **2.2.3. Units and Standards of the Metric System.** ‑ In the early
> metric system there were two fundamental or base units, the meter and
> the kilogram, for length and mass. The other units of length and mass,
> and all units of area, volume, and compound units such as density were
> derived from these two fundamental units.
>
> The meter was originally intended to be one ten‑millionth part of a
> meridional quadrant of the earth. The Meter of the Archives, the
> platinum length standard which was the standard for most of the 19th
> century, at first was supposed to be exactly this fractional part of
> the quadrant. More refined measurements over the earth\'s surface
> showed that this supposition was not correct. In 1889, a new
> international metric standard of length, the International Prototype
> Meter, a graduated line standard of platinum‑iridium, was selected
> from a group of bars because precise measurements found it to have the
> same length as the Meter of the Archives. The meter was then defined
> as the distance, under specified conditions, between the lines on the
> International Prototype Meter without reference to any measurements of
> the earth or to the Meter of the Archives, which it superseded.
> Advances in science and technology have made it possible to improve
> the definition of the meter and reduce the uncertainties associated
> with artifacts. From 1960 to 1983, the meter was defined as the length
> equal to 1 650 763.73 wavelengths in a vacuum of the radiation
> corresponding to the transition between the specified energy levels of
> the krypton 86 atom. Since 1983 the meter has been defined as the
> length of the path traveled by light in a vacuum during an interval of
> 1/299 792 458 of a second.
>
> The kilogram, originally defined as the mass of one cubic decimeter of
> water at the temperature of maximum density, was known as the Kilogram
> of the Archives. It was replaced after the International Metric
> Convention in 1875 by the International Prototype Kilogram which
> became the unit of mass without reference to the mass of a cubic
> decimeter of water or to the Kilogram of the Archives. Each country
> that subscribed to the International Metric Convention was assigned
> one or more copies of the international standards; these are known as
> National Prototype Meters and Kilograms.
>
> The liter is a unit of capacity or volume. In 1964, the 12th GCPM
> redefined the liter as being one cubic decimeter. By its previous
> definition \-- the volume occupied, under standard conditions, by a
> quantity of pure water having a mass of one kilogram \-- the liter was
> larger than the cubic decimeter by 28 parts per 1 000 000. Except for
> determinations of high precision, this difference is so small as to be
> of no consequence.
>
> The modern metric system (SI) includes two classes of units:
>
> [base]{.underline} units for length, mass, time, temperature, electric
> current, luminous intensity, and amount of substance; and
>
> [derived]{.underline} units for all other quantities (e.g., work,
> force, power) expressed in terms of the seven base units.
>
> For details, see NIST Special Publication 330 (2001), The
> International System of Units (SI) and NIST Special Publication
> 811(1995), Guide for the Use of the International System of Units.
>
> **2.2.4. International Bureau of Weights and Measures. ‑** The
> International Bureau of Weights and Measures (BIPM) was established at
> Sèvres, a suburb of Paris, France, by the International Metric
> Convention of May 20, 1875. The BIPM maintains the International
> Prototype Kilogram, many secondary standards, and equipment for
> comparing standards and making precision measurements. The Bureau,
> funded by assessment of the signatory governments, is truly
> international. In recent years the scope of the work at the Bureau has
> been considerably broadened. It now carries on researches in the
> fields of electricity, photometry and radiometry, ionizing radiations,
> and time and frequency besides its work in mass, length, and
> thermometry.
>
> **2.2.5. Status of the Metric System in the United States.** ‑ The use
> of the metric system in this country was legalized by Act of Congress
> in 1866, but was not made obligatory then or since.
>
> Following the signing of the Convention of the Meter in 1875, the
> United States acquired national prototype standards for the meter and
> the kilogram. U.S. Prototype Kilogram No. 20 continues to be the
> primary standard for mass in the United States. It is recalibrated
> from time to time at the BIPM. The prototype meter has been replaced
> by modern stabilized lasers following the most recent definition of
> the meter.
>
> From 1893 until 1959, the yard was defined as equal exactly to
> 3600/3937 meter.  In 1959, a small change was made in the definition
> of the yard to resolve discrepancies both in this country and abroad.
> Since 1959, we define the yard as equal exactly to 0.9144 meter; the
> new yard is shorter than the old yard by exactly two parts in a
> million.  At the same time, it was decided that any data expressed in
> feet derived from geodetic surveys within the United States would
> continue to bear the relationship as defined in 1893 (one foot equals
> 1200/3937 meter). We call this foot the U.S. Survey Foot, while the
> foot defined in 1959 is called the International Foot. Measurements
> expressed in U.S. statute miles, survey feet, rods, chains, links, or
> the squares thereof, and acres should be converted to the
> corresponding metric values by using pre‑1959 conversion factors if
> more than five significant figure accuracy is required.
>
> Since 1970, actions have been taken to encourage the use of metric
> units of measurement in the United States. A brief summary of actions
> by Congress is provided below as reported in the Federal Register
> Notice dated July 28, 1998.
>
> Section 403 of Public Law 93-380, the Education Amendment of 1974,
> states that it is the policy of the United States to encourage
> educational agencies and institutions to prepare students to use the
> metric system of measurement as part of the regular education program.
> Under both this act and the Metric Conversion Act of 1975, the
> \"metric system of measurement\" is defined as the International
> System of Units as established in 1960 by the General Conference on
> Weights and Measures and interpreted or modified for the United States
> by the Secretary of Commerce (sec. 4(4)- Pub. L. 94-168; sec.
> 403(a)(3)- Pub. L. 93-380). The Secretary has delegated authority
> under these subsections to the Director of the National Institute of
> Standards and Technology.
>
> Section 5164 of Public Law 100-418, the Omnibus Trade and
> Competitiveness Act of 1988, amends Public Law 94-168, The Metric
> Conversion Act of 1975. In particular, section 3 Metric Conversion Act
> is amended to read as follows:
>
> \"Sec. 3. It is therefore the declared policy of the United States--
>
> \(1\) to designate the metric system of measurement as the preferred
> system of weights and measures for United States trade and commerce;
>
> \(2\) to require that each Federal agency, by a date certain and to
> the extent economically feasible by the end of the fiscal year 1992,
> use the metric system of measurement in its procurements, grants, and
> other business-related activities, except to the extent that such use
> is impractical or is likely to cause significant inefficiencies or
> loss of markets to U.S. firms, such as when foreign competitors are
> producing competing products in non-metric units;
>
> \(3\) to seek ways to increase understanding of the metric system of
> measurement through educational information and guidance and in
> Government publications; and
>
> \(4\) to permit the continued use of traditional systems of weights
> and measures in nonbusiness activities.\"
>
> The Code of Federal Regulations makes the use of metric units
> mandatory for agencies of the Federal Government. (Federal Register,
> Vol. 56, No. 23, page 160, January 2, 1991.)

