Units of weight and volume are also considered

People have always found it necessary to measure time, distance, area, volume and weight, and have devised units that measure these quantities. For time, there is an absolute standard in the motions of the heavens, but for the other quantities the units have had to be chosen arbitrarily, and it is these that we talk about here. Only recently have we succeeded in creating natural units for length and time, but weight still is arbitrary. Let's now examine some of the ways things have been measured, concentrating on the measurement of length.

The metric system, originating in the French Revolution and propagated widely in the 19th century, has brought a dreary but convenient uniformity to units of measurement. Only the English foot-pound-second system put up resistance to this aggression, upholding an alternative comprehensive system. In engineering, English units were divided decimally just like metric ones, especially in the United States. The *kilopound* or *kip*, used in structural engineering, is an example, as is the decimal foot in engineering and geodetic surveying, which replaced all other units. Engineering has decided that the metre (if you go metric all the way, you must accept its spelling as well) is an inconvenient unit, and uses the millimetre as its standard instead. A small unit appears to be favored in practical use, without the definition of larger units. In all but precision engineering, one millimeter is sufficient accuracy.

The basic English unit of length was the *yard* of three feet, or the *fathom* of six. Each English foot was divided into 12 *inches*, and each inch into 3 *barleycorns* or 12 *lines*. Length units used in land surveying are discussed in the article "Chaining." Other units were defined for use in printing, textiles, and other areas of commerce. This proliferation of units for special applications was a very common phenomenon, and led to uncertainties in exact definitions. Eventually, one inch was defined as exactly 25.4 mm, which tied the English and metric units together. In the United States, a meter was sometimes defined as exactly 39.37 inches, which gave 1 inch = 25.40005 mm, just enough different to be annoying in geodesy. 12 such inches made a *survey foot*, used by the Coast and Geodetic Survey. 5 feet, 6 inches, and 7 lines was written 5' 6" 7"'. The single and double apostrophes have survived into modern times, but not the triple.

Before the metric system, the French unit of length was the *toise*, which is about 1949 mm. The toise was divided into 6 *Paris feet*, each of which was divided into 12 *pouces*, that were further subdivided into 12 *lignes*. The use of 'Paris' to modify the foot suggests that there were other feet, and there were. Differences between units of the same name at different locations were common in France, and the leading stimulus for the establishment of a uniform national system. The differences in units were sometimes used for swindling of consumers. The equality 1071 Paris feet = 1142 English feet, quoted by Poggendorff in his history, can be used to convert 17th century measurements into their equivalents. This makes the Paris *pouce* 27.1 mm. The *ligne usuel* was later defined as 0.09113 inches, and a toise as 78.74172 inches. The *livre* was 1.10258 pounds, the *boisseau* 2.7512 British gallons. The *litron* was 1.7608 British pint. These *Système Usuel* units were in use from 1812 to 1840, since the people refused to use the new metric units, and this at least gave some uniformity. In 1840 the use of metric units was compelled (compulsion is a great part of French liberté) under penalty, and the traditional names were forbidden. This is similar to what is happening in Britain at the present time. In the *Système Ancien*, used before 1812, the ligne was 0.0888 inches, the toise 6.3946 feet, the lieue was 2282 toise or 2.7637 miles. However, the old units varied from place to place in a confused way.

In Germany, the English and French feet competed as units of measurement of length, but there were local definitions as well. Germany was a patchwork of local jurisdictions at the time, each jealous of its individuality. The Prussian inch appears to have been 24.6 mm, since the wine gallon is reported to have been 254 cubic Prussian inches. It is difficult to assess the accuracy of old measurements because of the uncertainty in the size of the units, or even in their proper identification.

The units of length shown in the table were in use in Greek areas by the time of Herodotus (450 BC), who is our principal authority on units of his time. He is rather careful, and well aware of different definitions. The parasang is of Persian origin, the schoenus Egyptian. Herodotus explains how successively larger units are used by those richer and richer in land to survey their property. The relations between units are reliable, but the equivalents in modern units are only approximate. There were no universal standards in ancient times; indeed, there were none until recently. The fathom, for example, might more closely be 6 ft 1 in. The Greek foot was about 1% larger than the English foot.

