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Last updated 12/25/05

 

 

See images and analysis of ancient mathematical objects: IMAGE GRID

 

 

RAINER: (Greek) papyri

(as per E. G. Turner) See MPER.

http://www.anchist.mq.edu.au/222/222weekly.htm

 

P.Rain.Cent.: Festschrift zum 100-jährigen Bestehen der Papyrussammlung der Österreichischen Nationalbibliothek, Papyrus Erzherzog Rainer

P.Rain.Cent. 40. Official correpondence concerning lease of state land: (Greek; 257 bce?; Arsinoite)

With one line of Demotic

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0194

Math.

See RAINERI.

 

 

RAINERI: See Lugduno-Batava; [B_447=O_012,NO IMG,8.5]

 

 

RAMANUJAN: 19th century mathematician

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html

 

 

RAMESSEUM: (AE; OK) papyri and inscriptions

from Ramses III’s Temple complex

Across the Nile from Luxor and Karnak; Near the Colossi of Memnon; at the base of the VOK.

 

(as per AEB 96.0955) See Ramesseum Pap. III, IV, and V. (Medical).

 

(as per LEX) P. Ram. B is a dramatic text, see Kurt Sethe.

(as per LEX) P. Ram. C is Semnah Dispatches, see JEA 31 [B_303], (1945).

(as per LEX) P. Ram. E is Unique funerary Liturgy, see JEA 41 [B_303], (1955).

 

 

RAMSES: (AE; MK) Pharaohs

the strong links

[B_061,rvw,misc] CATNYP# *OBX 94-5894

“Le dernier pharaon : RAMSES III, ou, le crepuscule d’une civilisation / Francis Fevre.”

Paris, 1992.

See bibliography.

 

See MDAIK.

 

RAS SHAMRA: (Ugaritic) tablets

http://www.angelfire.com/nt/theology/Appx02.html

[Akkadian]

 

AKA Ras esh Shamra.

http://www.stormloader.com/flavin/abc.htm

See bibliography.

Notes and printed link to file with [B_289; for no reason].

 

[B_413,rvw] CATNYP# *OFXA 96-328

“Le Palais royal d’Ugarit, publie sous la direction de Claude F. –A. Schaeffer.”

Paris, 1970.

 

 

REALLEXIKON: publication

Reallexikon der Assyrologie. Unter Mitwirkung zahlreicher Fachgelehrter. Hrsg. Von Erich Ebeling [1886-1955] und Bruno Meissner [1868-1947]. Berlin 1932-_
Vol 3 has varied title=Reallexikon der Assyrologie und vorderasiatischen Archaologie. Editors vary. [B_289,see below]

 

[B_289,8.5’s,CUBIT; SARGON;GUDEA] CATNYP# *O-*OCK 86-872

Title Page and pages 457-530 from Reallexikon der Assyriologie VII [1990] (ed. D. O. Edzard et al.), Berlin and New York: De Gruyter.  (A paper by Marvin Powell, 'Masse und Gewichte').  Despite its title, the [very thorough] paper is in English.[and Akkadian tr. and German].

(9/9/01) Notes:

Barleycorn = 1/6 of finger

Finger = 1/30 Kus[h]

Kus[h] = Babylonian Cubit = B Cubit

B Cubit = ~550 mm.?; See Sacred HEBREW CUBIT also ~550 mm.?

 

Units: (see my “local” abbreviations; below)

A = Assyrian

AK = Akkadian

AK-OB=Akkadian-Old Babylonian

AU = Akkadian-Ur period(s)

B = Babylonian

K-NB = Kassite-Neo Babylonian

LB = Late Babylonian

M = Mesopotamian

MA = Middle Assyrian

MB = Middle Babylonian

MBAN = Middle Bronze Age Nuzi

NA = Neo [or New] Assyrian

NB = Neo [or New] Babylonian

NM = North Mesopotamian

NS = Non Standard

OA = Old Assyrian

OB = Old Babylonian

Post-OB = Post Old Babylonian

PRES = Pre-Sargonic [and also Pre-Old Babylonian]

ROG = Rule of Gudea [interpreted from statues B and F]

S = Sumerian

S-OB = Sumerian-Old Babylonian

SE = Seleucid period

ST = Standard

SY = Syrian

UM = Upper Mesopotamian

 

The following regional/period/matched groupings, are each in ascending size of unit order:

 

LENGTHS:

S/B/Post-OB 1/3 Cubit = S[h]u-du-a = S[h]izu? = 10 Fingers

Maybe two stacked fists or hands?

S/B/Post-OB 1/2 Cubit = Zipah = Utu?

S/B/Post-OB Cubit / Ell = forearm = Kus[h] =Ammatu =Gis[h]?

S/B Gis = AK Gis[h] Bad =

S/B/Post-OB Half-Reed = ?

S/B/Post-OB Reed = ?

S/B/Post-OB Rod [or Pole] = ?

S/B/Post-OB Half-Rope = ?

S/B/Post-OB Rope = ?

S/B/Post-OB Us[h] = ?

S/B/Post-OB Dana [or Beru?] = ?

 

K-NB Finger = ?

(K-NB system is independent?)

 

NB/LB Finger = ?

NB/LB Ubanu = ?

NB/LB 1/6 Ammatu = ?

NB/LB 1/3 Ammatu = ?

NB/LB 1/2 Ammatu = ?

NB/LB 2/3 Ammatu = ?

NB/LB Ammatu =

NB/LB Nikkas = ?

NB/LB Qanu = ?

NB/LB NINDA = ?

NB/LB Cubit of the King = ? = Royal AE Cubit (~523 mm.)?

NB/LB nikkas = ?

 

(NB/LB system independent units?)

NB/LB Reed Measure = ?

NB/LB Seed Measure = ?

See the “Seed Cubit” = Kus[h] Numun.

Seed Cubit = Two (2) Gis[h] Bad. = 60 Fingers =~1 meter

 

A/UM Ubanu = 1/6 [5/30] Sun diameters = 6 S[h]e = 5 minutes [of arc]?

A/UM Pus[h]ku = ?

A/UM Kisir Ammati = ?

A/UM S[h]izu = ?

A/UM Uutu (or?) Rutu = ?

A/UM Kabistu = ?

A/UM Kimsu = ?

A/UM Ammatu = 5 Sun diameters = 180 S[h]e = 2.5 degrees?

A/UM Nikkas = ?

A/UM Qanu = ?

A/UM Kumanu = ?

A/UM Suppu = ?

A/UM Iku = ?

A/UM Us[h] = ?

A/UM Beru = ?

 

AREA:

S-OB Barleycorn = ?

S-OB Little Shekel = ?

S-OB Little Mina = ?

S-OB Shekel = ?

S-OB Garden Plot = ?

S-OB 1/8 Dike = ?

S-OB 1/4 Dike = ?

S-OB 1/2 Dike = ?

S-OB Dike = ?

S-OB Rope = ?

S-OB Bur = ?

 

NM/SY Puridu = Leg = ?

NM/SY Hararnu = ?

NM/SY Kumanu = ?

NM/SY Awiharu = ?

NM/SY Iku = ?

NM/SY Imeru = ?

 

VOLUME:

S/B Barleycorn = ?

S/B Little Shekel = ?

S/B Little Mina = ?

S/B Shekel = ?

S/B Garden Plot = ?

S/B 1/8 Iku = ?

S/B 1/4 Iku = ?

S/B 1/2 Iku = ?

S/B Iku = ?

S/B Es[h]e = ?

S/B Bur = ?

 

S/B Brick Measure = ?

S/B Log Measure = ?

 

Post-OB/NB/LB system(s) = ?

 

A volume system(s) = ?

 

CAPACITY:

S/B Measure = ?

S/B Archaic Capacity Measure = ?

S/B Fara System = ?

S/B The standard Gur-Sag-Gal system = ?

S/B The Lagas[h] Gur-Sag-Gal system = ?

See [B_165,IMG].

 

AK-OB The standard system = ?

See the AK “Pace Cubit” or “Big Cubit” of 750 mm.

AK (“Pace Cubit” = Ammat Are) = (“Big Cubit” = Ammatu Rabitu)

 

K-NB The standard system = ?

 

NB/LB The standard system = ?

NB/LB Cubit = 24 Fingers.= ~.5 meters.

 

NM The standard capacity system = ?

 

Gasur/Nuzi capacity system = ?

 

1102 PFEIFFER, ROBERT H. & LACHEMAN, ERNEST R. Excavations at Nuzi Conducted by the Semitic Museum and the Fogg Art Museum of Harvard University, with the Cooperation of the American School of Oriental Research at Baghdad. Vol. IV: Miscellaneous Texts from Nuzi. Part I. (Harvard Semitic Series. 13.) xiv, (2), 104pp., 11 plates. 4to. Cloth. Ex-library.

Cambridge (Harvard University Press), 1942. $75.00

See: http://www.arslibri.com/cat124n.htm

 

See NUZZI tablets.

 

OA The “sack” system = ?

 

Mari system = ?


Imeru system(s) = ?

 

Achaemnid Royal Cubit = ?

 

VESSELS/Containers (S and NS):

Sila/Qui = ?

Ban/Situ = ?

Dug/Karpatu = ?

Ul = ?

