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Last updated 12/25/05
KAHUN: (AE; Hieratic) papyri
Hieratic Mathematical Papyrus (Dynasty [XI-] XIV).
Follow this link to a fragment (Kahun
IV, 3, An Arithmetic Progression) with some linear progression/number theory
notes by John Legon.
See same in DE 24 (1992).
http://www.legon.demon.co.uk/kahun.htm
Found in 1889 by (Professor) Sir
William Matthew Flinders Petrie and
Published with images by Francis
Llewellyn Griffith in:
[B_004,glyph
tr.,IMG,notes filed with] CATNYP# OBR+Griffith.
"The Petrie
Papyri: Hieratic Papyri from Kahun and Gurob",
London 1898.
See Kahun LV.4
40 x 3 = 120
1/10 x 120 = 12
1 divided by (3/4) = 4/3
4/3 x 12 = 16
square root of 16 = 4
3/4 x 4 = 3
[Answer] 10 of 4 x 3 rectangles/bandages?
Find the
3,4,5 triangle! Reciprocals and ratios!
Find
awareness of f(x): [((3/4)(sqrt2))^2]=(9/8)=1.125!
Find the
circle! [interpret considering AE Pi]:
Consider
16 unit squares forming a larger square.
Consider
the diagonal.
Diagonal=4(sqrt2)
3/4 of this diagonal=
(12(sqrt2))/(4)=3(sqrt2)=4.2426=~17/4=~4
1/4.
Note
.75(sqrt2)=1.06066
[Note
(1.06066)^2=1.124999=~9/8]
The difference of two perfect
squares may be lurking here.
A square of sides =1.125 has an
area of 1.125^2=1.265625
A square of sides =1.000 has an
area of 1.000^2=1.000000
1.265625/(1)= 1.265625 and guess
what! See it with my eyes.
The area of the largest possible inscribed circle within a square with sides = 1.125 would have its area determined via:
Circle Area=((8/9)diameter)^2
Interpreted AE Pi=256/81=~3.160493
Voila, a squared circle!
How close were they?
A circle as above, diameter=1.125,
Make the modern determination via
the obvious formula.
Now, using
Pi as 256/81, find how very close they were.
NO CATNYP, M. Cantor "Die Mathematischen Papyrus fragmente von Kahun" 1898, OLZ Volume 1 #10, 306-8.
Also mentioned
without image in:
[B_003,IGNR]
(CATNYP# OBM+Petrie) "Illahun, Kahun and Gurob
:1889-1890", by Petrie with chapters by Professor Sayce.
Follow these links to S. Fryer's hieratic samples from Kahun.
http://home.prcn.org/~sfryer/kahun_letter2.html
and,
http://home.prcn.org/~sfryer/Hieratic/Samples/Neni1.htm
(as per A.H. Gardiner) Gurob AKA Ghurab, which is near Hawara.
(as per AEB 96.0955)
See gynaecological P. Kahun.
See Kahun
8? Math.
(as per Y. Muffs)
See [W_012], ELEPHANTINE.
See
Papyrus Kahun II, I (MK; XII dynasty,
2000-1788 BCE.)
Legal
content. See DJE I, pp. II ff., *13.
(See
preliminary tr. by L. Bailey of Kahun LV.4)
See BUTO; reference from [B_406] to:
“Die chronologische Fixierung der agyptischen Mittleren Reich nach dem Tempelarchiv von Illahun.”
[Chronological adjustments of the MK]
See also CALENDAR.
(my analysis 051502):
Hieroglyph tr. of the Kahun fragment [IV.2].
From Clagett's V.3. [B_028]
[unbolded items demonstrate workings]
[note: the bracketed *[1/6] does not exist on the papyrus fragment!]
such distinctions except that some of the “workings” symbols are horus-eye parts.
Papyrus Kahun 13 which seems
to deal with debts mentions the word wAwA.
In an article published in 1973 Ray offered the suggestion that this
term means "interest". <snip>
http://www.cwru.edu/univlib/preserve/Etana/papyri_kahun_gurob_plates/plate8.pdf
KAIRO: See CAIRO
KAMARA; KATSESHNI: See NAVILLE.