The later milion, milion, was a Roman mile of 1000 paces, 1680 modern yards, divided into 8 stadia (corresponding to modern furlongs), each stadion consisting of 100 orguiai, or 6 plethra, pleqra. The stadion was from 200 to 210 yards in length, and would probably be laid out with a cord 10 orguiai (60 ft) in length, just as furlongs were later ten chains in length. The plethron was also a measure of area, a square one plethron on a side. It was common to express area measures in terms of the length of a square of equal area, so when reading one must determine whether length or area is meant. The square plethron was a little less than a quarter of an acre, equivalent to the Roman *jugerum* but not equal to it. Units are often translated as approximate equivalents, another source of uncertainty.

The modern feet are descended from the Roman measurement of the same name and approximate value. The Roman foot, 11.65 modern inches (29.6 cm), was usually divided into 16 inches, not 12, however (as four palms of four Roman inches, about 3 modern inches, each). Divisions by powers of 2 are specially useful, since they are *binary*, and much more adapted to computers than powers of 10. The English inch was later divided into halves, quarters, eighths, and so on, because of the utility and extendibility of this system, which completely replaced the use of lines. The Roman pound, *libra* (from which, lb.), was divided into 12 ounces, but the ounces were divided on the binary system down to 32nds, or *oboli*. Roman standards were relatively uniform, an interlude between times of confusion. Most Roman units of length survived in name or spirit in the English and other systems, even if changing somewhat in absolute value. For example, the *stadium*, which was 1/8 of a Roman mile, or 202 yards, became the *furlong*, 1/8 of an English mile, or 220 yards. The *cubit*, a forearm's length, was 1-1/2 Roman feet or 6 palms, and typically used in building. Some ancient cubits seem to have been longer than this, up to about 22 inches. *Hands* of 4 inches are still used to measure the height of a horse (at the shoulders). Note how length units were conveniently based on parts of the body used to measure distances.

The Roman mile consisted of 1000 double paces, or 5000 Roman feet, or 1480 metre, or 1619 yards. Distances on Roman roads were measured by odometers attached to carriage axles, as described in Vitruvius, and marked on mile stones. The English mile of 5280 feet is 1609 metre (a "metric mile" is, apparently, 1500 metre). As explained in the article "Chaining," it was defined as 80 chains of 66 ft each, and this is the reason for the odd number 5280. Gunter's chain of 100 links was a successful attempt to create a portable length standard that was not as stretchy as a cord. The English mile happened to come out a little larger than the Roman mile, to which it was intended to be an approximation. The nautical mile is 1852 metre, which corresponds to one minute of arc of latitude, approximately. The 'geographical' mile was 7420 metre, and the Prussian mile 7532 metre. These long miles were about five Roman miles. The *league* was another measure of journeys, usually 3 English miles. France had an assortment of leagues: 2000 toise for the *lieue de poste*, 3 Roman miles for the *lieue de terre*, 4 kilometers for the *lieue kilométrique*, and 3 nautical miles for the *lieue marine*. The Greeks had the *stadium* of 580-622 feet, and the *plethron* of 97-100 feet. The ancient Persian *parasang* was 3.25 to 3.3 miles, 30 Greek stadia. Any great accuracy in the size of old units is illusory unless a critical study is made. The standards have, of course, disappeared, and their magnitude can be determined only by remeasuring a distance in modern terms.

In the establishment of the metric system, the quadrant of the earth was measured as 5 130 738.62 toise, which was set equal to 10 000 000 metre. This was a bad choice for defining a unit of length, worse than simply making a couple of arbitrary scratches on a bar, since it involved a difficult and tedious procedure, especially when the news arrived in France that the earth was not a sphere to a sufficient approximation. At any rate, a scratched bar was later adopted as standard, and the earth's quadrant allowed to be whatever it turned out to be in terms of metres. A similar mistake was made in defining the kilogram as the weight of a cubic decimetre (a litre) of water, since the density of water changes with temperature, and weight can be measured more accurately than a decilitre can be measured. Thus, both the litre and the kilogram were defined arbitrarily. The clumsiness was reflected in the slight difference between a cubic centimetre and a millilitre that thereby arose. The second of time remained 1/86400 of a mean solar day, with common units of time not related decimally, but by factors of 24 and 60. The decimal calendar was a ludicrous failure. The French also defined a right angle as 100 grads, another superfluity. The units of time and angles were already uniform, and required no work. As has been mentioned above, the French population ignored the metric system until it was forced upon them.