Ba-ri-ga = ?

 

Kuli = ?

Kur = ?

Ninda-Banda = ?

Dug-Tur = ?

Gurgur = ?

Sadug = ?

Mud = ?

 

WEIGHT:

S/B Barleycorn = ?

S/B Little Shekel = ?

S/B Little Mina = ?

S/B Shekel = ?

S/B Mina = ?

S/B Talent = ?

 

(Post-OB subdivisions of a Shekel; See TORAH)

Post-OB Uttetu = ?

Post-OB Halleru = ?

Post-OB Giru (i. e. ~“Carat”) = ?

Post-OB Mahat = ?

Post-OB Bitqu = ?

Post-OB Suddu = ?

Post-OB Hummus[h]u = ?

Post-OB Rebutu = ?

Post-OB S[h]als[h]u = ?

Post-OB Zuzu = ?

 

NM Weight systems = ?

 

OA Weight systems = ?

 

Mari Weight systems = ?

 

Gasur/Nuzi Weight systems = ?

 

MBAN Weight systems = ?

 

MA/NA Weight systems = ?

 

Barleycorn = ? S[h]e; Uttetu ? = 1/6 Finger

 

Finger = S[h]u-si ? = Ubanu

 

See three different fingers: in PRES texts from Girsu; AK period; See JNES 5 p. 203-209; H.W.F. Sachs.

 

See Tables on pages 459 and 460

1 S[h]e=6 S[h]u-si=10 S[h]u-du-a=15 Zipah

=20 [*2?!] S[h]u-du-a?=30 Kus[h]

See TORAH; Numbers 13.11

“Gadi son of Susi”

APPROXIMATIONS!:

6 Barleycorns=1 Finger= 1.666 cm.

1 Stacked Hand = 10 Fingers = 16.666 cm.

1 Open [?] Hand = 15 Fingers = 25 cm.

2 Stacked Hands = 20 Fingers = 33.333 cm.

1 Forearm [Cubit]= 30 Fingers = 50 cm.

 

3 Ammutu=1 Nikkas

2 Nikkas= 1 Qanu

2 Qanu= 1 Nindanu

5 Nindanu= 1 Suppu

Note 3*2*2*5=60

2 Suppu= 1 As[h]lu

6 As[h]lu= 1 Us[h]

30 Us[h]= 1 Beru!

Note 2*6*30=360

1 Beru= 360*60 Ammatu= 21,600 Ammatu

 

page 460:

other weight conversion tables and sexagesimal quantities.

Much too vague!

 

See note page 461 that in 1st millenemium texts, the finger seems to represent 1/3 of the Sun’s diameter.

See: CALENDAR; ASTRONOMY

 

See Neugebauer 1975 text “History of Ancient Mathematical Astronomy”  which interprets an (~150 CE) Egyptian papyrus [as if its basis was Babylonian] with the following metrological relationships:

One “Sun Cubit”=7 1/2 palms = 30 fingers;

720 Sun diameters = 360 degrees [of arc].

1.5   degrees = 1 cubit = 5 Sun diameters = 30 fingers;

360 degrees = 144 cubits = 720 Sun diameters = 4320 fingers.

Suggests a link with the Beru system.

 

6 degrees = 12 Sun diameters = 6 x 12 fingers.

6 Fingers=36 Barleycorns=1 Sun diameter.

(See documented Chaldean uses in 1915 text by Neugebauer and Weidner)

“In the standard length system, the ratio between cubit and Nindan-rod is 12:1 [12 cubits=Nindan], but the ratio between the cubit of celestial distance and the Nindan of length-time [?] is 1:150.”

 

“seems to equate 1/2 cubit [12 Fingers of a 24 Finger cubit] with the disk of the moon”

 

“0,0,10=1 Susi”

 

“[late use?] 1 Barleycorn = 1/180 cubit”

[Implying only a 30 Finger cubit]

 

Half-reed=3 cubits=Nikkas?=Nakkasu?

=Nig-kas?

 

Reed=Gi?=Qanu?=6 cubits [NB/LB]

Above=7 cubits [Assyrian]

 

PRES evidence of Reed counting [p. 463]

Reed=6 cubits or

Nindan-reed=12 cubits?

 

Rod or Pole=Nindan?=Nindanu?=2 Reeds=12 cubits prior to NB/LB=in Assyrian context;

=14 cubits NB/LB

 

1 cubit of depth on a base of 1 square Nindan is 1,0 [3600] Gur of capacity

 

“In 1st Mill. Astronomical texts, Nindan is a length-time measure corresponding in our system to 1’ of arc or 4 seconds of time.”

 

ROG = [16 Finger] Rule Of Gudea

[B Cubit interpreted from “statue B” [incomplete] and “F”]

As per Jules Oppert; B Cubit = 30/16 x 270 mm

As per A. E. Berriman; B Cubit = 30/16 x 269 mm

As per M. Dieulafoy; B Cubit = 30/16 x 265.6 mm

As per F. Thureau-Dangin; B Cubit = 30/16 x 264.5 mm

 

Yielding a range of interpreted values:

495.9375 through 506.25 mm. Smaller than I anticipated as

compared to the AE [28 finger] Cubit of ~523 mm.

 

Nippur Cubit:

[interpreted (by U. Penn) from the southwest side of the EKUR temple.]

 

See Nuzi text references to the Cubit of the Gate;

See Shushan in HEBREW CUBITS.

 

See The Copper Cubit of the City Gate.

 

See Poetic description of the God Ningirsu measuring his Hero-to-be Eanatum with his handspan. See also STRABO XIII 2,3.

 

Chart VI on page 464 yields this info:

Nikkas=1/4 Nindan?=15 seconds [of arc]=1 second [of time]

 

Qanu=1/2 Nindan=30 seconds arc=2 seconds time

 

Qanu+Nikkas=3/4 Nindan=45 seconds=3 seconds time

 

Nindan=1 Nindan=1 minute arc=4 seconds time

 

Us[h]=60 Nindan=1 degree arc=4 minutes time

 

Beru=1800 Nindan=30 degrees arc=2 hours time

 

6 Beru=10,800 Nindan=180 degrees arc=12 hours time

 

12 Beru=21,600 Nindan=360 degrees arc=1 day

 

“units smaller than the Nikkas probably were not used because ancient technology was not capable of such precision measurement.”

 

AK=1/2 Rope=Suppu=5 Rods=10 Reeds=60 Cubits

 

See Julianos of Ascalon

[See also the astronomical treatise of Manilius (10-20 CE)]

4 Fingers=1 palm

4 Palms=1 foot

6 palms=1 cubit

6 cubits= 1 reed=[?akaina (AK/OB)]

10 reeds=1 plethron?

6 plethra?=1 Stadion?=

 

1 Stadion?=600 feet?=[360 cubits=1/60 Beru (AK/OB)]

 

1 day=720 Stadia=(21,600x12) cubits [in 12 Beru]

 

1 Rope=Es[h]e=As[h]lu=10 Nindan-rods=20 reeds=120 cubits=the side of a square which equals 1 Iku in area.

(i. e. = the square root of 1 Iku)

 

1 Rope=3600 fingers

Now on to p. 465 (figure 2)

Area of 1 Iku= a square which = 1.5 Ropes on a side=1/2 [Greek] Stadion a side=1/120 Beru a side = 180 Cubits a side= 3 Suppu a side= 3 Plethra a side=base of Ziggurat?=9 square Suppu in area= 9*(60^3) cubits squared in area= 1,944,000 sq cubits.

See Julianos of Ascalon.

1 square Suppu= a square which = 1 Suppu a side= 60 cubits a side= 1800 fingers a side= 1/2 a Rope a side= 1/4  Iku in area= (60^3) [216,000] sq. Cubits in area= 15*(60^4) [194,400,000] sq. Fingers in area.

AK root of Rope is As[h]lu or As[h]li?

AK: 10/20/30/40/50 Nindan= 1/2/3/4/5 Ropes

1 Aslu=”As[h]lu

2 Aslu=”[S[h]it]ta As[h]lu”

3 Aslu=”S[h]alas[h] As[h]lu”

4 Aslu=”arba As[h]lu”

5 Aslu=”hamis[h] As[h]lu”

*See Hebrew & Arabic number names.

6 Ropes=12 Suppu=60 Nindan-rods=120 Reeds= 720 cubits.

6 As[h]li?=”s[h]udus[h] As[h]li”

Counting in Ropes [rather than Cubits] may be as old as the Sumerians [? 1000 BCE].

P. 466 charts transliterations of units and format.

P. 467 notes: In 1st Millenium [BCE] astronomical texts; Us[h] denotes

what we now call one degree of Arc.

Dana=Danna=Beru=30 Us[h]=180 Ropes=1800 Nindan-Rods=3600 Reeds=21600 Cubits.=two hours march?=~10.8 Kilometers?

 

“In a S[h]ulgi hymn, the distance from Ur to Nippur is given as 15

Danna.

 

See Esarhaddon’s account of a march from Aphek to Raphia.

 

Danna/Beru=~2 Parasangs or 60 Greek Stadia.

 

See the “Diviner’s Manual”; JNES 33 [1974]. Showing how the [360] days are written.

 

Measuring time by weight [as in the water weight of a water clock].

