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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.”