§25.1Διὰ τί τὰς κλίνας ποιοῦσι διπλασιοπλεύρους, τὴν μὲν ἓξ ποδῶν καὶ μιαρῷ μείζω πλευράν, τὴν δὲ τριῶν;
Why do they make beds with one side double the other, one side being six feet (and a little longer), and the other three?
Καὶ διὰ τί ἐντείνουσιν οὐ κατὰ διάμετρον;
And why do they string them not diagonally?
Ἢ τὸ μὲν μέγεθος τηλικαύτας, ὅπως τοῖς σώμασιν ὦσι σύμμετροι;
Or is it that they make them of such dimensions in size, so that they may be commensurate with human bodies?
Γίνονται γὰρ οὕτω διπλασιόπλευροι, τετραπήχεις μὲν τὸ μῆκος, διπήχεις δὲ τὸ πλάτος.
For in this way they become twice as long as they are wide, four cubits in length and two cubits in width.
§25.2Ἐντείνουσι δὲ οὐ κατὰ διάμετρον ἀλλ’ ἀπ’ ἐναντίας, ὅπως τά τε ξύλα ᾖττον διασπᾶται· τάχιστα γὰρ σχίζεται κατὰ φύσιν διαιρούμενα ταύτῃ, καὶ ἑλκόμενα πονεῖ μάλιστα.
And they string them not diagonally but from opposite sides, so that the wood is less split; for wood is most quickly split when divided in this direction according to its nature, and when pulled it is strained most.
Ἔτι ἐπειδὴ δεῖ βάρος δύνασθαι τὰ σπαρτία φέρειν, οὕτως ἧττον πονέσει λοξοῖς τοῖς σπαρτίοις ἐπιτιθεμένου τοῦ βάρους ἢ πλαγίοις.
Furthermore, since the cords must be able to bear weight, in this way they will be less strained when the weight is placed on the diagonal cords than on the transverse ones.
Ἔτι δὲ ἔλαττον οὕτω σπαρτίον ἀναλίσκεται.
Moreover, less cord is consumed in this way.
§25.3Ἕστω γὰρ κλίνη ἡ ΑΖΗΙ, καὶ δίχα διῃρήσθω ἡ ΖΗ κατὰ τὸ Β. Ἴσα δὴ τρυπήματά ἐστιν ἐν τῇ ΖΒ καὶ ἐν τῇ ΖΑ. Καὶ γὰρ αἱ πλευραὶἴσαι εἰσίν· ἡ γὰρ ὅλη ΖΗ διπλασία ἐστίν.
For let the bed be AZHI, and let ZH be bisected at B. Then the holes in ZB are equal to those in ZA. For indeed the sides are equal; for the whole ZH is double [of ZA].
Ἐντείνουσι δ’ ὡς γέγραπται, ἀπὸ τοῦ Α ἐπὶ τὸ Β, εἶτα οὖ τὸ Γ, εἶτα οὗ τὸ Δ, εἶτα οὗ τὸ Θ, εἶτα οὖ τὸ Ε.
And they string it as has been diagrammed, from A to B, then to C, then to D, then to Th, then to E.
§25.4Καὶ οὕτως ἀεί, ἕως ἂν εἰς γωνίαν καταστρέψωσιν ἄλλην· δύο γὰρ ἔχουσι γωνίαι τὰς ἀρχὰς τοῦ σπαρτίου.
And so on always, until they wind up at another corner; for two corners contain the beginnings [and ends] of the cord.
Ἴσα δέ ἐστι τὰ σπαρτία κατὰ τὰς κάμψεις, τό τε ΑΒ καὶ ΒΓ τῷ ΓΔ καὶ ΔΘ. Καὶ τὰ ἄλλα δὲ τὰ τοιαῦτά ἐστιν, ὅτι οὕτωςἔχει ἡ αὐτὴ ἀπόδειξις, Ἡ μὲν γὰρ ΑΒ τῇ ΕΘ ἴση·
And the cords at the bends, namely AB and BC, are equal to CD and DTh. And the others of this kind are also equal, because the same proof holds for them.
ἴσαί γάρ εἰσιν αἱ πλευραὶ τοῦ ΒΗΚΑ χωρίου, καὶ τὰτρυπήματα ἴσα διέστηκεν.
For AB is equal to ETh; for the sides of the area BHKA are equal, and the holes are equally spaced.
