Ostr. Berlin P. 12609

TEI-XML-File: https://p612399.webspaceconfig.de/xml/elephantine_erc_db_016497.tei.xml

Metadata

Collection

Collection	Berlin, Ägyptisches Museum und Papyrussammlung, Staatliche Museen zu Berlin, SPK (P)
Inventory Number	Ostr. Berlin P. 12609
Publication Number	lit math
Current Location	exhibition \| Neues Museum, Berlin
Publication Permission Status	(unknown)
Publication Status	published

Origin / Provenance

Ancient Provenance Site	Elephantine (Ꜣbw; Yb; YbꜢ; YbꜤ; Ἐλεφαντίνη, יב , ⲉⲓⲏⲃ) [Trismegistos] Certainty:
Ancient Provenance District	Upper Egypt, 1st nome (Ombites) [Trismegistos]
Type of Discovery	archaeological excavation Certainty:
Finder (= First Purchaser)	Rubensohn, Otto (excavation director) Certainty:
District of Find / Purchase in Egypt	Upper Egypt, 1st nome (Ombites) [Trismegistos]
Type of Acquisition for the Intitution	partage
Date of Acquisition for the Intitution	between 1906 and 1906

Object

Object Type	ostracon
Color	pottery, yellow brown
Size (Height \| Width \| Thickness)	105 mm \| 91 mm \| 11 mm
Dating	between -300 and -201
Range of Preservation	incomplete

Text Basic Information

Localization of Text on Object	convex (outside)
Script, Primary	Greek
Language, Primary	Greek, Ancient
Comments on Handwriting	Zur Handschrift, vgl. F. A. J. Hoogendijk, Twelve Greek Ostraca from Elephantine, in: F. A. J. Hoogendijk - B. P. Muhs (Hgg.), Sixty-Five Papyrological Texts Presented to Klaas A. Worp on the Occasion of his 65th Birthday, Leiden 2008, 287; zu Z. 3 vgl. H. Cadell, Einleitung zu P.Coll.Youtie I 55–62, S. 337, Anm. 13.
Comments on Text Layout	19 lines, other side uninscribed
	recto	verso
Quantity of Lines	19

Text Content

Modern Title	Geometrische Aufgabe
Text Types	scientific \| math
Summary of Content	geometrical exercise: drawing of a regular icosahedron into a sphere of given diameter (cf. Euclid Stoicheia XIII 16)
Location of Composition	Elephantine Upper Egypt, 1st nome Egypt (Certainty: )
Multilingualism	Monolingual Script = Language

Text

Transcription

Translation

Pictures

convex

τῆι ηζἰσόπλευρον ἄρα τὸ ζρη. διὰ δὲ ταῦτα καὶ τὰ ησθ

2	θτι ιυκ κφζ. ἴση δὲ καὶ ἡ ρσ τῆ ι ηθ καὶ ἴσας ἑδείξαμεν
3	τὰς ρη σθ. ἰσόπλευρον ἄρ' ἑστὶν ἕκαστον τῶν μεταξὺ τριγώνων.
4	ἴσας γὰρ καὶ παραλλήλους τῆς χλ τὰς ζμ ην θξ ιο κπ ἐγράψαμεν. ἑξα-
5	γώνου ἄρα καὶ ἡ χρ, ἀλλὰ καὶ ἡ ψχδεκαγώνου, ὥστ' εἶναι τὰς ψρ ψσ
6	πενταγώνου. ἴση ἄρ' ἡ φρ τῆι ρσ, διὸ ἰσόπλευρά ἐστι τὰ ψφρ ψρσ
7	τρίγωνα. διὰ ταῦτα δὲ καὶ ἔκαστον τῶν τὸ ψ ὡς κορυφὴν
8	ἐχόντων ἰσόπλευρά ἐστιν. πάλιν ἐκβεβλήσθω
9	ἐπ' εὐθείας τῆι λψ ἡ λω, ἴση τῆι τοῦ δεκαγώνου πλευρᾶι.
10	ἤχθω δὴ ἀπὸ τοῦ λ κέντρου ἡ λζ ἑξαγώνου, δεκαγώνου δ' ἦν ἡ
11	λω, πενταγώνου ἄρ' ἡ ζω, διὰ ταῦτα δ' ἰσόπλευρόν ἐστι
12	τὰ τρίγωνον τὸ ωζη, διὰ ταὐτὰ τὸ ζηθικω στερεὸν σχἠμα ἐκ
13	πρὸς τῶι ω τὰς κορυφὰς ἐχόντων πέντε
14	τριγώνων ἰσοπλεύρων πεποίηται.
15	δύο δεκαγώνου πλευραὶ καὶ μία ἑξαγώνου τῆς
16	δοθείσης σφαίρας τὴν διάμετρον συμπληροῦσιν. καὶ ἡ τῆς
17	δοθείσης σφαίρας διάμετρος δυνάμει πενταπλασίων τῆς ἡμι-
18	διαμέτρου τοῦ κύκλου ἐστιν τοῦ τὸ πεντάγωνον
19	περιλαμβανομένου, ἀφ' οὗ τὸ εἰκοσάεδρον ἀναγέγραπται.