KARANIS: (Greek) papyri
(as per E. G. Turner) AKA Kom Aushim.
See GOODSPEED; Michigan.
P. Karanis Goodsp.: Papyri from Karanis.
P.Kar.Goodsp. 1.: (Greek; AD 158; from Karanis)
http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0142
Karanis=Karanidos
[O_060,rvw]
BOBST#
PA3352 .W52 Oversize
Wien, 1902.
Translations of Greek papyri by:
Carl Wessely, 1860-1931.
[B_456=O_025,8.5,IMG]
BOBST# PA3305 .M5 v.4
Schuman and Orsamus Merrill Pearl.”
“the 3 rolls together provided an unusually complete account of the daily collection of money taxes at
Karanis during 3 consecutive years in the second half of the second century A. D.” i.e. 150-200 CE
Lists of names; estates and payments received or due. Math. Accounts.
[Note about 100 years before Tetricus and his son the young prince Tetricus II abdicated Gaul to Aurelian.]
[B_469=O_039,IGNR]
BOBST# PA3305 .M5 vol.6
“Papyri and ostraca from Karanis <by> Youtie and Pearl.”
In the Humanistic Series [XLVII] of “Michigan Papyri.”
Edited and prepared by Herbert Chaim Youtie, 1904-?
Michigan, 1944 [-1951].
Translations of Greek papyri, No plates.
KARATEPE: (Semitic, Phoenician) inscriptions; statues
commemorative inscriptions in Northwest Semitic, such as the example found at Karatepe, Cilicia (in Turkey) in 1947.
From,
http://www.flavinscorner.com/4-6-01.htm
See Dr. Gordon; PARAÍBA
DNWSI promotion:
In the Karatepe
inscription, for example (KAI 26
ii 7-8, 13; cited in DNWSI, p. 473),
Azitiwada recounts several reasons why a Cilician city ought to be named after
him. One of these reasons is his ability to give its inhabitants a "pleasant
dwelling" (<sem>$bt n(mt</sem>).
DNWSI, citing this reference under the root <sem>y$b</sem>,
lists the possibility that the Phoenician term should be derived from <sem>$bt</sem>,
"to cease, rest," as well as <sem>y$b</sem>, "to sit, reside." In other
words, <sem>$bt</sem> at Karatepe poses the same morphological
and semantic ambiguities in its context as does htb#$
in Ruth 2:7. Seeing polysemantic ambivalence at Karatepe therefore better
helps explain the versional options and strengthens the possibility that polysemantic
variability might also be occurring in Ruth 2:7.
http://rosetta.reltech.org/TC/vol01/HoftijzerJongeling1996rev.html
KARATEPE: (Phoenician) inscribed statues:
On the back of the figure extending to below the waist was a 20-line Phoenician inscription, and scattered around were fragments of stone carved with hieroglyphics, suggesting the exciting possibility that this might be a bilingual inscription.
The next year (1947) excavations of this Late Hittite fort were led by Bossert and U. Bahadır Alkım. They discovered that the lion carved in relief was not actually a lion, but a bull - an animal held sacred by the Hittites - and that there were two of them. The figure which had stood upon the sacred bulls turned out to be the Storm God. But the bilingual inscription overshadowed all the other finds, and was to throw light on a little known period of Anatolia’s history. Comparison of the two texts, one in Phoenician and one in hieroglyphic Luwian, enabled the latter to be deciphered for the first time.
http://www.atamanhotel.com/karatepe.html
See Phoenicians; TRADE
KASR KARUN: (Greek; Ptolemaic) Temple at
See FAYUM; [B_075=O_002]
KELIM: ritual purifications
See TORAH; MISHNAH; TALMUD; CUBITS; HEBREW CUBITS
Notes from Encyclopaedia Judaica
V. 10 p. 899-900
1. Bava Kamma
2. Bava Mezia
3. Bava Batra
See also TARBIZ 16 (1945) p. 71 ff.
