projecting from the mouths; to which it is necessary to be wound with the tensioner by means of shoulder or back sinews of all animals except pigs, passing through the middle of the sockets for the tension of the turning, or with the thicker skeins of silk threads, or also with cords made of linen thread, receiving in the middle a long piece of wood inserted in the shape of a palintone arm, bent back behind and held by a pin, such as are the single-armed stone-throwers, 254 which some call slings. And it is necessary for an iron lever having a root-ring, inserted into the aforementioned mouth-pieces, to make the turning on the sockets forceful and to produce a violent tension. And the ram, being pushed from the ladders and brought against the wall, will hurl the released single-arm against the wall-defenders, and will effect a great capture of those standing on it. And the diagram has been drawn. This construction of the single-arm will also supply those who wish with catapult theory, contributing much 255 to long-range shooting with both straight-sprung and palintone engines, that is, with both stone-throwers and arrow-shooters; for the arrangement of the sockets makes the engine longer through the spring and, concerning the tension, produces a more violent discharge. For by as much as the spring is increased in length, by so much is the discharge of the missile naturally extended, with the bowstring effecting everything and sustaining the force of the discharge, whence they say its plaiting comes from the strongest and most exercised sinews of the animal; for example, of a deer, those in the shanks and feet, and of a bull, those on the neck; and that the sinews of the straight-sprung engines are round in their plaiting, so as to fall into the notches of the arrows; but those of the palintones are broad, like belts, having hooks at the ends, into which the arms are fitted, so that the stone, being struck by the middle of the broad bowstring, may be properly discharged without slipping off to the ends. They say that the spring in the arms is also plaited from women's hair; for being fine and long and nourished with much oil, when they are plaited, they acquire great elasticity, so that they are not inconsistent with the strength derived from sinews. But enough about these things for now. For in the treatise on the construction of missiles, in detail 256 for the more inquisitive, for construction, not only these things, but also the diameters of the holes in the engines, and the cylinders shown to be similar to each other in a triplicate ratio of the diameters