Ἀλλʼ ἐπὶ τὸν Ἵππαρχον πρότερον ἐπανιόντες τὰ ἑξῆς ἴδωμεν. πάλιν γὰρ πλάσας ἑαυτῷ λήμματα γεωμετρικῶς ἀνασκευάζει τὰ ὑπʼ ἐκείνου τυπωδῶς λεγόμενα. φησὶ γὰρ αὐτὸν λέγειν τὸ ἐκ Βαβυλῶνος εἰς μὲν Κασπίους πύλας διάστημα σταδίων ἑξακισχιλίων ἑπτακοσίων, εἰς δὲ τοὺς ὅρους τῆς Καρμανίας καὶ Περσίδος πλειόνων ἢ ἐνακισχιλίων, ὅπερ ἐπὶ γραμμῆς κεῖται πρὸς ἰσημερινὰς ἀνατολὰς εὐθείας ἀγομένης· γίνεσθαι δὲ ταύτην κάθετον ἐπὶ τὴν κοινὴν πλευρὰν τῆς τε δευτέρας καὶ τῆς τρίτης σφραγῖδος, ὥστε κατʼ αὐτὸν συνίστασθαι τρίγωνον ὀρθογώνιον ὀρθὴν ἔχον τὴν πρὸς τοῖς ὅροις τῆς Καρμανίας, καὶ τὴν ὑποτείνουσαν εἶναι ἐλάττω μιᾶς τῶν περὶ τὴν ὀρθὴν ἐχουσῶν· δεῖν οὖν τὴν Περσίδα τῆς δευτέρας ποιεῖν σφραγῖδος. πρὸς ταῦτα δʼ εἴρηται ὅτι οὔθʼ ἡ ἐκ Βαβυλῶνος εἰς τὴν Καρμανίαν ἐπὶ παραλλήλου λαμβάνεται, οὔθʼ ἡ διορίζουσα εὐθεῖα τὰς σφραγῖδας μεσημβρινὴ εἴρηται· ὥστʼ οὐδὲν εἴρηται πρὸς αὐτόν. οὐδὲ τὸ ἐπιφερόμενον · εἰρηκότος γὰρ ἀπὸ Κασπίων πυλῶν εἰς μὲν Βαβυλῶνα τοὺς λεχθέντας, εἰς δὲ Σοῦσα σταδίους εἶναι τετρακισχιλίους ἐνακοσίους, ἀπὸ δὲ Βαβυλῶνος τρισχιλίους τετρακοσίους, πάλιν ἀπὸ τῶν αὐτῶν ὁρμηθεὶς ὑποθέσεων ἀμβλυγώνιον τρίγωνον συνίστασθαί φησι πρός τε ταῖς Κασπίοις πύλαις καὶ Σούσοις καὶ Βαβυλῶνι, τὴν ἀμβλεῖαν γωνίαν ἔχον πρὸς Σούσοις, τὰ δὲ τῶν πλευρῶν μήκη τὰ ἐκκείμενα· εἶτʼ ἐπιλογίζεται, διότι συμβήσεται κατὰ τὰς ὑποθέσεις ταύτας τὴν διὰ Κασπίων πυλῶν μεσημβρινὴν γραμμὴν ἐπὶ τοῦ διὰ Βαβυλῶνος καὶ Σούσων παραλλήλου δυσμικωτέραν ἔχειν τὴν κοινὴν τομὴν τῆς κοινῆς τομῆς τοῦ αὐτοῦ παραλλήλου καὶ τῆς ἀπὸ Κασπίων πυλῶν καθηκούσης εὐθείας ἐπὶ τοὺς ὅρους τοὺς τῆς Καρμανίας καὶ τῆς Περσίδος πλείοσι τῶν τετρακισχιλίων καὶ τετρακοσίων· σχεδὸν δή τι πρὸς τὴν διὰ Κασπίων πυλῶν μεσημβρινὴν γραμμὴν ἡμίσειαν ὀρθῆς ποιεῖν γωνίαν τὴν διὰ Κασπίων πυλῶν καὶ τῶν ὅρων τῆς τε Καρμανίας καὶ τῆς