rule

=Rule=

toc This article defines the word rule in the sense of a linear measuring device, and clarifies its application and usefulness in scale studies.

=Overview= The ruler is highly significant to studies of scale. It figures as an embodied measurement, an operational definition of distance measure, and a metaphor for the act of measurement.

In this article we inventory aspects of rulers that are useful to superscale studies.

=Definition= A ruler, sometimes called a rule or line gauge, is an instrument used in tailoring, geometry, technical drawing, printing, building construction, and civil engineering to measure distances and/or to rule straight lines.

Strictly speaking, the ruler is essentially a straightedge used to rule lines, but typically the ruler also contains calibrated lines to measure distances that are shorter than the rule itself.

Rulers are also used to help people work with logarithms and exponents. These rulers must be calibrated with logarthmic scales.

=History of Rules=

Ancient India
Rulers made of ivory were in use by the Indus Valley civilization period prior to 1500 BC. Excavations at Lothal (dated to 2400 BC) yielded one such ruler calibrated to about 1⁄16 in. (1.6 mm). Ian Whitelaw holds that the Mohenjo-Daro ruler is divided into units corresponding to 1.32 in. (33.5 mm) and these are marked out in decimal subdivisions with amazing accuracy, to within 0.005 in (0.13 mm). Ancient bricks found throughout the region have dimensions that correspond to these units.

Ancient East Asia
Measuring rods for different purposes and sizes (construction, tailoring and land survey) have been found from China and elsewhere dating to the early 2nd millenium B.C.E.

The sun-god Shamash holding a ring of coiled rope and a rod. Assyrian Ruler holding a measuring rod. Public domain.

Measuring rod and coiled rope depicted in the Code of Hammarubi. The upper part of the stela of Hammurapis' code of laws. Public Domain.

Ur-Nammu-stela showing detail of rod, ring and beaded measuring flail. Ur Nammu with surveying instruments. Public domain.

Ancient Egypt
Flinders Petrie reported on a rod that shows a length of 520.5 mm, a few mm less than the Egyptian cubit. A slate measuring rod was also found, divided into fractions of a Royal Cubit and dating to the time of Akhenaten.

Further cubit rods have been found in the tombs of officials. Two examples are known from the tomb of Maya—the treasurer of Tutankhamun—in Saqqara. Another was found in the tomb of Kha (TT8) in Thebes. These cubits are ca 52.5 cm long and are divided into seven palms, each palm is divided into four fingers and the fingers are further subdivided. Another wooden cubit rod was found in Theban tomb TT40 (Huy) bearing the throne name of Tutankhamun (Nebkheperure).

Egyptian measuring rods also had marks for the Remen measurement of approximately 370mm, used in construction of the Pyramids. Two statues of Gudea of Lagash in the Louvre depict him sitting with a tablet on his lap, upon which are placed surveyors tools including a measuring rod.

Seal 154 recovered from Alalakh, now in the Biblioteque Nationale show a seated figure with a wedge shaped measuring rod. The Tablet of Shamash recovered from the ancient Babylonian city of Sippar and dated to the 9th century BC shows Shamash, the Sun God awarding the measuring rod and coiled rope to newly trained surveyors. (Hebrew reference [|here].)

A similar scene with measuring rod and coiled rope is shown on the top part of the diorite stele above the Code of Hammurabi in the Louvre, Paris, dating to ca. 1700 BC.

The Graeco-Egyptian God Serapis is also depicted in images and on coins with a measuring rod in hand and a vessel on his head. The most elaborate depiction is found on the Ur-Nammu-stela, where the winding of the cords has been detailed by the sculptor. This has also been described as a "staff and a chaplet of beads".

Ancient Greece
The "measuring rod" or tally stick is common in the iconography of Greek Goddess Nemesis.

Ancient Europe
An Oak rod from the Iron Age fortified settlement at Borre Fen in Denmark mearure 53.15 inches (135.0 cm), with marks dividing it up into eight parts of 6.64 inches (16.9 cm), corresponding quite closely to half a Doric Pous (a Greek foot). A Hazel measuring rod recovered from a Bronze Age burial mound in Borum Eshøj, East Jutland by P. V. Glob in 1875 mearured 30.9 inches (78 cm) corresponding remarkably to the traditional Danish foot. The megalithic structures of Great Britain has been hypothesized to have been built by a "Megalithic Yard", though some authorities believe these structures have been measured out by pacing. Several tentative Bronze age bone fragments have been suggested as being parts of a measuring rod for this hypothetical measurement.

