§1.1Πρῶτον μὲν οὖν τὰ συμβαίνοντα περὶ τὸν ζυγὸν ἀπορεῖται, διὰ τίνα αἰτίαν ἀκριβέστερά ἐστι τὰ ζυγὰ τὰ μείζω τῶν ἐλαττόνων.
First, then, questions are raised about what happens in the case of the balance: for what reason are larger balances more accurate than smaller ones?
Τούτου δὲ ἀρχή, διὰ τί ποτε ἐν τῷ κύκλῳ ἡ πλεῖον ἀφεστηκυῖα γραμμὴ τοῦ κέντρου τῆς ἐγγὺς τῇ αὐτῇ σχύϊ κινουμένης θᾶττον φέρεται τῆς ἐλάττονος;
The principle of this is: why is it that in a circle, the line which is further from the center is carried faster than the shorter one which is near it, when moved by the same force?
Τὸ γὰρ θᾶττον λέγεται διχῶς· ἄν τε γὰρ ἐν ἐλάττονι χρόνῳ ἴσὸν τόπον διεξέλθῃ, θᾶττον εἶναι λέγομεν, καὶ ἐὰν ἐν ἴσῳ πλείω.
For "faster" is used in two senses: we say a thing is faster if it traverses an equal distance in less time, and also if it traverses a greater distance in equal time.
§1.2Ἡδὲ μείζων ἐν ἲσῳ χρόνῳ γράφει μείζονα κύκλον· ὁ γὰρ ἐκτὸς μείζων τοῦ ἐντός.
And the larger line describes a larger circle in an equal time; for the outer circle is larger than the inner.
Αἴτιον δὲ τούτων ὅτι φέρεται δύο φορὰς ἡ γράφουσα τὸν κύκλον.
The cause of this is that the line describing the circle is carried by two motions.
Ὄταν μὲν οὖν ἐν λόγῳ τινὶ φέρηται, ἐπ’ εὐθείας ἀνάγκη φέρεσθαι τὸ φερόμενον, καὶ γίνεται διάμετρος αὐτὴ τοῦ σχήματος ὃ ποιοῦσιναἱ ἐν τούτῳ τῷ λόγῳ συντεθεῖσαι γραμμαί.
Therefore, when it is carried in a certain ratio, the moving body must be carried along a straight line, and this line becomes the diagonal of the figure formed by the lines combined in this ratio.
§1.3Ἔστω γὰρ ὁ λόγος ὃν φέρεται τὸ φερόμενον, ὃν ἔχει ἡ Α Β πρὸς τὴν ΑΓ· καὶ τὸ μὲν ΑΓ φερέσθω πρὸς τὸ Β, ἡ δὲ Α Β ὑποφερέσθω πρὸς τὴν Η Γ·
For let the ratio in which the moving body is carried be that which AB has to AC; and let AC be carried towards B, and AB be carried down towards HG.
ἐνηνέχθω δὲ τὸ μὲν Α πρὸς τὸ Δ, ἡ δὲ ἐφ’ ᾗ Α Β πρὸς τὸ Ε.Εἰ οὖν ἐπὶ τῆς φορᾶς ὁ λόγος ἦν ὅν ἡ Α Β ἔχει πρὸς τὴν Α Γ, ἀνάγκη καὶ τὴν Α Δ πρὸςτὴν Α Ε τοῦτον ἔχειν τὸ λόγον.
And let A be carried to D, and the point on AB to E. If, then, the ratio in the motion was that which AB has to AC, AD also must have this ratio to AE.
§1.4Ὄμοιον ἄρα ἐστὶ τῷ λόγῳ τὸ μικρὸν τετράπλευρον τῷ μείζονι, ὥστεκαὶ ἡ αὐτὴ διάμετρος αὐτῶν, καὶ τὸΑ ἔσται πρὸς Ζ. Τὸν αὐτὸν δὴ τρόπον δειχθήσεται κἂν ὁπουοῦν διαληφθῇ ἡ φορά· αἰεὶ γὰρ ἔσται ἐπὶ τῆς διαμέτρου.
Therefore, the small quadrilateral is similar in ratio to the larger one, so that their diagonal is also the same, and A will reach Z. In the same way it will be demonstrated, at whatever point the motion is interrupted; for it will always be on the diagonal.
Φανερὸν οὖν ὅτι τὸ κατὰ τὴν διάμετρον φερόμενον ἐν δύο φοραῖς ἀνάγκη τὸν τῶν πλευρῶν φέρεσθαι λόγον.
It is clear, therefore, that what is carried along the diagonal must, in two motions, be carried in the ratio of the sides.
§1.5Εἰ γὰρ ἄλλον τινά, οὐκοἰσθόσεται κατὰ τὴν διάμετρον.
For if it were in any other ratio, it will not be carried along the diagonal.
Ἐὰν δὲ ἐν μηδενὶ λόγῳ φέρηται δύο φορὰς κατὰ μηδένα χρόνον, ἀδύνατον εὐθεῖαν εἶναι τὴν φοράν.
And if a body is carried by two motions in no ratio during any time, it is impossible for the motion to be a straight line.
Ἕστω γὰρ εὐθεῖα.
For let it be a straight line.
Τεθείσης οὗν ταύτης διαμέτρου, καὶ παραπληρωθεισῶν τῶν πλευρῶν, ἀνάγκη τὸν τῶν πλευρῶν λόγον φέρεσθαι τὸ φερόμενον· τοῦτο γὰρ δέδεικται πρότερον.