Περσίδος, καὶ νεύειν αὐτὴν ἐπὶ τὰ μέσα τῆς τε μεσημβρίας καὶ τῆς ἰσημερινῆς ἀνατολῆς· ταύτῃ δʼ εἶναι παράλληλον τὸν Ἰνδὸν ποταμόν, ὥστε καὶ τοῦτον ἀπὸ τῶν ὀρῶν οὐκ ἐπὶ μεσημβρίαν ῥεῖν, ὥς φησιν Ἐρατοσθένης, ἀλλὰ μεταξὺ ταύτης καὶ τῆς ἰσημερινῆς ἀνατολῆς, καθάπερ ἐν τοῖς ἀρχαίοις πίναξι καταγέγραπται. τίς οὖν συγχωρήσει τὸ νῦν συσταθὲν τρίγωνον ἀμβλυγώνιον εἶναι, μὴ συγχωρῶν ὀρθογώνιον εἶναι τὸ περιέχον αὐτό; τίς δʼ ἐπὶ παραλλήλου κειμένην τὴν ἀπὸ Βαβυλῶνος εἰς Σοῦσα μίαν τῶν τὴν ἀμβλεῖαν περιεχουσῶν, τὴν ὅλην μὴ συγχωρῶν τὴν μέχρι Καρμανίας; τίς δὲ τῷ Ἰνδῷ παράλληλον τὴν ἀπὸ Κασπίων πυλῶν ἐπὶ τοὺς ὅρους τῆς Καρμανίας; ὧν χωρὶς κενὸς ἂν εἴη ὁ συλλογισμός. χωρὶς δὲ τούτων κἀκεῖνος εἴρηκεν, ὅτι ῥομβοειδές ἐστι τὸ σχῆμα τῆς Ἰνδικῆς· καὶ καθάπερ ἡ ἑωθινὴ πλευρὰ παρέσπασται πολὺ πρὸς ἕω, καὶ μάλιστα τῷ ἐσχάτῳ ἀκρωτηρίῳ, ὃ καὶ πρὸς μεσημβρίαν προπίπτει πλέον παρὰ τὴν ἄλλην ᾐόνα, οὕτω καὶ ἡ παρὰ τὸν Ἰνδὸν πλευρά. πάντα δὲ ταῦτα λέγει γεωμετρικῶς ἐλέγχων, οὐ πιθανῶς.
We will now return at once to Hipparchus, and see what comes next. Continuing to palm assumptions of his own [upon Eratosthenes], he goes on to refute, with geometrical accuracy, statements which that author had made in a mere general way. Eratosthenes, he says, estimates that there are 6700 stadia between Babylon and the Caspian Gates, and from Babylon to the frontiers of Carmania and Persia above 9000 stadia; this he supposes to lie in a direct line towards the equinoctial rising, and perpendicular to the common side of his second and third sections. Thus, according to his plan, we should have a right-angled triangle, with the right angle next to the frontiers of Carmania, and its hypotenuse less than one of the sides about the right angle! Consequently Persia should be included in the second section.
