§1## ΣΤΟΜΑΧΙΟΝ
Ἀρχιμήδους Στομάχιον
Τοῦ λεγομένου 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 καὶ ποῖαι γωνίαι σύνδυο λαμβανόμεναι καὶ θάς, εἴρηται πρὸς τὸ τὰς ἐναρμόσεις τῶν ἐξ αὐτῶν γεννωμένων σχαμάτων γιγνώσκεσθαι, εἴτε ἐπʼ 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 θεωρίᾳ λανθάνῃ, οὐ παρὰ τοῦτʼ ἐστὶν ἔκβλητα ἃ συνίσταται.
[STOMACHION] Archimedes' Stomachion Since the so-called Stomachion has a varied theory of the transposition of the figures from which it is composed, I deemed it necessary to set out, for those who occupy themselves [with this subject], into what it is divided, and what each of them is similar to, and further, what kinds of angles, taken two by two, [make a right angle]. This is said so that the fittings of the figures generated from them may be known, whether the sides generated in the figures are on a straight line, or whether, being slightly deficient, they escape [detection] in the theory; for such things are artistic, and even if there is a minimal deficiency, and it escapes [detection] in the theory, the things constructed are not on this account to be rejected.
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.
Thus, not a few figures are constructed from them, because of the transposition of [a piece] of equal and equiangular figure to another place.
Ἐνιότε δὲ καὶ δύο σχημάτων συνάμφω ἑνὶ σχήματι ἴσων ὄντων καὶ ὁμοίων τῷ ἑνὶ σχήματι ἢ καὶ δύο σχημάτων συνάμφω ἴσων τε καὶ ὁμοίων ὄντων δυσὶ σχήμασι συνάμφω πλείονα σχήματα συνίσταται ἐκ τῆς μεταθέσεως.
Sometimes, when two figures together are equal and similar to one figure, [they are transposed] to that one figure, or when two figures together are equal and similar to two figures together, [they are transposed] to the two figures together, more figures are constructed from the transposition.
Προγράφομεν οὖν τι θεώρημα εἰς αὐτὸ συντεῖνον.
We therefore pre-write a certain theorem contributing to this.
Ἔστω γὰρ παραλληλόγραμμον ὀρθογώνιον τὸ ΖΓ, καὶ δε. ι ω ἡ ΕΖ τῷ Κ, καὶ.
For let there be a rectangular parallelogram ZG, and let EZ [be divided] at K, and let GK, BE be drawn from G, B.
.
Let GK, BZ be produced from G and meet at D, and GH.
διήχθωσαν ἀπὸ τῶν Γ, Β αἱ ΓΚ, ΒΕ. ει ων τῶν Γ ἐκβεβλήσθωσαν αἱ ΓΚ, ΒΖ καὶ συμπιπτέτωσαν κατὰ τὸ △ ἡ ΓΗ. Ἐπεὶ ἴση ἐστὶν ἡ ΕΚ τῇ ΚΖ, ἴση καὶ ἡ ΓΕ, τουτέστιν ἡ ΒΖ, τῇ Ζ△ ὥστε μείζων ἡ ΓΖ τῆς Ζ△ καὶ γωνία ἄρα ἡ ὑπὸ τῶν Ζ△Γ τῆς ὑπὸ τῶν ΖΓ△ μείζων.
Since EK is equal to KZ, GE (that is, BZ) is also equal to ZD; so that GZ is greater than ZD, and therefore the angle ZDG is greater than ZGD.
Ἶσοι δέ εἰσιν αἱ ὑπὸ ΗΒ△, ΖΓΒ· ἡμίσεια γὰρ ὀρθῆς ἑκατέρα μείζων ἄρα καὶ ἡ ὑπὸ τῶν ΓΗΒ, ἐπεὶ ἡ ὑπὸ ΓΗΒ ἴση δυσὶ ταῖς ἐντὸς καὶ ἀπεναντίον ταῖς ὑπὸ ΗΒ△, Η△Β, τῆς ὑπὸ τῶν ΗΓΒ ὥστε μείζων ἐστὶν ἡ ΓΒ τῆς ΒΗ. Ἐὰν ἄρα δίχα τμηθῇ ἡ ΓΗ κατὰ Χ, ἔσται ἀμβλεῖα μὲν ἡ ὑπὸ 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.
And the angles HBD, ZGB are equal; for each is half of a right angle; therefore the angle GHB is also greater [than HGB], since the angle GHB is equal to the two interior and opposite angles HBD, HDB, [which is greater] than HGB; so that GB is greater than BH. If then GH is bisected at X, the angle GXB will be obtuse; for since GX is equal to XH, and XB is common, two [sides] are equal to two, and the base GB is greater than BH, therefore the angle also is greater than the angle.
Ἀμβλεῖα μὲν ἄρα ἡ ὑπὸ ΓΧΒ, ὀξεῖα δὲ ἡ ἐφεξῆς.
Therefore, the angle GXB is obtuse, and the adjacent angle is acute.
Ἡμίσεια δὲ ὀρθῆς ἡ ὑπὸ ΓΒΗ τοῦτο γάρ ἐστιν ὑποκείμενον τοῦ παραλληλογράμμου ὀξεῖα δὲ ἡ ὑπὸ ΒΧΗ. Καὶ.
And the angle GBH is half of a right angle; for this is the hypothesis of the parallelogram; and the angle BXH is acute.
τι δὴ ἴση ἡ λοιπαὶ ΓΒΗ καὶ συνίσταται καὶ διαιρεῖται τοῦτο ἐπ. ον τὸν βάσιος τι αστ α ἄρα ο ΑΒ αν ο τὴν ΓΑ νῶν έχον τὸ ἐπίλοιπ δύνασθαι ἀρ ξειν ἑκ τῶν τομῶν τῶν τάξιν ἐχοντ.
And indeed the remaining GBH is equal, and this is constructed and divided on the base [...] therefore AB [...] GA [...] [...] the remaining [...] to be able [...] from the cuts having order.
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 Γ△, ΒΕ, Ε△, καὶ τετμήσθωσαν δίχα αἱ ΓΗ, Ε△ κατὰ τὰ Θ, Χ, καὶ ἐπεζεύχθωσαν αἱ ΒΘ, ΧΖ, καὶ διὰ τῶν, Κ τῇ Β△ παράλληλοι ἤχθωσαν αἱ Κ, Ξ. Διὰ τὸ προκείμενον ἄρα θεώρημα τοῦ ΒΓΘ τριγώνου ἡ πρὸς τῷ Θ γωνία ἀμβλεῖα, ἡ δὲ λοιπὴ ὀξεῖα νερὸν φανερὸν δὲ ει.
Let GA be bisected at E, and through E let EZ be drawn parallel to BG; therefore ZG, ZA are squares. Let diagonals GD, BE, ED be drawn, and let GH, ED be bisected at Th, X, and let BTh, XZ be joined, and through [...], K let K[...], X (Ξ) be drawn parallel to BD. Therefore, because of the preceding theorem, the angle at Th of the triangle BGTh is obtuse, and the remaining angle is acute is clear.