§3.1#2πολλά δʼ ἐστὶν ἃ τοῖς ἀδιαιρέτοις οὐχ οἷόν τε ὑπάρχειν, τοῖς δὲ φυσοκοῖς ἀναγκαῖον, οἷον εἴ τί ἐστι διαιρετόν·
And there are many things which cannot belong to indivisible things, but which must necessarily belong to natural things, for example, if anything is divisible.
ἐν ἀδιαιρέτῳ γὰρ διαιρετὸν ἀδύνατον ὑπάρχειν, τὰ δὲ πάθη διαιρετὰ πάντα διχῶς· ἢ γὰρ κατʼ εἶδος ἢ κατὰ συμβεβηκός, κατʼ εἶδος μὲν οἷον χρώματος τὸ λευκὸν ἢ τὸ μέλαν, κατὰ συμβεβηκὸς δέ, ἂν ᾧ ὑπάρχει ᾖ διαιρετόν, ὥστε ὅσα ἀπλᾶ τῶν παθημάτων, πάντʼ ἐστὶ διαιρετὰ τοῦτον τὸν τρόπον.
For it is impossible for what is divisible to belong to an indivisible, and all affections are divisible in two ways: either in species or by accident, in species, as for instance white or black is of color, and by accident, if that to which it belongs is divisible, so that all those of the affections which are simple are all divisible in this way.
διὸ τὸ ἀδύνατον ἐν τοῖς τοιούτοις ἐπισκεπτέον.
Therefore we must consider the impossibility in such things.
εἰ δὴ τῶν ἀδυνάτων ἐστὶν ἑκατέρου μέρους μηδὲν ἔχοντος βάρος τὰ ἄμφω ἔχειν βάρος, τὰ δʼ αἰσθητὰ σώματα ἢ πάντα ἢ ἔνια βάρος ἔχει, οἷον ἡ γῆ καὶ τὸ ὕδωρ, ὡς κἂν αὐτοὶ φαῖεν, εἰ ἡ στιγμὴ μηδὲν ἔχει βάρος, δῆλον ὅτι οὐδʼ αἱ γραμμαί, εἰ δὲ μὴ αὗται, οὐδὲ τὰ ἐπίπεδα· ὥστʼ οὐδὲ τῶν σωμάτων οὐθέν.
If, then, it is among impossible things that, when neither of the two parts has weight, the two together should have weight, and yet sensible bodies either all or some of them have weight, such as earth and water, as even they themselves would say, then if the point has no weight, it is clear that neither do lines, and if these do not, neither do planes; so that none of the bodies would either.
ἀλλὰ μὴν ὅτι τὴν στιγμὴν οὐχ οἷόν τε βάρος ἔχειν, φανερόν.
But indeed, that it is impossible for the point to have weight is clear.
τὸ μὲν γὰρ βαρὺ ἅπαν καὶ βαρύτερον καὶ τὸ κοῦφον καὶ κουφότερον ἐνδέχεταί τινος εἶναι. τὸ δὲ βαρύτερον ἢ κουφότερον ἴσως οὐκ 299b ἀνάγκη βαρύ ἢ κοῦφον εἶναι, ὥσπερ καὶ τὸ μὲν μέγα μεῖζον, τὸ δὲ μεῖζον οὐ πάντως μέγα· πολλὰ γάρ ἐστιν ἃ μικρὰ ὄντα ἁπλῶς ὅμως μείζω ἑτέρων ἐστίν. εἰ δὴ ὃ ἂν βαρὺ ὂν βαρύτερον ᾖ, ἀνάγκη βάρει μεῖζον εἶναι, τὸ βαρὺ ἅπαν διαιρετὸν ἂν εἴη.
For everything heavy can be heavier, and everything light lighter, than something. (And yet, that which is heavier or lighter is perhaps not necessarily heavy or light, just as the 299b great is greater, but the greater is not always great; for there are many things which, though absolutely small, are nevertheless greater than others.) If, then, whatever is heavy and is heavier must be greater in weight, everything heavy must be divisible.
ἡ δὲ στιγμὴ ἀδιαίρετον ὑπόκειται.
But the point is assumed to be indivisible.