To this we reply, that the line drawn from Babylon to Carmania was never intended as a parallel, nor yet that which divides the two sections as a meridian, and that therefore nothing has been laid to his charge, at all events with any just foundation. In fact, Eratosthenes having stated the number of stadia from the Caspian Gates to Babylon as above given, [from the Caspian Gates] to Susa 4900 stadia, and from Babylon [to Susa] 3400 stadia, Hipparchus runs away from his former hypothesis, and says that [by drawing lines from] the Caspian Gates, Susa, and Babylon, an obtuse-angled triangle would be the result, whose sides should be of the length laid down, and of which Susa would form the obtuse angle. He then argues, that according to these premises, the meridian drawn from the Gates of the Caspian will intersect the parallel of Babylon and Susa 4400 stadia more to the west, than would a straight line drawn from the Caspian to the confines of Carmania and Persia; and that this last line, forming with the meridian of the Caspian Gates half a right angle, would lie exactly in a direction midway between the south and the equinoctial rising. Now as the course of the Indus is parallel to this line, it cannot flow south on its descent from the mountains, as Eratosthenes asserts, but in a direction lying between the south and the equinoctial rising, as laid down in the ancient charts. But who is there who will admit this to be an obtuse-angled triangle, without also admitting that it contains a right angle? Who will agree that the line from Babylon to Susa, which forms one side of this obtuse-angled triangle, lies parallel, without admitting the same of the whole line as far as Carmania? or that the line drawn from the Caspian Gates to the frontiers of Carmania is parallel to the Indus? Nevertheless, without this the reasoning [of Hipparchus] is worth nothing
Eratosthenes himself also states, [continues Hipparchus,] that the form of India is rhomboidal; and since the whole eastern border of that country has a decided tendency towards the east, but more particularly the extremest cape, which lies more to the south than any other part of the coast, the side next the Indus must be the same.
ταῦτα δὲ καὶ αὐτὸς ἑαυτῷ ἐπενέγκας ἀπολύεται φήσας, εἰ μὲν παρὰ μικρὰ διαστήματα ὑπῆρχεν ὁ ἔλεγχος, συγγνῶναι ἂν ἦν· ἐπειδὴ δὲ παρὰ χιλιάδας σταδίων φαίνεται διαπίπτων, οὐκ εἶναι συγγνωστά· καίτοι ἐκεῖνόν γε καὶ παρὰ τετρακοσίους σταδίους αἰσθητὰ ἀποφαίνεσθαι τὰ παραλλάγματα, ὡς ἐπὶ τοῦ διʼ Ἀθηνῶν παραλλήλου καὶ τοῦ διὰ Ῥόδου. ἔστι δὲ τὸ πρὸς αἴσθησιν οὐχ ἁπλοῦν, ἀλλὰ τὸ μὲν ἐν πλάτει μείζονι τὸ δʼ ἐν ἐλάττονι· μείζονι μέν, ἂν αὐτῷ τῷ ὀφθαλμῷ πιστεύωμεν ἢ καρποῖς ἢ κράσεσιν ἀέρων πρὸς τὴν τῶν κλιμάτων κρίσιν, ἐλάττονι δʼ, ἂν διʼ ὀργάνων γνωμονικῶν ἢ διοπτρικῶν. ὁ μὲν οὖν διʼ Ἀθηνῶν παράλληλος γνωμονικῶς ληφθεὶς καὶ ὁ διὰ Ῥόδου καὶ Καρίας, εἰκότως ἐν σταδίοις τοσούτοις αἰσθητὴν ἐποίησε τὴν διαφοράν. ὁ δʼ ἐν πλάτει μὲν τρισχιλίων σταδίων, μήκει δὲ καὶ τετρακισμυρίων ὄρους, πελάγους δὲ τρισμυρίων λαμβάνων τὴν ἀπὸ δύσεως ἐπʼ ἰσημερινὰς ἀνατολὰς γραμμήν, καὶ τὰ ἐφʼ ἑκάτερον τὸ μέρος τὰ μὲν νότια ὀνομάζων τὰ δὲ βόρεια, καὶ ταῦτα πλινθία καλῶν καὶ σφραγῖδας, νοείσθω πῶς καὶ ταῦτα λέγει καὶ πλευρὰ τὰ μὲν ἀρκτικὰ τὰ δὲ νότια, καὶ πῶς τὰ μὲν ἑσπέρια τὰ δὲ ἑωθινά· καὶ τὸ μὲν παρὰ πολὺ διαμαρτανόμενον παρορῶν ὑπεχέτω λόγον (δίκαιον γάρ), τὸ δὲ παρὰ μικρὸν οὐδὲ παριδὼν ἐλεγκτέος ἐστίν. ἐνταῦθα δʼ οὐδετέρως αὐτῷ προσάγεταί τις ἔλεγχος· οὔτε γὰρ τῶν ἐν τοσούτῳ πλάτει γεωμετρική τις δύναιτʼ ἂν ἀπόδειξις, οὔτʼ ἐν οἷς ἐπιχειρεῖ γεωμετρεῖν ὁμολογουμένοις χρῆται λήμμασιν, ἀλλʼ ἑαυτῷ πλάσας.
