§23.1Διὰ τί φερομένων δύο φορὰς ἐν τῷ ῥόμβῳ τῶν ἄκρων σημείων ἀμφοτέρων, οὐ τὴν ἴσην ἑκάτερον αὐτῶν εὐθεῖαν διέρχεται, ἀλλὰ πολλαπλασίαν θάτερον;
Why is it that, when both extreme points in a rhombus are moved with two motions, they do not each traverse an equal straight line, but one of them traverses many times as much?
Ὁ αὐτὸς δὲ λόγος καὶ διὰ τί τὸ ἐπὶ τῆς πλευρᾶς φερόμενον ἐλάττω διέρχεται τῆς πλευρᾶς.
And the same explanation holds also for why that which is carried on the side traverses less than the side.
Τὸ μὲν γὰρ τὴν διάμετροντὴν ἐλάττω, ἡ δὲ τὴν πλευρὰν τὴν μείζω, καὶ ἡ μὲν μίαν, τὸ δὲ δύο φέρεται φοράς.
For one traverses the shorter diagonal, and the other the longer side, and the one is moved with one motion, and the other with two.
§23.2Φερέσθω γὰρ ἐπὶ τῆς ΑΒ τὸ μὲν Α πρὸς τὸ Β, τὸ δὲ Β πρὸς τὸ Δ τῷ αὐτῷ τάχει· φερέσθω δὲ καὶ ἡ ΑΒ ἐπὶ τῆς ΑΓ παρὰ τὴν Γ Δ τῷ αὑτῷ τάχει τούτοις.
For let point A be carried along AB towards B, and B towards D at the same speed; and let AB itself also be carried along AΓ parallel to ΓΔ at the same speed as these.
Ἀνάγκη δὴ τὸ μὲν Α ἐπὶ τῆς ΑΔ διαμέτρου φέρεσθαι, τὸ δὲ Β ἐπὶ τῆς ΒΓ, καὶ ἅμα διεληλυθέναι ἑκατέραν, καὶ τὴν Α Β τὴνΓ πλευράν.
It is then necessary that A is carried along the diagonal AΔ, and B along BΓ, and that they have traversed each diagonal at the same time, and the side AB the side AΓ.
§23.3Ἐνηνέχθω γὰρ τὸ μὲν Α τὴν ΑΕ, ἡ δὲ ΑΒ τὴν ΑΖ, καὶ ἔστω ἐκβεβλημένη ἡ ΖΗ παρὰ τὴν ΑΒ, καὶ ἀπὸ τοῦ Ε πεπληρώσθω, Ὄμοιον οὖν γίνεται τὸ παραπληρωθὲν τῷ ὅλῳ.
For let A have traversed AE, and AB have traversed AZ, and let ZH be drawn parallel to AB, and let the figure be completed from E. The completed parallelogram is therefore similar to the whole.
Ἴση ἄρα ἡ ΑΖ τῇ ΑΕ,ὥστε τὸ Α ἐπὶ τῆς πλευρᾶς ἐνήνεκται τῆς ΑΕ. Ἡ δὲ ΑΒ τὴν ΑΖ εἴη ἂν ἐνηνεγμένη, Ἔσται ἄρα ἐπὶ τῆς διαμέτρου κατὰ τὸ Θ.
Therefore AZ is equal to AE, so that A has been carried along the side AE. And AB would have been carried the distance AZ, so it will be on the diagonal at Θ.
§23.4Καὶ αἰεὶ δὲ ἀνάγκη αὐτὸ φέρεσθαι κατὰ τὴν διάμετρον.
And it is always necessary for it to be carried along the diagonal.
Καὶ ἅμα ἡ πλευρὰ ἡ ΑΒ τὴν πλευρὰν τὴν ΑΓ δίεισι, καὶ τὸ Α τὴν διάμετρον δίεισι τὴν ΑΔ. Ὁμοίως δὲ δειχθήσεται καὶ τὸ Β ἐπὶ τῆς ΑΓ διαμέτρον φερόμενον.
And at the same time the side AB traverses the side AΓ, and A traverses the diagonal AΔ. In like manner it will also be shown that B is carried along the diagonal AΓ.
Ἴση γάρ ἐστιν ἡ ΒΕ τῇ ΒΗ. Παραπληρωθέντος οὖν ἀπὸ τοῦ Η, ὅμοιόν ἐστι τῷ ὅλῳ τὸ ἐντός.
For BE is equal to BH. Therefore, when completed from H, the inside figure is similar to the whole.
§23.5Καὶ τὸ Β ἐπὶ τῆς διαμέτρου ἔσται κατὰ τὴν σύναψιν τῶν πλευρῶν, καὶ ἅμα δίεσιν τε πλευρὰ τὴν πλευρὰν καὶ τὸ Β τὴν ΒΓ διάμετρον.
And B will be on the diagonal at the joint of the sides, and at the same time the side traverses the side, and B the diagonal BΓ.
