§1.11Ἕστω κύκλος ἐφ’ οὗ ΒΓΔΕ, καὶ ἄλλος ἐν τούτῳ ἐλάττων, ἐφ’ οὗ Χ Ν Μ Ξ, περὶ τὸ αὐτὸ κέντροντὸ Α· καὶ ἐκβεβλήσθωσαν αἱ διάμετροι, ἐν μὲν τῷ μεγάλῳ, ἐφ’ ὦν ΓΔ καὶ ΒΕ, ἐν δὲ τῷ ἐλάττονι αἱ Χ ΝΞ· καὶ τὸ ἑτερόμηκες παραπεπληρώσθω, τὸ ΔΨΡΓ.
Let there be a circle BCDE, and inside it another smaller one XNMO, about the same center A; and let the diameters be extended, in the larger circle, CD and BE, and in the smaller, XN and XO; and let the rectangle DYRG be completed.
§1.12Εἰ δὴ ἡ ΑΒ γράφουσα κύκλον ἥξει ἐπὶ τὸ αὐτὸ ὅθεν ὡρμώθη ἐπὶ τὴν Α Ε, δῆλον ὅτι φέρεται πρὸς αὑτήν.
If, then, AB describing a circle comes to the same point from which it started on AE, it is clear that it is carried towards itself.
Ὁμοίως δὲ καὶ ἡ ΑΧ πρὸς τὴν ΑΧ ἥξει.
In the same way AX also will come to AX.
Βραδύτερον δὲ φέρεται ἡ ΑΧ τῆς ΑΒ, ὥσπερ εἴρηται, διὰ τὸ γίνεσθαι μείζρνα τὴν ἔκκρουσιν καὶ ἀντισπᾶσθαι μᾶλλον τὴν ΑΧ.
But AX is carried more slowly than AB, as has been said, because the deflection becomes greater and AX is pulled back more.
§1.13Ἤχθω δὲ ἡ ΑΘΗ, καὶ ἀπὸ τοῦ Θ κάθετος ἐπὶ τὴν ΑΒ ἡ ΘΖ ἐν τῷ κύκλῳ, καὶ πάλιν ἀπὸ τοῦ Θ ἤχθω παρὰ τὴν ΑΒ ἡ ΘΩ, καὶ ὁ ΩΥ ἐπὶ τὴν ΑΒ κάθετον, καὶἡ ΗΚ. Αἱ δὴ ἐφ’ ὧν ΩΥ καὶ ΘΖ ἴσαι.
And let ATH be drawn, and from Th a perpendicular to AB, namely ThZ in the circle, and again from Th let ThO be drawn parallel to AB, and OY a perpendicular to AB, and also HK. Then OY and ThZ are equal.
Ἡ ἄρα ΒΥ ἐλάττων τῆς ΧΖ· αἱ γὰρ ἴσαι εὐθεῖαι ἐπ’ ἀνίσουςκύκλους ἐμβληθεῖσαι πρὸςόρθὰς τῇ διαμέτρῳ ἔλαττον τμῆμα ἀποτέμνουσι τῆς διαμέτρου ἐν τοῖς μείζοσι κύκλοις, ἔστι δὲ ἡ ΩΥ ἴση τῇ ΘΖ.
Therefore, BY is less than XZ; for equal straight lines, when inserted in unequal circles at right angles to the diameter, cut off a smaller segment of the diameter in the larger circles, and OY is equal to ThZ.
§1.14Ἐν ὅσῳ δἡ χρόνῳ ἡ ΑΘ τὴν ΧΘ ἐνηνέχθη, ἐν τοσούτῳ χρόνῳ ἐν τῷ κύκλῳτῷ μείζονι μείζονα τῆς ΒΩ ἐνήνεκται τὸ ἄκρον τῆς ΒΑ. μὲν γὰρ κατὰ φύσιν φορὰ ἴση, ἡ δὲ παρὰ φύσιν ἐλάττων· ἡ δὲ ΒΥ τῆς ΖΧ. Δεῖ δὲ ἀνάλογον εἶνας, ὡςτὸ κατὰ φύσιν πρὸς τὸ κατὰ φύσιν,τὸ παρὰ φύσιν πρὸς τὸ παρὰ φύσιν.
Therefore, in the same time as ATh has carried XTh, in the larger circle the extremity of BA has been carried through a distance greater than BO. For the motion in accordance with nature is equal, but that contrary to nature is less; namely, BY is less than ZX. But they must be proportional, as the motion in accordance with nature is to that in accordance with nature, so that contrary to nature is to that contrary to nature.
§1.15Μείζονα ἄρα περιφέρειαν διελήλυθε τὴν ΗΒ τῆς ΩΒ. Ἀνάγκη δὲ τὴν Η Β ἐν τούτῳ τῷ χρόνῳ διεληλυθέναι· ἐνταῦθα γὰρ ἔσται, ὅταν ἀνάλογον ἀμφοτέρως συμβαίνῃ τὸ παρὰ φύσιν πρὸς τὸ κατὰ φύσιν.
Therefore, it has traversed a greater arc HB than OB. And it must have traversed HB in this time; for it will be there when the motion contrary to nature is proportional in both cases to the motion in accordance with nature.
Εἰ δὴ μεῖζόν ἐστι τὸ κατὰ φύσιν ἐν τῇ μείζονι, καὶ τὸ παρὰ φύσιν μᾶλλον ἂν ἐνταῦθα συμπίπτοι μοναχῶς, ὥστε τὸ Β ἐνηνέχθαι ἄν τὴν ΒΗ ἐντῷ ἐφ’ οὗ Χ σημεῖον.