**2.3. British and United States Systems of Measurement.** ‑ In the
past, the customary system of weights and measures in the British
Commonwealth countries and that in the United States were very similar;
however, the SI metric system is now the official system of units in the
United Kingdom, while the customary units are still predominantly used
in the United States. Because references to the units of the old British
customary system are still found, the following discussion describes the
differences between the U.S. and British customary systems of units. 

After 1959, the U.S. and the British inch were defined identically for
scientific work and were identical in commercial usage. A similar
situation existed for the U.S. and the British pound, and many
relationships, such as 12 inches = 1 foot, 3 feet = 1 yard, and 1760
yards = 1 international mile, were the same in both countries; but there
were some very important differences.

In the first place, the U.S. customary bushel and the U.S. gallon, and
their subdivisions differed from the corresponding British Imperial
units.  Also the British ton is 2240 pounds, whereas the ton generally
used in the United States is the short ton of 2000 pounds.  The American
colonists adopted the English wine gallon of 231 cubic inches.  The
English of that period used this wine gallon and they also had another
gallon, the ale gallon of 282 cubic inches.  In 1824, the British
abandoned these two gallons when they adopted the British Imperial
gallon, which they defined as the volume of 10 pounds of water, at a
temperature of 62 F, which, by calculation, is equivalent to 277.42
cubic inches. At the same time, they redefined the bushel as 8 gallons.

In the customary British system the units of dry measure are the same as
those of liquid measure.  In the United States these two are not the
same, the gallon and its subdivisions are used in the measurement of
liquids; the bushel, with its subdivisions, is used in the measurement
of certain dry commodities. The U.S. gallon is divided into four liquid
quarts and the U.S. bushel into 32 dry quarts.  All the units of
capacity or volume mentioned thus far are larger in the customary
British system than in the U.S. system.  But the British fluid ounce is
smaller than the U.S. fluid ounce, because the British quart is divided
into 40 fluid ounces whereas the U.S. quart is divided into 32 fluid
ounces.