Ἄμα ἄρα καὶ τὸ Β τὴν πολλαπλασίαν τῆς Α Β δίεισι καὶ ἡ πλευρὰ τὴν ἐλάττονα πλευράν, τῷ αὐτῷ τάχει φερόμενα, καὶ ἡ πλευρὰ μείζω τοῦ Α διελήλυθε μίαν φορὰν φερομένη.
Therefore, at the same time, though carried at the same speed, B traverses many times the distance of AB, and the side traverses the shorter side, and the side has traversed a greater distance than A, which is moved with one motion.
§23.6Ὅσῳ γὰρ ἂν ὀξύτερος γένηται ὁ ῥόμβος, ἡ μὲν διάμετρος ἡ ἐλάττων γίνεται, ἡ δὲ ΒΓ μείζων, ἡ δὲ πλευρὰ τῆς Β Γ ἐλάττων.
For the more acute the rhombus becomes, the smaller the shorter diagonal becomes, and BΓ becomes larger, while the side becomes smaller than BΓ.
Ἄτοπον γάρ, ὥσπερ ἐλέχθη,τὸ δύο φορὰς φερόμενον ἐνίοτε βραδύτερον φέρεσθαι τοῦ μίαν, καὶ ἀμφοτέρων ἰσοταχῶν σημείων δοθέντων μείζω διεξιέναι θάτερον.
For it is strange, as has been said, that what is carried with two motions is sometimes carried more slowly than what is carried with one, and that when both points are given as equal in speed, one traverses a greater distance than the other.
§23.7Αἴτιον δὲ ὅτι τοῦ μὲν ἀπὸ τῆς ἀμβλείας φερομένου σχεδὸν ἐναντίαι ἀμφότεραι γίνονται, ἥν τε αὐτὴ φέρεται καὶ ἣν ὑπὸ τῆς πλευρᾶς ὑποφέρεται, τοῦ δὲ ἀπὸ τῆς ὀξείας συμβαίνει φέρεσθαι ἐπὶ τὸ αὐτό.
And the cause is that for that which is carried from the obtuse angle, both motions—the one it is carried by itself and the one it is carried under by the side—become almost opposite to each other; but for that which is carried from the acute angle, it happens to be carried in the same direction.
Συνεπουρίζει γὰρ ἡ τῆς πλευράς τὴν ἐπὶ τῆς διαμέτρου· καὶ ὅσῳ ἂν τὴν μὲν ὀξυτέραν ποιήσῃ, τὴν δὲ ἀμβλυτέραν, ἡ μὲν βραδυτέρα ἔσται, ἡ θάττων.
For the motion of the side helps to waft along the motion on the diagonal; and the more acute one angle is made, and the other more obtuse, the one motion will be slower, and the other faster.
§23.8Αἱ μὲν γὰρ ἐναντιώτεραι γίνονται διὰ τὸ ἀμβλυτέραν γίνεσθαι τὴν γωνίαν, αἱ δὲ μᾶλλον ἐπὶ τὰ αὐτὰ διὰ τὸ συνάγεσθαι τὰς γραμμάς.
For the motions become more opposite because the angle becomes more obtuse, and they become more in the same direction because the lines are drawn together.
Τὸ μὲν γὰρ Β σχεδὸν ἐπὶ τὸ αὐτὸ φέρεται κατ’ ἀμφοτέρας τὰς φοράς· συνεπουρίζεται οὖν ἡ ἑτέρα, καὶ ὅσῳ ἂν ὀξυτέρα γίνηται ἡ γωνία, τοσούτῳ μᾶλλον.
For B is carried in almost the same direction in both of its motions; therefore one helps to waft the other along, and the more acute the angle becomes, the more this is so.
§23.9Τὸ Α δὲ ἐπὶ τοὐναντίον· αὐτὸ μὲν γὰρ πρὸς τὸ Β φέρεται, ἡ δὲ πλευρὰ ὑποφέρει αὐτὸ πρὸς τὸ Δ. Καὶ ὅσῳ ἂν ἀμβλυτέρα ἡ γωνία ᾗ, ἐναντιώτεραι αἱ φοραὶ γίνονται·
But A is in the opposite case; for it is carried by itself towards B, but the side carries it down towards D.
εὐθυτέρα γὰρ ἡ γραμμὴ γίνεται.
And the more obtuse the angle is, the more opposite the motions become; for the line becomes straighter.
Εἰ δ’ ὅλως εὐθεῖα γένοιτο, παντελῶς ἂν εἴησαν ἐναντίαι.
And if it should become entirely straight, they would be completely opposite.
Ἡ δὲ πλευρὰ ὑπ’ οὐθενὸς κωλύεται μίαν φερομένη φοράν.
But the side is hindered by nothing, being carried with a single motion.
Εὐλόγως οὖν τὴν μείζω διέρχεται.
Reasonably, therefore, it traverses the greater distance.