If this is then assumed as a diagonal, and the sides are completed, the moving body must be carried in the ratio of the sides; for this has been shown before.
§1.6Οὐκ ἄρα ποιήσει εὐθεῖαν τὸ ἐν μηδενὶ λόγῳ φερόμενον μηδένα χρόνον.
Therefore, that which is carried in no ratio during any time will not make a straight line.
Ἐὰν γάρ τινα λόγον ἐνεχθῇ ἐνχρόνῳ τινί, τοῦτον ἀνάγκη τὸν χρόνον εὐθεῖαν εἶναι φορὰν διὰ τὰπροειρημένα.
For if it is carried in some ratio during some time, during this time the motion must be a straight line because of what was said before.
Ὤστε περιφερὲς γίνεται, δύο φερόμενον φορὰς ἐν μηθενὶ λόγῳ μηθένα χρόνον.
Consequently, it becomes a curve when it is carried by two motions in no ratio during any time.
§1.7Ὄτι μὲν τοίνυν ἡ τὸν κύκλον γράφουσα φέρεται δύο φορὰς ἅμα, φανερὸν ἔκ τε τούτων, καὶ ὅτιτὸ φερόμενον κατ’ εὐθεῖαν ἐπὶ τὴν κάθετον ἀφικνεῖται, ὥστε εἶναι πάλιν αὐτὴν ἀπὸ τοῦ κέντρου κάθετον.
That, then, the line describing the circle is carried by two motions at the same time is clear from these facts, and also because the body carried in a straight line reaches the perpendicular, so that it is again a perpendicular from the center.
Ἕστω κύκλος ὁ ΑΒΓ, τὸ δ’ ἄκρον τὸ ἐφ’ οὖ Β φερέσθω ἐπὶ τὸ Δ·
Let the circle be ABG, and let the extremity B be carried towards D; and let it at some time reach G.
ἀφικνεῖται δέποτε ἐπὶ τὸ Γ. Εἰ μὲν οὖν ἐν τῷ λόγῳ ἐφέρετο ὃν ἔχειἡ ΒΔ πρὸς τὴν ΔΓ, ἐφέρετο ἄν τὴν διάμετρον τὴν ἐφ’ ᾗ ΒΓ.
If, then, it were carried in the ratio which BD has to DG, it would be carried along the diagonal BG.
§1.8Νῦν δέ,ἐπείπερ ἐν οὐδενὶ λόγῳ, ἐπὶ τὴν περιφέρειανφέρεται τὴν ἐφ’ ᾗ ΒΕΓ. Ἐὰν δὲ δυοῖν φερομένοιν ἀπὸ τῆς αὑτῆς ἰσχύος τὸ μὲν ἐκκρούοιτο πλεῖον τὸ δὲ ἔλαττον, εὔλογον βραδύτερον κινηθῆναι τὸ πλεῖον ἐκκρουόμενον τοῦ ἔλαττον ἐκκρουομένου· ὃ δοκεῖ συμβαίνειν ἐπὶ τῆς μείζονος καὶ ἐλάττονος τῶν ἐκ τοῦ κέντρου γραφουσῶν τοὺς κύκλους.
But as it is, since it is in no ratio, it is carried along the circumference BEG. Now, if of two things carried by the same force, one is deflected more and the other less, it is reasonable that the one deflected more should move more slowly than the one deflected less; which seems to happen in the case of the larger and smaller of the lines describing circles from the center.
§1.9Διὰ γὰρ τὸ ἐγγύτερον εἶναι τοῦ μένοντος τῆς ἐλάττονος τὸ ἄκρον ἢ τὸ τῆς μείζονος, ὥσπερ ἀντισπώμενον εἰς τοὐναντίον, ἐπὶ τὸ μέσον βραδύτερον φέρεται τὸ τῆς ἐλάττονος ἄκρον.
For because the extremity of the smaller line is nearer than that of the larger to the stationary point [the center], being as it were pulled back in the contrary direction, the extremity of the smaller line is carried more slowly towards the center.
Πάσῃ μὲν οὖν κύκλον γραφούσῃ τοῦτο συμβαίνει, καὶ φέρεται τὴν μὲν κατὰ φύσιν κατὰ τὴν περιφέρειαν, τὴν δὲ παρὰ φύσιν εἰς τὸ πλάγιον καὶ τὸ κέντρον.
This happens to every line describing a circle, and it is carried in accordance with nature along the circumference, but contrary to nature sideways and towards the center.
§1.10Μείζω δ’ ἀεὶ τὴν παρὰ φύσιν ἡ ἐλάττων φέρεται·
And the smaller line is always carried with a greater motion contrary to nature; for because it is nearer to the center which pulls it back, it is more controlled.
διὰ γὰρ τὸ ἐγγύτερον εἶναι τοῦ κέντρου τοῦ ἀντισπῶντος κρατεῖται μᾶλλον, Ὄτι δὲ μεῖζον τὸ παρὰ φύσιν κινεῖται ἡ ἐλάττων τῆς μείζονος τῶν ἐκ τοῦ κέντρον γραφουσῶν τοὺς κύκλους, ἐκ τῶνδε δῆλον.
And that the smaller of the lines describing circles from the center moves with a greater motion contrary to nature than the larger, is clear from what follows.