1 Talent=12 Beru [on Earth / Ina qaqqari]

1 Talent=648000 Beru [in Heaven / Ina s[h]ame]

1 Mina?=1 Watch [of the night sky of which the OB had three watches per night?]=1 Massartu.

[Taken from AE]

[See also PLATO; ASTRONOMICAL considerations.]

 

“Post OB Length Measures”:

See the Esagila Tablet dated 229 BC (Louvre, AO 6555). [TCL 6, 32]; A Seleucid copy of older [Kassite?] material. Showing that the Pace-Cubit was 1.5 OB cubits.

http://www.metrum.org/measures/length_u.htm

 

The Big Kassite/NB Finger=45/24 or 1 7/7 OB Fingers.

 

The NB/LB Finger=30/24 or 5/4 OB Fingers.

 

NB/LB Cubit seems, by brick analysis, to measure 480-500 mm.

 

“Cubit of the King”=Ammat s[h]arri

 

See chart P. 472:

Pus[h]ku=Palm [of 4 Fingers?]

 

Rutu=Span=12 fingers or 1/2 of a 24 finger Cubit?

 

Kabistu=Footprint?

 

Kimsu [shin bone]=Esemtu [bone]=?3/4 Cubit?

P. 473:

..”more pertinent evidence from the time of Naram-Sin and S[h]ams[h]i-Adad I.

 

P. 475:

See the 10 Cubit thick walls of the Palace at Horsabad as described on Sargon’s Silver tablet.

 

See Sennacherib’s description of the moat at Niniveh.

 

See Esarhaddon’s given dimensions for the Ziqqurrat [Ziggarat] and the main wall of Babylon [Imgur-Enlil].

 

See reports by Nabonidus?

 

P476:

“The evidence as a whole seems to indicate the existence of an Assyrian Cubit of about [!] 32 standard [?] (OB) Fingers…”

 

Nikkas = Puridu / GIR = 3 Cubits

 

Qanu= 2 Nikkas = 6 Cubits

 

Suggests an Uruk period [or older more archaic] use of Reeds as primary measure.

 

Kumanu=10 Puridu?=30 Cubits.

 

P. 477:

?Kumanu=1/4 Iku?

[Nuzi] Awiharu=Furrow length?=Suppu=10 Reeds=60 Cubits.

MA As[h]lu; “land is measured in the King’s rope”

(ina as[h]al s[h]arri)

 

[!Probably] NA 1 Us[h]=6 Iku=6 As[h]lu=24 Kumanu=120 Reeds=240 Puridu=720 Cubits.

 

P. 478:

See metrological [S[h]ar=garden plot?] evidence of 2500-3000 BCE from Sumerian/OB texts discovered at Gamdat Nasr.

 

See PRES evidence from Girsu. A text dated to the 17th year of Entemena.

 

See OB math in JNES 5 [204-8] calculates an area 1 finger by 1/2 finger = 1/24 Barleycorn shown in Akkadian as 1/6 of 1/4 of a barleycorn.

 

P. 479

Little shekel=Gin-tur (Akkadian? = *Siqlu-sahru?)=1/60 Shekel

=1/3600 square Nindan = 3 [surface] S[h]e =

36 square fingers? Corresponding to the old Akkadian surface of a side of a cube containing 1 Sila [roughly a liter].

 

Little Mina=ma-na-tur/Manu sahru=1/3 shekel=60 surface S[h]e=720 square fingers.

 

Shekel=Gin/Siqlu=1/60 square Nindan (s/Sar)=180 surface S[h]e=2160 square fingers=36,0 square fingers.

 

Garden Plot (s/Sar) = Mus[h]aru = 60 Shekels=1/100 Iku=1 sq. Nindan=4 sq. Reeds=144 sq. Cubits

Hebrew/Aramaic=tpsr

 

Ki-la-bi=”Its weight”.

 

36 sq. Cubits [Kus] = 9,0,0 [21,600] sq. fingers= 1 sq. reed

=1/4 s/S[h]ar

 

PRES (from Lagas, Umma):

1/8 Dike= ½ sq. Suppu = 12.5 s/S[h]ar =50 sq. Reeds

=30,0 [1800] sq. Cubits

 

Calculations of an “irrigated land tax” suggest the above; see JESHO [B_380] 24; p 24; 114; 122.

 

P. 480

1/4 Dike= Ubu/ (ub/pu) =50 s/S[h]ar =200 sq. Reeds = 20,0,0 [72,000] sq. Cubits

 

AK/S: 1 Dike= Iku = 100 s/S[h]ar = 400 sq. Reeds = 1/6 Es[h]e = 1/18 Bur ?= 144,000 sq. Cubits ~=?Greek?AE Aroura~=Roman Jugerum ~=Iraqi [Persian?] Mes[h]ara and similar to the Donum and acre and hectare?

 

AK/S: 1 Rope (of land): Es[h]e / Eblu = 10,0 [600] s/S[h]ar = 6 Iku= 1/3 Bur

?Neru = 600

 

AK/S: 1 Bur/buru = 3 Es[h]e = 18 Iku = 1800 s/S[h]ar?

 

P. 481

S: 60 Bur = S[h]ar = ?”Ball”

S: 3600 Bur = S[h]argal= ?”Big Ball”

 

S/B: One bur of land is a strip 1 Nindan by 30,0 [1800] Nindan

[or 1 Danna]

See Strip of land of AE.

 

OB/Kassite-Neo Babylonian System:

Land Grant=Kuddurus

1 Gana= 3 Kus Gal-tu/tum = Is[h]ten Iku simid ammatu rabitu.

1 Iku (of land) = 1 Yoke (of seed) in the big cubit.

See p. 482 description of 1 Yoke as the amount of seed or land planted in one day’s work [by a Man and his Ox].

 

OB: 1 Bur (of land) = 240 Sila (of seed)

OB: 1 Gur (of land) = 240 Sila (of seed); See Fara texts.

 

P. 482

OB texts from Susa refer to the system.

 

NB/LB systems: 3 of them!

1. Reed system

2. Babylonian Seed system

3. Uruk Seed system

See Table XI:

1 surface finger=168 sq. fingers ~=729.166 sq. cm.

1 surface cubit =7 sq. cubits ~=1.75 sq. m.

1 surface nikkas= 24.5 sq. cubits ~= 6.125 sq. m.

1 surface reed= 49 sq. cubits ~= 12.25 sq. m.

 

P. 483

S/OB length and area systems:

See S[h]amas[h]-s[h]um-ukin (CT 44,70)

 

Seed measure system of the 7th or 8th century [BCE]:

See table XII:

Surface/capacity            Babylon            metric            Uruk                        metric

                        Sq. Cubits            Sq. M.            Sq. Cubits            Sq. M

1/10 akalu (108 s[h]e)            3            .75            2.777                        .69444

1 akalu (Ninda)                        30            7.5            27.777                        6.9444

1 qu (sila)                        300            75            277.777            69.444

1 sutu (ban)                        1800            450            1666.666            416.666

1 panu (bariga)                        10800            2700            10000                        2500

1 kurru (gur)                        54000            13500            50000                        12500

 

P. 484

1 Ubu (of land) = ¼ panu (containing 36 qu)?

See the Esagila tablet [Powell ZA 72, 114]

 

See figure 13, Uruk system:

1/3600 panu = 1/100 qu = 108 s[h]e = 1/10 akalu = 2.777 sq. cubits = 2;46,40 sq. cubits.

 

Northern (Mesopotamian and Syrian) systems:

Iku; Imeru (probably brought by the Amorites!)

Then integrated with the Hurrian hararnu; kumanu and awiharu..

 

P. 485

..Only roughly equivalent to the Northern systems.

See Table XIII:

Hurrian Nuzi                                    Alalah/Ugarit/MA

2 hararnu = 1 kumanu                        10 puridu = 1 kumanu

2 kumanu = 1 awiharu                        ?

2 awiharu = 1 iku                        4 kumanu = 1 iku

10 awiharu = 1 imeru                         5 iku = 1 imeru

 

“Independent customary values for Iku and Imeru” because:

 

Ratio of Imeru:Iku is 1:5 not 1:4.

 

The Northern Iku and the Babylonian Iku were “roughly” equivalent.

 

Awiharu may mean “furrow length”; from the Nuzi Hurrians.

 

Akkadian word Epinnu for “(Seeder) Plow”; from Awiharu again see Nuzi texts.

See Roman Actus; a furrow length of 120 Roman feet.

 

P. 486

Iku=SQUARE ROPE of 120 x 120 Cubits?

Iku=13+1/3 qu?

 

Mari measure Ugaru for “irrigation district”; capacity of:

10 Gur/Kurru of 120 Sila [qu?] each.

 

SQUARE ROPE=30 qu (sila)=1 Simdu=1 Karpatu=1/4 Narruqu=1/4 Mari Gur

See p. 486, fig. 15 for mensurational relationships in a Northern Iku.

 

P. 487

See “provisions by SENNACHERIB of land for the citizens of Niniveh in the amount of 2 Panu each” ~=6 Iku? ~=120 Sila or Qu?

