**LENGTH MEASUREMENT **

__Type of applied science__: Scientific measurement

__Field of study__: Metrology

** Length, being one of the seven fundamental quantities of
physical measure, has a rich history of attempts to quantify its extent. The
standardization of length measurements has many applications in science, technology, and
commerce. Its standardization has facilitated the communication of scientific
observations and the trade of goods and services world wide. **

__Principal terms__:

accuracy: A number that specifies the agreement of the result of measurement with the true value of the measured quantity.

gage: A device for determining the size or shape of an object. Gages are widely used to measure and control the dimensions of objects during manufacture and production.

measure of length: A distance between two points established according to some standard or reference.

metrology: The branch of science that deals with quantifying the measures of physical quantities.

precision: The repeatability of the measurement process. It refers to how well several measurements of the same quantity agree with each other.

primary standard: An unchanging physical construction or realization of a unit of measure. Primary standards provide the absolute basis of reference of the particular unit and are used to pass on the unit to all who need it.

unit: A quantity, value, or dimension adopted as a standard of measurement. Generally, a unit is fixed by definition. For example, the units of pound, bushel, and chain are used to express a fixed weight, capacity or volume, and length respectively.

**Overview of the Technology **

** The concept of length is intimately connected to the notion
of an event. An event is "a happening" at one point in space and one point in
time. A complicated phenomena, such as for example the collision between two billiard
balls, can be analyzed in terms of a series or succession of individual events. The first
event in this phenomena is the initial contact between the objects. The collision process
spreads out over many points in space and time as it evolves. **

** To determine the position of an event in space, a reference
position or origin must be established. The origin is the location of the "zero
position" of the coordinate system. A coordinate system may be imagined to consist of
a three-dimensional (3-D) grid of lines surrounding this origin. The location in space of
a particular event my simply be read off this grid of lines. Many phenomena and events are
conveniently represented using a 3-D Cartesian or rectangular coordinate system. This
coordinate system is based on three mutually perpendicular lines passing through the
origin. Two other coordinate systems are often used in physical measurement problems;
these are the spherical and cylindrical coordinate systems. **

** It is a observational fact that three coordinates are
necessary to locate the position of an event in the space in which we live. In addition,
it appears the space around us obeys a Euclidean geometry; that is, the postulates and
theorems of Euclid are valid in our world. One important theorem proved by Euclid asserts
that the sum of the interior angles if a plane triangle in space is equal to 180 degrees.
It is important to emphasize that the assumed three-dimensional character of space and its
Euclidean geometry are based only on empirical evidence. Empirical facts are based on
observation and experimentation; they do not explain the causes of what is observed, they
just describe. The validity of Euclid's triangle theorem has been demonstrated for plane
triangles on or near the Earth to within an uncertainty of a few tenths of a second of
arc. Thus, it appears that the ideas of Euclidean geometry are good to an accuracy of a
few parts in per million. **

** The meter is the fundamental unit of length in the Systeme
International (SI) or metric system of units. The meter was originally defined by the
French Academy of Sciences in the 1791 as one ten-millionth of the distance from the North
Pole to the Equator, along the meridian line passing through Paris. As one could imagine,
this definition of the meter was difficult to realize due the arduous nature of the
measurement. However, in 1798, these geodetic measurements were completed and in 1799, a
platinum prototype meter bar was constructed and housed in the Archives of the Republic in
Paris, France. Several iron copies of this standard meter were made and one, the
"committee meter", was brought to the United States, by Ferdinand Rudolph
Hassler, First Superintendent of the Coast Survey and Weights and Measures. This bar
served as the metric length standard in the United States throughout most of the 1800's. **

** In 1889, a new physical realization of the meter, the
International Prototype Meter was legalized by the 1'st General Conference on Weights and
Measures and constructed. This new realization, although constructed to agree in length
with the 1799 bar, was an arbitrary standard. That is, it was not required to conform to
any natural or absolute standard, rather, it was used to define the unit of length called
the meter. The 1889 legislation defined the meter as the distance, at 0 degrees Celsius,
between the center portions of two lines graduated on the polished surface of a particular
bar of platinum-iridium alloy. The material platinum-iridium was used because it is hard,
resists oxidization, takes a very high mirror polish, and has a low coefficient of thermal
expansion. The original International Prototype Meter is housed at the Bureau
International de Poids et Measures (BIPM) in Sevres, France. **

