Talk:Rhind Mathematical Papyrus

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Untitled[edit]

A copy of the (translated) 2/n Table should be included in the body of article. —Preceding unsigned comment added by 66.66.92.105 (talk) 17:55, 13 November 2008 (UTC)

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)
extended unreferenced numerological rant
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.

Remen as a Proportion to other units[edit]

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 Proportions to Greek Remen
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)

see also[edit]

Egyptian
Mesopotamian units,
Roman system

all the stuff about influences is OR - No, it's just unreferenced[edit]

needs to go--M a s (talk) 12:56, 8 March 2008 (UTC)

Whose original research? If the user(s) who wrote this read it in someone else's paper (e.g., http://www.mes3.learning.aau.dk/Plenaries/Powell.pdf), and never themselves looked at the papyrus with their own eyes, then it can't honestly be called original research now, can it? CompositeFan (talk) 13:39, 17 September 2008 (UTC)

category[edit]

2008 results[edit]

I rewrote part of the article because I find the following sentences too biased:

  • "Scholars from the 1880 to 1980 period claimed to have decoded the 2/n table, a task that was not completed until 2008. The intellectual basis and applications of Ahmes' red auxiliary numbers had been misunderstood for over 100 years."
  • "In 2008 one LCM method was proven to have created the RMP 2/n table."

There is no reference to this development that took place in 2008. And there has not passed enough time to show that this new proof is now generally accepted by the community. Such strong language is simply not suitable for Wikipedia without strong evidence that the consensus has shifted and now accepts the new proof. I also took out term Red auxiliary numbers for the same reason. It's not clear what that means, but they are usually called unit fractions. The only reference for the term "Red auxiliary numbers", according to the Wikipedia article, is Gardner's 2006 article in Gaṇita-Bhāratī. That's not enough evidence to change the language. -- Jitse Niesen (talk) 18:35, 29 October 2008 (UTC)

Book I, II, III[edit]

The inclusions of Book I, II and III are welcome additions. Overall the Wikipedia editorial pedagogy lacked humility in several areas. First, Peet's view of Book I, II and III needs to be expanded to include other scholars. Book I, II and III contents were improved upon by Griffith, an outline that Peet followed, a combined topic discussed by Spalinger in 1990.

In the current version there were not 40 RMP algebra problems, nor 20 arithmetic and 20 algebra problems. There were 10 algebra problems (RMP 25-34). The scribal algebra included scholarly discussions of 'trail and error' an open topic that scholars like Howard Eves reported false position related to scribal division for many years. Today, is it known that scribal algebra problems and/or the conversions of answers to Egyptian fraction series 'trail and error'? I hope that Wikipedia editors agree that humility is required to report the existence of this class of scholarly reviews of the RMP.

Second, Book I followed the 2/n table with six labor management problems (RMP 1-6). The labor management topic was "confirmed' by the Reisner Papyrus. The next grouping of RMP 7-20 included discussions of red auxiliary numbers; followed by RMP 21-24, set apart by Ahmes (Spalinger), as RMP 31 was set apart by Ahmes(Spalainger).

RMP 35-40 discussed hekat problems that were converted to 320 unit units (Clagett, 1999). RMP 40, 64 and a Kahun Papyrus arithmetic progression discussion was discussed by Robins-Shute (1987) and by John Legon (1992).

There are additional under reported math contents of Book II and III. Are Wikipedia editors interested in reporting these unsolved issues? I'd be happy to mention a small number of them, if requested.

Best Regards,Milogardner (talk) 16:28, 5 October 2010 (UTC)

Where actually is it kept?[edit]

From the introduction "The British Museum, where the [Rhind Mathematical Papyrus] papyrus is now kept ... The Rhind Mathematical Papryus is now at the Brooklyn Museum"? Aarghdvaark (talk) 13:10, 1 November 2012 (UTC)

The British Museum has two pages on it [1] [2] these seem to indicate that the majority are held there. The Brooklyn Museum's page [3] just has a few fragments. There is a comment on the Brit Mus page about the papyrus being broken up into two parts with part missing from the middle.--Salix (talk): 13:48, 1 November 2012 (UTC)
Thanks Aarghdvaark (talk) 14:13, 1 November 2012 (UTC)

Number of problems?[edit]

The article text makes it sound like Book III contained 84 problems. I believe that 84 may be the total number of problems in books II and III. Could someone verify this?

--Heath 198.82.20.227 (talk) 16:07, 29 August 2013 (UTC)