 

1 Imeru=”ass of land” = 200 Puridu = 40 Hararnu = 20 Kumanu = 10 Awiharu = 5 Iku

 

P. 488

“Imeru derives its name from the amount of land customarily sown with an assload of seed [in one day?]”

 

P. 489

“brick metrology”

“log metrology”

“The central feature of the Akkad reform seems to have been a shift from definition of volume in terms of cubic nindan to definition in terms of square nindan x 1 cubit of height/depth.”

 

The cubic nindan can be describes as a cube consisting of 60 layers of 60 Bariga each.

Each layer consisting of 3600 sila cubes.

Thus the cubic nindan holds 60^3 sila cubes.

 

Barleycorn (S[h]e) = 1/180 gin = 1/3 gintur = 1 + 2/3 sila = 360 cubic fingers = 1/10800 s/s[h]ar

~= 1.666 liters

See fig. 17. 1 Gintur of volume base 6 x 6 fingers; height 1 cubit.

 

Little Shekel: gin-tur = 3 s[h]e = 5 sila = 1/60 gur = 1080 cubic fingers = 1/3600 s/s[h]ar

~= 5 liters

 

Little mina: mana-tur = 60 s[h]e = 20 gintur = 100 sila = 1/3 gur

~=.8 cubic cubit = 1/180 s/s[h]ar ~= 100 liters

 

P. 490

See fig. 18

One Gin (shekel) of volume

Base 1 nindan x 6 fingers x [ht] 1 cubit.

 

Shekel: gin = 180 s[h]e = 300 sila = 1 gur = 2.4 cubic cubits = 1/60 s/s[h]ar ~= 300 liters

 

Garden plot: s/s[h]ar = 60 gin = 18000 sila = 60 gur = 144 cubic cubits = 1/100 iku

~= 18 cubic meters

=1 square nindan x 1 cubit hieght

 

The cubic nindan plays no role after the Akkad reform.

 

1/8 Iku = 12.5 s/s[h]ar = 750 gur ~= 225 cubic meters [Presargonic]

 

Es[h]e = 600 s/s[h]ar = 6 Iku = 36000 gur = 10 guru

~= 10800 cubic meters

 

1 Bur = 3 Es[h]e

 

Brick measure from OB to Akkad period:

Differentiated from Presargonic period [Reign of Entemena] by abandonement of plano-convex bricks in favor of rectangular solids

 

P. 491-2

See 5 Gana interpretation.

See Nalbanum i. e. prototypical brick.

 

Log measurement:

Its practical purpose was to express girth (circumference) of cylindrical or conical solids.

 

“Presupposes the use of 3 + 1/8 = Pi”

 

NB-LB period surface measure: Ground cubit = (Kus[h] qaq-qar)

Analagous to the Assyrian use of Mus[h]aru.

i. e. 144 cubic cubits? ~=18 cubic meters?

See Tukulti-Ninurta I’s inscriptions:

“I reached ground water at 20 Mus[h]aru down.”

 

P. 493

Capacity: “The archaic capacity system probably antedates the invention of writing.”

No satisfactory clarifications to date.

 

P. 494

See Akkad Sila versus Sumerian Sila.

“The symbols used in the basic system are probably derived from prototypes among the Late Uruk tokens.

 

P. 495-7

See Sumerian style symbols.

Granary: = 1,152,000 sila?

1 gurmah x 1 lidga x 10 sila?

See Obelisk of Manistusu

 

P. 498

Kassite-NB system:

The qu seems to be divided into 6 akalu

 

NB-LB system:

Akalu = amount of grain for a flatcake

 

P. 499

Gasur (3rd Millenium bce Nuzi) systems:

“a larger Gur”

 

OA “sack” system:

Sack (Naruqqu) = 4 pots (Karpatu) = ~ 30 Sila?

 

P. 500-501

See the Rimah text.

See texts from Cagar Bazar [in use during reign of Hammurapi]

 

P. 502

See inscribed jar from Tall al-Rimah

“This norm is distinct from the Babylonian norm, which is larger.”

 

P. 503-10

Discussion of the obvious hazards of the analysis of ancient metrology.

See the silver votive gurgur of Entemena = 4.15 liters (p. 506)

 

P. 511

Reign of DARIUS I linked systems:

1 Median Shekel = 2/3 Babylonian Shekel

1 Daric = 1 Babylonian Shekel

10 Shekels = 1 kars[h]a/kurs[h]am

6 kars[h]a/kurs[h]am = 1 Babylonian Mina

 

1 Gold Daric = [in value] to 20 Silver Median Shekels

See the Jewish papyri from ELEPHANTINE (Aramaic), listing similar conversions.

 

P. 512-3

Giru = carat = 1/24 shekel = seed of the carob tree

 

[Persian] Mahat = 1/12 shekel = 2 Giru

 

P. 513

See 3/4 = “S[h]ala[y]s[h] rebatu” See NUMBERS Hebrew and Arabic.

 

P. 514-7

More suggested metrologies by weight systems

Numerous references to Duck weights and other “standards”.

Statement of mission regarding unresolved systems.

 

P. 517-527 (German article suggesting similar analysis)

 

P. 527-530 (index of terms of Metrologies)

 

See also p. 539 for sketched image of:

Proto-Sumerian bread-and-beer text!

 

 

RECUEIL: (Greek; demotic) papyri

(as per Duke Univ.)

Recueil de textes démotiques et bilingues, ed. P.W. Pestman with J. Quaegebeur and R.L. Vos. Leiden 1977. Part I, Transcriptions (and notes): nos. 1-3, 7-10, Demotic; 4-6 Demotic with Greek subscriptions (Greek republished as P.Brook. 88-90); 11, a graffito, is an Egyptian text written in Greek letters; no. 12 a Greek inscription with some Egyptian words in Greek letters; 13 and 14 are Greek texts with Demotic subscriptions (no. 13 = P.Gen. I 32, no. 14 = Chrest.Wilck. 89); nos. 15-23 are mummy labels, (21 is Demotic, the others are bilingual). Part II, Traductions (and notes). Part III, Index et Planches. [o.e. EJB]

 

 

REIF: STEPHAN C., (Author and Fascinating Lecturer)

Lectured at Bartos Forum of NYPL, 11/27/01.

Professor of Medieval Studies

Director of Genizah Research

And Fellow of Saint John’s College

University of Cambridge.

See GENIZAH

 

http://www.lib.cam.ac.uk/Taylor-Schechter/GF/33/

On fragments from the Genizah at Ben Ezra’s Synagogue in Old Cairo.

Which is small and moderately ornate now that it’s been renovated.

I visited November, 1999.

 

See text:

[B_381,HOUSE, lent to M. Friedman]

“A Jewish Archive from Old Cairo/ A history of Cambridge University’s Genizah Collection.” Curzon Press, 2000. Stefan C. Reif

Solomon Schecter “discovered” Cairo Genizah.

Over 210,000 fragments recovered.

 

(As per S. Reif) Pursue the work on Eruvin and arithmetic in the Genizah. By Bernard Goldstein.

(as per Barnes and Noble reference staff) See publisher Springer-Verlag for:

“The astronomy of Levi ben Gerson (1288-1344)”

 

http://www.springer-ny.com/

Phone: (800) 777-4643

 

[B_396,rvw,JH] CATNYP# *PVE 75-9110

“The astronomical tables of Levi ben Gerson / by Bernard R. Goldstein.” New Haven, Archon Books, 1974.

Jewish ASTRONOMY.

Levi ben Gerson [1288-1344]

 

[B_415,rvw,SIBL] CATNYP# JSP 73-148

“Journal for the history of Astronomy [JHA]”

See B. R. Goldstein in:

“An Occultation of Venus observed by Abraham Zaent in 1476.”

See JHA 30,3 (1999), p. 187-200

 

“Levi Ben Gerson and the brightness of Mars.”

See JHA 27,4 (1996), p. 297-300

 

[B_416,rvw,SIBL] CATNYP# JSL 94-240

“Physis”

Firenze (Florence), 1958-1998.

See PHYSIS 35,1 (1998), p. 1-10

See B. R. Goldstein article:

“Some Astronomical tables of Abraham Zagut.”

 

 

 

REIN.: See Lugduno-Batava; [B_447=O_012,NO IMG,8.5]; REIN.=REINACH

 

 

REINACH: (Author); (Demotic and Greek) papyrus

(Administrative; late (cursive) Hieratic; demotic; Greek) papyri

Theodore Reinach 1860-1928.

 

[B_163,rvw] CATNYP# *OBS+(Reinach, T. Papyrus Th. Reinach. Papyrus grecs et demotiques), “Papyrus grecs et demotiques recueillis en Egypte et pub. Par Theodore Reinach, avec le concours de MM. W. Spiegelberg et S. De Ricci.” Paris, 1905.

 

(as per E. G. Turner) (Greek) P. Rein. = Papyrus grecs et demotiques recueillis en Egypte, ed. T. Reinach, W. Spiegelberg, and S. de Ricci, Paris, 1905.

Les Papyrus Theodore Reinach, t. ii, ed. P. Collart, Cairo, 1940.

[O_057,IGNR,no copy]

NO CATNYP.; BOBST# PA3306 .R44 1940

“Les Papyrus Theodore Reinach. Tome II. Publie par l’Institut de papyrologie de la Faculte des lettres de l’Universite de Paris sous la direction de Paul Collart [1902-?].”