** 29 copies of the International Prototype Meter were also
constructed at the BIPM and distributed to other countries. Prototype Meter No. 27 was
given to the United States and is now housed at the National Institute of Standards and
Technology (NIST) in Gaithersburg, Maryland. Accurate comparisons between the secondary
standards and objects of unknown length could be made using a longitudinal optical
comparator. The precision of optical comparisons is limited to approximately one part in
ten million (0.1 ppm). As science, technology, and commerce advanced in the twentieth
century this level of precision became inadequate. **

** The pioneering work of the American scientist Albert
Michelson, involving optical interferometric techniques, paved the way for a new and more
precise definition of the meter. An interferometer is a device used to measure accurately
the wavelength of light. In 1960, the 11'th International Conference on Weights and
Measures defined the meter to be 1,650,763.73 times the wavelength (in vacuum) of a
particular orange colored light emitted by an isotope with, atomic mass of 86, of the
element Krypton. **

** The advantage of the krypton standard is obvious. Since all
Kr-86 atoms are alike, this atomic length standard is universally accessible to any
suitably equipped scientific laboratory. It is not necessary to keep a "prototype
krypton-86 atom" at the BIPM for reference as was the case with 1889 International
Prototype Meter. Unknown lengths could be compared with the standard by the use of optical
interferometry. However, the wavelength of the emitted light is slightly uncertain due to
quantum mechanical effects that occur in the krypton atom during the emission process.
These uncertainties limit the absolute precision of the krypton-86 length standard to the
1 to 3 parts per billion level. Still, this is a clear improvement over the 1889 length
standard. **

** The current standard of length was defined during the 1983
International Conference on Weights and Measures. This standard of length is quite
different from the 1889 and 1960 standards since it is defined in terms of time. The 1983
standard defines the meter to be the distance traveled by light in vacuum in 1/299,792,458
of a second. The precision of this length standard is approximately one part per ten
trillion, a factor of one-million improvement over the 1889 standard. The basic rational
for this standard is the precision of time interval measurement provided by that the
current generation of atomic clocks. This level of precision of time measurements combined
with the assumed consistency of the speed of light in vacuum allow this
"natural" standard of length to be defined. The consistency of the speed of
light is vacuum was a fundamental postulate of Albert Einstein's theory of special
relativity. A consequence of this definition of the meter is that the speed of light in
vacuum is defined to have a value of 299,792,458 m/s. This result is in excellent accord
with the best experimental determinations of the speed of light in vacuum, (299,792,458 +-
1) m/s. **

** Practical realizations of the 1983 length standard using
frequency stabilized lasers allow the change in position (i.e. displacement) of an object
or event in the millimeter range to be measured with an uncertainty of one-picometer, the
change in position in the meter range to ten-nanometers, and the "length" (the
distance between the endpoints) of a sub-meter sized object to an uncertainty of
one-nanometer. The measurement of the length of objects to a lower uncertainty is
precluded by the slight deformations of the object under measure by the measuring
apparatus. **

**Uses of the Technology **

** One of the most important uses of length measurement
technology in industry involves the use of measuring instruments and gages to determine
and control the dimensions of manufactured parts. Mass production of goods requires
complex systems of metrology to evaluate critical dimensions. Components for a complex
manufactured object, such as an automobile, may be produced at several locations and then
brought together for final assembly. The standardization and control of length measurement
is absolutely critical in this context so that all the various parts will mesh with each
other as designed. **

** As an example of the demands placed on instruments used for
length measurement, imagine that it is desired to manufacture a part to some desired
dimension within a tolerance of one thousandth of an inch (.001"). In order that the
part conform to this level of tolerance, the instrument or gage that inspects it must be
accurate to one ten-thousandth of an inch (.0001"). The precision instrument that
checks the gage must be accurate to one hundred-thousandth of an inch (.00001"). The
working gage blocks that are used to set the precision instrument must be accurate to four
millionths of and inch (.000004") and the master blocks that calibrate the working
blocks must be accurate to one millionth of an inch (.000001"). In addition, if this
hierarchy of calibration structure is to have an absolute base of reference, the
dimensions of the master gage blocks must be derived from the primary length standard or
other standards derived from it. In the United States, master blocks must be certified to
length standards ultimately traceable to NIST. **

** Most manual instruments used for length measurement in the
manufacturing context are one of three basic types: line graduated measuring instruments,
fixed gages, and gage blocks. Line graduated measurement instruments are geometric objects
with graduation spacing representing known distances. These instruments may be used to
measure distances within their capacity range to some level of sensitivity or
discrimination. The discrimination level of a line graduated instrument is related to the
smallest increment of the scale graduation. Examples of line graduated measuring
instruments include: line graduated rules and tapes, line graduated bar standards, caliper
gages, micrometer gages, diffraction gratings, and line graduated angle measuring
instruments. Measurement errors that may occur when using line graduated measurement
instruments fall into two broad classes: instrument limitations and observational errors.
Instrument limitations include geometric deficiencies resulting from flatness or
parallelism errors and inaccuracies of scale graduations. Observational errors include
alignment deficiencies and parallax errors. Observational errors can in principle be
eliminated from the measurement process with proper instrument design and measurement
technique. **