ἔτι εἰ τὸ μὲν βαρὺ πυκνόν τι, τὸ δὲ κοῦφον μανόν, ἔστι δὲ πυκνὸν μανοῦ διαφέρον τῷ ἐν ἴσῳ ὄγκῳ πλεῖον ἐνυπάρχειν· εἰ οὖν ἐστὶ στιγμὴ βαρεῖα καὶ κούφη, ἔσται καὶ πυκνὴ καὶ μανή.
Furthermore, if the heavy is something dense, and the light rare, and the dense differs from the rare in having more contained in an equal volume; if, then, there is a heavy and light point, it will also be dense and rare.
ἀλλά τὸ μὲν πυκνὸν διαιρετόν, ἡ δὲ στιγμὴ ἀδιαίρετος.
But the dense is divisible, while the point is indivisible.
εἰ δὲ πᾶν τὸ βαρὺ ἢ μαλακὸν ἢ σκληρὸν ἀνάγκη εἶναι, ῥᾴδιον ἐκ τούτων ἀδύνατόν τι συναγαγεῖν.
And if everything heavy must be either soft or hard, it is easy to draw an impossible conclusion from these.
μαλακὸν μὲν γὰρ τὸ εἰς ἑαυτὸ ὑπεῖκον, σκληρὸν δὲ τὸ μὴ ὑπεῖκον. τὸ δὲ ὑπεῖκον διαιρετόν.
For the soft is that which yields into itself, and the hard is that which does not yield; and that which yields is divisible.
ἀλλά μὴν οὐδʼ ἐκ μὴ ἐχόντων βάρος ἔσται βάρος.
But indeed, neither will weight come to be from things that do not have weight.
τό τε γὰρ ἐπὶ πόσων συμβήσεται τοῦτο καὶ ἐπὶ ποίων, πῶς διοριοῦσι μὴ βουλόμενοι πλάττειν;
For how will they define on how many and on what kind of things this will happen, unless they wish to invent arbitrarily?
καὶ εἰ πᾶν μεῖζον βάρος βάρους βάρει, συμβήσεται καὶ ἕκαστον τῶν ἀμερῶν βάρος ἔχειν·
And if every greater weight is greater than a weight by a weight, it will follow that each of the impartibles also has weight.
εἰ γὰρ αἱ τέτταρες στιγμαὶ βάρος ἔχουσι, τὸ δʼ ἐκ πλειόνων ἢ τουδὶ βαρέος ὄντος βαρύτερον, ᾧ δὲ βαρέος βαρύτερον ἀνάγκη βαρὺ εἶναι, ὥσπερ καὶ ᾧ λευκοῦ λευκότερον λευκόν, ὥστε τὸ μεῖζον μιᾷ στιγμῇ 〈μιᾷ στιγμῇ〉 βαρύτερον ἔσται ἀφαιρεθέντος τοῦ ἴσου.
For if four points have weight, and that which consists of more than these is heavier than this heavy thing, and that which is heavier than a heavy thing must be heavy, just as that which is whiter than white is white, so that the thing which is greater by one point will be heavier by that one point when the equal is subtracted.
ὥστε καὶ ἡ μία στιγμὴ βάρος ἕξει.
So that the single point also will have weight.
ἔτι εἰ μὲν τά ἐπίπεδα μόνον κατὰ γραμμὴν ἐνδέχεται συντίθεσθαι, ἄτοπον· ὥσπερ γὰρ καὶ γραμμὴ πρὸς γραμμὴν ἀμφοτέρως συντίθεται, καὶ κατὰ μῆκος καὶ κατὰ πλάτος, δεῖ καὶ ἐπίπεδον ἐπιπέδῳ τὸν αὐτὸν τρόπον.
Furthermore, if it is only possible for planes to be composed along a line, it is absurd; for just as a line is composed with a line in both ways, both in length and in breadth, so must a plane be composed with a plane in the same way.
γραμμὴ δὲ δύναται γραμμῇ συντίθεσθαι κατὰ γραμμὴν ἐπιτιθεμένην, οὐ μὴν προστιθεμένην.
And a line can be composed with a line by being superimposed along the line, but not by being added to it.