From this we see that in the customary British system an avoirdupois
ounce of water at 62 F has a volume of one fluid ounce, because 10
pounds is equivalent to 160 avoirdupois ounces, and 1 gallon is
equivalent to 4 quarts, or 160 fluid ounces.  This convenient relation
does not exist in the U.S. system because a U.S. gallon of water at
62 F weighs about 8‑1/3 pounds, or 133‑1/3 avoirdupois ounces, and the
U.S. gallon is equivalent to 4 x 32, or 128 fluid ounces.

> 1 U.S. fluid ounce = 1.041 British fluid ounces
>
> 1 British fluid ounce = 0.961 U.S. fluid ounce
>
> 1 U.S. gallon = 0.833 British Imperial gallon
>
> 1 British Imperial gallon = 1.201 U.S. gallons

Among other differences between the customary British and the United
States measurement systems, we should note that they abolished the use
of the troy pound in England January 6, 1879, they retained only the
troy ounce and its subdivisions, whereas the troy pound is still legal
in the United States, although it is not now greatly used. We can
mention again the common use, for body weight, in England of the stone
of 14 pounds, this being a unit now unused in the United States,
although its influence was shown in the practice until World War II of
selling flour by the barrel of 196 pounds (14 stone).  In the apothecary
system of liquid measure the British add a unit, the fluid scruple,
equal to one third of a fluid drachm (spelled [dram]{.underline} in the
United States) between their minim and their fluid drachm.  In the
United States, the general practice now is to sell dry commodities, such
as fruits and vegetables, by their mass.

**2.4. Subdivision of Units.** ‑ In general, units are subdivided by one
of three methods: (a) decimal, into tenths; (b) duodecimal, into
twelfths; or (c) binary, into halves (twos).  Usually the subdivision is
continued by using the same method.  Each method has its advantages for
certain purposes, and it cannot properly be said that any one method is
\"best\" unless the use to which the unit and its subdivisions are to be
put is known.

For example, if we are concerned only with measurements of length to
moderate precision, it is convenient to measure and to express these
lengths in feet, inches, and binary fractions of an inch, thus 9 feet
4‑3/8 inches. However, if these lengths are to be subsequently used to
calculate area or volume, that method of subdivision at once becomes
extremely inconvenient.  For that reason, civil engineers, who are
concerned with areas of land, volumes of cuts, fills, excavations, etc.,
instead of dividing the foot into inches and binary subdivisions of the
inch, divide it decimally; that is, into tenths, hundredths, and
thousandths of a foot.

The method of subdivision of a unit is thus largely made based on
convenience to the user.  The fact that units have commonly been
subdivided into certain subunits for centuries does not preclude their
also having another mode of subdivision in some frequently used cases
where convenience indicates the value of such other method.  Thus, while
we usually subdivide the gallon into quarts and pints, most
gasoline‑measuring pumps, of the price‑computing type, are graduated to
show tenths, hundredths, or thousandths of a gallon.

Although the mile has for centuries been divided into rods, yards, feet,
and inches, the odometer part of an automobile speedometer shows tenths
of a mile.  Although we divide our dollar into 100 parts, we habitually
use and speak of halves and quarters.  An illustration of rather complex
subdividing is found on the scales used by draftsmen. These scales are
of two types: (a) architects, which are commonly graduated with scales
in which 3/32, 3/16, 1/8, 1/4, 3/8, ½, 3/4, 1, 1‑1/2, and 3 inches,
respectively, represent 1 foot full scale, and also having a scale
graduated in the usual manner to 1/16 inch; and (b) engineers, which are
commonly subdivided to 10, 20, 30, 40, 50, and 60 parts to the inch.

The dictum of convenience applies not only to subdivisions of a unit but
also to multiples of a unit.  Land elevations above sea level are given
in feet although the height may be several miles; the height of aircraft
above sea level as given by an altimeter is likewise given in feet, no
matter how high it may be.

On the other hand, machinists, toolmakers, gauge makers, scientists, and
others who are engaged in precision measurements of relatively small
distances, even though concerned with measurements of length only, find
it convenient to use the inch, instead of the tenth of a foot, but to
divide the inch decimally to tenths, hundredths, thousandths, etc., even
down to millionths of an inch.  Verniers, micrometers, and other
precision measuring instruments are usually graduated in this manner. 
Machinist scales are commonly graduated decimally along one edge and are
also graduated along another edge to binary fractions as small as 1/64
inch. The scales with binary fractions are used only for relatively
rough measurements.

