§1.5#15 'Aλλʼ ἐπεὶ δῆλον περὶ τούτων, περὶ τῶν λοιπῶν σκεπτέον, καὶ πρῶτον πότερον ἔστι τι σῶμα ἄπειρον, ὥσπερ οἱ πλεῖστοι τῶν ἀρχαίων φιλοσόφων ἄήθησαν, ἢ τοῦτʼ ἐστὶν ἕν τι τῶν ἀδυνάτων· τὸ γὰρ οὕτως ἢ ἐκείνως ἔχειν οὕ τι μικρὸν ἀλλʼ ὅλον διαφέρει καὶ πἂν πρὸς τήν τῆς ἀληθείας θεωρίαν.
5 But since these matters are clear, we must consider the remaining ones, and first whether there is any infinite body, as most of the ancient philosophers thought, or whether this is one of the impossible things; for whether the matter stands thus or otherwise is no small difference, but makes a whole and complete difference for the contemplation of truth.
σχεδὸν γὰρ αὕτη πασῶν ἀρχὴ τῶν ἔναντιώσεων τοῖς ἀποφηναμένοις τι περὶ τῆς ὅλης φύσεως καὶ γέγονε καὶ γένοιτʼ ἄν, εἴπερ καὶ τὸ μικρὸν παραβῆναι τῆς ἀληθείας ἀφισταμένοις γίνεται πόρροω μυριοπλάσιον, οἷον εἴ τις ἐλάχιστον εἶναί τι φαίη μέγεθος· οὗτος γὰρ τοὐλάχιστον εἰσαγαγών τὰ μέγιστα κινεῖ τῶν μαθηματικῶν.
For this has been and would be almost the starting-point of all oppositions for those who have declared anything about the whole of nature, if indeed even a small departure from the truth becomes multiplied ten-thousandfold as one goes further, as for instance if someone were to say that there is a certain smallest magnitude; for he, by introducing a smallest, shakes the greatest principles of mathematics.
τούτου δʼ αἴτιον ὅτι ἡ ἀρχὴ δυνάμει μείζων ἢ μεγέθει, διόπερ τὸ ἐν ἀρχ μικρὸν ἐν τῇ τελευτῇ γίνεται παμμέγεθες.
And the cause of this is that the starting-point (or principle) is greater in power than in size, wherefore that which is small at the start becomes immense at the end.
τὸ δʼ ἄπειρον καὶ ἀρχῆς ἔχει δύναμιν καὶ τοῦ ποσοῦ τὴν μεγίστην, ὥστʼ οὐδὲν ἄτοπον οὐδʼ ἄλογον τὸ θαυμαστὴν εἶναι τὴν διαφορὰν ἐκ τοῦ λαβεῖν ὡς ἔστι τι σῶμα ἄπειρον.
But the infinite has both the power of a starting-point and the greatest power of quantity, so that it is in no way absurd or unreasonable that the difference arising from assuming that there is some infinite body should be wonderful.
διὸ περὶ αὐτοῦ λεκτέον ἐξ ἀρχῆς ἀναλαβοῦσιν.
Therefore we must speak about it, taking up the matter from the beginning.
ἀνάγκη δὴ πᾶν σῶμα ἢ τῶν ἁπλῶν εἶναι ἢ τῶν συνθέτων, ὥστε καὶ τὸ ἄπειρον ἢ ἁπλοῦν εἶναι ἢ σύνθετον.
Indeed, every body must be either one of the simple or one of the composite bodies, so that the infinite also must be either simple or composite.
ἀλλὰ μὴν καὶ ὅτι γε πεπερασμένων τῶν ἁπλῶν ἀνάγκη πεπερασμένον εἶναι τὸ σύνθετον, δῆλον· τὸ γὰρ ἐκ πεπερασμένων καὶ πλήθει καὶ μεγέθει συγκείμενον πεπέρανται καὶ επλήθει καὶ μεγέθει· τοσοῦτον γάρ ἐστιν ἐξ ὅσων ἐστὶ συγκείμενον.
Furthermore, it is also clear that if the simple bodies are finite, the composite must of necessity be finite; for that which is composed of things finite both in number and in magnitude is itself finite both in number and in magnitude, since it is as great as the things of which it is composed.
λοιπὸν τοίνυνἰδεῖν πότερον ἐνδέχεταί τι τῶν ἁπλῶν ἄπειρον εἶναι τὸ μέγεθος, ἤ τοῦτʼ ἀδύνατον.
It remains, then, to see whether it is possible for any of the simple bodies to be infinite in magnitude, or whether this is impossible.
προχειρισάμενοι δὴ περὶ τοῦ πρώτου τῶν σωμάτων, οὕτω σκοπῶμεν καὶ περὶ τῶν λοιπῶν.
Let us then first discuss the first of the bodies, and in this way let us also consider the others.