KELLEYS ISLAND: (American Indian) pictographs in Ohio, near Lake Erie
From American Indians ~?1200-1600 CE.
http://www.ohiohistory.org/places/inscript/
KELLIS: (Greek) papyri
P. Kell.: Greek Papyri from Kellis.
P. Kell 1.1.:
(Greek; AD 293-94 ; from Kellis
http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0143
KEMET: full color; quarterly publication/ journal of AE
Inquire for publications to:
KENYON’S: (Greek) papyri
(as per D. Fowler)
See Kenyon’s
(Greek) Papyri ii , no. cclxv 40. Math.
(as per E. G. Turner) Kenyon, a classical scholar who joined
the British Museum in 1889. Pursue the bibliography of F. G. Kenyon,
“Egypt Exploration Fund Archaeological Reports”, 1892 and 1893, p 27.
(taken over in 1914 by JEA..;
[B_303])
(as per F. Hultsch; [B_358a])
See “Kenyon Greek Papyri in the British Museum.”
KESKINTO: (Greek) astronomical inscription
(as per personal correspondence to; M. Gardner; 102802)
Recent efforts to wrestle with the fragmented remaining phraseology found at Keskinto:
(Note, the numerical conversions from this portion of the inscription were made in the same style of
Milesian Accounting we previously discussed when I figured out the Hibeh i, 27 parapegma).
(Also note, the items in quotes are my desperate attempt to represent the Greek letters with our alphabet in order to follow my references on the copies I sent you earlier. This is followed by a poor phonetic representation of the Greek with my Brooklyn accent, and finally a definition [as per my helpful friends and Liddell and Scott's Lexicon]).
" o' " = owe = the [masculine]
"xuxlos" = keeklose = circle.
"moirvw" = mwahrohn = parts.
" tE' " = tau sigmeh = 300 [+] 60 =360 [degrees?]
"stigmwv" =steehmohn = points
"Dwx" = Theta Psi kappa = 9000 [+] 700 [+] 20 = 9720
[*27ths of a degree of a circle of 360 degrees]
" h' " = eee = the [feminine]
"moipa" = meerah = parts
"otigmwv" = steeghmohn = [secondary points or sub-points]
"xS" = kappa *stigma [or *digamma] = 27
[end of phrase 1]
"Deo]is" = Thayoh = God [may actually = monotheism]
"X[a]pistnpiov" = Kcharee-steeriohn = Thanks-offerings
[end of phrase 2]
[lost 35 symbols]…
"the circle parts 360 points 9720 the parts ['] secondary points 27."
[end of phrase 1]
[lost 12-18 symbols]
"Thanks-offerings [to] God!"
[end of phrase 2]
What an unusual choice.
A trinity of trinities.
3^3=27
If harmonized with the Moon or the Cubit or the Sexagesimal system, the degree ought to be divided into some other number of sub-points. Many other numbers would seem more reasonable. Don't you think?
divide 9720 by 60, find: 162
divide 162 by 27, find 6
[same numbers, slightly different perspective, no clue as to why 27 got here.]
Ennead; i.e. Theolological interference?]
Consider this a factor in early keskinto-ish recognition of the precession of the equinoxes
[Why not?]
If the solar day (variable length of sunrise to second sunrise) is 9720 parts
Each of our modern minutes (lepta) is 6.75 parts [exactly]
Each of our modern seconds (lepta/60) is 0.1125 parts [exactly]
Litterarvm Regiae Borvssicae Edidit Friderichvs Hiller de Gaertringen”
Melita, help!
Review the block Greek of the rest of [O_001] which interests me for astronomical pursuits.
KATAPLATOS
KATAMHKO[S]
KATABASOS
RYPOINTOS
RYPOENTOS
ThAI /// ONTOS
Th /// ESTONTOS
//////////E ODOI
IOIDOAKOI
TROPIKOI
REPOMAI?
ETCETERA!
See link below:
Avtor Prodolzhitel'nost' velikogo goda
Diogen
Stoik 6 480 000 let
Beros 2 160 000 let
Antioh 1 753 005 let
Nadpis' iz Keskinto 291 400 let [great year of 291,400 solar years]
Orfei 120 000 let
Geraklit (po Aetiyu) 18 000 let
Geraklit (po Cenzorinu) 10 800 let
Gippas 59 let
Paris, Durand, 1856.