It is seldom convenient or advisable to use binary subdivisions of the
inch that are smaller than 1/64.  In fact, 1/32‑, 1/16‑, or 1/8‑inch
subdivisions are usually preferable for use on a scale to be read with
the unaided eye.

**2.5.  Arithmetical Systems of Numbers.** ‑ The subdivision of units of
measurement is closely associated with arithmetical systems of numbers.
The systems of units used in this country for commercial and scientific
work, having many origins as has already been shown, naturally show
traces of the various number systems associated with their origins and
developments.  Thus, (a) the binary subdivision has come down to us from
the Hindus, (b) the duodecimal system of fractions from the Romans, (c)
the decimal system from the Chinese and Egyptians, some developments
having been made by the Hindus, and (d) the sexagesimal system (division
by 60) now illustrated in the subdivision of units of angle and of time,
from the ancient Babylonians. The use of decimal numbers in measurements
is becoming the standard practice.

**3.  Standards of Length, Mass, and Capacity or Volume**

**3.1. Standards of Length.** ‑ The meter, which is defined in terms of
the speed of light in a vacuum, is the unit on which all length
measurements are based.

> The yard is defined[^1] as follows:
>
> 1 yard = 0.914 4 meter

and the inch is exactly equal to 25.4 millimeters.

> **3.1.1. Calibration of Length Standards.** ‑ NIST calibrates
> standards of length including meter bars, yard bars, miscellaneous
> precision line standards, steel tapes, invar geodetic tapes, precision
> gauge blocks, micrometers, and limit gauges.  It also measures the
> linear dimensions of miscellaneous apparatus such as penetration
> needles, cement sieves, and hemacytometer chambers.  In general, NIST
> accepts for calibration only apparatus of such material, design, and
> construction as to ensure accuracy and permanence sufficient to
> justify calibration by the Institute. NIST performs calibrations in
> accordance with fee schedules, copies of which may be obtained from
> NIST.
>
> NIST does not calibrate carpenters' rules, machinist scales, draftsman
> scales, and the like.  Such apparatus, if they require calibration,
> should be submitted to State or local weights and measures officials.

**3.2. Standards of Mass.** ‑ The primary standard of mass for this
country is United States Prototype Kilogram 20, which is a
platinum‑iridium cylinder kept at NIST.  We know the value of this mass
standard in terms of the International Prototype Kilogram, a
platinum‑iridium standard which is kept at the International Bureau of
Weights and Measures.

In Colonial Times the British standards were considered the primary
standards of the United States.  Later, the U.S. avoirdupois pound was
defined in terms of the Troy Pound of the Mint, which is a brass
standard kept at the United States Mint in Philadelphia.  In 1911, the
Troy Pound of the Mint was superseded, for coinage purposes, by the Troy
Pound of the Institute.

The avoirdupois pound is defined in terms of the kilogram by the
relation:

> 1 avoirdupois pound = 0.453 592 37 kilogram.[^2]

These changes in definition have not made any appreciable change in the
value of the pound.

The grain is 1/7000 of the avoirdupois pound and is identical in the
avoirdupois, troy, and apothecary systems.  The troy ounce and the
apothecary ounce differ from the avoirdupois ounce but are equal to each
other, and equal to 480 grains.  The avoirdupois ounce is equal to 437.5
grains.