These arguments may be very geometrical, but they are not convincing. After having himself invented these various difficulties, he dismisses them, saying, Had [Eratosthenes] been chargeable for small distances only, he might have been excused; but since his mistakes involve thousands of stadia, we cannot pardon him, more especially since he has laid it down that at a mere distance of 400 stadia, such as that between the parallels of Athens and Rhodes, there is a sensible variation [of latitude]. But these sensible variations are not all of the same kind, the distance [involved therein] being in some instances greater, in others less; greater, when for our estimate of the climata we trust merely to the eye, or are guided by the vegetable productions and the temperature of the air; less, when we employ gnomons and dioptric instruments. Nothing is more likely than that if you measure the parallel of Athens, or that of Rhodes and Caria, by means of a gnomon, the difference resulting from so many stadia will be sensible. But when a geographer, in order to trace a line from west to east, 3000 stadia broad, makes use of a chain of mountains 40,000 stadia long, and also of a sea which extends still farther 30,000 stadia, and farther wishing to point out the situation of the different parts of the habitable earth relative to this line, calls some southern, others northern, and finally lays out what he calls the sections, each section consisting of divers countries, then we ought carefully to examine in what acceptation he uses his terms; in what sense he says that such a side [of any section] is the north side, and what other is the south, or east, or west side. If he does not take pains to avoid great errors, he deserves to be blamed, but should he be guilty merely of trifling inaccuracies, he should be forgiven. But here nothing shows thoroughly that Eratosthenes has committed either serious or slight errors, for on one hand what he may have said concerning such great distances, can never be verified by a geometrical test, and on the other, his accuser, while endeavouring to reason like a geometrician, does not found his arguments on any real data, but on gratuitous suppositions.
βέλτιον δὲ περὶ τῆς τετάρτης λέγει μερίδος, προστίθησι δὲ καὶ τοῦ φιλαιτίου καὶ τοῦ μένοντος ἐπὶ τῶν αὐτῶν ὑποθέσεων ἢ τῶν παραπλησίων. τοῦτο μὲν γὰρ ὀρθῶς ἐπιτιμᾷ διότι μῆκος ὀνομάζει τῆς μερίδος ταύτης τὴν ἀπὸ Θαψάκου μέχρι Αἰγύπτου γραμμήν, ὥσπερ εἴ τις παραλληλογράμμου τὴν διάμετρον μῆκος αὐτοῦ φαίη· οὐ γὰρ ἐπὶ τοῦ αὐτοῦ παραλλήλου κεῖται ἥ τε Θάψακος καὶ ἡ τῆς Αἰγύπτου παραλία, ἀλλʼ ἐπὶ διεστώτων πολὺ ἀλλήλων, ἐν δὲ τῷ μεταξὺ διαγώνιός πως ἄγεται καὶ λοξὴ ἡ ἀπὸ Θαψάκου εἰς Αἴγυπτον. τὸ δὲ θαυμάζειν, πῶς ἐθάρρησεν εἰπεῖν ἑξακισχιλίων σταδίων τὸ ἀπὸ Πηλουσίου εἰς Θάψακον, πλειόνων ὄντων ἢ ὀκτακισχιλίων, οὐκ ὀρθῶς. λαβὼν γὰρ διʼ ἀποδείξεως μέν, ὅτι ὁ διὰ Πηλουσίου παράλληλος