so that both the 60 cubits of the height and the 17 of the length of the base itself will have the same proportion to the feet, and likewise also the harmony in their ratios, because both are measured by a common measure. For 30 measures 90 three times, and 60 twice; and again 18 measures 24 three times, and 18 twice; and as 90 is to 60, so is 24 to 18; and as 24 is to 18, so also are the wheels to each other in number and in size, and the base to the base, as also three is to two. Thus it has been shown that the measuring numbers also have the same ratio to one another by proportion as the numbers being measured; so that Apollodorus and his school are found to be making the constructions of their mobile towers not only with symmetry, but also with harmony, in contrast to Diades and Charias and their school. And it is clear that the ancient engineers and most learned architects made the constructions of their engines scientifically and not irrationally. When the tower has thus been completed in its construction, if the place underlying its base should not be smooth and level but happens to be sloping, we shall make a substructure at the base of the tower with a similar framework to it, approaching the unevenness of the ground and broadening the underlying space below, so that in the engagement and turmoil of battle the tower may be kept unswerving in its position. And it will be protected against being burned by fire-bearing caltrops that are thrown and by kindled flames by being nailed over with planks, preferably of palm wood or those made from tough woods, except cedar, pine, and alder, because of their flammability, brittleness, and fragility. And let hides also be hung on the tower over the aforementioned surrounding galleries and walkways of the structure, not touching the planks, but standing a little way off from them, both on account of the fire-projectiles and so that the blows of the stone-throwers may become weak against them and feeble in yielding to the impact against what is pliant. And let the tower also be studded with nails from above, as was said before in the case of the tortoises, and let the space in between be filled up with rich and sticky clay. And in the parts of the tower exposed to fire-projectiles, instead of pipes for sending water, the intestines of oxen, prepared as if they were salted, are set up to pour water over. Under these intestines are placed wineskins full of water; and being squeezed and pressed, they send up the water. And if some hard-to-reach projection of the tower is burning, and there is not the device called a siphon, hollowed-out reeds, such