§1.5#2ἀλλʼ ἀδύνατον.
But this is impossible.
οὐκ ἄρα ἔστι κύκλῳ στραφῆναι τὸ ἄπειρον.
Therefore, it is impossible for the infinite to revolve in a circle.
ὥστʼ οὐδὲ τὸν κόσμον, εἰ ἦν ἄπειρος.
And consequently, neither could the world, if it were infinite.
Ἔτι δὲ καὶ ἐκ τῶνδε φανερόν, ὅτι τὸ ἄπειρον ἀδύνατον κινηθῆναι.
Furthermore, it is clear from the following that it is impossible for the infinite to be moved.
ἔστω γὰρ ἡ τὸ Α φερομένη παρὰ τὴν Β, πεπερασμένη παρὰ πεπερασμένην.
For let the line A in motion pass by the line B, a finite line passing by a finite line.
ἀνάγκη δὴ ἄμα τήν τε Α τῆς Β ἀπολελύσθαι καὶ τὴν Β τῆς Α· ὅσον γὰρ ἡ ἑτέρα ἐπιβάλλει τῆς ἑτέρας, καὶ ἡ ἑτέρα ἐκείνης τοσοῦτον.
It is necessary then that at the same time A should be clear of B and B clear of A; for as much as the one overlaps the other, so much also does the other overlap the first.
εἰ μὲν οὖν ἄμφω κινοῖντο είς τούναντίον, θᾶττονἂνἀπολύοιντο, εἰ δὲ παρὰ μένουσπν παραφέροιτο, βραδύτερον, τῷ αὐτῷ τάχει κινουμένου τοῷ παραφερομένου.
If indeed both were to move in opposite directions, they would get clear more quickly; but if one were to pass by a stationary one, they would get clear more slowly, provided that the moving one moves at the same speed.
ἀλλʼ ἐκεῖνό γε φανερόν, ὅτι ἀδύνατον τὴν ἄπειρον διελθεῖν ἐν πεπερασμένῳ χρόνφ.
But this at least is clear, that it is impossible to traverse the infinite line in a finite time.
ἐν ἀπείρῳ ἄρα· δέδεικται γὰρ τοὕτο πρότερον ἐν τοῖς περὶ κιἐπεὶ] νήσεως.
Therefore, it must be in an infinite time; for this has been shown before in the works on movement.
διαφέρει δέ γε οὐθὲν ἢ τὴν πεπερασμένην φέρεσθαι παρὰ τὴν ἄπειρον ἢ τὴν ἄπειρον παρʼ ἐκείνην· ὅταν γὰρ ἐκείνη παρʼ ἐκείνην, κἀκείνη παρʼ ἐκείνην, ὁμοίως κινουμένη 272b καὶ ἀκίνητος· πλὴν θἄττον, ἐἀν κινῶνται ἀμφότεραι, ἀπολυθήσονται.
And it makes no difference whether the finite line moves past the infinite, or the infinite past the finite; for when the one moves past the other, the other also moves past the one, whether it is moving or stationary, it is the same; except that they will get clear more quickly if both are in motion.
καίτοι ἐνίοτʼ οὐθὲν κωλύει τὴν κινουμένην παρʼ ἠρεμοῦσαν θᾶττον παρελθεῖν ἢ ἀντικινουμένην, ἐάν τις ποιήση τὰς μὲν ἀντικινουμένας ἀμφοτέρας φερομένας βραδέως, τὴν δὲ παρὰ τὴν ἠρεμοῦσαν πολλῷ ἐκείνων θᾶττον φερομένην.
And yet sometimes nothing prevents a moving line passing by a resting one more quickly than those moving in opposite directions, if one makes both of those in opposite directions move slowly, and the one passing by the resting one move much more quickly than those.
οὐδὲν οὖν πρὸς τὸν λόγον ἐμπόδιον ὅτι παρʼ ήρεμοῦσαν, ἐπείπερ κινουμένην ἐνδέχεται τὴν Α παρὰ κινουμένην τὴν Β βραόύτερον παρελθεῖν.
Therefore, its passing by a resting line is no obstacle to the argument, since it is possible for a moving line A to pass by a moving line B more slowly.
εἰ οὖν ἄπειρος ὁ χρόνος ὃν ἡ πεπερασμένη ἀπολύεται κινουμένη, καὶ ἐν ᾧ ἡ ἄπειρος τὴν πεπερασμένην ἐκινήθη, ἀνάγκη ἄπειρον εἶναι.
If, then, the time in which the finite line in motion gets clear is infinite, that in which the infinite line has moved past the finite line must also of necessity be infinite.
ἀδύνατον ἄρα τὸ ἄπειρον κινεῖσθαι ὅλως· ἐὰν γὰρ καὶ τοὐλάχιστον κινηθῇ, ἀνάγκη ἄπειρον γενέσθαι χρόνον.
Therefore, it is impossible for the infinite to move at all; for if it moves even the least bit, the time must of necessity be infinite.
