Ostomachion

Αρχιμήδους <Ὀ>στομάχιον (source: Greek palimpsest, from the Cloister of Saint Sabba [= Mar Saba], Jerusalem, then at Metochion of the Holy Sepulchre, Constantinopole, 1899 - The text is in Heiberg's edition of Archimedes, Opera, 2nd ed., Teubner, Leipzig, 1913, vol. II, pp. 415-424):

(1)
Τοῦ λεγομένου <Ὀ>στομαχίου ποικίλαν ἔχοντος τᾶς ἐξ  ὧν συνέστακε σχημάτων μεταθέσεως θεωρίαν ἀναγκαῖον ἡγησάμην πραττον του <...............> ρῶν ἐκθέσθαι, εἴς τε ἃ διαιρεῖται, ἕκαστόν τε αὐτῶν τίνι ἐστὶν ὁμοιούμενον, ἔτι δὲ καί, ποῖαι γωνίαι σύνδυο λαμβανόμεναι <. . .> καὶ <. . .> θάς, εἴηται πρὸς  τὸ τὰς ἐναρμόσεις τῶν ἐξ αὐτῶν γεννωμένων σχαμάτων γιγνώσκεσθαι, εἴτε ἐπ' εὐθείας εἰσὶν αἱ γεννώμεναι ἐν τοῖς σχάμασι πλευραί, εἴτε καὶ μικρῶς  λείπουσαι τᾶι θεωρίαι λανθάνουσιν· τὰ γὰρ τοιαῦτα  φιλότεχνα· καὶ ἐὰν ἐλάχιστον μὲν λείπηται, τᾶι δὲ θεωρία λανθάνη, οὐ καρὰ τοῦτ' ἐστιν ἔκβλητα, ἃ συνίσταται.

(2)
Ἔστι μὲν οὖν ἐξ αὐτῶν οὐκ ὀλίγων σχημάτων ....... ο .. διὰ τὸ ..... ν .. τον εἶναι εἰς ἕτερον τόπον τοῦ ἴσου καὶ ἰσογωνίου σχάματος μετατιθεμε ... καὶ ἑτέ ..... λαμβάνοντας. <ἐνίο>τε δὲ καὶ δύο σχημάτων συνάμφω ἑνὶ σχήματι ἴσων ὄντων καὶ ὁμοίων τῶι ἑνὶ σχήματι ἢ καὶ δύο σχημάτων συνάμφω ἴσον τε καὶ ὁμοίων ὄντων δυσὶ σχήμασι συνάμφω πλείονα σχήματα συνίσταται ἐκ τῆς μεταθέσεως. προγράφομεν οὖν τι θεώρημα εἰς αὐτὸ συντεῖνον.

(3)
Ἔστο γὰρ παραλληλόγραμμον ὀρθογώνιον τὸ ΖΓ, καὶ δε.ι.....ω ἡ ΕΖ τῶι Κ, καὶ .. διήχθωσαν ἀπὸ τῶν Γ, Β αἱ ΓΚ, ΒΕ .ει..ων ... τῶν ... Γ ........ ἐκ<βεβλή>σθωσαν αἱ ΓΚ, ΒΖ καὶ συμπιπτέ<τωσαν κατὰ τὸ Δ .......> ἡ ΓΗ. ἐπεὶ ἴση ἐστὶν ἡ ΕΚ τῆι ΚΖ, <ἴση> καὶ ἡ ΓΕ, τουτέστιν ἡ ΒΖ, τῆι ΖΔ· ὥ<στε> μείζων ἡ ΓΖ τῆς ΖΔ· καὶ γωνία <ἄρα> ἡ ὑπὸ τῶν ΖΔΓ τῆς ὑπὸ τῶν ΖΓΔ μείζων. ἴσαι δέ εἰσιν αἱ ὑπὸ ΗΒΔ, ΖΓΒ· ἡμίσεια γὰρ ὀρθῆς ἑκατέρα· μείζων ἄρα καὶ ἡ ὑπὸ τῶν ΓΗΒ, ἐπεὶ ἡ ὑπὸ ΓΗΒ ἴση δυσὶ ταῖς ἐντὸς καὶ ἀπεναντίον ταῖς ὑπὸ ΗΒΔ, ΗΔΒ, τῆς ὑπὸ τῶν ΗΓΒ· ὥστε μείζων ἐστὶν ἡ ΓΒ τῆς ΒΗ. ἐὰν ἄρα δίχα τμηθῆι ἡ ΓΗ κατὰ Χ, ἔσται ἀμβλεῖα μὲν ἡ ῦπὸ ΓΧΒ· ἐπεὶ γὰρ ἴση ἡ ΓΧ τῆι ΧΗ, καὶ κοινὴ ἡ ΧΒ, δύο δυσὶν ἴσαι· καὶ βάσις ἡ ΓΒ τῆς ΒΗ μείζων· καὶ ἡ γωνία ἄρα τῆς γωνίας μείζων. ἀμβλεῖα μὲν ἄρα ἡ ὑπὸ ΓΧΒ, ὀξεῖα δὲ ἡ ἐφεξῆς. ἡμίσεια δὲ ὀρθῆς ἡ ὑπὸ ΓΒΗ· τοῦτο γάρ ἐστιν ὑποκείμενον τοῦ παραλληλογράμμου· ὀξεῖα δὲ ἡ ὑπὸ ΒΧΗ. καὶ . τι δὴ ἴση ἡ λοιπαὶ ΓΒΗ καὶ συνίσταται καὶ διαιρεῖται τοῦτο επ. ον τον .... 