τοῦ διὰ Βαβυλῶνος πλείοσιν ἢ δισχιλίοις καὶ πεντακοσίοις σταδίοις νοτιώτερός ἐστι, κατʼ Ἐρατοσθένη δὲ (ὡς οἴεται), διότι τοῦ διὰ Βαβυλῶνος ὁ διὰ τῆς Θαψάκου ἀρκτικώτερος τετρακισχιλίοις ὀκτακοσίοις, συμπίπτειν φησὶ πλείους τῶν ὀκτακισχιλίων. πῶς οὖν κατʼ Ἐρατοσθένη δείκνυται ἡ τοσαύτη ἀπόστασις τοῦ διὰ Βαβυλῶνος παραλλήλου ἀπὸ τοῦ διὰ Θαψάκου, ζητῶ. ὅτι μὲν γὰρ ἀπὸ Θαψάκου ἐπὶ Βαβυλῶνα τοσοῦτόν ἐστιν, εἴρηκεν ἐκεῖνος· ὅτι δὲ καὶ ἀπὸ τοῦ διʼ ἑκατέρου παραλλήλου ἐπὶ τὸν διὰ θατέρου, οὐκ εἴρηκεν· οὐδὲ γάρ, ὅτι ἐπὶ ταὐτοῦ μεσημβρινοῦ ἐστιν ἡ Θάψακος καὶ ἡ Βαβυλών. τἀναντία γὰρ αὐτὸς ὁ Ἵππαρχος ἔδειξε κατʼ Ἐρατοσθένη πλείοσιν ἢ χιλίοις σταδίοις συμβαίνειν ἀνατολικωτέραν εἶναι τὴν Βαβυλῶνα τῆς Θαψάκου. ἡμεῖς τε παρετίθεμεν τὰς Ἐρατοσθένους ἀποφάσεις, ἐν αἷς τὸν Τίγριν καὶ τὸν Εὐφράτην ἐγκυκλοῦσθαι τήν τε Μεσοποταμίαν καὶ τὴν Βαβυλωνίαν, καὶ τὸ πλέον γε τῆς ἐγκυκλώσεως τὸν Εὐφράτην ποιεῖν· ἀπὸ γὰρ τῶν ἄρκτων ἐπὶ μεσημβρίαν ῥυέντα ἐπιστρέφειν πρὸς τὰς ἀνατολάς, ἐκπίπτειν δὲ ἐπὶ μεσημβρίαν. ἡ μὲν οὖν ἐπὶ μεσημβρίαν ἀπὸ τῶν ἄρκτων ὁδὸς ὡς ἂν μεσημβρινοῦ τινός ἐστιν, ἡ δʼ ἐπὶ τὰς ἀνατολὰς ἐπιστροφὴ καὶ ἐπὶ τὴν Βαβυλῶνα ἔκνευσίς τέ ἐστιν ἀπὸ τοῦ μεσημβρινοῦ καὶ οὐκ ἐπʼ εὐθείας διὰ τὴν ῥηθεῖσαν ἐγκύκλωσιν. τὴν δέ γε ὁδὸν εἴρηκε τετρακισχιλίων καὶ ὀκτακοσίων σταδίων τὴν ἐπὶ Βαβυλῶνα ἀπὸ Θαψάκου παρὰ τὸν Εὐφράτην προσθείς, καθάπερ ἐπίτηδες, τοῦ μή τινα εὐθεῖαν αὐτὴν δέξασθαι καὶ μέτρον τοῦ μεταξὺ δυεῖν παραλλήλων διαστήματος. μὴ διδομένου δὲ τούτου, κενόν ἐστι καὶ τὸ ἐφεξῆς δείκνυσθαι δοκοῦν, ὅτι συνισταμένου ὀρθογωνίου τριγώνου πρός τε Πηλουσίῳ καὶ Θαψάκῳ καὶ τῇ τομῇ τοῦ τε διὰ Πηλουσίου παραλλήλου καὶ τοῦ διὰ Θαψάκου μεσημβρινοῦ, μία τῶν περὶ τὴν ὀρθήν, ἡ ἐπὶ τοῦ μεσημβρινοῦ, μείζων ἔσται τῆς ὑπὸ τὴν ὀρθήν, τῆς ἀπὸ Θαψάκου εἰς Πηλούσιον. κενὸν δὲ καὶ τὸ συνάπτον τούτῳ, ἀπὸ μὴ συγχωρουμένου λήμματος κατασκευαζόμενον. οὐ γὰρ δὴ δίδοται τὸ ἀπὸ Βαβυλῶνος ἐπὶ τὸν διὰ Κασπίων πυλῶν μεσημβρινὸν εἶναι διάστημα τετρακισχιλίων ὀκτακοσίων. ἐλήλεγκται γὰρ ὑφʼ ἡμῶν ἐκ τῶν μὴ συγχωρουμένων ὑπʼ Ἐρατοσθένους κατεσκευακότα τοῦτο τὸν Ἵππαρχον· ἵνα δʼ ἀνίσχυρον ᾖ τὸ ὑπὸ ἐκείνου διδόμενον, λαβὼν τὸ εἶναι πλείους ἢ ἐννακισχιλίους ἐκ Βαβυλῶνος ἐπὶ τὴν ἐκ Κασπίων πυλῶν οὕτως ἀγομένην γραμμήν, ὡς ἐκεῖνος εἴρηκεν, ἐπὶ τοὺς ὅρους τῆς Καρμανίας, ἐδείκνυε τὸ αὐτό.
The fourth section Hipparchus certainly manages better, though he still maintains the same censorious tone, and obstinacy in sticking to his first hypotheses, or others similar. He properly objects to Eratosthenes giving as the length of this section a line drawn from Thapsacus to Egypt, as being similar to the case of a man who should tell us that the diagonal of a parallelogram was its length. For Thapsacus and the coasts of Egypt are by no means under the same parallel of latitude, but under parallels considerably distant from each other, and a line drawn from Thapsacus to Egypt would lie in a kind of diagonal or