Foot (unit)

The foot (pl. feet; abbreviation: ft; symbol: ′, the prime symbol) is a unit of length in the imperial and US customary systems of measurement. Since 1959, both units have been defined by international agreement as equivalent to 0.3048 meters exactly. In both systems, the foot comprises 12 inches and three feet compose a yard.

Historically the "foot" was a part of many local systems of units, including the Greek, Roman, Chinese, French, and English systems. It varied in length from country to country, from city to city, and sometimes from trade to trade. Its length was usually between 250 mm and 335 mm and was generally, but not always, subdivided into 12 inches or 16 digits.

The United States is the only industrialized nation that uses the international foot and the survey foot (a customary unit of length) in preference to the meter in its commercial, engineering, and standards activities.^[1] The foot is legally recognized in the United Kingdom; road signs must use imperial units (however distances on road signs are always marked in yards, not feet), while its usage is widespread among the British public as a measurement of height.^[2][3] The foot is recognized as an alternative expression of length in Canada^[4] officially defined as a unit derived from the meter^[5] although both the U.K. and Canada have partially metricated their units of measurement. The measurement of altitude in international aviation is one of the few areas where the foot is widely used outside the English-speaking world.

The length of the international foot corresponds to a human foot with shoe size of 13 (UK), 14 (US male), 15.5 (US female) or 46 (EU sizing).

Historical origin

Determination of the rod, using the length of the left foot of 16 randomly chosen people coming from church service. Woodcut published in the book Geometrei by Jakob Köbel (Frankfurt, c. 1536).

Historically the human body has been used to provide the basis for units of length.^[6] The foot of a white male is typically about 15.3% of his height,^[7] giving a person of 160 cm (5 ft 3 in) a foot of 245 mm and one of 180 cm (5 ft 11 in) a foot of 275 mm. These figures are less than the foot used in most cities over time, suggesting that the "foot" was actually a synonym for a "shoe".

Archeologists believe that the Egyptians, Ancient Indians and Mesopotamians preferred the cubit while the Romans and the Greeks preferred the foot. Under the Harappan linear measures, Indus cities during the Bronze Age used a foot of 13.2 inches (333.5 mm) and a cubit of 20.8 inches (528.3 mm).^[8] The Egyptian equivalent of the foot—a measure of four palms or 16 digits—was known as the djeser and has been reconstructed as about 30 cm (12 in).

The Greek foot (πούς, pous) had a length of  ¹⁄₆₀₀ of a stadion,^[9] one stadion being about 181.2 m,^[10] therefore a foot being at the time about 302 mm. Its exact size varied from city to city and could range as much as between 270 mm and 350 mm, but lengths used for temple construction appear to have been about 295 mm to 325 mm, the former being close to the size of the Roman foot.

The standard Roman foot (pes) was normally about 295.7 mm (97% of today's measurement), but in the provinces, the pes Drusianus (foot of Nero Claudius Drusus) was used, with a length of about 334 mm. (In reality, this foot predated Drusus.)^[11]

Originally both the Greeks and the Romans subdivided the foot into 16 digits, but in later years, the Romans also subdivided the foot into 12 unciae (from which both the English words "inch" and "ounce" are derived).

After the fall of the Roman Empire, some Roman traditions were continued but others fell into disuse. In AD 790 Charlemagne attempted to reform the units of measure in his domains. His units of length were based on the toise and in particular the toise de l'Écritoire, the distance between the fingertips of the outstretched arms of a man.^[12] The toise has 6 pieds (feet) each of 326.6 mm (12.86 in).

He was unsuccessful in introducing a standard unit of length throughout his realm: an analysis of the measurements of Charlieu Abbey shows that during the 9th century the Roman foot of 296.1 mm was used; when it was rebuilt in the 10th century, a foot of about 320 mm^{[Note 1]} was used. At the same time, monastic buildings used the Carolingian foot of 340 mm.^{[Note 1][13]}

The procedure for verification of the foot as described in the 16th century by Jacob Koebel in his book Geometrei. Von künstlichem Feldmessen und absehen is:^[14]

Stand at the door of a church on a Sunday and bid 16 men to stop, tall ones and small ones, as they happen to pass out when the service is finished; then make them put their left feet one behind the other, and the length thus obtained shall be a right and lawful rood to measure and survey the land with, and the 16th part of it shall be the right and lawful foot.