Cairo, 1940.

Studies include translations of Greek Papyri. No plates.

See also .[B_163]

[Bibliographical link:]

http://www.jack-travel.com/CoteAzur/CoteAzurhtml/Beaulieu_Villa_Kerylos.htm

Seek: Theodore Reinach, Historie des Israelites, p.221, a diversion.

He seems to have had a very strong aversion to the Cabalah.

 

See JEWS, TORAH.

 

P.Rein.: Papyrus grecs et démotiques recueillis en Égypte

P.Rein. 1.40.: = P.Dion. B. (114 bce; from Mochites)

=P.Dion.: Les archives privés de Dionysios, fils de Kephalas

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?lookup=P%2eDion%2e+9%25+736

 

 

REISNER: (Author); papyri

George Andrew Reisner 1867-1942.

 

The REISNER (Middle Kingdom) papyrus. Math.

At The Museum of Fine Arts, Boston, Cat# 38.2062, Dynasty XI or XII?

4 scrolls found by Dr. George A. Reisner from about 1880 BCE, 11.5 feet long.

Found in tomb N408 at Nag ed Deir.

 

[B_031=W_061,ALL,OS] (as per L. Bailey) CATNYP# *OBQK+ (Papyrus Reisner I). “The Records of a Building Project in the Reign of Sesostris I”, tr. & commentary, Boston, 1963 Simpson, William Kelly.

Big clear images. Numerous accountings.

An account of Man-days for given tasks. “Unit pricing.”

Purchase this excellent book. Especially volume one.

*If you can ever find it!

No copies to date, 10/9/01; 4/30/02!

Copied at NYPL via rm. 219 on 041705. Phew.

Harvest Book Company has yet to find a copy for sale.

(800) 563-1222

http://www.harvestbooks.com

 

Also available at BROWN University. See JOSIAH link.

http://library.brown.edu/search/Wreisner/Wreisner/1,16,16,B/frameset&F=Wreisner&12,12,

 

[W_062=B_031b,ALL,OS] also REISNER II accounts of dockyard workshop, 1965.

(as per AEB) Dockyard accounts from reign of Thutmosis III, see ZAS, S.R.K. Glanville.

[Stephen Ranulph Kingdon Glanville, 1900-1956]

P. # 10056.

Purchase this [REISNER II] too! *If you can ever find it!

CATNYP# *OBKQ+ (Papyrus Reisner II)

Copied at NYPL via rm. 219 on 041705.

[W_063=B_031c,ALL,OS] REISNER III records of a building project, 1969.

CATNYP# *OBKQ+ (Papyrus Reisner III)

Minimal arithmetical content. Copied at NYPL via rm. 219 on 041705.

 

 

[W_064=B_031d,HOUSE] also REISNER IV personnel accounts, 1986.

Significant quantity of arithmetical content of limited importance (long simple tallies).

“Personnel Accounts of the Early Twelfth Dynasty / Papyrus Reisner IV / Transcription and Commentary by William Kelly Simpson with Indices to Papyri Reisner I-IV and Paleography to Papyrus Resiner IV, Sections F, G / Prepared by Peter Der Manuelian / Pub. MFA Boston: 1986.”

Purchased via www.OXBOWbooks.com;  David Brown Book Company. $33.98 w/shipping.

Telephone (860) 945-9329; Fax (860) 945-9468

Toll free: 1 (800) 791-9354

 

http://www.mfa.org/giza/pages/overview.html

 

http://www.mfa.org/giza/pages/reisner.html

 

See BOSTON and GIZA.

 

See NIPPUR.

http://www.sag-online.de/pdf/mittsag3-8.pdf

 

REMUSAT: (Author) Historian

[Abel Remusat]

See CUBITS; JOMARD; JOURNAL ASIATIQUE; See JEWS; [B_561]

Abel-Rémusat, Jean Pierre. Foé Koué Ki, ou Relations des royaumes bouddhiques. Voyage dans la Tartarie, dans l'afghanistan et dan l'Inde, exécuté à la fin du IVe siècle par Chy Fa Hian. Tr. du chinois et commenté par Abel Rémusat. d'Eckstein, Baron. "Narasinha Oupanichat." (JA)

I did not realize Abel was so able.

 

REVENUE: (Greek) papyrus archive of Ptolemy Philadelphus

AKA REV. LAWS.

See:

[Bernard Pyne] Grenfell and Mahaffy, Revenue Laws of Ptolemy

Philadelphus.

 

See also ZENON; HIBEH

P.Rev.: Revenue Laws of Ptolemy Philadelphus

P.Rev. 2nded.: (Greek; 259 bce; Arsinoite)

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0196

 

(as per E. G. Turner) P. Rev. = Revue [!] Laws of Ptolemy Philadelphus, ed. B. P. Grenfell, Oxford, 1896. Re-edited by J. Bingen in SB, Beiheft i, Gottingen, 1952.

 

 

REVILLOUT: (Author); translator of Greek and Coptic; Demotic

Eugene Revillout 1843-1913.

 

(as per ZPE, D. Fowler) Seek work by E. Revillout,

on Coptic division tables. Math.

 

[B_090,rvw, BOTD] See Pamonth.

 

Seek:

Nouvelle Chrestomathie Démotique. Mission de 1878 Contrats de Berlin, Vienne, Leyde, etc. xii + 160 pp., 4to, top original wrp. laid on newer wrps. (uncut). Paris, Ernest Leroux, 1878; for sale at the link below:

http://www.archaeologia.com/egypt-r.htm

 

Seek:

Eugene REVILLOUT: Le Concile de Nicée d’après les textes coptes et les diverses collections canoniques. Paris, 1881.

 

 

REVUE ENCYCLOPEDIQUE: publication, RE

[B_197, Micro] CATNYP# ZP-PBM p. v. 36, No. 2.

“Observations sur un article de la Revue encyclopedique, dans lequel on examine le projet de traduire le Talmud de Babylone [Microform] : suivies du programme de la theorie du Judaisme appliquee a la reforme des Israelites de tous pays de l’Europe / par l’abbe L. Charini…”

Paris, Didot, 1829.

RE: Tome 38, pg. 20 (Avril 1828) housed the item discussed.

This was an article by M. Beugnot

The above analysis text [B_197] is not the RE.

 

The microformed review of the 1828 article makes a reference to:

“Bibliotheque Rabbinique de Bartolucci”, Pub Rome, ed. By Pope Innocent XI et Cardinals. Sounds interesting.

 

Because I checked this out, I learned the Call # at CATNYP for the true RE is: *DM.

See Below. See Metrology. See [B_149].

 

[B_220=S_001,cubits,8.5] CATNYP# *DM

This is not found in CATNYP, but they have it!

"Revue Encyclopedique, ou Analyse Raisonee des productions les plus remarquables dans la literature, les sciences et les arts, par un reunion de membres de l'Institut, et d'autres homme de lettres. (Quatrieme Annee.) Tome XVI"

See old catalog. Volume “Re”, p. 509, card 13.

Also available at Suny Buffalo and Princeton.

My thanks to Stanford University’s Librarians!

 

Page 432, center of page, begins this subheading:

"Note sur un Manuscrit egyptien sur papyrus, renfermant des plans de monumens, avec les mesures ecrites enchiffres hieroglyphiques."

 

The work shows one image of the Unit glyph (bent arm with thumb down) above the number 70 in glyphs (on page 433).

The only noted Author is Jomard.

 

Pub.: Paris, au Bureau Central de la Revue Encyclopedique, Rue d'Enfer-Saint-Michel, no 18. Et chez Arthus Bertrand, Rue Hautefeuille, no 23, Londres. -treuttel et wurtz, et dulau et c(ie?). Octobre 1822.

 

[B_220b,reject] CATNYP# GLP+ (Revue encyclopedique)

“Revue encyclopedique. No. 24 (Dec. 1, 1891)

[Containing a series of articles on Russia]

Paris, 1891.

 

RGVEDA: (Vedic) Fire Altars

See LALITAVISTARA and INDIAN MATHEMATICS and INDUS VALLEY SCRIPT.

 

(as per V. Gupta; 083104; personal correspondence)
Re: geometric construction of fire altars
http://www.math.ubc.ca/~cass/courses/m309-01a/kong/sulbasutra_geometry.htm

http://www.iish.org/publications1.htm

 

HERITAGE ACADEMIC FLOPPY DISC SERIES (401 - 500)
401.
Indian Heritage on mathematics
402.
Indian heritage on astronomy
403.
Bhoudayana sulbasutra (Vedic geomtry I)
404.
Apasthamba sulbasutra (Vedic geometry II)

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/References/Indian_sulbasutras.html

[Suggested] Books
B Datta, The science of the Sulba (Calcutta, 1932).
G G Joseph, The crest of the peacock (London, 1991).
Articles
R C Gupta, New Indian values of from the Manava sulba sutra, Centaurus 31 (2) (1988), 114-125.
R C Gupta, Baudhayana's value of 2, Math. Education 6 (1972), B77-B79.
S C Kak, Three old Indian values of , Indian J. Hist. Sci. 32 (4) (1997), 307-314.
R P Kulkarni, The value of known to Sulbasutrakaras, Indian J. Hist. Sci. 13 (1) (1978), 32-41.
G Kumari, Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals : its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.