By Victor Guerin (1821-1891)
Includes folding map and bibliography which
does not refer to the KESKINTO inscription.
inscription was discovered after publishing
of this text!
(as per D. Fowler)
See Rhodes, IG xiii(1) 913.
See the (Greek) division of a circle
into 360 parts. Sexagesimal math.
See HAMA ii, 590 ff. and 698 ff.
14. Astronomical inscription from Keskinto / Rhodos / Greece
"The Large Year takes 291,400 years" (i.e. 200 x
31 x 47 years). Annual details for sidereal and synodic circulation of the
planets read:
Merkur: 291400 / 918,700;
Mars: 154,920 / 136,480;
Jupiter 24,500 / 266,900;
Saturn 9,920 / 281,480.
(as per Gallica link below) See: “Comptes
rendus hebdomadaires des seances de l’Academie des sciences / Institut de
France.” Janvier 1895.
[B_356,IGNR,KESKINTO] CATNYP# *EO 1426 [indexes to title above] must be requested from the NYPL Annex, ughh.
Makes mention of this inscription as analyzed by M. Paul Tannery.
Note [B_356]; [1895] Gallica article by P. Tannery is the same as that in:
See also [B_387].
[B_387,no img,IGNR,SIBL] CATNYP# JSC 84-25
“Recherches sur l’histoire de l’astronomie ancienne / Paul Tannery.”
Hildesheim ; NY; Georg Olms, 1976.
No significant review of KESKINTO.
[B_388,IGNR,SIBL] CATNYP# *ZV-163
“Pour l’histoire de la science hellene [microform], par Paul Tannery. De Thales a Empedocle.” Paris, F. Alcan, 1887.
See also Math: prior to 1601.
[B_389,IGNR,KESKINTO,SIBL] CATNYP# OKA (Tannery, P. Geometrie Grecque)
“La geometrie grecque, comment son histoire nous est parvenue et ce que nous en savons. Essai critique par Paul Tannery. 1. ptie. Historie generale de la geometrie elementaire.”
Paris, 1887.
[B_390,8.5,KESKINTO,SIBL] CATNYP# OAL (Tannery, P.
Memoires scientifiques) Library has: v. 1-13.
L’Antiquite 1876-1884.”
NOTES FROM V1:
PAUL TANNERY
MEMOIRES SCIENTIFIQUES
PUBLIES
PAR
J.-L. HEIBERG & H.-G. ZEUTHEN
[Johan Ludvig Heiberg and Hieronymous Georg Zeuthen.]
II
SCIENCES EXACTES DANS L’ANTIQUITE
1876-1884
1912
Entries in this set of volumes
sometimes noted as:
“Extraits
des Memoires de la Societe des sciences physiques et naturelles de Bordeaux.”
Or,
“Extrait
de la Revue archeologique.”
Includes articles on the following:
1. [1876] Note sur le systeme astronomique d’Eudoxe [EUDOXUS].
[Assumed circular planetary orbits.]
2. [1876] Le nombre nuptial de Platon. [PLATO]
3. [1876] L’hypothese geometrique du Menon de Platon. [Plato]
4. [1878] Hippcrate de Chio et la quadrature des lunules [Hippocratus/Quadrature of the Lunes]
5. [1878]. Sur les solutions du probleme de Delos par Archytas et par Eudoxe
6. [1879]. A quelle epoque vivait Diophante. [DIOPHANTUS]
8. [1880] L’arithmetique des Grecs dans PAPPUS.
9. [1881] Sur l’age du pytharoricien Thymarids [PITAGORAS]
11. [1881] Sur le probleme des boeufs d’Archimede. [ARCHIMEDES]
14. [1882] Sur les fragments de Heron d’Alexandrie conserves par Proclus. [HERO; HULTSCH]
18. [1882] L’arithmetique des Grecs dans Heron d’Alexandrie. [HERO; HULTSCH]
19. [1882] Sur la mesure du cercle d’Archimede
20. [1882] De la solution geometrique des problemes du second degre avant Euclide. [EUCLID]
23. [1883] Sur une critique ancienne d’une demonstration d’Archimede.
24. [1883] Second note sur le systeme astronomique d’Eudoxe.
