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

• 1 digit or daktylos - plural : daktyloi (19.3 mm)
• 2 digits = 1 condylos
• 4 digits = 1 palaiste
• 8 digits = 1 dichas
• 12 digits = 1 spithame
• 16 digits = 1 pous or foot - plural podes (309 mm) (Variations such as Ionic foot 296 mm to the Doric foot 326 mm)
• 20 digits = 1 pygon (called “Remen” by the Egyptians)
• 24 digits = 1 pechya or pechys (πεχυα) ( small cubit)
• 40 digits = 1 bema
• 72 digits = 4.5 feet = 1 xylon

Also:

pechys (ephtymetrikos, neilos, histonikos, thrakikos) for (24, 28, 32, 34) daktyloi,

pygme 18 daktyloi

FOOT

• 1 foot pous – plural podes (309 mm) variants from Ionic to Doric foot (296-326 mm)
• 6 feet = 1 orgyia (1.854 m)
• 10 feet = 1 akaina
• 100 feet = 1 plethron
• 600 feet or 6 plethra = 1 stadion ( 185.4 m)

STADION

• 1 stadion (185.4 m) (Different variants of the stadion exist with around + - 30 m differences)
• 2 stadia = 1 diaulos
• 6 diauloi = 1 dolichos

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:

• Schoinion, a cord of some standard length, Schoinourgos, the land surveyor. Heron of Alexandria knew that for example the length of the schoinion could change by humidity and he gave some recipes how to keep the distance as constant as possible, by hanging weights or smearing the schoinion with wax. The schoinion was 100 cubits divided in 8 hammata (knots)
• halysis (metallic chain). This did not have the problems of the cord but it was probably expensive and too heavy to be used often as the schoinion.
• kalamos measuring rod from reed or wood.

For area measurements the aroura was used which is one schoinion square.

Triangulation

Triangles are the basis of many measurement techniques, from basic geodesic measurements performed in ancient Greece to more modern laser-based 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 triangulation-based 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' 16th-century telescopic theodolite, which was used in navigation, surveying and civil engineering to determine the direction of roads, tunnels or other structures

Reconstruction of Heron's dioptra from Schöne, Hero Alexandrinus. Heronis Alexandrini opera quae supersunt omnia. Vol. III: Rationes dimetiendi et commentation dioptrica. Griechisch und Deutsch H. Schöne. - Leipzig: B. G. Teubner, 1903.

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

Bematists

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 cubit-measure 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

Black figure amphora. Men weighing merchandise, Taleides 560 - 530 BC

Examples of standard weights and some Pb (Lead) weights used by merchants

Weight from Gela

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

Arkesilaos, king of Kyrene in North Africa, supervises the weighting of merchandise. Spartan cup of mid sixth century BC.

Psychostasia Image. Sometimes the gods are weighing the souls of two persons in a duel to determine who will survive, Athenian red-figure 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/rb-rolling-ball.html

Ancient Egyptian Calendar

More Information about Calenders

Pheidon of Argos The 'invention' of weights: A compilation of source material

Miscellaneous

Calendopaedia

References

M. 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