 

RHABDAS: (Author) Mathematician

Nicholas Rhabdas (d. 1350)

Mathematician from Smyrna who wrote treatises on arithmetic

such as the Ekphrasis [NO CATNYP] on finger-reckoning, among others.

http://www1.fhw.gr/chronos/10/en/lexiko/lexr.html

See FINGER COUNTING

 

See [B_390; KESKINTO,v4]

 

RHIND/AHMES: mathematical papyrus AKA the R.M.P.

The Rhind Mathematical Papyrus, Dynasty XIII origin.

(Scribe/Pharaoh’s son?) Ahmes copied it in Dynasty XVI.

British Museum # 10057, 10058.

A few small fragments (of minimal importance) are housed in the Brooklyn Museum.

 

Follow this link (By F. Lopez) to images of all of the RMP.

(except the fragments)

http://www.egiptomania.com/mitologia/ciencia/papiro_rhind_img.htm

 

The Periodicals Service Company sells the Thomas Eric Peet analysis:

 

[A_002=B_092,HOUSE,ACQRD 6/13/00]

CAIRO MUSEUM REF # 0, 33, FOI

"The Rhind Mathematical Papyrus", 1923, T. Eric Peet, Kraus reprint 1970.

See [B_092b,8.5,HOUSE] Notes from L. Bailey.

 

[B_001,HOUSE] CATNYP# OBK 94 2555 "Mathematics in the Time of the Pharaohs" by Richard J. Gillings, Dover 1982.

 

[A_003=B_002,IMG,ALL OS,and some 8.5]

CAIRO MUSEUM REF # 0, 38, FOI

CATNYP# OBR 84-12 “The Rhind Mathematical Papyrus, British Museum 10057 and 10058; photographic fascimile, heiroglyphic transliteration, literal translation, free translation, mathematical commentary, and bibliography.”

By Arnold Buffum Chace, published by the Mathematical Association of America 1927.[Oberlin, Ohio]

See the images of the RHIND P. in my archive with my corr. to F. Lopez.

See the repaired image by L. Bailey in my files here.

See also my copies from E. Garner.

See my complete full size copy acquired Thanksgiving 2001.

From the Library of the College of Mt. St. Joseph on the Ohio.

Includes the EMLR analysis and image!

 

[A_001=B_006,IMG,8.5] CAIRO MUSEUM REF # 0,249

CATNYP# OBN 97-2766 "The Rhind Mathematical Papyrus: an Ancient Egyptian Text" by Gay Robins and Charles Cameron Donald Shute.

 

See this link from Milo Gardner to the 2/n table for odd n(3-101).

http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptroll2-n.html

 

(as per AEB 91/2.2057) Sollman, W.C., Hoe rekenden de Egyptenaren? - 5', De Ibis, Amsterdam 14 (1989), 9-12. RMP review and analysis.

See (AEB 87.1014) by same.

 

(as per AEB 81.1162,HM) Gillings, R.J., The Recto of the RMP and the EMLR, Historia Mathematica, Toronto 6 (1979), 442-447.

 

(as per AEB 93.0943) Rampelberg, Doris, Du caractere general des formules mathematiques dans l'Egypte pharaonique, in: Individu, societe et spiritualite. Melanges Theodorides, 221-228.

 

(as per AEB 93.0945) Reineke, Walter F., Zur Entstehung der agyptischen Bruchrechnung, Altorientalische Forschungen, Berlin 19 (1992), 201-211.

 

(as per AEB 96.0946) Adjamagbo, Kossivi, Sur la measure du circle et de la sphere en Egypte ancienne, Ankh. Revue d'egyptologie et des civilisations africaines, (1995-6) 222-245. Includes also MMP.

 

(as per S. Lorber; W. Knorr) See work by B. L. Van der Waarden in:

“Die Entstehung der agyptischen Bruchrechnung.

 Quellen und Studien”, B4, 359-382., See [B_341,rvw again!]

 

(as per M. Gardner) Visit this excellent RMP link by Christos Obretenov.

http://www.math.sfu.ca/History_of_Math/Egypt/Ahmes/main.html

 

(as per M. Gardner) Study the Hultsch-Bruins (van der Waarden)

methods. Hultsch published in 1895. And others below.

2/pq= 2/A x A/pq; (p+1)

Where A=(p+1) or (p+q)            2/9= 2/4 x 4/9?

(p+q) [RMP 2/35 and 2/91 only]

2/35= 2/12 x 12/35?

 

2/pq= (1/p + 1/q) x {2/(p+1)}

 

2/pq= 1/(Arithmetic mean) * 1/(Harmonic mean)

 

2/p – 1/A= (2A-p) / Ap; 2/9 – 1/10= (2(10)-9) / 90

2/9= 1/10+11/90  = 1/9+1/10+1/90

also= 1/6 + 1/18

 

1/p=1/(p+1) + 1/(p*(p+1)) 1/6 = 1/7 +1/42

 

1/p= 1/p (1)= 1/p (1/2 + 1/3 + 1/6) See Gillings.

 

1/p= 1/A x A/p; 1/3= 1/5 x 5/3

 

1/pq= 1/A X A/pq 1/8= 1/25 x 25/8

(see the EMLR); 1/8= 1/5 x 5/8

1/8= 1/5 x (3/5 + 1/40)

1/8= 1/5 x (1/5 + 1/3 + 1/15 + 1/40)

 

n/pq= 1/A x A/pq

 

n/pq= 1/A + (nA-pq)/Apq

 

n/pq= 1/pr + 1/qr

Where r= (p+q)/n

 

(as per [B_226a]; RYLANDS)

Includes references to the works of Kurt Vogel (see JEA XVI, 1930; See also [B_303],)

Also; Vogel’s, “Die Grundlagen der agyptischen Arithmetik,”

pp. 52-53, 183-184. [B_296,8.5,all]

 

And Vogel’s work in, “Archeion”, xii, (1930), pp. 398-399.

And Vogel’s work in, “Archeion”, xii, (1930), pp. 152. Titled

“Die Algebra der Agypter des mittleren Reiches. [B_367]

 

[B_367,8.5] CATNYP# OA (Archivio di storia della scienza) v. 1-21; Mar. 1919-1938. Rome. AKA Archeion!

Numerous past affiliations and supplements and irregular publishing.

Archeion 12 (1930) intro p. 1-4 by Abel Rey.

“Histoire de la Science ou Histoire des Sciences”

Archeion 12 (1930) intro p. 126-162 by Kurt Vogel.

“Die Algebra der Agypter des mittleren Reiches.”

See P. 148-149 for review of RMP #72.

See P. 150-151 for review of Kahun LV.4.

See P. 151-153 for review of Berlin P. #6619.

Making numerous references to Olivier Gillain:

See [B_336,SIBL].

And, suggesting:

F. E. Robbins: “P. Mich 620 : A Series of Arithmetical Problems.”

(1929), S. 321-329.

 

Archeion 12 (1930) intro p. 397-400 book review by Kurt Vogel.

Review of: A. B. Chace, H. P. Manning, L. Bull, R. C. Archibald, The Rhind Mathematical Papyrus. Ohio, 1927.

 

[B_296,8.5,all] CATNYP# *OBKQ (Vogel, K. Grundlagen der agyptischen

Arithmetik in ihrem Zusammenhang) [an inaugural dissertation.]

“Die Grundlagen der agyptischen Arithmetik in ihrem Zusammenhang, mit der 2:n-Tabelle des Papyrus Rhind, von Dr. Kurt Vogel. Studienrat am Maximilians-Gymnasium Munchen”

Munich, 1929.

Magnificent Bibliography includes these goodies (and others):

1. R. C. Archibald, Besprechung zu Peet (2). The American Math. Monthly 31 (1924) S. 246/251

2. V. Bobynin, Sur le procede employe dans le Papyrus de Rhind pour reduire les fractions en quantiemes. Bibl. Math. 4[2?] (1890) S. 109-122.

3. L. Borchardt (I), Wie wurden die Boschungen der Pyramiden bestimmt? AZ [ZAS] 31 (1893), S. 9-17.

4. L. Borchardt (2), Der inhalt der Halbkugel nach einem Papyrusfragment des mitteleren ReichesAZ [ZAS] 35 (1897), S. 150-152.

5. C. A. Bretschneider, Die Geometrie und die Geometer vor Euklides, Leipzig 1870, IV + 184 S.

6. A. Favaro, Sulla interpretazione matematica del Papiro Rhind. Mem. Della Accad. Di szienze, lett. Ed arti in Modena, 19 (1879), S. 89-143.

7. F. L. Griffith (1), The Rhind Mathematical Papyrus, PSBA, 13 (1891), S. 328-332.

8. F. L. Griffith (2), The Metrology of the Medical Papyrus Ebers, PSBA, 13 (1891), S. 392-406 und S. 526-538.

9. F. L. Griffith (3), The Rhind Math. Papyrus, PSBA, 14 (1892), S. 26-31.

10. F. L. Griffith (4), Notes on Egyptian Weights and Measures, PSBA, 14 (1892), S. 403-450.

11. F. L. Griffith (5), Notes on Egyptian Weights and Measures, PSBA, 15 (1893), S. 301-316.

12. F. L. Griffith (6), The Rhind Math. Pap., PSBA, 16 (1894), S. 164-173, 201-208, 230-248.

13. L. C. Karpinski (1), Algebraical Developments among the Egyptians and Babylonians. The Amer. Math. Monthly 24 (1917), S. 257-265.