oblique direction between them. But he is wrong when he expresses his surprise that Eratosthenes should dare to state the distance between Pelusium and Thapsacus at 6000 stadia, when he says there are above 8000. In proof of this he advances that the parallel of Pelusium is south of that of Babylon by more than 2500 stadia, and that according to Eratosthenes (as he supposes) the latitude of Thapsacus is above 4800 stadia north of that of Babylon; from which Hipparchus tells us it results that [between Thapsacus and Pelusium] there are more than 8000 stadia. But I would inquire how he can prove that Eratosthenes supposed so great a distance between the parallels of Babylon and Thapsacus? He says, indeed, that such is the distance from Thapsacus to Babylon, but not that there is this distance between their parallels, nor yet that Thapsacus and Babylon are under the same meridian. So much the contrary, that Hipparchus has himself pointed out, that, according to Eratosthenes, Babylon ought to be east of Thapsacus more than 2000 stadia. We have before cited the statement of Eratosthenes, that Mesopotamia and Babylon are encircled by the Tigris and Euphrates, and that the greater portion of the Circle is formed by this latter river, which flowing north and south takes a turn to the east, and then, returning to a southerly direction, discharges itself [into the sea]. So long as it flows from north to south, it may be said to follow a southerly direction; but the turning towards the east and Babylon is a decided deviation from the southerly direction, and it never recovers a straight course, but forms the circuit we have mentioned above. When he tells us that the journey from Babylon to Thapsacus is 4800 stadia, he adds, following the course of the Euphrates, as if on purpose lest any one should understand such to be the distance in a direct line, or between the two parallels. If this be not granted, it is altogether a vain attempt to show that if a right-angled triangle were constructed by lines drawn from Pelusium and Thapsacus to the point where the parallel of Thapsacus intercepts the meridian of Pelusium, that one of the lines which form the right angle, and is in the direction of the meridian, would be longer than that forming the hypotenuse drawn from Thapsacus to Pelusium. Worthless, too, is the argument in connexion with this, being the inference from a proposition not admitted; for Eratosthenes never asserts that from Babylon to the meridian of the Caspian Gates is a distance of 4800 stadia. We have shown that Hipparchus deduces this from data not admitted by Eratosthenes; but desirous to controvert every thing advanced by that writer, he assumes that from Babylon to the line drawn from the Caspian Gates to the mountains of Carmania, according to Eratosthenes’ description, there are above 9000 stadia, and from thence draws his conclusions.