England

The unofficial public imperial measurement standards erected at the Royal Observatory in Greenwich in the 19th century

The measures of Iron Age Britain are uncertain and proposed reconstructions such as the Megalithic Yard are controversial. Later Welsh legend credited Dyfnwal Moelmud with the establishment of their units, including a foot of 9 inches. The Belgic or North German foot of 335 mm (13.2 inches) was introduced to England either by the Belgic Celts during their invasions prior to the Romans or by the Anglo-Saxons in the 5th & 6th century.

Roman units were introduced following their invasion in AD 43. The Roman foot had been previously standardized by Agrippa at around 296 mm or 11.65 inches. Following the Roman withdrawal and Saxon invasions, the Roman foot continued to be used in the construction crafts while the Belgic foot was used for land measurement. Both the Welsh and Belgic feet seem to have been based on multiples of the barleycorn, but by as early as 950 the English kings seem to have (ineffectually) ordered measures to be based upon an iron yardstick at Winchester and then London. Henry I was said to have ordered a new standard to be based upon his own arm and, by the c. 1300 Act concerning the Composition of Yards and Perches^[15] traditionally credited to Edward I or II, the statute foot was a different measure exactly  ¹⁰⁄₁₁ of the old foot. The barleycorn, inch, ell, and yard were likewise shrunk, while rods and furlongs remained the same.^[16] The ambiguity over the state of the mile was resolved by the 1593 Act against Converting of Great Houses into Several Tenements and for Restraint of Inmates and Inclosures in and near about the City of London and Westminster, which codified the statute mile as comprising 5,280 feet. The differences among the various physical standard yards around the world, revealed by increasingly powerful microscopes, eventually led to the 1959 adoption of the international foot defined in terms of the meter.

Definition

International foot

The international yard and pound agreement of July 1959 defined the length of the international yard in the United States and countries of the Commonwealth of Nations as exactly 0.9144 meters. Consequently, the international foot is defined to be equal to exactly 0.3048 meters. This was 2 ppm shorter than the previous U.S. definition and 1.7 ppm longer than the previous British definition.^[17]

The international standard symbol for a foot is "ft" (see ISO 31-1, Annex A). In some cases, the foot is denoted by a prime, which is often marked by an apostrophe, and the inch by a double prime; for example, 2 feet 4 inches is sometimes denoted as 2′−4″, 2′ 4″ or 2′4″. (See 'minute' for another case where prime and double prime symbols are used to denote first and second cuts in refining measurement.)

Pre-1959

In the United States, the foot was defined as 12 inches, with the inch being defined by the Mendenhall Order of 1893 by 39.37 inches = 1 m. In Imperial units, the foot was defined as  ¹⁄₃ yard, with the yard being realized as a physical standard (separate from the standard meter).

The yard standards of the different Commonwealth countries were periodically compared with one another.^[18] The value of the United Kingdom primary standard of the yard was determined in terms of the meter by the National Physical Laboratory in 1964 as 6999914396900000000♠0.9143969 m,^[19] implying a pre-1959 foot in the UK of approximately 6999304798966667000♠0.304798966667 m.

Survey foot

When the international foot was defined in 1959, a great deal of survey data was already available based on the former definitions, especially in the United States and in India. The small difference between the survey and the international foot would not be detectable on a survey of a small parcel, but becomes significant for mapping, or when the state plane coordinate system (SPCS) is used in the US, because the origin of the system may be hundreds of thousands of feet (hundreds of miles) from the point of interest. Hence the previous definitions continued to be used for surveying in the United States and India for many years, and are denoted survey feet to distinguish them from the international foot. The United Kingdom was unaffected by this problem, as the retriangulation of Great Britain (1936–62) had been done in meters.

US survey foot

The United States survey foot is defined as exactly  ¹²⁰⁰⁄₃₉₃₇ meters, approximately 6999304800609601000♠0.304800609601 m.^[20] Out of the 50 states, 24 have legislated that surveying measures should be based on the U.S. survey foot, eight have legislated that they be made on the basis of the international foot, and 18 have not specified the conversion factor from metric units.^[21]

In 1986 the National Geodetic Survey (NGS) released the North American Datum of 1983, which underlies the state plane coordinate systems and is entirely defined in meters. An NGS policy from 1991 has this to say about the units used with the new datum to define the SPCS 83:

In preparation for the adjustment of the North American Datum of 1983, 31 states enacted legislation for the State Plane Coordinate System of 1983 (SPCS 83). All states defined SPCS 83 with metric parameters. Within the legislation, the U.S. Survey Foot was specified in 11 states and the International Foot was specified in 6 states. In all other states the meter is the only referenced unit of measure in the SPCS 83 legislation. The remaining 19 states do not yet have any legislation concerning SPCS 83.^[22]

Since then, 42 states have abandoned the non-metric versions of SPCS 83: seven states continue to keep location data in survey feet as well as in meters, while one state keeps data in international feet as well as in meters.^[21] State legislation is also important for determining the conversion factor to be used for everyday land surveying and real estate transactions, although the difference (2 ppm) is of no practical significance given the precision of normal surveying measurements over short distances (usually much less than a mile).