14. L. C. Karpinski (2), Michigan Mathematical Papyrus Nr. 621. Isis 5 (1923) Nr. 13, S. 20-25.

15. L. C. Karpinski (3), The Origin and Development of Algebra. School Science and Mathematics 23 (1923), S. 54-64.

16. L. C. Karpinski (4), An Egyptian Mathematical Papyrus in Moscow. Science new series 57 (1923), S. 528-529.

17. M. L. d’OOge Nichomachus of Gerasa. Introduction to Arithmetic. With Studies in Greek Arithmetic by F. E. Robbins and L. C. Karpinski, NY 1926, S. 1-318.

18. A[bel], Rey (1), Coup d’oeil sur la Mathematique Egyptienne, Rev. de Synth. Hist. 41 (1926), S. 19-62.

19. A[bel], Rey (2), Nouveau coup d’oeil sur la Mathematique Egyptienne, Rev. de Synth. Hist. 43 (1927), S. 27-35.

20. L. Rodet (1) Sur un Manuel du calculateur decouvert dans un papyrus egyptien, Bull. de la Soc. mathem. De France, 6 (1878), S. 139-149. See also [B_352,8.5].

21. J. J. Sylvester, On a Point in the Theory of Vulgar Fractions. Amer. Journ. Of Mathem., 3 (1880), S. 332-335 und 388-389.

22. B. Touraeff, The Volume of the truncated Pyramid in Egyptian. Mathematics Ancient Egypt 1917, S. 100-102.

23. Q. Vetter (1), Egyptske deleni. Mem. De la Soc. Royale des Sciences de Boheme Classe des Sciences, Jahrg. 1921/22, Prag. 1923, XIV. Abh., S. 1-25.

24. Q. Vetter (2), Le Progressioni arithmetiche presso gli Egiziani. Bolletino di Matematica 4 (1923) [I?].

 

See also [B_296b,8.5,LB] NO CATNYP; NO WATSONLINE

“Aegyptisk Matematik.” Erik Lundsgaard; Kobenhavn, 1945.

 

(as per M. Gardner) See also F. Gnaedinger link:

http://www.seshat.ch/index.htm

 

(as per A. Rey, [B_353]) See:

“Fascimile of the RHIND Mathematical Papyrus in the British Museum.”

Londres, 1898.

 

(my personal corr.)
Do note this is a very suspect translation [By Eisenlohr].
“…all secret things…”
It [RMP problem 85; an upside down; fist sized patch to the 20 foot long papyrus] is a version of the hieroglyphs converted and coded in a script form.
This writing generally is more pictographic than the usual writing styles in that the first sylabble of each pictogram represents a sound and then a word or concept which disguises the meaning.
Similar in practice to: ICU812 = "I see you ate one too."
This is not the standard for writing of any Egyptian period. This writing is known as enigmatic script and it ran parallel to the Egyptian's civilization from probably not much earlier than Ahmes the Scribes' writing of the
RMP to Ptolemaic times. See ENIGMATIC; EDFU.

http://www.digitalegypt.ucl.ac.uk/lahun/uc32159.html

http://www.pupress.princeton.edu/books/maor/sidebar_a.pdf
[problem 79 = seven wives …]

 

RHODES or RHODOS: Island between Greece and Turkey.

See KESKINTO.

[B_460,rvw,out of service]

CATNYP# J-10 27 (Rottiers, B. E. A. Description des monumens de Rhodes)

By Colonel Bernard Eugene Antoine Rottiers, 1771-1858.

“Description des monumens de Rhodes / par le colonel Rottiers.” Bruxelles, 1830, prior to discovery of the KESKINTO astronomical inscription.

 

[B_461,rvw,out of service]

CATNYP# J-10 27 (Rottiers, B. E. A. Monumens de Rhodes)

“Monumens de Rhodes / Lith. Belge de H. Delpierre.”Bruxelles, 1828 (also prior to discovery of KESKINTO Astronomical Inscription)

 

 

RMP: See also RHIND

(as per M. Gardner) See also F. Gnaedinger link:

http://www.thebritishmuseum.ac.uk/egyptian/ea/gall/rhind.html

[B_527, SIBL=Y_016,IGNR]

CATNYP# JSE 84-604

SUMMIT# QA300 .S77 1983

“Primer of modern analysis : directions for knowing all dark things, Rhind Papyrus, 1800 B.C. / Kennan T. Smith.”

New York, 1983.

I requested this at hometown library via interlibrary loan on 8/22/02.

It reached me on 9/3/02. At my viewing on 9/4/02 I realized this text has no mention and no relation to the Rhind P. with the exception of the title!

IGNORE this text!

 

[B_527b, SIBL=Y_016b,IGNR]

CATNYP# JSE 72-50

SUMMIT# QA300 .S77

“Primer of modern analysis : directions for knowing all dark things, Rhind Papyrus, 1800 B.C. / Kennan T. Smith.”

New York, 1971.

Kennan T. Smith [1926-?].

This text has no mention and no relation to the Rhind P. with the exception of the title!

IGNORE this text!

RO: (AE METROLOGICAL) unit and module function

See BERICHTE ZUR WISSENSCHAFTSGESCHICHTE
See ACHMIM

 

 

RODAH: (Islamic reforms noted) nilometer at Egyptian island of

[B_182b, ARCH. IMG,IGNR] CATNYP# *OBL, Institut Francais d’Arch. A la Bibliotheque nationale de Paris, Le Caire, 1931, Tome 31 (1-2), plates

No math or useful metrological data in this information.

Only architectural and historical interests are served.

A recent nilometer. ~1700 CE.

 

AKA ROUDAH, ROUDEH.

 

 

ROGERS: (AE) papyrus

NO CATNYP

 

(as per Y. Koenig, Bulaq) Seek the Rogers Papyrus, ORACULAR.

 

WATSONLINE Yields nothing.

 

 

ROLLIN: (AE; Hieratic) papyrus

[B_055,rvw] CATNYP# *OBKQ++ (Pleyte,W. Papyrus Rollin de la Bibliotheque Imperiale de Paris), "Papyrus Rollin", 1868.

 

(as per 2terres; NO CATNYP) Barre, J. Y. Un Complot contre Ramse III; also Lee I & II?

 

[B_019,OS1] CATNYP# OBKQ++Spiegelberg, W., Rechnungen aus der zeit Setis I, "Papyrus Rollin".

See also BULAK 10?

Numerous transcriptions from fragmented papyri:

Pap. Biblitheque Nationale 206-209.

 

(as per LEX) See Theodule Deveria, Le Papyrus judiciaire de Turin et le P. Lee et Rollin, Paris 1868.

 

(as per AEB) Accounts. [math]

 

 

ROSETTA: (Heiroglyphic; Demotic; Greek) stone; trilingual

http://www.angelfire.com/nt/theology/Appx02.html

 

See BRITISH MUSEUM.

 

(as per EEF; M. Tilgner; 072904)
Hierolyphic text: Urk. II, 166-198
http://snipurl.com/83px
English translation of the Greek Section
http://www.fordham.edu/halsall/ancient/rosetta-stone-translation.html
German translation:
http://www.meritneith.de/text_rosette.htm
French translation by François Chabas, Oeuvres diverses, tome 3,
Paris, 1903, pp. 95-168, 2 pls. [original publication: 1867] - one
Plate is showing the hieroglyphic part only
http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55537
photograph - 650 KB
http://www.hi.is/~marj/kennsla/rosetta.jpg
photograph - 785 KB
http://snipurl.com/83pz
drawing - 570 KB
http://snipurl.com/83q1

 

ROSS. –GEORG.: (Greek) papyri

(as per E. G. Turner) P. Ross.-Georg = Papyri russicher und georgischer Sammlungen, ed. G. Zereteli, O. Kruger, and P. Jernstedt, Tiflis, 1925-35. Reprint 1965.

 

Vol i, Literarische Texte, ed. G Zereteli and O. Kruger, 1925.

 

Vol ii, Ptolemaische und Fruhromische Texte, ed. O. Kruger, 1929.

 

Vol iii, Spatromische und Byzantanische Texte, ed. G. Zereteli and P. Jernstedt, 1930.

 

Vol. iv, Die Kome-Aphrodite Papyri der Sammlung Lichacov, ed. P. Jernstedt, 1927.

 

Vol v, Varia, ed. G. Zereteli and P. Jernstedt, 1935.

 

P.Ross.Georg.: Papyri russischer und georgischer Sammlungen

P.Ross.Georg. 2.1. Loan and date formula of a contract:

(Greek; 244 bce)

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0197

 

 

RUNIC: (old Fluthark/ Blekinge) inscriptions

http://nora.hd.uib.no/FMProCGI/RuneFindLinks.html

 

http://home1.swipnet.se/~w-93783

The oldest known runic inscriptions of Blekinge are those of Björketorp, Flegehall, Gummarp, Istaby, Stentoften, Sölvesborg and Tjurkö. From a Semitic Fertility cult.

