Talk:Rhind Mathematical Papyrus
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There are many images of the Rhind Papyrus on the Internet. I'd imagine at least some of them are common property, esp. this one: http://fr.wikipedia.org/wiki/Papyrus_Rhind
Can someone with more wiki-aptitude than me upload a related image?
Dan McCarty 17:10, 16 January 2007 (UTC)
- done Thanatosimii 21:12, 16 January 2007 (UTC)
-
- The Egyptian mathematics found in the Rhind Papyrus were picked up upon and used by other cultures such as the Greeks and Romans in setting the standards for commodities such as bread and beer, the size of containers, architectural proportions, the area of fields and the doubling system by which standards of length were related to standards of area and volume to ease calculations.
The Egyptian use of proportion in calculation is briefly discussed in Gillings. In particular the use of the Remen which has two values is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling is the basis of most of the unit fraction calculations, up to and including the calculations of circles with dimensions given in khet, perhaps looking at how the remen is used will provide some insights.
The Remen is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. In its earliest form it is the diagonal of a square, with its sides a cubit. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.
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[edit] Remen as a Proportion to other units
The proportion of foot to remen can be either 4:5 making it the hypotenuse or 3:4 making it the side of a right triangle. If the remen is the hypotenuse of a 3:4:5 triangle then the foot is one side and the quarter another so the proportions are 3:4 quarter to foot, 4:5 foot to remen and 3:5 quarter to Remen. The quarter is 1/4 yard. The foot is 1/3 yard.
The remen may also be the side of a square whose diagonal is a cubit The proportion of remen to cubit is 4:5
- The proportion of palm to remen is 1:5
- The proportion of hand to remen is 1:4
- The proportion of palm to foot is 1:4
- The proportion of hand to foot is 1:3
The table below demonstrates a harmonious system of proportion much like the musical scales, with fourths and fifths, and other scales based on geometric divisions, diameters, circumferences, diagonals, powers, and series coordinated with the canons of architectural proportion, Pi, phi and other mathematical constants.
In Mesopotamia and Egypt the Remen could be divided into different proportions as a similar triangle with sides as fingers, palms, or hands. The Egyptians thought of the Remen as proportionate to the cubit or mh foot and palm. They used it as the diagonal of a unit rise or run like a modern framing square. Their relatedseked gives a slope. Its convenient to think of remen as intermediate to both large and small scale elements.
Even before the Greeks like Solon, Herodotus, Pythagorus, Plato, Ptolomy, Aristotle, Eratosthenes, and the Romans like Vitruvius, there seems to be a concept that all things should be related to one another proportionaly.
Its not certain whether the ideas of proportionality begin with studies of the elements of the body as they relate to scaling architecture to the needs of humans, or the divisions of urban planning laying out cities and fields to the needs of surveyors with their knotted cords.
In all cultures the canons of proportion are proportional to reproducable standards. In ancient cultures the standards are divisions of a degree of the earths circumference into mia chillioi, mille passus, and stadia. Stadia, are used to lay out city blocks, roads, large public buildings and fields. Remen are the most common subdivision of land measure
Fields are divided into acres using as their sides, furlongs, perches, cords, rods, fathoms, paces, yards, cubits, and remen which are proportional to miles and stadia
Buildings are divided into feet, hands, palms and fingers, which are also systematized to the sides of agricultural units.
Inside buildings the elements of the architectural design follow the canons of proportion of the the inscription grids based on body measures and the orders of architectural components.
In manufacturing the same unit fraction proportions are systematized to the length and width of boards, cloth and manufactured goods.
The unit fractions used are generally the best sexigesimal factors, three quarters, halves, 3rds, fourths, fifths, sixths, sevenths, eighths, tenths, unidecimals, sixteenths and their inverses used as a doubling system.
Greek Remen generally have long, median and short forms with their sides related geometrically as arithmetric series or geometric series based on hands and feet.