Indian survey foot

The Indian survey foot is defined as exactly 6999304799600000000♠0.3047996 m,^[23] presumably derived from a measurement of the previous Indian standard of the yard. The current National Topographic Database of the Survey of India is based on the metric WGS-84 datum,^[24] which is also used by the Global Positioning System.

Historical use

Page from Austrian Lehrbuch des gesammten Rechnens für die vierte Classe der Hauptschulen in den k.k. Staaten – 1848^[25]

Metric foot

An ISO 2848 measure of 3 basic modules (30 cm) is called a "metric foot", but there were earlier distinct definitions of a metric foot during metrication in France and Germany.

France

In 1799 the meter became the official unit of length in France. This was not fully enforced, and in 1812 Napoleon introduced the system of mesures usuelles which restored the traditional French measurements in the retail trade, but redefined them in terms of metric units. The foot, or pied métrique, was defined as one third of a meter. This unit continued in use until 1837.^[26]

Germany

In southwestern Germany in 1806, the Confederation of the Rhine was founded and three different reformed feet were defined, all of which were based on the metric system:^[27]

• In Hesse, the Fuß (foot) was redefined as 25 cm.
• In Baden, the Fuß was redefined as 30 cm.
• In the Palatinate, the Fuß was redefined as being  33 ¹⁄₃ cm (as in France).

Other obsolete feet

Prior to the introduction of the metric system, many European cities and countries used the foot, but it varied considerably in length: the voet in Ieper, Belgium, was 273.8 millimetres (10.78 in) while the piede in Venice was 347.73 millimetres (13.690 in). Lists of conversion factors between the various units of measure were given in many European reference works including:

• Traité, Paris – 1769^[28]
• Palaiseau – Bordeaux: 1816 ^[29]
• de Gelder, Amsterdam and The Hague – 1824^[30]
• Horace, Brussels – 1840^[31]
• Noback & Noback (2 volumes), Leipzig – 1851^[32][33]
• Bruhns, Leipzig – 1881^[34]

Many of these standards were peculiar to a particular city, especially in Germany (which, before German Unification in 1871, consisted of many kingdoms, principalities, free cities and so on). In many cases the length of the unit was not uniquely fixed: for example, the English foot was stated as 11 pouces 2.6 lignes (French inches and lines) by Picard, 11 pouces 3.11 lignes by Maskelyne and 11 pouces 3 lignes by D'Alembert.^[35]

Most of the various feet in this list ceased to be used when the countries adopted the metric system. The Netherlands and modern Belgium adopted the metric system in 1817, having used the mesures usuelles under Napoleon^[36] and the newly formed German Empire adopted the metric system in 1871.^[37]

The palm (typically 200 mm to 280 mm) was used in many Mediterranean cities instead of the foot. Horace Doursther, whose reference was published in Belgium which had the smallest foot measurements, grouped both units together, while J.F.G. Palaiseau devoted three chapters to units of length: one for linear measures (palms and feet), one for cloth measures (ells) and one for distances traveled (miles and leagues). In the table below, arbitrary cut-off points of 270 mm and 350 mm have been chosen.

(In Belgium, the words pied (French) and voet (Dutch) would have been used interchangeably.)

Notes

1. 1 2 3 4 The source document used pre-metric French units (pied, pouce and lignes)
2. ↑ The original meter was computed using pre-metric French Units
3. ↑ The Norwegian fot was defined in 1824 as the length of a (theoretical) pendulum that would have a period of  ¹²⁄₃₈ seconds at 45° from the equator
4. ↑ Prior to 1835, the pé or foot was not used in Portugal – instead a palm was used. In 1835 the size of the palm was increased from 217.37 mm (according to Palaiseau) to 220 mm
5. ↑ The Scots foot ceased to be legal after the Act of Union in 1707

See also

• Anthropic units
• English units
• Mermin's foot
• Metric foot
• History of measurement
• Imperial units
• International System of Units
• Pous
• Systems of measurement
• United States customary units
• Units of measurement

Notes

1. 1 2 The original reference was given in a round number of centimeters

References