§20.1Διὰ τί αἱ φάλαγγες τὰ κρέα ἔστάσιν ἀπὸ μικροῦ ἀρτήματος μεγάλα βάρη, τοῦ ὅλουἡμιζυγίου ὄντος;
Why is it that steelyards weigh great weights, such as meat, from a small hanging weight, when the whole is a half-balance?
Οὗ μὲν γὰρ τὸ βάρος ἐντίθεται, κατήρτηται μόνον ἡ πλάστιγξ, ἐπὶ θάτερον δὲ ἡ φάλαγξ ἐστὶ μόνον.
For where the weight is placed, only the scale-pan is suspended, and on the other side there is only the steelyard itself.
Ἢ ὅτι ἃμα συμβαίνει ζυγὸν καὶ μοχλὸν εἶναι τὴν φάλαγγα;
Or is it because the steelyard happens to be at once a balance and a lever?
Ζυγὸν μὲν γὰρ, ᾖ τῶν σπαρτίων ἕκαστον γίνεται τὸ κέντρον τῆς φάλαγγος.
It is a balance in so far as each of the cords becomes the center of the steelyard.
§20.2Τὸ μὲν οὖν ἐπὶ θάτερα ἔχει πλάστιγγα, τὸ δὲ ἐπὶ θάτερα ἀντὶ τῆς πλάστιγγος τὸ σφαίρωμα, ὃ τῷ ζυγῷ ἔγκεινται, ὥσπερ εἴ τις τὴν ἑτέραν πλάστιγγα καὶ τὸν σταθμὸν ἐπιθείη ἐπὶ τὸ ἄκρον τῆς πλάστιγγος· δῆλον γὰρ ὅτι ἕλκει τοσοῦτον βάρος ἐν τῆ ἑτέρᾳ κείμενον πλάστιγγι.
Therefore, on one side it has a scale-pan, and on the other side, instead of the scale-pan, the spherical weight which is attached to the balance, just as if one were to place the other scale-pan and the counterweight upon the extremity of the scale-pan; for it is clear that it pulls (balances) as much weight as is placed in the other scale-pan.
§20.3Ὅπως δὲ τὸἓν ζυγὸν πολλὰ ᾖ ζυγά, τοιαῦτα τὰ σπαρτία πολλὰ ἔγκειται ἐν τῷ τοιούτῳ ζυγῷ, ὦν ἑκάστου τὸ ἐπὶ τάδε ἐπὶ τὸ σφαίρωμα τὸ ἣμισν τῆς φάλαγγός ἐστι, καὶ ὁ σταθμὸς δι’ ἴσου τῶν ἀπ’ ἀλλήλων τῶν σπαρτίων κινουμένων, ὥστε συμμετρεῖσθαι πόσον βάρος ἕλκει τὸ ἐν τῇ πλάστιγγι κείμενον· ὥστε γινώσκειν, ὅταν ὀρθὴ ἡ φάλαγξ ᾖ, ἀπὸ ποίου σπάρτου πόσον βάρος ἔχει ἢ πλάστιγξ, καθάπερ εἴρηται.
In order that the single balance may be many balances, many such cords are attached to such a balance, of each of which the part on this side towards the spherical weight is half of the steelyard, and the weight [functions] as the cords are moved at equal intervals from one another, so that it can be measured how much weight the object lying in the scale-pan pulls; so that, when the steelyard is horizontal, one may know from which cord how much weight the scale-pan has, as has been said.
§20.4Ὅλως μέν ἐστι τοῦτο ζυγόν, ἔχον μίαν μὲν πλάστιγγα, ἐν ᾗ σταται τὸ βάρος, τὴν δ’ ἑτέραν, ἐν ᾗτὸ σταθμὸν ἐν τῇ φάλαγγι.
On the whole, this is a balance, having on one side the scale-pan in which the weight is weighed, and on the other, the counterweight in the steelyard.
Διὸ σφαίρωμά ἐστιν ἡ φάλαγξ ἐπὶ θάτερον.
Therefore, the steelyard has a spherical weight on the other side.
Τοιοῦτον δὲ ὄν πολλὰζυγά ἐστι, καὶ τοσαῦτα ὅσαπέρ ἐστι τὰ σπαρτία.
Being such, it is many balances, as many indeed as the cords are.
§20.5Ἀεὶ δὲ τὸ ἐγγύτερον σπαρτίον τῆς πλάστιγγος καὶ τοῦ ἑσταμένου βάρους μεῖζον ἕλκει βάρος, διὰ τὸ γίνεσθαι τὴν μὲν φάλαγγα πάσαν μοχλὸν ἀνεστραμμένον ὑπομόχλιον μὲν γὰρ τὸ σπαρτίον ἕκαστον ἄνωθεν ὅν, τὸ δὲ βάρος τὸ ἐνὸν ἐν τῆ πλάστιγγι), ὅσῳ δ’ ἄν μακρότερον τὸ μῆκος τοῦ μοχλοῦτοῦ ἀπὸ τοῦ ὑπομοχλίου, τοσούτῳ ἐκεῖ μὲν ῥᾷον κινεῖ, ἐνταῦθα δὲ σήκωμα ποιεῖ, καὶ ἵστησι τὸ πρὸς τὸ σφαίρωμα βάρος τῆς φάλαγγος.
And the cord that is always closer to the scale-pan and the weight being weighed pulls a greater weight, because the whole steelyard becomes an inverted lever (for each cord is the fulcrum, being above, and the weight in the scale-pan is the active weight), and by as much as the length of the lever from the fulcrum is longer, by so much in that case it moves more easily, and in this case it produces a counterpoise, and balances the weight of the steelyard towards the spherical weight.