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Last updated 12/25/05

 

 

 

See images and analysis of ancient mathematical objects: IMAGE GRID

 

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]

 

This is a 2/3 through 2/21 fragment (only) for 2/odd numbers.

 

My modern style tr. follows in the same order as the hieratic fragment.

 

[answer]           preceeds                                               [question]

[unbolded items demonstrate workings]

 

2   2/3                                                                                       3

 

1/3  1/15  2/3  1  1/3                                                                   5

 

1/4  1/28  1/4  1/2  1  1/4                                                            7

 

1/2  1/18  1/2  1  1/6                                                                   9

 

1/6  1/66  1/6  2/3  1  1/6                                                            11

 

1/8  1/104  1/4  1/52  1/8  1/2  1  1/8                                13

 

1/2  1/30  1/2  1  1/10                                                     15

 

1/4  1/68  1/3  1/51  1/12  1/3  1  1/12                              17

[note: 17*12=204; 204/68=3; 204/51=4; 204/12=17]

 

*[1/6]  1/114  1/4  1/76  1/12  1/2  1  1/12                         19

[note: 19*12=228; 228/114=2; 228/76=3; 228/12=19]

[note: the bracketed  *[1/6]  does not exist on the papyrus fragment!]

 

1/2  1/42  1/2  1  1/14                                                     21

 

Although I have formatted this to bring attention to specific fractions, the hieratic does not include any

such distinctions except that some of the “workings” symbols are horus-eye parts.

 

See EL-LAHUN; LAHUN

 

http://members.aol.com/brucefriedmandcg/page26.html

 

http://www.cwru.edu/UL/preserve/Etana/papyri_kahun_gurob_text/ chapter4.pdf

For the text to [B_004]

See pages 15-18. Re: Math

(As per M. Silver; EEF; 021604)
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>

Images of the crucial plate #8 from text:
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

See NYU; [B_452=O_020,8.5,IMG]

Seek information of Julius Septimius Sabinus (~300 CE, Censitor)

 

See Soknopaiu Nesos.

[O_060,rvw]

NO CATNYP

BOBST# PA3352 .W52 Oversize

“Karanis and Soknopaiu Nesos, studien zur geschichte antiker cultur-und personenverhaltnisse. Von Dr. Carl Wessely.”

Wien, 1902.

Translations of Greek papyri by:

Carl Wessely, 1860-1931.

 

[B_456=O_025,8.5,IMG]

CATNYP# STG (Michigan. University. University of Michigan Studies, Humanistic Series. V. 42-43)

BOBST# PA3305 .M5 v.4

“Tax rolls from KARANIS; edited by Herbert Chayyim Youtie, with the collaboration of Verne Brinson

Schuman and Orsamus Merrill Pearl.”

Ann Arbor, Mich., 1936-.

See BOBST Archive: O 1

Includes a collection of tax accounts

Roll 223: P. Cairo 57187

Roll 224: P. Mich Inv. 3159; 4171; 4910; P. Iand. VII, 141 [over 6400 lines]

*JANDA papyrus identified as a missing portion of this roll.

Roll 225: P. Mich Inv. 4172

“The 1st of three rolls unearthed at Kom Aushim (Karanis) by sebbakhin in 1924…”

“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.]

 

See YOUTIE; AURELIUS ISODORUS

[B_469=O_039,IGNR]

CATNYP# STG (Michigan. University. University of Michigan studies, Humanistic series. v.47)

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.

 

See also Giss.; Cairo; Chicago; Columbia:

 

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

 

Kelim (vessels) is the first section of the tractate Mishnah Tohorot.

1. Bava Kamma

2. Bava Mezia

3. Bava Batra

 

See Sidrei Tohorot (1873) by R. Gershon Hanokh Leiner

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:

KMT Communications, Inc.

P.O. Box 1475, Sebastopol, CA 95473-1475

 

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]).