(4)
........ βάσιος . τι .... ...... αστ.α. ἄρα ο... ΑΒ ... .αν..ο... τῆν ΓΑ .....νῶν ...... έχον .... τὸ ἐπίλοιπ.... .................... .. δύνασθαι ἀρ.....ξειν εκ.... τῶν τομῶν ... τῶν τάξιν ἐχοντ.

(5)
τετμήσθω ἡ ΓΑ δίχα κατὰ τὸ Ε, καὶ διὰ τοῦ Ε τῆι ΒΓ παράλληλος ἤχθω ἡ ΕΖ· ἔστιν οὖν τετράγωνα τὰ ΓΖ, ΖΑ. ἤχθωσαν διάμετροι αἱ ΓΔ, ΒΕ, ΕΔ, καὶ τετμήσθωσαν δίχα αἱ ΓΗ, ΕΔ κατὰ τὰ Θ, Χ, καὶ ἐπεζεύχθωσαν αἱ ΒΘ, ΧΖ, καὶ διὰ τῶν ., Κ τῆι ΒΔ παράλληλοι ἤχθωσαν αἱ Κ., .Ξ. διὰ τὸ προκείμενον ἄρα θεώρημα τοῦ ΒΓΘ τριγώνου ἡ πρὸς τῶι Θ γωνία ἀμπλεῖα, ἡ δὲ λοιπὴ ὀξεῖα. .... νερὸν φανερὸν δὲ ...ει....

 

(This text is rather incomplete and describes the relationships between various angles of the pieces. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces of paper could be assembled into the shape of a square and thus represents an example of an early problem in combinatorics)

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D. M. Ausonii 'Cento nuptialis' (bibliographical source: edited & translated by H. G. Evelyn White. Loeb Classical Library. Vol. I, Book XVII: Cento Nuptialis):
[...] Accipe igitur opusculum de inconexis continuum, de diversis unum, de seriis ludicrum, de alieno nostrum, ne in sacris et fabulis aut Thyonianum mireris aut Virbium, illum de Dionyso, hunc de Hippolyto reformatum. Et si pateris, ut doceam docendus ipse, cento quid sit absolvam. Variis de locis sensibusque quaedam carminis structura solidatur, in unum versum ut coeant aut caesi duo aut unus et sequens cum medio, nam duos iunctim locare ineptum est et tres una serie merae nugae. Diffinduntur autem per caesuras omnes, quas recipit versus heroicus, convenire ut possit aut penthemimeris cum reliquo anapaestico aut trochaice cum posteriore segmento aut septem semipedes cum anapaestico chorico aut <ponatur> post dactylum atque semipedem quidquid restat hexametro, simile ut dicas ludicro, quod Graeci «ostomachion» vocavere. Ossicula ea sunt: ad summam quattuordecim figuras geometricas habent. Sunt enim aequaliter triquetra vel extentis lineis vel <eiusdem> frontis, <vel aequicruria vel aequilatera, vel rectis> angulis vel obliquis: isoskele ipsi vel isopleura vocant, orthogonia quoque et scalena. Harum verticularum variis coagmentis simulantur species mille formarum: helephantus belua aut aper bestia, anser volans et mirmillo in armis, subsidens venator et latrans canis, quin et turris et cantharus et alia huiusmodi innumerabilium figurarum, quae alius alio scientius variegant. Sed peritorum concinnatio miraculum est, imperitorum iunctura ridiculum. [...]
(Ausonius compares here the Ostomachion to a form of poetry in which various meters are jumbled together)