convex

1	ΗΖ ... So the triangle ZΡH is equilateral. Thus also the triangles ΗΣΘ,
2	ΘΤΙ, ΙΥΚ, ΚΦΖ. But equal is also ΡΣ to the line ΗΘ, and as equal we have shown
3	the lines ΡΗ and ΣΘ. So each of the triangles lying between (scil. the circles) is equilateral,
4	because we had drawn the perpendiculars ΖΜ, ΗΝ, ΘΞ, ΙΟ, ΚΠ equal and parallel to the line XΛ.
5	That is why XP is also (scil. equal to one) hexagonal side. But also ΨΧ was (scil. equal to a) decagonal side, so that ΨΡ and ΨΣ
6	are pentagonal sides. Equal was <also> ΦΡ to side ΡΣ, therefore ΨΦΡ and ΨΡΣ are equilateral
7	triangles. Therefore each of the triangles whose common apex is Ψ is also
8	equilateral. Furthermore, let the distance
9	ΛΩ be drawn as an extension of the distance ΛΨ, equal to the side of the decagon.
11	Let the radius ΛΖ equal to the hexagon side be drawn from the centre of the circle Λ, but (scil. equal) to the decagon side was
12	ΛΩ , so ΖΩ is a pentagon side. Therefore equilateral is
13	the triangle ΩΖΗ. For the same reason, the spatial figure ΖΗΘΙΚΩ is formed by
14	five equilateral triangles having their apex in Ω.
15	Two decagonal sides and one hexagonal side together
16	give the diameter of the given sphere ¹cf. Euclid El. XIII 16 corollary,
17	and the diameter of the given sphere squared is five times
18	the radius of the given circle squared,
19	drawn around the pentagon on which the icosahedron was built. (after Müller/Mau 1960)

¹ cf. Euclid El. XIII 16 corollary

Places (read out from edition)

Dates

RulerID	Regnal Year	MonthID	Day Date of the Text	Gregorian Date	dating_comment
				-299 BCE -200 BCE	palaeographical dating;

Literature

Dorandi, T., La tradizione papiracea degli „Elementi“ di Euclide (1992). (311)
Fowler, D., The Mathematics of Plato’s Academy. A New Reconstruction (1999). (209-211)
Horak, u., Illuminierte Papyri, Pergamente und Papiere I (1992). (253)
Müller, W. & Mau, J., Mathematische Ostraka aus der Berliner Sammlung (1960). (1-10, no. 1)

DatasetID	16497
last Change	29.07.2022
Berlpap	16497
Trismegistos	65672
Author	BerlPap-Team; Daniel Werning, Philip Schmitz
Dataset License	Creative Commons: Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA)
Data set citation	Data set 16497 (= Ostr. Berlin P. 12609), ERC-Project ELEPHANTINE: BerlPap-Team; Daniel Werning, Philip Schmitz.