§7.5Ἐπεὶ δὲ τὸ κινοῦν κινεῖ τι ἀεὶ καὶ ἔν τινι καὶ μέχρι του (λέγω δὲ τὸ μὲν ἔν τινι, ὅτι ἐν χρόνῳ, τὸ δὲ μέχρι του, ὅτι ποσόν τι μῆκος· ἀεὶ γὰρ ἅμα κινεῖ καὶ κεκίνηκεν, ὥστε ποσόν τι ἔσται ὅ ἐκινήθη, καὶ ἐν ποσῷ), εἰ δὴ τὸ μὲν A τὸ κινοῦν, τὸ δὲ B τὸ κινούμενον, ὅσον δὲ κεκίνηται μῆκος τὸ Γ, ἐν ὅσῳ δέ, ὁ χρόνος, ἐφʼ οὗ τὸ Δ, ἐν δὴ τῷ ἴσῳ χρόνῳ ἡ κη δύναμις ἡ ἐφʼ οὗ τὸ A τὸ ἥμισυ τοῦ B διπλασίαν τῆς Γ κινήσει, τὴν δὲ τὸ Γ ἐν τῷ ἡμίσει τοῦ Δ·
Since that which moves always moves something, and in something (i.e., in time), and up to something (i.e., a certain quantity of length) — for it always moves and has moved at the same time, so that there will be a certain quantity which has been moved, and in a certain quantity of time — if indeed A is the mover, B the thing moved, Γ the distance (length) over which it has been moved, and Δ the time in which [it was moved], then in the equal time the force, namely A (the force on which is A), will move half of B over twice Γ, and over Γ in half of Δ; for in this way it will be proportional.
οὕτω γὰρ ἀνάλογον ἔσται. καὶ εἰ ἡ αὐτὴ δύναμις τὸ αὐτὸ ἐν τῳδὶ τῷ χρόνῳ τοσήνδε κινεῖ καὶ τὴν ἡμίσειαν ἐν τῷ ἡμίσει, καὶ ἡ ἡμίσεια ἰσχὺς τὸ ήμισυ κινήσει ἐν τῷ ἴσῳ χρόνῳ τὸ ἴσον.
And if the same force moves the same thing in this time over so much, and half of it in half the time, then half the strength will move half [the weight] in an equal time over an equal distance.
οον τῆς A δυνάμεως ἔστω ἡμίσεια ἡ τὸ E καὶ τοῦ B τὸ ἥμισυ· ὁμοίως δὴ ἔχουσι καὶ ἀνάλογον ἡ ἰσχὺς πρὸς τὸ βάρος, ὥστε ἴσον ἐν ἴσῳ χρόνῳ κινήσουσιν.
For example, let E be half of the force A, and Z half of B; they then stand in the same relation and proportionally, the strength to the weight, so they will move an equal distance in an equal time.
καὶ εἰ τὸ E τὸ Z κινεῖ ἐν τῷ Δ τὴν Γ, οὐκ ἀνάγκη ἐν τῷ ἴσῳ χρόνῳ τὸ ἐφʼ οὗ E τὸ διπλάσιον τοῦ Ζ κινεῖν τὴν ἡμίσειαν τῆς Γ εἰ δὴ τὸ Α τὴν τὸ B κινεῖ ἐν τῷ Δ ὅσην ἡ τὸ Γ, τὸ ἥμισυ τοῦ Α τὸ ἐφʼ E τὴν τὸ B οὐ κινήσει ἐν τῷ χρόνῳ ἐφʼ ᾧ τὸ Δ οὐδʼ ἔν τινι τοῦ Δ τι τῆς Γ ἀνάλογον πρὸς τὴν ὅλην τὴν Γ ὡς τὸ A πρὸς τὸ Ε ὅλως γὰρ εἰ ἔτυχεν οὐ κινήσει οὐδέν·
And if E moves Z in [time] Δ over [distance] Γ, it is not necessary that in the equal time, the force E should move twice Z over half of Γ. If indeed A moves B in Δ over [a distance] as much as Γ, then half of A, which is E, will not move B in the time Δ, nor in any part of Δ will it move any part of Γ proportional to the whole Γ as A is to E.
οὐ γὰρ εἰ ἡ ὅλη ἰσχὺς τοσήνδε ἐκίνησεν, ἡ ἡμίσεια οὐ κινήσει οὔτε ποσὴν οὔτʼ ἐν ὁποσροῦν·
For in general, it may happen that it will not move it at all; for it does not follow that if the whole strength moved so much, the half will move [the weight] either some amount or in any time whatever.