ὅτι μὲν τοίνυν ἀνάγκη τὸ σῶμα τὸ κύκλῳ φερόμενον πεπεράνθαι πἄν, ἐκ τῶνδε δῆλον.
That, therefore, the body which is carried in a circle must of necessity be entirely finite, is clear from the following.
εἰ γὰρ ἄπειρον τὸ κύκλῳ φερόμεσῶμα, νον ἄπειροι ἔσονται αἱ ἀπὸ τοῦ μέσου ἐκβαλλόμεναι.
For if the body carried in a circle is infinite, the straight lines drawn from the center will be infinite.
τῶν δʼ ἀπείρων τὸ διαάστημα ἄπειρον· διάστημα γὰρ λέγω τῶν γραμμῶν, οὖ μηδὲν ἔστιν ἔξω λαβεῖν μέγεθος ἁπτόμενον τῶν γραμμῶν.
And the distance between infinite lines is infinite; for by the "distance" of the lines I mean that outside of which it is impossible to take any magnitude that touches the lines.
τοῦτʼ οὖν ἀνάγκη ἄπειρον εἶναι· τῶν γὰρ πεπερασμένων ἀεὶ ἔσται πεπερασμένον.
This, therefore, must of necessity be infinite; for the distance between finite lines will always be finite.
ἐπεὶ δʼ ἕει ἔστι τοῦ δοθέντος μεῖζον λαβεῖν, ὥστε καθάπερ ἀριθμὸν λέγομεν ἄπειρον, ὅτι μέγιστος οὐκ ἔστιν, ὁ αὐτὸς λόγος καὶ περὶ τοῦ διαστήματος, εἰ οὖν τὸ μὲν ἄπειρον μὴ ἔστι διελθεῖν, ἀπείρου δʼ ὄντος ἀνάγκη ἄπειρον τὸ διάστημα εἶναι, οὐκ ἄν ἐνδέχοιτο κινηθῆναι κύκλῳ·
And since it is possible to take a magnitude greater than any given magnitude, so that just as we say that number is infinite because there is no greatest number, the same reasoning holds also for distance, if then it is impossible to traverse the infinite, and if, when the body is infinite, the distance must of necessity be infinite, it would be impossible for it to move in a circle.
τὸν δʼ οὐρανὸν ὁρῶμεν κύκλῳ στρεφόμενον, καὶ τῷ λόγῳ δὲ διωρίσαμεν ὅτι ἐστί τινος ἡ κύκλῳ κίνησις.
But we see the heaven revolving in a circle, and we have also defined by reason that circular motion belongs to some body.
ἔτι ἀπὸ πεπερασμένου χρόνου ἐἀν ἀφέλῃς πεπερασμένον, ἀνάγκη καὶ τὸν λοιπὸν εἶναι πεπερασμένον καὶ ἔχειν ἀρχήν.
Furthermore, if you subtract a finite time from a finite time, the remaining time must also of necessity be finite and have a beginning.
εἰ δʼ ὁ χρόνος ὁ τῆς βαδίσεως ἔχει ἀρχήν, ἔστιν ἀρχὴ καὶ τῆς κινήσεως, ὥστε καὶ τοῦ μεγέθους ὃ βεβάδωιεν.
And if the time of traversing has a beginning, the movement also has a beginning, and consequently also the magnitude which has been traversed.
ὁμοίως δὲ τοῦτο καὶ ἐπὶ τῶν ἄλλων.
And this is similarly true in other cases as well.
ἔστω δή γραμμὴ ἄπειρος, ἐφʼ ᾖ ΑΓΕ, ἐπὶ θάτερα, ἧ τὸ E· ἡ δʼ ἐφʼ ἧ τὰ ΒΒ, ἐπʼ ἀμφότερα ἄπειρος.
Let there be, then, an infinite line, AGE, in one direction, namely toward E; and let another line, BB, be infinite in both directions.
εἰ δὴ γράψει κύκλον ἡ τὸ ΑΓE ἀπὸ τοῦ Γ’ κέντρου, τέμνουσά ποτε οἰσθήσεται κύκλῳ τὴν ΒΒ ἡ τὸ ΑΓΕ πεπερασμένον χρόνον· ὁ γὰρ πᾶς χρόνος ἐν ὅσῳ κύκλῳ ἠνέχθη ὁ οὐρανός, πεπερασμένος.
If, then, the line AGE describes a circle from the center Γ, the line AGE, being carried in a circle, will at some time cut the line BB in a finite time; for the entire time in which the heaven is carried in a circle is finite.
καὶ ὁ ἀφῃρημένος ἄρα, ὃν ἡ τέμνουσα ἐφέρετο.
Therefore, the subtracted time also, during which the cutting line was moving, will be finite.
ἔσται ἄρα τις ἀρχὴ ἡ πρῶτον ἡ τὰ E τὴν τὰ ΒΒ ἔτεμεν.
Consequently, there will be some starting-point at which E first cut the line BB.