§9.26a.1οἱ δὲ πλεῖστοι τῶν ἀνθρώπων ἐξ αὐτῆς τῆς περιμέτρου τεκμαίρονται τὰ μεγέθη τῶν προειρημένων. §9.26a.2λοιπὸν ὅταν εἴπῃ τις τὴν μὲν τῶν Μεγαλοπολιτῶν πόλιν πεντήκοντα σταδίων ἔχειν τὸν περίβολον, τὴν δὲ τῶν Λακεδαιμονίων ὀκτὼ καὶ τετταράκοντα, τῷ δὲ μεγέθει διπλῆν εἶναι τὴν Λακεδαίμονα τῆς Μεγάλης πόλεως, ἄπιστον αὐτοῖς εἶναι δοκεῖ τὸ λεγόμενον. §9.26a.3ἂν δὲ καὶ συναυξῆσαί τις βουλόμενος τὴν ἀπορίαν εἴπῃ διότι δυνατόν ἐστι τετταράκοντα σταδίων πόλιν ἢ στρατοπεδείαν ἔχουσαν τὴν περιγραφὴν διπλασίαν γίνεσθαι τῆς ἑκατὸν σταδίων ἐχούσης τὴν περίμετρον, τελέως ἐκπληκτικὸν αὐτοῖς φαίνεται τὸ λεγόμενον. §9.26a.4τοῦτο δʼ ἐστὶν αἴτιον ὅτι τῶν ἐν τοῖς παιδικοῖς μαθήμασι παραδιδομένων ἡμῖν διὰ τῆς γεωμετρίας οὐ μνημονεύομεν. §9.26a.5περὶ μὲν οὖν τούτων προήχθην εἰπεῖν διὰ τὸ μὴ μόνον τοὺς πολλοὺς ἀλλὰ καὶ τῶν πολιτευομένων καὶ τῶν ἐν ταῖς ἡγεμονίαις ἀναστρεφομένων τινὰς ἐκπλήττεσθαι, θαυμάζοντας ποτὲ μὲν εἰ δυνατόν ἐστι μείζω τὴν τῶν Λακεδαιμονίων πόλιν εἶναι, καὶ πολλῷ μείζω, τῆς τῶν Μεγαλοπολιτῶν, τὸν περίβολον ἔχουσαν ἐλάττω, §9.26a.6ποτὲ δὲ τὸ πλῆθος τῶν ἀνδρῶν τεκμαίρεσθαι, στοχαζομένους ἐξ αὐτῆς τῆς περιμέτρου τῶν στρατοπεδειῶν. §9.26a.7παραπλήσιον δέ τι καὶ ἕτερον ἀδίκημα συμβαίνει περὶ τὰς τῶν πόλεων ἐπιφάσεις.
Most people, however, estimate the size of the aforementioned things from their perimeter alone. Consequently, when someone says that the city of Megalopolis has a circuit of fifty stades, and that of Lacedaemon forty-eight, but that Lacedaemon is twice the size of Megalopolis, what is said seems incredible to them. And if someone, wishing to increase their perplexity, further says that it is possible for a city or camp with a boundary of forty stades to be twice as large as one with a perimeter of a hundred stades, what is said appears completely astonishing to them. The reason for this is that we do not remember the lessons delivered to us in our childhood studies through geometry. I was led to speak about these matters because not only the common people, but also some of those engaged in politics and active in commands are astonished, wondering at one time whether it is possible for the city of Lacedaemon to be larger, and much larger, than that of Megalopolis, while having a smaller circuit, and at another time estimating the number of men by guessing from the perimeter of the camps alone. A similar error also occurs regarding the surfaces of cities.
οἱ γὰρ πολλοὶ τῶν ἀνθρώπων τὰς περικεκλασμένας καὶ βουνώδεις πλείους οἰκίας ὑπολαμβάνουσι κατέχεσθαι τῶν ἐπιπέδων. §9.26a.8τὸ δʼ οὐκ ἔστι τοιοῦτον διὰ τὸ τὰ τειχία τῶν οἰκοδομιῶν μὴ τοῖς ἐγκλίμασι τῶν ἐδάφων ἀλλὰ τοῖς ὑποκειμένοις ἐπιπέδοις οἰκοδομεῖσθαι πρὸς ὀρθάς, ἐφʼ ὧν καὶ τοὺς λόφους αὐτοὺς βεβηκέναι συμβαίνει. §9.26a.9γνοίη δʼ ἄν τις ἐκ τοῦ φαινομένου παιδικῶς ὅμως τὸ λεγόμενον. §9.26a.10εἰ γὰρ νοήσαι τις εἰς ὕψος ἀνατεταμένας τὰς ἐν τοῖς κλίμασιν οἰκίας οὕτως ὥστε πάσας ἰσοϋψεῖς ὑπάρχειν, φανερὸν ὡς ἑνὸς ἐπιπέδου γενομένου τοῦ κατὰ τὰ τέγη τῶν οἰκιῶν, ἴσον καὶ παράλληλον ἔσται τοῦτο τῷ τοῖς λόφοις ὑποκειμένῳ καὶ τοῖς τῶν τειχῶν θεμελίοις ἐπιπέδῳ. §9.26a.11περὶ μὲν οὖν τῶν ἡγεῖσθαι καὶ πολιτεύεσθαι βουλομένων, ἀγνοούντων δὲ τὰ τοιαῦτα
For the majority of people suppose that rugged and hilly areas contain more houses than flat ones. But this is not the case, because the walls of buildings are built at right angles not to the slope of the ground, but to the underlying flat plane, upon which the hills themselves also happen to stand. One could understand what is said, however, even from a simple, visual example. For if one were to imagine the houses on the slopes extended upward in height so that they were all of equal height, it is clear that when the plane at the roofs of the houses becomes one, this will be equal and parallel to the flat plane underlying the hills and the foundations of the walls. Now, concerning those who wish to lead and engage in politics, but are ignorant of such matters...