εἶς γὰρ ἂν κινοίη τὸ πλοῖον, εἴπερ ἥ τε τῶν νεωλκῶν τέμνεται ἰσχὺς εἰς τὸν ἀριθμὸν καὶ τὸ μῆκος ὃ πάντες ἐκίνησαν.
For [otherwise] a single person could move a ship, if the strength of the ship-haulers is divided into their number and the length which all of them moved.
διὰ τοῦτο ὁ ήνωνος λόγος οὐκ ἀληθής, ὡς ψοφεῖ τῆς κέγχρου ὁτιοῦν μέρος· οὐδὲν γὰρ κωλύει μὴ κινεῖν τὸν ἀέρα ἐν μηθενὶ χρόνῳ τοῦτον ὃν ἐκίνησεν πεσών ὁ ὅλος μέδιμνος.
Because of this, Zeno's argument is not true, that any part whatever of a millet-seed makes a sound; for nothing prevents it from not moving the air in any time [at all], which [air] the whole bushel moved when falling.
οὐδὲ δὴ τοσοῦτον μόριον, ὅσον ἂν κινήσειεν τοῦ ὅλου εἰ εἶη καθʼ αὐτὸ τοῦτο, οὐ κινεῖ.
Indeed, it does not even move that part of the whole [air] which it would move if this [part of the seed] were by itself.
οὐδὲ γὰρ οὐδὲν ἔστιν ἀλλʼ ἢ δυνάμει ἐν τῷ ὅλῳ.
For there is nothing but only potentially in the whole.
εἰ δὲ τὰ κινοῦντα δύο, καὶ ἑκατέρον τῶνδε ἑκάτερον κινεῖ τὸ τοσόνδε ἐν τοσῶδε, καὶ συντιθέμεναι αἱ δυνάμεις τὸ σύνθετον ἐκ τῶν βαρῶν τὸ ἴσον κινήσουσιν μῆκος καὶ ἐν ἴσῳ χρόνῳ· ἀνάλογον γάρ.
But if there are two movers, and each of these moves each of those [weights] by so much in so much [time], then the combined forces will move the combined weight over an equal length and in an equal time; for it is proportional.
ἆρʼ οὖν οὕτω καὶ ἐπʼ ἀλλοιώσεως καὶ ἐπʼ οὐ. ξήσεως;
Is it, then, also thus in the case of alteration and increase?
τὶ μὲν γὰρ τὸ. αὕξον, τὶ δὲ τὸ αὐξανόμενον, ἐν ποσῷ δὲ χρόνῳ καὶ ποσὸν τὸ μὲν αὔξει τὸ δὲ αὐξάνεται.
For there is something that increases, and something that is increased, and in how much time, and by how much the one increases and the other is increased.
καὶ τὸ ἀλλοιοῦν καὶ τὸ ἀλλοιούμενον ὡσαύτως—τὶ καὶ ποσὸν κατὰ τὸ μᾶλλον καὶ ἧττον ἠλλοίωται, καὶ ἐν ποσῷ χρόνν, ἐν διπλασίῳ διπλάσιον, καὶ τὸ διπλάσιον ἐν διπλασίῳ· τὸ δʼ ἥμισυ ἐν ἡμίσει χρόνῳ (ἢ ἐν ἡμίσει ήμιμσυ), ἢ ἐν ἴσῳ διπλάσιον.
And that which alters and that which is altered likewise: something and a certain quantity in respect of more and less is altered, and in a certain amount of time — in double [time] double, and double in double [time]; and half in half the time (or in half [time] half), or in an equal [time] double.
εἰ δὲ τὸ ἀλλοιοῦν ἢ αὐξον τὸ τοσόνδε ἐν τῷ τοσῶδε αὔξει ἢ ἀλλοιοῖ, οὐκ ἀνάγκη καὶ τὸ ἥμισυ ἐν ἡμίσει καὶ ἐν ἡμίσει ήμισυ, ἀλλʼ οὐδέν, εἰ ἔτυχεν, ἀλλοιώσει ἢ αὐξήσει, ὥσπερ καὶ ἐπὶ τοῦ βάρους.
But if that which alters or increases increases or alters so much in so much [time], it is not necessary that half also [does so] in half [time], and in half [time] half, but it may happen that it alters or increases nothing at all, just as in the case of weight.