Britain introduced Imperial Units, based on the yard, pound, and second, in the 19th century to resist the metric system. The gallon was newly defined as the volume occupied by ten pounds of water, in stupid imitation of the metric system, which made the Imperial gallon (eight pints) somewhat larger than the previous, and American, wine gallon of 231 cubic inches. An Imperial gallon is 1.2032 US gallon, more exactly. A pint, which was now 20 ounces instead of 16, contained 25% more beer. In the Imperial system 1 gallon = 2 pottles = 4 quarts = 8 pints = 32 gills. Spirits were dispensed in gills. The basis of the US dry measure was the old Winchester struck bushel of 2150.42 cubic inches, equal to 4 pecks = 8 gallons = 32 quarts = 64 pints. British dry measure was slightly different, a bushel being 2218.192 cubic inches, but divided like the American. Measures of weight are the same in Britain and the US. A *grain* is the same in troy (used for precious metals), avoirdupois and apothecary's systems. In the troy system, 5760 grains = 12 ounces = 1 pound (troy). An avoirdupois pound is 7000 grains or 16 ounces. Apothecary's weight is the same as troy, except that the ounce is divided into 8 drams of 3 scruples each, with 20 grains to the scruple. The *stone* of 14 pounds is used to measure the weights of people in Britain.

A hundredweight is usually 112 pounds, and twenty hundredweight, 2240 pounds, is a ton. The units used to measure weights of coal in Britain were extremely various and local, and the hundredweight is the sole surviving representative of such units. One might assume that a hundredweight would be 100 pounds, and a ton 2000 pounds, as in fact they are in the United States. The reason for the additional eighth seems to have originated to compensate for the stone and slack that would be included with a miner's coal. Then, 2240 pounds (or gross ton) weight delivered by the miner would be paid for as 2000 pounds (or net ton) of coal. The heavier ton was in common use in the United States before 1900, though the net ton is now understood when tons are mentioned. On the other hand, 112 pounds is exactly 8 stone, and stones did not originate in mining measurements. The gross ton is probably the original unit, the net ton derived from it by subtracting about an eighth.

The Russian foot was the same as the English. The *sachine* was 7 feet, and the *verst* 500 sachine or 3500 feet (not far from a kilometer). The *pood* was 36.114 pounds.

The Castilian *vara* was 32.8748 inches. In California, it was legally 33.372 inches, and in Texas, I think it was 33 inches exactly. Land originally settled from Spain was surveyed in varas. The *legua* was 5000 varas, about 2.6335 miles. Many of the traditional units differed from locality to locality, which was the big problem with traditional units. The *cántara* of wine was 4.1 US gallons in Havana, 4.263 gallons in Castile. For precious metals, 1 marco = 50 castellanos = 400 tómine = 4800 Spanish gold grains. The *castellano* was from 71.07 to 71.04 grains avoirdupois. Three US gold dollars weighed 1.1 castellano. The *marco* was very close to 1/2 pound avoirdupois, but the figure varied from .50 to .54 pound. For larger weights, the *libra* was 1.0161 pounds, the Madrid *arroba* 25.4025 pounds (25 libras), the *quintal* 101.61 pounds (100 libras), and the Castilian *tonelada* 2032.2 pounds (2000 libras).

Length is now defined in terms of the speed of light, assumed to be an exact value (299792458 m/s), and time by the frequency of atomic vibrations, to a rather great precision. Weight is still defined by an arbitrary kilogram, or rather a collection of them. It is called *mass*, with undeserved dignity, since macroscopic comparisons are made by weighing. Masses on an atomic scale can be compared to better than one part in 10^{8}, which is better than weighing, so some atomic definition of the unit of mass is probably overdue so that uncertainties in the standard do not affect the values.

R. A. Young and T. J. Glover, *Measure for Measure* (Littleton, CO: Blue Willow, 1996). An extensive table of equivalents, including many local units and curiosities.

If you have a lot of converting to do, you might visit Online Length Unit Converter

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Composed by J. B. Calvert

Created 3 July 1999

Last Revised 13 May 2010