- The Egyptian bd is 300 mm and its remen is 375 mm. the proportion is 1:1.25
- The Ionian pous and Roman pes are a short foot measuring 296 mm their remen is 370 mm
- the Old English foot is 3 hands (15 digits of 20.32 mm) = 304.8 mm and its remen is 381 mm
- The Modern English foot is 12 inches of 25.4 mm = 304.8 mm and its remen is 381 mm (15")
- The Attic pous measures 308.4 mm its remen is 385.5
- The Athenian pous measures 316 mm and is considered of median length its remen is 395 mm
- Long pous are actually Remen (4 hands) and pygons
- See cubit for the discussion of the choice of division into hands or palms
- See the table below for proportions relative to other ancient Mediterranean units
Roman Remen generally have long, and short forms with their sides related geometrically as arithmetric or geometric series based on fingers palms and feet.
By Roman times the Remen is standardized as the diagonal of a 3:4:5 triangle with one side a palmus and another a pes. The Remen and similar forms of sacred geometry formed the basis of the later system of Roman architectural proportions as described by Vitruvius.
Generally the sexagesimal (base-six) or decimal (base-ten) multiples have Mesopotamian origins while the septenary (base-seven) multiples have Egyptian origins.
| Unit | Finger | Culture | Metric | Palm | Hand | Foot | Remen | Pace | Fathom | |
|---|---|---|---|---|---|---|---|---|---|---|
| (1 ŝuŝi | 1 (little finger) | Mesop | 14.49 mm | .2 | 0.067 | 0.05 | ||||
| 1 ŝushi | 1 (ring finger) | Mesop | 16.67 mm | .2 | 0.67 | 0.05 | ||||
| 1 shushi | 1 (ring finger) | Mesop | 17 mm | .2 | 0.67 | 0.05 | ||||
| 1 digitus | 1 (long finger) | Roman | 18.5 mm | .25 | 0.0625 | 0.04 | ||||
| 1 dj | 1 (long finger) | Egyptian | 18.75 mm | .25 | 0.0625 | 0.04 | ||||
| 1 daktylos | 1 (index finger) | Greek | 19.275 mm | .2 | 0.067 | 0.04 | ||||
| 1 uban | 1 (index finger) | Mesop | .2 | .2 | 0.067 | 0.04 | ||||
| 1 finger | 1 (index finger) | Old English | 20.32 mm | .2 | 0.067 | 0.045 | ||||
| 1 inch | (thumb) | English | 25.4 mm | 0.083 | .067 | |||||
| 1 uncia | (thumb or inch) | Roman | 24.7 mm | .25 | 0.083 | .067 | ||||
| 1 condylos | 2 (daktylos) | Greek | 38.55 mm | .5 | 2 | .1 | ||||
| 1 palaiste, palm | 4 (daktylos) | Greek | 77.1 mm | 1 | 0.25 | .2 | ||||
| 1 palaistos, hand | 5 (daktylos) | Greek | 96.375 mm | 1 | 0.333 | .25 | ||||
| 1 hand | 5 (fingers) | English | 101.6mm | 1 | 0.333 | .25 | ||||
| 1 dichas, | 8 (daktylos) | Greek | 154.2 mm | 2 | 0.5 | .4 | ||||
| 1 spithame | 12 (daktylos) | Greek | 231.3 mm | 3 | .75 | .6 | ||||
| 1 pous, foot of 4 palms | 16 (daktylos) | Ionian Greek | 296 mm | 4 | 1 | .8 | ||||
| 1 pes, foot | 16 (digitus) | Roman | 296.4 mm | 4 | 1 | .8 | ||||
| 1 uban, foot | 15 (uban) | Mesop | 300 mm | 3 | 1 | .75 | ||||
| 1 bd, foot | 16 (dj) | Egyptian | 300 mm | 4 | 1 | .8 | ||||
| 1 foote(3 hands) | 15 (fingers) | Old English | 304.8 mm | 3 | 1 | .75 | ||||
| 1 foot, (12 inches) | 16 (inches) | English | 308.4 mm | 3 | 1 | .75 | ||||