§25.5Ἡ δὲ Β Η ἴση τῇ ΚΑ·
And BH is equal to KA; for angle B is equal to [angle at] H.
ἡ γὰρΒ γωνία ἴση τῇ Η. Ἐν ἴσοις γὰρ ἡ μὲν ἐκτός, ἡ δὲ ἐντός· καὶ ἡ μὲν Β ἐστὶν ἡμίσεια ὀρθῆς· ἡ γὰρ ΖΒ ἴση τῇ ΖΑ· καὶ γωνία δὲ ἡ κατὰ τὸ Ζ ὁρθή.
For among equal angles, one is external and the other internal; and angle B is half of a right angle; for ZB is equal to ZA, and the angle at Z is a right angle.
Ἠ δὲ Β γωνία ἴση τῇ κατὰ τὸ Η· ἡ γὰρ κατὰ τὸ Ζ ὁρθή, ἐπειδὴ διπλασιόπλευρον τὸ ἑτερόμηκες καὶ πρὸς μέσον κέκλασται.
And angle B is equal to that at H; for that at Z is a right angle, since the oblong is twice as long as wide and is bent towards the middle.
§25.6Ὠστε ἡ ΑΓ τῇ ΕΗ ἴση.
Consequently, AC is equal to EH.
Ταύτῃ δὲ ἡ Κ Θ· παράλληλος γάρ.
And KTh is equal to this; for it is parallel.
Ὤστε ἡ ΒΓ ἴση τῇ ΚΘ. Ἡδὲ ΓΕ τῇ ΔΘ. Ὁμοίως δὲ καὶ αἰ ἄλλαι δείκνυνται ὅτι ἴσαί εἰσὶν αἱ κατὰ τὰς κάμψεις δύο ταῖς δυσίν.
Therefore, BC is equal to KTh, and CE to DTh. Similarly, the others are also shown to be equal, namely that the cords at the bends, two by two, are equal to each other.
Ὤστε δῆλον ὅτι τὰ τηλικαῦτασπαρτία ὅσον τὸ Α Β, τέσσαρα τοσαῦτ’ ἔνεστιν ἐν τῇ κλίνῃ· ὅσον δ’ ἐστὶ τὸ πλῆθος τῶν ἐν τῇ ΖΗ πλευρᾷ τρυπημάτων, καὶ ἐν τῷ ἡμίσειτῷ Ζ Β τὰ ἡμίση.
So it is clear that there are four such cords of the size of AB in the bed; and as is the number of holes on the side ZH, there are half that number in the half ZB.
§25.7Ὤστε ἐν τῇ ἡμισείᾳ κλίνῃ τηλικαῦτα μεγέθη σπαρτίων ἐστὶν ὅσοντῷ ΒΑ ἔνεστι, τοσαῦτα δὲ τὸ πλῆθος ὅσαπερ ἐν τῷ ΒΗ τρυπήματα.
Consequently, in the half bed, there are cords of such size as BA, and as many in number as there are holes in BH.
Γαῦτα δὲ οὐδὲν διαφέρει λέγειν ἢ ὅσα ἐν τῇ ΑΖ καὶ ΒΖ τὰ συνάμφω.
This is no different from saying as many as those in AZ and BZ combined.
εἰ δὲ κατὰ διάμετρον ἐνταθῇ τὰ σπαρτία, ὡς ἐν τῇ Α Β Γ Δ κλίνῃ ἔχει, τὰ ἡμίσεά εἰσιν οὐ τοσαῦτα ὅσα αἱ πλευραὶ ἀμφοῖν, αἱ Α Ζ Ζ Η· τὰ ἴσα δέ, ὅσα ἐν τῷ ΖΒΖΑ τρυπήματα ἔνεστιν.
But if the cords are strung diagonally, as in the bed ABCD, the halves are not as much as the two sides combined, AZ and ZH; but equal to the number of holes in ZB and ZA.
Μείζονες δέ εἰσιν αἰ ΑΖ ΒΖ δύο οὖσαι τῆς ΑΒ. Ὤστε καὶ τὸ σπαρτίον μεῖζον τοσούτῳ ὅσον αἱ πλευραὶ ἄμφω μείζους εἰσὶ τῆς διαμέτρου.
But AZ and BZ, being two, are greater than AB. Consequently, the cord is also larger by as much as the two sides are greater than the diagonal.