in the bases, and the much-talked-of duplication of the cube, by which the sizes of the missiles for any given engines, becoming double or even triple, are discharged to the same distances. There everything has been investigated mathematically and mechanically for proof by Hero of Alexandria. There is also another type of wheeled ladder, itself also constructed from strong and light woods, to which another, as in the manner of a gangway, inserted from above in the middle, is joined by a small axle, having cylindrical and smooth ends turned on a lathe, which are joined to the side-pieces, so that, being brought down easily against the other, it may be closed, and again may be brought back up to it, erect, held by ropes from behind, being let out and drawn in commensurately by means of engines called polyspasts, that is, smooth-running pulleys, imparting a powerful force for the 257 slackening and tensioning of the ropes through the integral arrangement of the adjacent pulleys. And let there be little stops or some projections on top of the standing ladder, to receive the ladder being raised as it turns back, lest, being raised suddenly to the top of the other, it should rather threaten a fall upon the workers. And if the ladder intended to stand below should be of a single beam, let it have the rungs, one by one, projecting from the side-pieces and pierced at the projection, so that a strong, taut rope passing through the holes may keep it from breaking; but if composite

τῶν στομίων παρεξέχοντα· πρὸς ἃ νεύροις ὠμιαίοις ἢ νωτιαίοις πάντων ζῴων πλὴν συῶν διὰ μέσου τῶν χοινικίδων διερχομένοις ἐπὶ τῇ τῆς στροφῆς τάσει δεῖ τῷ ἐντονίῳ περιειλεῖσθαι, ἢ τοῖς ἐκ νημάτων σηρικῶν ἁδροτέροις μαλάθοις, ἢ καὶ σχοινίοις ἐκ λίνου νηματικοῖς, κατὰ μέσον εἰσδεχομένοις ξύλον μακρὸν ἐμβαλλόμενον ἐν σχήματι παλιντόνου ἀγκῶνος ἀντεστραμμένον ὄπισθεν καὶ κατακλεῖδι κρατούμενον, οἷοί εἰσιν οἱ λιθοβόλοι μονάγκωνες, 254 οὕς τινες σφενδόνας καλοῦσι. Μοχλὸν δὲ δεῖ σιδηροῦν ῥιζοκρίκιν ἔχοντα, πρὸς τὰς ῥηθείσας περιστομίδας ἐμβαλλόμενον, βιαίαν τὴν ἐπιστροφὴν ἐπὶ τῶν χοινικίδων ποιεῖν καὶ σφοδρὰν τὴν τάσιν ἀπεργάζεσθαι. Ὁ δὲ κριὸς ἀπὸ τῶν κλιμάκων ὠθούμενος καὶ τῷ τείχει ἐπιφερόμενος ἐπιρρίψει τοῖς τειχοφύλαξιν ἀπολυθέντα τὸν μονάγκωνα, καὶ πολλὴν ἐργάσεται τῶν ἐφεστώτων ἅλωσιν. Καὶ τὸ σχῆμα καταγέγραπται. Αὕτη ἡ τοῦ μονάγκωνος κατασκευὴ καὶ πρὸς καταπαλτικὴν θεωρίαν τοὺς βουλομένους ἐφοδιάσει, πολλὰ συμβαλλομένη 255 πρὸς τὸ μακροβολεῖν εὐθυτόνοις τε καὶ παλιντόνοις ὀργάνοις, ἤτοι λιθοβόλοις τε καὶ ὀξυβελέσιν· ἡ γὰρ τῶν χοινικίδων παράθεσις ἐπιμηκέστερον διὰ τοῦ τόνου τὸ ὄργανον