| 1 pous, foot of 4 palms | 16 (daktylos) | Attic Greek | 308.4 mm | 4 | 1 | .8 | ||||
| 1 pous, foot of 3 hands | 15 (daktylos) | Athenian Greek | 316 mm | 4 | 1 | .8 | ||||
| 1 pygon, remen | 20 (daktylos) | Greek | 385.5 mm | 5 | 1.25 | 1.25 | 1 | |||
| 1 pechya, cubit | 24 (daktylos) | Greek | 462.6 mm | 6 | 1.5 | 1.1 | ||||
| 1 cubit of 17.6" 6 palms | 25 (fingers) | Egyptian | 450 mm | 6 | 1.5 | 1.3 | ||||
| 1 cubit of 19.2" 5 hands | 25 (fingers) | English | 480 mm | 5 | 1.62 | 1.3 | ||||
| 1 mh royal cubit | 28 (dj) | Egyptian | 525 mm | 7 | 2.33 | 1.4 | ||||
| 1 bema | 40 (daktylos) | Greek | 771 mm | 10 | 2.5 | 2 | ||||
| 1 yard | 48 (finger) | English | 975.36 mm | 12 | 3 | 2.4 | ||||
| 1 xylon | 72 (daktylos) | Greek | 1.3878 m | 18 | 4.55 | 3.64 | ||||
| 1 passus pace | 80 (digitus) | Roman | 1.542 m | 20 | 5 | 4 | 1 | |||
| 1 orguia | 96 (daktylos) | Greek | 1.8504 m | 24 | 6 | 5 | 1 | |||
| 1 akaina | 160 (daktylos) | Greek | 3.084 m | 40 | 10 | 8 | 2 | |||
| 1 English rod | 264 (fingers) | English | 5.365 m | 66 | 16.5 | 13.2 | ||||
| 1 hayt | 280 (dj) | Egyptian | 5.397 m | 70 | 17.5 | 14 | 3 | |||
| 1 perch | 1,056 (fingers) | English | 20.3544 m | 264 | 66 | 53.4 | 11 | |||
| 1 plethron | 1,600 (daktylos) | Greek | 30.84 m | 400 | 100 | 80 | 20 | |||
| 1 actus | 1,920 (digitus) | Roman | 37.008 m | 480 | 120 | 96 | 24 | 20 | ||
| khet side of 100 royal cubits | 2,800 (dj) | Egyptian | 53.97 m | 700 | 175 | 140 | 35 | |||
| iku side | 3,600 (ŝushi) | Mesop | 60m | 720 | 240 | 180 | 48 | 40 | ||
| acre side | 3,333 (daktylos) | English | 64.359 m | 835 | 208.71 | 168.9 | ||||
| 1 stade of Eratosthenes | 8,400 (dj) | Egyptian | 157.5 m | 2100 | 525 | 420 | 84 | 70 | ||
| 1 stade | 8,100 (shushi) | Persian | 162 m | 2700 | 900 | 525 | 85 | |||
| 1 minute | 9,600 (daktylos) | Egyptian | 180 m | 2400 | 600 | 480 | 96 | 80 | ||
| 1 stadion 600 pous | 9,600 (daktylos) | Greek | 185 m | 2400 | 600 | 480 | 96 | 80 | ||
| 1 stadium625 pes | 9,600 (daktylos) | Roman | 185 m | 2400 | 625 | 500 | 100 | |||
| 1 furlong 625 pes | 10,000 (digitus) | Roman | 185.0 m | 2640 | 660 | 528 | 132 | 88 | ||
| 1 furlong 600 pous | 9900 (daktylos) | English | 185.0 m | 1980 | 660 | 528 | 132 | 88 | ||
| 1 Olympic Stadion 600 pous | 10,000 (daktylos) | Greek | 192.8 m | 2500 | 625 | 500 | 100 | |||
| 1 furlong 625 fote | 10,000(fingers) | Old English | 203.2 m | 2500 | 635 | 500 | 100 | |||
| 1 stade | 11,520 (daktylos) | Persian | 222 m | 2880 | 720 | 576 | 144 | 120 | ||
| 1 cable | 11,520 (daktylos) | English | 222 m | 2880 | 720 | 576 | 144 | 120 | ||
| 1 furlong 660 feet | 10,560 (inches) | English | 268.2 m | 2640 | 660 | 528 | 132 | 110 | ||
| 1 diaulos | 19,200 (daktylos) | Greek | 370 m | 4800 | 1,200 | 960 | 192 | 160 | ||
| 1 English myle | 75,000(fingers) | Old English | 1.524 km | 15000 | 5,000 | 4000 | 800 | |||
| 1 mia chilioi | 80,000 (daktylos) | Greek | 1.628352 km | 20,000 | 5,000 | 1000 | ||||
| 1 mile | 84,480 (fingers) | English | 1.628352 km | 21,120 | 5,280 | 4224 | 1056 | 880 | ||