25. [1883] Le fragment d’Eudeme sur la quadrature des lunules [Hippocratus/Quadrature of the Lunes]
26. [1883] Aristarque de Samos [ARISTARCHUS]
27. [1883] Stereometrie de Heron d’Alexandrie. [HERO; HULTSCH]
28. [1883] Etudes heroniennes. [HERO; HULTSCH]
29. [1883] Sur le <<modius castrensis>>. [MODIUS; ASTRONOMY]
See p. 464-5
Notes from article 1:
Note sur
le systeme astronomique d’Eudoxe.
About 400 BCE the influence of systems of astronomy similar to or yielded from the works of Eudoxus arise. Concentric orbits are considered perfect and beautiful and holy and as God is a perfect being he would build the heavens with circles. Sidereal and diurnal conflicts were noticed but not resolved by this theory. Callipus noted lunar movements inconsistent with the circle. Retrograde conflicts [mercury] further disproved circular orbits. A trigonometric lesson in how to attain cognizance of the fact that the orbits do not conform to circles and to find the quadrature of the Lunes [if they did], presented by Tannery and Schiaparelli is given on page 3 and ff.
Notes from article 29.
Sur le
<<modius castrensis>>. Copies of pages 464-5 only.
Modius castrensis versus Modius Ordinaire may be resolved in:
Greichische und romische Metrologie, by Freidrich Hultsch, 1882.
Sextarius castrensis
NOTES FROM V2:
PAUL TANNERY
MEMOIRES SCIENTIFIQUES
PUBLIES
PAR
J.-L. HEIBERG & H.-G. ZEUTHEN
II
SCIENCES EXACTES DANS L’ANTIQUITE
1883-1898
1912
Includes articles on the following:
31. [1884] Sur l’authenticite des axiomes d’Euclide [EUCLID]
32. [1884] Sur les manuscrits de Diophante a Paris [DIOPHANTUS]
33. [1884] La pert de sept livres de Diophante [DIOPHANTUS]
34. [1884] Sur la langue mathematique de Platon [PLATO]
38. [1885] Sur l’arithmetique pythagoricienne [PITAGORAS]
43. [1886] La coudee [CUBIT] astronomique et les anciennes divisions du cercle. [CALENDAR]
46. [1888] La grande annee d’Aristarque de Samos [Aristarchus; CALENDAR; CHINESE REMAINDER THEOREM]
47. [1887-8] Etudes sur Diophante. [DIOPHANTUS]
48. [1889] Le hypothese geometrique du Menon de Platon. [Plato]
49. [1889] L’art d’Eudoxe. [EUDOXUS]
50. [[1891] Les manuscrits de Diphante a l’Escorial. [DIOPHANTUS]
51. [1891] Sur une epigramme attribue a Diophante [DIOPHANTUS]
51b. [1892] Note sur un passage de Theon de Smyrne, De Musica
[See DIOPHANTUS]
53. [1894] Un fragment des Metriques de Heron. [HERO, HULTSCH]
54. [1894] Sur un fragment inedit des Metriques de Heron d’Alexandrie. [HERO, HULTSCH]
55. [1894] Sur Theon de Smyrne
56. [1895] Sur un passage de Theon de Smyrne
58. [1895] L’inscription astronomique de KESKINTO.
59. [1895] Sur l’inscription astronomique de KESKINTO.
60. [1895]
Une inscription grecque astronomique.
61. [1895] Sur les subdivisions de l’heurs dans antiquite.
62. [[1896] Sur la religion des derniers mathematiques de l’antiquite.
Notes
from article 58. [copy all]
L’inscription
astronomique de KESKINTO.
The Astronomical inscription at Keskinto [Rhodes]
p. 487
Eudoxus is from Cnide?
The Keskinto [town] inscription at Rhodes [island] has been dated by its content in that the theories of heavenly bodies, to be interpreted from the accurate data [via astrolabe or similar devices] inscribed, is not still suggestive of a belief that the motions of these bodies lied in concentric circles. The introduction of eccentric epicycles about the time of Hipparchus abandoned Eudoxus and others’ principles in an effort to better resolve this refined data and harmonize the perpetual calendar. This change and this inscription therefore occurred after 250 BCE [in Greek territory].
[