as bird-catchers have, are joined together, and where it is needed, they send out the water; for wineskins, as in the case of the intestines, are placed underneath, carrying it out through the reeds to the place that is on fire. And no small benefit will be afforded to the tower also by cushions hung on the outside, filled with chaff soaked in vinegar, or nets of wet seaweed or of what is called sea-leek, as they are able not only to fight against the fires from the fire-projectiles but also to withstand the blows from the stone-throwers. And one must know that all the [effects] from the fire-bearing projectiles and kindled flames ... ...................... ... parts for the ram to be carried while supported. And from the third step of the upper ladder to the third and equally high step of the other, they are roofed over with planks or woven withes, with both being smeared with clay or covered with the hides of newly slaughtered oxen on account of both the fire-projectiles and the shots of stones. And lower than the third step, after 18 or even 20 feet, the intervening steps receive another roof, not across the whole width of the steps; for the space for the ascent will be left unroofed. And the pins inserted into the side-beams of the ladders must extend out further, so that the space of the ladders may be roofed more widely; for there the ram will operate most effectively, suspended from the roof above by two supports of nearly equal height, so that also those pushing the ram, standing on the roofed deck

ὥστε καὶ αἱ τοῦ ὕψους πήχεις ˉξ καὶ αὐτῆς βάσεως τοῦ μήκους ˉιˉζ τὴν αὐτὴν ἀναλογίαν πρὸς τοὺς πόδας ἕξουσιν, ὡσαύτως δὲ καὶ τὴν ἐν λόγοις συμφωνίαν, ὅτι κοινῷ μέτρῳ ἀμφότεροι μετροῦνται. Ὁ γὰρ ˉλ τρὶς μὲν μετρεῖ τὸν ˉ, δὶς δὲ τὸν ˉξ· καὶ πάλιν ὁ ˉη τρὶς μὲν τὸν ˉκˉδ μετρεῖ, δὶς δὲ τὸν ˉιˉ· καὶ ἔστιν ὡς ˉ πρὸς ˉξ, οὕτως ˉκˉδ πρὸς ˉιˉ· καὶ ὡς ˉκˉδ πρὸς ˉιˉ, οὕτως καὶ οἱ τροχοὶ πρὸς ἀλλήλους κατ' ἀριθμόν τε καὶ μέγεθος, καὶ ἡ βάσις πρὸς τὴν βάσιν, ὡς καὶ τὰ τρία πρὸς δύο. ἐδείχθησαν ἄρα καὶ οἱ με246 τροῦντες πρὸς ἀλλήλους τὸν αὐτὸν τοῖς μετρουμένοις κατ' ἀναλογίαν ἔχοντες λόγον· ὥστε οὐ μόνον πρὸς συμμετρίαν, ἀλλὰ καὶ συμφωνίαν τῶν φορητῶν πύργων κατασκευὰς οἱ περὶ Ἀπολλόδωρον πρὸς τοὺς περὶ ∆ιάδην καὶ Χαρίαν εὑρίσκονται ποιοῦντες. Καὶ φανερὸν ὅτι οἱ πάλαι μηχανικοὶ καὶ πολυμαθέστατοι ἀρχιτέκτονες ἐπιστημονικῶς καὶ οὐκ ἀλόγως τὰς τῶν μηχανημάτων κατασκευὰς ἐποίουν. Τοῦ πύργου οὕτως ἐπὶ τῇ κατασκευῇ τελεσθέντος, ἐὰν μὴ ὁμαλὸς καὶ ἰσοπέδιος πρὸς τὴν βάσιν ὑποκείμενος εἴη τόπος ἀλλ' ἀνωφερὴς τυγχάνῃ, ποιήσομεν ὑπόθημα πρὸς τῇ βάσει τοῦ πύργου τῇ ὁμοίᾳ αὐτοῦ συμπλοκῇ ἐπὶ τῷ ἀνωμάλῳ τῆς γῆς προσερχομένῃ καὶ πλατυνούσῃ τὸν ὑποκείμενον κάτωθεν τόπον, ὅπως ἐπὶ τῇ συμβολῇ καὶ τῷ κλόνῳ τῆς μάχης ἀκλινὴς πρὸς τὴν στάσιν ὁ πύργος συντηρῆται. ∆ιαφυλαχθήσεται δὲ πρὸς τὸ μὴ καίεσθαι ἐκ τῶν πεμπομένων πυροφόρων τριβόλων καὶ ἀναπτομένων φλογῶν προσηλούμενος σανίσι, μάλιστα μὲν φοινικίναις ἢ ταῖς ἐξ εὐτόνων ξύλων γινομέναις, πλὴν κεδρίνων πευκίνων τε καὶ κληθρίνων, διὰ τὸ ἔκπυρον αὐτῶν εὔκλαστόν τε καὶ εὔθραυστον. Ἐπικρεμάσθωσαν δὲ καὶ δέρματα τῷ πύργῳ ἐπὶ τοῖς ῥηθεῖσιν ἐπὶ τῇ κατασκευῇ περιπτέροις τε καὶ περιδρόμοις, μὴ προσεγγίζοντα 247 ταῖς σανίσιν, ἀλλὰ μικρὸν ἔξωθεν ἀπέχοντα διά τε τὰ πυροβόλα καὶ ὅπως ἀσθενεῖς πρὸς αὐτὰ καὶ ἔκλυτοι ἐπὶ τῇ τῆς βολῆς ἐνδόσει αἱ τῶν λιθοβόλων πρὸς τὸ χαῦνον γίνωνται πληγαί. Προσηλούσθω δὲ καὶ ἥλοις ὁ πύργος ἄνωθεν, ὡς προείρηται καὶ ἐπὶ τῶν χελωνῶν, καὶ πηλοῦ λιπαροῦ καὶ κολλώδους ἀναγεμισθήτω ὁ διὰ μέσου τόπος. Εἰς δὲ τὰ προκείμενα τοῖς πυροβόλοις μέρη τοῦ πύργου ἀντὶ σωλήνων τῶν τὸ ὕδωρ πεμπόντων ἔντερα βοῶν εἰργασμένα ὡσὰν τεταριχευμένα παρατίθενται ὕδωρ ἐπιχέοντα. Τούτοις τοῖς ἐντέροις ἀσκοὶ πλήρεις ὕδατος ὑποτίθενται· ἐκθλιβόμενοι δὲ καὶ πιεζόμενοι ἀναφέρουσι τὸ ὕδωρ. Εἰ δὲ καὶ ἀκρωτήριόν που τοῦ πύργου δυσδιάβατον καίεται, μὴ ἔστι δὲ ὄργανον ὃ καλεῖται σίφων, καλαμοὶ τετρυπημένοι δι' ὅλου ἔσωθεν οἵους οἱ ἰξευταὶ ἔχουσι πρὸς ἀλλήλους συμβάλλονται, καὶ ὅπου δέῃ ἐκπέμπουσι τὸ ὕδωρ· ἀσκοὶ γὰρ ὡς καὶ ἐπὶ τῶν ἐντέρων ὑπόκεινται ἐκφέροντες αὐτὸ διὰ τῶν καλάμων ἐπὶ τὸν ἐμπυριζόμενον τόπον. Οὐ μικρὰν δὲ ὠφέλειαν τῷ πύργῳ ἐμπαρέξουσι καὶ τύλια ἔξωθεν κρεμαμένα ἀχύροις ὄξει βεβρεγμένοις γεμισθέντα, ἢ δίκτυα ἐνύγρων βρύων ἢ τοῦ καλουμένου θαλασσοπράσου, ὡς δυνάμενα 248 μὴ μόνον ταῖς ἐκ τῶν πυροβόλων ἀντιμάχεσθαι ἐμπρήσεσιν ἀλλὰ καὶ πρὸς τὰς τῶν λιθοβόλων ἀντέχεσθαι πληγάς. Καὶ δεῖ εἰδέναι ὅτι πᾶσαι αἱ ἐκ τῶν πυροφόρων καὶ ἀναπτομένων φλογῶν ... ...................... ... μέρη τὸν κριὸν βασταζόμενον φέρεσθαι. Ἀπὸ δὲ τοῦ τρίτου βαθμοῦ τῆς ἄνωθεν κλίμακος ἐπὶ τὸν τῆς ἑτέρας τρίτον καὶ ἰσοϋψῆ διὰ σανίδων στεγάζονται ἢ βεργῶν πεπλεγμένων, ἀμφοτέρων δὲ πηλῷ ἐπικεχρισμένων ἢ βύρσαις βοῶν νεοσφαγῶν σκεπομένων διά τε τὰ πυροβόλα καὶ τὰς τῶν λίθων βολάς. Κατωτέρω δὲ τοῦ 249 τρίτου βαθμοῦ μετὰ πόδας ˉιˉη ἢ καὶ ˉκ οἱ τυχόντες βαθμοὶ ἄλλην στέγην λαμβάνουσιν, οὐ πρὸς ὅλον τὸ πλάτος τῶν βαθμῶν· ἀστέγαστος γὰρ ὁ πρὸς τὴν ἀνάβασιν παραλειφθήσεται τόπος. Χρὴ δὲ τὰς ἐμβαλλομένας περόνας ἐπὶ τοὺς τῶν κλιμάκων μηροὺς πλέον παρεξέρχεσθαι, ἵνα πλατύτερον τῶν κλιμάκων τὸ χωρίον στεγάζηται· ἐκεῖ γὰρ ὁ κριὸς ἐνεργήσει μάλιστα ἀπὸ τῆς ἄνωθεν ἠρτημένος στέγης δυσὶ βαστάγμασιν ἰσοϋψέσι παρὰ μικρόν τι, ἵνα καὶ οἱ τὸν κριὸν ὠθοῦντες ἐπὶ τοῦ καταστρώματος τοῦ ἐστεγασμένου ἑστῶτες