οὐ τοῦτο οὖν λεκτέον πρὸς τὸν Ἐρατοσθένη, ἀλλʼ ὅτι τῶν ἐν πλάτει λεγομένων καὶ μεγεθῶν καὶ σχημάτων εἶναί τι δεῖ μέτρον καὶ ὅπου μὲν μᾶλλον ὅπου δὲ ἔλαττον, συγχωρητέον. ληφθέντος γὰρ τοῦ τῶν ὀρῶν πλάτους τῶν ἐπὶ τὰς ἰσημερινὰς ἀνατολὰς ἐκτεινομένων τρισχιλίων σταδίων, ὁμοίως δὲ καὶ τοῦ τῆς θαλάττης τῆς μέχρι στηλῶν, μᾶλλον ἄν τις συγχωρήσειεν ὡς ἐπὶ μιᾶς γραμμῆς ἐξετάζεσθαι τὰς παραλλήλους ἐκείνης ἐν τῷ αὐτῷ πλάτει ἀγομένας ἢ τὰς συμπιπτούσας, καὶ τῶν συμπιπτουσῶν τὰς ἐν αὐτῷ ἐκείνῳ τῷ πλάτει τὴν σύμπτωσιν ἐχούσας ἢ τὰς ἐκτός· ὡσαύτως καὶ τὰς διισταμένας μέχρι τοῦ μὴ ἐκβαίνειν τοῦ πλάτους ἢ τὰς ἐκβαινούσας, καὶ τὰς ἐν μείζονι μήκει μᾶλλον ἢ τὰς ἐν ἐλάττονι. καὶ γὰρ ἡ ἀνισότης τῶν μηκῶν συγκρύπτοιτʼ ἂν μᾶλλον καὶ ἡ ἀνομοιότης τῶν σχημάτων· οἷον ἐν τῷ πλάτει τοῦ Ταύρου παντὸς καὶ τῆς μέχρι στηλῶν θαλάττης, ὑποκειμένων τρισχιλίων σταδίων, νοεῖται ἕν τι παραλληλόγραμμον χωρίον, τὸ περιγράφον τό τε ὄρος ἅπαν καὶ τὴν λεχθεῖσαν θάλατταν. ἐὰν οὖν διέλῃς εἰς πλείω παραλληλόγραμμα τὸ μῆκος, καὶ τὴν διάμετρον ὅλου τε τούτου λάβῃς καὶ τῶν μερῶν, ῥᾷον ἂν ἡ τοῦ ὅλου διάμετρος ἡ αὐτὴ λογισθείη τῇ κατὰ τὸ μῆκος πλευρᾷ ἤπερ ἡ ἐν τοῖς μέρεσι· καὶ ὅσῳ γʼ ἂν ἔλαττον ᾖ τὸ παραλληλόγραμμον τὸ ληφθὲν ἐν μέρει, τοσῷδε μᾶλλον τοῦτʼ ἂν συμβαίνοι. ἥ τε γὰρ λοξότης τῆς διαμέτρου ἧττον ἀπελέγχεται καὶ ἡ ἀνισότης τοῦ μήκους ἐν τοῖς μεγάλοις, ὥστʼ οὐδʼ ἂν ὀκνήσειας ἐπʼ αὐτῶν τὴν διάμετρον εἰπεῖν μῆκος τοῦ σχήματος. ἐὰν οὖν τὴν διάμετρον λοξώσῃς μᾶλλον, ὥστε ἐκπεσεῖν ἔξω τῶν πλευρῶν ἑκατέρας ἢ τῆς γε ἑτέρας, οὐκ ἂν ὁμοίως ἔτι ταῦτα συμβαίνοι· τοιοῦτον δή τι λέγω τὸ μέτρον τῶν ἐν πλάτει λεγομένων. ὁ δʼ ἀπὸ τῶν Κασπίων πυλῶν τὴν μὲν διʼ αὐτῶν τῶν ὀρῶν λαμβάνων ὡς ἂν ἐπὶ ταὐτοῦ παραλλήλου μέχρι στηλῶν ἀγομένην, τὴν δʼ ἀπονεύουσαν εἰς Θάψακον εὐθὺς ἔξω πολὺ τῶν ὀρῶν, καὶ πάλιν ἐκ Θαψάκου προσεκβάλλων ἄλλην μέχρι Αἰγύπτου τοσοῦτον ἐπιλαμβάνουσαν πλάτος, εἶτα τῷ μήκει τῷ ταύτης καταμετρῶν τὸ τοῦ χωρίου μῆκος, διαμέτρῳ τετραγώνου καταμετρεῖν ἂν δόξειε τὸ τοῦ τετραγώνου μῆκος. ὅταν δὲ μηδὲ διάμετρος ᾖ ἀλλὰ κεκλασμένη ἡ γραμμή, πολὺ μᾶλλον ἂν δόξειε πλημμελεῖν· κεκλασμένη γάρ ἐστιν ἡ ἀπὸ Κασπίων πυλῶν διὰ Θαψάκου πρὸς τὸν Νεῖλον ἀγομένη. πρὸς μὲν Ἐρατοσθένη ταῦτα.
Eratosthenes cannot, therefore, be found fault with on these grounds; what may be objected against him is as follows. When you wish to give a general outline of size and configuration, you should devise for yourself some rule which may be adhered to more or less. After having laid down that the breadth of the space occupied by the mountains which run in a direction due east, as well as by the sea which reaches to the Pillars of Hercules, is 3000 stadia, would you pretend to estimate different lines, which you may draw within the breadth of