ἀλλὰ μὴν εἴ γε καὶ κατὰ πλάτος ἐνδέχεται συντίθεσθαι, ἔσται τι σῶμα ὃ οὕτε στοιχεῖον οὕτε ἐκ στοιχείων συντιθέμενον ἐκ τῶν οὕτω συντιθεμένων ἐπιπέδων, ἔτι εἰ μὲν πλήθει βαρύτερα τὰ σώματα τὰ τῶν ἐπιπέδων, ὥσπερ ἐν τῷ Τιμαίῳ διώρισται, δῆλον ὡς ἕξει καὶ ἡ γραμμὴ καὶ ἡ στιγμὴ βάρος·
But indeed, if they can also be composed in breadth, there will be some body, composed of the planes thus composed, which is neither an element nor composed of elements.
ἀνάλογον γὰρ πρὸς ἄλληλα ἔχουσιν, ὥσπερ καὶ πρότερον εἰρήκαμεν.
Furthermore, if bodies are heavier by the number of their planes, as is defined in the Timaeus, it is clear that both the line and the point will have weight; for they stand in analogical relation to one another, just as we have also said before.
εἰ δὲ μὴ τοῦτον διαφέρει τὸν τρόπον ἀλλὰ τῷ τὴν μὲν γῆν εἶναι βαρύ τὸ δὲ πῦρ κοῦφον, ἔσται καὶ τῶν ἐπιπέδων τὸ μὲν κοῦφον τὸ δὲ βαρύ.
But if they do not differ in this way, but by earth being heavy and fire light, then some of the planes will be light and others heavy.
καὶ τῶν γραμμῶν δὴ καὶ τῶν στιγμῶν ὡσαύτως· τὸ γὰρ τῆς γῆς ἐπίπεδον ἔσται βαρύτερον ἢ τὸ τοῦ πυρός.
And likewise with lines and points; for the plane of earth will be heavier than that of fire.
ὅλως δὲ συμβαίνει ἢ μηδέν ποτʼ εἶναι μέγεθος, ἢ δύνασθαί γε ἀναιρεθῆναι, εἴπερ ὁμοίως ἔχει στιγμὴ μὲν πρὸς γραμμήν, γραμμὴ δὲ πρὸς ἐπίπεδον, τοῦτο δὲ πρὸς σῶμα· πάντα γὰρ εἰς ἄλληλα ἀναλυόμενα εἰς τὰ πρῶτα ἀναλυθήσεται· ὥστʼ ἐνδέχοιτʼ ἄν στιγμὰς μόνον εἶναι, σῶμα δὲ μηθέν.
In general, it follows either that there is never any magnitude, or at least that it can be abolished, if indeed the relation of point to line is the same as that of line to plane, and of this to body; for all things being resolved into one another will be resolved into the first things; so that it would be possible for there to be only points, and no body at all.
πρὸς δὲ τούτοις καὶ εἰ ὁ χρόνος ὁμοίως ἔχει, ἀναιροῖτʼ ἄν ποτε ἢ ἐνδέχοιτʼ ἀναιρεθῆναι· τὸ γὰρ νῶν τὸ ἄτομον οἷον στιγμὴ γραμμῆς ἐστίν.
In addition to these, if time is also in like case, it would at some time be abolished or could be abolished; for the indivisible 'now' is like a point of a line.
τὸ δʼ αὐτὸ συμβαίνει καὶ τοῖς ἐξ ἀριθμῶν συντιστᾶσι τὸν οὐρανόν· ἔνιοι γὰρ τὴν φύσιν ἐξ ἀριθμῶν συνιστᾶσιν, ὥσπερ τῶν Πυθαγορείων τινές· τὰ μὲν γὰρ φυσικὰ σώματα φαίνεται βάρος ἔχοντα καὶ κουφότητα, τὰς δὲ μονάδας οὔτε σῶμα ποιεῖν οἷόν τε συντιθεμένας οὕτε βάρος ἔχειν.
And the same thing happens also to those who compose the heaven from numbers; for some compose nature from numbers, as some of the Pythagoreans do; for natural bodies appear to have weight and lightness, but it is impossible for monads, when composed, either to make a body or to have weight.