καὶ περὶ τὴν τάσιν σφοδρο τέραν τὴν ἐξαποστολὴν ἀπεργάζεται. Ὅσῳ γὰρ κατὰ μῆκος ὁ τόνος παραυξάνεται, ἐπὶ τοσούτῳ καὶ ἡ τοῦ βέλους ἐξαποστολὴ παρεκτείνεσθαι πέφυκε, τῆς τοξίτιδος νευρᾶς πάντα ἀπεργαζομένης καὶ τὴν τῆς ἐξαποστολῆς ὑπομενούσης βίαν, ὅθεν τὴν ταύτης πλοκὴν ἐκ τῶν εὐτονωτέρων καὶ ἐπίπλεον γυμναζομένων νεύρων τοῦ ζῴου λέγουσι γίνεσθαι· οἷον ἐλάφου μὲν τῶν ἐν τοῖς σκέλεσι καὶ ποσὶ, ταύρου δὲ τῶν ἐπὶ τοῦ αὐχένος· καὶ τὰς μὲν τῶν εὐθυτόνων νευρὰς στρογγύλας περὶ τὴν πλοκὴν εἶναι, διὰ τὸ πρὸς τὰς τῶν ὀϊστῶν ἐμπίπτειν χηλάς· τὰς δὲ τῶν παλιντόνων πλατείας, καθάπερ ζώνας, ἐξ ἄκρων ἀγκύλας ἐχούσας, πρὸς ἃς οἱ ἀγκῶνες ἐμβιβάζονται, ὅπως ὁ λίθος ὑπὸ τοῦ μέσου τῆς τοξίτιδος κατὰ πλάτος τυπτόμενος καλῶς ἐξαποστολῇ μὴ παρεκπίπτων πρὸς τὰ ἄκρα. Τὸν δὲ ἐν τοῖς ἀγκῶσι τόνον καὶ ἐκ τριχῶν φασι γυναικείων πλέκεσθαι· λεπταὶ γὰρ οὖσαι καὶ μακραὶ καὶ πολλῷ ἐλαίῳ τραφεῖσαι, ὅταν πλακῶσι, πολλὴν εὐτονίαν λαμβάνουσιν, ὥστε μὴ ἀπᾴδειν αὐτὰς τῆς διὰ τῶν νεύρων ἰσχύος. Ἀλλὰ περὶ μὲν τούτων νῦν ἅλις. Ἐν γὰρ τῇ βελοποιϊκῇ πραγματείᾳ λεπτομερῶς 256 τοῖς ζητητικωτέροις πρὸς κατασκευὴν οὐ μόνον ταῦτα, ἀλλὰ καὶ αἱ τῶν τρημάτων ἐν τοῖς ὀργάνοις διάμετροι, οἵ τε ἐν τριπλασίονι λόγῳ τῶν ἐν ταῖς βάσεσι διαμέτρων ὅμοιοι πρὸς ἀλλήλους δεικνύμενοι κύλινδροι, καὶ ὁ πολυθρύλητος τοῦ κύβου διπλασιασμός, δι' ὧν τὰ μεγέθη τῶν βελῶν πρὸς τὰ τῶν τυχόντων ὀργάνων διπλάσια ἢ καὶ τριπλάσια γινόμενα ἐπὶ τοῖς αὐτοῖς ἐξαποστέλλονται μήκεσι. Μαθηματικῶς ἐκεῖ καὶ ὀργανικῶς πάντα τῷ Ἀλεξανδρεῖ Ἥρωνι πρὸς ἀπόδειξιν διεξήτασται. Γίνεται καὶ ἑτέρα τῷ εἴδει κλῖμαξ ὑπότροχος, ἀπὸ ξύλων εὐτόνων τε καὶ ἐλαφρῶν καὶ αὐτὴ κατασκευαζομένη, πρὸς ἣν ἄλλη ὡς ἐν τάξει ἐπιβάθρας κατὰ μέσον ἄνωθεν ἐμβαλλομένη ζευγνύεται δι' ἀξονίου, κυλινδρικὰ καὶ λεῖα ἀπὸ τόρνου ἔχοντος τὰ πρὸς τοῖς μηροῖς συμβαλλόμενα ἄκρα, ὅπως εὐλύτως πρὸς τὴν ἑτέραν καταφερομένη ἐπικλείηται, καὶ πάλιν πρὸς αὐτὴν ἐπανάγηται πάρορθος ὑπὸ σχοινίων ὄπισθεν κρατουμένη, ἐπιδιδομένη τε καὶ ἐφελκομένη συμμέτρως διὰ μαγγάνων τῶν λεγο μένων πολυσπάστων, ἤτοι καρείων εὐτρόχων, δύναμιν βιαίαν ἐπὶ τῇ 257 ἀνέσει καὶ τάσει τῶν σχοινίων ἐμποιούντων διὰ τῆς τῶν παρακειμένων τροχίλων συμφυοῦς παραθέσεως. Κωλυμάτια δὲ ἄνωθεν ἐπὶ τῇ ἱσταμένῃ γινέσθωσαν ἢ παρεξοχαί τινες, ὄπισθεν ἀναστρεφομένην τὴν ἀνορθουμένην ὑποδεχόμεναι, μήπως ἀθρόως ἐπὶ τῇ κορυφῇ τῆς ἑτέρας ἀναχθεῖσα πτῶσιν ἐπὶ τοῖς ἐργαζομένοις μᾶλλον ἀπειλήσει. Καί, εἰ μὲν μονόξυλος εἴη ἡ κάτωθεν ἵστασθαι μέλλουσα κλῖμαξ, τὰς βαθμίδας ἐχέτω μίαν περὶ μίαν τῶν μηρῶν παρεξεχούσας καὶ τετρημένας ἐπὶ τῇ ἐξοχῇ, ὅπως διὰ τῶν τρημάτων σχοινίον εὔτονον ἐκτετανυσμένον διερχόμενον ἀδιάκλαστον αὐτὴν φυλάξῃ· σύνθετος δὲ