| 1 dolichos | 115,200 (daktylos) | Greek | 2.22 km | 28,800 | 7,200 | 5760 | 4800 | |||
| 1 stadia of Xenophon | 280,000 (daktylos) | Greek | 5.397 km | 70,000 | 17,500 | 1400 | 3500 | |||
| 1/10 degree | 560,000 (daktylos) | Greek | 10.797 km | 140,000 | 35,000 | 2800 | 7000 | |||
| 1 schϓnus | 576,000 (daktylos)Z | Greek | 11.1 km | 144,000 | 36,000 | 288000 | 28800 | 24000 | ||
| 1 stathmos | 1,280,000 (daktylos) | Greek | 24.672 km | 320,000 | 80,000 | 64000 | 16000 | |||
| 1 degree | 5,760,000 (digitus) | Roman | 111 km | 1,440,000 | 360,000 | 288000 | 72000 | 60000 |
- 1 daktulos (pl. daktuloi), digit := 1/16 pous
- 1 condulos := 1/8 pous
- 1 palaiste, palm := ¼ pous
- 1 dikhas := ½ pous
- 1 spithame, span := ¾ pous
- 1 pous (pl. podes), foot :≈ 316 mm, said to be 3/5 Egyptian royal cubit. There are variations, from 296 mm (Ionic) to 326 mm (Doric)
- 1 pugon, Homeric cubit := 1¼ podes
- 1 pechua, cubit := 1½ podes ≈ 47.4 cm
- 1 bema, pace := 2½ podes
- 1 khulon := 4½ podes
- 1 orguia, fathom := 6 podes
- 1 akaina := 10 podes
- 1 plethron (pl. plethra) := 100 podes, a cord measure
- 1 stadion (pl. stadia) := 6 plethra = 600 podes ≈ 185.4 m
- 1 diaulos (pl. diauloi) := 2 stadia, only used for the Olympic footrace introduced in 724 BC
- 1 dolikhos := 6 or 12 diauloi. Only used for the Olympic foot race introduced in 720 BC
- 1 parasanges := 30 stadia ≈ 5.5 km. Persian measure used by Xenophon, for instance
- 1 skhoinos (pl. skhoinoi, lit. "reefs") := 60 stadia ≈ 11.1 km (usually), based on Egyptian river measure iter or atur, for variants see there
- 1 stathmos :≈ 25 km, one day's journey. May have been variable, dependent on terrain
- For variant, the stadion at Olympia measures 192.3 m. With a widespread use throughout antiquity, there were many variants of a stadion, from as short as 157.5 m up to 222 m,
- The common Greed stadion is 185 m.
- The Greek root stadios means 'to have standing'. Stadions are used to measure the sides of fields.
In the time of Herodotus, the standard Attic stadion used for distance measure is 600 pous of 308.4 mm equal to 185 m. so that 600 stadia equal one degree and are combined at 8 to a mia chilioi or thousand which measures the boustredon or path of yoked oxen as a distance of a thousand orguia, taken as one orguia wide which defines an aroura or thousand of land and at 10 agros or chains equal to one nautical mile of 1850 m.
Several centuries later, Marinus and Ptolemy used 500 stadia to a degree, but their stadia were composed of 600 Remen of 370 mm and measured 222 m, so the measuRement of the degree was the same.
The same is also true for Eratosthenes, who used 700 stadia of 157.5 m or 300 Egyptian royal cubits to a degree, and for Aristotle, Posidonius, and Archimedes, whose stadia likewise measured the same degree.
The 1771 Encyclopædia Britannica mentions a measure named acæna which was a rod ten (Greek) feet long used in measuring land. Rktect 12:24, 1 August 2007 (UTC)
[edit] see also
[edit] all the stuff about influences is OR
needs to go--M a s (talk) 12:56, 8 March 2008 (UTC)