ἀλλὰ μὴν ὅ γ’ οὐρανὸς περιέρχεται καὶ στρέφεται ὅλως κύκλῳ ἐν πεπερασμένφ χρόνῳ, ὥστε περίεισιν ἄπασαν τὴν ἐντός, οἶον τὴν ἈΒ πεπερασμένην.
But indeed the heaven revolves and rotates entirely in a circle in a finite time, so that it goes around all that is within, as for instance the finite line AB.
ἀδύνατον ἄρα ἄπειρον εἶναι τὸ κύκλῳ.
Therefore, it is impossible for that which moves in a circle to be infinite.
ἔτι ὥσπερ γραμμὴν ἦς πέρας ἐστὶν ἀδύνατον εἶναι ἄπειρον, ἀλλʼ εἴπερ, ἐπὶ μῆκος, καὶ ἐπίπεδον ὡσαύτως ἦ πέρας οὐκ ἐνδέχεται·
Furthermore, just as it is impossible for a line that has a limit to be infinite—or if it is, it can only be so in length—so likewise it is impossible for a surface to have a limit and be infinite; but when it is bounded, it is in no way possible for it to be infinite, as for instance an infinite square, or circle, or sphere, just as a line of a foot long cannot be infinite.
ὅταν δʼ ὁρισθῇί οὐθαμῇί οἷον τετράγωνον νον ἄπειρον ἢ κύκλον ἢ σφαῖραν, ὥσπερ οὐδὲ ποδιαίαν ἄπειεἰ οὖν μήτε σφαῖρα μήτε τετράγωνον μήτε κύκλος ἐστὶν ἄπειρος, μὴ ὄντος δὲ κύκλου οὐδʼ ἔν ἡ κύκλῳ εἴη φορά, ὁμοίως δὲ μηδʼ ἀπείρου ὄντος οὐκ ἂν εἴη ἄπειρος, εἰ μηδʼ ὁ κύκλος ἄπειρός ἐστιν, οὐκ ἄν κινοῖτο κυκλικῶς ἄπειρον σῶμα.
If, then, neither a sphere, nor a square, nor a circle is infinite, and if where there is no circle there could be no circular motion, and likewise where there is no infinite circle there could be no infinite circular motion, then if the circle itself is not infinite, an infinite body could not move in a circle.
ἔτι εἰ τὸ T κέντρον, ἡ δὲ τὸ ΑΒ ἄπειρος καὶ ἡ τὸ E πρὸς ὀρθὴν ἄπειρος καὶ ἡ τὸ ΓΔ κινουμένη, οὐδέποτʼ ἀπολυθήσεται τῆς Ε, ἀλλʼ ἀεὶ ἵξει ὥσπερ ἡ ΓE·
Furthermore, if Γ is the center, and the line AB is infinite, and the line E perpendicular to it is infinite, and the line ΓΔ is in motion, it will never get clear of E, but will always remain like ΓE; for it cuts it at Z.
τέμνει γὰρ ᾖ τὸ Ζ. οὐκ ἄρα περίιισι κύκλῳ ἡ ἄπειρος.
Therefore, the infinite line does not revolve in a circle.
ἔτι εἴπερ ἄπειρος ὁ οὐρανός, κινεῖται δὲ κύκλῳ, ἐν πεπερασμένῳ χρόνῳ ἄπειρον ἔσται διεληλυθώς.
Furthermore, if the heaven is indeed infinite, and moves in a circle, it will have traversed an infinite distance in a finite time.
ἔστω γὰρ ὁ μὲν μένων οὐρανὸς ἄπειρος, ὁ δʼ ἐντούτῳ κινούμενος ἴσος.
For let the stationary heaven be infinite, and the heaven moving within it be equal to it.
ὥστʼ εἴπερ περιελήλυθε κύκλῳ ἄπειρος ὤν, ἄπειρον τὸ ἴσον αὑτῷ διελήλυθεν ἐν πεπερασμένῳ 273a χρόνῳ.
Consequently, if indeed, being infinite, it has revolved in a circle, it has traversed an infinite distance equal to itself in a finite time.
ἀλλά τοῦτʼ ἦν ἀδύνατον.
But this was impossible.
ἔστι δὲ καὶ ἀντεστραμμένως εἰπεῖν, ὅτι εἰ πεπερασμένος ὁ χρόνος ἐν ᾧ περιεστράφη, καὶ τὸ μέγεθος ὃ διελήλυθεν ἀνάγκη εἶναι πεπερασμένον· ἴσον δʼ αὐτῷ διελήλυθεν· πεπέρανται ἄρα καὶ αὐτός.
And it is also possible to state the converse: that if the time in which it revolved is finite, the magnitude which it has traversed must also of necessity be finite; but it has traversed a magnitude equal to itself; therefore, it too is finite.
ὅτι μὲν οὖν τὸ κύκλῳ κινούμενον οὐκ ἔστιν ἀτελεύτητον οὐδʼ ἄπειρον, ἀλλʼ ἔχει τέλος, φανερόν.
That, therefore, the body which moves in a circle is not without end nor infinite, but has a limit, is clear.