If indeed the motion in accordance with nature is greater in the larger circle, the motion contrary to nature also would coincide there in the only possible way, so that the point B has been carried through BH in the same time as the point X has been carried through XTh.
§1.16Ἐνταῦθα γὰρ κατὰ φύσιν μὲν γίνεται τῷ Β σημείῳ τὸ κέντρον (ἔστι γὰρ αὐτὴ ἀπὸ τοῦ ΗΚ κάθετος), παρὰ φύσιν δὲ ἐς τὸ ΚΒ. Ἕστι δὲ ὡς τὸ ΗΚ πρὸς τὸ ΚΒ. τὸ ΘΖ πρὸς τὸ ΖΧ. Φανερὸν δὲ ἐὰν ἐπιζευχθῶσιν ἀπὸ τῶν ΒΧ ἐπὶ τὰ Η Θ. Εἰ δὲ ἐλάττων ἢ μείζων τῆς Η Β ἔσται, ἢν ἐνέχθη τὸ Β, οὐχ ὁμοίως ἔσται οὐδὲ ἀνάλογον ἐν ἀμφοῖντὸ κατὰ φύσιν πρὸς τὸ παρὰ φύσιν.
For there, the motion in accordance with nature for the point B is towards the center (for it is the perpendicular from HK), while that contrary to nature is towards KB. And as HK is to KB, so is ThZ to ZX. This is clear if lines are joined from B and X to H and Th. But if the distance through which B was carried is less or greater than HB, the motion in accordance with nature will not be similar or proportional to that contrary to nature in both.
§1.17Δι’ ἢν μὲν τοίνυν αἰτίαν ἀπὸ τῆς αὐτῆς ἰσχύος φέρεται θᾶττον τὸ πλέον ἀπέχον τοῦ κέντρουσημεῖον, δῆλον διὰ τῶν εἰρημένων· διότι δὲ τὰ μὲν μείζω ζυγὰ ἀκριβέστερά ἐστι τῶν ἐλαττόνων, φανερὸν ἐκ τούτων.
For what reason, therefore, the point further from the center is carried faster by the same force is clear from what has been said; and that larger balances are more accurate than smaller ones is manifest from this.
Γίνεται γὰρ τὸ μὲν σπάρτον κέντρον (μένει γὰρ τοῦτο), τὸ δὲ ἐπὶ ἑκάτερον μέρος τῆς πλάστιγγος αἱ ἐκ τοῦ κέντρου.
For the cord becomes the center (for this remains stationary), and the parts of the beam on either side become the lines from the center.
§1.18Ἀπὸ οὖν τοῦ αὐτοῦ βάρους ἀνάγκη θᾶττον κινεῖσθαι τὸ ἄκρον τῆς πλάστιγγος, ὅσῳ ἂν πλεῖον ἀπέχῃ τοῦ σπάρτου, καὶ ἔνια μὲν μὴ δῆλα εἶναι ἐν τοῖς μικροῖς ζυγοῖς πρὸς τὴναἴσθησιν ἐπιτιθέμενα βάρη, ἐν δὲ τοῖς μεγάλοις δῆλα·
Therefore, by the same weight, the extremity of the beam must move faster the further it is from the cord; and some weights placed upon small balances are not clear to the senses, but on large ones they are clear.
οὐθὲν γὰρ κωλύει ἔλαττονκινηθῆναι μέγεθος ἢ ὥστε εἶναι τῇ ὄψει φανερόν.
For nothing prevents the magnitude from being moved less than is visible to the sight.
§1.19Ἐπι δὲ τῆς μεγάλης πλάστιγγος ποιεῖ ὁρατὸν τὸ αὐτὸ βάρος μέγεθος. Ἔνια δὲ δῆλα μὲν ἐπ’ ἀμφοῖν ἐστίν, ἀλλὰ πολλῷ μᾶλλον ἐπὶ τῶν μειζόνων διὰ τὸ πολλῷ μεῖζον γίνεσθαι τὸ μέγεθος τῆς ῥοπῆς ὑπὸ τοῦ αὐτοῦ βάρους ἐν τοῖς μείζοσι.
But on the large beam, the same weight makes the magnitude visible.
§1.20Καὶ διὰ τοῦτο τεχνάζουσιν οἱ ἁλουργοπῶλαι πρὸς τὸ παρακρούεσθαι ἱστάντες, τό τε σπάρτον οὐκ ἐν μέσῳ τιθέντες, καὶ μόλυβδον τῆς φάλαγγος εἰς θάτερον μέροςἐγχέοντες, ἢ τοῦ ξύλου τὸ πρὸς τὴν ῥίζαν πρὸς ὃ βούλονται ῥέπειν ποιοῦντες, ἢ ἐἀν ἔχῃ ὄζον·
And some weights are clear on both, but much more so on the larger because the magnitude of the turn becomes much greater by the same weight in the larger ones.
βαρύτερον γὰρ ἐν ὦ μέρος ἢ ῥίζα τοῦ ξύλου ἐστίν, ὁ δὲ ὄζος ῥίζα τίς ἐστιν.
And for this reason purple-sellers devise tricks to deceive when weighing, putting the cord not in the middle, and pouring lead into one part of the beam, or making the part of the wood near the root be on the side they want to incline, or if it has a knot; for that part of the wood where the root is is heavier, and a knot is a kind of root.