Roman Empire
Large public works and imperial expansion, particularly the large network of Roman roads and the many milecastles, made the measuring rod an indispensable part of both the military and civilian aspects of Roman life. Republican Rome used several measures, including the various Greek feet measurements and the Oscan foot of 23,7 cm. Standardization was introduced by Agrippa in 29 BC, replacing all previous measurements by a Roman foot of 29.6 cm, which became the foot of Imperial Rome.

The Roman measuring rod was 10 Roman feet long, and hence called a decempeda, Latin for ‘ten feet’. It was usually of square section caped at both ends by a metal shoe, and painted in alternating colors. Together with the groma and Dioptra the decempeda formed the basic kit for the Roman surveyors. The measuring rod is frequently found depicted in roman art showing the surveyors at work. A shorter folding yardstick one Roman foot long is known from excavations of a Roman fort in Niederburg, Germany.

Middle Ages
In the Middle Ages, bars were used as standards of length when surveying land. These bars often used a unit of measure called a Rod (unit) of length equal to 5.5 yards, 5.0292 metres, 16.5 feet, or 1⁄320 of a statute mile. A rod is the same length as a perch or a pole. In old English, the term lug is also used. The length is equal to the standardized length of the ox goad used for teams of eight oxen by medieval English plowmen. The lengths of the perch (one rod unit) and chain (four rods) were standardized in 1607 by Edmund Gunter.

Modern
Anton Ullrich invented the folding ruler in 1851.

=Rules in Mythology=

Ancient
Early use of tally sticks is mentioned in the debate between sheep and grain along with cubits measured with cords.

Inanna
The myth of Inanna's descent to the nether world describes how the goddess dresses and prepares herself: "She held the lapis-lazuli measuring rod and measuring line in her hand."

Lechesis
Lachesis in Greek mythology was one of the three Moirae (or fates) and "allotter" (or drawer of lots). She measured the thread of life allotted to each person with her measuring rod. Her Roman equivalent was Decima (the 'Tenth').

Serapis
Serapis on crocodile: “The Alexandrian Serapis” Etching from Goeree, Willem (1700) Mosaize Historie der Hebreeuwse Kerke, reprinted in Hall, Manly P. (1928) The Secret Teachings of All Ages. Photo by Fuzzypeg (all rights released). Serapis holding a measuring rod. Public domain.

Varuna
Varuna in the Rg Veda, is described as using the Sun as a measuring rod to lay out space in a creation myth. W. R. Lethaby has commented on how the measurers were seen as solar deities and noted how Vishnu "measured the regions of the Earth".

=Rules in the Bible=

Measuring rods or reeds are mentioned many times in the Bible.

A measuring rod and line is seen in a vision of Yahweh in Ezekiel 40:2-3: In visions of God he took me to the land of Israel and set me on a very high mountain, on whose south side were some buildings that looked like a city. He took me there, and I saw a man whose appearance was like bronze; he was standing in the gateway with a linen cord and a measuring rod in his hand.

Another example is Revelation 11:1: I was given a reed like a measuring rod and was told, "Go and measure the temple of God and the altar, and count the worshipers there". The measuring rod also appears in connection with foundation stone rites in Revelation 20:14-15: And the wall of the city had twelve foundation stones, and on them were the twelve names of the twelve apostles of the Lamb. The one who spoke with me had a gold measuring rod to measure the city, and its gates and its wall.

=The Traditional Rule==

The traditional ruler is a linear, calibrated stick.

=Logarithmic Rule=

Logarithmic scale rule is most common in slide rules, see below.

=Duchamp’s Rule=

The artist Duchamp created unusual rules by dropping one meter threads from a one meter height onto wood, then cutting and varnishing the rules to the outline of the thread. The rules have no calibrated scale.

Slide Rule
A tool for calculation that is based on the mapping of additive numbers to multiplicative numbers.

The logarithmic slide rule was configured by William Oughtred (1574–1660) as early as 1621. The dilemmas of computation in the recent decades have prefigured in the construction and worldwide use of tens (if not hundreds) of millions of slide rules, linear, circular, cylindrical, or hybrid, wooden, metal, or plastic, handmade or mass produced, cheap or expensive, with accessories such as cursors and magnifying glasses to increase the accuracy without increasing the size.