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Caesii Bassi sive Atilii Fortunatiani fragmentum 'De Metris' (bibliographical source: H. Keil, Grammatici Latini, Volume VI, Hildesheim, 1961):

[...] ut quodcumque metrum novum aliquis se invenisse iudicarit, ad haec quae enumeravi utique referatur, cum omnia metra varientur aut adiectione aut detractione aut concinnatione aut permutatione [...] habet autem metrorum contemplatio, si exercitatio accessit, in cognoscendo voluptatem, cum et quaecumque dicuntur metra celeriter intellegamus unde sint et qua ratione composita, et multa ipsi nova excogitare possimus. nam si loculus ille | Archimedius, <qui> quattuordecim eboreas lamellas, quarum varii anguli sunt, in quadratam formam inclusas habet, componentibus nobis aliter atque aliter modo galeam, modo sicam, alias columnam, alias navem figurat et innumerabiles efficit species, solebatque nobis pueris hic loculus ad confirmandam memoriam prodesse plurimum, quanto maiorem potest nobis adferre voluptatem quantoque pleniorem utilitatem carmina inter manus habentibus metrorum varia tractatio, cum subinde apud poetas ea quae fallunt imperitos metra inserta numeris et intermixta carminibus || hac arte deprehendemus? de quibus in his libris explicabimus, quos de melicis poetis et de tragicis choris scripturi videmur, quibus necesse erit etiam graeca interponere exempla, quod ne faceremus in hoc libro elaboravimus. nam si volumus de omni ratione carminum dicere, necesse est ad eorum copias manum porrigamus, qui cum essent non tantum poetae perfectissimi sed etiam musici, sine magno labore praeparatis utebantur facultatibus. [...]
(Bassus reports that the combinatorial puzzle game was played by children as it was supporting their memory)

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Marii Victorini 'Ars grammatica' (bibliographical source: p. 100 in the edition of Keil. Given in Archimedes Opera II, 2nd ed., 1913, p. 417):
[...] ut ille Loculus Archimedius e quattuordecim crustis eburneis, nunc quadratis, nunc triangulis, nunc ex utraque specie varie figuratis, et velut quibusdam membris artis struendae causa compositus proditur. Nam ut in illo praefinito ac determinato crustarum numero multiplici earundem varietatum specie nunc navis, nunc gladius, nunc arbuscula et siqua alia figurantur, ita metrorum certo ex origine numero intra decem prototypa comprehenso multiplex admodum varietatis copia propagatur. [...]

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Lucretius 'De rerum natura' (II. 772-787 - Rose, 1956):
[...] quod si caeruleis constarent aequora ponti
seminibus, nullo possent albescere pacto ;
nam quocumque modo perturbes caerula quae sint,
[775]
numquam in marmoreum possunt migrare colorem.
sin alio atque alio sunt semina tincta colore,
quae maris efficiunt unum purumque nitorem,
ut saepe ex aliis formis variisque figuris
efficitur quiddam quadratum unaque figura,
[780]
conveniebat, ut in quadrato cernimus esse
dissimiles formas, ita cernere in aequore ponti
aut alio in quovis uno puroque nitore
dissimiles longe inter se variosque colores.
praeterea nihil officiunt obstantque figurae
[785]
dissimiles, quo quadratum minus omne sit extra ;
at varii rerum inpediunt prohibentque colores,
quominus esse uno possit res tota nitore.
(Lucretius argues that apparently uniform visible colors are composed of diverse atoms/elements, as the pieces of a geometric puzzle fit together to form a square)

 

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M. Felix Ennodius 'De stomachio eburneo' (carm. II 133 Hartel = 340 Vogel):
[...] sollicitata levi marcescunt corda virorum
tormento: fas est ludere virginibus.
frangunt Marmaricis elefans quod misit ab arvis
per micas sparsum mox solidatur opus.
[5]
de poena tenerae discunt cum fraude iocari :
nam ridere necis munere femineum est.
angusta norunt res mille includere capsa :
omne ebur haec, mulier, pectoris arca tui est.
(The first two verses may testify to the annoyance to which the Ostomachion puzzle could give rise, while the next two apparently refer in a rather elaborate way to the composition of figures from bits of ivory. It is also possible that with this epigram Ennodius wanted to poetically compare the heart of women with the Ostomachion and to assimilate the many deceptions concealed by nature in the feminine soul to the endless figurative variations of the puzzle.)

Related Links

Ostomachion puzzle The Loculus of Archimedes, solved Tangram puzzle Leonardo's rebuses Alcuin's puzzles Symposius' riddles Fibonacci's numbers

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