 

[lost 35 symbols]

" 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]

 

[lost 12-18 symbols]

"Deo]is"            = Thayoh = God [may actually = monotheism]

"X[a]pistnpiov"   = Kcharee-steeriohn = Thanks-offerings

[end of phrase 2]

 

Read phrase 1 - literally as a string:

[lost 35 symbols]…

"the circle parts 360 points 9720 the parts ['] secondary points 27."

[end of phrase 1]

 

Read phrase 2 - as noted:

[lost 12-18 symbols]

"Thanks-offerings [to] God!"

[end of phrase 2]

 

Milo, all I can scratch my head about is  - why divide a degree into 27ths?

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?

 

(as per personal correspondence; M. Gardner; 102902)

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?]

 

Other Bruce thoughts:

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]

 

[O_001=Y_001(SYRACUSE!),8.5,TR ONLY,NO IMG]

NO CATNYP. BOBST# PA 27.I7 vol. 12 fasc. 1-3

Y_001=SUMMIT# CN360 I6 v.12 (sent to me as a loan 8/20/02)

The primary source text for analysis of the Keskinto astronomical inscriptions:

and Binding reads: “Inscriptiones Graecae XII 1-3 SVPPLEMENTVM”

Internal cover page heading reads: “Inscriptiones Insvlarum Maris Aegaei Praeter Delvm”

Part one: Rhodi [Island of Rhodes] = Fasciculus I

Fasciculus I = “Inscriptiones Rhodi Chalces Carpathi Cvm Saro Casi Cosilio et Avctoritate Academie

Litterarvm Regiae Borvssicae Edidit Friderichvs Hiller de Gaertringen”

“Berolini Apvd Georgivm Reimervm”

“MDCCCXCV”

[I believe this scholarly Tome was bound with Fasciculus II and Fasciculus III in 1895 [2 years after the 1893 discovery/archeological voyage to Rhodes] and/or 1939.

Item 913: on page 148-149 and its corrigenda referenced to item 913 on page 207!

 

This was harder to find than love.

 

KESKINTO notes and annus vagus considerations filed with [O_001].

 

Melita, help!

Review the block Greek of the rest of [O_001] which interests me for astronomical pursuits.

 

As follows:

KATAPLATOS

 

KA ‘ ASXHMA

KATAMHKO[S]

KATABASOS

RYPOINTOS

RYPOENTOS

ThAI /// ONTOS

Th /// ESTONTOS

 

//////////APOMAI

//////////E ODOI

IOIDOAKOI

TROPIKOI

REPOMAI?

ETCETERA!

 

See link below:

http://www.bbaw.de/forschung/ig/pub.html

See IG; [O_001]

 

See PALMISTRY; PETERHOUSE; CLOCKS; work by Derek Price.

 

http://encyclopedia.astrologer.ru:8005/cgi-bin/guard/V/veliky_god.html

web link above includes reference to KESKINTO

Investigate and translate

See sexagesimal considerations and Sidereal and synodic discussion.

Avtor Prodolzhitel'nost' velikogo goda

Diogen Stoik 6 480 000 let

Kassandr 3 600 000 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

 

[B_439; 8.5;NO KESKINTO]

CATNYP# BVX (Guerin, V. Voyage dans l’ile de Rhodes et description)

“Voyage dans l’ile de Rhodes et description de cette ile.”

Paris, Durand, 1856.

By Victor Guerin (1821-1891)

Includes folding map and bibliography which does not refer to the KESKINTO inscription.

7/6/02 copied map (1) and cover only – original crumbling and bibliography not useful as I determined the

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.

http://www.calendersign.ric.at/english/turnofage1.htm

 

(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.

http://gallica.bnf.fr/Fonds_Tables/000/M0003076.htm

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:

[B_390] V2: “Sur l’inscription astronomique de Keskinto.”

The only Keskinto study I do not have at hand is that by Derek Price.

I have yet to identify the text, if any, Price's study appears 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.

“Memoires scientifiques, publies Par J.-L. Heiberg & H.-G. Zenthen   I   Sciences Exactes Dans

L’Antiquite 1876-1884.” Toulouse, 1912-

Johan Ludvig Heiberg and Hieronymous Georg Zeuthen.

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].

 

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