
Part 1 Moreover, they collected from the members of the human body the proportionate dimensions which appear necessary in all building operations; the finger or inch, the palm, the foot, the cubit. and these they grouped into the perfect number which the Greeks call teleon. Now the ancients determined as perfect the number which is called ten. For from the hands they took the number of the inches; from the palm, the foot was discovered. Now while in the two palms with their fingers, ten inches are naturally complete, Plato considered that number perfect, for the reason that from the individual things which are called monades among the Greeks, the decad is perfect. But as soon as they are made eleven or twelve, because they are in excess, they cannot be perfect until they reach the second decad. For individual things are minor parts of that number. Vitruvius, BOOK III, CHAPTER 1 The planning of temples See: http://www.vitruvius.be/boek3h1.htm Jakob Köbel, Geometry, Frankfurt, Germany in1616, Definition of the rute as16 feet used as a length standard.
Greeks used different systems of measuring distances and weights, partly taken by Egyptian such as in the Hellenistic age where Greek and Egypt cultures mixed. The same word could mean different lengths such as the stadion length depending when and where it is used. This produced confusion of what the earth radius was estimated by Eratosthenes or Posidonius. But even today different systems may be used a tragic such event is the failure of a Mars probe using a mixture of the international metric system and the American systems of foots and inches. It is difficult to give real accurate distances and lengths and the following gives only a rough idea of what units were used. I will try to provide more detailed information and therefore this is only a preliminary version. Distance Measurements
Also: pechys (ephtymetrikos, neilos, histonikos, thrakikos) for (24, 28, 32, 34) daktyloi, pygme 18 daktyloi
FOOT
STADION
The cubit was a commonly used unit of length in many kingdoms; The Egyptian hieroglyphic symbol for the cubit is a picture of the forearm, indicating its derivation from the human body. It was the distance from the peak of the elbow to the tip of the middle finger, and was the basis of other units based on the human body. The digit was the width of a thumb, and a palm was the width of a hand. In most lands, four digits made a palm, and five palms made a cubit. Fractions of a cubit were used also extensively. Wood and stone cubit standards dating from 2400 BC to the first century AD and the variation over a long period of these standards for example for the cubit varied less than 5 %. The cubit had an extremely long life, and was used in some countries as late as the l960s before being replaced by metric measures. While it may be funny to think that distances were used based on human anatomy we should not forget that the foot is still used in the United States. Between 1868 and 1883 Potagos Panagiotis , a Greek explorer, visited Afghanistan. He reported that there was still the Stadion unit used since it was introduced by Alexander the Great . In the Hellenistic era different methods were used to measure distances described for example by Heron of Alexandria. Lengths were measured by:
For area measurements the aroura was used which is one schoinion square. The Rivergod Nilus with the 16 babies as described by Pliny the Elder represent the 16 cubits to which the Nile river in Egypt rose annually. Triangulation Triangles are the basis of many measurement techniques, from basic geodesic measurements performed in ancient Greece to more modern laserbased 3D cameras. The Greek philosopher Thales (6th century BC) has been credited with the discovery of five theorems in geometry. Two of which are used to solve the triangulation equations in a triangulationbased range camera: opposite angles of intersecting straight lines and the cosines law. Devices used: Dioptra is a surveying device using triangulation long before English mathematician Leonard Digges' 16thcentury telescopic theodolite, which was used in navigation, surveying and civil engineering to determine the direction of roads, tunnels or other structures
Odometer, a device used in the late Hellenistic time and by Romans for indicating distance traveled by a vehicle. Vitruvius around 27 and 23 BC describes such a device although the actual invention may have been by Archimedes during the First Punic War. Chariots with wheels of 4 feet diameter turns exactly 400 times in one Roman mile. For each revolution, a pin on the axle engage a 400 tooth cogwheel, thus making one complete revolution per mile. This engages another gear with holes along the circumference, where pebbles (calculus) are located, that drop one by one into a box. The number of miles travelled is given simply by counting the number of pebbles. Whether this instrument was actually built is disputed. Leonardo da Vinci tried to build it according to the description, but failed.
See also Leveling Chorobates an instrument that in Greek means “land racer” or “land ranger”. See a rendered image of a Chorobates But it is with us as if some one else measured us and we came to know how big we are by seeing that he applied the cubitmeasure to such and such a fraction of us. But Protagoras says ‘man is the measure of all things’, as if he had said ‘the man who knows’ or ‘the man who perceives’; and these because they have respectively knowledge and perception, which we say are the measures of objects. Such thinkers are saying nothing, then, while they appear to be saying something remarkable. Aristotle Weights
Weight from Gela Weighing of Silphium in the presence of King Arcesilaus II of Cyrene 560c. 550 BC The Egyptians and the Greeks used a wheat seed as the smallest unit of weight, a standard that was very uniform and accurate for the times. The grain is still in limited use as a standard weight. However, wheat seeds are no longer actually put in the pan of the balance scale. Instead, a weight that is practically the same as that of an average grain of wheat is arbitrarily assigned to the grain. Greeks measured dry capacity by the medimnos (25 kilogram) and liquid capacity by the metretes (34 liters). Each measure was based on the common unit of the kotyle (plural kotylai) so that the liquid metretes was 4.5 times greater in volume than the dry medimnos. DRY MEASURES 1 medimnos = 48 choenikes 1 choenix = 4 kotylai 1 kotyle = 6 kyathoi LIQUID MEASURES 1 metretes = 12 choes = 144 kotylai = 864 kyathoi 1 kotyle = 6 kyathoi 1 chous = 12 kotylai Images Psychostasia Image. Sometimes the gods are weighing the souls of two persons in a duel to determine who will survive, Athenian redfigure vase, ca. 460 BC. Paris. Musée du Louvre G399 Links The Lead Weigths from Akrotiri: The Archaeological Record Palamedes, The mythical Origin of Greek measurements and weight units SURVEYING AND ENGINEERING IN ANCIENT ROME http://www.database.com/~lemur/rbrollingball.html More Information about Calenders Pheidon of Argos The 'invention' of weights: A compilation of source material Miscellaneous ReferencesM. J. T. Lewis, Surveying Instruments of Greece and Rome,Cambridge University Press, 2001, 410 pages, 100 figures, ISBN: 0521792975 Andrι Wegener Sleeswyk, "Vitruvius' Odometer", Scientific American October 1981 Dieter Lelgemann, Recovery of the Ancient System of Foot/Cubit/Stadion – Length Units (PDF File) with some interesting ideas about Greek units and their relation to Egyptian units and to astronomical lenghts. Robert R. Stieglitz , Classical Greek Measures and the Builder’s Instruments from the Ma‘agan Mikhael Shipwreck , Volume 110, Issue 2, Apr 2006 Ancient Greek texts on measurement Preliminary version