> **3.2.1. Mass and Weight.** ‑ The mass of a body is a measure of its
> inertial property or how much matter it contains. The weight of a body
> is a measure of the force exerted on it by gravity or the force needed
> to support it. Gravity on earth gives a body a downward acceleration
> of about 9.8 m/s^2^. (In common parlance, weight is often used as a
> synonym for mass as in weights and measures.) The incorrect use of
> weight in place of mass should be phased out, and the term mass used
> when mass is meant.
>
> Standards of mass are ordinarily calibrated by comparison to a
> reference standard of mass.  If two objects are compared on a balance
> and give the same balance indication, they have the same \"mass\"
> (excluding the effect of air buoyancy).  The forces of gravity on the
> two objects are balanced.  Even though the value of the acceleration
> of gravity, [g]{.underline}, is different from location to location,
> because the two objects of equal mass in the same location (where both
> masses are acted upon by the same [g]{.underline}) will be affected in
> the same manner and by the same amount by any change in the value of
> [g]{.underline}, the two objects will balance each other under any
> value of [g]{.underline}.
>
> However, on a spring balance the mass of a body is not balanced
> against the mass of another body.  Instead, the gravitational force on
> the body is balanced by the restoring force of a spring.  Therefore,
> if a very sensitive spring balance is used, the indicated mass of the
> body would be found to change if the spring balance and the body were
> moved from one locality to another locality with a different
> acceleration of gravity.  But a spring balance is usually used in one
> locality and is adjusted or calibrated to indicate mass at that
> locality.
>
> **3.2.2.  Effect of Air Buoyancy.** ‑ Another point that must be taken
> into account in the calibration and use of standards of mass is the
> buoyancy or lifting effect of the air.  A body immersed in any fluid
> is buoyed up by a force equal to the force of gravity on the displaced
> fluid. Two bodies of equal mass, if placed one on each pan of an
> equal‑arm balance, will balance each other in a vacuum.  A comparison
> in a vacuum against a known mass standard gives \"true mass.\" If
> compared in air, however, they will not balance each other unless they
> are of equal volume. If of unequal volume, the larger body will
> displace the greater volume of air and will be buoyed up by a greater
> force than will the smaller body, and the larger body will appear to
> be of less mass than the smaller body.
>
> The greater the difference in volume, and the greater the density of
> the air in which we make the comparison weighing, the greater will be
> the apparent difference in mass.  For that reason, in assigning a
> precise numerical value of mass to a standard, it is necessary to base
> this value on definite values for the air density and the density of
> the mass standard of reference.
>
> The apparent mass of an object is equal to the mass of just enough
> reference material of a specified density (at 20 C) that will produce
> a balance reading equal to that produced by the object if the
> measurements are done in air with a density of 1.2 mg/cm^3^ at 20 C. 
> The original basis for reporting apparent mass is apparent mass versus
> brass. The apparent mass versus a density of 8.0 g/cm^3^ is the more
> recent definition, and is used extensively throughout the world.  The
> use of apparent mass versus 8.0 g/cm^3^ is encouraged over apparent
> mass versus brass.  The difference in these apparent mass systems is
> insignificant in most commercial weighing applications.
>
> A full discussion of this topic is given in NIST Monograph 133, Mass
> and Mass Values, by Paul E. Pontius \[for sale by the National
> Technical Information Service, 5285 Port Royal Road, Springfield, VA
> 22161 (COM 7450309).\]
>
> **3.2.3. Calibrations of Standards of Mass.** ‑ Standards of mass
> regularly used in ordinary trade should be tested by State or local
> weights and measures officials.  NIST calibrates mass standards
> submitted, but it does not manufacture or sell them. Information
> regarding the mass calibration service of NIST and the regulations
> governing the submission of standards of mass to NIST for calibration
> are contained in NIST Special Publication 250, Calibration and Related
> Measurement Services of NIST, latest edition.

**3.3. Standards of Capacity.** ‑ Units of capacity or volume, being
derived units, are in this country defined in terms of linear units. 
Laboratory standards have been constructed and are maintained at NIST.
These have validity only by calibration with reference either directly
or indirectly to the linear standards.  Similarly, NIST has made and
distributed standards of capacity to the several States.  Other
standards of capacity have been verified by calibration for a variety of
uses in science, technology, and commerce.

> **3.3.1. Calibrations of Standards of Capacity.** ‑ NIST makes
> calibrations on capacity or volume standards that are in the customary
> units of trade; that is, the gallon, its multiples, and submultiples,
> or in metric units. Further, NIST calibrates precision-grade
> volumetric glassware which is normally in metric units.  NIST makes
> calibrations in accordance with fee schedules, copies of which may be
> obtained from NIST.

**3.4. Maintenance and Preservation of Fundamental Standard of Mass.** ‑
It is a statutory responsibility of NIST to maintain and preserve the
national standard of mass at NIST and to realize all the other base
units. The U.S. Prototype Kilogram maintained at NIST is fully protected
by an alarm system.  All measurements made with this standard are
conducted in special air‑conditioned laboratories to which the standard
is taken a sufficiently long time before the observations to ensure that
the standard will be in a state of equilibrium under standard conditions
when the measurements or comparisons are made.  Hence, it is not
necessary to maintain the standard at standard conditions, but care is
taken to prevent large changes of temperature.  More important is the
care to prevent any damage to the standard because of careless handling.