Recent debates over special versus general purpose software and over competing software operating systems was long rehearsed in debates over choice of general or special purpose slide rule scales and scale system standards. Moreover, as scales of all sorts slid beside each other or within each other, the logarithmic slide rule turned out to be only one of innumerable versions of slide rules. Material culture scholars see no end to collection of calculation wheels devised to compute phenomena ranging from a menstrual cycle to a baseball season. (Calculation and Computation, encyc. History of Ideas, p.421)

Logarithmic Scale

Logarithmic Scale for Slide Rule

Multi-rule Linear Slide Rule Scale

Multi-rule Circular Slide Rule Scale

Single Rule Circular Slide Rule Scale

Vernier
A vernier scale is an additional scale which allows a distance or angle measurement to be read more precisely than directly reading a uniformly-divided straight or circular measurement scale. It is a sliding secondary scale that is used to indicate where the measurement lies when it is in between two of the marks on the main scale.

Verniers are common on sextants used in navigation, scientific instruments used to conduct experiments, machinists' measuring tools (all sorts, but especially calipers and micrometers) used to work materials to fine tolerances and on theodolites used in surveying. When a measurement is taken by mechanical means using one of the above mentioned instruments, the measure is read off a finely marked data scale (the "fixed" scale, in the diagram). The measure taken will usually be between two of the smallest graduations on this scale. The indicating scale ("vernier" in the diagram) is used to provide an even finer additional level of precision without resorting to estimation.

The vernier scale was invented in its modern form in 1631 by the French mathematician Pierre Vernier (1580–1637). In some langauges, this device is called a nonius. It was also commonly called a nonius in English until the end of the 18th century. Nonius is the Latin name of the Portuguese astronomer and mathematician Pedro Nunes (1502–1578) who in 1542 invented a related but different system for taking fine measurements on the astrolabe that was a precursor to the vernier.

=Cartesian Plane Rules=

The Cartesian plane or Cartesian scale is divided into abscissa (x-axis) and ordinate (y-axis). Each is calibrated to a scale.

Graph Paper
Graph paper is a type of Cartesian plane rule.

Semi-logarithmic Cartesian Rule
Semi-logarithmic graph paper is graph paper that has an arithmetic scale on one axis and a logarithmic scale on the other axis. Semi-log graph paper comes in several different ranges that cover different cycles or orders of magnitude. For example, paper with divisions from 1 to 10 is 1-cycle paper, from 1 to 100 is 2-cycle paper and paper from 0.01 to 1,000 is 5-cycle paper. Divisions on individual cycles are often labeled from 1 to 9 so that the user can decide the scan of the scales, but individual divisions must not be renumbered.

When using semi-log paper, it is important to choose the paper with an appropriate number of cycles. For instance, if your data spans from 0.13 to 7.9, then you should use two-cycle rather than five-cycle paper.

The minor divisions (those between 1 and 10) on the logarithmic scale are also separated by logarithmic divisions. Thus the distance between 2 and 4 is the same as the division between 4 and 8. The position of the lines is in proportion to the logarithm of the number that they represent. Using this scale, 3 lies about half way between 1 and 10, because log 3 = 0.4771. On the arithmetic scale 3 lies about one-third of the way between 1 and 10.

If you are using a 3-cycle paper, and the first cycle is used to represent numbers between 1 and 10, then the next minor division represents 20, the following 30 and so on until the last division of the second cycle, which represents 100. The next division represents 200 and so on up to 1,000.

On the logarithmic scale, you should notice that successive doublings of a number (for example from 1 to 2, from 2 to 4 or from 4 to 8, etc.) result in identical changes in distance. In fact, successive multiplication by any number always results in identical changes on the log scale. This is the principle of the old-fashioned slide-rule.

Semi-logarithmic graph paper allows you to plot your 'raw' data directly without the need first to transform it into logs numerically. To illustrate the use of semi-logarithmic graph paper, consider the data obtained from a culture of Escherichia coli, observed over a four hour period at half hour intervals.

=Links and References=

For other pages about types of rules, see architect's scale, linear encoder, nomogram or engineer's scale.

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