** The next general class of manual measurement instruments are
fixed gages. A fixed gage is either a direct or reverse physical replica of the object
dimension to be measured. Fixed gages may be constructed to represent the part dimension
in its desired or nominal dimension - the master gage, or it may be used to check its
limit conditions resulting from the tolerance specified on the dimension - the limit gage.
Fixed gages are very useful in the role of inspection of dimensions of manufactured parts.
They are critical to the success of the interchangeable part manufacturing system.
Examples of fixed gages include: limit length gages, adjustable limit snap gages,
cylindrical limit gages, taper gages, multiple dimension gages, screw thread gages, and
contour gages. Some advantages associated with using fixed gages in the production
environment include the following: **

i) Fixed gages are free from errors due to the drift of the original adjustment. However, they are not free of errors due to the use and subsequent wear of the gage.

ii) Limit gages provide a definite yes/no answer to the acceptability of the inspected part.

iii) Fixed gages can be transported to the place needed and usually require no additional set-up.

iv) The cost of fixed gages is relatively modest and thus makes this type of inspection economical.

** Limit gages are made to sizes which are identical to the
design limit sizes of the dimension to be inspected; i.e. the nominal dimension plus or
minus the dimension tolerance. A limit gage that deems a part acceptable for assembly is
termed a "GO" gage. If the "GO" limit gage can enter or be entered by
the part, then it is acceptable. A limit gage that deems a part unacceptable for assembly
is termed a "NO-GO" gage. If a "NO-GO" gage can enter or be entered by
the part, then its dimension measured is incorrect and the part may be rejected. **

** The design, construction, and dimensioning of fixed gages is
a critical application of length measurement technology. Physical properties of the gage
material, such as long-term dimensional stability, thermal stability, and wear resistance
must be considered carefully along with the intended application when designing fixed
gages. The tolerances of the dimensions of fixed gages must often be at the working gage
block or even the master block level of the calibration hierarchy described above; a task
that would be nearly impossible without universal standards of length. **

** Gage blocks are the third class of manual measurement
instruments. A gage block is a length standard with rectangular, round, or square cross
sections having flat parallel opposing gaging surfaces. Gage blocks are indeed the
"master gages" of the machine shop; they are true secondary standards of length
and have a calibration that is often traceable to one of the three primary standards of
length described in the first section of this article. Since individual gage blocks are
often combined together to make a standard of specific length, they must meet the
following requirements: **

i) The individual blocks must be available in sizes needed to construct a set able to achieve any desired size and graduation.

ii) The accuracy of each individual element of the set must be within a known and accepted tolerance limit.

iii) In the built-up combinations, the individual blocks must be attached so closely so that the length of the combination is for all practical purposes equal to the sum of the lengths of the individual elements.

iv) The attachment of the individual blocks to each other must be firm enough to allow for a reasonable amount of handling of the combination, but should not harm or prevent the reuse of the blocks in any way.

** The technical requirements for gage block sets that meet all
four of the requirements listed above is outlined in the first entry of the bibliography.
Gage block sets for length measurement exist in both English unit and metric unit versions
and are also manufactured in several tolerance grades. For English unit gage blocks with
nominal size less that one inch, the length tolerance for grades 0.5, 1, 2, and 3 are
respectively: +- 1, +-2, +4 to -2, and +8 to -4 millionths of an inch. Notice that the
length uncertainty of a one inch grade 0.5 gage block is two parts per million. This is
only a factor of 20 less than the ultimate precision of the 1889 International Prototype
Meter; grade 0.5 gage blocks are truly master gages! **

** The bilateral nature of the tolerance for each block usually
results in the total tolerance for a stack of blocks to be much less than the sum of the
tolerances for the individual blocks. For example, if 30 one-half inch grade 0.5 blocks
were combined together, the cumulative tolerance would amount to .000030 inches (30
micro-inches). However, the actual length of this stack would be much closer to
15.000000" than the 15.000030" one would expect if the individual tolerances
were additive. **

** Some applications of grade 0.5 and grade 1 gage blocks in
length measurement technology include: providing a readily accessible length measurement
standard to those needing it, providing reference for gage calibration, and calibration of
precision measuring instruments. Grade 2 gage blocks are used to: check limit gages, set
adjustable limit gages, and measure setting gages. Grade 3 gage blocks are used routinely
for measurement tasks including: direct measurement of distances between parallel
surfaces, checking and adjusting mechanics' measuring tools, and precision layout of work
places. **