**4.  Specialized Use of the Terms \"Ton\" and \"Tonnage\"**

As weighing and measuring are important factors in our everyday lives,
it is quite natural that questions arise about the use of various units
and terms and about the magnitude of quantities involved.  For example,
the words \"ton\" and \"tonnage\" are used in widely different senses,
and a great deal of confusion has arisen regarding the application of
these terms.

The ton is used as a unit of measure in two distinct senses:  (1) as a
unit of mass, and (2) as a unit of capacity or volume.

In the first sense, the term has the following meanings:

> a\. The [short]{.underline}, or [net]{.underline} ton of 2000 pounds.
>
> b\. The [long]{.underline}, [gross]{.underline}, or
> [shipper\'s]{.underline} ton of 2240 pounds.
>
> c\. The [metric]{.underline} ton of 1000 kilograms, or 2204.6 pounds.

In the second sense (capacity), it is usually restricted to uses
relating to ships and has the following meaning:

> a\. The [register]{.underline} ton of 100 cubic feet.
>
> b\. The [measurement]{.underline} ton of 40 cubic feet.
>
> c\. The [English water]{.underline} ton of 224 British Imperial
> gallons.

In the United States and Canada the ton (mass) most commonly used is the
[short]{.underline} ton. In Great Britain, it is the [long]{.underline}
ton, and in countries using the metric system, it is the
[metric]{.underline} ton.  The [register]{.underline} ton and the
[measurement]{.underline} ton are capacity or volume units used in
expressing the tonnage of ships.  The [English water]{.underline} ton is
used, chiefly in Great Britain, in statistics dealing with petroleum
products.

There have been many other uses of the term ton such as the
[timber]{.underline} ton of 40 cubic feet and the [wheat]{.underline}
ton of 20 bushels, but their use has been local and the meanings have
not been consistent from one place to another.

Properly, the word \"tonnage\" is used as a noun only in respect to the
capacity or volume and dimensions of ships, and to the amount of the
ship\'s cargo. There are two distinct kinds of tonnage; namely, [vessel
tonnage]{.underline} and [cargo tonnage]{.underline} and each of these
is used in various meanings.

The several kinds of [vessel tonnage]{.underline} are as follows:

[Gross tonnage]{.underline}, or [gross register tonnage]{.underline}, is
the total cubical capacity or volume of a ship expressed in register
tons of 100 cubic feet, or 2.83 cubic meters, less such space as
hatchways, bakeries, galleys, etc., as are exempted from measurement by
different governments.  There is some lack of uniformity in the gross
tonnages as given by different nations due to lack of agreement on the
spaces that are to be exempted.  Official merchant marine statistics of
most countries are published in terms of the [gross register
tonnage]{.underline}.  Press references to ship [tonnage]{.underline}
are usually to the [gross tonnage]{.underline}.

The [net tonnage]{.underline}, or [net register tonnage]{.underline}, is
the [gross tonnage]{.underline} less the different spaces specified by
maritime nations in their measurement rules and laws.  The spaces
deducted are those totally unavailable for carrying cargo, such as the
engine room, coal bunkers, crew quarters, chart and instrument room,
etc.  The [net tonnage]{.underline} is used in computing how much cargo
that can be loaded on a ship. It is used as the basis for wharfage and
other similar charges.

The [register under‑deck tonnage]{.underline} is the cubical capacity of
a ship under her tonnage deck expressed in register tons.  In a vessel
having more than one deck, the tonnage deck is the second from the keel.

There are several variations of [displacement tonnage]{.underline}.

The [dead weight tonnage]{.underline} is the difference between the
\"loaded\" and \"light\" [displacement tonnages]{.underline} of a
vessel.  It is expressed in terms of the long ton of 2240 pounds, or the
metric ton of 2204.6 pounds, and is the weight of fuel, passengers, and
cargo that a vessel can carry when loaded to its maximum draft.

The second variety of tonnage, [cargo tonnage]{.underline}, refers to
the weight of the particular items making up the cargo.  In overseas
traffic it is usually expressed in long tons of 2240 pounds or metric
tons of 2204.6 pounds. The short ton is only occasionally used. 
Therefore, the [cargo tonnage]{.underline} is very distinct from [vessel
tonnage]{.underline}.

[^1]: See Federal Register for July 1, 1959. See also next to last
    paragraph of 2.2.5

[^2]: See Federal Register for July 1, 1959.