 

Proto-Bulgarian (Runic) Calendar?

http://members.tripod.com/~Groznijat/pb_lang/index.html

 

Where the Special Names in the Calendar of Isperih come from:

[Pamirs and the Hindu-Kush]

http://members.tripod.com/~Groznijat/pb_lang/pbl_3_3.html

 

 

RYLANDS: papyri collection

Moeller reference.

 

[B_093=C_002,8.5’s,IMG] CATNYP# ZAC (Smith, G. E. Evolution of the Dragon)

"The evolution of the dragon / by G. Elliot Smith, M.A., M.D., F.R.S. / Professor of Anatomy in the University of Manchester / Illustrated.”

Manchester, 1919.

Text was stolen from NYPL.

Sir Grafton Elliot Smith 1871-1937.

Only available text found at Columbia University Library in NY City. My day pass in index file.

Found 5/1/02 at Barnard’s Wollman Library [C_002] Call# BL313.S6 [Third floor stacks].

Preface describes author’s lectures at the John RYLANDS Library.

“In the earliest records from Egypt and Babylonia it is customary to portray a king’s benificence by representing him initiating irrigation works.”

Fravashi=Ka=placenta=Homunculus?

Fravashi affiliated with Great Mother; Water-god; “Good Dragon”.

Fravashi identified with Warrior Sun-god.

Fravashi associated with Dragon

AE “didi” NOT = mandrake; see TORAH! [dudayim] etymology obscure. =Mother Pot=Hathor?

Refers to Mr W. J. Perry’s works.

 

Ch. 1: “The dragon was primarily a personification of the life-giving and life-destroying powers of water.”

Aegean influences may be seen in the stupas and dagabas of Ceylon

Mr Blackman compares the placenta to the Ka.

AE contact as of the 1st dynasty with: southern Arabia, Arabia, Sumer, and Elam.

“the diffusion of the Sumerian and Elamite culture in very early times at least as far north as Russian Turkestan and as far east as Baluchistan.”

See “The Origin of Early Siberian Civilization,” in Memoirs and Proceedings of the Manchester Literary and Philosophical Society. Circa 1920.

“in Susa [Persian Capitol], where the earliest pictorial representation of a real Dragon developed..”

See Fig 9. Dragon from the Ishtar [see ASTARTE] Gate of Babylon

Dragon Hathor + Horus and = OB Tiamat

See: “Representations of Dieties of the MAYA Manuscripts,” Papers of the PEABODY Museum, vol. Iv., 1904.

[p. 84] See Seler’s articles on the CODEX VATICANUS. In ZFE and Peabody Museum Papers.

See Fig. 11 on Mayan Codex Traono. See MAYAN elephants.

See Fig. 13 on the 36th page of the Dresden MAYA Codex.

[p. 90] Tlaloc=Chac

[p. 91] Chinese dragon’s origins in AE Cerastes

[p. 91] Algonquin and Iriquois legends: the horned serpent is a water serpent and an enemy of the thunder bird.

[p. 110] refers to Erman’s: “Life in Ancient Egypt.”

[p. 133] Paurvas=Pliedes

Fig 18: NARMER palette.

Fig. 19 MAYAN Stela B from Copan

[p. 155] See Fig. 4: ASTARTE.

[p. 164-5] Fig. 21: “The ECUADOR APHRODITE.” Bas relief from Cerro Jaboncillo.

[p. 178-9] Shells as evidence of migration and cultural exchange.

Fig. 25 (e) Mycenaean vase from Old SALAMIS. Late image of Hathor?

[p. 201] Pliny quoted: “a stem two CUBITS in length…”

[p. 206] “It was the similarity of periodic phases of the moon and of womankind…”

[p. 206] “But incidentally the moon determined the earliest subdivision of time into months ; and the moon-goddess lenct the sanctity of her divine attributes to the number twenty-eight.”

[p. 234; in closing.] “The dragon was originally a concrete expression of the divine powers of life-giving ; but with the development of a higher conception of religious ideals it became regulated to a baser role, and eventually became the symbol of the powers of evil.”

See bibliography for further RYLANDS publications.

See COLUMBIA and see archive file for the Campus directory at [B_093]

 

[B_059,rvw] CATNYP# YIZ+ (Fawtier, R. Bible historiee), "La Bible Historiee; tout figuree de la John Rylands Library. Reproduction integrale du Manuscript French 5, accompagnee d'une etude par Robert Fawtier, Assistant Keeper of Manuscripts.", Paris, 1924.

 

(as per B.P. Grenfell,[B_051],HIBEH) see P.2 F. Ll. Griffith.?

 

(as per D. Fowler) Seek P. Ryl. [inv.] 666 = P. Ryl. iv 589, dated 180 BCE.

P. Ryl. iv 589=Ledger of Debts and Scheme of Lunar Months

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0198&layout=&loc=589

 

See work by Turner and Neugebauer, “GDNM”.

Includes: Accounts; Lunar Calendar.

See also P. Carlsberg 9.

 

(as per E. G. Turner) (Greek) P. Ryl. = Catalog of the Greek Papyri in the John Rylands Library, Manchester, by A. S. Hunt, J. de M. Johnson, V. Martin, C. H. Roberts, and E. G. Turner, Manchester,

1911-1952, 4 volumes as of 1965.

 

(as per G. Robins): See T.E. Peet’s article on AE math.

[B_226a,8.5all,LB] CATNYP# *GX (John Rylands Library, Manchester. Bulletin) 1903-

“Bulletin of the John Rylands University Library of Manchester”

Manchester, 1903-1996. Edited by the Librarian Henry Guppy.

"Mathematics in Ancient Egypt" by T.E. Peet,

Bulletin of the John Rylands Library 15 (1931), 409-41.

Includes references to the works of Kurt Vogel (see JEA XVI, 1930; See also [B_303])

Also; Vogel’s, “Die Grundlagen der agyptischen Arithmetik,”

pp. 52-53, 183-184. [B_296;8.5,all] See RHIND.

And Vogel’s work in, “Archeion”, xii, (1930), pp. 398-399.

And Vogel’s work in, “Archeion”, xii, (1930), pp. 152. Titled

“Die Algebra der Agypter des mittleren Reiches.

See RHIND [B_367,8.5]

 

See also Abel Rey’s, “La Science orientale avant les Grecs”, pp. 251 ff., pp. 281 ff. See [B_353,8.5]

See MATH: prior to 1601

 

See Kurt Sethe’s:

[B_340,8.5,all] CATNYP# *OBO (Sethe, K. H. Von Zahlen und Zahlworten bei den Agyptern)

“Von Zahlen und Zahlworten bei den Agyptern und was fur andere Volker und Sprachen daraus zu lernern ist: Ein Beitrag zur Geschichte von Rechenkunst und Sprache / von Kurt Sethe.” Schriften der Wissenschaftlichen Gesellschaft Strasburg, Volume [Heft] 25.

Karl J. Trubner. Strassburg, 1916, pp. 91 ff.

See three plates:

1. showing number forms in hieratic; hieroglyphs and demotic.

2. showing horus eye number forms

3. showing AE glyph and composite ligatures for 5/6, 3/4…

See RYLANDS.

 

Sethe refers to Hultsch’s, “Die Elemente der agyptischen Theilungsrechnung,” in Abhandlungen der Koniglichen Sachs. Gesellschaft d. Wiss. [Wissenschaften], phil-hist. [Philologische-Historische] Classe, Band xvii, Leipzig, 1897, see pp. 9-10. [B_358a]

 

(as per A. Rey [B_353]) See pages 1-192 from the text cited below.

 

See [B_358a]; ASAW

See CATNYP# *EE S122= [also?!] ASAW

I ordered this from the annex on 10/20/01.

On 11/3/01 The Annex Staff sent me the wrong volume.

I have re-requested them as of 11/3/01 and 12/3/01.

At 12/8/01 the request came back with further queries attached. Ugh!

Still digging.

1/12/02 item found! Thanks to Todd at room 219.

 

 

See Rodet in Journal Asiatique [JA], 1881, pp. 196-215. [B_352]

 

See Olivier Gillain’s work, “La Science egyptienne : L’Arithmetique au Moyen Empire”, Brussels, 1927,

pp 245-250.

AKA AEME See [B_336,SIBL]

 

See Gunn in JEA xii, p. 134. and JEA xv, pp. 184-5.

See [B_303]

 

See Gunn-Peet in JEA xv, pp. 167-185. See [B_303]

 

[B_226b,rvw alt source]

CATNYP# *XL-817 v. 1-44; Apr./June 1903-1962.

“Bulletin of the John Rylands University Library of Manchester”

 

P.Ryl.: Catalogue of the Greek and Latin Papyri in the John Rylands Library, Manchester.

P.Ryl. 2.65. Judicial Sentence: (Greek; 67 bce; Oxyrhynchus)

http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0198

See Lugduno-Batava; [B_447=O_012,NO IMG,8.5]

 

Robert K. Ritner. "The End of the Libyan Anarchy in Egypt:
P. Rylands IX. cols. 11-12." Enchoria 17 (1990), 101-108.
See TRADE

 

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