**Context **

** The early role of man as a creator of physical structure and
the shaper of his environment necessitated the need for dimensional measurement. Body
measurements were probably the most convenient references for early length measurement.
The cubit, devised by the Egyptians about 3000 BC, is generally regarded as the most
important length standard in the ancient Mediterranean world. It appears that the cubit
represented the length of the forearm, from the elbow to fingertips. The Egyptian cubit
was standardized by a master "royal cubit" made of black granite, against which
all other cubit sticks used in Egypt were compare with. The present system of comparing
units of measure with a standard physical realization of it follows directly from this
Egyptian custom. The royal cubit, whose length is 524 millimeters, was subdivided in a
complicated way. The smallest division, the digit, represented the width of a finger.
There are 28 digits in a royal cubit. Four digits was equal to a palm and five digits to a
hand. Twelve digits equaled a small span and 14 digits equaled as large span. Twenty-four
digits was a small cubit. The digit was in turn subdivided into parts, the smallest being
1/16 part of a digit or 1/448 of a royal cubit. The accuracy of the royal cubit and the
Egyptian system of length standards is realized in their Great Pyramid's. The lengths of
the bases of the sides of the Great Pyramid of Giza vary by less that 0.05 percent from
the mean length of 230.364 meters, a truly remarkable feat. **

** The historical progression of units, on the European
continent at least, has followed a generally westward direction. The units of the ancient
nations, such as Egypt, traveled most likely as the result of trade to the Greek and then
the Roman empires, then to Britain via the Roman conquest and finally to America. **

** Today, the total standardization of measurement throughout
other world still has not be realized. However, in the industrial world, systems of
metrology conform to either the inch or metric systems whose basis are precisely defined
primary measurement standards. The standardization of measurement allows individual
countries to produce and consume goods and services in the world economy, and exchange
important technology and scientific information worldwide; an obvious benefit to all
mankind. **

**Bibliography **

**The American Society of Mechanical Engineers. Precision Inch Gage Blocks for
Length Measurement (Through 20 inches). New York: The American Society of Mechanical
Engineers, 1974. **

This small book specifies the American National Standard for gage blocks up to and including 20" in length. This standard is of critical importance to industry and commerce since gage blocks are the most widely used transfer standards for length measurement. This book specifies the physical properties, tolerance grades, flatness, parallelism, and surface texture requirements for gage blocks.

**American Society of Tool and Manufacturing Engineers. Handbook of Industrial
Metrology. Englewood Cliffs, NJ: Prentice-Hall Inc, 1967. **

This book is an extensive reference on principles, techniques, and instrumentation design and applications for physical measurement in the manufacturing industries. Topics ranging from mathematical concepts of metrology and general principles of measurement to methods of measuring gear and screw thread forms are discussed. Contains many illustrations and photographs of measuring instruments.

**Cochrane, Rexmond C. Measurement For Progress: A History of the National
Bureau Of Standards. Washington D.C.: U.S. Department Of Commerce, 1966. **

A historical account of the role of the National Bureau of Standards in developing and providing measurement standards. Gives many examples of the uses of standards in commerce and industry. This book also includes a very informative appendix that describes the French origin of the metric system and some details of the construction of the 1889 meter and kilogram primary standards.

**Farago, Francis T. Handbook of Dimensional Measurement. 2'nd edition.
New York: Industrial Press Inc, 1982. **

A comprehensive and readable guide to advanced dimensional measurement technology. This book contains both a theoretical discussion and practical information relating to the tools and techniques of length measurement.

**Kibbe, Richard R., et. al. Machine Tool Practices. New York: John Wiley
& Sons, 1979. **

This is an introductory textbook for beginning machinists. Section C of this text, entitled "Dimensional Measurement", discusses the use of various measuring instruments, such as: steel rules, vernier calipers, micrometer instruments, and gage blocks in a manufacturing or production context.

**Sydenham, P.H. Measuring instruments: tools of knowledge and control.
Stevenage, UK: Peter Peregrinus Ltd., 1979. **

A largely historical account of the design, development, construction, and use of measuring instruments by man. This book traces this subject from ancient times to the mid-twentieth century.

**Zebrowski, E. Jr. Fundamentals of Physical Measurement. North Seituate,
MA: Duxbury Press, 1979. **

A concise introduction to the fundamental concepts of units and standards in relation to measurement theory.

**Copyright ©1993 Ben Shaevitz and Salem Press **

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