that space, as one and the same line? We should be more willing to grant you the power of doing so with respect to the lines which run parallel to that space than with those which fall upon it; and among these latter, rather with respect to those which fall within it than to those which extend without it; and also rather for those which, in regard to the shortness of their extent, would not pass out of the said space than for those which would. And again, rather for lines of some considerable length than for any thing very short, for the inequality of lengths is less perceptible in great extents than the difference of configuration. For example, if you give 3000 stadia for the breadth at the Taurus, as well as for the sea which extends to the Pillars of Hercules, you will form a parallelogram entirely enclosing both the mountains of the Taurus and the sea; if you divide it in its length into several other parallelograms, and draw first the diagonal of the great parallelogram, and next that of each smaller parallelogram, surely the diagonal of the great parallelogram will be regarded as a line more nearly parallel and equal to the side forming the length of that figure than the diagonal of any of the smaller parallelograms: and the more your lesser parallelograms should be multiplied, the more will this become evident. Certainly, it is in great figures that the obliquity of the diagonal and its difference from the side forming the length are the less perceptible, so that you would have but little scruple in taking the diagonal as the length of the figure. But if you draw the diagonal more inclined, so that it falls beyond both sides, or at least beyond one of the sides, then will this no longer be the case; and this is the sense in which we have observed, that when you attempted to draw even in a very general way the extents of the figures, you ought to adopt some rule. But Eratosthenes takes a line from the Caspian Gates along the mountains, running as it were in the same parallel as far as the Pillars, and then a second line, starting directly from the mountains to touch Thapsacus; and again a third line from Thapsacus to the frontiers of Egypt, occupying so great a breadth. If then in proceeding you give the length of the two last lines [taken together] as the measure of the length of the district, you will appear to measure the length of one of your parallelograms by its diagonal. And if, farther, this diagonal should consist of a broken line, as that would be which stretches from the Caspian Gates to the embouchure of the Nile, passing by Thapsacus, your error will appear much greater. This is the sum of what may be alleged against Eratosthenes.
Cite This Text